Chapter 4 SelfConsistent Field Ground State Methods
4.1 Introduction
4.1.1 Overview of Chapter
Theoretical chemical models [6] involve two principal
approximations. One must specify the type of atomic orbital basis set used (see
Chapters 7 and 8), and one must specify the way in
which the instantaneous interactions (or correlations) between electrons are
treated. Selfconsistent field (SCF) methods are the simplest and most widely
used electron correlation treatments, and contain as special cases all
KohnSham density functional methods and the HartreeFock method. This
Chapter summarizes QChem's SCF capabilities, while the next Chapter
discusses more complex (and computationally expensive!) wavefunctionbased
methods for describing electron correlation. If you are new to quantum
chemistry, we recommend that you also purchase an introductory textbook on the
physical content and practical performance of standard
methods [6,[7,[8].
This Chapter is organized so that the earlier sections provide a mixture of
basic theoretical background, and a description of the minimum number of
program input options that must be specified to run SCF jobs. Specifically,
this includes the sections on:
 HartreeFock theory
 Density functional theory. Note that all basic input options described in
the HartreeFock also apply to density functional calculations.
Later sections introduce more specialized options that can be consulted as
needed:
 Large molecules and linear scaling methods. A short overview of the ideas
behind methods for very large systems and the options that control them.
 Initial guesses for SCF calculations. Changing the default initial guess
is sometimes important for SCF calculations that do not converge.
 Converging the SCF calculation. This section describes the iterative
methods available to control SCF calculations in QChem. Altering the
standard options is essential for SCF jobs that have failed to converge
with the default options.
 Unconventional SCF calculations. Some nonstandard SCF methods
with novel physical and mathematical features. Explore further if you are
interested!
 SCF Metadynamics. This can be used to locate multiple solutions to the SCF
equations and help check that your solution is the lowest minimum.
4.1.2 Theoretical Background
In 1926, Schrödinger [9] combined the wave nature of the
electron with the statistical knowledge of the electron viz.
Heisenberg's Uncertainty Principle [10] to formulate an
eigenvalue equation for the total energy of a molecular system. If we focus on
stationary states and ignore the effects of relativity, we have the
timeindependent, nonrelativistic equation
where the coordinates R and r refer to nuclei and electron position
vectors respectively and H is the Hamiltonian operator. In atomic units,
H=− 
1
2


N ∑
i=1

∇_{i}^{2} − 
1
2


M ∑
A=1


1
M_{A}

∇_{A}^{2} − 
N ∑
i=1


M ∑
A=1


Z_{A}
r_{iA}

+ 
N ∑
i=1


N ∑
j > i


1
r_{ij}

+ 
M ∑
A=1


M ∑
B > A


Z_{A} Z_{B}
R_{AB}


 (4.2) 
where ∇^{2} is the Laplacian operator,
∇^{2} ≡ 
∂^{2}
∂x^{2}

+ 
∂^{2}
∂y^{2}

+ 
∂^{2}
∂z^{2}


 (4.3) 
In Eq. ,
Z is the nuclear charge, M_{A} is the ratio of the mass of nucleus A to
the mass of an electron, R_{AB} = R_{A} − R_{B} is the distance between
the Ath and Bth nucleus, r_{ij} = r_{i} − r_{j}
is the distance between the ith and jth electrons, r_{iA} =  r_{i} − R_{A} is the distance between the ith electron and the Ath
nucleus, M is the number of nuclei and N is the number of
electrons. E is an eigenvalue of H, equal to the total energy, and the wave
function Ψ, is an eigenfunction of H.
Separating the motions of the electrons from that of the nuclei, an idea
originally due to Born and Oppenheimer [11], yields the electronic
Hamiltonian operator:
H_{elec} = − 
1
2


N ∑
i=1

∇_{i}^{2} − 
N ∑
i=1


M ∑
A=1


Z_{A}
r_{iA}

+ 
N ∑
i=1


N ∑
j > i


1
r_{ij}


 (4.4) 
The solution of the corresponding electronic Schrödinger equation,
H_{elec} Ψ_{elec} = E_{elec} Ψ_{elec} 
 (4.5) 
gives the total electronic energy, E_{elec}, and electronic wave
function, Ψ_{elec}, which describes the motion of the electrons for a
fixed nuclear position. The total energy is obtained by simply adding the
nuclearnuclear repulsion energy [the fifth term in Eq. (4.2)] to the
total electronic energy:
E_{tot} = E_{elec} +E_{nuc} 
 (4.6) 
Solving the eigenvalue problem in Eq. (4.5) yields a set of eigenfunctions
(Ψ_{0}, Ψ_{1}, Ψ_{2} …) with corresponding eigenvalues
(E_{0}, E_{1}, E_{2}…) where E_{0} ≤ E_{1} ≤ E_{2} ≤ ….
Our interest lies in determining the lowest eigenvalue and associated
eigenfunction which correspond to the ground state energy and wavefunction
of the molecule. However, solving Eq. (4.5) for other than the most trivial
systems is extremely difficult and the best we can do in practice is to find
approximate solutions.
The first approximation used to solve Eq. (4.5) is that electrons move
independently within molecular orbitals (MO), each of which describes the
probability distribution of a single electron. Each MO is determined by
considering the electron as moving within an average field of all the other
electrons. Ensuring that the wavefunction is antisymmetric upon electron
interchange, yields the well known Slaterdeterminant wavefunction [12,[13],
where χ_{i}, a spin orbital, is the product of a molecular orbital ψ_{i}
and a spin function (α or β).
One obtains the optimum set of MOs by variationally minimizing the energy in
what is called a "selfconsistent field" or SCF approximation to the
manyelectron problem. The archetypal SCF method is the HartreeFock
approximation, but these SCF methods also include KohnSham Density Functional
Theories (see Section 4.3). All SCF methods lead to equations of the form
where the Fock operator f(i) can be written
f(i)=− 
1
2

∇_{i}^{2} +υ^{eff}(i) 
 (4.9) 
Here x_{i} are spin and spatial coordinates of the ith electron,
χ are the spin orbitals and υ^{eff} is the effective potential
"seen" by the ith electron which depends on the spin orbitals of the
other electrons. The nature of the effective potential υ^{eff} depends on
the SCF methodology and will be elaborated on in further sections.
The second approximation usually introduced when solving Eq. (4.5), is the
introduction of an Atomic Orbital (AO) basis. AOs (ϕ_{μ}) are usually
combined linearly to approximate the true MOs. There are many standardized,
atomcentered basis sets and details of these are discussed in Chapter 7.
After eliminating the spin components in Eq. (4.8) and introducing a finite basis,
Eq. (4.8) reduces to the RoothaanHall matrix equation,
where F is the Fock matrix, C is a square matrix of molecular
orbital coefficients, S is the overlap matrix with elements
S_{μν} =  ⌠ ⌡

ϕ_{μ} (r) ϕ_{ν} (r)dr 
 (4.12) 
and ε is a diagonal matrix of the orbital energies.
Generalizing to an unrestricted formalism by introducing separate spatial
orbitals for α and β spin in Eq. (4.7) yields the
PopleNesbet [14] equations
Solving Eq. (4.11) or Eq. (4.13) yields the restricted or unrestricted
finite basis HartreeFock approximation. This approximation inherently
neglects the instantaneous electronelectron correlations which are averaged
out by the SCF procedure, and while the chemistry resulting from HF
calculations often offers valuable qualitative insight, quantitative energetics
are often poor. In principle, the DFT SCF methodologies are able to capture
all the correlation energy (the difference in energy between the HF energy and
the true energy). In practice, the best currently available density functionals
perform well, but not perfectly and conventional HFbased approaches to
calculating the correlation energy are still often required. They are discussed
separately in the following Chapter.
In selfconsistent field methods, an initial guess is calculated for the MOs
and, from this, an average field seen by each electron can be calculated. A new
set of MOs can be obtained by solving the RoothaanHall or PopleNesbet
eigenvalue equations. This procedure is repeated until the new MOs differ
negligibly from those of the previous iteration.
Because they often yield acceptably accurate chemical predictions at a
reasonable computational cost, selfconsistent field methods are the corner
stone of most quantum chemical programs and calculations. The formal costs of
many SCF algorithms is O(N^{4}), that is, they grow with the fourth power of
the size, N, of the system. This is slower than the growth of the cheapest
conventional correlated methods but recent work by QChem, Inc. and its
collaborators has dramatically reduced it to O(N), an improvement that now
allows SCF methods to be applied to molecules previously considered beyond the
scope of ab initio treatment.
In order to carry out an SCF calculation using QChem, three $rem variables
need to be set:
BASIS  to specify the basis set (see Chapter 7). 
EXCHANGE  method for treating Exchange. 
CORRELATION  method for treating Correlation (defaults to NONE)

Types of ground state energy calculations currently available in QChem
are summarized in Table 4.1.
Calculation  $rem Variable JOBTYPE  
Single point energy (default)  SINGLE_POINT, SP 
Force  FORCE 
Equilibrium Structure Search  OPTIMIZATION, OPT 
Transition Structure Search  TS 
Intrinsic reaction pathway  RPATH 
Frequency  FREQUENCY, FREQ 
NMR Chemical Shift  NMR 
Table 4.1: The type of calculation to be run by QChem is controlled by the
$rem variable JOBTYPE.
4.2 HartreeFock Calculations
4.2.1 The HartreeFock Equations
As with much of the theory underlying modern quantum chemistry, the
HartreeFock approximation was developed shortly after publication of the
Schrödinger equation, but remained a qualitative theory until the advent of
the computer. Although the HF approximation tends to yield qualitative chemical
accuracy, rather than quantitative information, and is generally inferior to
many of the DFT approaches available, it remains as a useful tool in the
quantum chemist's toolkit. In particular, for organic chemistry, HF predictions
of molecular structure are very useful.
Consider once more the RoothaanHall equations, Eq. (4.11), or the
PopleNesbet equations, Eq. (4.13), which can be traced back to the
integrodifferential Eq. (4.8) in which the effective potential
υ^{eff} depends on the SCF methodology. In a restricted HF (RHF)
formalism, the effective potential can be written as
υ^{eff}= 
N/2 ∑
a

[ 2J_{a} (1)−K_{a} (1) ] − 
M ∑
A=1


Z_{A}
r_{1A}


 (4.14) 
where the Coulomb and exchange operators are defined as
J_{a} (1)=  ⌠ ⌡

ψ_{a}^{∗} (2) 
1
r_{12}

ψ_{a} (2)dr_{ 2} 
 (4.15) 
and
K_{a} (1)ψ_{i} (1)=  ⎡ ⎣
 ⌠ ⌡

ψ_{a}^{∗} (2) 
1
r_{12}

ψ_{i} (2)dr_{ 2}  ⎤ ⎦

ψ_{a} (1) 
 (4.16) 
respectively. By introducing an atomic orbital basis, we obtain Fock matrix elements
F_{μν} = H_{μν}^{core} +J_{μν} −K_{μν} 
 (4.17) 
where the core Hamiltonian matrix elements
H_{μν}^{core} = T_{μν} +V_{μν} 
 (4.18) 
consist of kinetic energy elements
T_{μν} =  ⌠ ⌡

ϕ_{μ} (r)  ⎡ ⎣

− 
1
2

∇^{2}  ⎤ ⎦

ϕ_{ν} (r)dr 
 (4.19) 
and nuclear attraction elements
V_{μν} =  ⌠ ⌡

ϕ_{μ} (r)  ⎡ ⎣

− 
∑
A


Z_{A}
 R_{ A} −r 
 ⎤ ⎦

ϕ_{ν} (r)dr 
 (4.20) 
The Coulomb and Exchange elements are given by
J_{μν} = 
∑
λσ

P_{λσ} ( μνλσ ) 
 (4.21) 
and
K_{μν} = 
1
2


∑
λσ

P_{λσ} ( μλνσ ) 
 (4.22) 
respectively, where the density matrix elements are
P_{μν} = 2 
N/2 ∑
a=1

C_{μa} C_{νa} 
 (4.23) 
and the two electron integrals are
( μνλσ ) =  ⌠ ⌡

 ⌠ ⌡

ϕ_{μ} (r_{ 1} )ϕ_{ν} (r_{ 1} )  ⎡ ⎣

1
r_{12}
 ⎤ ⎦

ϕ_{λ} (r_{ 2} )ϕ_{σ} (r_{2} )dr_{ 1} dr_{ 2} 
 (4.24) 
Note:
The formation and utilization of twoelectron integrals is a topic
central to the overall performance of SCF methodologies. The performance of the
SCF methods in new quantum chemistry software programs can be quickly estimated
simply by considering the quality of their atomic orbital integrals packages.
See Appendix B for details of QChem's AOINTS package. 
Substituting the matrix element in Eq. (4.17) back into the RoothaanHall
equations, Eq. (4.11), and iterating until selfconsistency is achieved will
yield the Restricted HartreeFock (RHF) energy and wavefunction.
Alternatively, one could have adopted the unrestricted form of the wavefunction
by defining an alpha and beta density matrix:
 


n_{α} ∑
a=1

C_{μa}^{α} C_{νa}^{α} 
 
 


n_{β} ∑
a=1

C_{μa}^{β} C_{νa}^{β} 
  (4.25) 

The total electron density matrix P^{T} is simply the sum of the alpha
and beta density matrices. The unrestricted alpha Fock matrix,
F_{μν}^{α} = H_{μν}^{core} +J_{μν} −K_{μν}^{α} 
 (4.26) 
differs from the restricted one only in the exchange contributions where the
alpha exchange matrix elements are given by
K_{μν}^{α} = 
N ∑
λ


N ∑
σ

P_{λσ}^{α} ( μλνσ ) 
 (4.27) 
4.2.2 Wavefunction Stability Analysis
At convergence, the SCF energy will be at a stationary point with respect to
changes in the MO coefficients. However, this stationary point is not
guaranteed to be an energy minimum, and in cases where it is not, the
wavefunction is said to be unstable. Even if the wavefunction is at a minimum,
this minimum may be an artifact of the constraints placed on the form of the
wavefunction. For example, an unrestricted calculation will usually give a
lower energy than the corresponding restricted calculation, and this can give
rise to a RHF→UHF instability.
To understand what instabilities can occur, it is useful to consider the
most general form possible for the spin orbitals:
χ_{i} (r,ζ)=ψ_{i}^{α} (r)α(ζ)+ψ_{i}^{β} (r)β(ζ) 
 (4.28) 
Here, the ψ's are complex functions of the Cartesian coordinates
r, and α and β are spin eigenfunctions of the
spinvariable ζ. The first constraint that is almost universally
applied is to assume the spin orbitals depend only on one or other of the
spinfunctions α or β. Thus, the spinfunctions take the form
χ_{i}(r,ζ)=ψ_{i}^{α}(r)α(ζ) or χ_{i}(r,ζ)=ψ_{i}^{β} (r)β(ζ) 
 (4.29) 
where the ψ's are still complex functions. Most SCF packages,
including QChem's, deal only with real functions, and this places an
additional constraint on the form of the wavefunction. If there exists a
complex solution to the SCF equations that has a lower energy, the
wavefunction will exhibit either a RHF → CRHF or a UHF → CUHF
instability. The final constraint that is commonly placed on the
spinfunctions is that ψ_{i}^{α} = ψ_{i}^{β}, i.e., the spatial
parts of the spinup and spindown orbitals are the same. This gives the
familiar restricted formalism and can lead to a RHF→ UHF instability as
mentioned above. Further details about the possible instabilities can be
found in Ref. .
Wavefunction instabilities can arise for several reasons, but frequently occur if
 There exists a singlet biradical at a lower energy then the closedshell
singlet state.
 There exists a triplet state at a lower energy than the lowest singlet
state.
 There are multiple solutions to the SCF equations, and the calculation
has not found the lowest energy solution.
If a wavefunction exhibits an instability, the seriousness of it can be judged
from the magnitude of the negative eigenvalues of the stability matrices. These
matrices and eigenvalues are computed by QChem's Stability Analysis package,
which was implemented by Dr Yihan Shao. The package is invoked by setting the
STABILITY_ANALYSIS $rem variable is set to TRUE. In order
to compute these stability matrices QChem must first perform a CIS
calculation. This will be performed automatically, and does not require any
further input from the user. By default QChem computes only the lowest
eigenvalue of the stability matrix. This is usually sufficient to determine if
there is a negative eigenvalue, and therefore an instability. Users wishing to
calculate additional eigenvalues can do so by setting the
CIS_N_ROOTS $rem variable to a number larger than 1.
QChem's Stability Analysis package also seeks to correct internal
instabilities (RHF→RHF or UHF→UHF). Then, if such an instability is
detected, QChem automatically performs a unitary transformation of the
molecular orbitals following the directions of the lowest eigenvector, and
writes a new set of MOs to disk. One can read in these MOs as an initial guess
in a second SCF calculation (set the SCF_GUESS $rem variable to
READ), it might also be desirable to set the SCF_ALGORITHM
to GDM. In cases where the lowestenergy SCF solution breaks the
molecular pointgroup symmetry, the SYM_IGNORE $rem should be set
to TRUE.
Note:
The stability analysis package can be used to analyze both DFT and HF
wavefunctions. 
4.2.3 Basic HartreeFock Job Control
In brief, QChem supports the three main variants of the HartreeFock
method. They are:
 Restricted HartreeFock (RHF) for closed shell molecules. It is typically
appropriate for closed shell molecules at their equilibrium geometry,
where electrons occupy orbitals in pairs.
 Unrestricted HartreeFock (UHF) for open shell molecules. Appropriate for
radicals with an odd number of electrons, and also for molecules with
even numbers of electrons where not all electrons are paired (for example
stretched bonds and diradicaloids).
 Restricted open shell HartreeFock (ROHF) for open shell molecules, where
the alpha and beta orbitals are constrained to be identical.
Only two $rem variables are required in order to run HartreeFock (HF)
calculations:

EXCHANGE must be set to HF.
 A valid keyword for BASIS must be specified (see Chapter
7).
In slightly more detail, here is a list of basic $rem variables associated
with running HartreeFock calculations. See Chapter 7 for
further detail on basis sets available and Chapter 8 for
specifying effective core potentials.
JOBTYPE
Specifies the type of calculation. 
TYPE:
DEFAULT:
OPTIONS:
SP  Single point energy. 
OPT  Geometry Minimization. 
TS  Transition Structure Search. 
FREQ  Frequency Calculation. 
FORCE  Analytical Force calculation. 
RPATH  Intrinsic Reaction Coordinate calculation. 
NMR  NMR chemical shift calculation. 
BSSE  BSSE calculation. 
EDA  Energy decomposition analysis. 
RECOMMENDATION:

 EXCHANGE
Specifies the exchange level of theory. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use HF for HartreeFock calculations. 



BASIS
Specifies the basis sets to be used. 
TYPE:
DEFAULT:
OPTIONS:
General, Gen  User defined ($basis keyword required). 
Symbol  Use standard basis sets as per Chapter 7. 
Mixed  Use a mixture of basis sets (see Chapter 7). 
RECOMMENDATION:
Consult literature and reviews to aid your selection. 

 PRINT_ORBITALS
Prints orbital coefficients with atom labels in analysis part of output. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Do not print any orbitals. 
TRUE  Prints occupied orbitals plus 5 virtuals. 
NVIRT  Number of virtuals to print. 
RECOMMENDATION:
Use TRUE unless more virtuals are desired. 



THRESH
Cutoff for neglect of two electron integrals. 10^{−THRESH} (THRESH
≤ 14). 
TYPE:
DEFAULT:
8  For single point energies. 
10  For optimizations and frequency calculations. 
14  For coupledcluster calculations. 
OPTIONS:
n  for a threshold of 10^{−n}. 
RECOMMENDATION:
Should be at least three greater than SCF_CONVERGENCE. Increase for
more significant figures, at greater computational cost. 

 SCF_CONVERGENCE
SCF is considered converged when the wavefunction error is less that
10^{−SCF_CONVERGENCE}. Adjust the value of THRESH at the same
time. Note that in QChem 3.0 the DIIS error is measured by the maximum error
rather than the RMS error as in previous versions. 
TYPE:
DEFAULT:
5  For single point energy calculations. 
7  For geometry optimizations and vibrational analysis. 
8  For SSG calculations, see Chapter 5. 
OPTIONS:
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger
values provide more significant figures, at greater computational cost. 



UNRESTRICTED
Controls the use of restricted or unrestricted orbitals. 
TYPE:
DEFAULT:
FALSE  (Restricted) Closedshell systems. 
TRUE  (Unrestricted) Openshell systems. 
OPTIONS:
TRUE  (Unrestricted) Openshell systems. 
FALSE  Restricted openshell HF (ROHF). 
RECOMMENDATION:
Use default unless ROHF is desired. Note that for unrestricted calculations on
systems with an even number of electrons it is usually necessary to break
alpha/beta symmetry in the initial guess, by using SCF_GUESS_MIX or
providing $occupied information (see Section 4.5 on
initial guesses). 

4.2.4 Additional HartreeFock Job Control Options
Listed below are a number of useful options to customize a HartreeFock
calculation. This is only a short summary of the function of these $rem
variables. A full list of all SCFrelated variables is provided in
Appendix C.
A number of other specialized topics (large molecules, customizing initial
guesses, and converging the calculation) are discussed separately in
Sections 4.4, 4.5, and 4.6, respectively.
INTEGRALS_BUFFER
Controls the size of incore integral storage buffer. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default, or consult your systems administrator for hardware limits. 

 DIRECT_SCF
TYPE:
DEFAULT:
OPTIONS:
TRUE  Forces direct SCF. 
FALSE  Do not use direct SCF. 
RECOMMENDATION:
Use default; direct SCF switches off incore integrals. 



METECO
Sets the threshold criteria for discarding shellpairs. 
TYPE:
DEFAULT:
2  Discard shellpairs below 10^{−THRESH}. 
OPTIONS:
1  Discard shellpairs four orders of magnitude below machine precision. 
2  Discard shellpairs below 10^{−THRESH}. 
RECOMMENDATION:

 STABILITY_ANALYSIS
Performs stability analysis for a HF or DFT solution. 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Perform stability analysis. 
FALSE  Do not perform stability analysis. 
RECOMMENDATION:
Set to TRUE when a HF or DFT solution is suspected to be unstable. 



SCF_PRINT
Controls level of output from SCF procedure to QChem output file. 
TYPE:
DEFAULT:
0  Minimal, concise, useful and necessary output. 
OPTIONS:
0  Minimal, concise, useful and necessary output. 
1  Level 0 plus component breakdown of SCF electronic energy. 
2  Level 1 plus density, Fock and MO matrices on each cycle. 
3  Level 2 plus twoelectron Fock matrix components (Coulomb, HF exchange 
 and DFT exchangecorrelation matrices) on each cycle. 
RECOMMENDATION:
Proceed with care; can result in extremely large output files at level 2 or higher.
These levels are primarily for program debugging. 

 SCF_FINAL_PRINT
Controls level of output from SCF procedure to QChem output file at the
end of the SCF. 
TYPE:
DEFAULT:
OPTIONS:
0  No extra print out. 
1  Orbital energies and breakdown of SCF energy. 
2  Level 1 plus MOs and density matrices. 
3  Level 2 plus Fock and density matrices. 
RECOMMENDATION:
The breakdown of energies is often useful (level 1). 



DIIS_SEPARATE_ERRVEC
Control optimization of DIIS error vector in unrestricted calculations. 
TYPE:
DEFAULT:
FALSE  Use a combined alpha and beta error vector. 
OPTIONS:
FALSE  Use a combined alpha and beta error vector. 
TRUE  Use separate error vectors for the alpha and beta spaces. 
RECOMMENDATION:
When using DIIS in QChem a convenient optimization for unrestricted calculations is to sum
the alpha and beta error vectors into a single vector which is used for extrapolation. This
is often extremely effective, but in some pathological systems with symmetry breaking, can lead
to false solutions being detected, where the alpha and beta components of the error vector
cancel exactly giving a zero DIIS error. While an extremely uncommon occurrence, if it is suspected,
set DIIS_SEPARATE_ERRVEC to TRUE to check. 

4.2.5 Examples
Provided below are examples of QChem input files to run ground state,
HartreeFock single point energy calculations.
Example 4.0 Example QChem input for a single point energy calculation on
water. Note that the declaration of the single point $rem variable and level
of theory to treat correlation are redundant because they are the same as the
QChem defaults.
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.2
hoh = 120.0
$end
$rem
JOBTYPE sp Single Point energy
EXCHANGE hf Exact HF exchange
CORRELATION none No correlation
BASIS sto3g Basis set
$end
$comment
HF/STO3G water single point calculation
$end
Example 4.0 UHF/6311G calculation on the Lithium atom. Note that
correlation and the job type were not indicated because QChem defaults
automatically to no correlation and single point energies. Note also that,
since the number of alpha and beta electron differ, MOs default to an
unrestricted formalism.
$molecule
0,2
3
$end
$rem
EXCHANGE HF HartreeFock
BASIS 6311G Basis set
$end
Example 4.0 ROHF/6311G calculation on the Lithium atom. Note again that
correlation and the job type need not be indicated.
$molecule
0,2
3
$end
$rem
EXCHANGE hf HartreeFock
UNRESTRICTED false Restricted MOs
BASIS 6311G Basis set
$end
Example 4.0 RHF/631G stability analysis calculation on the singlet state of
the oxygen molecule. The wavefunction is RHF→UHF unstable.
$molecule
0 1
O
O 1 1.165
$end
$rem
EXCHANGE hf HartreeFock
UNRESTRICTED false Restricted MOs
BASIS 631G(d) Basis set
STABILITY_ANALYSIS true Perform a stability analysis
$end
4.2.6 Symmetry
Symmetry is a powerful branch of mathematics and is often exploited in quantum
chemistry, both to reduce the computational workload and to classify the final
results obtained [16,[17,[18]. QChem is able
to determine the point group symmetry of the molecular nuclei and, on
completion of the SCF procedure, classify the symmetry of molecular orbitals,
and provide symmetry decomposition of kinetic and nuclear attraction energy
(see Chapter 10).
Molecular systems possessing point group symmetry offer the possibility of
large savings of computational time, by avoiding calculations of integrals
which are equivalent i.e., those integrals which can be mapped on to one
another under one of the symmetry operations of the molecular point group.
The QChem default is to use symmetry to reduce computational time, when
possible.
There are several keywords that are related to symmetry, which causes frequent
confusion. SYM_IGNORE controls symmetry throughout all modules.
The default is FALSE. In some cases it may be desirable to turn off
symmetry altogether, for example if you do not want QChem to reorient the
molecule into the standard nuclear orientation, or if you want to turn it off for
finite difference calculations. If the SYM_IGNORE $rem is set to
TRUE then the coordinates will not be altered from the input, and the
point group will be set to C_{1}.
The SYMMETRY (an alias for ISYM_RQ) keyword controls symmetry in
some integral routines.
It is set to FALSE by default.
Note that setting it to FALSE
does not turn point group symmetry off, and does not disable
symmetry in the coupledcluster suite (CCMAN and CCMAN2), which is controlled
by CC_SYMMETRY (see Chapters 5
and 6), although we noticed that sometimes it may mess up
the determination of orbital symmetries, possibly due to numeric noise.
In some cases, SYMMETRY=TRUE
can cause problems (poor convergence and crazy SCF energies) and
turning it off can help.
Note:
The user should be aware about different conventions for defining
symmetry
elements.
The arbitrariness affects, for example, C_{2v} point group.
The specific choice affects how the irreps in the affected groups are labeled.
For example, b_{1} and b_{2} irreps in C_{2v} are flipped when using different
conventions. QChem uses nonMulliken symmetry convention.
See http://iopenshell.usc.edu/howto/symmetry
for detailed explanations. 
 SYMMETRY
Controls the efficiency through the use of point group symmetry for
calculating integrals. 
TYPE:
DEFAULT:
TRUE  Use symmetry for computing integrals. 
OPTIONS:
TRUE  Use symmetry when available. 
FALSE  Do not use symmetry. This is
always the case for RIMP2 jobs 
RECOMMENDATION:
Use default unless benchmarking.
Note that symmetry usage is disabled for RIMP2, FFT, and QM/MM jobs. 



SYM_IGNORE
Controls whether or not QChem determines the point group of the molecule and reorients the molecule to the standard orientation. 
TYPE:
DEFAULT:
FALSE  Do determine the point group (disabled for RIMP2 jobs). 
OPTIONS:
RECOMMENDATION:
Use default unless you do not want the molecule to be reoriented.
Note that symmetry usage is disabled for RIMP2 jobs. 

 SYM_TOL
Controls the tolerance for determining point group symmetry. Differences in
atom locations less than 10^{−SYM_TOL} are treated as zero. 
TYPE:
DEFAULT:
5  corresponding to 10^{−5}. 
OPTIONS:
RECOMMENDATION:
Use the default unless the molecule has high symmetry which is not being
correctly identified. Note that relaxing this tolerance too much may introduce
errors into the calculation. 



4.3 Density Functional Theory
4.3.1 Introduction
In recent years, Density Functional Theory [19,[20,[21,[22]
has emerged as an accurate alternative firstprinciples approach to quantum
mechanical molecular investigations. DFT currently accounts for approximately
90% of all quantum chemical calculations being performed, not only because
of its proven chemical accuracy, but also because of its relatively cheap
computational expense. These two features suggest that DFT is likely to remain
a leading method in the quantum chemist's toolkit well into the future.
QChem contains fast, efficient and accurate algorithms for all popular
density functional theories, which make calculations on quite large molecules
possible and practical.
DFT is primarily a theory of electronic ground state structures based on the
electron density, ρ(r), as opposed to the manyelectron wavefunction
Ψ(r_{1},…,r_{N}) There are a number of distinct similarities and
differences to traditional wavefunction approaches and modern DFT
methodologies. Firstly, the essential building blocks of the many electron
wavefunction are singleelectron orbitals are directly analogous to the
KohnSham (see below) orbitals in the current DFT framework. Secondly, both the
electron density and the manyelectron wavefunction tend to be constructed
via a SCF approach that requires the construction of matrix elements
which are remarkably and conveniently very similar.
However, traditional approaches using the many electron wavefunction as a
foundation must resort to a postSCF calculation (Chapter 5)
to incorporate correlation effects, whereas DFT
approaches do not. PostSCF methods, such as perturbation theory or coupled
cluster theory are extremely expensive relative to the SCF procedure. On the
other hand, the DFT approach is, in principle, exact, but in practice relies on
modeling the unknown exact exchange correlation energy functional. While more
accurate forms of such functionals are constantly being developed, there is no
systematic way to improve the functional to achieve an arbitrary level of
accuracy. Thus, the traditional approaches offer the possibility of achieving
an arbitrary level of accuracy, but can be computationally demanding, whereas
DFT approaches offer a practical route but the theory is currently incomplete.
4.3.2 KohnSham Density Functional Theory
The Density Functional Theory by Hohenberg, Kohn and Sham [23,[24]
stems from the original work of
Dirac [25], who found that the exchange energy of a uniform electron gas
may be calculated exactly, knowing only the charge density. However, while the
more traditional DFT constitutes a direct approach and the necessary equations
contain only the electron density, difficulties associated with the kinetic
energy functional obstructed the extension of DFT to anything more than a crude
level of approximation. Kohn and Sham developed an indirect approach to the
kinetic energy functional which transformed DFT into a practical tool for
quantum chemical calculations.
Within the KohnSham formalism [24], the ground state
electronic energy, E, can be written as
E = E_{T}^{} + E_{V}^{} + E_{J}^{} + E_{XC}^{} 
 (4.30) 
where E_{T}^{} is the kinetic energy, E_{V}^{} is the electronnuclear
interaction energy, E_{J}^{} is the Coulomb selfinteraction of the
electron density ρ(r) and E_{XC}^{} is the exchangecorrelation
energy. Adopting an unrestricted format, the alpha and beta total electron
densities can be written as
 


n_{α} ∑
i=1

ψ_{i}^{α}^{2} 
 
 


n_{β} ∑
i=1

ψ_{i}^{β} ^{2} 
  (4.31) 

where n_{α} and n_{β} are the number of alpha and beta electron
respectively and, ψ_{i} are the KohnSham orbitals. Thus, the total
electron density is
ρ(r) = ρ_{α} (r) + ρ_{β}(r) 
 (4.32) 
Within a finite basis set [26], the density is represented by
ρ(r) = 
∑
μν

P_{μν}^{T} ϕ_{μ} (r) ϕ_{ν} (r) 
 (4.33) 
The components of Eq. (4.28) can now be written as
 


n_{α} ∑
i=1



ψ_{i}^{α}  ⎢ ⎢

− 
1
2

∇^{2}  ⎢ ⎢

ψ_{i}^{α} 

+ 
n_{β} ∑
i=1



ψ_{i}^{β}  ⎢ ⎢

− 
1
2

∇^{2}  ⎢ ⎢

ψ_{i}^{β} 


 
 


∑
μν

P_{μν}^{T} 

ϕ_{μ}(r)  ⎢ ⎢

− 
1
2

∇^{2}  ⎢ ⎢

ϕ_{ν}(r) 


  (4.34) 
 

− 
M ∑
A=1

Z_{A} 
ρ(r)
r−R_{A}

dr 
 
 

− 
∑
μν

P_{μν}^{T} 
∑
A



ϕ_{μ}(r)  ⎢ ⎢

Z_{A}
r−R_{A}
 ⎢ ⎢

ϕ_{ν}(r) 


  (4.35) 
 


1
2



ρ(r_{1})  ⎢ ⎢

1
r_{1} − r_{2}
 ⎢ ⎢

ρ(r_{2}) 


 
 


1
2


∑
μν


∑
λσ

P_{μν}^{T} P_{λσ}^{T} (μνλσ) 
  (4.36) 
 

  (4.37) 

Minimizing E with respect to the unknown KohnSham orbital coefficients
yields a set of matrix equations exactly analogous to the UHF case

F^{α} C^{α} = ε^{α} SC^{α} 
  (4.38) 
 F^{β} C^{β} = ε^{β} SC^{β} 
  (4.39) 

where the Fock matrix elements are generalized to

F_{μν}^{α} = H_{μν}^{core} + J_{μν} − F_{μν}^{XCα} 
  (4.40) 
 F_{μν}^{β} = H_{μν}^{core} + J_{μν} − F_{μν}^{XCβ} 
  (4.41) 

where F_{μν}^{XCα} and F_{μν}^{XCβ} are the
exchangecorrelation parts of the Fock matrices dependent on the
exchangecorrelation functional used. The PopleNesbet equations are obtained
simply by allowing
F_{μν}^{XCα} = K_{μν}^{α} 
 (4.42) 
and similarly for the beta equation. Thus, the density and energy are obtained
in a manner analogous to that for the HartreeFock method. Initial guesses are
made for the MO coefficients and an iterative process applied until self
consistency is obtained.
4.3.3 ExchangeCorrelation Functionals
There are an increasing number of exchange and correlation functionals and
hybrid DFT methods available to the quantum chemist, many of which are very
effective. In short, there are nowadays five basic working types of functionals
(five rungs on the Perdew's "Jacob`s Ladder"): those based on
the local spin density approximation (LSDA) are on the first rung,
those based on generalized gradient approximations (GGA) are on the second rung.
Functionals that include not only density gradient
corrections (as in the GGA functionals), but also a dependence on
the electron kinetic energy density and/or
the Laplacian of the electron density, occupy
the third rung of the Jacob`s Ladder and are known as "metaGGA"
functionals. The latter lead to a systematic,
and often substantial improvement over GGA for thermochemistry
and reaction kinetics. Among the metaGGA functionals, a particular attention
deserve the VSXC functional [27],
the functional of Becke and Roussel for exchange [28],
and for correlation [29] (the BR89B94 metaGGA
combination [29]). The latter functional did not receive enough
popularity until recently, mainly because it was not representable
in an analytic form. In QChem, BR89B94 is implemented now selfconsistently
in a fully analytic form, based on the recent work [30].
The one and only nonempirical metaGGA functional,
the TPSS functional [31],
was also implemented recently in QChem [32].
Each of the above mentioned "pure" functionals can be
combined with a fraction of exact (HartreeFock) nonlocal
exchange energy replacing a similar fraction
from the DFT local exchange energy.
When a nonzero amount of HartreeFock exchange
is used (less than a 100%), the corresponding functional is a hybrid extension
(a global hybrid) of the parent "pure" functional.
In most cases a hybrid functional would have one or more
(up to 21 so far) linear mixing parameters that are fitted
to experimental data. An exception is the hybrid extension
of the TPSS metaGGA functional, the nonempirical TPSSh scheme, which is
also implemented now in QChem [32].
The forth rung of functionals ("hyperGGA" functionals)
involve occupied KohnSham orbitals as additional
nonlocal variables [33,[34,[35,[36].
This helps tremendously in describing cases of strong inhomogeneity
and strong nondynamic correlation, that are evasive for global
hybrids at GGA and metaGGA levels of the theory. The success is
mainly due to one novel feature of these functionals:
they incorporate a 100% of exact (or HF) exchange
combined with a hyperGGA model correlation.
Employing a 100% of exact exchange has been a long standing dream in DFT,
but most previous attempts were unsuccessful. The correlation models
used in the hyperGGA schemes B05 [33] and
PSTS [36], properly compensate the spuriously
high nonlocality of the exact exchange hole,
so that cases of strong nondynamic correlation become treatable.
In addition to some GGA and metaGGA variables, the B05 scheme employs a new
functional variable, namely, the exactexchange energy density:
e^{HF}_{X}(r) = − 
1
2

