Molecular potential energy surfaces rely on the BornOppenheimer separation of nuclear and electronic motion. Minima on such energy surfaces correspond to the classical picture of equilibrium geometries, and transition state structures correspond to firstorder saddle points. Both equilibrium and transitionstate structures are stationary points for which the energy gradient vanishes. Characterization of such critical points requires consideration of the eigenvalues of the Hessian (second derivative matrix): minimumenergy, equilibrium geometries possess Hessians whose eigenvalues are all positive, whereas transitionstate structures are defined by a Hessian with precisely one negative eigenvalue. (The latter is therefore a local maximum along the reaction path between minimumenergy reactant and product structures, but a minimum in all directions perpendicular to this reaction path.
The quality of a geometry optimization algorithm is of major importance; even the fastest integral code in the world will be useless if combined with an inefficient optimization algorithm that requires excessive numbers of steps to converge. QChem incorporates a geometry optimization package (Optimize—see Appendix A) developed by the late Jon Baker over more than ten years.
The key to optimizing a molecular geometry successfully is to proceed from the starting geometry to the final geometry in as few steps as possible. Four factors influence the path and number of steps:
starting geometry
optimization algorithm
quality of the Hessian (and gradient)
coordinate system
QChem controls the last three of these, but the starting geometry is solely determined by the user, and the closer it is to the converged geometry, the fewer optimization steps will be required. Decisions regarding the optimization algorithm and the coordinate system are generally made by the Optimize package (i.e., internally, within QChem) to maximize the rate of convergence. Although users may override these choices in many cases, this is not generally recommended.
Level of Theory 
Analytical 
Maximum Angular 
Analytical 
Maximum Angular 
(Algorithm) 
Gradients 
Momentum Type 
Hessian 
Momentum Type 
DFT 
✓ 

✓ 

HF 
✓ 

✓ 

ROHF 
✓ 

✗ 

MP2 
✓ 

✗ 

(V)OD 
✓ 

✗ 

(V)QCCD 
✓ 

✗ 

CIS (except RO) 
✓ 

✓ 

CFMM 
✓ 

✗ 
Another consideration when trying to minimize the optimization time concerns the quality of the gradient and Hessian. A higherquality Hessian (i.e., analytical versus approximate) will in many cases lead to faster convergence, in the sense of requiring fewer optimization steps. However, the construction of an analytical Hessian requires significant computational effort and may outweigh the advantage of fewer optimization cycles. Currently available analytical gradients and Hessians are summarized in Table 9.1.
Features of QChem’s geometry and transitionstate optimization capabilities include:
Cartesian, Zmatrix or internal coordinate systems
Eigenvector Following (EF) or GDIIS algorithms
Constrained optimizations
Equilibrium structure searches
Transition structure searches
Hessianfree characterization of stationary points
Initial Hessian and Hessian update options
Reaction pathways using intrinsic reaction coordinates (IRC)
Optimization of minimumenergy crossing points (MECPs) along conical seams
Obviously a level of theory, basis set, and starting molecular geometry must be specified to begin a geometry optimization or transitionstructure search. These aspects are described elsewhere in this manual, and this section describes jobcontrol variables specific to optimizations.
JOBTYPE
Specifies the calculation.
TYPE:
STRING
DEFAULT:
Default is singlepoint, which should be changed to one of the following options.
OPTIONS:
OPT
Equilibrium structure optimization.
TS
Transition structure optimization.
RPATH
Intrinsic reaction path following.
RECOMMENDATION:
Applicationdependent.
GEOM_OPT_HESSIAN
Determines the initial Hessian status.
TYPE:
STRING
DEFAULT:
DIAGONAL
OPTIONS:
DIAGONAL
Set up diagonal Hessian.
READ
Have exact or initial Hessian. Use as is if Cartesian, or transform
if internals.
RECOMMENDATION:
An accurate initial Hessian will improve the performance of the optimizer, but is expensive to compute.
GEOM_OPT_COORDS
Controls the type of optimization coordinates.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0
Optimize in Cartesian coordinates.
1
Generate and optimize in internal coordinates, if this fails abort.
1
Generate and optimize in internal coordinates, if this fails at any stage of the
optimization, switch to Cartesian and continue.
2
Optimize in matrix coordinates, if this fails abort.
2
Optimize in matrix coordinates, if this fails during any stage of the
optimization switch to Cartesians and continue.
RECOMMENDATION:
Use the default; delocalized internals are more efficient.
GEOM_OPT_TOL_GRADIENT
Convergence on maximum gradient component.
TYPE:
INTEGER
DEFAULT:
300
tolerance on maximum gradient component.
OPTIONS:
Integer value (tolerance = ).
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.
GEOM_OPT_TOL_DISPLACEMENT
Convergence on maximum atomic displacement.
TYPE:
INTEGER
DEFAULT:
1200 tolerance on maximum atomic displacement.
OPTIONS:
Integer value (tolerance = ).
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.
GEOM_OPT_TOL_ENERGY
Convergence on energy change of successive optimization cycles.
TYPE:
INTEGER
DEFAULT:
100 tolerance on maximum (absolute) energy change.
OPTIONS:
Integer value (tolerance = value ).
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.
GEOM_OPT_MAX_CYCLES
Maximum number of optimization cycles.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
User defined positive integer.
RECOMMENDATION:
The default should be sufficient for most cases. Increase if the initial guess geometry is poor, or for systems with shallow potential wells.
GEOM_OPT_PRINT
Controls the amount of Optimize print output.
TYPE:
INTEGER
DEFAULT:
3
Error messages, summary, warning, standard information and gradient print out.
OPTIONS:
0
Error messages only.
1
Level 0 plus summary and warning print out.
2
Level 1 plus standard information.
3
Level 2 plus gradient print out.
4
Level 3 plus Hessian print out.
5
Level 4 plus iterative print out.
6
Level 5 plus internal generation print out.
7
Debug print out.
RECOMMENDATION:
Use the default.
GEOM_OPT_SYMFLAG
Controls the use of symmetry in Optimize.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Make use of point group symmetry.
FALSE
Do not make use of point group symmetry.
RECOMMENDATION:
Use the default.
GEOM_OPT_MODE
Determines Hessian mode followed during a transition state search.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Mode following off.
Maximize along mode .
RECOMMENDATION:
Use the default, for geometry optimizations.
GEOM_OPT_MAX_DIIS
Controls maximum size of subspace for GDIIS.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Do not use GDIIS.
1
Default size = min(NDEG, NATOMS, 4) NDEG = number of molecular
degrees of freedom.
Size specified by user.
RECOMMENDATION:
Use the default or do not set too large.
GEOM_OPT_DMAX
Maximum allowed step size. Value supplied is multiplied by 10.
TYPE:
INTEGER
DEFAULT:
300
= 0.3
OPTIONS:
Userdefined cutoff.
RECOMMENDATION:
Use the default.
GEOM_OPT_UPDATE
Controls the Hessian update algorithm.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1
Use the default update algorithm.
0
Do not update the Hessian (not recommended).
1