 ⌠ ⌡

dr^{′} 
n(r,r^{′})^{2}
r−r^{′}

, 
 (4.43) 
where
n(r,r^{′}) = 
1
ρ(r)


occ ∑
i

φ_{i}^{ks}(r)φ_{i}^{ks}(r^{′}) . 
 (4.44) 
This new variable enters the correlation energy component in a
rather sophisticated nonlinear manner [33]:
This presents a huge challenge for the practical implementation
of such functionals, since they require a HartreeFock
type of calculation at each grid point, which renders the
task impractical. Significant progress in implementing
efficiently the B05 functional was reported only
recently [37,[38]. This new implementation
achieves a speedup of the B05 calculations by a factor of 100
based on resolutionofidentity (RI) technique (the RIB05 scheme) and
analytical interpolations. Using this methodology, the PSTS
hyperGGA was also implemented in QChem more recently [32].
For the time being only singlepoint SCF calculations are
available for RIB05 and RIPSTS (the energy gradient will be available soon).
In contrast to B05 and PSTS, the forthrung functional MCY employs a
100% global exact exchange, not only as a separate energy component of the functional,
but also as a nonlinear variable used the MCY correlation energy
expression [34,[35]. Since this variable is
the same at each grid point, it has to be calculated
only once per SCF iteration. The form of the MCY correlation functional
is deduced from known adiabatic connection and coordinate
scaling relationships which, together with a few fitting
parameters, provides a good correlation match to the exact exchange.
The MCY functional [34] in its MCY2 version [35] is now
implemented in QChem, as described in Ref. [32].
The fifthrung functionals include not only occupied KohnSham orbitals, but
also unoccupied orbitals, which improves further the quality of the
exchangecorrelation energy. The practical application so far of these
consists of adding empirically a small fraction of correlation energy obtained
from MP2like postSCF calculation [39,[40].
Such functionals are known as "doublehybrids". A more detailed
description of some these as implemented in QChem is
given in Subsections 4.3.9 and 4.3.4.3.
In summary, QChem includes the following exchange and correlation functionals:
LSDA functionals:
 SlaterDirac (Exchange) [25]
 VoksoWilkNusair (Correlation) [41]
 PerdewZunger (Correlation) [42]
 Wigner (Correlation) [43]
 PerdewWang 92 (Correlation) [44]
GGA functionals:
 Becke86 (Exchange) [45]
 Becke88 (Exchange) [46]
 PW86 (Exchange) [47]
 refit PW86 (Exchange) [48]
 Gill96 (Exchange) [49]
 GilbertGill99 (Exchange [50]
 LeeYangParr (Correlation) [51]
 Perdew86 (Correlation) [52]
 GGA91 (Exchange and correlation) [53]
 PBE (Exchange and correlation) [54,[55]
 revPBE (Exchange) [56]
 B3LYP (Exchange and correlation within a hybrid scheme) [57]
 Becke97 (Exchange and correlation within a hybrid scheme) [58,[55]
 Becke971 (Exchange and correlation within a hybrid scheme) [59,[55]
 Becke972 (Exchange and correlation within a hybrid scheme) [60,[55]
 HCTH (Exchange correlation within a hybrid scheme) [59,[55]
 HCTH120 (Exchange correlation within a hybrid scheme) [61,[55]
 HCTH147 (Exchange correlation within a hybrid scheme) [61,[55]
 HCTH407 (Exchange correlation within a hybrid scheme) [62,[55]
 The ωB97X functionals developed by Chai and Gordon [63]
(Exchange and correlation within a hybrid scheme, with longrange correction,
see further in this manual for details)
 BNL (Exchange GGA functional) [64,[65]
 BOP (Becke88 exchange plus the "oneparameter progressive" correlation
functional, OP) [66]
 PBEOP (PBE Exchange plus the OP correlation functional) [66]
 SOGGA (Exchange plus the PBE correlation functional) [67]
 SOGGA11 (Exchange and Correlation) [68]
 SOGGA11X (Exchange and Correlation within a hybrid scheme, with reoptimized SOGGA11 parameters) [69]
Note:
The OP correlation functional used in BOP has been parameterized for use with
Becke88 exchange, whereas in the PBEOP functional, the same correlation ansatz
is reparameterized for use with PBE exchange. These two versions of OP correlation
are available as the correlation functionals (B88)OP and (PBE)OP. The BOP
functional, for example, consists of (B88)OP correlation combined with Becke88 exchange. 
MetaGGA functionals involving the kinetic energy
density (τ), and or the Laplacian of the electron density:
 VSXC (Exchange and Correlation) [27]
 TPSS (Exchange and Correlation in a single nonempirical
scheme) [31,[32]
 TPSSh (Exchange and Correlation within a nonempirical hybrid scheme) [70]
 BMK (Exchange and Correlation within a hybrid scheme) [71]
 M05 (Exchange and Correlation within a hybrid scheme) [72,]
 M052X (Exchange and Correlation within a hybrid scheme) [74,[73]
 M06L (Exchange and Correlation) [75,[73]
 M06HF (Exchange and Correlation within a hybrid scheme) [76,[73]
 M06 (Exchange and Correlation within a hybrid scheme) [77,[73]
 M062X (Exchange and Correlation within a hybrid scheme) [77,[73]
 M08HX (Exchange and Correlation within a hybrid scheme) [78]
 M08SO (Exchange and Correlation within a hybrid scheme) [78]
 M11L (Exchange and Correlation) [79]
 M11 (Exchange and Correlation within a hybrid scheme, with longrange correction) [80]
 BR89 (Exchange) [28,[81]
 B94 (Correlation) [29,[81]
 PK06 (Correlation) [82]
HyperGGA functionals:
 B05 (A full exactexchange KohnSham scheme of Becke that accounts for static corrrelation
via realspace corrections) [33,[37,[38]
 mB05 (Modified B05 method that has simpler functional form and SCF potential) [83]
 PSTS (HyperGGA functional of PerdewStaroverovTaoScuseria) [36]
 MCY2 (The adiabatic connectionbased MCY2 functional) [34,[35,[32]
Fifthrung, doublehybrid (DH) functionals:
 ωB97X2 (Exchange and Correlation within a DH generalization of the
LC corrected ωB97X scheme) [40]
 B2PLYP (another DH scheme proposed by Grimme, based on GGA exchange and correlation
functionals) [84]
 XYG3 and XYGJOS (an efficient DH scheme based on generalization of B3LYP) [85]
In addition to the above functional types, QChem contains the
Empirical Density Functional 1 (EDF1), developed by Adamson, Gill and
Pople [86]. EDF1 is a combined exchange and correlation
functional that is specifically adapted to yield good results with the
relatively modestsized 631+G* basis set, by direct fitting to thermochemical
data. It has the interesting feature that exact exchange mixing was not found
to be helpful with a basis set of this size. Furthermore, for this basis set,
the performance substantially exceeded the popular B3LYP functional,
while the cost of the calculations is considerably lower because there is no
need to evaluate exact (nonlocal) exchange. We recommend consideration of
EDF1 instead of either B3LYP or BLYP for density functional calculations on
large molecules, for which basis sets larger than 631+G* may be too
computationally demanding.
EDF2, another Empirical Density Functional, was developed by Ching Yeh Lin and
Peter Gill [87] in a similar vein to EDF1, but is specially designed for
harmonic frequency calculations. It was optimized using the ccpVTZ basis
set by fitting into experimental harmonic frequencies and is designed
to describe the potential energy curvature well. Fortuitously, it also
performs better than B3LYP for thermochemical properties.
A few more words deserve the hybrid functionals [57], where several
different exchange and correlation functionals can be combined linearly to form a
hybrid functional. These have proven successful in a number of reported applications.
However, since the hybrid functionals contain HF exchange they are more expensive
that pure DFT functionals. QChem has incorporated two of the most popular
hybrid functionals, B3LYP [88] and B3PW91 [28],
with the additional option for users to define their own hybrid functionals via
the $xc_functional keyword (see userdefined functionals in Section
4.3.17, below). Among the latter, a recent new hybrid combination
available in QChem is the 'B3tLap' functional, based on Becke's B88
GGA exchange and the "tLap" (e.g., PK06) metaGGA
correlation [82,[30]. This
hybrid combination is on average more accurate than B3LYP, BMK, and M06
functionals for thermochemistry and better than B3LYP for reaction
barriers, while involving only five fitting parameters.
Another hybrid functional in QChem that deserves attention
is the hybrid extension of the BR89B94 metaGGA
functional [29,[30].
This hybrid functional yields a very good thermochemistry results, yet has
only three fitting parameters.
In addition, QChem now includes the M05 and M06 suites of density
functionals. These are designed to be used only with
certain definite percentages of HartreeFock exchange. In
particular, M06L [75] is designed to be used with no HartreeFock
exchange (which reduces the cost for large molecules), and M05 [72],
M052X [74], M06, and M062X [77] are designed to be used with
28%, 56%, 27%, and 54% HartreeFock exchange. M06HF [76] is designed to
be used with 100% HartreeFock exchange, but it still contains some
local DFT exchange because the 100% nonlocal HartreeFock
exchange replaces only some of the local exchange.
Note:
The hybrid functionals are not simply a pairing of an exchange
and correlation functional, but are a combined exchangecorrelation functional
(i.e., BLYP and B3LYP vary in the correlation contribution in addition to the
exchange part). 
4.3.4 LongRangeCorrected DFT
As pointed out in Ref. and elsewhere, the description of
chargetransfer excited states within density functional theory
(or more precisely, timedependent DFT, which is discussed in
Section 6.3) requires full (100%)
nonlocal HartreeFock exchange, at least in the limit of large
donoracceptor distance.
Hybrid functionals such as B3LYP [88] and
PBE0 [90] that are wellestablished and in widespread use,
however, employ only 20%
and 25% HartreeFock exchange, respectively. While these functionals
provide excellent results for many groundstate properties, they cannot
correctly describe the distance dependence of chargetransfer excitation
energies, which are enormously underestimated by most common density
functionals. This is a serious problem in any case, but it is a
catastrophic
problem in large molecules and in clusters, where TDDFT often predicts a
nearcontinuum of of spurious, lowlying charge transfer states [91,[92].
The problems with TDDFT's description of
charge transfer are not limited to large donoracceptor distances, but have
been observed at ∼ 2 Å separation, in systems as small as
uracil(H_{2}O)_{4} [91]. Rydberg excitation energies
also tend to be substantially underestimated by standard TDDFT.
One possible avenue by which to correct such problems is to parameterize
functionals
that contain 100% HartreeFock exchange. To date, few such functionals exist,
and those that do (such as M06HF) contain a very large number of
empirical adjustable parameters. An alternative option is to
attempt to preserve the form of common GGAs and hybrid functionals at
short range (i.e., keep the 25% HartreeFock exchange in PBE0)
while incorporating 100% HartreeFock exchange
at long range. Functionals along these lines are known variously
as "Coulombattenuated" functionals, "rangeseparated"
functionals, or (our preferred designation) "longrangecorrected"
(LRC) density functionals. Whatever the nomenclature, these functionals
are all based upon
a partition of the electronelectron Coulomb potential into long and
shortrange components, using the error function (erf):

1
r_{12}^{}

≡ 
1−erf(ωr_{12}^{})
r_{12}

+ 
erf(ωr_{12}^{})
r_{12}


 (4.45) 
The first term on the right in Eq. (4.45)
is singular but shortrange, and decays
to zero on a length scale of ∼ 1/ω, while the second term
constitutes a nonsingular, longrange background. The basic idea of LRCDFT
is to utilize the shortrange component of the Coulomb operator in
conjunction with standard DFT exchange (including any component of
HartreeFock exchange, if the functional is a hybrid), while at the same
time incorporating full HartreeFock exchange using the longrange
part of the Coulomb operator. This provides a
rigorously correct description of the longrange distance dependence of
chargetransfer excitation energies, but aims to avoid contaminating shortrange
exchangecorrelation effects with extra HartreeFock exchange.
Consider an exchangecorrelation functional of the form
E_{XC}^{} = E_{C}^{} + E_{X}^{GGA} + C_{HF}^{} E_{X}^{HF} 
 (4.46) 
in which E_{C}^{} is the correlation energy, E_{X}^{GGA}
is the (local) GGA exchange energy, and E_{X}^{HF} is the (nonlocal)
HartreeFock exchange energy. The constant C_{HF}^{} denotes
the fraction of HartreeFock exchange in the functional, therefore
C_{HF}^{} = 0 for GGAs, C_{HF}^{} = 0.20 for B3LYP,
C_{HF}^{} = 0.25 for PBE0, etc..
The LRC version of the generic functional in Eq. (4.46) is
E_{XC}^{LRC} = E_{C}^{} + E_{X}^{GGA, SR} + C_{HF}^{} E_{X}^{HF, SR} + E_{X}^{HF, LR} 
 (4.47) 
in which the designations "SR" and "LR" in the various exchange
energies indicate that these components
of the functional are evaluated using either the shortrange (SR) or the
longrange (LR) component of the Coulomb operator. (The correlation
energy E_{C}^{} is evaluated using the full Coulomb operator.)
The LRC functional in
Eq. (4.47) incorporates full HartreeFock exchange in the
asymptotic limit via the final term, E_{X}^{HF, LR}. To fully
specify the LRC functional, one must choose a value for the range
separation parameter ω in Eq. (4.45);
in the limit ω→ 0, the LRC functional in
Eq. (4.47) reduces to the original functional in
Eq. (4.46), while the ω→∞ limit
corresponds to a new functional, E_{XC}^{} = E_{C}^{} + E_{X}^{HF}. It is well known that full HartreeFock exchange
is inappropriate for use with most contemporary GGA correlation functionals,
so the latter limit is expected to perform quite poorly. Values of
ω > 1.0 bohr^{−1} are probably not worth considering [93,[94].
Evaluation of the short and longrange HartreeFock exchange energies
is straightforward [95], so the crux of LRCDFT rests upon
the form of the shortrange GGA exchange energy. Several different
shortrange GGA exchange functionals are available in QChem, including
shortrange variants of B88 and PBE exchange described by Hirao and
coworkers [96,[97], an alternative formulation of
shortrange PBE exchange proposed by Scuseria and coworkers [98],
and several shortrange variants of B97 introduced by
Chai and HeadGordon [63,[99,[100,[40].
The reader is referred to these papers for additional methodological details.
These LRCDFT functionals have been shown to remove
the nearcontinuum of spurious chargetransfer excited states that
appear in largescale TDDFT calculations [93].
However, certain results depend sensitively upon the rangeseparation
parameter ω [92,[93,[94,[101], and the results
of LRCDFT calculations must therefore be interpreted with caution,
and probably for a range of ω values. In two recent benchmark
studies of several LRC density functionals,
Rohrdanz and Herbert [94,[101]
have considered the errors engendered, as a function of ω, in
both groundstate properties and also TDDFT vertical excitation energies.
In Ref. , the sensitivity of valence excitations versus chargetransfer
excitation energies in TDDFT was considered, again as a function of ω.
A careful reading of these references is suggested prior to
performing any LRCDFT calculations.
Within QChem 3.2, there are three ways to perform LRCDFT calculations.
4.3.4.1 LRCDFT with the μB88, μPBE, and ωPBE exchange functionals
The form of E_{X}^{GGA, SR} is different for each different
GGA exchange functional, and shortrange versions of B88
and PBE exchange are available in QChem through the efforts of
the Herbert group. Versions of B88 and PBE, in which the Coulomb attenuation
is performed according to the procedure of Hirao [97], are denoted
as μB88 and μPBE, respectively (since μ, rather than ω, is
the Hirao group's notation for the rangeseparation parameter). Alternatively,
a shortrange version of PBE exchange called ωPBE is available, which is
constructed according to the prescription of Scuseria and coworkers [98].
These shortrange exchange functionals can be used in the absence of longrange
HartreeFock exchange, and using a combination of
ωPBE exchange and PBE correlation, a user could, for example, employ the
shortrange hybrid functional recently described by Heyd, Scuseria, and
Ernzerhof [102]. Shortrange hybrids appear to be most appropriate for
extended systems, however. Thus, within QChem, shortrange GGAs should
be used with longrange HartreeFock exchange, as in Eq. 4.47.
Longrange HartreeFock exchange is requested by setting LRC_DFT to
TRUE.
LRCDFT is thus available
for any functional whose exchange component consists of some combination
of HartreeFock, B88, and PBE exchange (e.g., BLYP, PBE, PBE0,
BOP, PBEOP, and various userspecified combinations, but not B3LYP or other
functionals whose exchange components are more involved). Having
specified such a functional via the EXCHANGE and
CORRELATION variables, a user may request the
corresponding LRC functional by setting
LRC_DFT to TRUE. Longrangecorrected variants of PBE0,
BOP, and PBEOP must be obtained through the appropriate userspecified
combination of exchange and correlation functionals (as demonstrated in the
example below). In any case,
the value of ω must also be
specified by the user. Analytic energy gradients are available but
analytic Hessians are not. TDDFT vertical excitation energies are also
available.
LRC_DFT
Controls the application of longrangecorrected DFT 
TYPE:
DEFAULT:
OPTIONS:
FALSE  (or 0) Do not apply longrange correction. 
TRUE  (or 1) Use the longrangecorrected version of the requested functional.

RECOMMENDATION:
Longrange correction is available for any combination of HartreeFock,
B88, and PBE exchange (along with any standalone correlation functional).


 OMEGA
Sets the Coulomb attenuation parameter ω. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to ω = n/1000, in units of bohr^{−1} 
RECOMMENDATION:



Example 4.0 Application of LRCBOP to (H_{2}O)_{2}^{−}.
$comment
To obtain LRCBOP, a shortrange version of BOP must be specified,
using muB88 shortrange exchange plus (B88)OP correlation, which is
the version of OP parameterized for use with B88.
$end
$molecule
1 2
O 1.347338 .017773 .071860
H 1.824285 .813088 .117645
H 1.805176 .695567 .461913
O 1.523051 .002159 .090765
H .544777 .024370 .165445
H 1.682218 .174228 .849364
$end
$rem
EXCHANGE GEN
BASIS 631(1+,3+)G*
LRC_DFT TRUE
OMEGA 330 ! = 0.330 a.u.
$end
$xc_functional
C (B88)OP 1.0
X muB88 1.0
$end
Regarding the choice of functionals and ω values, it has been found that
the Hirao and Scuseria ansatz afford virtually identical TDDFT
excitation energies, for all values of ω [101]. Thus,
functionals based on μPBE versus ωPBE provide the same excitation
energies, as a function of ω. However, the ωPBE functional appears
to be somewhat superior in the sense that it can provide accurate TDDFT
excitation energies and accurate groundstate properties using the
same value of ω [101], whereas this does not seem
to be the case for functionals based on μB88 or μPBE [94].
Recently, Rohrdanz et al. [101] have published a thorough
benchmark study of both ground and excitedstate properties, using the
"LRCωPBEh" functional, a hybrid (hence the "h")
that contains a fraction of shortrange
HartreeFock exchange in addition to full longrange HartreeFock exchange:
E_{XC}^{}(LRC−ωPBEh) = E_{C}^{}(PBE) + E_{X}^{SR}(ωPBE) + C_{HF}^{} E_{X}^{SR}(HF) + E_{X}^{LR}(HF) 
 (4.48) 
The statisticallyoptimal parameter set, consider both groundstate properties and
TDDFT excitation energies together, was found to be C_{HF}^{} = 0.2 and
ω = 0.2 bohr^{−1} [101]. With these parameters, the
LRCωPBEh functional outperforms the traditional hybrid functional PBE0
for groundstate atomization energies and barrier heights. For TDDFT excitation
energies corresponding to localized excitations, TDPBE0 and TDLRCωPBEh
show similar statistical errors of ∼ 0.3 eV, but the latter functional also
exhibits only ∼ 0.3 eV errors for chargetransfer excitation energies, whereas
the statistical error for TDPBE0 chargetransfer excitation energies is 3.0 eV!
Caution is definitely warranted in the case of chargetransfer excited states,
however, as these excitation energies are very sensitive to the precise value of
ω [92,[101]. It was later found that the parameter
set (C_{HF}^{} = 0, ω = 0.3 bohr^{−1}) provides similar
(statistical) performance to that described above, although the predictions for
specific chargetransfer excited states can be somewhat different as compared to
the original parameter set [92].
Example 4.0 Application of LRCωPBEh to the C_{2}H_{4}C_{2}F_{4}
heterodimer at 5 Å separation.
$comment
This example uses the "optimal" parameter set discussed above.
$end
$molecule
0 1
C 0.670604 0.000000 0.000000
C 0.670604 0.000000 0.000000
H 1.249222 0.929447 0.000000
H 1.249222 0.929447 0.000000
H 1.249222 0.929447 0.000000
H 1.249222 0.929447 0.000000
C 0.669726 0.000000 5.000000
C 0.669726 0.000000 5.000000
F 1.401152 1.122634 5.000000
F 1.401152 1.122634 5.000000
F 1.401152 1.122634 5.000000
F 1.401152 1.122634 5.000000
$end
$rem
EXCHANGE GEN
BASIS 631+G*
LRC_DFT TRUE
OMEGA 200 ! = 0.2 a.u.
CIS_N_ROOTS 4
CIS_TRIPLETS FALSE
$end
$xc_functional
C PBE 1.00
X wPBE 0.80
X HF 0.20
$end
4.3.4.2 LRCDFT with the BNL Functional
The BaerNeuhauserLivshits (BNL) functional [64,[65]
is also based on the range separation of the Coulomb
operator in Eq. 4.45.
Its functional form resembles Eq. 4.47:
E_{XC}^{} = E_{C}^{} + C_{GGA,X} E_{X}^{GGA, SR} + E_{X}^{HF, LR} 
 (4.49) 
where the recommended GGA correlation functional is LYP.
The recommended GGA exchange functional is BNL,
which is described by a local functional [103].
For ground state properties, the optimized value for
C_{GGA,X} (scaling factor for the BNL exchange functional) was found to be 0.9.
The value of ω in BNL calculations can be
chosen in several different ways.
For example, one can use the optimized value ω=0.5 bohr^{−1}.
For calculation of excited states and properties related to orbital energies,
it is strongly recommend to tune ω as described
below[104,[105].
Systemspecific optimization of
ω is based on Koopmans conditions that would be satisfied for
the exact functional[104], that is, ω is varied
until the Koopmans IE/EA for the
HOMO/LUMO is equal to ∆E IE/EA.
Based on published
benchmarks [65,[106], this systemspecific
approach yields the most accurate values of IEs and excitation
energies.
The script that optimizes ω is called
.pl@ and is located in the $QC/bin directory.
The script optimizes ω in the range 0.10.8 (100800).
See the script for the instructions how
to modify the script to optimize in a broader range.
To execute the script, you need to create three inputs for a BNL
job using the same geometry and basis set for a neutral molecule (.in@),
anion (.in@), and cation (.in@), and then type
'OptOmegaIPEA.pl >& optomega'.
The script will run creating outputs for each step (_*@, _*@,
_*@) writing the optimization output into .
A similar script, .pl@, will optimize ω to satisfy
the Koopmans condition for the IP only. This script minimizes
J=(IP+ϵ_{HOMO})^{2}, not the absolute values.
Note:
(i) If the system does not have positive EA, then the tuning should be done
according to the IP condition only. The IPEA script will
yield a wrong value of ω in such cases.
(ii) In order for the scripts to work, one must
specify SCF_FINAL_PRINT=1 in the inputs.
The scripts look for specific regular expressions and will not work correctly
without this keyword.
(iii) When tuning omega we recommend taking the amount of X BNL in the
XC part as 1.0 and not 0.9. 
The $xc_functional keyword for a BNL calculation reads:
$xc_functional
X HF 1.0
X BNL 0.9
C LYP 1.0
$end
and the $rem keyword reads
$rem
EXCHANGE GENERAL
SEPARATE_JK TRUE
OMEGA 500 != 0.5 Bohr$^{1}$
DERSCREEN FALSE !if performing unrestricted calcn
IDERIV 0 !if performing unrestricted Hessian evaluation
$end
4.3.4.3 LRCDFT with ωB97, ωB97X, ωB97XD, and ωB97X2 Functionals
Also available in QChem are the ωB97 [63], ωB97X [63],
ωB97XD [99], and ωB97X2 [40] functionals,
recently developed by Chai and HeadGordon. These authors have proposed a very simple ansatz to extend any E_{X}^{GGA} to
E_{X}^{GGA,SR}, as long as the SR operator has considerable spatial
extent [63,[100]. With the use of flexible GGAs, such
as Becke97 functional [58], their new LRC hybrid
functionals [63,[99,[100]
outperform the corresponding global hybrid
functionals (i.e., B97) and popular hybrid functionals (e.g., B3LYP) in thermochemistry, kinetics, and noncovalent interactions, which has not been easily
achieved by the previous LRC hybrid functionals. In addition, the qualitative failures of the commonly used hybrid density functionals in
some "difficult problems", such as dissociation of symmetric radical cations and longrange chargetransfer excitations, are significantly reduced
by these new functionals [63,[99,[100]. Analytical gradients and analytical Hessians are available for ωB97, ωB97X, and ωB97XD.
Example 4.0 Application of ωB97 functional to nitrogen dimer.
$comment
Geometry optimization, followed by a TDDFT calculation.
$end
$molecule
0 1
N1
N2 N1 1.1
$end
$rem
jobtype opt
exchange omegaB97
basis 631G*
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97
basis 631G*
scf_guess READ
cis_n_roots 10
rpa true
$end
Example 4.0 Application of ωB97X functional to nitrogen dimer.
$comment
Frequency calculation (with analytical Hessian methods).
$end
$molecule
0 1
N1
N2 N1 1.1
$end
$rem
jobtype freq
exchange omegaB97X
basis 631G*
$end
Among these new LRC hybrid functionals, ωB97XD is a DFTD
(density functional theory with empirical dispersion corrections) functional, where
the total energy is computed as the sum of a DFT part and an empirical
atomicpairwise dispersion correction. Relative to ωB97 and ωB97X,
ωB97XD is significantly superior for nonbonded interactions,
and very similar in performance for bonded interactions. However, it should be noted
that the remaining shortrange selfinteraction error is somewhat
larger for ωB97XD than for ωB97X than for ωB97.
A careful reading of Refs.
is suggested prior to performing any DFT and TDDFT calculations based on variations
of ωB97 functional. ωB97XD functional automatically involves two keywords for the
dispersion correction, DFT_D and DFT_D_A, which are described in Section 4.3.6.
Example 4.0 Application of ωB97XD functional to methane dimer.
$comment
Geometry optimization.
$end
$molecule
0 1
C 0.000000 0.000323 1.755803
H 0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 1.024959 1.393014
H 0.000000 0.001084 2.842908
C 0.000000 0.000323 1.755803
H 0.000000 0.001084 2.842908
H 0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 1.024959 1.393014
$end
$rem
jobtype opt
exchange omegaB97XD
basis 631G*
$end
Similar to the existing doublehybrid density functional theory
(DHDFT) [39,[107,[108,[109,[85],
which is described in Section 4.3.9,
LRCDFT can be extended to include nonlocal orbital correlation energy from secondorder
MøllerPlesset perturbation theory (MP2) [110], that includes a samespin (ss) component
E_{c}^{ss}, and an oppositespin (os) component E_{c}^{os} of PT2 correlation energy. The two scaling
parameters, c_{ss} and c_{os}, are introduced to avoid doublecounting
correlation with the LRC hybrid functional.
E_{total} = E_{LRC−DFT} + c_{ss} E_{c}^{ss} + c_{os} E_{c}^{os} 
 (4.50) 
Among the ωB97 series, ωB97X2 [40]
is a longrange corrected doublehybrid (DH) functional, which can
greatly reduce the selfinteraction errors (due to its
high fraction of HartreeFock exchange), and has been shown
significantly superior for systems with bonded and nonbonded interactions. Due to the sensitivity of PT2
correlation energy with respect to the choices of basis sets, ωB97X2
was parameterized with two different basis sets. ωB97X2(LP) was parameterized with
the 6311++G(3df,3pd) basis set (the large Pople type basis set), while
ωB97X2(TQZ) was parameterized with the TQ extrapolation to the basis set limit.
A careful reading of Ref. is thus highly advised.
ωB97X2(LP) and ωB97X2(TQZ) automatically
involve three keywords for the PT2 correlation energy, DH, DH_SS and DH_OS,
which are described in Section 4.3.9. The PT2 correlation
energy can also be computed with the efficient resolutionofidentity
(RI) methods (see Section 5.5).
Example 4.0 Application of ωB97X2(LP) functional to LiH molecules.
$comment
Geometry optimization and frequency calculation on LiH, followed by
singlepoint calculations with nonRI and RI approaches.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype opt
exchange omegaB97X2(LP)
correlation mp2
basis 6311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype freq
exchange omegaB97X2(LP)
correlation mp2
basis 6311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X2(LP)
correlation mp2
basis 6311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X2(LP)
correlation rimp2
basis 6311++G(3df,3pd)
aux_basis rimp2augccpvtz
$end
Example 4.0 Application of ωB97X2(TQZ) functional to LiH molecules.
$comment
Singlepoint calculations on LiH.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype sp
exchange omegaB97X2(TQZ)
correlation mp2
basis ccpvqz
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X2(TQZ)
correlation rimp2
basis ccpvqz
aux_basis rimp2ccpvqz
$end
4.3.4.4 LRCDFT with the M11 Family of Functionals
The Minnesota family of functional by Truhlar's group has been recently updated
by adding two new functionals: M11L [79] and M11 [80].
The M11 functional is a longrange corrected metaGGA,
obtained by using the LRC scheme of Chai and HeadGordon (see above),
with the successful parameterization of the Minnesota metaGGA functionals:
E^{M11}_{xc} =  ⎛ ⎝

X
100
 ⎞ ⎠

E^{SR−HF}_{x} +  ⎛ ⎝

1 − 
X
100
 ⎞ ⎠

E^{SR−M11}_{x} + E^{LR−HF}_{x} + E^{M11}_{c} 
 (4.51) 
with the percentage of HartreeFock exchange at short range X being 42.8.
An extension of the LRC scheme to local functional (no HF exchange) was
introduced in the M11L functional by means of the dualrange exchange:
E^{M11−L}_{xc} = E^{SR−M11}_{x} + E^{LR−M11}_{x} + E^{M11−L}_{c} 
 (4.52) 
The correct longrange scheme is selected automatically with the input keywords.
A careful reading of the references [79,[80]
is suggested prior to performing any calculations with the M11 functionals.
Example 4.0 Application of M11 functional to water molecule
$comment
Optimization of H2O with M11
$end
$molecule
0 1
O 0.000000 0.000000 0.000000
H 0.000000 0.000000 0.956914
H 0.926363 0.000000 0.239868
$end
$rem
jobtype opt
exchange m11
basis 631+G(d,p)
$end
4.3.5 Nonlocal Correlation Functionals
QChem includes four nonlocal correlation functionals that describe longrange
dispersion (i.e. van der Waals) interactions:
 vdWDF04, developed by Langreth, Lundqvist, and coworkers [111,[112]
and implemented as described in Ref. [113];
 vdWDF10 (also known as vdWDF2), which is a reparameterization [114]
of vdWDF04, implemented in the same way as its precursor [113];
 VV09, developed [115] and implemented [116] by Vydrov and Van Voorhis;
 VV10 by Vydrov and Van Voorhis [117].
All these functionals are implemented selfconsistently and analytic gradients with respect
to nuclear displacements are available [113,[116,[117]. The nonlocal
correlation is governed by the $rem variable NL_CORRELATION, which can be set to
one of the four values: vdWDF04, vdWDF10, VV09, or VV10.
Note that vdWDF04, vdWDF10, and VV09 functionals are used in combination with LSDA correlation,
which must be specified explicitly. For instance, vdWDF10 is invoked by the following keyword
combination:
CORRELATION PW92
NL_CORRELATION vdWDF10
VV10 is used in combination with PBE correlation, which must be added explicitly.
In addition, the values of two parameters, C and b must be specified for VV10.
These parameters are controlled by the $rem variables NL_VV_C and
NL_VV_B, respectively. For instance, to invoke VV10 with C = 0.0093
and b = 5.9, the following input is used:
CORRELATION PBE
NL_CORRELATION VV10
NL_VV_C 93
NL_VV_B 590
The variable NL_VV_C may also be specified for VV09, where it has the same
meaning. By default, C = 0.0089 is used in VV09 (i.e. NL_VV_C is set to
89). However, in VV10 neither C nor b are assigned a default value and must
always be provided in the input.
As opposed to local (LSDA) and semilocal (GGA and metaGGA) functionals, evaluated as a single 3D
integral over space [see Eq. (4.37)], nonlocal functionals require double integration over the spatial
variables:
E_{c}^{nl} =  ⌠ ⌡

f(r,r′) dr dr′. 
 (4.53) 
In practice, this double integration is performed numerically on a quadrature
grid [113,[116,[117].
By default, the SG1 quadrature (described in Section 4.3.13 below) is used
to evaluate E_{c}^{nl}, but a different grid can be requested via the $rem variable
NL_GRID. The nonlocal energy is rather insensitive to the fineness of the grid, so that
SG1 or even SG0 grids can be used in most cases. However, a finer grid may be required for the (semi)local
parts of the functional, as controlled by the XC_GRID variable.
Example 4.0 Geometry optimization of the methane dimer using VV10 with rPW86 exchange.
$molecule
0 1
C 0.000000 0.000140 1.859161
H 0.888551 0.513060 1.494685
H 0.888551 0.513060 1.494685
H 0.000000 1.026339 1.494868
H 0.000000 0.000089 2.948284
C 0.000000 0.000140 1.859161
H 0.000000 0.000089 2.948284
H 0.888551 0.513060 1.494685
H 0.888551 0.513060 1.494685
H 0.000000 1.026339 1.494868
$end
$rem
JobType Opt
BASIS augccpVTZ
EXCHANGE rPW86
CORRELATION PBE
XC_GRID 75000302
NL_CORRELATION VV10
NL_GRID 1
NL_VV_C 93
NL_VV_B 590
$end
In the above example, an EML(75,302) grid is used to evaluate the rPW86 exchange and PBE
correlation, but a coarser SG1 grid is used for the nonlocal part of VV10.
4.3.6 DFTD Methods
4.3.6.1 Empirical dispersion correction from Grimme
Thanks to the efforts of the Sherrill group, the popular empirical
dispersion corrections due to Grimme [84] are
now available in QChem. Energies, analytic gradients, and analytic
second derivatives are available. Grimme's empirical dispersion
corrections can be added to any of the density functionals
available in QChem.
DFTD methods add an extra term,
 

−s_{6} 
∑
A


∑
B < A


C_{6}^{AB}
R_{AB}^{6}

f_{dmp}(R_{AB}) 
  (4.54) 
 

  (4.55) 
 

  (4.56) 

where s_{6} is a global scaling parameter (near unity), f_{dmp} is a
damping parameter meant to help avoid doublecounting correlation effects at
short range, d is a global scaling parameter for the damping function,
and R_{AB}^{0} is the sum of the van der Waals radii of atoms A and B.
DFTD using Grimme's parameters may be turned on using
DFT_D EMPIRICAL_GRIMME
Grimme has suggested scaling factors s_{6} of 0.75 for PBE, 1.2 for BLYP,
1.05 for BP86, and 1.05 for B3LYP; these are the default values of s_{6} when
those functionals are used. Otherwise, the default value of s_{6} is 1.0.
It is possible to specify different values of s_{6}, d, the atomic C_{6}
coefficients, or the van der Waals radii by using the
$empirical_dispersion keyword; for example:
$empirical_dispersion
S6 1.1
D 10.0
C6 Ar 4.60 Ne 0.60
VDW_RADII Ar 1.60 Ne 1.20
$end
Any values not specified explicitly will default to the values in Grimme's
model.
4.3.6.2 Empirical dispersion correction from Chai and HeadGordon
The empirical dispersion correction from Chai and HeadGordon [99]
employs a different damping function and can be activated by using
DFT_D EMPIRICAL_CHG
It uses another keyword DFT_D_A to control the strength of dispersion corrections.
DFT_D
Controls the application of DFTD or DFTD3 scheme. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  (or 0) Do not apply the DFTD or DFTD3 scheme 
EMPIRICAL_GRIMME  dispersion correction from Grimme 
EMPIRICAL_CHG  dispersion correction from Chai and HeadGordon 
EMPIRICAL_GRIMME3  dispersion correction from Grimme's DFTD3 method 
 (see Section 4.3.8)

RECOMMENDATION:

 DFT_D_A
Controls the strength of dispersion corrections in the ChaiHeadGordon DFTD scheme in Eq.(3) of Ref. . 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to a = n/100. 
RECOMMENDATION:



4.3.7 XDM DFT Model of Dispersion
While standard DFT functionals describe chemical bonds relatively
well, one major deficiency is their inability
to cope with dispersion interactions, i.e., van der Waals (vdW) interactions.
Becke and Johnson have proposed a conceptually simple yet accurate
dispersion model called the exchangedipole model (XDM) [33,[118].
In this model the dispersion attraction emerges from the interaction between the instant
dipole moment of the exchange hole in one molecule and the induced
dipole moment in another. It is a conceptually simple but powerful approach
that has been shown to yield very accurate dispersion coefficients without
fitting parameters. This allows the calculation of both intermolecular
and intramolecular dispersion interactions within a single DFT framework.
The implementation and validation of this method in the QChem code
is described in Ref. .
Fundamental to the XDM model is the calculation of the norm of the dipole
moment of the exchange hole at a given point:
d_{σ}(r)=−  ⌠ ⌡

h_{σ}(r,r^{′})r^{′}d^{3}r^{′}−r 
 (4.57) 
where σ labels the spin and h_{σ}(r,r^{′}) is
the exchangehole function. The XDM version that is implemented
in QChem employs the BeckeRoussel (BR) model exchangehole function.
It was not given in an analytical form and one had to determine
its value at each grid point numerically.
QChem has developed for the first time an analytical
expression for this function based on nonlinear interpolation and spline
techniques, which greatly improves efficiency as well as the numerical
stability [28].
There are two different damping functions used in the XDM model of
Becke and Johnson. One of them uses only the intermolecular C6 dispersion
coefficient. In its QChem implementation it is denoted as "XDM6". In this version
the dispersion energy is computed as
E_{vdW}= 
∑
 E_{vdW},_{ij}=− 
∑
i > j


C_{6,ij}
R_{ij}^{6}+kC_{6,ij}/(E_{i}^{C}+E_{j}^{C})


 (4.58) 
where k is a universal parameter, R_{ij}^{} is the distance
between atoms i and j, and E_{ij}^{C} is the sum of the absolute values of
the correlation energy of free atoms i and j. The dispersion
coefficients C_{6,ij} is computed as
C_{6,ij}= 
〈d_{X}^{2}〉_{i}〈d_{X}^{2}〉_{j}α_{i}α_{j}
〈d_{X}^{2}〉_{i}α_{j}+〈d_{X}^{2}〉_{j}α_{i}


 (4.59) 
where 〈d_{X}^{2}〉_{i} is the exchange hole dipole moment of
the atom, and α_{i} is the effective polarizability of the atom i in
the molecule.
The XDM6 scheme is further generalized to include higherorder dispersion
coefficients, which leads to the "XDM10" model in QChem implementation. The
dispersion energy damping function used in XDM10 is
E_{vdW}=− 
∑
i > j

 ⎛ ⎝

C_{6,ij}
R_{vdW,ij}^{6}+R_{ij}^{6}

+ 
C_{8,ij}
R_{vdW,ij}^{8}+R_{ij}^{8}

+ 
C_{10,ij}
R_{vdW,ij}^{10}+R_{ij}^{10}
 ⎞ ⎠


 (4.60) 
where C_{6,ij}, C_{8,ij} and C_{10,ij} are dispersion coefficients
computed at higherorder multipole (including dipole, quadrupole and octopole)
moments of the exchange hole [120]. In above,
R_{vdW,ij}^{} is the sum of the effective
vdW radii of atoms i and j, which is a linear function of the
so called critical distance R_{C,ij}^{} between atoms i and j:
R_{vdW,ij}^{}=a_{1}R_{C,ij}^{}+a_{2} 
 (4.61) 
The critical distance, R_{C,ij}^{}, is computed by averaging these three distances:
R_{C,ij}^{} = 
1
3

 ⎡ ⎣
 ⎛ ⎝

C_{8,ij}
C_{6,ij}
 ⎞ ⎠

1/2

+  ⎛ ⎝

C_{10,ij}
C_{6,ij}
 ⎞ ⎠

1/4

+  ⎛ ⎝

C_{10,ij}
C_{8,ij}
 ⎞ ⎠

1/2
 ⎤ ⎦


 (4.62) 
In the XDM10 scheme there are two universal parameters, a_{1} and a_{2}. Their
default values of 0.83 and 1.35, respectively, are
due to Johnson and Becke [118], determined by
least square fitting to the binding energies of a set
of intermolecular complexes. Please keep in mind that these values are not
the only possible optimal set to use with XDM.
Becke's group has suggested later on several
other XC functional combinations with XDM
that employ different a_{1} and a_{2}
values. The user is advised to consult their recent papers for more details
(e.g., Refs. ).
The computed vdW energy is added as a postSCF correction. In addition, QChem
also has implemented the first and second nuclear derivatives of vdW energy
correction in both the XDM6 and XDM10 schemes.
Listed below are a number of useful options to customize the vdW calculation
based on the XDM DFT approach.
DFTVDW_JOBNUMBER
TYPE:
DEFAULT:
OPTIONS:
0  Do not apply the XDM scheme. 
1  add vdW gradient correction to SCF. 
2  add VDW as a DFT functional and do full SCF. 
RECOMMENDATION:
This option only works with C6 XDM formula 

 DFTVDW_METHOD
Choose the damping function used in XDM 
TYPE:
DEFAULT:
OPTIONS:
1  use Becke's damping function including C6 term only. 
2  use Becke's damping function with higherorder (C8,C10) terms. 
RECOMMENDATION:



DFTVDW_MOL1NATOMS
The number of atoms in the first monomer in dimer calculation 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:

 DFTVDW_KAI
Damping factor K for C6 only damping function 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:



DFTVDW_ALPHA1
Parameter in XDM calculation with higherorder terms 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:

 DFTVDW_ALPHA2
Parameter in XDM calculation with higherorder terms. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:



DFTVDW_USE_ELE_DRV
Specify whether to add the gradient correction to the XDM energy.
only valid with Becke's C6 damping function
using the interpolated BR89 model. 
TYPE:
DEFAULT:
OPTIONS:
1  use density correction when applicable (default). 
0  do not use this correction (for debugging purpose) 
RECOMMENDATION:

 DFTVDW_PRINT
Printing control for VDW code 
TYPE:
DEFAULT:
OPTIONS:
0 no printing. 
1  minimum printing (default) 
2  debug printing 
RECOMMENDATION:



Example 4.0 Below is a sample input
illustrating a frequency calculation
of a vdW complex consisted of He atom and N2 molecule.
$molecule
0 1
He .0 .0 3.8
N .000000 .000000 0.546986
N .000000 .000000 0.546986
$end
$rem
JOBTYPE FREQ
IDERIV 2
EXCHANGE B3LYP
!default SCF setting
INCDFT 0
SCF_CONVERGENCE 8
BASIS 631G*
XC_GRID 1
SCF_GUESS SAD
!vdw parameters setting
DFTVDW_JOBNUMBER 1
DFTVDW_METHOD 1
DFTVDW_PRINT 0
DFTVDW_KAI 800
DFTVDW_USE_ELE_DRV 0
$end
One should note that the XDM option can be used in
conjunction with different GGA, metaGGA pure or hybrid functionals,
even though the original implementation of Becke and Johnson
was in combination with HartreeFock exchange, or with a specific
metaGGA exchange and correlation (the BR89 exchange and the BR94
correlation described in previous sections above). For example,
encouraging results were obtained using the XDM option with the
popular B3LYP [119]. Becke has found more recently
that this model can be efficiently combined with the old GGA
exchange of Perdew 86 (the P86 exchange option in QChem), and
with his hyperGGA functional B05. Using XDM together
with PBE exchange plus LYP correlation, or PBE exchange plus
BR94 correlation has been also found fruitful.
4.3.8 DFTD3 Methods
Recently, Grimme proposed DFTD3 method [123] to
improve his previous DFTD method [84] (see Section 4.3.6).
Energies and analytic gradients of DFTD3 methods are available in QChem.
Grimme's DFTD3 method can be combined with any of the
density functionals available in QChem.
The total DFTD3 energy is given by
E_{DFT−D3} = E_{KS−DFT} + E_{disp} 
 (4.63) 
where E_{KSDFT} is the total energy from KSDFT and E_{disp} is the dispersion correction
as a sum of two and threebody energies,
E_{disp} = E^{(2)}+E^{(3)} 
 (4.64) 
DFTD3 method can be turned on by five keywords, DFT_D, DFT_D3_S6, DFT_D3_RS6, DFT_D3_S8 and DFT_D3_3BODY.
DFT_D
Controls the application of DFTD3 or DFTD scheme. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  (or 0) Do not apply the DFTD3 or DFTD scheme 
EMPIRICAL_GRIMME3  dispersion correction from Grimme's DFTD3 method 
EMPIRICAL_GRIMME  dispersion correction from Grimme (see Section 4.3.6) 
EMPIRICAL_CHG  dispersion correction from Chai and HeadGordon (see Section 4.3.6)

RECOMMENDATION:

Grimme suggested three scaling factors s_{6}, s_{r,6} and s_{8} that were optimized for several functionals
(see Table IV in Ref. ).
For example, s_{r,6} of 1.217 and s_{8} of 0.722 for PBE, 1.094 and 1.682 for BLYP, 1.261 and 1.703 for B3LYP, 1.532 and 0.862 for PW6B95,
0.892 and 0.909 for BECKE97, and 1.287 and 0.928 for PBE0; these are the QChem
default values of s_{r,6} and s_{8}. Otherwise, the default values
of s_{6}, s_{r,6} and s_{8} are 1.0.
DFT_D3_S6
Controls the strength of dispersion corrections, s_{6}, in Grimme's DFTD3 method (see Table IV in
Ref. ). 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to s_{6} = n/1000. 
RECOMMENDATION:

 DFT_D3_RS6
Controls the strength of dispersion corrections, s_{r6}, in the Grimme's DFTD3 method (see Table IV in Ref. ). 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to s_{r6} = n/1000. 
RECOMMENDATION:



DFT_D3_S8
Controls the strength of dispersion corrections, s_{8}, in Grimme's DFTD3 method (see Table IV in
Ref. ). 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to s_{8} = n/1000. 
RECOMMENDATION:

The threebody interaction term, mentioned in Ref. , can also be turned on, if needed.
DFT_D3_3BODY
Controls whether the threebody interaction in Grimme's DFTD3 method should be applied
(see Eq. (14) in Ref. ). 
TYPE:
DEFAULT:
OPTIONS:
FALSE  (or 0) Do not apply the threebody interaction term 
TRUE  Apply the threebody interaction term

RECOMMENDATION:

Example 4.0 Applications of B3LYPD3 to a methane dimer.
$comment
Geometry optimization, followed by singlepoint calculations with a larger basis set.
$end
$molecule
0 1
C 0.000000 0.000323 1.755803
H 0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 1.024959 1.393014
H 0.000000 0.001084 2.842908
C 0.000000 0.000323 1.755803
H 0.000000 0.001084 2.842908
H 0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 1.024959 1.393014
$end
$rem
jobtype opt
exchange B3LYP
basis 631G*
DFT_D EMPIRICAL_GRIMME3
DFT_D3_S6 1000
DFT_D3_RS6 1261
DFT_D3_S8 1703
DFT_D3_3BODY FALSE
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange B3LYP
basis 6311++G**
DFT_D EMPIRICAL_GRIMME3
DFT_D3_S6 1000
DFT_D3_RS6 1261
DFT_D3_S8 1703
DFT_D3_3BODY FALSE
$end
4.3.9 DoubleHybrid Density Functional Theory
The recent advance in doublehybrid density functional theory
(DHDFT) [39,[107,[108,[109,[85],
has demonstrated its great potential for approaching the chemical accuracy with a
computational cost comparable to the secondorder MøllerPlesset perturbation theory (MP2).
In a DHDFT, a KohnSham (KS) DFT
calculation is performed first, followed by a treatment of
nonlocal orbital correlation energy at the level of secondorder MøllerPlesset perturbation
theory (MP2) [110]. This MP2 correlation correction includes a
a samespin (ss) component, E_{c}^{ss}, as well as an
oppositespin (os) component, E_{c}^{os}, which are added to the total energy obtained from
the KSDFT calculation. Two scaling parameters, c_{ss} and c_{os}, are introduced
in order to avoid doublecounting correlation:
E_{DH−DFT} = E_{KS−DFT} + c_{ss} E_{c}^{ss} + c_{os} E_{c}^{os} 
 (4.65) 
Among DH functionals, ωB97X2 [40], a longrange corrected
DH functional, is described in Section 4.3.4.3.
There are three keywords for turning on DHDFT as below.
DH
Controls the application of DHDFT scheme. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  (or 0) Do not apply the DHDFT scheme 
TRUE  (or 1) Apply DHDFT scheme 
RECOMMENDATION:

 DH_SS
Controls the strength of the samespin component of PT2 correlation energy. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to c_{ss} = n/1000000 in Eq. (4.65). 
RECOMMENDATION:



DH_OS
Controls the strength of the oppositespin component of PT2 correlation energy. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to c_{os} = n/1000000 in Eq. (4.65). 
RECOMMENDATION:

For example, B2PLYP [84], which involves 53% HartreeFock exchange,
47% Becke 88 GGA exchange, 73% LYP GGA correlation and 27% PT2
orbital correlation, can be called with the following input file.
The PT2 correlation energy can also be computed with the
efficient resolutionofidentity (RI) methods
(see Section 5.5).
Example 4.0 Applications of B2PLYP functional to LiH molecule.
$comment
Geometry optimization and frequency calculation on LiH, followed by
singlepoint calculations with nonRI and RI approaches.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype opt
exchange general
correlation mp2
basis ccpvtz
DH 1
DH_SS 270000 !0.27 = 270000/1000000
DH_OS 270000 !0.27 = 270000/1000000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype freq
exchange general
correlation mp2
basis ccpvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange general
correlation mp2
basis ccpvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange general
correlation rimp2
basis ccpvtz
aux_basis rimp2ccpvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
A more detailed gist of one particular class of DH functionals, the
XYG3 & XYGJOS functionals follows below thanks to Dr Yousung Jung
who implemented these functionals in QChem.
A starting point of these DH functionals is the adiabatic connection formula
which provides a rigorous way to define them. One considers an adiabatic
path between the fictitious noninteracting KohnSham system (λ= 0)
and the real physical system (λ= 1) while holding the electron
density fixed at its physical state for all λ:
E_{XC} [ρ]=  ⌠ ⌡

1
0

U_{XC,λ} [ρ]dλ , 
 (4.66) 
where U_{XC,λ} is the exchange correlation potential energy at
a coupling strength λ. If one assumes a linear model of the latter:
one obtains the popular hybrid functional that includes the HartreeFock exchange
(or occupied orbitals) such as B3LYP. If one further uses the GorlingLevy's
perturbation theory (GL2) to define the initial slope at λ= 0,
one obtains the doubly hybrid functional (see Eq. 4.65) that includes
MP2 type perturbative terms (PT2) involving virtual KohnSham orbitals:
U_{XC,λ} = 
∂U_{XC,λ}
λ
 ⎢ ⎢

λ = 0

=2E_{C}^{GL2} . 
 (4.68) 
In the DH functional XYG3, as implemented in QChem, the B3LYP orbitals
are used to generate the density and evaluate the PT2 terms. This is different
from P2PLYP, an earlier doubly hybrid functional by Grimme. P2PLYP
uses truncated KohnSham orbitals while XYG3 uses converged KS orbitals
to evaluate the PT2 terms. The performance of XYG3 is not only comparable
to that of the G3 or G2 theory for thermochemistry, but barrier heights
and noncovalent interactions, including stacking interactions,
are also very well described by XYG3 [85].
The recommended basis set for XYG3 is 6311+G(3df,2p).
Due to the inclusion of PT2 terms, XYG3 or all other forms of doubly
hybrid functionals formally scale as the 5th power of system size as
in conventional MP2, thereby not applicable to large systems and
partly losing DFT's cost advantages. With the success of SOSMP2 and
local SOSMP2 algorithms developed in QChem, the natural extension
of XYG3 is to include only oppositespin correlation contributions,
ignoring the samespin parts. The resulting DH functional is XYGJOS
also implemented in QChem. It has 4 parameters that are optimized
using thermochemistry data. This new functional is both accurate
(comparable or even slightly better than XYG3) and faster. If the local
algorithm is applied, the formal scaling of XYGJOS is cubic, without
the locality, it has still 4th order scaling.
Example 1: XYG3 calculation of N2. XYG3 invokes automatically
the B3LYP calculation first, and use the resulting orbitals
for evaluating the MP2type correction terms. One can also
use XYG3 combined with RI approximation for the PT2 terms; use
EXCHANGE = XYG3RI to do so, along with an appropriate
choice of auxiliary basis set.
Example 4.0 XYG3 calculation of N2
$molecule
0 1
N 0.00000000 0.00000000 0.54777500
N 0.00000000 0.00000000 0.54777500
$end
$rem
exchange xyg3
basis 6311+G(3df,2p)
$end
Example 2: XYGJOS calculation of N2. Since it uses
the RI approximation by default, one must define the auxiliary basis.
Example 4.0 XYGJOS calculation of N2
$molecule
0 1
N 0.00000000 0.00000000 0.54777500
N 0.00000000 0.00000000 0.54777500
$end
$rem
exchange xygjos
basis 6311+G(3df,2p)
aux_basis rimp2ccpVtZ
purecart 1111
time_mp2 true
$end
Example 3: Local XYGJOS calculation of N2. The same as XYGJOS,
except for the use of the attenuated Coulomb metric to solve
the RI coefficients. Omega determines the locality of the metric.
Example 4.0 Local XYGJOS calculation of N2
$molecule
0 1
N 0.000 0.000 0.54777500
N 0.000 0.000 0.54777500
$end
$rem
exchange lxygjos
omega 200
basis 6311+G(3df,2p)
aux_basis rimp2ccpVtZ
purecart 1111
$end
4.3.10 Asymptotically Corrected ExchangeCorrelation Potentials
It is well known that no gradientcorrected exchange functional can simultaneously produce the correct
contribution to the exchange energy density and exchange potential in the
asymptotic region of molecular systems [124]. Existing GGA exchangecorrelation
(xc) potentials decay much faster than the correct −1/r xc potential in the asymptotic
region [125]. Highlying occupied orbitals and lowlying virtual orbitals are therefore
too loosely bounded from these GGA functionals, and the minus HOMO energy becomes
much less than the exact ionization potential (as required by the exact
DFT) [126,[127].
Moreover, response properties could be poorly predicted from TDDFT calculations
with GGA functionals [127]. Longrange corrected hybrid DFT (LRCDFT), described in
Section 4.3.4, has greatly remedied this situation. However, due to the
use of longrange HF exchange, LRCDFT is computationally more expensive than KSDFT with GGA functionals.
To circumvent this, van Leeuwen and Baerends proposed an asymptotically corrected (AC) exchange potential [124]:
v_{x}^{LB} = −β 
x^{2}
1+3 βsinh^{−1}(x)


 (4.69) 
that will reduce to −1/r, for an exponentially decaying density, in the
asymptotic region of molecular systems, where x = [( ∇ρ)/(ρ^{4/3})] is the reduced
density gradient. The LB94 xc potential is formed by a linear combination of LDA
xc potential and the LB exchange potential [124]:
v_{xc}^{LB94} = v_{xc}^{LDA} + v_{x}^{LB} 
 (4.70) 
The parameter β was determined by fitting the LB94 xc potential to the beryllium atom.
As mentioned in Ref. , for TDDFT and TDDFT/TDA calculations,
it is sufficient to include the AC xc potential for groundstate calculations followed by TDDFT
calculations with an adiabatic LDA xc kernel. The implementation of LB94 xc potential
in QChem thus follows this; using LB94 xc potential for groundstate calculations, followed
by TDDFT calculations with an adiabatic LDA xc kernel. This TDLDA/LB94 approach
has been widely applied to study excitedstate properties of large molecules in literature.
Since the LB exchange potential does not come from the functional derivative of some exchange
functional, we use the LevyPerdew virial relation [130] (implemented in QChem)
to obtain its exchange energy:
E_{x}^{LB} = −  ⌠ ⌡

v_{x}^{LB}([ρ],r)[3ρ(r)+r∇ρ(r)]dr 
 (4.71) 
LB94_BETA
Set the β parameter of LB94 xc potential 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to β = n/10000. 
RECOMMENDATION:
Use default, i.e., β = 0.05 

Example 4.0 Applications of LB94 xc potential to N_{2} molecule.
$comment
TDLDA/LB94 calculation is performed for excitation energies.
$end
$molecule
0 1
N 0.0000 0.0000 0.0000
N 1.0977 0.0000 0.0000
$end
$rem
jobtype = sp
exchange = lb94
basis = 6311(2+,2+)G**
cis_n_roots = 30
rpa = true
$end
4.3.11 DFT Numerical Quadrature
In practical DFT calculations, the forms of the approximate
exchangecorrelation functionals used are quite complicated, such that the
required integrals involving the functionals generally cannot be evaluated
analytically. QChem evaluates these integrals through numerical quadrature
directly applied to the exchangecorrelation integrand (i.e., no fitting of
the XC potential in an auxiliary basis is done). QChem provides a standard
quadrature grid by default which is sufficient for most purposes.
The quadrature approach in QChem is generally similar to that found in many
DFT programs. The multicenter XC integrals are first partitioned into
"atomic" contributions using a nuclear weight function. QChem uses the
nuclear partitioning of Becke [131], though without the atomic
size adjustments". The atomic integrals are then evaluated through standard
onecenter numerical techniques.
Thus, the exchangecorrelation energy E_{XC} is obtained as
E_{XC} = 
∑
A


∑
i

w_{Ai} f( r_{Ai} ) 
 (4.72) 
where the first summation is over the atoms and the second is over the
numerical quadrature grid points for the current atom. The f function is the
exchangecorrelation functional. The w_{Ai} are the quadrature weights, and
the grid points r_{Ai} are given by
where R_{A} is the position of nucleus A, with the
r_{i} defining a suitable onecenter integration grid, which is
independent of the nuclear configuration.
The singlecenter integrations are further separated into radial and angular
integrations. Within QChem, the radial part is usually treated by the
EulerMaclaurin scheme proposed by Murry et al. [132]. This scheme
maps the semiinfinite domain [0,∞)→ [0,1) and applies the extended
trapezoidal rule to the transformed integrand. Recently Gill and
Chien [133] proposed a radial scheme based on a Gaussian quadrature on
the interval [0,1] with weight function ln^{2}x. This scheme is exact for
integrands that are a linear combination of a geometric sequence of exponential
functions, and is therefore well suited to evaluating atomic integrals. The
authors refer to this scheme as MultiExp.
4.3.12 Angular Grids
Angular quadrature rules may be characterized by their degree, which is the
highest degree of spherical harmonics for which the formula is exact, and
their efficiency, which is the number of spherical harmonics exactly
integrated per degree of freedom in the formula. QChem supports the
following types of angular grids:
Lebedev
These are specially constructed grids for quadrature on the surface of a
sphere [134,[135,[136] based on the octahedral group.
Lebedev grids of the following degrees are available:
Degree  3rd  5th  7th  9th  11th  15th  17th  19th  23rd  29th 
Points  6  18  26  38  50  86  110  146  194  302 
Additional grids with the following number of points are also available: 74,
170, 230, 266, 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702,
3074, 3470, 3890, 4334, 4802, 5294. Lebedev grids typically have
efficiencies near one, with efficiencies greater than one in some cases.
GaussLegendre
These are spherical product rules separating the two angular dimensions
θ and ϕ. Integration in the θ dimension is carried out
with a Gaussian quadrature rule derived from the Legendre polynomials
(orthogonal on [−1,1] with weight function unity), while the ϕ
integration is done with equally spaced points.
A GaussLegendre grid is selected by specifying the total number of points,
2N^{2}, to be used for the integration. This gives a grid with
2N_{ϕ} ϕpoints, N_{θ} θpoints, and a degree of 2N−1.
In contrast with Lebedev grids, GaussLegendre grids have efficiency of only
2/3 (hence more GaussLegendre points are required to attain the same
accuracy as Lebedev). However, since GaussLegendre grids of general degree
are available, this is a convenient mechanism for achieving arbitrary
accuracy in the angular integration if desired.
Combining these radial and angular schemes yields an intimidating selection
of threedimensional quadratures. In practice, is it useful to standardize
the grids used in order to facilitate the comparison of calculations at
different levels of theory.
4.3.13 Standard Quadrature Grids
Both the SG0 [137] and SG1 [138] standard quadrature
grids were designed to yield the performance of a large, accurate quadrature
grid, but with as few points as possible for the sake of computational
efficiency. This is accomplished by reducing the number of angular points in
regions where sophisticated angular quadrature is not necessary, such as near
the nuclei where the charge density is nearly spherically symmetric, while
retaining large numbers of angular points in the valence region where angular
accuracy is critical.
The SG0 grid was derived in this fashion from a MultiExpLebedev(23,170),
(i.e., 23 radial points and 170 angular points per radial point). This grid
was pruned whilst ensuring the error in the computed exchange energies for the
atoms and a selection of small molecules was not larger than the corresponding
error associated with SG1. In our evaluation, the RMS error associated with
the atomization energies for the molecules in the G1 data set is 72
microhartrees. While relative energies are expected to be reproduced well by
this scheme, if absolute energies are being sought, a larger grid is
recommended.
The SG0 grid is implemented in QChem from H to micro Hartrees, excepted He and Na; in
this scheme, each atom has around 1400point, and SG1 is used for those
their SG0 grids have not been defined. It should be noted that, since
the SG0 grid used for H has been reoptimized in this version of QChem
(version 3.0), quantities calculated in this scheme may not reproduce those
generated by the last version (version 2.1).
The SG1 grid is derived from a EulerMaclaurinLebedev(50,194) grid
(i.e., 50 radial points, and 194 angular points per radial point). This grid
has been found to give numerical integration errors of the order of 0.2 kcal/mol
for mediumsized molecules, including particularly demanding test
cases such as isomerization energies of alkanes. This error is deemed
acceptable since it is significantly smaller than the accuracy typically
achieved by quantum chemical methods. In SG1 the total number of points is
reduced to approximately 1/4 of that of the original EML(50,194) grid, with
SG1 generally giving the same total energies as EML(50,194) to within a few
microhartrees (0.01 kcal/mol). Therefore, the SG1 grid is relatively
efficient while still maintaining the numerical accuracy necessary for chemical
reliability in the majority of applications.
Both the SG0 and SG1 grids were optimized so that the error in the energy
when using the grid did not exceed a target threshold. For single point
calculations this criterion is appropriate. However, derivatives of the energy
can be more sensitive to the quality of the integration grid, and it is
recommended that a larger grid be used when calculating these. Special care
is required when performing DFT vibrational calculations as imaginary
frequencies can be reported if the grid is inadequate. This is more of a
problem with lowfrequency vibrations. If imaginary frequencies are found, or
if there is some doubt about the frequencies reported by QChem, the
recommended procedure is to perform the calculation again with a larger grid
and check for convergence of the frequencies. Of course the geometry must be
reoptimized, but if the existing geometry is used as an initial guess, the
geometry optimization should converge in only a few cycles.
4.3.14 Consistency Check and Cutoffs for Numerical Integration
Whenever QChem calculates numerical density functional integrals, the
electron density itself is also integrated numerically as a test on the
quality of the quadrature formula used. The deviation of the numerical
result from the number of electrons in the system is an indication of the
accuracy of the other numerical integrals. If the relative error in the
numerical electron count reaches 0.01%, a warning is printed; this is an
indication that the numerical XC results may not be reliable. If the warning
appears at the first SCF cycle, it is probably not serious, because the
initialguess density matrix is sometimes not idempotent, as is the case
with the SAD guess and the density matrix taken from a different geometry in
a geometry optimization. If that is the case, the problem will be corrected
as the idempotency is restored in later cycles. On the other hand, if the
warning is persistent to the end of SCF iterations, then either a finer grid
is needed, or choose an alternative method for generating the initial guess.
Users should be aware, however, of the potential flaws that have been
discovered in some of the grids currently in use. Jarecki and Davidson [139],
for example, have recently shown that correctly
integrating the density is a necessary, but not sufficient, test of grid
quality.
By default, QChem will estimate the magnitude of various XC contributions on
the grid and eliminate those determined to be numerically insignificant.
QChem uses specially developed cutoff procedures which permits evaluation of
the XC energy and potential in only O(N) work for large molecules, where N
is the size of the system. This is a significant improvement over the formal
O(N^{3}) scaling of the XC cost, and is critical in enabling DFT calculations
to be carried out on very large systems. In very rare cases, however, the
default cutoff scheme can be too aggressive, eliminating contributions that
should be retained; this is almost always signaled by an inaccurate numerical
density integral. An example of when this could occur is in calculating anions
with multiple sets of diffuse functions in the basis. As mentioned above, when
an inaccurate electron count is obtained, it maybe possible to remedy the
problem by increasing the size of the quadrature grid.
Finally we note that early implementations of quadraturebased KohnSham DFT
employing standard basis sets were plagued by lack of rotational invariance.
That is, rotation of the system yielded a significantly energy change.
Clearly, such behavior is highly undesirable. Johnson et al. rectified the
problem of rotational invariance by completing the specification of the grid
procedure [140] to ensure that the computed XC energy is the same
for any orientation of the molecule in any Cartesian coordinate system.
4.3.15 Basic DFT Job Control
Three $rem variables are required to run a DFT job: EXCHANGE,
CORRELATION and BASIS. In addition, all of the basic input
options discussed for HartreeFock calculations in Section
4.2.3, and the extended options discussed in Section
4.2.4 are all valid for DFT calculations. Below we list
only the basic DFTspecific options (keywords).
 EXCHANGE
Specifies the exchange functional or exchangecorrelation functional for hybrid. 
TYPE:
DEFAULT:
No default exchange functional 
OPTIONS:
NAME  Use EXCHANGE = NAME, where NAME is 
 one of the exchange functionals listed in Table 4.2. 
RECOMMENDATION:
Consult the literature to guide your selection. 



CORRELATION
Specifies the correlation functional. 
TYPE:
DEFAULT:
OPTIONS:
None  No correlation 
VWN  VoskoWilkNusair parameterization #5 
LYP  LeeYangParr (LYP) 
PW91, PW  GGA91 (PerdewWang) 
PW92  LSDA 92 (Perdew and Wang) [44] 
LYP(EDF1)  LYP(EDF1) parameterization 
Perdew86, P86  Perdew 1986 
PZ81, PZ  PerdewZunger 1981 
PBE  PerdewBurkeErnzerhof 1996 
TPSS  The correlation component of the TPSS functional 
B94  Becke 1994 correlation in fully analytic form 
B94hyb  Becke 1994 correlation as above, but readjusted for use only within 
 the hybrid scheme BR89B94hyb 
PK06  ProynovKong 2006 correlation (known also as "tLap" 
(B88)OP  OP correlation [66], optimized for use with B88 exchange 
(PBE)OP  OP correlation [66], optimized for use with PBE exchange 
Wigner  Wigner 
RECOMMENDATION:
Consult the literature to guide your selection. 


EXCHANGE =  Description  

HF  Fock exchange 
Slater, S  Slater (Dirac 1930) 
Becke86, B86  Becke 1986 
Becke, B, B88  Becke 1988 
muB88  Shortrange Becke exchange, as formulated by Song et al. [97] 
Gill96, Gill  Gill 1996 
GG99  Gilbert and Gill, 1999 
Becke(EDF1), B(EDF1)  Becke (uses EDF1 parameters) 
PW86,  PerdewWang 1986 
rPW86,  Refitted PW86 [48] 
PW91, PW  PerdewWang 1991 
PBE  PerdewBurkeErnzerhof 1996 
TPSS  The nonempirical exchangecorrelation scheme of Tao, 
 Perdew, Staroverov, and Scuseria (requires also that the user 
 specify "TPSS" for correlation) 
TPSSH  The hybrid version of TPSS (with no input line for correlation) 
PBE0, PBE1PBE  PBE hybrid with 25% HF exchange 
revPBE  revised PBE exchange [56] 
PBEOP  PBE exchange + oneparameter progressive correlation 
wPBE  Shortrange ωPBE exchange, as formulated by
Henderson et al. [98] 
muPBE  Shortrange μPBE exchange, as formulated by
Song et al. [97] 
B97  Becke97 XC hybrid 
B971  Becke97 reoptimized by Hamprecht et al. (1998) 
B972  Becke971 optimized further by Wilson et al. (2001) 
B3PW91, Becke3PW91, B3P  B3PW91 hybrid 
B3LYP, Becke3LYP  B3LYP hybrid 
B3LYP5  B3LYP based on correlation functional #5 of Vosko, Wilk, 
 and Nusair (rather than their functional #3) 
HCTH  HCTH hybrid 
HCTH120  HCTH120 hybrid 
HCTH147  HCTH147 hybrid 
HCTH407  HCTH407 hybrid 
BOP  B88 exchange + oneparameter progressive correlation 
EDF1  EDF1 
EDF2  EDF2 
VSXC  VSXC metaGGA, not a hybrid 
BMK  BMK hybrid 
M05  M05 hybrid 
M052X  M052X hybrid 
M06L  M06L hybrid 
M06HF  M06HF hybrid 
M06  M06 hybrid 
M062X  M062X hybrid 
M08HX  M08HX hybrid 
M08SO  M08SO hybrid 
M11L  M11L hybrid 
M11  M11 longrange corrected hybrid 
SOGGA  SOGGA hybrid 
SOGGA11  SOGGA11 hybrid 
SOGGA11X  SOGGA11X hybrid 
BR89  BeckeRoussel 1989 represented in analytic form 
omegaB97  ωB97 longrange corrected hybrid 
omegaB97X  ωB97X longrange corrected hybrid 
omegaB97XD  ωB97XD longrange corrected hybrid with dispersion 
 corrections 
omegaB97X2(LP)  ωB97X2(LP) longrange corrected doublehybrid 
omegaB97X2(TQZ)  ωB97X2(TQZ) longrange corrected doublehybrid 
MCY2  The MCY2 hyperGGA exchangecorrelation (with no 
 input line for correlation) 
B05  Full exactexchange hyperGGA functional of Becke 05 with 
 RI approximation for the exactexchange energy density 
BM05  Modified B05 hyperGGA scheme with RI approximation for 
 the exactexchange energy density used as a variable. 
General, Gen  User defined combination of K, X and C (refer to the next 
 section) 

Table 4.2: DFT exchange functionals available within QChem. 

NL_CORRELATION
Specifies a nonlocal correlation functional that includes nonempirical dispersion. 
TYPE:
DEFAULT:
None  No nonlocal correlation. 
OPTIONS:
None  No nonlocal correlation 
vdWDF04  the nonlocal part of vdWDF04 
vdWDF10  the nonlocal part of vdWDF10 (aka vdWDF2) 
VV09  the nonlocal part of VV09 
VV10  the nonlocal part of VV10 
RECOMMENDATION:
Do not forget to add the LSDA correlation (PW92 is recommended) when
using vdWDF04, vdWDF10, or VV09. VV10 should be used with PBE
correlation. Choose exchange functionals carefully: HF, rPW86, revPBE, and
some of the LRC exchange functionals are among the recommended choices. 

 NL_VV_C
Sets the parameter C in VV09 and VV10. This parameter is fitted to asymptotic
van der Waals C_{6} coefficients. 
TYPE:
DEFAULT:
89  for VV09 
No default  for VV10 
OPTIONS:
n  Corresponding to C = n/10000 
RECOMMENDATION:
C = 0.0093 is recommended when a semilocal exchange functional is used.
C = 0.0089 is recommended when a longrange corrected (LRC) hybrid functional is used.
See further details in Ref. [117]. 



NL_VV_B
Sets the parameter b in VV10. This parameter controls the short range behavior
of the nonlocal correlation energy. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to b = n/100 
RECOMMENDATION:
The optimal value depends strongly on the exchange functional used.
b = 5.9 is recommended for rPW86. See further details in Ref. [117]. 

 FAST_XC
Controls direct variable thresholds to accelerate exchange correlation (XC) in
DFT. 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Turn FAST_XC on. 
FALSE  Do not use FAST_XC. 
RECOMMENDATION:
Caution: FAST_XC improves the speed of a DFT calculation, but
may occasionally cause the SCF calculation to diverge. 



XC_GRID
Specifies the type of grid to use for DFT calculations. 
TYPE:
DEFAULT:
OPTIONS:
0  Use SG0 for H, C, N, and O, SG1 for all other atoms. 
1  Use SG1 for all atoms. 
2  Low Quality. 
mn  The first six integers correspond to m radial points and the second six 
 integers correspond to n angular points where possible numbers of Lebedev 
 angular points are listed in section 4.3.11. 
−mn  The first six integers correspond to m radial points and the second six 
 integers correspond to n angular points where the number of GaussLegendre 
 angular points n = 2N^{2}. 
RECOMMENDATION:
Use default unless numerical integration problems arise. Larger grids may be
required for optimization and frequency calculations. 

 XC_SMART_GRID
Uses SG0 (where available) for early SCF cycles, and switches to the
(larger) grid specified by XC_GRID (which defaults to SG1, if not
otherwise specified) for final cycles of the SCF. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The use of the smart grid can save some time on initial SCF cycles. 



NL_GRID
Specifies the grid to use for nonlocal correlation. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless computational cost becomes prohibitive, in which case SG0 may be used.
XC_GRID should generally be finer than NL_GRID. 

4.3.16 Example
Example 4.0 QChem input for a DFT single point energy calculation on
water.
$comment
BLYP/STO3G water single point calculation
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.2
hoh = 120.0
$end.
$rem
EXCHANGE Becke Becke88 exchange
CORRELATION lyp LYP correlation
BASIS sto3g Basis set
$end
4.3.17 UserDefined Density Functionals
The format for entering userdefined exchangecorrelation density functionals
is one line for each component of the functional. Each line requires three
variables: the first defines whether the component is an exchange or
correlation functional by declaring an X or C, respectively. The second
variable is the symbolic representation of the functional as used for the
EXCHANGE and CORRELATION $rem variables. The final variable
is a real number corresponding to the contribution of the component to the
functional. HartreeFock exchange contributions (required for hybrid density
functionals) can be entered using only two variables (K, for HF exchange)
followed by a real number.
$xc_functional
X exchange_symbol coefficient
X exchange_symbol coefficient
...
C correlation_symbol coefficient
C correlation_symbol coefficient
...
K coefficient
$end
Note:
(1) Coefficients are real.
(2) A userdefined functional does not require all X, C and K components. 
Examples of userdefined XCs: these are XC options that
for the time being can only be invoked via the user defined XC input section:
Example 4.0 QChem input of water with B3tLap.
$comment
water with B3tLap
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 0.97
hoh = 120.0
$end
$rem
EXCHANGE gen
CORRELATION none
XC_GRID 000120000194 ! recommended for high accuracy
BASIS G3LARGE ! recommended for high accuracy
THRESH 14 ! recommended for high accuracy and better convergence
$end
$xc_functional
X Becke 0.726
X S 0.0966
C PK06 1.0
K 0.1713
$end
Example 4.0 QChem input of water with BR89B94hyb.
$comment
water with BR89B94hyb
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 0.97
hoh = 120.0
$end
$rem
EXCHANGE gen
CORRELATION none
XC_GRID 000120000194 ! recommended for high accuracy
BASIS G3LARGE ! recommended for high accuracy
THRESH 14 ! recommended for high accuracy and better convergence
$end
$xc_functional
X BR89 0.846
C B94hyb 1.0
K 0.154
$end
More specific is the use of the RIB05 and RIPSTS functionals. In this release
we offer only singlepoint SCF calculations with these functionals.
Both options require a converged LSD or HF solution as initial guess,
which greatly facilitates the convergence. It
also requires specifying a particular auxiliary basis set:
Example 4.0 QChem input of H_{2} using RIB05.
$comment
H2, example of SP RIB05.
First do a wellconverged LSD, G3LARGE is the basis of choice
for good accuracy. The input lines
purecar 222
SCF_GUESS CORE
are obligatory for the time being here.
$end
$molecule
0 1
H 0. 0. 0.0
H 0. 0. 0.7414
$end
$rem
JOBTYPE SP
SCF_GUESS CORE
EXCHANGE SLATER
CORRELATION VWN
BASIS G3LARGE
purcar 222
THRESH 14
MAX_SCF_CYCLES 80
PRINT_INPUT TRUE
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
SCF_CONVERGENCE 9
$end
@@@@
$comment
For the time being the following input lines are obligatory:
purcar 22222
AUX_BASIS riB05ccpvtz
dft_cutoffs 0
1415 1
MAX_SCF_CYCLES 0
JOBTYPE SP
$end
$molecule
READ
$end
$rem
JOBTYPE SP
SCF_GUESS READ
EXCHANGE B05
! EXCHANGE PSTS ! use this line for RIPSTS
purcar 22222
BASIS G3LARGE
AUX_BASIS riB05ccpvtz ! The aux basis for RIB05 and RIPSTS
THRESH 14
PRINT_INPUT TRUE
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
MAX_SCF_CYCLES 0
dft_cutoffs 0
1415 1
$end
Besides postLSD, the RIB05 option can be used as postHartreeFock
method as well, in which case one first does a wellconverged HF
calculation and uses it as a guess read in the consecutive RIB05
run.
4.4 Large Molecules and Linear Scaling Methods
4.4.1 Introduction
Construction of the effective Hamiltonian, or Fock matrix, has traditionally
been the ratedetermining step in selfconsistent field calculations, due
primarily to the cost of twoelectron integral evaluation, even with the
efficient methods available in QChem (see Appendix B). However, for
large enough molecules, significant speedups are possible by employing
linearscaling methods for each of the nonlinear terms that can arise. Linear
scaling means that if the molecule size is doubled, then the computational
effort likewise only doubles. There are three computationally significant
terms:
 Electronelectron Coulomb interactions, for which QChem incorporates
the Continuous Fast Multipole Method (CFMM) discussed in section
4.4.2
 Exact exchange interactions, which arise in hybrid DFT calculations and
HartreeFock calculations, for which QChem incorporates the LinK
method discussed in section 4.4.3 below.
 Numerical integration of the exchange and correlation functionals in DFT
calculations, which we have already discussed in section 4.3.11.
QChem supports energies and efficient analytical gradients for all three of
these high performance methods to permit structure optimization of large
molecules, as well as relative energy evaluation. Note that analytical second
derivatives of SCF energies do not exploit these methods at present.
For the most part, these methods are switched on automatically by the program
based on whether they offer a significant speedup for the job at hand.
Nevertheless it is useful to have a general idea of the key concepts behind
each of these algorithms, and what input options are necessary to control them.
That is the primary purpose of this section, in addition to briefly describing
two more conventional methods for reducing computer time in large calculations
in Section 4.4.4.
There is one other computationally significant step in SCF calculations, and
that is diagonalization of the Fock matrix, once it has been constructed.
This step scales with the cube of molecular size (or basis set size), with a
small prefactor. So, for large enough SCF calculations (very roughly in the
vicinity of 2000 basis functions and larger), diagonalization becomes the
ratedetermining step. The cost of cubic scaling with a small prefactor at
this point exceeds the cost of the linear scaling Fock build, which has a
very large prefactor, and the gap rapidly widens thereafter. This sets an
effective upper limit on the size of SCF calculation for which QChem is
useful at several thousand basis functions.
4.4.2 Continuous Fast Multipole Method (CFMM)
The quantum chemical Coulomb problem, perhaps better known as the DFT
bottleneck, has been at the forefront of many research efforts throughout the
1990s. The quadratic computational scaling behavior conventionally seen in the
construction of the Coulomb matrix in DFT or HF calculations has prevented the
application of ab initio methods to molecules containing many hundreds
of atoms. QChem, Inc., in collaboration with White and HeadGordon at the
University of California at Berkeley, and Gill now at the Australian National
University, were the first to develop the generalization of Greengard's Fast
Multipole Method (FMM) [141] to Continuous charged matter
distributions in the form of the CFMM, which is the first linear scaling
algorithm for DFT calculations. This initial breakthrough has since lead to an
increasing number of linear scaling alternatives and analogies, but for Coulomb
interactions, the CFMM remains state of the art. There are two computationally
intensive contributions to the Coulomb interactions which we discuss in turn:
 Longrange interactions, which are treated by the CFMM
 Shortrange interactions, corresponding to overlapping charge
distributions, which are treated by a specialized "Jmatrix engine"
together with QChem's stateofthe art twoelectron integral methods.
The Continuous Fast Multipole Method was the first implemented linear scaling
algorithm for the construction of the J matrix. In collaboration with
QChem, Inc., Dr. Chris White began the development of the CFMM by more
efficiently deriving [142] the original Fast Multipole Method
before generalizing it to the CFMM [143]. The generalization
applied by White et al. allowed the principles underlying the success of the
FMM to be applied to arbitrary (subject to constraints in evaluating the
related integrals) continuous, but localized, matter distributions. White and
coworkers further improved the underlying CFMM algorithm [144,[145]
then implemented it efficiently [146],
achieving performance that is an order of magnitude faster than some competing
implementations.
The success of the CFMM follows similarly with that of the FMM, in that the
charge system is subdivided into a hierarchy of boxes. Local charge
distributions are then systematically organized into multipole representations
so that each distribution interacts with local expansions of the potential due
to all distant charge distributions. Local and distant distributions are
distinguished by a wellseparated (WS) index, which is the number of boxes that
must separate two collections of charges before they may be considered distant
and can interact through multipole expansions; nearfield interactions must be
calculated directly. In the CFMM each distribution is given its own WS index
and is sorted on the basis of the WS index, and the position of their space
centers. The implementation in QChem has allowed the efficiency gains of
contracted basis functions to be maintained.
The CFMM algorithm can be summarized in five steps:
 Form and translate multipoles.
 Convert multipoles to local Taylor expansions.
 Translate Taylor information to the lowest level.
 Evaluate Taylor expansions to obtain the farfield potential.
 Perform direct interactions between overlapping distributions.
Accuracy can be carefully controlled by due consideration of tree depth,
truncation of the multipole expansion and the definition of the extent of
charge distributions in accordance with a rigorous mathematical error bound.
As a rough guide, 10 poles are adequate for single point energy calculations,
while 25 poles yield sufficient accuracy for gradient calculations. Subdivision
of boxes to yield a onedimensional length of about 8 boxes works quite well
for systems of up to about one hundred atoms. Larger molecular systems, or
ones which are extended along one dimension, will benefit from an increase in
this number. The program automatically selects an appropriate number of boxes
by default.
For the evaluation of the remaining shortrange interactions, QChem
incorporates efficient Jmatrix engines, originated by White and HeadGordon [147].
These are analytically exact methods that are based on
standard twoelectron integral methods, but with an interesting twist. If one
knows that the twoelectron integrals are going to be summed into a Coulomb
matrix, one can ask whether they are in fact the most efficient intermediates
for this specific task. Or, can one instead find a more compact and
computationally efficient set of intermediates by folding the density matrix
into the recurrence relations for the twoelectron integrals. For integrals
that are not highly contracted (i.e., are not linear combinations of more than a
few Gaussians), the answer is a dramatic yes. This is the basis of the
Jmatrix approach, and QChem includes the latest algorithm developed by
Yihan Shao working with Martin HeadGordon at Berkeley for this purpose.
Shao's Jengine is employed for both energies [148] and forces [149]
and gives substantial speedups relative to the use of
twoelectron integrals without any approximation (roughly a factor of 10
(energies) and 30 (forces) at the level of an uncontracted dddd shell quartet,
and increasing with angular momentum). Its use is automatically selected for
integrals with low degrees of contraction, while regular integrals are employed
when the degree of contraction is high, following the state of the art PRISM
approach of Gill and coworkers [150].
The CFMM is controlled by the following input parameters:
CFMM_ORDER
Controls the order of the multipole expansions in CFMM calculation. 
TYPE:
DEFAULT:
15  For single point SCF accuracy 
25  For tighter convergence (optimizations) 
OPTIONS:
n  Use multipole expansions of order n 
RECOMMENDATION:

 GRAIN
Controls the number of lowestlevel boxes in one dimension for CFMM. 
TYPE:
DEFAULT:
1  Program decides best value, turning on CFMM when useful 
OPTIONS:
1  Program decides best value, turning on CFMM when useful 
1  Do not use CFMM 
n ≥ 8  Use CFMM with n lowestlevel boxes in one dimension 
RECOMMENDATION:
This is an expert option; either use the default, or use a value of 1 if CFMM
is not desired. 



4.4.3 Linear Scaling Exchange (LinK) Matrix Evaluation
HartreeFock calculations and the popular hybrid density functionals such as
B3LYP also require twoelectron integrals to evaluate the exchange energy
associated with a single determinant. There is no useful multipole expansion
for the exchange energy, because the bra and ket of the twoelectron integral
are coupled by the density matrix, which carries the effect of exchange.
Fortunately, density matrix elements decay exponentially with distance for
systems that have a HOMOLUMO gap [151]. The better the
insulator, the more localized the electronic structure, and the faster the rate
of exponential decay. Therefore, for insulators, there are only a linear number
of numerically significant contributions to the exchange energy. With
intelligent numerical thresholding, it is possible to rigorously evaluate the
exchange matrix in linear scaling effort. For this purpose, QChem contains
the linear scaling K (LinK) method [152] to evaluate both
exchange energies and their gradients [153] in linear scaling
effort (provided the density matrix is highly sparse). The LinK method
essentially reduces to the conventional direct SCF method for exchange in the
small molecule limit (by adding no significant overhead), while yielding large
speedups for (very) large systems where the density matrix is indeed highly
sparse. For full details, we refer the reader to the original
papers [152,[153]. LinK can be explicitly requested by the
following option (although QChem automatically switches it on when the
program believes it is the preferable algorithm).
LIN_K
Controls whether linear scaling evaluation of exact exchange (LinK) is used. 
TYPE:
DEFAULT:
Program chooses, switching on LinK whenever CFMM is used. 
OPTIONS:
TRUE  Use LinK 
FALSE  Do not use LinK 
RECOMMENDATION:
Use for HF and hybrid DFT calculations with large numbers of atoms. 

4.4.4 Incremental and Variable Thresh Fock Matrix Building
The use of a variable integral threshold, operating for the first few cycles of
an SCF, is justifiable on the basis that the MO coefficients are usually of
poor quality in these cycles. In QChem, the integrals in the first iteration
are calculated at a threshold of 10^{−6} (for an anticipated final integral
threshold greater than, or equal to 10^{−6} to ensure the error in the first
iteration is solely sourced from the poor MO guess. Following this, the
integral threshold used is computed as
tmp_thresh = varthresh×DIIS_error 
 (4.74) 
where the DIIS_error is that calculated from the previous cycle, varthresh
is the variable threshold set by the program (by default) and tmp_thresh is
the temporary threshold used for integral evaluation. Each cycle requires
recalculation of all integrals. The variable integral threshold procedure has
the greatest impact in early SCF cycles.
In an incremental Fock matrix build [154], F is computed
recursively as
F^{m}=F^{m−1}+∆J^{m−1}− 
1
2

∆K^{m−1} 
 (4.75) 
where m is the SCF cycle, and ∆J^{m} and ∆K^{m} are computed using the difference density
Using Schwartz integrals and elements of the difference density, QChem is
able to determine at each iteration which ERIs are required, and if necessary,
recalculated. As the SCF nears convergence, ∆P^{m} becomes sparse and
the number of ERIs that need to be recalculated declines dramatically, saving
the user large amounts of computational time.
Incremental Fock matrix builds and variable thresholds are only used when the
SCF is carried out using the direct SCF algorithm and are clearly complementary
algorithms. These options are controlled by the following input parameters,
which are only used with direct SCF calculations.
INCFOCK
Iteration number after which the incremental Fock matrix algorithm is
initiated 
TYPE:
DEFAULT:
1  Start INCFOCK after iteration number 1 
OPTIONS:
Userdefined (0 switches INCFOCK off) 
RECOMMENDATION:
May be necessary to allow several iterations before switching on INCFOCK. 

 VARTHRESH
Controls the temporary integral cutoff threshold. tmp_thresh = 10^{−VARTHRESH}×DIIS_error 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
3 has been found to be a practical level, and can slightly speed up SCF
evaluation. 