MurtaghSargent update.
2
Powell update.
3
Powell/MurtaghSargent update (TS default).
4
BFGS update (OPT default).
5
BFGS with safeguards to ensure retention of positive definiteness
(GDISS default).
RECOMMENDATION:
Use the default.
GEOM_OPT_LINEAR_ANGLE
Threshold for near linear bond angles (degrees).
TYPE:
INTEGER
DEFAULT:
165 degrees.
OPTIONS:
Userdefined level.
RECOMMENDATION:
Use the default.
FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference.
TYPE:
INTEGER
DEFAULT:
100
Corresponding to 0.001 . For calculating second derivatives.
OPTIONS:
Use a step size of .
RECOMMENDATION:
Use the default except in cases where the potential surface is very flat, in which case a larger value should be used. See FDIFF_STEPSIZE_QFF for third and fourth derivatives.
Example 9.208 As outlined, the rate of convergence of the iterative optimization process is dependent on a number of factors, one of which is the use of an initial analytic Hessian. This is easily achieved by instructing QChem to calculate an analytic Hessian and proceed then to determine the required critical point
$molecule 0 1 O H 1 oh H 1 oh 2 hoh oh = 1.1 hoh = 104 $end $rem JOBTYPE freq Calculate an analytic Hessian METHOD hf BASIS 631g(d) $end $comment Now proceed with the Optimization making sure to read in the analytic Hessian (use other available information too). $end @@@ $molecule read $end $rem JOBTYPE opt METHOD hf BASIS 631g(d) SCF_GUESS read GEOM_OPT_HESSIAN read Have the initial Hessian $end
QChem allows the user to characterize the stationary point found by a geometry optimization or transition state search without performing a full analytical Hessian calculation, which is sometimes unavailable or computationally unaffordable. This is achieved via a finite difference Davidson procedure developed by Sharada et al.[535] For a geometry optimization, it solves for the lowest eigenvalue of the Hessian () and checks if (a negative indicates a saddle point); for a TS search, it solves for the lowest two eigenvalues, and and indicate a transition state. The lowest eigenvectors of the updated PRFO (approximate) Hessian at convergence are used as the initial guess for the Davidson solver.
The cost of this Hessianfree characterization method depends on the rate of convergence of the Davidson solver. For example, to characterize an energy minimum, it requires total energy + gradient calculations, where is the number of iterations that the Davidson algorithm needs to converge, and “2" is for forward and backward displacements on each iteration. According to Ref. Sharada:2014, this method can be much more efficient than exact Hessian calculation for substantially large systems.
Note: At the moment, this method does not support QM/MM or systems with fixed atoms.
GEOM_OPT_CHARAC
Use the finite difference Davidson method to characterize the resulting energy minimum/transition state.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE
do not characterize the resulting stationary point.
TRUE
perform a characterization of the stationary point.
RECOMMENDATION:
Set it to TRUE when the character of a stationary point needs to be verified, especially for a transition structure.
GEOM_OPT_CHARAC_CONV
Overide the builtin convergence criterion for the Davidson solver.
TYPE:
INTEGER
DEFAULT:
0 (use the builtin default value 10)
OPTIONS:
Set the convergence criterion to 10.
RECOMMENDATION:
Use the default. If it fails to converge, consider loosening the criterion with caution.
Example 9.209 Geometry optimization of a triflate anion that converges to an eclipsed conformation, which is a first order saddle point. This is verified via the finite difference Davidson method by setting GEOM_OPT_CHARAC to TRUE.
$molecule 1 1 C 0.00000 0.00078 0.98436 F 1.09414 0.63166 1.47859 S 0.00000 0.00008 0.94745 O 1.25831 0.72597 1.28972 O 1.25831 0.72597 1.28972 O 0.00000 1.45286 1.28958 F 1.09414 0.63166 1.47859 F 0.00000 1.26313 1.47663 $end $rem JOBTYPE opt METHOD BP86 GEOM_OPT_DMAX 50 BASIS 6311+G* SCF_CONVERGENCE 8 THRESH 14 SYMMETRY FALSE SYM_IGNORE TRUE GEOM_OPT_TOL_DISPLACEMENT 10 GEOM_OPT_TOL_ENERGY 10 GEOM_OPT_TOL_GRADIENT 10 GEOM_OPT_CHARAC TRUE $end
Example 9.210 TS search for alanine dipeptide rearrangement reaction beginning with a guess structure converges correctly. The resulting TS structure is verified using the finite difference Davidson method.
$molecule 0 1 C 3.21659 1.41022 0.26053 C 2.16708 0.35258 0.59607 N 1.21359 0.16703 0.41640 C 0.11616 0.82394 0.50964 C 1.19613 0.03585 0.74226 N 2.18193 0.02502 0.18081 C 3.43891 0.74663 0.01614 O 2.19596 0.25708 1.63440 C 0.11486 1.96253 0.53088 O 1.29658 0.59392 1.85462 H 3.25195 2.14283 1.08721 H 3.06369 1.95423 0.67666 H 4.20892 0.93714 0.22851 H 1.24786 0.78278 1.21013 H 0.25990 1.31404 1.47973 H 2.02230 0.38818 1.10143 H 3.60706 1.48647 0.76756 H 4.29549 0.06423 0.04327 H 3.36801 1.25875 0.98106 H 0.68664 2.66864 0.27269 H 0.01029 1.65112 1.56461 H 1.06461 2.50818 0.45885 $end $rem JOBTYPE freq EXCHANGE B3LYP BASIS 631G SCF_MAX_CYCLES 250 SYMMETRY false SYM_IGNORE true $end @@@ $rem JOBTYPE ts SCF_GUESS read GEOM_OPT_DMAX 100 GEOM_OPT_MAX_CYCLES 1500 EXCHANGE B3LYP BASIS 631G MAX_SCF_CYCLES 250 GEOM_OPT_HESSIAN read SYMMETRY false SYM_IGNORE true GEOM_OPT_CHARAC true $end $molecule read $end