4.4.5 Incremental DFT
Incremental DFT (IncDFT) uses the difference density and functional values to improve the
performance of the DFT quadrature procedure by providing a better screening of negligible
values. Using this option will yield improved efficiency at each successive iteration
due to more effective screening.
INCDFT
Toggles the use of the IncDFT procedure for DFT energy calculations. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Do not use IncDFT 
TRUE  Use IncDFT 
RECOMMENDATION:
Turning this option on can lead to faster SCF calculations, particularly
towards the end of the SCF. Please note that for some systems use of this
option may lead to convergence problems. 

 INCDFT_DENDIFF_THRESH
Sets the threshold for screening density matrix values in the IncDFT procedure. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to a threshold of 10^{−n}. 
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. 



INCDFT_GRIDDIFF_THRESH
Sets the threshold for screening functional values in the IncDFT procedure 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to a threshold of 10^{−n}. 
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. 

 INCDFT_DENDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening density matrix
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_DENDIFF_THRESH is reached. This means this
value must be set lower than INCDFT_DENDIFF_THRESH. 
TYPE:
DEFAULT:
0  Variable threshold is not used. 
OPTIONS:
n  Corresponding to a threshold of 10^{−n}. 
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. 



INCDFT_GRIDDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening the functional
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_GRIDDIFF_THRESH is reached. This means that
this value must be set lower than INCDFT_GRIDDIFF_THRESH. 
TYPE:
DEFAULT:
0  Variable threshold is not used. 
OPTIONS:
n  Corresponding to a threshold of 10^{−n}. 
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. 

4.4.6 Fourier Transform Coulomb Method
The Coulomb part of the DFT calculations using `ordinary' Gaussian
representations can be sped up dramatically using plane waves as a secondary
basis set by replacing the most costly analytical electron repulsion integrals
with numerical integration techniques. The main advantages to keeping the
Gaussians as the primary basis set is that the diagonalization step is much
faster than using plane waves as the primary basis set, and all electron
calculations can be performed analytically.
The Fourier Transform Coulomb (FTC) technique [155,[156] is
precise and tunable and all results are practically identical with the
traditional analytical integral calculations. The FTC technique is at least
23 orders of magnitude more accurate then other popular plane wave based
methods using the same energy cutoff. It is also at least 23 orders of
magnitude more accurate than the density fitting (resolution of identity)
technique. Recently, an efficient way to implement the forces of the Coulomb
energy was introduced [157], and a new technique to localize
filtered core functions. Both of these features have been implemented within
QChem and contribute to the efficiency of the method.
The FTC method achieves these spectacular results by replacing the analytical
integral calculations, whose computational costs scales as O(N^{4}) (where
N is the number of basis function) with procedures that scale as only
O(N^{2}). The asymptotic scaling of computational costs with system size is
linear versus the analytical integral evaluation which is quadratic. Research
at QChem Inc. has yielded a new, general, and very efficient implementation
of the FTC method which work in tandem with the Jengine and the CFMM
(Continuous Fast Multipole Method) techniques [158].
In the current implementation the speedups arising from the FTC technique are
moderate when small or medium Pople basis sets are used. The reason is that the
Jmatrix engine and CFMM techniques provide an already highly efficient
solution to the Coulomb problem. However, increasing the number of
polarization functions and, particularly, the number of diffuse functions
allows the FTC to come into its own and gives the most significant
improvements. For instance, using the 6311G+(df,pd) basis set for a mediumtolarge
size molecule is more affordable today then before. We found also significant
speed ups when nonPople basis sets are used such as ccpvTZ. The FTC energy
and gradients calculations are implemented to use up to ftype basis
functions.
FTC
Controls the overall use of the FTC. 
TYPE:
DEFAULT:
OPTIONS:
0  Do not use FTC in the Coulomb part 
1  Use FTC in the Coulomb part 
RECOMMENDATION:
Use FTC when bigger and/or diffuse basis sets are used. 

 FTC_SMALLMOL
Controls whether or not the operator is evaluated on a large grid and stored in
memory to speed up the calculation. 
TYPE:
DEFAULT:
OPTIONS:
1  Use a big precalculated array to speed up the FTC calculations 
0  Use this option to save some memory 
RECOMMENDATION:
Use the default if possible and use 0 (or buy some more memory) when
needed. 



FTC_CLASS_THRESH_ORDER
Together with FTC_CLASS_THRESH_MULT, determines the cutoff
threshold for included a shellpair in the dd class, i.e., the class that
is expanded in terms of plane waves. 
TYPE:
DEFAULT:
5  Logarithmic part of the FTC classification threshold. Corresponds to 10^{−5} 
OPTIONS:
RECOMMENDATION:

 FTC_CLASS_THRESH_MULT
Together with FTC_CLASS_THRESH_ORDER, determines the cutoff
threshold for included a shellpair in the dd class, i.e., the class that
is expanded in terms of plane waves. 
TYPE:
DEFAULT:
5  Multiplicative part of the FTC classification threshold. Together with 
 the default value of the FTC_CLASS_THRESH_ORDER this leads to 
 the 5×10^{−5} threshold value. 
OPTIONS:
RECOMMENDATION:
Use the default. If diffuse basis sets are used and the molecule is relatively
big then tighter FTC classification threshold has to be used. According to our
experiments using Popletype diffuse basis sets, the default 5×10^{−5} value provides accurate result for an alanine5 molecule while 1×10^{−5} threshold value for alanine10 and 5×10^{−6} value for
alanine15 has to be used. 



4.4.7 Multiresolution ExchangeCorrelation (mrXC) Method
MrXC (multiresolution exchangecorrelation) [159,[160,[161]
is a new method developed by the QChem development
team for the accelerating the computation of exchangecorrelation
(XC) energy and matrix originated from the XC functional.
As explained in 4.4.6, the XC
functional is so complicated that the integration of it is usually
done on a numerical quadrature. There are two basic types of
quadrature. One is the atomcentered grid (ACG), a superposition of
atomic quadrature described in 4.4.6.
ACG has high density of points near the nucleus to handle the
compact core density and low density of points in the valence and
nonbonding region where the electron density is smooth. The other
type is evenspaced cubic grid (ESCG), which is typically used
together with pseudopotentials and planewave basis functions where
only the e electron density is assumed smooth. In quantum chemistry,
ACG is more often used as it can handle accurately allelectron
calculations of molecules. MrXC combines those two integration
schemes seamlessly to achieve an optimal computational efficiency
by placing the calculation of the smooth part of the density and XC
matrix onto the ESCG. The computation associated with the smooth
fraction of the electron density is the
major bottleneck of the XC part of a DFT
calculation and can be done at a much faster rate on the ESCG due
to its low resolution. Fast Fourier transform and Bspline
interpolation are employed for the accurate transformation between
the two types of grids such that the final results remain the
same as they would be on the ACG alone. Yet, a speedup of several times for
the calculations of electrondensity and XC matrix is achieved. The smooth
part of the calculation with mrXC can also be combined with FTC
(see section 4.4.6) to achieve further gain of efficiency.
MRXC
Controls the use of MRXC. 
TYPE:
DEFAULT:
OPTIONS:
0  Do not use MRXC 
1  Use MRXC in the evaluation of the XC part 
RECOMMENDATION:
MRXC is very efficient for medium and large molecules,
especially when medium and large basis sets are used. 

The following two keywords control the smoothness precision.
The default value is carefully selected to maintain high accuracy.
MRXC_CLASS_THRESH_MULT
Controls the of smoothness precision 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
a prefactor in the threshold for mrxc error control:
im*10.0^{−io} 

 MRXC_CLASS_THRESH_ORDER
Controls the of smoothness precision 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The exponent in the threshold of the mrxc error control:
im*10.0^{−io} 



The next keyword controls the order of the
Bspline interpolation:
LOCAL_INTERP_ORDER
Controls the order of the Bspline 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The default value is sufficiently accurate 

4.4.8 Examples
Example 4.0 QChem input for a large single point energy calculation. The
CFMM is switched on automatically when LinK is requested.
$comment
HF/321G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
EXCHANGE HF HF exchange
BASIS 321G Basis set
LIN_K TRUE Calculate K using LinK
$end
Example 4.0 QChem input for a large single point energy calculation. This
would be appropriate for a mediumsized molecule, but for truly large
calculations, the CFMM and LinK algorithms are far more efficient.
$comment
HF/321G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
exchange hf HF exchange
basis 321G Basis set
incfock 5 Incremental Fock after 5 cycles
varthresh 3 1.0d03 variable threshold
$end
4.5 SCF Initial Guess
4.5.1 Introduction
The RoothaanHall and PopleNesbet equations of SCF theory are nonlinear in
the molecular orbital coefficients. Like many mathematical problems involving
nonlinear equations, prior to the application of a technique to search for a
numerical solution, an initial guess for the solution must be generated. If the
guess is poor, the iterative procedure applied to determine the numerical
solutions may converge very slowly, requiring a large number of iterations, or
at worst, the procedure may diverge.
Thus, in an ab initio SCF procedure, the quality of the initial guess
is of utmost importance for (at least) two main reasons:
 To ensure that the SCF converges to an appropriate ground state. Often
SCF calculations can converge to different local minima in wavefunction
space, depending upon which part of that space the initial guess places
the system in.
 When considering jobs with many basis functions requiring the
recalculation of ERIs at each iteration, using a good initial guess that
is close to the final solution can reduce the total job time
significantly by decreasing the number of SCF iterations.
For these reasons, sooner or later most users will find it helpful to have some
understanding of the different options available for customizing the initial
guess. QChem currently offers five options for the initial guess:
 Superposition of Atomic Density (SAD)
 Core Hamiltonian (CORE)
 Generalized WolfsbergHelmholtz (GWH)
 Reading previously obtained MOs from disk. (READ)
 Basis set projection (BASIS2)
The first three of these guesses are builtin, and are briefly described in
Section 4.5.2. The option of reading MOs from disk is described in
Section 4.5.3. The initial guess MOs can be modified, either by
mixing, or altering the order of occupation. These options are discussed in
Section 4.5.4. Finally, QChem's novel basis set projection
method is discussed in Section 4.5.5.
4.5.2 Simple Initial Guesses
There are three simple initial guesses available in QChem. While they are
all simple, they are by no means equal in quality, as we discuss below.
 Superposition of Atomic Densities (SAD): The SAD guess is almost
trivially constructed by summing together atomic densities that have been
spherically averaged to yield a trial density matrix. The SAD guess is far
superior to the other two options below, particularly when large basis
sets and/or large molecules are employed. There are three issues
associated with the SAD guess to be aware of:
 No molecular orbitals are obtained, which means that SCF algorithms
requiring orbitals (the direct minimization methods discussed in
Section 4.6) cannot directly use the SAD guess,
and,
 The SAD guess is not available for general (readin) basis sets. All
internal basis sets support the SAD guess.
 The SAD guess is not idempotent and thus requires at least
two SCF iterations to ensure proper SCF convergence (idempotency of
the density).
 Generalized WolfsbergHelmholtz (GWH): The GWH guess
procedure [162] uses a combination of the overlap matrix elements
in Eq. (4.12), and the diagonal elements of the Core Hamiltonian matrix
in Eq. (4.18). This initial guess is most satisfactory in small basis
sets for small molecules. It is constructed according to the relation
given below, where c_{x} is a constant.
H_{μυ} = c_{x} S_{μυ} (H_{μμ} +H_{υυ} )  / 
(H_{μμ} +H_{υυ} ) 2 2 
 (4.77) 
 Core Hamiltonian: The core Hamiltonian guess simply obtains the
guess MO coefficients by diagonalizing the core Hamiltonian matrix in
Eq. (4.18). This approach works best with small basis sets, and
degrades as both the molecule size and the basis set size are increased.
The selection of these choices (or whether to read in the orbitals) is
controlled by the following $rem variables:
SCF_GUESS
Specifies the initial guess procedure to use for the SCF. 
TYPE:
DEFAULT:
SAD  Superposition of atomic density (available only with standard basis
sets) 
GWH  For ROHF where a set of orbitals are required. 
FRAGMO  For a fragment MO calculation 
OPTIONS:
CORE  Diagonalize core Hamiltonian 
SAD  Superposition of atomic density 
GWH  Apply generalized WolfsbergHelmholtz approximation 
READ  Read previous MOs from disk 
FRAGMO  Superimposing converged fragment MOs 
RECOMMENDATION:
SAD guess for standard basis sets. For general basis sets, it is best to use
the BASIS2 $rem. Alternatively, try the GWH or core Hamiltonian
guess. For ROHF it can be useful to READ guesses from an SCF calculation on the
corresponding cation or anion. Note that because the density is made spherical,
this may favor an undesired state for atomic systems, especially transition
metals. Use FRAGMO in a fragment MO calculation. 

 SCF_GUESS_ALWAYS
Switch to force the regeneration of a new initial guess for each series of
SCF iterations (for use in geometry optimization). 
TYPE:
DEFAULT:
OPTIONS:
False  Do not generate a new guess for each series of SCF iterations in an 
 optimization; use MOs from the previous SCF calculation for the guess, 
 if available. 
True  Generate a new guess for each series of SCF iterations in a geometry 
 optimization. 
RECOMMENDATION:
Use default unless SCF convergence issues arise 



4.5.3 Reading MOs from Disk
There are two methods by which MO coefficients can be used from a previous
job by reading them from disk:
 Running two independent jobs sequentially invoking qchem with
three command line variables:.
localhost1> qchem job1.in job1.out save
localhost2> qchem job2.in job2.out save
Note:
(1) The $rem variable SCF_GUESS must be set to READ
in job2.in.
(2) Scratch files remain in $QCSCRATCH/save on exit. 
 Running a batch job where two jobs are placed into a single input file
separated by the string @@@ on a single line.
Note:
(1) SCF_GUESS must be set to
READ in the second job of the batch file.
(2) A third qchem command line variable is not necessary.
(3) As for the SAD guess, QChem requires at least two SCF cycles to
ensure proper SCF convergence (idempotency of the density). 
Note:
It is up to the user to make sure that the basis
sets match between the two jobs. There is no internal checking for this,
although the occupied orbitals are reorthogonalized in the current basis after
being read in. If you want to project from a smaller basis into a larger basis,
consult section 4.5.5. 
4.5.4 Modifying the Occupied Molecular Orbitals
It is sometimes useful for the occupied guess orbitals to be other than the
lowest N_{α} (or N_{β}) orbitals. Reasons why one may need
to do this include:
 To converge to a state of different symmetry or orbital occupation.
 To break spatial symmetry.
 To break spin symmetry, as in unrestricted calculations on molecules with
an even number of electrons.
There are two mechanisms for modifying a set of guess orbitals: either by
SCF_GUESS_MIX, or by specifying the orbitals to occupy. QChem
users may define the occupied guess orbitals using the $occupied or
$swap_occupied_virtual
keywords. In the former, occupied guess orbitals are defined by listing the
alpha orbitals to be occupied on the first line and beta on the second. In the
former, only pair of orbitals that needs to be swapped is specified.
Note:
To prevent QChem to change orbital occupation during SCF procedure,
MOMSTART option is often used in combination with
$occupied or $swap_occupied_virtual keywords. 
Note:
The need for orbitals renders these options incompatible with the SAD
guess. Most often, they are used with SCF_GUESS=READ. 
Example 4.0 Format for modifying occupied guess orbitals.
$occupied
1 2 3 4 ... nalpha
1 2 3 4 ... nbeta
$end
Example 4.0 Alternative format for modifying occupied guess orbitals.
$swap_occupied_virtual
<spin> <io1> <iv1>
<spin> <io2> <iv2>
$end
Example 4.0 Example of swapping guess orbitals.
$swap_occupied_virtual
alpha 5 6
beta 6 7
$end
This is identical to:
Example 4.0 Example of specifying occupied guess orbitals.
$occupied
1 2 3 4 6 5 7
1 2 3 4 5 7 6
$end
The other $rem variables related to altering the orbital occupancies are:
SCF_GUESS_PRINT
Controls printing of guess MOs, Fock and density matrices. 
TYPE:
DEFAULT:
OPTIONS:
0  Do not print guesses. 
SAD  
1  Atomic density matrices and molecular matrix. 
2  Level 1 plus density matrices. 
CORE and GWH  
1  No extra output. 
2  Level 1 plus Fock and density matrices and, MO coefficients and 
 eigenvalues. 
READ  
1  No extra output 
2  Level 1 plus density matrices, MO coefficients and eigenvalues. 
RECOMMENDATION:

 SCF_GUESS_MIX
Controls mixing of LUMO and HOMO to break symmetry in the initial guess. For
unrestricted jobs, the mixing is performed only for the alpha orbitals. 
TYPE:
DEFAULT:
0 (FALSE)  Do not mix HOMO and LUMO in SCF guess. 
OPTIONS:
0 (FALSE)  Do not mix HOMO and LUMO in SCF guess. 
1 (TRUE)  Add 10% of LUMO to HOMO to break symmetry. 
n  Add n×10% of LUMO to HOMO (0 < n < 10). 
RECOMMENDATION:
When performing unrestricted calculations on molecules with an even number of
electrons, it is often necessary to break alpha/beta symmetry in the initial
guess with this option, or by specifying input for $occupied. 



4.5.5 Basis Set Projection
QChem also includes a novel basis set projection method developed by Dr Jing
Kong of QChem Inc. It permits a calculation in a large basis set to
bootstrap itself up via a calculation in a small basis set that is
automatically spawned when the user requests this option. When basis set
projection is requested (by providing a valid small basis for BASIS2),
the program executes the following steps:
 A simple DFT calculation is performed in the small basis,
BASIS2, yielding a converged density matrix in this basis.
 The large basis set SCF calculation (with different values of
EXCHANGE and CORRELATION set by the input) begins by
constructing the DFT Fock operator in the large basis but with the
density matrix obtained from the small basis set.
 By diagonalizing this matrix, an accurate initial guess for the density
matrix in the large basis is obtained, and the target SCF calculation
commences.
Two different methods of projection are available and can be set using the
BASISPROJTYPE $rem. The OVPROJECTION option expands the MOs
from the BASIS2 calculation in the larger basis, while the
FOPPROJECTION option constructs the Fock matrix in the larger basis
using the density matrix from the initial, smaller basis set calculation. Basis
set projection is a very effective option for general basis sets, where the SAD
guess is not available. In detail, this initial guess is controlled by the
following $rem variables:
BASIS2
Sets the small basis set to use in basis set projection. 
TYPE:
DEFAULT:
No second basis set default. 
OPTIONS:
Symbol. Use standard basis sets as per Chapter 7. 
BASIS2_GEN  General BASIS2 
BASIS2_MIXED  Mixed BASIS2 
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using
a basis larger than a minimal basis when BASIS2 is used for initial guess purposes.
Larger, standardized BASIS2 options are available for dualbasis calculations
(see Section 4.7). 

 BASISPROJTYPE
Determines which method to use when projecting the density matrix of
BASIS2 
TYPE:
DEFAULT:
FOPPROJECTION (when DUAL_BASIS_ENERGY=false) 
OVPROJECTION (when DUAL_BASIS_ENERGY=true) 
OPTIONS:
FOPPROJECTION  Construct the Fock matrix in the second basis 
OVPROJECTION  Projects MO's from BASIS2 to BASIS. 
RECOMMENDATION:



Note:
BASIS2 sometimes messes up postHartreeFock calculations. It is recommended
to split such jobs into two subsequent one, such that in the first job a desired HartreeFock solution is found using BASIS2, and in the second job, which performs a
postHF calculation, SCF_GUESS=READ is invoked. 
4.5.6 Examples
Example 4.0 Input where basis set projection is used to generate a good
initial guess for a calculation employing a general basis set, for which the
default initial guess is not available.
$molecule
0 1
O
H 1 r
H 1 r 2 a
r 0.9
a 104.0
$end
$rem
EXCHANGE hf
CORRELATION mp2
BASIS general
BASIS2 sto3g
$end
$basis
O 0
S 3 1.000000
3.22037000E+02 5.92394000E02
4.84308000E+01 3.51500000E01
1.04206000E+01 7.07658000E01
SP 2 1.000000
7.40294000E+00 4.04453000E01 2.44586000E01
1.57620000E+00 1.22156000E+00 8.53955000E01
SP 1 1.000000
3.73684000E01 1.00000000E+00 1.00000000E+00
SP 1 1.000000
8.45000000E02 1.00000000E+00 1.00000000E+00
****
H 0
S 2 1.000000
5.44717800E+00 1.56285000E01
8.24547000E01 9.04691000E01
S 1 1.000000
1.83192000E01 1.00000000E+00
****
$end
Example 4.0 Input for an ROHF calculation on the OH radical. One SCF cycle
is initially performed on the cation, to get reasonably good initial guess
orbitals, which are then read in as the guess for the radical. This avoids the
use of QChem's default GWH guess for ROHF, which is often poor.
$comment
OH radical, part 1. Do 1 iteration of cation orbitals.
$end
$molecule
1 1
O 0.000 0.000 0.000
H 0.000 0.000 1.000
$end
$rem
BASIS = 6311++G(2df)
EXCHANGE = hf
MAX_SCF_CYCLES = 1
THRESH = 10
$end
@@@
$comment
OH radical, part 2. Read cation orbitals, do the radical
$end
$molecule
0 2
O 0.000 0.000 0.000
H 0.000 0.000 1.000
$end
$rem
BASIS = 6311++G(2df)
EXCHANGE = hf
UNRESTRICTED = false
SCF_ALGORITHM = dm
SCF_CONVERGENCE = 7
SCF_GUESS = read
THRESH = 10
$end
Example 4.0 Input for an unrestricted HF calculation on H_{2} in the
dissociation limit, showing the use of SCF_GUESS_MIX = 2
(corresponding to 20% of the alpha LUMO mixed with the alpha HOMO).
Geometric direct minimization with DIIS is used to converge the SCF, together
with MAX_DIIS_CYCLES = 1 (using the default value for
MAX_DIIS_CYCLES, the DIIS procedure just oscillates).
$molecule
0 1
H 0.000 0.000 0.0
H 0.000 0.000 10.0
$end
$rem
UNRESTRICTED = true
EXCHANGE = hf
BASIS = 631g**
SCF_ALGORITHM = diis_gdm
MAX_DIIS_CYCLES = 1
SCF_GUESS = gwh
SCF_GUESS_MIX = 2
$end
4.6 Converging SCF Calculations
4.6.1 Introduction
As for any numerical optimization procedure, the rate of convergence of the SCF
procedure is dependent on the initial guess and on the algorithm used to step
towards the stationary point. QChem features a number of alternative SCF
optimization algorithms, which are discussed in the following sections, along
with the $rem variables that are used to control the calculations. The main
options are discussed in sections which follow and are, in brief:
 The highly successful DIIS procedures, which are the default, except for
restricted openshell SCF calculations.
 The new geometric direct minimization (GDM) method, which is highly
robust, and the recommended fallback when DIIS fails. It can also be
invoked after a few initial iterations with DIIS to improve the initial
guess. GDM is the default algorithm for restricted openshell SCF
calculations.
 The older and less robust direct minimization method (DM). As for GDM,
it can also be invoked after a few DIIS iterations (except for RO jobs).
 The maximum overlap method (MOM) which ensures that DIIS always occupies
a continuous set of orbitals and does not oscillate between different
occupancies.
 The relaxed constraint algorithm (RCA) which guarantees that the energy
goes down at every step.
4.6.2 Basic Convergence Control Options
See also more detailed options in the following sections, and note that the SCF
convergence criterion and the integral threshold must be set in a compatible
manner, (this usually means THRESH should be set to at least 3 higher
than SCF_CONVERGENCE).
MAX_SCF_CYCLES
Controls the maximum number of SCF iterations permitted. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase for slowly converging systems such as those containing transition
metals. 

 SCF_ALGORITHM
Algorithm used for converging the SCF. 
TYPE:
DEFAULT:
OPTIONS:
DIIS  Pulay DIIS. 
DM  Direct minimizer. 
DIIS_DM  Uses DIIS initially, switching to direct minimizer for later iterations 
 (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). 
DIIS_GDM  Use DIIS and then later switch to geometric direct minimization 
 (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). 
GDM  Geometric Direct Minimization. 
RCA  Relaxed constraint algorithm 
RCA_DIIS  Use RCA initially, switching to DIIS for later iterations (see 
 THRESH_RCA_SWITCH and MAX_RCA_CYCLES described 
 later in this chapter) 
ROOTHAAN  Roothaan repeated diagonalization. 
RECOMMENDATION:
Use DIIS unless performing a restricted openshell calculation, in which case GDM is recommended.
If DIIS fails to find a reasonable approximate solution in the initial iterations,
RCA_DIIS is the recommended fallback option.
If DIIS approaches the correct solution but fails to finally converge,
DIIS_GDM is the recommended fallback. 



SCF_CONVERGENCE
SCF is considered converged when the wavefunction error is less that
10^{−SCF_CONVERGENCE}. Adjust the value of THRESH at the same
time. Note that in QChem 3.0 the DIIS error is measured by the maximum error
rather than the RMS error. 
TYPE:
DEFAULT:
5  For single point energy calculations. 
7  For geometry optimizations and vibrational analysis. 
8  For SSG calculations, see Chapter 5. 
OPTIONS:
n  Corresponding to 10^{−n} 
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger
values provide more significant figures, at greater computational cost. 

In some cases besides the total SCF energy, one needs its separate energy components, like
kinetic energy, exchange energy, correlation energy, etc. The values of these
components are printed at each SCF cycle if one specifies in the input:
SCF_PRINT 1 .
4.6.3 Direct Inversion in the Iterative Subspace (DIIS)
The SCF implementation of the Direct Inversion in the Iterative Subspace
(DIIS) method [163,[164] uses the property of
an SCF solution that requires the density matrix to commute with the Fock matrix:
During the SCF cycles, prior to achieving selfconsistency, it is therefore possible
to define an error vector e_{i}, which is nonzero except at convergence:
SP_{i} F_{i} −F_{i} P_{i} S=e_{i} 
 (4.79) 
Here, P_{i} is obtained from diagonalization of ∧F_{i} , and

^
F

k

= 
k−1 ∑
j=1

c_{j} F_{j} 
 (4.80) 
The DIIS coefficients c_{k}, are obtained by a leastsquares constrained
minimization of the error vectors, viz
Z=  ⎛ ⎝

∑
k

c_{k} e_{k}  ⎞ ⎠

·  ⎛ ⎝

∑
k

c_{k} e_{k}  ⎞ ⎠


 (4.81) 
where the constraint
is imposed to yield a set of linear equations, of dimension N+1:
 ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝


 ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠

 ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝


 ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠

=  ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝


 ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠


 (4.83) 
Convergence criteria requires the largest element of the Nth error vector
to be below a cutoff threshold, usually 10^{−5} for single point energies,
often increased to 10^{−8} for optimizations and frequency calculations.
The rate of convergence may be improved by restricting the number of previous
Fock matrices (size of the DIIS subspace, $rem variable
DIIS_SUBSPACE_SIZE) used for determining the DIIS coefficients:

^
F

k

= 
k−1 ∑
j=k−(L+1)

c_{j} F_{j} 
 (4.84) 
where L is the size of the DIIS subspace. As the Fock matrix nears
selfconsistency, the linear matrix equations in Eq. (4.83) tend to become
severely illconditioned and it is often necessary to reset the DIIS subspace
(this is automatically carried out by the program).
Finally, on a practical note, we observe that DIIS has a tendency to converge
to global minima rather than local minima when employed for SCF calculations.
This seems to be because only at convergence is the density matrix in the DIIS
iterations idempotent. On the way to convergence, one is not on the "true"
energy surface, and this seems to permit DIIS to "tunnel" through barriers in
wavefunction space. This is usually a desirable property, and is the motivation
for the options that permit initial DIIS iterations before switching to direct
minimization to converge to the minimum in difficult cases.
The following $rem variables permit some customization of the DIIS iterations:
DIIS_SUBSPACE_SIZE
Controls the size of the DIIS and/or RCA subspace during the SCF. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:

 DIIS_PRINT
Controls the output from DIIS SCF optimization. 
TYPE:
DEFAULT:
OPTIONS:
0  Minimal print out. 
1  Chosen method and DIIS coefficients and solutions. 
2  Level 1 plus changes in multipole moments. 
3  Level 2 plus Multipole moments. 
4  Level 3 plus extrapolated Fock matrices. 
RECOMMENDATION:



Note:
In QChem 3.0 the DIIS error is determined by the maximum error rather
than the RMS error. For backward compatibility the RMS error can be forced by
using the following $rem 
 DIIS_ERR_RMS
Changes the DIIS convergence metric from the maximum to the RMS error. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default, the maximum error provides a more reliable criterion. 



4.6.4 Geometric Direct Minimization (GDM)
Troy Van Voorhis, working at Berkeley with Martin HeadGordon, has developed a
novel direct minimization method that is extremely robust, and at the same time
is only slightly less efficient than DIIS. This method is called geometric
direct minimization (GDM) because it takes steps in an orbital rotation space
that correspond properly to the hyperspherical geometry of that space. In other
words, rotations are variables that describe a space which is curved like a
manydimensional sphere. Just like the optimum flight paths for airplanes are
not straight lines but great circles, so too are the optimum steps in orbital
rotation space. GDM takes this correctly into account, which is the origin of
its efficiency and its robustness. For full details, we refer the reader to
Ref. . GDM is a good alternative to DIIS
for SCF jobs that exhibit convergence difficulties with DIIS.
Recently, Barry Dunietz, also working at Berkeley with Martin HeadGordon, has
extended the GDM approach to restricted openshell SCF calculations. Their
results indicate that GDM is much more efficient than the older
direct minimization method (DM).
In section 4.6.3, we discussed the fact that DIIS can efficiently
head towards the global SCF minimum in the early iterations. This can be true
even if DIIS fails to converge in later iterations. For this reason, a hybrid
scheme has been implemented which uses the DIIS minimization procedure to
achieve convergence to an intermediate cutoff threshold. Thereafter, the
geometric direct minimization algorithm is used. This scheme combines the
strengths of the two methods quite nicely: the ability of DIIS to recover from
initial guesses that may not be close to the global minimum, and the ability of
GDM to robustly converge to a local minimum, even when the local surface
topology is challenging for DIIS. This is the recommended procedure with which
to invoke GDM (i.e., setting SCF_ALGORITHM = DIIS_GDM). This
hybrid procedure is also compatible with the SAD guess, while GDM itself is
not, because it requires an initial guess set of orbitals. If one wishes to
disturb the initial guess as little as possible before switching on GDM, one
should additionally specify MAX_DIIS_CYCLES = 1 to obtain only a
single Roothaan step (which also serves up a properly orthogonalized set of
orbitals).
$rem options relevant to GDM are SCF_ALGORITHM which should be
set to either GDM or DIIS_GDM and the following:
MAX_DIIS_CYCLES
The maximum number of DIIS iterations before switching to (geometric) direct
minimization when SCF_ALGORITHM is DIIS_GDM or
DIIS_DM. See also THRESH_DIIS_SWITCH. 
TYPE:
DEFAULT:
OPTIONS:
1  Only a single Roothaan step before switching to (G)DM 
n  n DIIS iterations before switching to (G)DM. 
RECOMMENDATION:

 THRESH_DIIS_SWITCH
The threshold for switching between DIIS extrapolation and direct minimization
of the SCF energy is 10^{−THRESH_DIIS_SWITCH} when
SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See
also MAX_DIIS_CYCLES 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:



4.6.5 Direct Minimization (DM)
Direct minimization (DM) is a less sophisticated forerunner of the geometric
direct minimization (GDM) method discussed in the previous section. DM does not
properly step along great circles in the hyperspherical space of orbital
rotations, and therefore converges less rapidly and less robustly than GDM, in
general. It is retained for legacy purposes, and because it is at present the
only method available for restricted open shell (RO) SCF calculations in
QChem. In general, the input options are the same as for GDM, with the
exception of the specification of SCF_ALGORITHM, which can be either
DIIS_DM (recommended) or DM.
PSEUDO_CANONICAL
When SCF_ALGORITHM = DM, this controls the way the initial
step, and steps after subspace resets are taken. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Use Roothaan steps when (re)initializing 
TRUE  Use a steepest descent step when (re)initializing 
RECOMMENDATION:
The default is usually more efficient, but choosing TRUE sometimes
avoids problems with orbital reordering. 

4.6.6 Maximum Overlap Method (MOM)
In general, the DIIS procedure is remarkably successful. One difficulty that
is occasionally encountered is the problem of an SCF that occupies two
different sets of orbitals on alternating iterations, and therefore
oscillates and fails to converge. This can be overcome by choosing orbital
occupancies that maximize the overlap of the new occupied orbitals with the set
previously occupied. QChem contains the maximum overlap method (MOM) [166],
developed by Andrew Gilbert and Peter Gill now at the
Australian National University.
MOM is therefore is a useful adjunct to DIIS in convergence problems involving
flipping of orbital occupancies. It is controlled by the $rem variable
MOM_START, which specifies the SCF iteration on which the MOM
procedure is first enabled. There are two strategies that are useful in setting
a value for MOM_START. To help maintain an initial configuration it
should be set to start on the first cycle. On the other hand, to assist
convergence it should come on later to avoid holding on to an initial
configuration that may be far from the converged one.
The MOMrelated $rem variables in full are the following:.
MOM_PRINT
Switches printing on within the MOM procedure. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Printing is turned off 
TRUE  Printing is turned on. 
RECOMMENDATION:

 MOM_START
Determines when MOM is switched on to stabilize DIIS iterations. 
TYPE:
DEFAULT:
OPTIONS:
0 (FALSE)  MOM is not used 
n  MOM begins on cycle n. 
RECOMMENDATION:
Set to 1 if preservation of initial orbitals is desired. If MOM is to be
used to aid convergence, an SCF without MOM should be run to determine when
the SCF starts oscillating. MOM should be set to start just before the
oscillations. 



4.6.7 Relaxed Constraint Algorithm (RCA)
The relaxed constraint algorithm (RCA) is an ingenious and simple means of minimizing the SCF energy
that is particularly effective in cases where the initial guess is poor.
The latter is true, for example, when employing a userspecified basis
(when the Core or GWH guess must be employed) or when neardegeneracy effects imply
that the initial guess will likely occupy the wrong orbitals relative to the desired converged solution.
Briefly, RCA begins with the SCF problem as a constrained minimization of the energy
as a function of the density matrix, E(P) [167,[168].
The constraint is that the density matrix be idempotent, P ·P=P,
which basically forces the occupation numbers to be either zero or one.
The fundamental realization of RCA is that this constraint can be relaxed to
allow subidempotent density matrices, P ·P ≤ P.
This condition forces the occupation numbers to be between zero and one.
Physically, we expect that any state with fractional occupations can
lower its energy by moving electrons from higher energy orbitals to lower ones.
Thus, if we solve for the minimum of E(P) subject to the relaxed subidempotent constraint,
we expect that the ultimate solution will nonetheless be idempotent.
In fact, for HartreeFock this can be rigorously proven.
For density functional theory, it is possible that the minimum will have
fractional occupation numbers but these occupations have a physical interpretation
in terms of ensemble DFT.
The reason the relaxed constraint is easier to deal with is that it is easy to prove
that a linear combination of subidempotent matrices is also
subidempotent as long as the linear coefficients are between zero and one.
By exploiting this property, convergence can be accelerated in a way
that guarantees the energy will go down at every step.
The implementation of RCA in QChem closely follows the "Energy DIIS" implementation of
the RCA algorithm [169].
Here, the current density matrix is written as a linear combination of the previous density matrices:
To a very good approximation (exact for HartreeFock) the energy for P(x)
can be written as a quadratic function of x:
E(x) = 
∑
i

E_{i} x_{i}+ 
1
2


∑
i

x_{i}(P_{i}− P_{j}) ·(F_{i}− F_{j}) x_{j} 
 (4.86) 
At each iteration, x is chosen to minimize E(x) subject to the constraint
that all of the x_{i} are between zero and one.
The Fock matrix for P(x) is further written as a linear combination of the previous Fock matrices,
F(x) = 
∑
i

x_{i} F_{i} + δF_{xc}(x) 
 (4.87) 
where δF_{xc}(x) denotes a (usually quite small) change in the exchangecorrelation part
that is computed once x has been determined.
We note that this extrapolation is very similar to that used by DIIS.
However, this procedure is guaranteed to reduce the energy E(x) at every iteration, unlike DIIS.
In practice, the RCA approach is ideally suited to difficult convergence situations
because it is immune to the erratic orbital swapping that can occur in DIIS.
On the other hand, RCA appears to perform relatively poorly near convergence,
requiring a relatively large number of steps to improve the precision of a "good" approximate solution.
It is thus advantageous in many cases to run RCA for the initial steps and
then switch to DIIS either after some specified number of iterations or
after some target convergence threshold has been reached.
Finally, note that by its nature RCA considers the energy as a function of the density matrix.
As a result, it cannot be applied to restricted open shell calculations which are explicitly orbitalbased.
Note: RCA interacts poorly with INCDFT, so INCDFT is disabled by default
when an RCA or RCA_DIIS calculation is requested. To enable INCDFT with
such a calculation, set INCDFT = 2 in the $rem section. RCA may also have
poor interactions with INCFOCK; if RCA fails to converge, disabling INCFOCK
may improve convergence in some cases.
RCA options are:
RCA_PRINT
Controls the output from RCA SCF optimizations. 
TYPE:
DEFAULT:
OPTIONS:
0  No print out 
1  RCA summary information 
2  Level 1 plus RCA coefficients 
3  Level 2 plus RCA iteration details 
RECOMMENDATION:

 MAX_RCA_CYCLES
The maximum number of RCA iterations before switching to DIIS when SCF_ALGORITHM is RCA_DIIS. 
TYPE:
DEFAULT:
OPTIONS:
N  N RCA iterations before switching to DIIS 
RECOMMENDATION:



THRESH_RCA_SWITCH
The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS. 
TYPE:
DEFAULT:
OPTIONS:
N  Algorithm changes from RCA to DIIS when Error is less than 10^{−N}. 
RECOMMENDATION:

Please see next section for an example using RCA.
4.6.8 Examples
Example 4.0 Input for a UHF calculation using geometric direct minimization
(GDM) on the phenyl radical, after initial iterations with DIIS. This example
fails to converge if DIIS is employed directly.
$molecule
0 2
c1
x1 c1 1.0
c2 c1 rc2 x1 90.0
x2 c2 1.0 c1 90.0 x1 0.0
c3 c1 rc3 x1 90.0 c2 tc3
c4 c1 rc3 x1 90.0 c2 tc3
c5 c3 rc5 c1 ac5 x1 90.0
c6 c4 rc5 c1 ac5 x1 90.0
h1 c2 rh1 x2 90.0 c1 180.0
h2 c3 rh2 c1 ah2 x1 90.0
h3 c4 rh2 c1 ah2 x1 90.0
h4 c5 rh4 c3 ah4 c1 180.0
h5 c6 rh4 c4 ah4 c1 180.0
rc2 = 2.672986
rc3 = 1.354498
tc3 = 62.851505
rc5 = 1.372904
ac5 = 116.454370
rh1 = 1.085735
rh2 = 1.085342
ah2 = 122.157328
rh4 = 1.087216
ah4 = 119.523496
$end
$rem
BASIS = 631G*
EXCHANGE = hf
INTSBUFFERSIZE = 15000000
SCF_ALGORITHM = diis_gdm
SCF_CONVERGENCE = 7
THRESH = 10
$end
Example 4.0 An example showing how to converge a ROHF calculation on the ^{3}A_{2}
state of DMX. Note the use of reading in orbitals from a previous closedshell
calculation and the use of MOM to maintain the orbital occupancies. The ^{3}B_{1} is
obtained if MOM is not used.
$molecule
+1 1
C 0.000000 0.000000 0.990770
H 0.000000 0.000000 2.081970
C 1.233954 0.000000 0.290926
C 2.444677 0.000000 1.001437
H 2.464545 0.000000 2.089088
H 3.400657 0.000000 0.486785
C 1.175344 0.000000 1.151599
H 2.151707 0.000000 1.649364
C 0.000000 0.000000 1.928130
C 1.175344 0.000000 1.151599
H 2.151707 0.000000 1.649364
C 1.233954 0.000000 0.290926
C 2.444677 0.000000 1.001437
H 2.464545 0.000000 2.089088
H 3.400657 0.000000 0.486785
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 631+G*
SCF_GUESS core
$end
@@@
$molecule
read
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 631+G*
SCF_GUESS read
MOM_START 1
$end
$occupied
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28
$end
@@@
$molecule
1 3
... <as above> ...
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 631+G*
SCF_GUESS read
$end
Example 4.0 RCA_DIIS algorithm applied a radical
$molecule
0 2
H 1.004123 0.180454 0.000000
O 0.246002 0.596152 0.000000
O 1.312366 0.230256 0.000000
$end
$rem
UNRESTRICTED true
EXCHANGE hf
BASIS ccpVDZ
SCF_GUESS gwh
SCF_ALGORITHM RCA_DIIS
DIIS_SUBSPACE_SIZE 15
THRESH 9
$end
4.7 DualBasis SelfConsistent Field Calculations
The dualbasis approximation [170,[171,[172,[173,[174,[175]
to selfconsistent field (HF or DFT) energies provides an
efficient means for obtaining large basis set effects at vastly less cost
than a full SCF calculation in a large basis set.
First, a full SCF calculation is performed in a
chosen small basis (specified by BASIS2). Second, a single SCFlike
step in the larger, target basis (specified, as usual, by BASIS) is used to
perturbatively approximate the large basis energy. This correction amounts to a
firstorder approximation in the change in density matrix, after the single
largebasis step:
E_{total} = E_{small basis} + Tr[(∆P)·F]_{large basis} 
 (4.88) 
where F (in the large basis) is built from the converged (small basis)
density matrix. Thus, only a single Fock build is required in the large basis
set. Currently, HF and DFT energies (SP) as well as analytic first derivatives
(FORCE or OPT) are available. [Note: As of version 4.0,
first derivatives of unrestricted dualbasis DFT energiesthough correctrequire a
codeefficiency fix. We do not recommend use of these derivatives until
this improvement has been made.]
Across the G3 set [176,[177,[178] of 223 molecules, using ccpVQZ, dualbasis
errors for B3LYP are 0.04 kcal/mol (energy) and 0.03 kcal/mol (atomization
energy per bond) and are at least an order of magnitude less than using a smaller
basis set alone. These errors are obtained at roughly an order of magnitude savings
in cost, relative to the full, targetbasis calculation.
4.7.1 DualBasis MP2
The dualbasis approximation can also be used for the reference energy
of a correlated secondorder MøllerPlesset (MP2) calculation [171,[175].
When activated, the dualbasis HF energy is first calculated as described above;
subsequently, the MO coefficients and orbital energies are used to calculate
the correlation energy in the large basis. This technique is particularly
effective for RIMP2 calculations (see Section 5.5), in which the
cost of the underlying SCF calculation often dominates.
Furthermore, efficient analytic gradients
of the DBRIMP2 energy have been developed [173] and added to QChem.
These gradients allow for the optimization of molecular structures with RIMP2 near the
basis set limit. Typical computational savings are on the order of 50% (augccpVDZ) to 71% (augccpVTZ).
Resulting dualbasis errors are only 0.001 Å in molecular structures and are, again,
significantly less than use of a smaller basis set alone.
4.7.2 Basis Set Pairings
We recommend using basis pairings in which the small basis set is a proper subset of the target basis
(631G into 631G*, for example).
They not only produce more accurate results; they also lead to more efficient integral
screening in both energies and gradients. Subsets for many standard basis sets (including Dunningstyle
ccpVXZ basis sets and their augmented analogs)
have been developed and thoroughly tested for these purposes. A summary of the
pairings is provided in Table 4.3; details of these truncations are provided in
Figure 4.1.
A new pairing for 631G*type calculations is also available. The 64G subset (named r64G in QChem)
is a subset by primitive functions and provides a smaller, faster alternative for this
basis set regime [174]. A casedependent switch in the projection code
(still OVPROJECTION) properly handles 64G. For DBHF, the calculations proceed as described
above. For DBDFT, empirical scaling factors (see Ref. for details) are applied
to the dualbasis correction. This scaling is handled automatically by the code and prints accordingly.
As of QChem version 3.2, the basis set projection code has also been adapted to properly account for
linear dependence [175], which can often be problematic for large, augmented
(augccpVTZ, etc..) basis set calculations. The same standard keyword (LINDEPTHRESH)
is utilized for linear dependence in the projection code. Because of the scheme utilized to account
for linear dependence, only propersubset pairings are now allowed.
Like singlebasis calculations, userspecified general or mixed basis sets may be
employed (see Chapter 7) with dualbasis calculations. The target basis specification
occurs in the standard $basis section. The smaller, secondary basis is placed in a similar
$basis2 section; the syntax within this section is the same as the syntax for $basis.
General and mixed small
basis sets are activated by BASIS2=BASIS2_GEN and BASIS2=BASIS2_MIXED, respectively.
BASIS  BASIS2  
ccpVTZ  rccpVTZ 
ccpVQZ  rccpVQZ 
augccpVDZ  raccpVDZ 
augccpVTZ  raccpVTZ 
augccpVQZ  raccpVQZ 
631G*  r64G, 631G 
631G**  r64G, 631G 
631++G**  631G* 
6311++G(3df,3pd)  6311G*, 6311+G* 
Table 4.3: Summary and nomenclature of recommended dualbasis pairings
#1Structure of the truncated basis set pairings for ccpV(T,Q)Z and augccpV(D,T,Q)Z.
The most compact functions are listed at the top. Primed functions depict aug (diffuse) functions.
Dashes indicate eliminated functions, relative to the paired standard basis set. In each case, the
truncations for hydrogen and heavy atoms are shown, along with the nomenclature used in QChem.
4.7.3 Job Control
DualBasis calculations are controlled with the following $rem.
DUAL_BASIS_ENERGY turns on the DualBasis approximation. Note that
use of BASIS2 without DUAL_BASIS_ENERGY only uses
basis set projection to generate the initial guess and does not invoke the
DualBasis approximation (see Section 4.5.5). OVPROJECTION is used as the default
projection mechanism for DualBasis calculations; it is not recommended that
this be changed. Specification of SCF variables
(e.g., THRESH) will apply to calculations in both basis sets.
DUAL_BASIS_ENERGY
Activates dualbasis SCF (HF or DFT) energy correction. 
TYPE:
DEFAULT:
OPTIONS:
Analytic first derivative available for HF and DFT (see JOBTYPE) 
Can be used in conjunction with MP2 or RIMP2 
See BASIS, BASIS2, BASISPROJTYPE 
RECOMMENDATION:
Use DualBasis to capture largebasis effects at smaller basis cost.
Particularly useful with RIMP2, in which HF often dominates. Use only proper
subsets for smallbasis calculation. 

4.7.4 Examples
Example 4.0 Input for a DualBasis B3LYP singlepoint calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE b3lyp
BASIS 6311++G(3df,3pd)
BASIS2 6311G*
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a DualBasis B3LYP singlepoint calculation with a minimal 64G small basis.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE b3lyp
BASIS 631G*
BASIS2 r64G
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a DualBasis RIMP2 singlepoint calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS rimp2ccpVQZ
BASIS ccpVQZ
BASIS2 rccpVQZ
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a DualBasis RIMP2 geometry optimization.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE opt
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS rimp2augccpVDZ
BASIS augccpVDZ
BASIS2 raccpVDZ
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a DualBasis RIMP2 singlepoint calculation with mixed basis sets.
$molecule
0 1
H
O 1 1.1
H 2 1.1 1 104.5
$end
$rem
JOBTYPE opt
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS aux_mixed
BASIS mixed
BASIS2 basis2_mixed
DUAL_BASIS_ENERGY true
$end
$basis
H 1
ccpVTZ
****
O 2
augccpVTZ
****
H 3
ccpVTZ
****
$end
$basis2
H 1
rccpVTZ
****
O 2
raccpVTZ
****
H 3
rccpVTZ
****
$end
$aux_basis
H 1
rimp2ccpVTZ
****
O 2
rimp2augccpVTZ
****
H 3
rimp2ccpVTZ
****
$end
4.7.5 DualBasis Dynamics
The ability to compute SCF and MP2 energies and forces at reduced cost makes dualbasis calculations
attractive for ab initio molecular dynamics simulations. Dualbasis BOMD has
demonstrated [179]
savings of 58%, even relative to stateoftheart, Fockextrapolated BOMD. Savings are further
increased to 71% for dualbasis RIMP2 dynamics. Notably, these timings outperform estimates
of extendedLagrangian (CarParrinello) dynamics, without detrimental energy conservation
artifacts that are sometimes observed in the latter [180].
Two algorithmic factors make modest but worthwhile improvements to dualbasis dynamics. First,
the iterative, smallbasis calculation can benefit from Fock matrix extrapolation [180].
Second,
extrapolation of the response equations (the socalled "Zvector" equations) for
nuclear forces further increases efficiency [181] . Both sets of keywords
are described in Section 9.7, and the code automatically
adjusts to extrapolate in the proper basis set when DUAL_BASIS_ENERGY
is activated.
4.8 HartreeFock and DensityFunctional Perturbative Corrections
4.8.1 HartreeFock Perturbative Correction
An HFPC [182,[183] calculation consists of an iterative HF calculation in a small primary
basis followed by a single Fock matrix formation, diagonalization, and energy
evaluation in a larger, secondary basis. We denote a conventional HF
calculation by HF/basis, and a HFPC calculation by HFPC/primary/secondary.
Using a primary basis of n functions, the restricted HF matrix elements for a
2melectron system are
F_{μν} = h_{μν} + 
n ∑
λσ

P_{λσ}  ⎡ ⎣

(μνλσ) − 
1
2

(μλνσ)  ⎤ ⎦


 (4.89) 
Solving the RoothaanHall equation in the primary basis results in
molecular orbitals and an associated density matrix, P. In an HFPC
calculation, P is subsequently used to build a new Fock matrix,
F^{[1]}, in a larger secondary basis of N functions
F_{ab}^{[1]} = h_{ab} + 
n ∑
λσ

P_{λσ}  ⎡ ⎣

(abλσ) − 
1
2

(aλbσ)  ⎤ ⎦


 (4.90) 
where λ, σ indicate primary basis functions and a, b
represent secondary basis functions. Diagonalization of F^{[1]} yields
improved molecular orbitals and an associated density matrix P^{[1]}. The
HFPC energy is given by
E^{HFPC} = 
N ∑
ab

P^{[1]}_{ab} h_{ab} + 
1
2


N ∑
abcd

P^{[1]}_{ab}P^{[1]}_{cd} [2(abcd) − (acbd)] 
 (4.91) 
where a, b, c and d represent secondary basis functions. This
differs from the DBHF energy evaluation where P P^{[1]}, rather than
P^{[1]}P^{[1]}, is used. The inclusion of contributions that are
quadratic in P^{[1]} is the key reason for the fact that HFPC is more
accurate than DBHF.
Unlike DBHF, HFPC does not require proper subset/superset
basis set combinations and is therefore able to jump between
any two basis sets. Benchmark study of HFPC on a large and diverse data set of
total and reaction energies show that, for a range of primary/secondary basis set
combinations the HFPC scheme can reduce the error of the primary calculation by
around two orders of magnitude at a cost of about one third that of the full
secondary calculation.
4.8.2 Density Functional Perturbative Correction (Density Functional "Triple Jumping")
Density Functional Perturbation Theory (DFPC) [184] seeks to combine the low cost of pure calculations using small bases and grids with the high accuracy of hybrid calculations using large bases and grids. Our method is motivated by the dual functional method of Nakajima and
Hirao [185] and the dual grid scheme of Tozer et al. [186]
We combine these with dual basis ideas to obtain a triple perturbation in the functional,
grid and basis directions.
4.8.3 Job Control
HFPC/DFPC calculations are controlled with the following $rem.
HFPT turns on the HFPC/DFPC approximation. Note that
HFPT_BASIS specifies the secondary basis set.
HFPT
Activates HFPC/DFPC calculation. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use DualBasis to capture largebasis effects at smaller basis cost. See reference for recommended basis set, functional, and grid pairings. 

 HFPT_BASIS
Specifies the secondary basis in a HFPC/DFPC calculation. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. 



DFPT_XC_GRID
Specifies the secondary grid in a HFPC/DFPC calculation. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. 

 DFPT_EXCHANGE
Specifies the secondary functional in a HFPC/DFPC calculation. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. 



4.8.4 Examples
Example 4.0 Input for a HFPC singlepoint calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE hf
BASIS ccpVDZ !primary basis
HFPT_BASIS ccpVQZ !secondary basis
PURECART 1111 ! set to purecart of the target basis
HFPT true
$end
Example 4.0 Input for a DFPC singlepoint calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE blyp !primary functional
DFPT_EXCHANGE b3lyp !secondary functional
DFPT_XC_GRID 00075000302 !secondary grid
XC_GRID 0 !primary grid
HFPT_BASIS 6311++G(3df,3pd) !secondary basis
BASIS 6311G* !primary basis
PURECART 1111
HFPT true
$end
4.9 Constrained Density Functional Theory (CDFT)
Under certain circumstances,
it is desirable to apply constraints to the electron density during a selfconsistent calculation.
For example, in a transition metal complex it may be desirable to constrain the net spin density
on a particular metal atom to integrate to a value consistent with the M_{S} value expected from ligand field theory.
Similarly, in a donoracceptor complex one may be interested in constraining the total density
on the acceptor group so that the formal charge on the acceptor is either neutral or negatively charged,
depending as the molecule is in its neutral or charge transfer configuration.
In these situations, one is interested in controlling the average value of some density observable,
O(r), to take on a given value, N:
There are of course many states that satisfy such a constraint,
but in practice one is usually looking for the lowest energy such state.
To solve the resulting constrained minimization problem,
one introduces a Lagrange multiplier, V, and solves for the stationary point of
V[ρ, V] = E[ρ] − V(  ⌠ ⌡

ρ(r) O(r) d^{3}r − N ) 
 (4.93) 
where E[ρ] is the energy of the system described using density functional theory (DFT).
At convergence, the functional W gives the density, ρ, that satisfies the constraint exactly
(i.e., it has exactly the prescribed number of electrons on the acceptor or spins on the metal center)
but has the lowest energy possible.
The resulting selfconsistent procedure can be efficiently solved by ensuring at every SCF step the constraint is satisfied exactly.
The QChem implementation of these equations closely parallels those in Ref. .
The first step in any constrained DFT calculation is the specification of the constraint operator, O(r).
Within QChem, the user is free to specify any constraint operator
that consists of a linear combination of the Becke's atomic partitioning functions:
O(r) = 
∑
A,σ

C^{σ}_{A} w_{A}(r) 
 (4.94) 
Here the summation runs over the atoms in the system (A) and over the electron spin
(σ = α, β).
Note that each weight function is designed to be nearly 1 near the nucleus of atom A
and rapidly fall to zero near the nucleus of any other atom in the system.
The specification of the C_{A}^{σ} coefficients is accomplished using
$cdft
CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...
$end
Here, each CONSTRAINT_VALUE is a real number that specifies the desired average value (N) of
the ensuing linear combination of atomic partition functions.
Each COEFFICIENT specifies the coefficient (C_{α}) of a partition function
or group of partition functions in the constraint operator O.
For each coefficient, all the atoms between the integers FIRST_ATOM and LAST_ATOM
contribute with the specified weight in the constraint operator.
Finally, TYPE specifies the type of constraint being appliedeither "CHARGE" or "SPIN".
For a CHARGE constraint the spin up and spin down densities contribute equally
(C_{A}^{α}=C_{A}^{β} = C_{A}) yielding the total number of electrons on the atom A.
For a SPIN constraint, the spin up and spin down densities contribute with opposite sign
(C_{A}^{α}−C_{A}^{β} = C_{A}) resulting in a measure of the net spin on the atom A.
Each separate CONSTRAINT_VALUE creates a new operator whose average is to be
constrainedfor instance, the example above includes several independent constraints:
X, Y, …. QChem can handle an arbitrary number of constraints
and will minimize the energy subject to all of these constraints simultaneously.
In addition to the $cdft input section of the input file,
a constrained DFT calculation must also set the CDFT flag to TRUE for the calculation to run.
If an atom is not included in a particular operator,
then the coefficient of that atoms partition function is set to zero for that operator.
The TYPE specification is optional, and the default is to perform a charge constraint.
Further, note that any charge constraint is on the net atomic charge.
That is, the constraint is on the difference between the average number of electrons
on the atom and the nuclear charge.
Thus, to constrain CO to be negative, the constraint value would be 1 and not 15.
The choice of which atoms to include in different constraint regions
is left entirely to the user and in practice must be based somewhat on chemical intuition.
Thus, for example, in an electron transfer reaction the user must specify which atoms
are in the "donor" and which are in the "acceptor".
In practice, the most stable choice is typically to make the constrained region
as large as physically possible.
Thus, for the example of electron transfer again, it is best to assign every atom
in the molecule to one or the other group (donor or acceptor),
recognizing that it makes no sense to assign any atoms to both groups.
On the other end of the spectrum, constraining the formal charge on a single atom
is highly discouraged.
The problem is that while our chemical intuition tells us that the lithium atom in LiF
should have a formal charge of +1, in practice the quantum mechanical charge is much closer
to +0.5 than +1.
Only when the fragments are far enough apart do our intuitive pictures of formal charge
actually become quantitative.
Finally, we note that SCF convergence is typically more challenging in constrained DFT calculations
as compared to their unconstrained counterparts.
This effect arises because applying the constraint typically leads to a broken symmetry, biradicallike state.
As SCF convergence for these cases is known to be difficult even for unconstrained states,
it is perhaps not surprising that there are additional convergence difficulties in this case.
Please see the section on SCF convergence for ideas on how to improve convergence for constrained calculations.
[Special Note: The direct minimization methods are not available for constrained calculations.
Hence, some combination of DIIS and RCA must be used to obtain convergence.
Further, it is often necessary to break symmetry in the initial guess (using SCF_GUESS_MIX)
to ensure that the lowest energy solution is obtained.]
Analytic gradients are available for constrained DFT calculations [188].
Second derivatives are only available by finite difference of gradients.
For details on how to apply constrained DFT to compute magnetic exchange couplings,
see Ref. .
For details on using constrained DFT to compute electron transfer parameters,
see Ref. .
CDFT options are:
CDFT
Initiates a constrained DFT calculation 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Perform a Constrained DFT Calculation 
FALSE  No Density Constraint 
RECOMMENDATION:
Set to TRUE if a Constrained DFT calculation is desired. 

 CDFT_POSTDIIS
Controls whether the constraint is enforced after DIIS extrapolation. 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Enforce constraint after DIIS 
FALSE  Do not enforce constraint after DIIS 
RECOMMENDATION:
Use default unless convergence problems arise,
in which case it may be beneficial to experiment with setting CDFT_POSTDIIS to FALSE.
With this option set to TRUE, energies should be variational after the first iteration. 



CDFT_PREDIIS
Controls whether the constraint is enforced before DIIS extrapolation. 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Enforce constraint before DIIS 
FALSE  Do not enforce constraint before DIIS 
RECOMMENDATION:
Use default unless convergence problems arise, in which case it may be beneficial to experiment with setting
CDFT_PREDIIS to TRUE. Note that it is possible to enforce the constraint
both before and after DIIS by setting both CDFT_PREDIIS and CDFT_POSTDIIS to TRUE. 

 CDFT_THRESH
Threshold that determines how tightly the constraint must be satisfied. 
TYPE:
DEFAULT:
OPTIONS:
N  Constraint is satisfied to within 10^{−N}. 
RECOMMENDATION:
Use default unless problems occur. 



CDFT_CRASHONFAIL
Whether the calculation should crash or not if the constraint iterations do not converge. 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Crash if constraint iterations do not converge. 
FALSE  Do not crash. 
RECOMMENDATION:

 CDFT_BECKE_POP
Whether the calculation should print the Becke atomic charges at convergence 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Print Populations 
FALSE  Do not print them 
RECOMMENDATION:
Use default. Note that the Mulliken populations printed at the end of an SCF run will not typically add up to the prescribed constraint value. Only the Becke populations are guaranteed to satisfy the userspecified constraints. 



Example 4.0 Charge separation on FAAQ
$molecule
0 1
C 0.64570736 1.37641945 0.59867467
C 0.64047568 1.86965826 0.50242683
C 1.73542663 1.01169939 0.26307089
C 1.48977850 0.39245666 0.15200261
C 0.17444585 0.86520769 0.27283957
C 0.91002699 0.02021483 0.46970395
C 3.07770780 1.57576311 0.14660056
C 2.57383948 1.35303134 0.09158744
C 3.93006075 0.78485926 0.20164558
C 4.16915637 0.61104948 0.08827557
C 5.48914671 1.09087541 0.20409492
H 5.64130588 2.16192921 0.11315072
C 6.54456054 0.22164774 0.42486947
C 6.30689287 1.16262761 0.53756193
C 5.01647654 1.65329553 0.42726664
H 1.45105590 2.07404495 0.83914389
H 0.85607395 2.92830339 0.61585218
H 0.02533661 1.93964850 0.19096085
H 7.55839768 0.60647405 0.51134530
H 7.13705743 1.84392666 0.71043613
H 4.80090178 2.71421422 0.50926027
O 2.35714021 2.57891545 0.20103599
O 3.29128460 2.80678842 0.23826460
C 2.29106231 0.63197545 0.53957285
O 2.55084900 1.72562847 0.95628300
N 3.24209015 0.26680616 0.03199109
H 2.81592456 1.08883943 0.45966550
C 4.58411403 0.11982669 0.15424004
C 5.28753695 1.14948617 0.86238753
C 5.30144592 0.99369577 0.39253179
C 6.65078185 1.06387425 1.01814801
H 4.73058059 1.98862544 1.26980479
C 6.66791492 1.05241167 0.21955088
H 4.76132422 1.76584307 0.92242502
C 7.35245187 0.03698606 0.47966072
H 7.18656323 1.84034269 1.55377875
H 7.22179827 1.89092743 0.62856041
H 8.42896369 0.10082875 0.60432214
$end
$rem
JOBTYPE FORCE
EXCHANGE B3LYP
BASIS 631G*
SCF_PRINT TRUE
CDFT TRUE
$end
$cdft
2
1 1 25
1 26 38
$end
Example 4.0 Cu2Ox High Spin
$molecule
2 3
Cu 1.4674 1.6370 1.5762
O 1.7093 0.0850 0.3825
O 0.5891 1.3402 0.9352
C 0.6487 0.3651 0.1716
N 1.2005 3.2680 2.7240
N 3.0386 2.6879 0.6981
N 1.3597 0.4651 3.4308
H 2.1491 0.1464 3.4851
H 0.5184 0.0755 3.4352
H 1.3626 1.0836 4.2166
H 1.9316 3.3202 3.4043
H 0.3168 3.2079 3.1883
H 1.2204 4.0865 2.1499
H 3.8375 2.6565 1.2987
H 3.2668 2.2722 0.1823
H 2.7652 3.6394 0.5565
Cu 1.4674 1.6370 1.5762
O 1.7093 0.0850 0.3825
O 0.5891 1.3402 0.9352
C 0.6487 0.3651 0.1716
N 1.2005 3.2680 2.7240
N 3.0386 2.6879 0.6981
N 1.3597 0.4651 3.4308
H 2.6704 3.4097 0.1120
H 3.6070 3.0961 1.4124
H 3.5921 2.0622 0.1485
H 0.3622 3.1653 3.2595
H 1.9799 3.3721 3.3417
H 1.1266 4.0773 2.1412
H 0.5359 0.1017 3.4196
H 2.1667 0.1211 3.5020
H 1.3275 1.0845 4.2152
$end
$rem
JOBTYPE SP
EXCHANGE B3LYP
BASIS 631G*
SCF_PRINT TRUE
CDFT TRUE
$end
$cdft
2
1 1 3 s
1 17 19 s
$end
4.10 Configuration Interaction with Constrained Density Functional Theory (CDFTCI)
There are some situations in which a system is not welldescribed by a
DFT calculation on a single configuration. For example, transition
states are known to be poorly described by most functionals, with
the computed barrier being too low. We can, in particular, identify
homolytic dissociation of diatomic species as situations where static
correlation becomes extremely important. Existing DFT functionals
have proved to be very effective in capturing dynamic correlation,
but frequently exhibit difficulties in the presence of strong
static correlation. Configuration Interaction, well known in
wavefunction methods, is a multireference method that is quite
wellsuited for capturing static correlation; the CDFTCI technique
allows for CI calculations on top of DFT calculations, harnessing
both static and dynamic correlation methods.
Constrained DFT is used to compute densities (and KohnSham wavefunctions)
for two or more diabaticlike states; these states are then used to
build a CI matrix. Diagonalizing this matrix yields energies
for the ground and excited states within the configuration space.
The coefficients of the initial diabatic states are printed, to
show the characteristics of the resultant states.
Since DensityFunctional Theory only gives converged densities,
not actual wavefunctions, computing the offdiagonal coupling
elements H_{12} is not completely straightforward, as the physical
meaning of the KohnSham wavefunction is not entirely clear.
We can, however, perform the following manipulation [191]:
 


1
2

[〈1H+V_{C1}ω_{C1}−V_{C1}ω_{C1} 2〉+ 〈1H+V_{C2}ω_{C2}−V_{C2}ω_{C2}2〉] 
 
 


1
2

[(E_{1}+V_{C1}N_{C1}+E_{2}+V_{C2}N_{C2}) 〈12〉−V_{C1}〈1ω_{C1}2〉 −V_{C2}〈1ω_{C2}2〉] 
 

(where the converged states i〉 are assumed to be the
ground state of H+V_{Ci}ω_{Ci} with eigenvalue E_{i}+V_{Ci}N_{Ci}).
This manipulation eliminates the twoelectron integrals from the
expression, and experience has shown that the use of Slater determinants
of KohnSham orbitals is a reasonable approximation for the
quantities 〈12〉 and 〈1ω_{Ci}2〉.
We note that since these constrained states are eigenfunctions
of different Hamiltonians (due to different constraining potentials),
they are not orthogonal states, and we must set up our
CI matrix as a generalized eigenvalue problem. Symmetric orthogonalization
is used by default, though the overlap matrix and Hamiltonian
in nonorthogonal basis are also printed at higher print levels
so that other orthogonalization schemes can be used afterthefact.
In a limited number of cases, it is possible to find an orthogonal
basis for the CDFTCI Hamiltonian, where a physical interpretation
can be assigned to the orthogonal states. In such cases, the
matrix representation of the Becke weight operator is diagonalized,
and the (orthogonal) eigenstates can be characterized [192].
This matrix is printed as the "CDFTCI Population Matrix" at
increased print levels.
In order to perform a CDFTCI calculation, the N interacting
states must be defined; this is done in a very similar fashion
to the specification for CDFT states:
$cdft
STATE_1_CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
STATE_1_CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...

STATE_2_CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
STATE_2_CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...
...
$end
Each state is specified with the CONSTRAINT_VALUE and the
corresponding weights on sets of atoms whose average value should
be the constraint value.
Different states are separated by a single line containing three or more
dash characters.
If it is desired to use an unconstrained state as one of the
interacting configurations, charge and spin constraints of
zero may be applied to the atom range from 0 to 0.
It is MANDATORY to specify a spin constraint corresponding
to every charge constraint (and it must be immediately following
that charge constraint in the input deck), for reasons described below.
In addition to the $cdft input section of the input file,
a CDFTCI calculation must also set the CDFTCI flag
to TRUE for the calculation to run. Note, however, that the
CDFT flag is used internally by CDFTCI, and should
not be set in the input deck. The variable CDFTCI_PRINT
may also be set manually to control the level of output. The default
is 0, which will print the energies and weights (in the diabatic
basis) of the N CDFTCI states. Setting it to 1 or above will
also print the CDFTCI overlap matrix, the CDFTCI Hamiltonian matrix
before the change of basis, and the CDFTCI Population matrix.
Setting it to 2 or above
will also print the eigenvectors and eigenvalues of the CDFTCI
Population matrix. Setting it to 3 will produce more output
that is only useful during application debugging.
For convenience, if CDFTCI_PRINT is not set in the
input file, it will be set to the value of SCF_PRINT.
As mentioned in the previous section, there is a disparity
between our chemical intuition of what charges should be
and the actual quantummechanical charge. The example was given
of LiF, where our intuition gives the lithium atom a formal
charge of +1; we might similarly imagine performing a
CDFTCI calculation on H_{2}, with two ionic states
and two spinconstrained states. However, this would result
in attempting to force both electrons of H_{2} onto
the same nucleus, and this calculation is impossible to converge
(since by the nature of the Becke weight operators, there will
be some nonzero amount of the density that gets proportioned
onto the other atom, at moderate internuclear separations).
To remedy problems such as this, we have adopted a mechanism by
which to convert the formal charges of our chemical intuition
into reasonable quantummechanical charge constraints.
We use the formalism of "promolecule" densities, wherein
the molecule is divided into fragments (based on the partitioning
of constraint operators), and a DFT calculation is performed on
these fragments, completely isolated from each other [192].
(This step is why both spin and charge constraints are required, so
that the correct partitioning of electrons for each fragment
may be made.) The resulting promolecule densities, converged
for the separate fragments, are then added together, and the
value of the various weight operators as applied to this
new density, is used as a constraint for the actual CDFT
calculations on the interacting states.
The promolecule density method compensates for the effect
of nearby atoms on the actual density that will be constrained.
The comments about SCF convergence for CDFT calculations also
apply to the calculations used for CDFTCI, with the addition
that if the SCF converges but CDFT does not, it may be necessary
to use a denser integration grid or reduce the value of CDFT_THRESH.
Analytic gradients are not available. For details on using CDFTCI
to calculate reaction barrier heights, see Ref. .
CDFTCI options are:
 CDFTCI
Initiates a constrained DFTconfiguration interaction calculation 
TYPE:
DEFAULT:
OPTIONS:
TRUE  Perform a CDFTCI Calculation 
FALSE  No CDFTCI 
RECOMMENDATION:
Set to TRUE if a CDFTCI calculation is desired. 



CDFTCI_PRINT
Controls level of output from CDFTCI procedure to QChem output file. 
TYPE:
DEFAULT:
OPTIONS:
0  Only print energies and coefficients of CDFTCI final states 
1  Level 0 plus CDFTCI overlap, Hamiltonian, and population matrices 
2  Level 1 plus eigenvectors and eigenvalues of the CDFTCI population matrix 
3  Level 2 plus promolecule orbital coefficients and energies 
RECOMMENDATION:
Level 3 is primarily for program debugging; levels 1 and 2 may be useful
for analyzing the coupling elements 

 CDFT_LAMBDA_MODE
Allows CDFT potentials to be specified directly, instead of being
determined as Lagrange multipliers. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Standard CDFT calculations are used. 
TRUE  Instead of specifying target charge and spin constraints, use the values 
 from the input deck as the value of the Becke weight potential

RECOMMENDATION:
Should usually be set to FALSE. Setting to TRUE can be useful to
scan over different strengths of charge or spin localization, as
convergence properties are improved compared to regular CDFT(CI) calculations. 



CDFTCI_SKIP_PROMOLECULES
Skips promolecule calculations and allows fractional charge and spin
constraints to be specified directly. 
TYPE:
DEFAULT:
OPTIONS:
FALSE  Standard CDFTCI calculation is performed. 
TRUE  Use the given charge/spin constraints directly, with no
promolecule calculations. 
RECOMMENDATION:
Setting to TRUE can be useful for scanning over constraint values. 

Note that CDFT_LAMBDA_MODE and CDFTCI_SKIP_PROMOLECULES
are mutually incompatible.
CDFTCI_SVD_THRESH
By default, a symmetric orthogonalization is performed on the CDFTCI
matrix before diagonalization. If the CDFTCI overlap matrix is nearly
singular (i.e., some of the diabatic states are nearly degenerate), then
this orthogonalization can lead to numerical instability. When computing
→S^{−1/2}, eigenvalues smaller than 10^{−CDFTCI_SVD_THRESH}
are discarded. 
TYPE:
DEFAULT:
OPTIONS:
n  for a threshold of 10^{−n}. 
RECOMMENDATION:
Can be decreased if numerical instabilities are encountered in the
final diagonalization. 

 CDFTCI_STOP
The CDFTCI procedure involves performing independent SCF calculations
on distinct constrained states. It sometimes occurs that the same
convergence parameters are not successful for all of the states of
interest, so that a CDFTCI calculation might converge one of these
diabatic states but not the next. This variable allows a user to
stop a CDFTCI calculation after a certain number of states have
been converged, with the ability to restart later on the next state,
with different convergence options. 
TYPE:
DEFAULT:
OPTIONS:
n  stop after converging state n (the first state is state 1) 
0  do not stop early 
RECOMMENDATION:
Use this setting if some diabatic states converge but others do not. 



CDFTCI_RESTART
To be used in conjunction with CDFTCI_STOP, this variable
causes CDFTCI to read alreadyconverged states from disk and begin
SCF convergence on later states. Note that the same $cdft section
must be used for the stopped calculation and the restarted calculation. 
TYPE:
DEFAULT:
OPTIONS:
n  start calculations on state n+1 
RECOMMENDATION:
Use this setting in conjunction with CDFTCI_STOP. 

Many of the CDFTrelated rem variables are also applicable to CDFTCI calculations.
4.11 Unconventional SCF Calculations
4.11.1 CASE Approximation
The Coulomb Attenuated Schrödinger Equation (CASE) [194]
approximation follows from the KWIK [195] algorithm in which
the Coulomb operator is separated into two pieces using the error
function, Eq. (4.45). Whereas in Section 4.3.4
this partition of the Coulomb operator was used to incorporate longrange
HartreeFock exchange into DFT, within the CASE approximation it is used
to attenuate all occurrences of the Coulomb operator in
Eq. (4.2), by
neglecting the longrange portion of the identity in
Eq. (4.45). The parameter ω in
Eq. (4.45) is used to tune the level of attenuation.
Although the
total energies from Coulomb attenuated calculations are significantly different
from nonattenuated energies, it is found that relative energies, correlation
energies and, in particular, wavefunctions, are not, provided a reasonable
value of ω is chosen.
By virtue of the exponential decay of the attenuated operator, ERIs can be
neglected on a proximity basis yielding a rigorous O(N) algorithm for single
point energies. CASE may also be applied in geometry optimizations and
frequency calculations.
OMEGA
Controls the degree of attenuation of the Coulomb operator. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to ω = n/1000, in units of bohr^{−1} 
RECOMMENDATION:

 INTEGRAL_2E_OPR
Determines the twoelectron operator. 
TYPE:
DEFAULT:
OPTIONS:
1  Apply the CASE approximation. 
2  Coulomb Operator. 
RECOMMENDATION:
Use default unless the CASE operator is desired. 



4.11.2 Polarized Atomic Orbital (PAO) Calculations
Polarized atomic orbital (PAO) calculations are an interesting unconventional
SCF method, in which the molecular orbitals and the density matrix are not
expanded directly in terms of the basis of atomic orbitals. Instead, an
intermediate moleculeoptimized minimal basis of polarized atomic orbitals
(PAOs) is used [196]. The polarized atomic orbitals are defined by
an atomblocked linear transformation from the fixed atomic orbital basis,
where the coefficients of the transformation are optimized to minimize the
energy, at the same time as the density matrix is obtained in the PAO
representation. Thus a PAOSCF calculation is a constrained variational
method, whose energy is above that of a full SCF calculation in the same basis.
However, a molecule optimized minimal basis is a very compact and useful
representation for purposes of chemical analysis, and it also has potential
computational advantages in the context of MP2 or local MP2 calculations, as
can be done after a PAOHF calculation is complete to obtain the PAOMP2
energy.
PAOSCF calculations tend to systematically underestimate binding energies
(since by definition the exact result is obtained for atoms, but not for
molecules). In tests on the G2 database, PAOB3LYP/6311+G(2df,p) atomization
energies deviated from full B3LYP/6311+G(2df,p) atomization energies by
roughly 20 kcal/mol, with the error being essentially extensive with the number
of bonds. This deviation can be reduced to only 0.5 kcal/mol
with the use of a simple noniterative second order
correction for "beyondminimal basis" effects [197]. The second
order correction is evaluated at the end of each PAOSCF calculation, as it
involves negligible computational cost. Analytical gradients are available
using PAOs, to permit structure optimization. For additional discussion of the
PAOSCF method and its uses, see the references cited above.
Calculations with PAOs are determined controlled by the following $rem
variables. PAO_METHOD = PAO invokes PAOSCF calculations, while the
algorithm used to iterate the PAO's can be controlled with
PAO_ALGORITHM.
PAO_ALGORITHM
Algorithm used to optimize polarized atomic orbitals (see PAO_METHOD) 
TYPE:
DEFAULT:
OPTIONS:
0  Use efficient (and riskier) strategy to converge PAOs. 
1  Use conservative (and slower) strategy to converge PAOs. 
RECOMMENDATION:

 PAO_METHOD
Controls evaluation of polarized atomic orbitals (PAOs). 
TYPE:
DEFAULT:
EPAO  For local MP2 calculations Otherwise no default. 
OPTIONS:
PAO  Perform PAOSCF instead of conventional SCF. 
EPAO  Obtain EPAO's after a conventional SCF. 
RECOMMENDATION:



4.12 SCF Metadynamics
As the SCF equations are nonlinear in the electron density, there are in
theory very many solutions (i.e., sets of orbitals where the energy is stationary
with respect to changes in the orbital subset). Most often sought is the
solution with globally minimal energy as this is a variational upper bound
to the true eigenfunction in this basis.
The SCF methods available in QChem allow the user to converge upon an
SCF solution, and (using STABILITY_ANALYSIS) ensure it is a minimum,
but there is no known method of ensuring that the found solution is a global
minimum; indeed in systems with many lowlying energy levels the solution
converged upon may vary considerably with initial guess.
SCF metadynamics [198] is a technique which can be used to
locate multiple SCF solutions, and thus gain some confidence that
the calculation has converged upon the global minimum.
It works by searching out a solution to the SCF equations. Once found,
the solution is stored, and a biasing potential added so as
to avoid reconverging to the same solution.
More formally, the distance between two solutions, w and x,
can be expressed as d_{wx}^{2}=〈^{w}Ψ ^{w}∧ρ− ^{x}∧ρ  ^{w}Ψ〉,
where ^{w}Ψ is a Slater determinant formed from the orthonormal orbitals,
^{w}ϕ_{i}, of solution w, and ^{w}∧ρ is the oneparticle
density operator for ^{w}Ψ. This definition is equivalent to
d_{wx}^{2}=N−^{w}P^{μν}S_{νσ}·^{x}P^{στ}S_{τμ}. and is easily calculated.
d_{wx}^{2} is bounded by 0 and the number of electrons, and can be taken as the
distance between two solutions. As an example, any singly excited
determinant from an SCF determinant (which will not in general be
another SCF solution), would be a distance 1 away from it.
In a manner analogous to classical metadynamics, to bias against the set of
previously located solutions, x, we create a new Lagrangian,
 
~
E

=E+ 
∑
x

N_{x} e^{−λx d0x2} 
  (4.95) 

where 0 represents the present density.
From this we may derive a new effective Fock matrix,
 
~
F

μν

=F_{μν}+ 
x ∑
x

P_{μν} N_{x} λ_{x} e^{−λx d0x2} 
  (4.96) 

This may be used with very little modification within a standard DIIS procedure to locate multiple solutions.
When close to a new solution, the biasing potential is removed so the location of that solution is not affected by it.
If the calculation ends up reconverging to the same solution, N_{x} and λ_{x} can be modified to avert this.
Once a solution is found it is added to the list of solutions, and the orbitals mixed
to provide a new guess for locating a different solution.
This process can be customized by the REM variables below.
Both DIIS and GDM methods can be used, but it is advisable
to turn on MOM when using DIIS to maintain the orbital ordering.
PostHF correlation methods can also be applied. By default
they will operate for the last solution located, but this can be changed with the
SCF_MINFIND_RUNCORR variable.
The solutions found through metadynamics also appear to be good approximations to diabatic surfaces
where the electronic structure does not significantly change with geometry.
In situations where there are such multiple electronic states close in energy, an adiabatic state
may be produced by diagonalizing a matrix of these states  Configuration Interaction. As they are distinct solutions of the
SCF equations, these states are nonorthogonal (one cannot be constructed as a single determinant
made out of the orbitals of another), and so the CI is a little more complicated and is a NonOrthogonal CI.
For more information see the NOCI section in Chapter 6
SCF_SAVEMINIMA
Turn on SCF Metadynamics and specify how many solutions to locate. 
TYPE:
DEFAULT:
OPTIONS:
0  Do not use SCF Metadynamics 
n  Attempt to find n distinct SCF solutions. 
RECOMMENDATION:
Perform SCF Orbital metadynamics and attempt to locate
n different SCF solutions. Note that these may not all be minima. Many saddle points are often located.
The last one located will be the one used in any postSCF treatments.
In systems where there are infinite point groups, this procedure
cannot currently distinguish between spatial rotations of different
densities, so will likely converge on these multiply. 

 SCF_READMINIMA
Read in solutions from a previous SCF Metadynamics calculation 
TYPE:
DEFAULT:
OPTIONS:
n  Read in n previous solutions and attempt to locate them all. 
−n  Read in n previous solutions, but only attempt to locate solution n. 
RECOMMENDATION:
This may not actually locate all solutions required and will probably
locate others too. The SCF will also stop when the number of
solutions specified in SCF_SAVEMINIMA are found.
Solutions from other geometries may also be read in and used as starting orbitals.
If a solution is found and matches one that is read in within
SCF_MINFIND_READDISTTHRESH, its orbitals are saved in
that position for any future calculations.
The algorithm works by restarting from the orbitals and density
of a the minimum it is attempting to find. After 10 failed
restarts (defined by SCF_MINFIND_RESTARTSTEPS), it moves
to another previous minimum and attempts to locate that instead.
If there are no minima to find, the restart does random mixing
(with 10 times the normal random mixing parameter).




SCF_MINFIND_WELLTHRESH
Specify what SCF_MINFIND believes is the basin of a solution 
TYPE:
DEFAULT:
OPTIONS:
n for a threshold of 10^{−n} 
RECOMMENDATION:
When the DIIS error is less than 10^{−n}, penalties are switched
off to see whether it has converged to a new solution. 

 SCF_MINFIND_RESTARTSTEPS
Restart with new orbitals if no minima have been found within this many steps 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If the SCF calculation spends many steps not finding a
solution, lowering this number may speed up solutionfinding.
If the system converges to solutions very
slowly, then this number may need to be raised. 



SCF_MINFIND_INCREASEFACTOR
Controls how the height of the penalty function
changes when repeatedly trapped at the same solution 
TYPE:
DEFAULT:
OPTIONS:
abcde  corresponding to a.bcde 
RECOMMENDATION:
If the algorithm converges to a solution which corresponds
to a previously located solution, increase both the
normalization N and the width lambda of the penalty function there. Then do a restart. 

 SCF_MINFIND_INITLAMBDA
Control the initial width of the penalty function. 
TYPE:
DEFAULT:
OPTIONS:
abcde  corresponding to ab.cde 
RECOMMENDATION:
The initial inversewidth (i.e., the inversevariance) of the
Gaussian to place to fill solution's well. Measured in electrons^{(}−1).
Increasing this will repeatedly converging on the same solution. 



SCF_MINFIND_INITNORM
Control the initial height of the penalty function. 
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde 
RECOMMENDATION:
The initial normalization of the Gaussian to place to fill a well. Measured in Hartrees. 

 SCF_MINFIND_RANDOMMIXING
Control how to choose new orbitals after locating a solution 
TYPE:
DEFAULT:
00200 meaning .02 radians 
OPTIONS:
abcde corresponding to a.bcde radians 
RECOMMENDATION:
After locating an SCF solution, the orbitals are mixed
randomly to move to a new position in orbital space. For each
occupied and virtual orbital pair picked at random and rotate
between them by a random angle between 0 and this. If this
is negative then use exactly this number, e.g., −15708 will
almost exactly swap orbitals. Any number < −15708 will cause the orbitals to be swapped exactly. 



SCF_MINFIND_NRANDOMMIXES
Control how many random mixes to do to generate new orbitals 
TYPE:
DEFAULT:
OPTIONS:
n  Perform n random mixes. 
RECOMMENDATION:
This is the number of occupied/virtual pairs to attempt to mix,
per separate density (i.e., for unrestricted calculations both
alpha and beta space will get this many rotations). If this
is negative then only mix the highest 25% occupied and lowest 25% virtuals. 

 SCF_MINFIND_READDISTTHRESH
The distance threshold at which to consider two solutions the same 
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde 
RECOMMENDATION:
The threshold to regard a minimum as the same as a read in
minimum. Measured in electrons. If two minima are closer
together than this, reduce the threshold to distinguish them. 



SCF_MINFIND_MIXMETHOD
Specify how to select orbitals for random mixing 
TYPE:
DEFAULT:
OPTIONS:
0  Random mixing: select from any orbital to any orbital. 
1  Active mixing: select based on energy, decaying with distance from the Fermi level. 
2  Active Alpha space mixing: select based on energy, decaying with distance from the 
 Fermi level only in the alpha space. 
RECOMMENDATION:
Random mixing will often find very high energy solutions.
If lower energy solutions are desired, use 1 or 2. 

 SCF_MINFIND_MIXENERGY
Specify the active energy range when doing Active mixing 
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde 
RECOMMENDATION:
The standard deviation of the Gaussian distribution used
to select the orbitals for mixing (centered on the Fermi level). Measured in Hartree.
To find lessexcited solutions, decrease this value 



SCF_MINFIND_RUNCORR
Run postSCF correlated methods on multiple SCF solutions 
TYPE:
DEFAULT:
OPTIONS:
If this is set > 0, then run correlation methods for all found SCF solutions. 
RECOMMENDATION:
PostHF correlation methods should function correctly with excited
SCF solutions, but their convergence is often much more difficult owing to intruder states. 

4.13 Ground State Method Summary
To summarize the main features of QChem's ground state selfconsistent field
capabilities, the user needs to consider:
 Input a molecular geometry ($molecule keyword)
 Cartesian
 Zmatrix
 Read from prior calculations
 Declare the job specification ($remkeyword)

JOBTYPE
 Single point
 Optimization
 Frequency
 See Table 4.1 for further options
 BASIS
 Refer to Chapter 7 (note: $basis keyword for user defined basis sets)
 Effective core potentials, as described in Chapter 8
 EXCHANGE
 Linear scaling algorithms for all methods
 Arsenal of exchange density functionals
 User definable functionals and hybrids
 CORRELATION
 DFT or wavefunctionbased methods
 Linear scaling (CPU and memory) incorporation of correlation with DFT
 Arsenal of correlation density functionals
 User definable functionals and hybrids
 See Chapter 5 for wavefunctionbased correlation methods.
 Exploit QChem's special features
 CFMM, LinK large molecule options
 SCF rate of convergence increased through improved guesses and
alternative minimization algorithms
 Explore novel methods if
desired: CASE approximation, PAOs.
Chapter 5 WavefunctionBased Correlation Methods
5.1 Introduction
The HartreeFock procedure, while often qualitatively correct, is frequently
quantitatively deficient. The deficiency is due to the underlying assumption of
the HartreeFock approximation: that electrons move independently
within molecular orbitals subject to an averaged field imposed by the remaining
electrons. The error that this introduces is called the correlation energy and
a wide variety of procedures exist for estimating its magnitude. The purpose of
this Chapter is to introduce the main wavefunctionbased methods available in
QChem to describe electron correlation.
Wavefunctionbased electron correlation methods concentrate on the design of
corrections to the wavefunction beyond the meanfield HartreeFock
description. This is to be contrasted with the density functional theory
methods discussed in the previous Chapter. While density functional methods
yield a description of electronic structure that accounts for electron
correlation subject only to the limitations of presentday functionals (which,
for example, omit dispersion interactions), DFT cannot be systematically
improved if the results are deficient. Wavefunctionbased approaches for
describing electron correlation [201,[202] offer this main
advantage. Their main disadvantage is relatively high computational cost,
particularly for the higherlevel theories.
There are four broad classes of models for describing electron correlation that
are supported within QChem. The first three directly approximate the full
timeindependent Schrödinger equation. In order of increasing accuracy, and
also increasing cost, they are:
 Perturbative treatment of pair correlations between electrons, typically
capable of recovering 80% or so of the correlation energy in stable
molecules.
 Selfconsistent treatment of pair correlations between electrons (most often
based on coupledcluster theory),
capable of recovering on the order of 95% or so of the correlation
energy.
 Noniterative corrections for higher than double substitutions, which
can account for more than 99% of the correlation energy. They are the
basis of many modern methods that are capable of yielding chemical
accuracy for ground state reaction energies, as exemplified by the
G2 [176] and G3 methods [177].
These methods are discussed in the following subsections.
There is also a fourth class of methods supported in QChem, which have a
different objective. These active space methods aim to obtain a balanced
description of electron correlation in highly correlated systems, such as
biradicals, or along bondbreaking coordinates. Active space methods are
discussed in Section 5.8. Finally, equationofmotion (EOM)
methods provide tools for describing openshell and electronically excited
species. Selected configuration interaction (CI) models are also available.
In order to carry out a wavefunctionbased electron correlation calculation
using QChem, three $rem variables need to be set:

BASIS to specify the basis set (see Chapter 7)
 CORRELATION method for treating Correlation (defaults to
NONE)
 N_FROZEN_CORE frozen core electrons (0 default, optionally
FC, or n)
Additionally, for EOM or CI calculations the number of target states of each type
(excited, spinflipped, ionized, attached, etc.) in each
irreducible representation (irrep) should be specified (see Section
6.6.7). The level of correlation of the target EOM states
may be different from that used for the reference, and can be specified by
EOM_CORR keyword.
Note that for wavefunctionbased correlation methods, the default option for
EXCHANGE is HF (HartreeFock). It can therefore be omitted
from the input. If desired, correlated calculations can employ DFT
orbitals by setting EXCHANGE to a specific DFT method (see
Section 5.10).
The full range of ground state wavefunctionbased correlation methods
available (i.e. the recognized options to the CORRELATION keyword)
are as follows:.
CORRELATION
Specifies the correlation level of theory, either DFT or wavefunctionbased. 
TYPE:
DEFAULT:
OPTIONS:
MP2  Sections 5.2 and 5.3 
Local_MP2  Section 5.4 
RILMP2  Section 5.5.1 
ZAPT2  A more efficient restricted openshell MP2 method [203]. 
MP3  Section 5.2 
MP4SDQ  Section 5.2 
MP4  Section 5.2 
CCD  Section 5.6 
CCD(2)  Section 5.7 
CCSD  Section 5.6 
CCSD(T)  Section 5.7 
CCSD(2)  Section 5.7 
CCSD(fT)  Section 5.7.3 
CCSD(dT)  Section 5.7.3 
QCISD  Section 5.6 
QCISD(T)  Section 5.7 
OD  Section 5.6 
OD(T)  Section 5.7 
OD(2)  Section 5.7 
VOD  Section 5.8 
VOD(2)  Section 5.8 
QCCD  Section 5.6 
QCCD(T)  
QCCD(2)  
VQCCD  Section 5.8 
RECOMMENDATION:
Consult the literature for guidance. 

5.2 MøllerPlesset Perturbation Theory
5.2.1 Introduction
MøllerPlesset Perturbation Theory [110] is a widely used
method for approximating the correlation energy of molecules. In particular,
second order MøllerPlesset perturbation theory (MP2) is one of the
simplest and most useful levels of theory beyond the HartreeFock
approximation. Conventional and local MP2 methods available in QChem are
discussed in detail in Sections 5.3 and 5.4
respectively. The MP3 method is still occasionally used, while MP4 calculations
are quite commonly employed as part of the G2 and G3 thermochemical
methods [176,[177]. In the remainder of this section, the
theoretical basis of MøllerPlesset theory is reviewed.
5.2.2 Theoretical Background
The HartreeFock wavefunction Ψ_{0} and energy E_{0} are
approximate solutions (eigenfunction and eigenvalue) to the exact
Hamiltonian eigenvalue problem or Schrödinger's electronic wave equation,
Eq. (4.5). The HF wavefunction and energy are, however, exact solutions for the
HartreeFock Hamiltonian H_{0} eigenvalue problem. If we assume that the
HartreeFock wavefunction Ψ_{0} and energy E_{0} lie near the exact wave
function Ψ and energy E, we can now write the exact Hamiltonian
operator as
where V is the small perturbation and λ is a dimensionless parameter.
Expanding the exact wavefunction and energy in terms of the HF wavefunction
and energy yields
E=E^{(0)}+λE^{(1)}+λ^{2}E^{(2)}+λ^{3}E^{(3)}+… 
 (5.2) 
and
Ψ = Ψ_{0} +λΨ^{(1)}+λ^{2}Ψ^{(2)}+λ^{3}Ψ^{(3)}+… 
 (5.3) 
Substituting these expansions into the Schrödinger equation and collecting terms according
to powers of λ yields
H_{0} Ψ_{0} = E^{(0)}Ψ_{0} 
 (5.4) 
H_{0} Ψ^{(1)}+VΨ_{0} = E^{(0)}Ψ^{(1)}+E^{(1)}Ψ_{0} 
 (5.5) 
H_{0} Ψ^{(2)}+VΨ^{(1)}=E^{(0)}Ψ^{(2)}+E^{(1)}Ψ^{(1)}+E^{(2)}Ψ_{0} 
 (5.6) 
and so forth. Multiplying each of the above equations by Ψ_{0} and
integrating over all space yields the following expression for the nthorder
(MPn) energy:
E^{(0)}=〈Ψ_{0} H_{0}  Ψ_{0} 〉 
 (5.7) 
E^{(1)}=〈Ψ_{0} V Ψ_{0} 〉 
 (5.8) 
E^{(2)}=〈 Ψ_{0} V Ψ^{(1)} 〉 
 (5.9) 
Thus, the HartreeFock energy
E_{0} = 〈 Ψ_{0} H_{0} +V Ψ_{0} 〉 
 (5.10) 
is simply the sum of the zeroth and first order energies
The correlation energy can then be written
E_{corr} = E_{0}^{(2)} +E_{0}^{(3)} +E_{0}^{(4)} +… 
 (5.12) 
of which the first term is the MP2 energy.
It can be shown that the MP2 energy can be written (in terms of spinorbitals) as
E_{0}^{(2)} = − 
1
4


virt ∑
ab


occ ∑
ij


 〈 ab  ij 〉 ^{2}
ε_{a} +ε_{b} −ε_{i} −ε_{j}


 (5.13) 
where
〈 ab ij 〉 = 〈 ab  ab ij ij 〉 −〈 ab  ab ji ji 〉 
 (5.14) 
and
〈 ab  ab cd cd 〉 =  ⌠ ⌡

ψ_{a} (r_{1} )ψ_{c} (r_{1} )  ⎡ ⎣

1
r_{12}
 ⎤ ⎦

ψ_{b} (r_{2} )ψ_{d} (r_{2} )dr_{1} dr_{2} 
 (5.15) 
which can be written in terms of the twoelectron repulsion integrals
〈 ab  ab cd cd 〉 = 
∑
μ


∑
ν


∑
λ


∑
σ

C_{μa} C_{νc} C_{λb} C_{σd} ( μνλσ ) 
 (5.16) 
Expressions for higher order terms follow similarly, although with much greater
algebraic and computational complexity. MP3 and particularly MP4 (the third and
fourth order contributions to the correlation energy) are both occasionally
used, although they are increasingly supplanted by the coupledcluster methods
described in the following sections. The disk and memory requirements for MP3
are similar to the selfconsistent pair correlation methods discussed in
Section 5.6 while the computational cost of MP4 is similar to the
"(T)" corrections discussed in Section 5.7.
5.3 Exact MP2 Methods
5.3.1 Algorithm
Second order MøllerPlesset theory (MP2) [110] probably the
simplest useful wavefunctionbased electron correlation method. Revived in
the mid1970s, it remains highly popular today, because it offers systematic
improvement in optimized geometries and other molecular properties relative to
HartreeFock (HF) theory [6]. Indeed, in a recent comparative
study of small closedshell molecules [204], MP2 outperformed
much more expensive singles and doubles coupledcluster theory for such
properties! Relative to stateoftheart KohnSham density functional
theory (DFT) methods, which are the most economical methods to account for
electron correlation effects, MP2 has the advantage of properly incorporating
longrange dispersion forces. The principal weaknesses of MP2 theory are for
open shell systems, and other cases where the HF determinant is a poor starting
point.
QChem contains an efficient conventional semidirect method to evaluate the
MP2 energy and gradient [205]. These methods require OVN memory
(O, V, N are the numbers of occupied, virtual and total orbitals,
respectively), and disk space which is bounded from above by OVN^{2}/2. The
latter can be reduced to IVN^{2}/2 by treating the occupied orbitals in batches
of size I, and reevaluating the twoelectron integrals O/I times. This
approach is tractable on modern workstations for energy and gradient
calculations of at least 500 basis functions or so, or molecules of between 15
and 30 first row atoms, depending on the basis set size. The computational cost
increases between the 3rd and 5th power of the size of the molecule,
depending on which part of the calculation is timedominant.
The algorithm and implementation in QChem is improved over earlier
methods [206,[207], particularly in the following areas:
 Uses pure functions, as opposed to Cartesians, for all fifthorder steps.
This leads to large computational savings for basis sets containing pure
functions.
 Customized loop unrolling for improved efficiency.
 The sortless semidirect method avoids a read and write operation
resulting in a large I/O savings.
 Reduction in disk and memory usage.
 No extra integral evaluation for gradient calculations.
 Full exploitation of frozen core approximation.
The implementation offers the user the following alternatives:
 Direct algorithm (energies only).
 Diskbased sortless semidirect algorithm (energies and gradients).
 Local occupied orbital method (energies only).
The semidirect algorithm is the only choice for gradient calculations. It is
also normally the most efficient choice for energy calculations. There are two
classes of exceptions:
 If the amount of disk space available is not significantly larger than
the amount of memory available, then the direct algorithm is preferred.
 If the calculation involves a very large basis set, then the local
orbital method may be faster, because it performs the transformation in a
different order. It does not have the large memory requirement (no
OVN array needed), and always evaluates the integrals four times. The
AO2MO_DISK option is also ignored in this algorithm, which
requires up to O^{2}VN megabytes of disk space.
There are three important options that should be wisely chosen by the user in
order to exploit the full efficiency of QChem's direct and semidirect MP2
methods (as discussed above, the LOCAL_OCCUPIED method has different
requirements).

MEM_STATIC : The value specified for this $rem variable must be
sufficient to permit efficient integral evaluation (1080Mb) and to hold
a large temporary array whose size is OVN, the product of the number of
occupied, virtual and total numbers of orbitals.
 AO2MO_DISK: The value specified for this $rem variable should be
as large as possible (i.e., perhaps 80% of the free space on your
$QCSCRATCH partition where temporary job files are held). The
value of this variable will determine how many times the twoelectron
integrals in the atomic orbital basis must be reevaluated, which is a
major computational step in MP2 calculations.
 N_FROZEN_CORE: The computational requirements for MP2 are
proportional to the number of occupied orbitals for some steps, and the
square of that number for other steps. Therefore the CPU time can be
significantly reduced if your job employs the frozen core approximation.
Additionally the memory and disk requirements are reduced when the frozen
core approximation is employed.
5.3.2 The Definition of Core Electron
The number of core electrons in an atom is relatively well defined, and
consists of certain atomic shells, (note that ECPs are available in
`smallcore' and `largecore' varieties, see Chapter 8 for
further details). For example, in phosphorus the core consists of 1s, 2s,
and 2p shells, for a total of ten electrons. In molecular systems, the core
electrons are usually chosen as those occupying the n/2 lowest energy
orbitals, where n is the number of core electrons in the constituent atoms.
In some cases, particularly in the lower parts of the periodic table, this
definition is inappropriate and can lead to significant errors in the
correlation energy. Vitaly Rassolov has implemented an alternative definition
of core electrons within QChem which is based on a Mulliken population
analysis, and which addresses this problem [208].
The current implementation is restricted to nkl type basis sets such as
321 or 631, and related bases such as 631+G(d). There are essentially two
cases to consider, the outermost 6G functions (or 3G in the case of the 321G
basis set) for Na, Mg, K and Ca, and the 3d functions for the elements GaKr.
Whether or not these are treated as core or valence is determined by the
CORE_CHARACTER $rem, as summarized in Table 5.3.2.
CORE_CHARACTER  Outermost 6G (3G)  3d (GaKr) 
 for Na, Mg, K, Ca   
1  valence  valence 
2  valence  core 
3  core  core 
4  core  valence 
Table 5.1: A summary of the effects of different core definitions
5.3.3 Algorithm Control and Customization
The direct and semidirect integral transformation algorithms used by QChem
(e.g., MP2, CIS(D)) are limited by available disk space, D, and memory, C,
the number of basis functions, N, the number of virtual orbitals, V and the
number of occupied orbitals, O, as discussed above. The generic description
of the key $rem variables are:
MEM_STATIC
Sets the memory for Fortran AO integral calculation and transformation modules. 
TYPE:
DEFAULT:
64  corresponding to 64 Mb. 
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
For direct and semidirect MP2 calculations, this must exceed OVN +
requirements for AO integral evaluation (32160 Mb), as discussed above. 

 MEM_TOTAL
Sets the total memory available to QChem, in megabytes. 
TYPE:
DEFAULT:
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
Use default, or set to the physical memory of your machine. Note that if more
than 1GB is specified for a CCMAN job, the memory is allocated as
follows 
12%  MEM_STATIC 
50%  CC_MEMORY 
35%  Other memory requirements:





AO2MO_DISK
Sets the amount of disk space (in megabytes) available for MP2 calculations. 
TYPE:
DEFAULT:
2000  Corresponding to 2000 Mb. 
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
Should be set as large as possible, discussed in Section 5.3.1. 

 CD_ALGORITHM
Determines the algorithm for MP2 integral transformations. 
TYPE:
DEFAULT:
OPTIONS:
DIRECT  Uses fully direct algorithm (energies only). 
SEMI_DIRECT  Uses diskbased semidirect algorithm. 
LOCAL_OCCUPIED  Alternative energy algorithm (see 5.3.1). 
RECOMMENDATION:
Semidirect is usually most efficient, and will normally be chosen by default. 



N_FROZEN_CORE
Sets the number of frozen core orbitals in a postHartreeFock calculation. 
TYPE:
DEFAULT:
OPTIONS:
FC  Frozen Core approximation (all core orbitals frozen). 
n  Freeze n core orbitals. 
RECOMMENDATION:
While the default is not to freeze orbitals, MP2 calculations are more
efficient with frozen core orbitals. Use FC if possible. 

 N_FROZEN_VIRTUAL
Sets the number of frozen virtual orbitals in a postHartreeFock
calculation. 
TYPE:
DEFAULT:
OPTIONS:
n  Freeze n virtual orbitals. 
RECOMMENDATION:



CORE_CHARACTER
Selects how the core orbitals are determined in the frozencore
approximation. 
TYPE:
DEFAULT:
OPTIONS:
0  Use energybased definition. 
14  Use Mullikenbased definition (see Table 5.3.2 for details). 
RECOMMENDATION:
Use default, unless performing calculations on molecules with heavy elements. 

 PRINT_CORE_CHARACTER
Determines the print level for the CORE_CHARACTER option. 
TYPE:
DEFAULT:
OPTIONS:
0  No additional output is printed. 
1  Prints core characters of occupied MOs. 
2  Print level 1, plus prints the core character of AOs. 
RECOMMENDATION:
Use default, unless you are uncertain about what the core character is. 



5.3.4 Example
Example 5.0 Example of an MP2/631G* calculation employing the frozen core
approximation. Note that the EXCHANGE $rem variable will default to
HF
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.01
hoh = 105
$end
$rem
CORRELATION mp2
BASIS 631g*
N_FROZEN_CORE fc
$end
5.4 Local MP2 Methods
5.4.1 Local Triatomics in Molecules (TRIM) Model
The development of what may be called "fast methods" for evaluating electron
correlation is a problem of both fundamental and practical importance, because
of the unphysical increases in computational complexity with molecular size
which afflict "exact" implementations of electron correlation methods.
Ideally, the development of fast methods for treating electron correlation
should not impact either model errors or numerical errors associated with the
original electron correlation models. Unfortunately this is not possible at
present, as may be appreciated from the following rough argument. Spatial
locality is what permits reformulations of electronic structure methods that
yield the same answer as traditional methods, but faster. The oneparticle
density matrix decays exponentially with a rate that relates to the HOMOLUMO
gap in periodic systems. When length scales longer than this characteristic
decay length are examined, sparsity will emerge in both the oneparticle
density matrix and also pair correlation amplitudes expressed in terms of
localized functions. Very roughly, such a length scale is about 5 to 10 atoms
in a line, for good insulators such as alkanes. Hence sparsity emerges beyond
this number of atoms in 1D, beyond this number of atoms squared in 2D, and
this number of atoms cubed in 3D. Thus for threedimensional systems,
locality only begins to emerge for systems of between hundreds and thousands of
atoms.
If we wish to accelerate calculations on systems below this size regime, we
must therefore introduce additional errors into the calculation, either as
numerical noise through looser tolerances, or by modifying the theoretical
model, or perhaps both. QChem's approach to local electron correlation is
based on modifying the theoretical models describing correlation with an
additional welldefined local approximation. We do not attempt to accelerate
the calculations by introducing more numerical error because of the
difficulties of controlling the error as a function of molecule size, and the
difficulty of achieving reproducible significant results. From this
perspective, local correlation becomes an integral part of specifying the
electron correlation treatment. This means that the considerations necessary
for a correlation treatment to qualify as a welldefined theoretical model
chemistry apply equally to local correlation modeling. The local approximations
should be
 Sizeconsistent: meaning that the energy of a supersystem of two
noninteracting molecules should be the sum of the energy obtained from
individual calculations on each molecule.
 Uniquely defined: Require no input beyond nuclei, electrons, and
an atomic orbital basis set. In other words, the model should be uniquely
specified without customization for each molecule.
 Yield continuous potential energy surfaces: The model
approximations should be smooth, and not yield energies that exhibit
jumps as nuclear geometries are varied.
To ensure that these model chemistry criteria are met, QChem's local MP2
methods [209,[210] express the double substitutions (i.e., the
pair correlations) in a redundant basis of atomlabeled functions. The
advantage of doing this is that local models satisfying model chemistry
criteria can be defined by performing an atomic truncation of the double
substitutions. A general substitution in this representation will then involve
the replacement of occupied functions associated with two given atoms by empty
(or virtual) functions on two other atoms, coupling together four different
atoms. We can force one occupied to virtual substitution (of the two that
comprise a double substitution) to occur only between functions on the same
atom, so that only three different atoms are involved in the double
substitution. This defines the triatomics in molecules (TRIM) local
model for double substitutions. The TRIM model offers the potential for
reducing the computational requirements of exact MP2 theory by a factor
proportional to the number of atoms. We could also force each occupied to
virtual substitution to be on a given atom, thereby defining a more drastic
diatomics in molecules (DIM) local correlation model.
The simplest atomcentered basis that is capable of spanning the occupied
space is a minimal basis of core and valence atomic orbitals on each
atom. Such a basis is necessarily redundant because it also contains sufficient
flexibility to describe the empty valence antibonding orbitals necessary to
correctly account for nondynamical electron correlation effects such as
bondbreaking. This redundancy is actually important for the success of the
atomic truncations because occupied functions on adjacent atoms to some extent
describe the same part of the occupied space. The minimal functions we use to
span the occupied space are obtained at the end of a large basis set
calculation, and are called extracted polarized atomic orbitals
(EPAOs) [211]. We discuss them briefly below. It is even possible to
explicitly perform an SCF calculation in terms of a moleculeoptimized minimal
basis of polarized atomic orbitals (PAOs) (see Chapter 4).
To span the virtual space, we use the full set of atomic orbitals,
appropriately projected into the virtual space.
We summarize the situation. The number of functions spanning the occupied
subspace will be the minimal basis set dimension, M, which is greater than
the number of occupied orbitals, O, by a factor of up to about two. The
virtual space is spanned by the set of projected atomic orbitals whose number
is the atomic orbital basis set size N, which is fractionally greater than
the number of virtuals VNO. The number of double substitutions in such a
redundant representation will be typically three to five times larger than the
usual total. This will be more than compensated by reducing the number of
retained substitutions by a factor of the number of atoms, A, in the local
triatomics in molecules model, or a factor of A^{2} in the diatomics in
molecules model.
The local MP2 energy in the TRIM and DIM models are given by the following
expressions, which can be compared against the full MP2 expression given
earlier in Eq. (5.13). First, for the DIM model:
E_{DIM MP2} = − 
1
2


∑
―P―Q


 ⎛ ⎝


P

 

Q
 ⎞ ⎠

 ⎛ ⎝


P

 

Q
 ⎞ ⎠

∆_{―P} +∆_{―Q}


 (5.17) 
The sums run over the linear number of atomic single excitations after they
have been canonicalized. Each term in the denominator is thus an energy
difference between occupied and virtual levels in this local basis.
Similarly, the TRIM model corresponds to the following local MP2 energy:
E_{TRIM MP2} = − 
∑
―Pbj


 ⎛ ⎝


P

 jb  ⎞ ⎠

 ⎛ ⎝


P

 jb  ⎞ ⎠

∆_{―P} +ε_{b} −ε_{j}

− E_{DIM MP2} 
 (5.18) 
where the sum is now mixed between atomic substitutions ―P, and nonlocal
occupied j to virtual b substitutions. See
Refs. for a full derivation and discussion.
The accuracy of the local TRIM and DIM models has been tested in a series of
calculations [209,[210]. In particular, the TRIM model
has been shown to be quite faithful to full MP2 theory via the following tests:
 The TRIM model recovers around 99.7% of the MP2 correlation energy for
covalent bonding. This is significantly higher than the roughly 9899%
correlation energy recovery typically exhibited by the SaeboPulay local
correlation method [212]. The DIM model recovers around
95% of the correlation energy.
 The performance of the TRIM model for relative energies is very robust,
as shown in Ref. for the challenging case of
torsional barriers in conjugated molecules. The RMS error in these
relative energies is only 0.031 kcal/mol, as compared to around 1
kcal/mol when electron correlation effects are completely neglected.
 For the water dimer with the augccpVTZ basis, 96% of the MP2
contribution to the binding energy is recovered with the TRIM model, as
compared to 62% with the SaeboPulay local correlation method.
 For calculations of the MP2 contribution to the G3 and G3(MP2) energies
with the larger molecules in the G399 database [178],
introduction of the TRIM approximation results in an RMS error relative
to full MP2 theory of only 0.3 kcal/mol, even though the absolute
magnitude of these quantities is on the order of tens of kcal/mol.
5.4.2 EPAO Evaluation Options
When a local MP2 job (requested by the LOCAL_MP2 option for
CORRELATION) is performed, the first new step after the SCF
calculation is converged is to extract a minimal basis of polarized atomic
orbitals (EPAOs) that spans the occupied space. There are three valid choices
for this basis, controlled by the PAO_METHOD and
EPAO_ITERATE keywords described below.
 Uniterated EPAOs: The initial guess EPAOs are the default for
local MP2 calculations, and are defined as follows. For each atom, the
covariant density matrix (SPS) is diagonalized, giving eigenvalues which
are approximate natural orbital occupancies, and eigenvectors which are
corresponding atomic orbitals. The m eigenvectors with largest
populations are retained (where m is the minimal basis dimension for
the current atom). This nonorthogonal minimal basis is symmetrically
orthogonalized, and then modified as discussed in
Ref. to ensure that these functions rigorously span the
occupied space of the full SCF calculation that has just been performed.
These orbitals may be denoted as EPAO(0) to indicate that no iterations
have been performed after the guess. In general, the quality of the local
MP2 results obtained with this option is very similar to the EPAO option
below, but it is much faster and fully robust. For the example of the
torsional barrier calculations discussed above [209], the
TRIM RMS deviations of 0.03 kcal/mol from full MP2 calculations are
increased to only 0.04 kcal/mol when EPAO(0) orbitals are employed rather
than EPAOs.
 EPAOs: EPAOs are defined by minimizing a localization functional
as described in Ref. . These functions were designed
to be suitable for local MP2 calculations, and have yielded excellent
results in all tests performed so far. Unfortunately the functional is
difficult to converge for large molecules, at least with the algorithms
that have been developed to this stage. Therefore it is not the default,
but is switched on by specifying a (large) value for
EPAO_ITERATE, as discussed below.
 PAO: If the SCF calculation is performed in terms of a
moleculeoptimized minimal basis, as described in Chapter 4, then the
resulting PAOSCF calculation can be corrected with either conventional
or local MP2 for electron correlation. PAOSCF calculations alter the
SCF energy, and are therefore not the default. This can be enabled by
specifying PAO_METHOD as PAO, in a job which also requests
CORRELATION as LOCAL_MP2
PAO_METHOD
Controls the type of PAO calculations requested. 
TYPE:
DEFAULT:
EPAO  For local MP2, EPAOs are chosen by default. 
OPTIONS:
EPAO  Find EPAOs by minimizing delocalization function. 
PAO  Do SCF in a moleculeoptimized minimal basis. 
RECOMMENDATION:

 EPAO_ITERATE
Controls iterations for EPAO calculations (see PAO_METHOD). 
TYPE:
DEFAULT:
0  Use uniterated EPAOs based on atomic blocks of SPS. 
OPTIONS:
n  Optimize the EPAOs for up to n iterations. 
RECOMMENDATION:
Use default. For molecules that are not too large, one can test the
sensitivity of the results to the type of minimal functions by the use of
optimized EPAOs in which case a value of n=500 is reasonable. 



EPAO_WEIGHTS
Controls algorithm and weights for EPAO calculations (see PAO_METHOD). 
TYPE:
DEFAULT:
115  Standard weights, use 1^{st} and 2^{nd} order optimization 
OPTIONS:
15  Standard weights, with 1^{st} order optimization only. 
RECOMMENDATION:
Use default, unless convergence failure is encountered. 

5.4.3 Algorithm Control and Customization
A local MP2 calculation (requested by the LOCAL_MP2 option for
CORRELATION) consists of the following steps:
 After the SCF is converged, a minimal basis of EPAOs are obtained.
 The TRIM (and DIM) local MP2 energies are then evaluated (gradients are
not yet available).
Details of the efficient implementation of the local MP2 method described above
are reported in the recent thesis of Dr. Michael Lee [213]. Here we
simply summarize the capabilities of the program. The computational advantage
associated with these local MP2 methods varies depending upon the size of
molecule and the basis set. As a rough general estimate, TRIM MP2 calculations
are feasible on molecule sizes about twice as large as those for which
conventional MP2 calculations are feasible on a given computer, and this is
their primary advantage. Our implementation is well suited for large basis set
calculations. The AO basis twoelectron integrals are evaluated four times.
DIM MP2 calculations are performed as a byproduct of TRIM MP2 but no
separately optimized DIM algorithm has been implemented.
The resource requirements for local MP2 calculations are as follows:
 Memory: The memory requirement for the integral transformation
does not exceed OON, and is thresholded so that it asymptotically grows
linearly with molecule size. Additional memory of approximately 32N^{2}
is required to complete the local MP2 energy evaluation.
 Disk: The disk space requirement is only about 8OVN, but is
not governed by a threshold. This is a very large reduction from the case
of a full MP2 calculation, where, in the case of four integral
evaluations, OVN^{2}/4 disk space is required. As the local MP2 disk
space requirement is not adjustable, the AO2MO_DISK keyword is
ignored for LOCAL_MP2 calculations.
The evaluation of the local MP2 energy does not require any further
customization. An adequate amount of MEM_STATIC (80 to 160 Mb) should
be specified to permit efficient AO basis twoelectron integral evaluation,
but all large scratch arrays are allocated from MEM_TOTAL.
5.4.4 Examples
Example 5.0 A relative energy evaluation using the local TRIM model for MP2
with the 6311G** basis set. The energy difference is the internal rotation
barrier in propenal, with the first geometry being planar trans, and the second
the transition structure.
$molecule
0 1
C
C 1 1.32095
C 2 1.47845 1 121.19
O 3 1.18974 2 123.83 1 180.00
H 1 1.07686 2 121.50 3 0.00
H 1 1.07450 2 122.09 3 180.00
H 2 1.07549 1 122.34 3 180.00
H 3 1.09486 2 115.27 4 180.00
$end
$rem
CORRELATION local_mp2
BASIS 6311g**
$end
@@@
$molecule
0 1
C
C 1 1.31656
C 2 1.49838 1 123.44
O 3 1.18747 2 123.81 1 92.28
H 1 1.07631 2 122.03 3 0.31
H 1 1.07484 2 121.43 3 180.28
H 2 1.07813 1 120.96 3 180.34
H 3 1.09387 2 115.87 4 179.07
$end
$rem
CORRELATION local_mp2
BASIS 6311g**
$end
5.5 Auxiliary Basis Set (ResolutionofIdentity) MP2 Methods
For a molecule of fixed size, increasing the number of basis functions
per atom, n, leads to O(n^{4}) growth in the number of significant
fourcenter twoelectron integrals, since the number of nonnegligible
product charge distributions, μν〉, grows as O(n^{2}). As a result,
the use of large (highquality) basis expansions is computationally costly.
Perhaps the most practical way around this "basis set quality" bottleneck is
the use of auxiliary basis expansions [214,[215,[216].
The ability to use auxiliary
basis sets to accelerate a variety of electron correlation methods, including
both energies and analytical gradients, is one of the major new features of
QChem 3.0.
The auxiliary basis {K〉} is used to approximate products of Gaussian
basis functions:
μν〉 ≈  
~
μν

〉 = 
∑
K

K〉C_{μν}^{K} 
 (5.19) 
Auxiliary basis expansions were introduced long ago, and are now widely
recognized as an effective and powerful approach, which is sometimes
synonymously called resolution of the identity (RI) or density fitting (DF).
When using auxiliary basis expansions, the rate of growth of computational cost
of largescale electronic structure calculations with n is reduced to
approximately n^{3}.
If n is fixed and molecule size increases, auxiliary basis expansions reduce
the prefactor associated with the computation, while not altering the
scaling. The important point is that the prefactor can be reduced by 5 or 10
times or more. Such large speedups are possible because the number of
auxiliary functions required to obtain reasonable accuracy, X, has been shown
to be only about 3 or 4 times larger than N.
The auxiliary basis expansion coefficients, C, are determined by
minimizing the deviation between the fitted distribution and the actual
distribution, 〈μν−~μν  μν−~μν〉, which leads to the following set of linear equations:

∑
L

〈 K L 〉 C_{μν}^{L} = 〈 K μν 〉 
 (5.20) 
Evidently solution of the fit equations requires only two and threecenter
integrals, and as a result the (fourcenter) twoelectron integrals can be
approximated as the following optimal expression for a given choice of
auxiliary basis set:
〈μνλσ〉 ≈ 〈 
~
μν

 
~
λσ

〉 = 
∑
 K,LC_{μ}^{L}〈LK 〉C_{λσ}^{K} 
 (5.21) 
In the limit where the auxiliary basis is complete (i.e. all products of
AOs are included), the fitting procedure described above will be exact.
However, the auxiliary basis is invariably incomplete (as mentioned above,
X ≈ 3N) because this is essential for obtaining increased computational
efficiency. Standardized auxiliary basis sets have been developed by the
Karlsruhe group for second order perturbation (MP2) calculations [217,[218]
of the correlation energy. With these basis
sets, small absolute errors (e.g., below 60 μHartree per atom in MP2) and
even smaller relative errors in computed energies are found, while the
speedup can be 330 fold. This development has made the routine use of
auxiliary basis sets for electron correlation calculations possible.
Correlation calculations that can take advantage of auxiliary basis expansions
are described in the remainder of this section (MP2, and MP2like methods) and
in Section 5.13 (simplified active space coupled cluster methods
such as PP, PP(2), IP, RP). These methods automatically employ auxiliary basis
expansions when a valid choice of auxiliary basis set is supplied using the
AUX_BASIS keyword which is used in the same way as the BASIS
keyword.
The PURECART $rem is no longer needed here, even if using
a auxiliary basis that does not have a predefined value. There
is a builtin automatic procedure that provides the effect
of the PURECART $rem in these cases by default.
5.5.1 RIMP2 Energies and Gradients.
Following common convention, the MP2 energy evaluated approximately using an
auxiliary basis is referred to as "resolution of the identity" MP2, or RIMP2
for short. RIMP2 energy and gradient calculations are enabled simply by
specifying the AUX_BASIS keyword discussed above. As discussed above,
RIMP2 energies [214] and gradients [219,[220]
are significantly faster than the best
conventional MP2 energies and gradients, and cause negligible loss of accuracy,
when an appropriate standardized auxiliary basis set is employed. Therefore
they are recommended for jobs where turnaround time is an issue. Disk
requirements are very modest; one merely needs to hold various 3index
arrays. Memory requirements grow more slowly than our conventional MP2
algorithmsonly quadratically with molecular size. The minimum memory
requirement is approximately 3X^{2}, where X is the number of auxiliary
basis functions, for both energy and analytical gradient evaluations, with some
additional memory being necessary for integral evaluation and other small
arrays.
In fact, for molecules that are not too large (perhaps no more than 20 or 30
heavy atoms) the RIMP2 treatment of electron correlation is so efficient that
the computation is dominated by the initial HartreeFock calculation. This is
despite the fact that as a function of molecule size, the cost of the RIMP2
treatment still scales more steeply with molecule size (it is just that the
prefactor is so much smaller with the RI approach). Its scaling remains
5th order with the size of the molecule, which only dominates the
initial SCF calculation for larger molecules. Thus, for RIMP2 energy
evaluation on moderate size molecules (particularly in large basis sets), it is
desirable to use the dual basis HF method to further improve execution times
(see Section 4.7).
5.5.2 Example
Example 5.0 QChem input for an RIMP2 geometry optimization.
$molecule
0 1
O
H 1 0.9
F 1 1.4 2 100.
$end
$rem
JOBTYPE opt
CORRELATION rimp2
BASIS ccpvtz
AUX_BASIS rimp2ccpvtz
SYMMETRY false
$end
For the size of required memory, the followings need to be considered.
 MEM_STATIC
Sets the memory for AOintegral evaluations and their transformations. 
TYPE:
DEFAULT:
64  corresponding to 64 Mb. 
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
For RIMP2 calculations, 150(ON + V) of MEM_STATIC is required.
Because a number of matrices with N^{2} size also need to be
stored, 32160 Mb of additional MEM_STATIC is needed. 



MEM_TOTAL
Sets the total memory available to QChem, in megabytes. 
TYPE:
DEFAULT:
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
Use default, or set to the physical memory of your machine.
The minimum requirement is 3X^{2}. 

5.5.3 OpenMP Implementation of RIMP2
An experimental OpenMP code can be invoked by using CORR=primp2.
Only RHF/riMP2 energies are available in 4.0.1.
Example 5.0 Example of OpenMPparallel riMP2 job.
$molecule
0 1
C1
H1 C1 1.0772600000
H2 C1 1.0772600000 H1 131.6082400000
$end
$rem
jobtype SP
exchange HF
correlation pRIMP2
basis ccpVTZ
aux_basis rimp2ccpVTZ
purecart 11111
symmetry false
thresh 12
scf_convergence 8
max_sub_file_num 128
!time_mp2 true
$end
5.5.4 GPU Implementation of RIMP2
5.5.4.1 Requirements
QChem currently offers the possibility of accelerating RIMP2 calculations
using graphics processing units (GPUs).
Currently, this is implemented for CUDAenabled NVIDIA graphics
cards only, such as (in historical order from 2008) the GeForce,
Quadro, Tesla and Fermi cards. More information about
CUDAenabled cards is available at
 ://www.nvidia.com/object/cuda_gpus.html@

://www.nvidia.com/object/cuda_gpus.html@
It should be noted that these GPUs have specific power and motherboard requirements.
Software requirements include the installation of the appropriate NVIDIA
CUDA driver (at least version 1.0, currently 3.2) and linear algebra library,
CUBLAS (at least version 1.0, currently 2.0). These can be downloaded jointly
in NVIDIA's developer website:
 ://developer.nvidia.com/object/cuda_3_2_downloads.html@

://developer.nvidia.com/object/cuda_3_2_downloads.html@
We have implemented a mixedprecision algorithm in order to
get better than single precision when users only have
singleprecision GPUs. This is accomplished by noting that RIMP2 matrices
have a large fraction of numerically "small" elements and a
small fraction of numerically "large" ones. The latter can
greatly affect the accuracy of the calculation in singleprecision only
calculations, but calculation
involves a relatively small number of compute cycles. So, given a threshold
value δ, we perform a separation between "small" and "large"
elements and accelerate the former computeintensive operations
using the GPU (in singleprecision) and compute the latter
on the CPU (using doubleprecision). We are thus able to
determine how much "doubleprecision" we desire by tuning
the δ parameter, and tailoring the balance between computational speed and accuracy.
5.5.4.2 Options
CUDA_RIMP2
Enables GPU implementation of RIMP2 
TYPE:
DEFAULT:
OPTIONS:
FALSE  GPUenabled MGEMM off 
TRUE  GPUenabled MGEMM on 
RECOMMENDATION:
Necessary to set to 1 in order to run GPUenabled RIMP2 

 USECUBLAS_THRESH
Sets threshold of matrix size sent to GPU
(smaller size not worth sending to GPU). 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default value. Anything less can
seriously hinder the GPU acceleration 



USE_MGEMM
Use the mixedprecision matrix scheme (MGEMM)
if you want to make calculations in your card in singleprecision
(or if you have a singleprecisiononly GPU), but leave some parts
of the RIMP2 calculation in double precision) 
TYPE:
DEFAULT:
OPTIONS:
0  MGEMM disabled 
1  MGEMM enabled 
RECOMMENDATION:
Use when having singleprecision cards 

 MGEMM_THRESH
Sets MGEMM threshold to determine the separation
between "large" and "small" matrix elements.
A larger threshold value will result in a value closer
to the singleprecision result. Note that the desired factor
should be multiplied by 10000 to ensure an integer value. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
For small molecules and basis sets up to tripleζ, the
default value suffices to not deviate too much from the
doubleprecision values. Care should be taken to reduce
this number for larger molecules and also larger basissets. 



5.5.4.3 Input examples
Example 5.0 RIMP2 doubleprecision calculation
$comment
RIMP2 doubleprecision example
$end
$molecule
0 1
c
h1 c 1.089665
h2 c 1.089665 h1 109.47122063
h3 c 1.089665 h1 109.47122063 h2 120.
h4 c 1.089665 h1 109.47122063 h2 120.
$end
$rem
jobtype sp
exchange hf
correlation rimp2
basis ccpvdz
aux_basis rimp2ccpvdz
cuda_rimp2 1
$end
Example 5.0 RIMP2 calculation with MGEMM
$comment
MGEMM example
$end
$molecule
0 1
c
h1 c 1.089665
h2 c 1.089665 h1 109.47122063
h3 c 1.089665 h1 109.47122063 h2 120.
h4 c 1.089665 h1 109.47122063 h2 120.
$end
$rem
jobtype sp
exchange hf
correlation rimp2
basis ccpvdz
aux_basis rimp2ccpvdz
cuda_rimp2 1
USE_MGEMM 1
mgemm_thresh 10000
$end
5.5.5 OppositeSpin (SOSMP2, MOSMP2, and O2) Energies and Gradients
The accuracy of MP2 calculations can be significantly improved by
semiempirically scaling the oppositespin and samespin correlation components
with separate scaling factors, as shown by Grimme [221]. Results
of similar quality can be obtained by just scaling the opposite spin
correlation (by 1.3), as was recently demonstrated [222].
Furthermore this SOSMP2 energy can be evaluated using the RI approximation
together with a Laplace transform technique, in effort that scales only with
the 4th power of molecular size. Efficient algorithms for the energy [222]
and the analytical gradient [223] of this method
are available in QChem 3.0, and offer advantages in speed over MP2 for
larger molecules, as well as statistically significant improvements in
accuracy.
However, we note that the SOSMP2 method does systematically underestimate
longrange dispersion (for which the appropriate scaling factor is 2 rather
than 1.3) but this can be accounted for by making the scaling factor
distancedependent, which is done in the modified opposite spin variant
(MOSMP2) that has recently been proposed and tested [224]. The
MOSMP2 energy and analytical gradient are also available in QChem 3.0 at a
cost that is essentially identical with SOSMP2. Timings show that the
4thorder implementation of SOSMP2 and MOSMP2 yields substantial
speedups over RIMP2 for molecules in the 40 heavy atom regime and larger. It is
also possible to customize the scale factors for particular applications, such
as weak interactions, if required.
A fourth order scaling SOSMP2/MOSMP2 energy calculation can be invoked by
setting the CORRELATION keyword to either SOSMP2 or
MOSMP2. MOSMP2 further requires the specification of the $rem
variable OMEGA, which tunes the level of attenuation of the MOS
operator [224]:
g_{ω}(r_{12}) = 
1
r_{12}

+c_{MOS} 
erf( ωr_{12} )
r_{12}


 (5.22) 
The recommended OMEGA value is ω = 0.6 a.u. [224].
The fast algorithm makes use of auxiliary basis expansions
and therefore, the keyword AUX_BASIS should be
set consistently with the user's choice of BASIS. Fourthorder
scaling analytical gradient for both SOSMP2 and MOSMP2 are also available and
is automatically invoked when JOBTYPE is set to OPT or
FORCE. The minimum memory requirement is 3X^{2}, where X = the
number of auxiliary basis functions, for both energy and analytical gradient
evaluations. Disk space requirement for closed shell calculations is ∼ 2OVX for energy evaluation and ∼ 4OVX for analytical gradient evaluation.
More recently, Brueckner orbitals (BO) are introduced into SOSMP2 and MOSMP2 methods to
resolve the problems of symmetry breaking and spin contamination
that are often associated with HartreeFock orbitals.
So the molecular orbitals are optimized with the meanfield energy
plus a correlation energy taken as the oppositespin component of the secondorder
manybody correlation energy, scaled by an empirically chosen parameter.
This "optimized secondorder oppositespin" abbreviated as O2 method [225]
requires fourthorder computation on each orbital iteration. O2 is shown to yield predictions of
structure and frequencies for closedshell molecules that are very similar to scaled MP2
methods. However, it yields substantial improvements for openshell molecules,
where problems with spin contamination and symmetry breaking are shown to be greatly
reduced.
Summary of key $rem variables to be specified:
CORRELATION  SOSMP2 
 MOSMP2 
JOBTYPE  sp (default) single point energy evaluation 
 opt geometry optimization with analytical gradient 
 force force evaluation with analytical gradient 
BASIS  user's choice (standard or userdefined: GENERAL or
MIXED) 
AUX_BASIS  corresponding auxiliary basis (standard or userdefined: 
 AUX_GENERAL or AUX_MIXED 
OMEGA  no default n; use ω = n/1000. The recommended value is 
 n=600 (ω = 0.6 a.u.) 
N_FROZEN_CORE  Optional 
N_FROZEN_VIRTUAL  Optional

5.5.6 Examples
Example 5.0 Example of SOSMP2 geometry optimization
$molecule
0 3
C1
H1 C1 1.07726
H2 C1 1.07726 H1 131.60824
$end
$rem
JOBTYPE opt
CORRELATION sosmp2
BASIS ccpvdz
AUX_BASIS rimp2ccpvdz
UNRESTRICTED true
SYMMETRY false
$end
Example 5.0 Example of MOSMP2 energy evaluation with frozen core
approximation
$molecule
0 1
Cl
Cl 1 2.05
$end
$rem
JOBTYPE sp
CORRELATION mosmp2
OMEGA 600
BASIS ccpVTZ
AUX_BASIS rimp2ccpVTZ
N_FROZEN_CORE fc
THRESH 12
SCF_CONVERGENCE 8
$end
Example 5.0 Example of O2 methodology applied to O(N^{4}) SOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED TRUE
JOBTYPE FORCE Options are SP/FORCE/OPT
EXCHANGE HF
DO_O2 1 O2 with O(N^4) SOSMP2 algorithm
SOS_FACTOR 100 Opposite Spin scaling factor = 100/100 = 1.0
SCF_ALGORITHM DIIS_GDM
SCF_GUESS GWH
BASIS sto3g
AUX_BASIS rimp2vdz
SCF_CONVERGENCE 8
THRESH 14
SYMMETRY FALSE
PURECART 1111
$end
Example 5.0 Example of O2 methodology applied to O(N^{4}) MOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED TRUE
JOBTYPE FORCE Options are SP/FORCE/OPT
EXCHANGE HF
DO_O2 2 O2 with O(N^4) MOSMP2 algorithm
OMEGA 600 Omega = 600/1000 = 0.6 a.u.
SCF_ALGORITHM DIIS_GDM
SCF_GUESS GWH
BASIS sto3g
AUX_BASIS rimp2vdz
SCF_CONVERGENCE 8
THRESH 14
SYMMETRY FALSE
PURECART 1111
$end
5.5.7 RITRIM MP2 Energies
The triatomics in molecules (TRIM) local correlation approximation to MP2
theory [209] was described in detail in Section 5.4.1 which
also discussed our implementation of this approach based on conventional
fourcenter twoelectron integrals. QChem 3.0 also includes an auxiliary
basis implementation of the TRIM model. The new RITRIM MP2 energy algorithm [226]
greatly accelerates these local correlation calculations
(often by an order of magnitude or more for the correlation part), which scale
with the 4th power of molecule size. The electron correlation part of
the calculation is speeded up over normal RIMP2 by a factor proportional to
the number of atoms in the molecule. For a hexadecapeptide, for instance, the
speedup is approximately a factor of 4 [226]. The TRIM model
can also be applied to the scaled opposite spin models discussed above. As for
the other RIbased models discussed in this section, we recommend using
RITRIM MP2 instead of the conventional TRIM MP2 code whenever runtime of the
job is a significant issue. As for RIMP2 itself, TRIM MP2 is invoked by
adding AUX_BASIS $rems to the input deck,
in addition to requesting CORRELATION = RILMP2.
Example 5.0 Example of RITRIM MP2 energy evaluation
$molecule
0 3
C1
H1 C1 1.07726
H2 C1 1.07726 H1 131.60824
$end
$rem
CORRELATION rilmp2
BASIS ccpVDZ
AUX_BASIS rimp2ccpVDZ
PURECART 1111
UNRESTRICTED true
SYMMETRY false
$end
5.5.8 DualBasis MP2
The successful computational cost speedups of the previous sections often leave the
cost of the underlying SCF calculation dominant. The dualbasis method provides a means
of accelerating the SCF by roughly an order of magnitude, with minimal associated error
(see Section 4.7). This dualbasis reference energy may be combined
with RIMP2 calculations for both energies [171,[175] and
analytic first derivatives [173]. In the latter case, further savings
(beyond the SCF alone) are demonstrated in the gradient due to the ability to solve
the response (Zvector) equations in the smaller basis set. Refer to
Section 4.7 for details and job control options.
5.6 CoupledCluster Methods
The following sections give short summaries of the various coupledcluster based
methods
available in QChem, most of which are variants of coupledcluster
theory. The basic objectoriented tools necessary to permit the implementation
of these methods in QChem was accomplished by Profs. Anna Krylov and David
Sherrill, working at Berkeley with Martin HeadGordon, and then continuing
independently at the University of Southern California and Georgia Tech,
respectively. While at Berkeley, Krylov and Sherrill also developed the
optimized orbital coupledcluster method, with additional assistance from Ed
Byrd. The extension of this code to MP3, MP4, CCSD and QCISD is the work of Prof.
Steve Gwaltney at Berkeley, while the extensions to QCCD were implemented by Ed
Byrd at Berkeley.
The original tensor library and CC/EOM suite of methods are
handled by the CCMAN module of QChem. Recently, a new code (termed CCMAN2)
has been developed in Krylov group by Evgeny Epifanovsky and others,
and a gradual transition from CCMAN to CCMAN2 has began.
During the transition time, both
codes will be available for users via the CCMAN2 keyword.
CORRELATION
Specifies the correlation level of theory handled by CCMAN/CCMAN2. 
TYPE:
DEFAULT:
OPTIONS:
CCMP2  Regular MP2 handled by CCMAN/CCMAN2 
MP3  CCMAN 
MP4SDQ  CCMAN 
MP4  CCMAN 
CCD  CCMAN 
CCD(2)  CCMAN 
CCSD  CCMAN and CCMAN2 
CCSD(T)  CCMAN and CCMAN2 
CCSD(2)  CCMAN 
CCSD(fT)  CCMAN 
CCSD(dT)  CCMAN 
QCISD  CCMAN 
QCISD(T)  CCMAN 
OD  CCMAN 
OD(T)  CCMAN 
OD(2)  CCMAN 
VOD  CCMAN 
VOD(2)  CCMAN 
QCCD  CCMAN 
QCCD(T)  CCMAN 
QCCD(2)  CCMAN 
VQCCD  CCMAN 
VQCCD(T)  CCMAN 
VQCCD(2)  CCMAN 
RECOMMENDATION:
Consult the literature for guidance. 

5.6.1 Coupled Cluster Singles and Doubles (CCSD)
The standard approach for treating pair correlations selfconsistently are
coupledcluster methods where the cluster operator contains all single and
double substitutions [227], abbreviated as CCSD. CCSD yields
results that are only slightly superior to MP2 for structures and frequencies
of stable closedshell molecules. However, it is far superior for reactive
species, such as transition structures and radicals, for which the performance
of MP2 is quite erratic.
A full textbook presentation of CCSD is beyond the scope of this manual, and
several comprehensive references are available. However, it may be useful to
briefly summarize the main equations. The CCSD wavefunction is:
 Ψ_{CCSD} 〉 = exp  ⎛ ⎝

^
T

1

+ 
^
T

2
 ⎞ ⎠

 Φ_{0} 〉 
 (5.23) 
where the single and double excitation operators may be defined by their
actions on the reference single determinant (which is normally taken as the
HartreeFock determinant in CCSD):

^
T

1

 Φ_{0} 〉 = 
occ ∑
i


virt ∑
a

t_{i}^{a}  Φ_{i}^{a} 〉 
 (5.24) 

^
T

2

 Φ_{0} 〉 = 
1
4


occ ∑
ij


virt ∑
ab

t_{ij}^{ab}  Φ_{ij}^{ab} 〉 
 (5.25) 
It is not feasible to determine the CCSD energy by variational minimization of
〈E 〉_{CCSD} with respect to the singles and doubles amplitudes
because the expressions terminate at the same level of complexity as full
configuration interaction (!). So, instead, the Schrödinger equation is
satisfied in the subspace spanned by the reference determinant, all single
substitutions, and all double substitutions. Projection with these functions
and integration over all space provides sufficient equations to determine the
energy, the singles and doubles amplitudes as the solutions of sets of
nonlinear equations. These equations may be symbolically written as follows:
 

 
 



Φ_{0}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

1

+ 
1
2


^
T

2 1

+ 
^
T

2
 ⎞ ⎠

Φ_{0} 

C


  (5.26) 
 



Φ_{i}^{a}  ⎢ ⎢

^
H

−E_{CCSD}  ⎢ ⎢

Ψ_{CCSD} 


 
 



Φ_{i}^{a}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

1

+ 
1
2


^
T

2 1

+ 
^
T

2

+ 
^
T

1


^
T

2

+ 
1
3!


^
T

3 1
 ⎞ ⎠

Φ_{0} 

C


  (5.27) 
 



Φ_{ij}^{ab}  ⎢ ⎢

^
H

−E_{CCSD }  ⎢ ⎢

Ψ_{CCSD} 


 
 



Φ_{ij}^{ab}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

1

+ 
1
2


^
T

2 1

+ 
^
T

2

+ 
^
T

1


^
T

2

+ 
1
3!


^
T

3 1


 
 

+ 
1
2


^
T

2 2

+ 
1
2


^
T

2 1


^
T

2

+ 
1
4!


^
T

4 1
 ⎞ ⎠

Φ_{0} 

C


  (5.28) 

The result is a set of equations which yield an energy that is not necessarily
variational (i.e., may not be above the true energy), although it is strictly
sizeconsistent. The equations are also exact for a pair of electrons, and, to
the extent that molecules are a collection of interacting electron pairs, this
is the basis for expecting that CCSD results will be of useful accuracy.
The computational effort necessary to solve the CCSD equations can be shown to
scale with the 6th power of the molecular size, for fixed choice of basis
set. Disk storage scales with the 4th power of molecular size, and
involves a number of sets of doubles amplitudes, as well as twoelectron
integrals in the molecular orbital basis. Therefore the improved accuracy
relative to MP2 theory comes at a steep computational cost. Given these
scalings it is relatively straightforward to estimate the feasibility (or
non feasibility) of a CCSD calculation on a larger molecule (or with a larger
basis set) given that a smaller trial calculation is first performed.
QChem supports both energies and analytic gradients for CCSD for
RHF and UHF references (including frozencore). For ROHF, only energies and
unrelaxed properties are available.
5.6.2 Quadratic Configuration Interaction (QCISD)
Quadratic configuration interaction with singles and doubles (QCISD) [228]
is a widely used alternative to CCSD, that shares its main
desirable properties of being sizeconsistent, exact for pairs of electrons,
as well as being also non variational. Its computational cost also scales in the
same way with molecule size and basis set as CCSD, although with slightly
smaller constants. While originally proposed independently of CCSD based on
correcting configuration interaction equations to be sizeconsistent, QCISD is
probably best viewed as approximation to CCSD. The defining equations are given
below (under the assumption of HartreeFock orbitals, which should always be
used in QCISD). The QCISD equations can clearly be viewed as the CCSD equations
with a large number of terms omitted, which are evidently not very numerically
significant:
E_{QCISD} = 

Φ_{0}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

2
 ⎞ ⎠

Φ_{0} 

C


 (5.29) 
0= 

Φ_{i}^{a}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

^
T

1

+ 
^
T

2

+ 
^
T

1


^
T

2
 ⎞ ⎠

Φ_{0} 

C


 (5.30) 
0= 

Φ_{ij}^{ab}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

1

+ 
^
T

2

+ 
1
2


^
T

2
2
 ⎞ ⎠

Φ_{0} 

C


 (5.31) 
QCISD energies are available in QChem, and are requested with the QCISD
keyword. As discussed in Section 5.7, the non iterative QCISD(T)
correction to the QCISD solution is also available to approximately incorporate
the effect of higher substitutions.
5.6.3 Optimized Orbital Coupled Cluster Doubles (OD)
It is possible to greatly simplify the CCSD equations by omitting the single
substitutions (i.e., setting the T_{1} operator to zero). If the same single
determinant reference is used (specifically the HartreeFock determinant),
then this defines the coupledcluster doubles (CCD) method, by the following
equations:
 



Φ_{0}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

2
 ⎞ ⎠

Φ_{0} 

C


  (5.32) 
 



Φ_{ij}^{ab}  ⎢ ⎢

^
H
 ⎢ ⎢

 ⎛ ⎝

1+ 
^
T

2

+ 
1
2


^
T

2 2
 ⎞ ⎠

Φ_{0} 

C


  (5.33) 

The CCD method cannot itself usually be recommended because while pair
correlations are all correctly included, the neglect of single substitutions
causes calculated energies and properties to be significantly less reliable
than for CCSD. Single substitutions play a role very similar to orbital
optimization, in that they effectively alter the reference determinant to be
more appropriate for the description of electron correlation (the HartreeFock
determinant is optimized in the absence of electron correlation).
This suggests an alternative to CCSD and QCISD that has some additional
advantages. This is the optimized orbital CCD method (OOCCD), which we
normally refer to as simply optimized doubles (OD) [229]. The
OD method is defined by the CCD equations above, plus the additional set of
conditions that the cluster energy is minimized with respect to orbital
variations. This may be mathematically expressed by
where the rotation angle θ_{i}^{a} mixes the ith occupied orbital
with the ath virtual (empty) orbital. Thus the orbitals that define
the single determinant reference are optimized to minimize the coupledcluster
energy, and are variationally best for this purpose. The resulting orbitals are
approximate Brueckner orbitals.
The OD method has the advantage of formal simplicity (orbital variations and
single substitutions are essentially redundant variables). In cases where
HartreeFock theory performs poorly (for example artificial symmetry
breaking, or nonconvergence), it is also practically advantageous to use the OD
method, where the HF orbitals are not required, rather than CCSD or QCISD.
QChem supports both energies and analytical gradients using the OD method.
The computational cost for the OD energy is more than twice that of the CCSD or
QCISD method, but the total cost of energy plus gradient is roughly similar,
although OD remains more expensive. An additional advantage of the OD method is
that it can be performed in an active space, as discussed later, in
Section 5.8.
5.6.4 Quadratic Coupled Cluster Doubles (QCCD)
The non variational determination of the energy in the CCSD, QCISD, and OD
methods discussed in the above subsections is not normally a practical problem.
However, there are some cases where these methods perform poorly. One such
example are potential curves for homolytic bond dissociation, using closed
shell orbitals, where the calculated energies near dissociation go
significantly below the true energies, giving potential curves with unphysical
barriers to formation of the molecule from the separated fragments [230].
The Quadratic Coupled Cluster Doubles (QCCD) method [231]
recently proposed by Troy Van Voorhis at Berkeley uses a
different energy functional to yield improved behavior in problem cases of this
type. Specifically, the QCCD energy functional is defined as
E_{QCCD} = 

Φ_{0}  ⎛ ⎝

1+ 
^
Λ

2

+ 
1
2


^
Λ

2
2
 ⎞ ⎠

 ⎢ ⎢

^
H
 ⎢ ⎢

exp  ⎛ ⎝

^
T

2
 ⎞ ⎠

Φ_{0} 

C


 (5.35) 
where the amplitudes of both the ∧T_{2} and ∧Λ _{2}
operators are determined by minimizing the QCCD energy functional.
Additionally, the optimal orbitals are determined by minimizing the QCCD
energy functional with respect to orbital rotations mixing occupied and
virtual orbitals.
To see why the QCCD energy should be an improvement on the OD energy, we first
write the latter in a different way than before. Namely, we can write a CCD
energy functional which when minimized with respect to the ∧T_{2} and
∧Λ_{2} operators, gives back the same CCD equations defined earlier.
This energy functional is
E_{CCD} = 

Φ_{0}  ⎛ ⎝

1+ 
^
Λ

2
 ⎞ ⎠

 ⎢ ⎢

^
H
 ⎢ ⎢

exp  ⎛ ⎝

^
T

2
 ⎞ ⎠

Φ_{0} 

C


 (5.36) 
Minimization with respect to the ∧Λ_{2} operator gives the equations
for the ∧T_{2} operator presented previously, and, if those equations are
satisfied then it is clear that we do not require knowledge of the ∧Λ_{2} operator itself to evaluate the energy.
Comparing the two energy functionals, Eqs. (5.35) and (5.36), we see
that the QCCD functional includes up through quadratic terms of the Maclaurin
expansion of exp(∧Λ_{2}) while the conventional CCD functional
includes only linear terms. Thus the bra wavefunction and the ket wavefunction
in the energy expression are treated more equivalently in QCCD than in CCD.
This makes QCCD closer to a true variational treatment [230]
where the bra and ket wavefunctions are treated precisely equivalently, but
without the exponential cost of the variational method.
In practice QCCD is a dramatic improvement relative to any of the conventional
pair correlation methods for processes involving more than two active electrons
(i.e., the breaking of at least a double bond, or, two spatially close single
bonds). For example calculations, we refer to the original paper [231],
and the followup paper describing the full implementation [232].
We note that these improvements carry a computational price.
While QCCD scales formally with the 6th power of molecule size like
CCSD, QCISD, and OD, the coefficient is substantially larger. For this reason,
QCCD calculations are by default performed as OD calculations until they are
partly converged. QChem also contains some configuration interaction models
(CISD and CISDT). The CI methods are inferior to CC due to sizeconsistency
issues, however, these models may be useful for benchmarking and development
purposes.
5.6.5 Job Control Options
There are a large number of options for the coupledcluster singles and
doubles methods. They are documented in Appendix C, and, as the reader will
find upon following this link, it is an extensive list indeed. Fortunately,
many of them are not necessary for routine jobs. Most of the options for
nonroutine jobs concern altering the default iterative procedure, which is
most often necessary for optimized orbital calculations (OD, QCCD), as well as
the active space and EOM methods discussed later in Section
5.8. The more common options relating to convergence control
are discussed there, in Section 5.8.5. Below we list the options
that one should be aware of for routine calculations.
The RI approximation (see Section 5.5)
can be used in coupledcluster calculations, which substantially
reduces the cost of integral transformation and disk storage requirements.
The RI is invoked when AUX_BASIS is specified.
Note:
RI is available for all CCMAN/CCMAN2 methods. CCMAN requires that the
unrestricted reference be used, CCMAN2 does not have this limitation.
In addition, while RI is available for jobs that need analytical gradients,
only energies are computed using RI. Energy derivatives are calculated
using regular electron repulsion integral derivatives. 
For memory options and parallel execution, see Section 5.12.
CC_CONVERGENCE
Overall convergence criterion for the coupledcluster codes. This is designed
to ensure at least n significant digits in the calculated energy, and
automatically sets the other convergencerelated variables
(CC_E_CONV, CC_T_CONV, CC_THETA_CONV,
CC_THETA_GRAD_CONV) [10^{−n}]. 
TYPE:
DEFAULT:
OPTIONS:
n  Corresponding to 10^{−n} convergence criterion. Amplitude convergence is set 
 automatically to match energy convergence. 
RECOMMENDATION:

 CC_DOV_THRESH
Specifies minimum allowed values for the coupledcluster energy denominators.
Smaller values are replaced by this constant during early iterations only, so
the final results are unaffected, but initial convergence is improved when the
HOMOLUMO gap is small or when nonconventional references are used. 
TYPE:
DEFAULT:
OPTIONS:
abcde  Integer code is mapped to abc×10^{−de}, e.g.,
2502 corresponds to 0.25 
RECOMMENDATION:
Increase to 0.25, 0.5 or 0.75 for non convergent coupledcluster calculations. 



CC_SCALE_AMP
If not 0, scales down the step for updating coupledcluster amplitudes in cases of problematic convergence. 
TYPE:
DEFAULT:
OPTIONS:
abcd  Integer code is mapped to abcd×10^{−2}, e.g.,
90 corresponds to 0.9 
RECOMMENDATION:
Use 0.9 or 0.8 for non convergent coupledcluster calculations. 

 CC_MAX_ITER
Maximum number of iterations to optimize the coupledcluster energy. 
TYPE:
DEFAULT:
OPTIONS:
n  up to n iterations to achieve convergence. 
RECOMMENDATION:



CC_PRINT
Controls the output from postMP2 coupledcluster module of QChem 
TYPE:
DEFAULT:
OPTIONS:
0→7  higher values can lead to deforestation... 
RECOMMENDATION:
Increase if you need more output and don't like trees 

5.6.6 Example
Example 5.0 A series of jobs evaluating the correlation energy (with core
orbitals frozen) of the ground state of the NH_{2} radical with three methods
of coupledcluster singles and doubles type: CCSD itself, OD, and QCCD.
$molecule
0 2
N
H1 N 1.02805
H2 N 1.02805 H1 103.34
$end
$rem
CORRELATION ccsd
BASIS 631g*
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION od
BASIS 631g*
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION qccd
BASIS 631g*
N_FROZEN_CORE fc
$end
5.7 Noniterative Corrections to Coupled Cluster Energies
5.7.1 (T) Triples Corrections
To approach chemical accuracy in reaction energies and related properties, it
is necessary to account for electron correlation effects that involve three
electrons simultaneously, as represented by triple substitutions relative to
the mean field single determinant reference, which arise in MP4. The best
standard methods for including triple substitutions are the CCSD(T) [233]
and QCISD(T) methods [228] The accuracy of
these methods is welldocumented for many cases [234], and in
general is a very significant improvement relative to the starting point
(either CCSD or QCISD). The cost of these corrections scales with the
7th power of molecule size (or the 4th power of the number of
basis functions, for a fixed molecule size), although no additional disk resources
are required relative to the starting coupledcluster calculation. QChem
supports the evaluation of CCSD(T) and QCISD(T) energies, as well as the
corresponding OD(T) correction to the optimized doubles method discussed in the
previous subsection. Gradients and properties are not yet available for
any of these (T) corrections.
5.7.2 (2) Triples and Quadruples Corrections
While the (T) corrections discussed above have been extraordinarily successful,
there is nonetheless still room for further improvements in accuracy, for at
least some important classes of problems. They contain judiciously chosen terms
from 4th and 5thorder MøllerPlesset perturbation
theory, as well as higher order terms that result from the fact that the
converged cluster amplitudes are employed to evaluate the 4th and 5thorder
order terms. The (T) correction therefore depends upon the bare
reference orbitals and orbital energies, and in this way its effectiveness
still depends on the quality of the reference determinant. Since we are
correcting a coupledcluster solution rather than a single determinant, this is
an aspect of the (T) corrections that can be improved. Deficiencies of the (T)
corrections show up computationally in cases where there are neardegeneracies
between orbitals, such as stretched bonds, some transition states, open shell
radicals, and biradicals.
Prof. Steve Gwaltney, while working at Berkeley with Martin HeadGordon, has
suggested a new class of non iterative correction that offers the prospect of
improved accuracy in problem cases of the types identified above [235].
QChem contains Gwaltney's implementation of this new
method, for energies only. The new correction is a true second order correction
to a coupledcluster starting point, and is therefore denoted as (2). It is
available for two of the cluster methods discussed above, as OD(2) and
CCSD(2) [235,[236]. Only energies are available at present.
The basis of the (2) method is to partition not the regular Hamiltonian into
perturbed and unperturbed parts, but rather to partition a
similaritytransformed Hamiltonian, defined as ~H=e^{−∧T}∧He^{∧T}. In the truncated space (call it the pspace)
within which the cluster problem is solved (e.g., singles and doubles for
CCSD), the coupledcluster wavefunction is a true eigenvalue of ~H.
Therefore we take the zero order Hamiltonian, ~H^{(0)},
to be the full ~H in the pspace, while in the space of
excluded substitutions (the qspace) we take only the onebody part of ~H
(which can be made diagonal). The fluctuation potential describing
electron correlations in the qspace is ~H−~H^{(0)},
and the (2) correction then follows from second order perturbation
theory.
The new partitioning of terms between the perturbed and unperturbed
Hamiltonians inherent in the (2) correction leads to a correction that shows
both similarities and differences relative to the existing (T) corrections.
There are two types of higher correlations that enter at second order: not only
triple substitutions, but also quadruple substitutions. The quadruples are
treated with a factorization ansatz, that is exact in 5th order
MøllerPlesset theory [237], to reduce their computational
cost from N^{9} to N^{6}. For large basis sets this can still be larger
than the cost of the triples terms, which scale as the 7th power of
molecule size, with a factor twice as large as the usual (T) corrections.
These corrections are feasible for molecules containing between four and ten
first row atoms, depending on computer resources, and the size of the basis set
chosen. There is early evidence that the (2) corrections are superior to the
(T) corrections for highly correlated systems [235]. This shows
up in improved potential curves, particularly at long range and may also extend
to improved energetic and structural properties at equilibrium in problematical
cases. It will be some time before sufficient testing on the new (2)
corrections has been done to permit a general assessment of the performance of
these methods. However, they are clearly very promising, and for this reason
they are available in QChem.
5.7.3 (dT) and (fT) corrections
Alternative inclusion of noniterative N^{7} triples corrections is described in
Section 6.6.17. These methods called (dT) and (fT) are of similar
accuracy to other triples corrections. CCSD(dT) and CCSD(fT) are equivalent to
the CRCCSD(T)_{L} and CRCCSD(T)_{2} methods of Piecuch and coworkers.
5.7.4 Job Control Options
The evaluation of a non iterative (T) or (2) correction after a coupledcluster
singles and doubles level calculation (either CCSD, QCISD or OD) is controlled
by the correlation keyword, and the specification of any frozen orbitals via
N_FROZEN_CORE (and possibly N_FROZEN_VIRTUAL).
There is only one additional job control option. For the (2) correction, it is
possible to apply the frozen core approximation in the reference coupled
cluster calculation, and then correlate all orbitals in the (2) correction.
This is controlled by CC_INCL_CORE_CORR, described below.
The default is to include core and corevalence correlation automatically in
the CCSD(2) or OD(2) correction, if the reference CCSD or OD calculation was
performed with frozen core orbitals. The reason for this choice is that core
correlation is economical to include via this method (the main cost increase is
only linear in the number of core orbitals), and such effects are important to
account for in accurate calculations. This option should be made false if a job
with explicitly frozen core orbitals is desired. One good reason for freezing
core orbitals in the correction is if the basis set is physically inappropriate
for describing core correlation (e.g., standard Pople basis sets, and Dunning
ccpVxZ basis sets are designed to describe valenceonly correlation effects).
Another good reason is if a direct comparison is desired against another method
such as CCSD(T) which is always used in the same orbital window as the CCSD
reference.
CC_INCL_CORE_CORR
Whether to include the correlation contribution from frozen core orbitals in
non iterative (2) corrections, such as OD(2) and CCSD(2). 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless no corevalence or core correlation is desired (e.g., for
comparison with other methods or because the basis used cannot describe core
correlation). 

5.7.5 Example
Example 5.0 Two jobs that compare the correlation energy calculated via the
standard CCSD(T) method with the new CCSD(2) approximation, both using the
frozen core approximation. This requires that CC_INCL_CORE_CORR
must be specified as FALSE in the CCSD(2) input.
$molecule
0 2
O
H O 0.97907
$end
$rem
CORRELATION ccsd(t)
BASIS ccpvtz
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION ccsd(2)
BASIS ccpvtz
N_FROZEN_CORE fc
CC_INCL_CORE_CORR false
$end
Example 5.0 Water: Ground state CCSD(dT) calculation using RI
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.957
HOH = 104.5
$end
$rem
JOBTYPE SP
BASIS ccpvtz
AUX_BASIS rimp2ccpvtz
CORRELATION CCSD(dT)
$end
5.8 Coupled Cluster Active Space Methods
5.8.1 Introduction
Electron correlation effects can be qualitatively divided into two classes.
The first class is static or nondynamical correlation: long wavelength
lowenergy correlations associated with other electron configurations that are
nearly as low in energy as the lowest energy configuration. These correlation
effects are important for problems such as homolytic bond breaking, and are the
hardest to describe because by definition the single configuration
HartreeFock description is not a good starting point. The second class is
dynamical correlation: short wavelength highenergy correlations associated
with atomiclike effects. Dynamical correlation is essential for
quantitative accuracy, but a reasonable description of static
correlation is a prerequisite for a calculation being qualitatively
correct.
In the methods discussed in the previous several subsections, the objective was
to approximate the total correlation energy. However, in some cases, it is
useful to model directly the nondynamical and dynamical correlation
energies separately. The reasons for this are pragmatic: with approximate
methods, such a separation can give a more balanced treatment of electron
correlation along bondbreaking coordinates, or reaction coordinates that
involve diradicaloid intermediates. The nondynamical correlation energy is
conveniently defined as the solution of the Schrödinger equation within a
small basis set composed of valence bonding, antibonding and lone pair
orbitals: the socalled full valence active space. Solved exactly, this is
the socalled full valence complete active space SCF (CASSCF) [238],
or equivalently, the fully optimized reaction space (FORS) method [239].
Full valence CASSCF and FORS involve computational complexity which increases
exponentially with the number of atoms, and is thus unfeasible beyond systems
of only a few atoms, unless the active space is further restricted on a
casebycase basis. QChem includes two relatively economical methods that
directly approximate these theories using a truncated coupledcluster doubles
wavefunction with optimized orbitals [240]. They are active space
generalizations of the OD and QCCD methods discussed previously in
Sections 5.6.3 and 5.6.4, and are discussed in the following two
subsections. By contrast with the exponential growth of computational cost with
problem size associated with exact solution of the full valence CASSCF problem,
these cluster approximations have only 6thorder growth of computational
cost with problem size, while often providing useful accuracy.
The full valence space is a welldefined theoretical chemical model. For these
active space coupledcluster doubles methods, it consists of the union of
valence levels that are occupied in the single determinant reference,
and those that are empty. The occupied levels that are to be replaced can only
be the occupied valence and lone pair orbitals, whose number is defined by the
sum of the valence electron counts for each atom (i.e., 1 for H, 2 for He, 1 for
Li, etc..). At the same time, the empty virtual orbitals to which the double
substitutions occur are restricted to be empty (usually antibonding) valence
orbitals. Their number is the difference between the number of valence atomic
orbitals, and the number of occupied valence orbitals given above. This
definition (the full valence space) is the default when either of the
"valence" active space methods are invoked (VOD or VQCCD)
There is also a second useful definition of a valence active space, which we
shall call the 1:1 or perfect pairing active space. In this definition, the
number of occupied valence orbitals remains the same as above. The number of
empty correlating orbitals in the active space is defined as being exactly the
same number, so that each occupied orbital may be regarded as being associated
1:1 with a correlating virtual orbital. In the water molecule,
for example, this means that the lone pair electrons as well as the
bondorbitals are correlated. Generally the 1:1 active space recovers more
correlation for molecules dominated by elements on the right of the periodic
table, while the full valence active space recovers more correlation for
molecules dominated by atoms to the left of the periodic table.
If you wish to specify either the 1:1 active space as described above, or
some other choice of active space based on your particular chemical problem,
then you must specify the numbers of active occupied and virtual orbitals.
This is done via the standard "window options", documented earlier in the
Chapter.
Finally we note that the entire discussion of active spaces here leads only
to specific numbers of active occupied and virtual orbitals. The orbitals
that are contained within these spaces are optimized by minimizing the trial
energy with respect to all the degrees of freedom previously discussed: the
substitution amplitudes, and the orbital rotation angles mixing occupied and
virtual levels. In addition, there are new orbital degrees of freedom to be
optimized to obtain the best active space of the chosen size, in the sense
of yielding the lowest coupledcluster energy. Thus rotation angles mixing
active and inactive occupied orbitals must be varied until the energy is
stationary. Denoting inactive orbitals by primes and active orbitals without
primes, this corresponds to satisfying
Likewise, the rotation angles mixing active and inactive virtual orbitals
must also be varied until the coupledcluster energy is minimized with
respect to these degrees of freedom:
5.8.2 VOD and VOD(2) Methods
The VOD method is the active space version of the OD method described earlier
in Section 5.6.3. Both energies and gradients are available for VOD, so
structure optimization is possible. There are a few important comments to make
about the usefulness of VOD. First, it is a method that is capable of
accurately treating problems that fundamentally involve 2 active electrons in a
given local region of the molecule. It is therefore a good alternative for
describing single bondbreaking, or torsion around a double bond, or some
classes of diradicals. However it often performs poorly for problems where
there is more than one bond being broken in a local region, with the
non variational solutions being quite possible. For such problems the newer
VQCCD method is substantially more reliable.
Assuming that VOD is a valid zero order description for the electronic
structure, then a second order correction, VOD(2), is available for energies
only. VOD(2) is a version of OD(2) generalized to valence active spaces. It
permits more accurate calculations of relative energies by accounting for
dynamical correlation.
5.8.3 VQCCD
The VQCCD method is the active space version of the QCCD method described
earlier in Section 5.6.3. Both energies and gradients are available for
VQCCD, so that structure optimization is possible. VQCCD is applicable to a
substantially wider range of problems than the VOD method, because the modified
energy functional is not vulnerable to non variational collapse. Testing to
date suggests that it is capable of describing double bond breaking to similar
accuracy as full valence CASSCF, and that potential curves for triple
bondbreaking are qualitatively correct, although quantitatively in error by a
few tens of kcal/mol. The computational cost scales in the same manner with
system size as the VOD method, albeit with a significantly larger prefactor.
5.8.4 Local Pair Models for Valence Correlations Beyond Doubles
Working with Prof. HeadGordon at Berkeley, John Parkhill has developed
implementations for pair models which couple 4 and 6 electrons together
quantitatively. Because these truncate the coupled cluster equations at
quadruples and hextuples respectively they have been termed the "Perfect Quadruples"
and "Perfect Hextuples" models. These can be viewed as local approximations
to CASSCF. The PQ and PH models are executed through an extension of QChem's
coupled cluster code, and several options defined for those models will
have the same effects although the mechanism may be different
(CC_DIIS_START, CC_DIIS_SIZE, CC_DOV_THRESH, CC_CONV, etc..).
In the course of implementation, the nonlocal coupled cluster models
were also implemented up to ∧T_{6}. Because the algorithms are
explicitly sparse their costs relative to the existing implementations
of CCSD are much higher (and should never be used in lieu
of an existing CCMAN code), but this capability may be useful for
development purposes, and when computable, models above CCSDTQ are
highly accurate.
To use PQ, PH, their dynamically correlated "+SD"
versions or this machine generated cluster code set: "CORRELATION MGC".
MGC_AMODEL
Choice of approximate cluster model. 
TYPE:
DEFAULT:
Determines  how the CC equations are approximated: 
OPTIONS:
0%  Local ActiveSpace Amplitude iterations. 
 (precalculate GVB orbitals with 
 your method of choice (RPP is good)). 
 
7%  OptimizeOrbitals using the VOD 2step solver. 
 (Experimental only use with MGC_AMPS = 2, 24 ,246) 
 
8%  Traditional Coupled Cluster up to CCSDTQPH. 
9%  MRCC version of the PairModels. (Experimental) 
RECOMMENDATION:

 MGC_NLPAIRS
Number of local pairs on an amplitude. 
TYPE:
DEFAULT:
OPTIONS:
Must be greater than 1, which corresponds to the PP model.
2 for PQ, and 3 for PH. 
RECOMMENDATION:



MGC_AMPS
Choice of Amplitude Truncation 
TYPE:
DEFAULT:
OPTIONS:
2 ≤ n ≤ 123456, a sorted list of integers for every amplitude 
which will be iterated. Choose 1234 for PQ and 123456 for PH 
RECOMMENDATION:

 MGC_LOCALINTS
Pair filter on an integrals. 
TYPE:
DEFAULT:
OPTIONS:
Enforces a pair filter on the 2electron integrals, significantly 
reducing computational cost. Generally useful. for more than 1 pair locality. 
RECOMMENDATION:



MGC_LOCALINTER
Pair filter on an intermediate. 
TYPE:
DEFAULT:
OPTIONS:
Any nonzero value enforces the pair constraint on intermediates, 
significantly reducing computational cost. Not recommended for ≤ 2 pair locality 
RECOMMENDATION:

5.8.5 Convergence Strategies and More Advanced Options
These optimized orbital coupledcluster active space methods enable the use
of the full valence space for larger systems than is possible with
conventional complete active space codes. However, we should note at the
outset that often there are substantial challenges in converging valence
active space calculations (and even sometimes optimized orbital coupled
cluster calculations without an active space). Active space calculations
cannot be regarded as "routine" calculations in the same way as SCF
calculations, and often require a considerable amount of computational trial
and error to persuade them to converge. These difficulties are largely
because of strong coupling between the orbital degrees of freedom and the
amplitude degrees of freedom, as well as the fact that the energy surface is
often quite flat with respect to the orbital variations defining the active
space.
Being aware of this at the outset, and realizing that the program has nothing
against you personally is useful information for the uninitiated user of these
methods. What the program does have, to assist in the struggle to achieve a
converged solution, are accordingly many convergence options, fully documented
in Appendix C. In this section, we describe the basic options and the ideas
behind using them as a starting point. Experience plays a critical role,
however, and so we encourage you to experiment with toy jobs that give rapid
feedback in order to become proficient at diagnosing problems.
If the default procedure fails to converge, the first useful option to employ
is CC_PRECONV_T2Z, with a value of between 10 and 50. This is useful
for jobs in which the MP2 amplitudes are very poor guesses for the converged
cluster amplitudes, and therefore initial iterations varying only the
amplitudes will be beneficial:
CC_PRECONV_T2Z
Whether to preconverge the cluster amplitudes before beginning orbital
optimization in optimized orbital cluster methods. 
TYPE:
DEFAULT:
0  (FALSE) 
10  If CC_RESTART, CC_RESTART_NO_SCF or
CC_MP2NO_GUESS are TRUE 
OPTIONS:
0  No preconvergence before orbital optimization. 
n  Up to n iterations in this preconvergence procedure. 
RECOMMENDATION:
Experiment with this option in cases of convergence failure. 

Other options that are useful include those that permit some damping of step
sizes, and modify or disable the standard DIIS procedure. The main choices are
as follows.
CC_DIIS
Specify the version of Pulay's Direct Inversion of the Iterative Subspace
(DIIS) convergence accelerator to be used in the coupledcluster code. 
TYPE:
DEFAULT:
OPTIONS:
0  Activates procedure 2 initially, and procedure 1 when gradients are smaller 
 than DIIS12_SWITCH. 
1  Uses error vectors defined as differences between parameter vectors from 
 successive iterations. Most efficient near convergence. 
2  Error vectors are defined as gradients scaled by square root of the 
 approximate diagonal Hessian. Most efficient far from convergence. 
RECOMMENDATION:
DIIS1 can be more stable. If DIIS problems are encountered in the early stages
of a calculation (when gradients are large) try DIIS1. 

 CC_DIIS_START
Iteration number when DIIS is turned on. Set to a large number to disable
DIIS. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Occasionally DIIS can cause optimized orbital coupledcluster calculations to
diverge through large orbital changes. If this is seen, DIIS should be
disabled. 



CC_DOV_THRESH
Specifies minimum allowed values for the coupledcluster energy denominators.
Smaller values are replaced by this constant during early iterations only, so
the final results are unaffected, but initial convergence is improved when the
guess is poor. 
TYPE:
DEFAULT:
2502  Corresponding to 0.25 
OPTIONS:
abcde  Integer code is mapped to abc×10^{−de} 
RECOMMENDATION:
Increase to 0.5 or 0.75 for non convergent coupledcluster calculations. 

 CC_THETA_STEPSIZE
Scale factor for the orbital rotation step size. The optimal rotation steps
should be approximately equal to the gradient vector. 
TYPE:
DEFAULT:
OPTIONS:
abcde  Integer code is mapped to abc×10^{−de} 
 If the initial step is smaller than 0.5, the program will increase step 
 when gradients are smaller than the value of THETA_GRAD_THRESH, 
 up to a limit of 0.5. 
RECOMMENDATION:
Try a smaller value in cases of poor convergence and very large orbital
gradients. For example, a value of 01001 translates to 0.1 



An even strongerand moreorless last resortoption permits iteration of the
cluster amplitudes without changing the orbitals:
CC_PRECONV_T2Z_EACH
Whether to preconverge the cluster amplitudes before each change of the
orbitals in optimized orbital coupledcluster methods. The maximum number of
iterations in this preconvergence procedure is given by the value of this
parameter. 
TYPE:
DEFAULT:
OPTIONS:
0  No preconvergence before orbital optimization. 
n  Up to n iterations in this preconvergence procedure. 
RECOMMENDATION:
A very slow last resort option for jobs that do not converge. 

5.8.6 Examples
Example 5.0 Two jobs that compare the correlation energy of the water
molecule with partially stretched bonds, calculated via the two coupledcluster
active space methods, VOD, and VQCCD. These are relatively "easy" jobs to
converge, and may be contrasted with the next example, which is not easy to
converge. The orbitals are restricted.
$molecule
0 1
O
H 1 r
H 1 r a
r = 1.5
a = 104.5
$end
$rem
CORRELATION vod
EXCHANGE hf
BASIS 631G
$end
@@@
$molecule
read
$end
$rem
CORRELATION vqccd
EXCHANGE hf
BASIS 631G
$end
Example 5.0 The water molecule with highly stretched bonds, calculated via
the two coupledcluster active space methods, VOD, and VQCCD. These are
"difficult" jobs to converge. The convergence options shown permitted the job
to converge after some experimentation (thanks due to Ed Byrd for this!). The
difficulty of converging this job should be contrasted with the previous
example where the bonds were less stretched. In this case, the VQCCD method
yields far better results than VOD!.
$molecule
0 1
O
H 1 r
H 1 r a
r = 3.0
a = 104.5
$end
$rem
CORRELATION vod
EXCHANGE hf
BASIS 631G
SCF_CONVERGENCE 9
THRESH 12
CC_PRECONV_T2Z 50
CC_PRECONV_T2Z_EACH 50
CC_DOV_THRESH 7500
CC_THETA_STEPSIZE 3200
CC_DIIS_START 75
$end
@@@
$molecule
read
$end
$rem
CORRELATION vqccd
EXCHANGE hf
BASIS 631G
SCF_CONVERGENCE 9
THRESH 12
CC_PRECONV_T2Z 50
CC_PRECONV_T2Z_EACH 50
CC_DOV_THRESH 7500
CC_THETA_STEPSIZE 3200
CC_DIIS_START 75
$end
5.9 Frozen Natural Orbitals in CCD, CCSD, OD, QCCD and QCISD Calculations
Large computational savings are possible if the virtual space is truncated
using the frozen natural orbital (FNO) approach.
For example, using a fraction f of the full virtual space results in a 1/(1−f)^{4}fold speed up for each CCSD iteration
(CCSD scales with the forth power of the virtual space size).
FNObased truncation for groundstates CC methods was introduced by Bartlett
and coworkers [241,[242,[243].
Extension of the FNO approach to ionized states within
EOMCC formalism was recently introduced and benchmarked [244]
(see Section 6.6.6).
The FNOs are computed as the eigenstates of the virtualvirtual block of the MP2 density matrix
[O(N^{5}) scaling], and the eigenvalues are the occupation numbers associated with the respective FNOs.
By using a userspecified threshold, the FNOs with the smallest
occupations are frozen in CC calculations. This could be done in
CCSD, CCSD(T), CCSD(2), CCSD(dT), CCSD(fT) as well as CCD, OD,QCCD, VQCCD, and all
possible triples corrections for these wavefunctions.
The truncation can be performed using two different schemes.
The first approach is to simply specify the total number of virtual
orbitals to retain, e.g., as the percentage of total virtual orbitals,
as was done in Refs. .
The second approach is to specify the percentage of total
natural occupation (in the virtual space) that
needs to be recovered in the truncated space.
These two criteria are referred to as the
POVO (percentage of virtual orbitals) and OCCT (occupation threshold)
cutoffs, respectively [244].
Since the OCCT criterion is based on the correlation in a specific molecule,
it yields more consistent results than POVO.
For ionization energy calculations employing 9999.5% natural occupation
threshold should yields errors (relative to the full virtual space values)
below 1 kcal/mol [244].
The errors decrease linearly as a function of
the total natural occupation recovered, which
can be exploited by extrapolating truncated calculations to the full virtual space values.
This extrapolation scheme is called the extrapolated FNO
(XFNO) procedure [244].
The linear behavior is exhibited by the total energies of the ground
and the ionized states as a function of OCCT.
Therefore, the XFNO scheme can be employed even when the two states are not
calculated on the same level, e.g., in adiabatic energy differences and EOMIPCC(2,3)
calculations (more on this in Ref. ).
The FNO truncation often causes slower convergence of the CCSD and EOM procedures.
Nevertheless, despite larger number of iterations, the FNObased truncation of orbital space
reduces computational cost considerably, with a negligible decline in accuracy [244].
5.9.1 Job Control Options
 CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO) 
TYPE:
DEFAULT:
OPTIONS:
range  000010000 
abcd  Corresponding to ab.cd% 
RECOMMENDATION:



CC_FNO_USEPOP
Selection of the truncation scheme 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:

5.9.2 Example
Example 5.0 CCSD(T) calculation using FNO with POVO=65%
$molecule
0 1
O
H 1 1.0
H 1 1.0 2 100.
$end
$rem
correlation = CCSD(T)
basis = 6311+G(2df,2pd)
CC_fno_thresh 6500 65% of the virtual space
CC_fno_usepop 0
$end
5.10 NonHartreeFock Orbitals in Correlated Calculations
In cases of problematic openshell references, e.g., strongly spincontaminated doublet radicals,
one may choose to use DFT orbitals. This can be achieved by first doing DFT calculation and then
reading the orbitals and turning the HartreeFock procedure off. A more convenient way is just to
specify EXCHANGE, e.g., EXCHANGE=B3LYP means that B3LYP orbitals will be
computed and used in the CCMAN/CCMAN2 module.
5.10.1 Example
Example 5.0 CCSD calculation of triplet methylene using B3LYP orbitals
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
jobtype SP single point
exchange b3lyp
LEVCOR ccsd
BASIS ccpVDZ
N_FROZEN_CORE 1
$end
5.11 Analytic Gradients and Properties for CoupledCluster Methods
Analytic gradients are available for CCSD, OOCCD/VOD, CCD, and QCCD/VQCCD methods for
both closed and openshell references (UHF and RHF only), including frozen
core and/or virtual functionality. In addition, gradients for selected GVB models are
available.
For the CCSD and OOCCD wavefunctions, QChem can also calculate
dipole moments, 〈R^{2}〉 (as well as XX, YY and ZZ components separately, which is
useful for assigning different Rydberg states, e.g., 3p_{x} vs. 3s, etc.), and the
〈S^{2}〉 values. Interface of the CCSD and (V)OOCCD codes with the NBO 5.0
package is also available.
This code is closely related to EOMCCSD properties/gradient calculations
(Section 6.6.10).
Solvent models available for CCSD are described in Chapter 10.2.
Limitations: Gradients and fully relaxed properties
for ROHF and nonHF (e.g., B3LYP) orbitals as well as RI approximation are not yet available.
Note:
If gradients or properties are computed with frozen core/virtual, the algorithm will
replace frozen orbitals to restricted. This will not affect the energies, but will change the
orbital numbering in the CCMAN printout. 
5.11.1 Job Control Options
CC_REF_PROP
Whether or not the nonrelaxed (expectation value) or full response (including
orbital relaxation terms) oneparticle CCSD
properties will be calculated. The properties currently include permanent
dipole moment, the second moments 〈X^{2}〉, 〈Y^{2}〉, and
〈Z^{2}〉 of electron density, and the total
〈R^{2}〉
= 〈X^{2}〉+〈Y^{2}〉+〈Z^{2}〉 (in atomic units).
Incompatible with JOBTYPE=FORCE, OPT, FREQ. 
TYPE:
DEFAULT:
FALSE  (no oneparticle properties will be calculated) 
OPTIONS:
RECOMMENDATION:
Additional equations need to be solved (lambda CCSD equations) for properties
with the cost approximately the same as CCSD equations. Use default if you do
not need properties. The cost of the properties calculation itself is low. The
CCSD oneparticle density can be analyzed with NBO package by specifying
NBO=TRUE, CC_REF_PROP=TRUE and JOBTYPE=FORCE.


 CC_REF_PROP_TE
Request for calculation of nonrelaxed twoparticle CCSD properties. The
twoparticle properties currently include 〈S^{2}〉. The oneparticle
properties also will be calculated, since the additional cost of the
oneparticle properties calculation is inferior compared to the cost of
〈S^{2}〉. The variable CC_REF_PROP must be also set to
TRUE. 
TYPE:
DEFAULT:
FALSE  (no twoparticle properties will be calculated) 
OPTIONS:
RECOMMENDATION:
The twoparticle properties are computationally expensive, since
they require calculation and use of the twoparticle density matrix (the cost
is approximately the same as the cost of an analytic gradient calculation). Do
not request the twoparticle properties unless you really need them. 



CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed.
The variable CC_REF_PROP must be also set to TRUE. 
TYPE:
DEFAULT:
FALSE  (no orbital response will be calculated) 
OPTIONS:
RECOMMENDATION:
Not available for non UHF/RHF references and for the methods that do not have
analytic gradients (e.g., QCISD). 

5.11.2 Examples
Example 5.0 CCSD geometry optimization
of HHeF followed up by properties calculations
$molecule
0 1
H .000000 .000000 1.886789
He .000000 .000000 1.093834
F .000000 .000000 .333122
$end
$rem
JOBTYPE OPT
CORRELATION CCSD
BASIS augccpVDZ
GEOM_OPT_TOL_GRADIENT 1
GEOM_OPT_TOL_DISPLACEMENT 1
GEOM_OPT_TOL_ENERGY 1
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE SP
CORRELATION CCSD
BASIS augccpVDZ
SCF_GUESS READ
CC_REF_PROP 1
CC_FULLRESPONSE 1
$end
Example 5.0 CCSD on 1,2dichloroethane gauche conformation using SCRF solvent model
$molecule
0 1
C 0.6541334418569877 0.3817051480045552 0.8808840579322241
C 0.6541334418569877 0.3817051480045552 0.8808840579322241
Cl 1.7322599856434779 0.0877596094659600 0.4630557359272908
H 1.1862455146007043 0.1665749506296433 1.7960750032785453
H 0.4889356972641761 1.4444403797631731 0.8058465784063975
Cl 1.7322599856434779 0.0877596094659600 0.4630557359272908
H 1.1862455146007043 0.1665749506296433 1.7960750032785453
H 0.4889356972641761 1.4444403797631731 0.8058465784063975
$end
$rem
JOBTYPE SP
EXCHANGE HF
CORRELATION CCSD
BASIS 631g**
N_FROZEN_CORE FC
CC_SAVEAMPL 1 Save CC amplitudes on disk
SOLVENT_METHOD SCRF
SOL_ORDER 15 L=15 Multipole moment order
SOLUTE_RADIUS 36500 3.65 Angstrom Solute Radius
SOLVENT_DIELECTRIC 89300 8.93 Dielectric (Methylene Chloride)
$end
5.12 Memory Options and Parallelization of CoupledCluster Calculations
The coupledcluster suite of methods, which includes groundstate methods
mentioned earlier in this Chapter and excitedstate methods in the next Chapter,
has been parallelized to take advantage of the multicore architecture.
The code is parallelized at the level of the tensor library such that
the most time consuming operation, tensor contraction, is performed on
different processors (or different cores of the same processor) using
shared memory and shared scratch disk space.
Parallelization on multiple CPUs or CPU cores is achieved by breaking down
tensor operations into batches and running each batch in a separate thread.
Because each thread occupies one CPU core entirely, the maximum number of
threads must not exceed the total available number of CPU cores. If multiple
computations are performed simultaneously, they together should not run
more threads than available cores. For example, an eightcore node can
accommodate one eightthread calculation, two fourthread calculations,
and so on.
The number of threads to be used in the calculation is specified as a command
line option ( nt nthreads) Here nthreads should be given a
positive integer value. If this option is not specified, the job will run
in serial mode using single thread only.
Note:
The use of QCTHREADS environment variable to specify the number of parallel
threads in coupledcluster calculatoin is obsolete. For QChem release 4.0.1 and above,
the number of threads to be used in coupledcluster calculations must be explicitly
specified with command line option 'nt' or it defaults to singlethread execution. 
Setting the memory limit correctly is also very important for high performance
when running larger jobs. To estimate the amount of memory required for
coupledclusters and related calculations, one can use the following formula:
Memory = 
(Number of basis set functions)^{4}
131072

Mb 
 (5.39) 
If the new code (CCMAN2) is used and the calculation is based on a RHF reference,
the amount of memory needed is a half of that given by the formula.
In addition, if gradients are calculated, the amount should be multiplied
by two.
Because the size of data increases steeply with the size of the molecule
computed, both CCMAN and CCMAN2 are able to use disk space to supplement
physical RAM if so required. The strategies of memory management in older
CCMAN and newer CCMAN2 slightly differ, and that should be taken into account
when specifying memory related keywords in the input file.
The MEM_STATIC keyword specifies the amount of memory in megabytes
to be made available to routines that run prior to coupledclusters
calculations: HartreeFock and electronic repulsion integrals evaluation.
A safe recommended value is 500 Mb.
The value of MEM_STATIC should rarely exceed 10002000 Mb even
for relatively large jobs.
The memory limit for coupledclusters calculations is set by
CC_MEMORY. When running older CCMAN, its value is used as
the recommended amount of memory, and the calculation can in fact use less
or run over the limit. If the job is to run exclusively on a node,
CC_MEMORY should be given 50% of all RAM. If the calculation runs
out of memory, the amount of CC_MEMORY should be reduced
forcing CCMAN to use memory saving algorithms.
CCMAN2 uses a different strategy. It allocates the entire amount of RAM given
by CC_MEMORY before the calculation and treats that as a strict
memory limit. While that significantly improves the stability of larger jobs,
it also requires the user to set the correct value of CC_MEMORY
to ensure high performance.
The default value of approximately 1.5 Gb is not appropriate for large
calculations, especially if the node has more resources available. When running
CCMAN2 exclusively on a node, CC_MEMORY should be set to
7580% of the total available RAM.
Note:
When running small jobs, using too large CC_MEMORY in CCMAN2
is not recommended because QChem will allocate more resources than needed for
the calculation, which will affect
other jobs that you may wish to run on the same node. 
In addition, the user should verify
that the disk and RAM together have enough space by using the above formula.
In cases when CC_MEMORY set up is in conflict with the
available space on a particular platform, the CC job may segfault
at run time. In such cases readjusting the CC_MEMORY
value in the input is necessary so as to eliminate the segfaulting.
In addition to memory settings, the user may need to adjust
MAX_SUB_FILE_NUM which determines the maximum size
of tmp files.
MEM_STATIC
Sets the memory for individual Fortran program modules 
TYPE:
DEFAULT:
240  corresponding to 240 Mb or 12% of MEM_TOTAL 
OPTIONS:
n  Userdefined number of megabytes. 
RECOMMENDATION:
For direct and semidirect MP2 calculations, this must exceed OVN +
requirements for AO integral evaluation (32160 Mb). Up to 2000 Mb for large
coupledclusters calculations. 

 CC_MEMORY
Specifies the maximum size, in Mb, of the buffers for incore storage of
blocktensors in CCMAN and CCMAN2. 
TYPE:
DEFAULT:
50% of MEM_TOTAL. If MEM_TOTAL is not set, use 1.5 Gb.
A minimum of 
192 Mb is hardcoded. 
OPTIONS:
RECOMMENDATION:
Larger values can give better I/O performance and are recommended for systems
with large memory (add to your .qchemrc file. When running
CCMAN2 exclusively on a node, CC_MEMORY should be set to
7580% of the total available RAM. ) 



MAX_SUB_FILE_NUM
Sets the maximum number of sub files allowed. 
TYPE:
DEFAULT:
16 Corresponding to a total of 32Gb for a given file. 
OPTIONS:
n  Userdefined number of gigabytes. 
RECOMMENDATION:
Leave as default, or adjust according to your system limits. 

5.13 Simplified CoupledCluster Methods Based on a PerfectPairing
Active Space.
The methods described below are related to valence bond theory and are handled by
the GVBMAN module. The following models are available:
CORRELATION
Specifies the correlation level in GVB models handled by GVBMAN. 
TYPE:
DEFAULT:
OPTIONS:
PP  
GVB_IP  
GVB_SIP  
GVB_DIP  
OP  
NP  
2P  
RECOMMENDATION:
Consult the literature for guidance. 

Molecules where electron correlation is strong are characterized by small
energy gaps between the nominally occupied orbitals (that would comprise the
HartreeFock wavefunction, for example) and nominally empty orbitals. Examples
include socalled diradicaloid molecules [245], or molecules with
partly broken chemical bonds (as in some transitionstate structures). Because the
energy gap is small, electron configurations other than the reference
determinant contribute to the molecular wavefunction with considerable
amplitude, and omitting them leads to a significant error.
Including all possible configurations however, is a vast overkill. It is common to
restrict the configurations that one generates to be constructed not from all
molecular orbitals, but just from orbitals that are either "core" or
"active". In this section, we consider just one type of active space, which
is composed of two orbitals to represent each electron pair: one nominally
occupied (bonding or lone pair in character) and the other nominally empty, or
correlating (it is typically antibonding in character). This is usually called
the perfect pairing active space, and it clearly is wellsuited to represent
the bondingantibonding correlations that are associated with bondbreaking.
The quantum chemistry within this (or any other) active space is given by a
Complete Active Space SCF (CASSCF) calculation, whose exponential cost growth
with molecule size makes it prohibitive for systems with more than about 14
active orbitals. One welldefined coupled cluster (CC) approximation based
on CASSCF is to include only double substitutions in the valence space
whose orbitals are then optimized. In the framework of conventional
CC theory, this defines the valence optimized doubles (VOD) model [240],
which scales as O(N^{6}) (see Section 5.8.2).
This is still too expensive to be readily applied to large molecules.
The methods described in this section bridge the gap between sophisticated but
expensive coupled cluster methods and inexpensive methods such as DFT, HF and
MP2 theory that may be (and indeed often are) inadequate for describing
molecules that exhibit strong electron correlations such as diradicals. The
coupled cluster perfect pairing (PP) [246,[247], imperfect
pairing (IP) [248] and restricted coupled cluster
(RCC) [249] models are local approximations to VOD that include only a
linear and quadratic number of double substitution amplitudes respectively.
They are close in spirit to generalized valence bond (GVB)type
wavefunctions [250], because in fact they are all coupled cluster models for
GVB that share the same perfect pairing active space. We shall therefore
sometimes collectively refer to PP, IP and RCC as GVB methods in the
remainder of this section.
To be more specific, the coupled cluster PP wavefunction is written as
Ψ〉 = exp  ⎛ ⎝

n_{active} ∑
i=1

t_{i} 
^
a

† i∗


^
a

† ―i∗


^
a

―i


^
a

i
 ⎞ ⎠

 Φ〉 
 (5.40) 
where n_{active} is the number of active electrons, and the t_{i} are the
linear number of unknown cluster amplitudes, corresponding to exciting the two
electrons in the ith electron pair from their bonding orbital pair to their
antibonding orbital pair. In addition to t_{i}, the core and the active orbitals
are optimized as well to minimize the PP energy. The algorithm used for this is
a slight modification of the GDM method, described for SCF calculations in
Section 4.6.4. Despite the simplicity of the PP wavefunction, with
only a linear number of correlation amplitudes, it is still a useful
theoretical model chemistry for exploring strongly correlated systems.
This is because it is exact for a single electron pair in the PP active space,
and it is also exact for a collection of noninteracting electron pairs in
this active space. Molecules, after all, are in a sense a collection of
interacting electron pairs! In practice, PP on molecules recovers between 60%
and 80% of the correlation energy in its active space.
Cases where PP needs improvement include molecules with several strongly
correlated electron pairs that are all localized in the same region of space,
and therefore involve significant interpair, as well as intrapair
correlations. For this purpose, we have the IP and RCC wavefunctions. The
expressions for the IP and RCC wavefunctions includes
an additional quadratic number of cluster amplitudes, t_{ij} that describes
the correlation of an electron in the ith pair with an electron in the
jth pair. IP and RCC are physically virtually identical. Generally, IP
should be used unless bonds are completely broken with restricted
orbitals. In that case RCC is preferred as it has been constructed to
eliminate the tendency of restricted coupled cluster methods to become
nonvariational in the dissociation limit. IP and RCC typically recover
between 80% and 95% of the correlation energy in their perfect pairing
active spaces.
In QChem, the unrestricted and restricted GVB methods are
implemented with a resolution of the identity (RI) algorithm
that makes them computationally very efficient [251,[252].
They can be applied to systems with more than 100 active electrons, and both
energies and analytical gradients are available. These methods are requested
via the standard CORRELATION keyword. If
AUX_BASIS is not specified, the calculation uses fourcenter
twoelectron integrals by default. Much faster auxiliary basis algorithms (see
5.5 for an introduction), which are used for the correlation energy
(not the reference SCF energy), can be enabled by specifying a valid string for
AUX_BASIS. The example below illustrates a simple IP calculation.
Example 5.0 Imperfect pairing with auxiliary basis set for geometry
optimization.
$molecule
0 1
H
F 1 1.0
$end
$rem
JOBTYPE opt
CORRELATION gvb_ip
BASIS ccpVDZ
AUX_BASIS rimp2ccpVDZ
% PURECART 11111
$end
If further improvement in the orbitals are needed, the GVB_SIP, GVB_DIP, OP, NP
and 2P models are also included [253]. The GVB_SIP model
includes all the amplitudes of GVB_IP plus a set of quadratic
amplitudes the represent the single ionization of a pair. The GVB_DIP
model includes the GVB_SIP models amplitudes and the doubly ionized
pairing amplitudes which are analogous to the correlation of the occupied
electrons of the ith pair exciting into the virtual orbitals of
the jth pair. These two models have the implementation limit of no
analytic orbital gradient, meaning that a slow finite differences calculation
must be performed to optimize their orbitals, or they must be computed using
orbitals from a different method. The 2P model is the same as the GVB_DIP model,
except it only allows the amplitudes to couple via integrals that span only two
pairs. This allows for a fast implementation of it's analytic orbital gradient
and enables the optimization of it's own orbitals. The OP method is like the
2P method except it removes the "direct"like IP amplitudes and all of the
samespin amplitudes. The NP model is the GVB_IP model with the DIP amplitudes.
This model is the one that works best with the symmetry breaking corrections
that will be discussed later. All GVB methods except GVB_SIP and
GVB_DIP have an analytic nuclear gradient implemented for both regular
and RI fourcenter twoelectron integrals.
There are often considerable challenges in converging the orbital optimization
associated with these GVBtype calculations. The situation is somewhat
analogous to SCF calculations but more severe because there are more orbital
degrees of freedom that affect the energy (for instance, mixing occupied active
orbitals amongst each other, mixing active virtual orbitals with each other, mixing
core and active occupied, mixing active virtual and inactive virtual).
Furthermore, the energy changes associated with many of these new orbital
degrees of freedom are rather small and delicate. As a consequence, in cases
where the correlations are strong, these GVBtype jobs often require many more
iterations than the corresponding GDM calculations at the SCF level. This is a
reflection of the correlation model itself. To deal with convergence issues, a
number of REM values are available to customize the calculations, as listed
below.
GVB_ORB_MAX_ITER
Controls the number of orbital iterations allowed in GVBCC calculations.
Some jobs, particularly unrestricted PP jobs can require 5001000 iterations. 
TYPE:
DEFAULT:
OPTIONS:
Userdefined number of iterations. 
RECOMMENDATION:
Default is typically adequate, but some jobs, particularly UPP jobs, can
require 5001000 iterations if converged tightly. 

 GVB_ORB_CONV
The GVBCC wavefunction is considered converged when the rootmeansquare
orbital gradient and orbital step sizes are less than
10^{−GVB_ORB_CONV}. Adjust THRESH simultaneously. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use 6 for PP(2) jobs or geometry optimizations.
Tighter convergence (i.e. 7 or higher) cannot always be reliably achieved. 



GVB_ORB_SCALE
Scales the default orbital step size by n/1000. 
TYPE:
DEFAULT:
1000  Corresponding to 100% 
OPTIONS:
RECOMMENDATION:
Default is usually fine, but for some stretched geometries it
can help with convergence to use smaller values. 

 GVB_AMP_SCALE
Scales the default orbital amplitude iteration step size by n/1000 for IP/RCC.
PP amplitude equations are solved analytically, so this parameter does not
affect PP. 
TYPE:
DEFAULT:
1000  Corresponding to 100% 
OPTIONS:
RECOMMENDATION:
Default is usually fine, but in some highlycorrelated
systems it can help with convergence to use smaller values. 



GVB_RESTART
Restart a job from previouslyconverged GVBCC orbitals. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Useful when trying to converge to the same GVB
solution at slightly different geometries, for example. 

 GVB_REGULARIZE
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spinunrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term (c/10000)(e^{tijp}−1)/(e^{1}−1) 
TYPE:
DEFAULT:
0 for restricted  1 for unrestricted 
OPTIONS:
RECOMMENDATION:
Should be increased if unrestricted amplitudes do not converge or
converge slowly at dissociation. Set this to zero to remove
all dynamicallyvalued amplitude regularization. 



GVB_POWER
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spinunrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term included in
the energy denominator: (c/10000)(e^{tijp}−1)/(e^{1}−1) 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should be decreased if unrestricted amplitudes do not converge or
converge slowly at dissociation, and should be kept even valued. 

 GVB_SHIFT
Value for a statically valued energy shift in the energy
denominator used to solve the coupled cluster amplitude equations, n/10000. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Default is fine, can be used in lieu of the dynamically
valued amplitude regularization if it does not aid convergence. 



Another issue that a user of these methods should be aware of is the fact that
there is a multiple minimum challenge associated with GVB
calculations. In SCF calculations it is sometimes possible to converge to more
than one set of orbitals that satisfy the SCF equations at a given geometry.
The same problem can arise in GVB calculations, and based on our experience to
date, the problem in fact is more commonly encountered in GVB calculations than
in SCF calculations. A user may therefore want to (or have to!) tinker with the
initial guess used for the calculations. One way is to set GVB_RESTART
= TRUE (see above), to replace the default initial guess (the converged SCF
orbitals which are then localized). Another way is to change the localized
orbitals that are used in the initial guess, which is controlled by the
GVB_LOCAL variable, described below. Sometimes different localization
criteria, and thus different initial guesses, lead to different converged
solutions. Using the new amplitude regularization keywords enables some control
over the solution GVB optimizes [254].
A calculation can be performed with amplitude regularization
to find a desired solution, and then the calculation can be rerun with GVB_RESTART
= TRUE and the regularization turned off to remove the energy penalty of regularization.
GVB_LOCAL
Sets the localization scheme used in the initial guess wavefunction. 
TYPE:
DEFAULT:
OPTIONS:
0  No Localization 
1  Boys localized orbitals 
2  PipekMezey orbitals 
RECOMMENDATION:
Different initial guesses can sometimes lead to different solutions.
It can be helpful to try both to ensure the global minimum has been found. 

 GVB_DO_SANO
Sets the scheme used in determining the active virtual orbitals
in a UnrestrictedinActive Pairs GVB calculation. 
TYPE:
DEFAULT:
OPTIONS:
0  No localization or Sano procedure 
1  Only localizes the active virtual orbitals 
2  Uses the Sano procedure 
RECOMMENDATION:
Different initial guesses can sometimes lead to different
solutions. Disabling sometimes can aid in finding more nonlocal solutions for the orbitals. 



If the calculation is perfect pairing (CORRELATION = PP), it is
possible to look for unrestricted solutions in addition to restricted ones.
Indeed there is no implementation of restricted open shell orbitals for PP in
QChem 3.0. Unrestricted orbitals are the default for molecules with odd
numbers of electrons, but can also be specified for molecules with even numbers
of electrons. This is accomplished by setting GVB_UNRESTRICTED =
TRUE. Given a restricted guess, this will, however usually converge to a
restricted solution anyway, so additional REM variables should be specified to
ensure an initial guess that has broken spin symmetry. This can be accomplished
by using an unrestricted SCF solution as the initial guess, using the
techniques described in Chapter 4. Alternatively a restricted set
of guess orbitals can be explicitly symmetry broken just before the calculation
starts by using GVB_GUESS_MIX, which is described below.
There is also the implementation of UnrestrictedinActive Pairs (UAP) [253]
which is the default unrestricted implementation
for GVB methods. This method simplifies the process of unrestriction by
optimizing only one set of ROHF MO coefficients and a single rotation
angle for each occupiedvirtual pair. These angles are used to construct
a series of 2x2 Given's rotation matrices which are applied to the ROHF
coefficients to determine the α spin MO coefficients and their
transpose is applied to the ROHF coefficients to determine the β
spin MO coefficients. This algorithm is fast and eliminates many of the
pathologies of the unrestricted GVB methods near the dissociation limit.
To generate a full potential curve we find it is best to start at the
desired UHF dissociation solution as a guess for GVB and follow it
inwards to the equilibrium bond distance.
GVB_UNRESTRICTED
Controls restricted versus unrestricted PP jobs. Usually handled
automatically. 
TYPE:
DEFAULT:
same value as UNRESTRICTED 
OPTIONS:
RECOMMENDATION:
Set this variable explicitly only to do a UPP job from an RHF
or ROHF initial guess. Leave this variable alone and specify
UNRESTRICTED=TRUE to access the new UnrestrictedinActivePairs
GVB code which can return an RHF or ROHF solution if used with GVB_DO_ROHF 

 GVB_DO_ROHF
Sets the number of UnrestrictedinActive Pairs to be kept restricted. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If n is the same value as GVB_N_PAIRS returns the ROHF solution
for GVB, only works with the UNRESTRICTED=TRUE
implementation of GVB with GVB_OLD_UPP=0 (it's default value) 



GVB_OLD_UPP
Which unrestricted algorithm to use for GVB. 
TYPE:
DEFAULT:
OPTIONS:
0  Use UnrestrictedinActive Pairs 
1  Use Unrestricted Implementation described in Ref. 
RECOMMENDATION:
Only works for Unrestricted PP and no other GVB model. 

 GVB_GUESS_MIX
Similar to SCF_GUESS_MIX, it breaks alpha/beta symmetry for UPP by
mixing the alpha HOMO and LUMO orbitals according to the userdefined fraction
of LUMO to add the HOMO. 100 corresponds to a 1:1 ratio of HOMO and LUMO in
the mixed orbitals. 
TYPE:
DEFAULT:
OPTIONS:
n  Userdefined, 0 ≤ n ≤ 100 
RECOMMENDATION:
25 often works well to break symmetry without overly
impeding convergence. 



Other $rem variables relevant to GVB calculations are given below. It is
possible to explicitly set the number of active electron pairs using the
GVB_N_PAIRS variable. The default is to make all valence electrons
active. Other reasonable choices are certainly possible. For instance all
electron pairs could be active (n_{active} = n_{β}). Or
alternatively one could make only formal bonding electron pairs active
(n_{active} = N_{STO−3G} − n_{α}). Or in some cases, one might
want only the most reactive electron pair to be active (n_{active} = 1).
Clearly making physically appropriate choices for this variable is essential
for obtaining physically appropriate results!
GVB_N_PAIRS
Alternative to CC_REST_OCC and CC_REST_VIR for setting
active space size in GVB and valence coupled cluster methods. 
TYPE:
DEFAULT:
PP active space (1 occ and 1 virt for each valence electron pair) 
OPTIONS:
RECOMMENDATION:
Use the default unless one wants to study a special active space. When using
small active spaces, it is important to ensure that the proper orbitals are
incorporated in the active space. If not, use the $reorder_mo feature to adjust
the SCF orbitals appropriately. 

 GVB_PRINT
Controls the amount of information printed during a GVBCC job. 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should never need to go above 0 or 1. 



GVB_TRUNC_OCC
Controls how many pairs' occupied orbitals are truncated from the GVB active space 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing the n
lowest energy occupied orbitals from the GVB active space
while leaving their paired virtual orbitals in the active space.
Only the models including the SIP and DIP amplitudes (ie NP and 2P)
benefit from this all other models this equivalent to just
reducing the total number of pairs. 

 GVB_TRUNC_VIR
Controls how many pairs' virtual orbitals are truncated from the GVB active space 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing
the n highest energy occupied orbitals from the GVB
active space while leaving their paired virtual orbitals
in the active space. Only the models including the SIP
and DIP amplitudes (ie NP and 2P) benefit from this all
other models this equivalent to just reducing the total number of pairs. 



GVB_REORDER_PAIRS
Tells the code how many GVB pairs to switch around 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for the user to change the order the active
pairs are placed in after the orbitals are read in or are
guessed using localization and the Sano procedure. Up to 5
sequential pair swaps can be made, but it is best to leave this alone. 

 GVB_REORDER_1
Tells the code which two pairs to swap first 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified
in GVB_REORDER_PAIRS ≥ 1. 



GVB_REORDER_2
Tells the code which two pairs to swap second 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified in GVB_REORDER_PAIRS ≥ 2. 

 GVB_REORDER_3
Tells the code which two pairs to swap third 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002.
Must be specified in GVB_REORDER_PAIRS ≥ 3. 



GVB_REORDER_4
Tells the code which two pairs to swap fourth 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must
be specified in GVB_REORDER_PAIRS ≥ 4. 

 GVB_REORDER_5
Tells the code which two pairs to swap fifth 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must be
specified in GVB_REORDER_PAIRS ≥ 5. 



it is known that symmetry breaking of the orbitals to favor
localized solutions over nonlocal solutions is an issue with
GVB methods in general. A combined coupledcluster perturbation
theory approach to solving symmetry breaking (SB) using perturbation
theory level double amplitudes that connect up to three pairs
has been examined in the literature [255,[256],
and it seems to alleviate the SB problem to a large extent.
It works in conjunction with the GVB_IP, NP, and 2P levels of
correlation for both restricted and unrestricted wavefunctions
(barring that there is no restricted implementation of the 2P model,
but setting GVB_DO_ROHF to the same number as the number of pairs in the system is equivalent).
GVB_SYMFIX
Should GVB use a symmetry breaking fix 
TYPE:
DEFAULT:
OPTIONS:
0  no symmetry breaking fix 
1  symmetry breaking fix with virtual orbitals spanning the active space 
2  symmetry breaking fix with virtual orbitals spanning the whole virtual space 
RECOMMENDATION:
It is best to stick with type 1 to get a symmetry breaking correction
with the best results coming from CORRELATION=NP and GVB_SYMFIX=1. 

 GVB_SYMPEN
Sets the prefactor for the amplitude regularization term for the SB amplitudes 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the prefactor for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e^{(c*100)*t2}−1). 



GVB_SYMSCA
Sets the weight for the amplitude regularization term for the SB amplitudes 
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the weight for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e^{(c*100)*t2}−1). 

The PP and IP models are potential replacements for HF theory as a zero order
description of electronic structure and can be used as a starting point for
perturbation theory. They neglect all correlations that involve electron
configurations with one or more orbitals that are outside the active space.
Physically this means that the socalled "dynamic correlations", which
correspond to atomiclike correlations involving high angular momentum virtual
levels are neglected. Therefore, the GVB models may be not so accurate
for describing energy differences that are sensitive to this neglected
correlation energy, e.g., atomization energies. It is desirable to
correct them for this neglected correlation in a way that is similar to how the
HF reference is corrected via MP2 perturbation theory.
For this purpose, the leading (second order) correction to the PP model, termed
PP(2) [257], has been formulated and efficiently implemented for
restricted and unrestricted orbitals (energy only). PP(2) improves upon many of
the worst failures of MP2 theory (to which it is analogous), such as for open
shell radicals. PP(2) also greatly improves relative energies relative to PP
itself. PP(2) is implemented using a resolution of the identity (RI) approach
to keep the computational cost manageable. This cost scales in the same 5thorder
way with molecular size as RIMP2, but with a prefactor
that is about 5 times larger. It is therefore vastly cheaper than CCSD or
CCSD(T) calculations which scale with the 6th and 7th powers
of system size respectively. PP(2) calculations are requested with
CORRELATION = PP(2). Since the only available algorithm uses auxiliary
basis sets, it is essential to also provide a valid value for
AUX_BASIS to have a complete input file.
The example below shows a PP(2) input file for the challenging case of the N2
molecule with a stretched bond. For this reason a number of the nonstandard
options outlined earlier for orbital convergence are enabled here. First, this
case is an unrestricted calculation on a molecule with an even number of
electrons, and so it is essential to break the alpha/beta spin symmetry in
order to find an unrestricted solution. Second, we have chosen to leave the
lone pairs uncorrelated, which is accomplished by specifying
GVB_N_PAIRS.
Example 5.0 A nonstandard PP(2) calculation. UPP(2) for stretched N2 with
only 3 correlating pairs Try Boys localization scheme for initial guess.
$molecule
0 1
N
N 1 1.65
$end
$rem
UNRESTRICTED true
CORRELATION pp(2)
EXCHANGE hf
BASIS ccpvdz
AUX_BASIS rimp2ccpvdz must use RI with PP(2)
% PURECART 11111
SCF_GUESS_MIX 10 mix SCF guess 100{\%}
GVB_GUESS_MIX 25 mix GVB guess 25{\%} also!
GVB_N_PAIRS 3 correlate only 3 pairs
GVB_ORB_CONV 6 tighter convergence
GVB_LOCAL 1 use Boys initial guess
$end
We have already mentioned a few issues associated with the GVB calculations: the
neglect of dynamic correlation [which can be remedied with PP(2)], the
convergence challenges and the multiple minimum issues. Another weakness of
these GVB methods is the occasional symmetrybreaking artifacts that are a
consequence of the limited number of retained pair correlation amplitudes. For
example, benzene in the PP approximation prefers D_{3h} symmetry over
D_{6h} by 3 kcal/mol (with a 2 distortion), while in IP, this difference
is reduced to 0.5 kcal/mol and less than 1 [248]. Likewise
the allyl radical breaks symmetry in the unrestricted PP model [247],
although to a lesser extent than in restricted open shell
HF. Another occasional weakness is the limitation to the perfect pairing active
space, which is not necessarily appropriate for molecules with expanded valence
shells, such as in some transition metal compounds (e.g. expansion from 4s3d
into 4s4p3d) or possibly hypervalent molecules (expansion from 3s3p into
3s3p3d). The singlet strongly orthogonal geminal method (see the next
section) is capable of dealing with expanded valence shells and could be used
for such cases. The perfect pairing active space is satisfactory for most
organic and first row inorganic molecules.
To summarize, while these GVB methods are powerful and can yield much insight
when used properly, they do have enough pitfalls for not to be
considered true "black box" methods.
5.14 Geminal Models
5.14.1 Reference wavefunction
Computational models that use single reference wavefunction describe molecules
in terms of independent electrons interacting via mean Coulomb and exchange
fields. It is natural to improve this description by using correlated electron
pairs, or geminals, as building blocks for molecular wavefunctions.
Requirements of computational efficiency and size consistency constrain
geminals to have S_{z}=0 [258], with each geminal spanning its own
subspace of molecular orbitals [259]. Geminal wavefunctions were
introduced into computational chemistry by Hurley, LennardJones, and Pople [260].
An excellent review of the history and properties of geminal
wavefunctions is given by Surjan [261].
We implemented a size consistent model chemistry based on Singlet type Strongly
orthogonal Geminals (SSG). In SSG, the number of molecular orbitals in each
singlet electron pair is an adjustable parameter chosen to minimize total
energy. Open shell orbitals remain uncorrelated. The SSG wavefunction is
computed by setting SSG $rem variable to 1. Both spinrestricted
(RSSG) and spinunrestricted (USSG) versions are available, chosen by the
UNRESTRICTED $rem variable.
The wavefunction has the form
 


^
A

[ψ_{1}(r_{1},r_{2}) … ψ_{nβ}(r_{2nβ−1},r 
 
