Excited states may be obtained from density functional theory by time-dependent density functional theory [324, 325], which calculates poles in the response of the ground state density to a time-varying applied electric field. These poles are Bohr frequencies or excitation energies, and are available in Q-Chem [326], together with the CIS-like Tamm-Dancoff approximation [147]. TDDFT is becoming very popular as a method for studying excited states because the computational cost is roughly similar to the simple CIS method (scaling as roughly the square of molecular size), but a description of differential electron correlation effects is implicit in the method. The excitation energies for low-lying valence excited states of molecules (below the ionization threshold, or more conservatively, below the first Rydberg threshold) are often remarkably improved relative to CIS, with an accuracy of roughly 0.1–0.3 eV being observed with either gradient corrected or local density functionals.

However, standard density functionals do not yield a potential with the correct long-range Coulomb tail, owing to the so-called self-interaction problem, and therefore excitation energies corresponding to states that sample this tail (*e.g.*, diffuse Rydberg states and some charge transfer excited states) are not given accurately [146, 327, 109]. The extent to which a particular excited state is characterized by charge transfer can be assessed using an a spatial overlap metric proposed by Peach, Benfield, Helgaker, and Tozer (PBHT) [328]. (However, see Ref. Richard:2011 for a cautionary note regarding this metric.)

It is advisable to only employ TDDFT for low-lying valence excited states that are below the first ionization potential of the molecule. This makes radical cations a particularly favorable choice of system, as exploited in Ref. Hirata:2001. TDDFT for low-lying valence excited states of radicals is in general a remarkable improvement relative to CIS, including some states, that, when treated by wavefunction-based methods can involve a significant fraction of double excitation character [326]. The calculation of the nuclear gradients of full TDDFT and within the Tamm-Dancoff approximation is also implemented [331].

Standard TDDFT also does not yield a good description of static correlation effects (see Section 5.9), because it is based on a single reference configuration of Kohn-Sham orbitals. Recently, a new variation of TDDFT called spin-flip density functional theory (SFDFT) was developed by Yihan Shao, Martin Head-Gordon and Anna Krylov to address this issue [66]. SFDFT is different from standard TDDFT in two ways:

The reference is a high-spin triplet (quartet) for a system with an even (odd) number of electrons;

One electron is spin-flipped from an alpha Kohn-Sham orbital to a beta orbital during the excitation.

SF-DFT can describe the ground state as well as a few low-lying excited states, and has been applied to bond-breaking processes, and di- and tri-radicals with degenerate or near-degenerate frontier orbitals. Recently, we also implemented[332] a SFDFT method with a non-collinear exchange-correlation potential from Tom Ziegler *et al.* [333, 334], which is in many case an improvement over collinear SFDFT [66]. Recommended functionals for SF-DFT calculations are 5050 and PBE50 (see Ref. [332] for extensive benchmarks). See also Section 6.7.3 for details on wavefunction-based spin-flip models.

Much of chemistry and biology occurs in solution or on surfaces. The molecular environment can have a large effect on electronic structure and may change chemical behavior. Q-Chem is able to compute excited states within a local region of a system through performing the TDDFT (or CIS) calculation with a reduced single excitation subspace [335]. This allows the excited states of a solute molecule to be studied with a large number of solvent molecules reducing the rapid rise in computational cost. The success of this approach relies on there being only weak mixing between the electronic excitations of interest and those omitted from the single excitation space. For systems in which there are strong hydrogen bonds between solute and solvent, it is advisable to include excitations associated with the neighboring solvent molecule(s) within the reduced excitation space.

The reduced single excitation space is constructed from excitations between a subset of occupied and virtual orbitals. These can be selected from an analysis based on Mulliken populations and molecular orbital coefficients. For this approach the atoms that constitute the solvent needs to be defined. Alternatively, the orbitals can be defined directly. The atoms or orbitals are specified within a *$solute* block. These approach is implemented within the TDA and has been used to study the excited states of formamide in solution [336], CO on the Pt(111) surface [337], and the tryptophan chromophore within proteins [338].

Input for time-dependent density functional theory calculations follows very closely the input already described for the uncorrelated excited state methods described in the previous section (in particular, see Section 6.2.5). There are several points to be aware of:

The exchange and correlation functionals are specified exactly as for a ground state DFT calculation, through EXCHANGE and CORRELATION.

If RPA is set to TRUE, a full TDDFT calculation will be performed. This is not the default. The default is RPA = FALSE, which leads to a calculation employing the Tamm-Dancoff approximation (TDA), which is usually a good approximation to full TDDFT.

If SPIN_FLIP is set to TRUE when performing a TDDFT calculation, a SFDFT calculation will also be performed. At present, SFDFT is only implemented within TDDFT/TDA so RPA must be set to FALSE. Remember to set the spin multiplicity to 3 for systems with an even-number of electrons (

*e.g.*, diradicals), and 4 for odd-number electron systems (*e.g.*, triradicals).

TRNSS

Controls whether reduced single excitation space is used

TYPE:

LOGICAL

DEFAULT:

FALSE

Use full excitation space

OPTIONS:

TRUE

Use reduced excitation space

RECOMMENDATION:

None

TRTYPE

Controls how reduced subspace is specified

TYPE:

INTEGER

DEFAULT:

1

OPTIONS:

1

Select orbitals localized on a set of atoms

2

Specify a set of orbitals

3

Specify a set of occupied orbitals, include excitations to all virtual orbitals

RECOMMENDATION:

None

N_SOL

Specifies number of atoms or orbitals in

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

User defined

RECOMMENDATION:

None

CISTR_PRINT

Controls level of output

TYPE:

LOGICAL

DEFAULT:

FALSE

Minimal output

OPTIONS:

TRUE

Increase output level

RECOMMENDATION:

None

CUTOCC

Specifies occupied orbital cutoff

TYPE:

INTEGER: CUTOFF=CUTOCC/100

DEFAULT:

50

OPTIONS:

0-200

RECOMMENDATION:

None

CUTVIR

Specifies virtual orbital cutoff

TYPE:

INTEGER: CUTOFF=CUTVIR/100

DEFAULT:

0

No truncation

OPTIONS:

0-100

RECOMMENDATION:

None

PBHT_ANALYSIS

Controls whether overlap analysis of electronic excitations is performed.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE

Do not perform overlap analysis

TRUE

Perform overlap analysis

RECOMMENDATION:

None

PBHT_FINE

Increases accuracy of overlap analysis

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE

TRUE

Increase accuracy of overlap analysis

RECOMMENDATION:

None

SRC_DFT

Selects form of the short-range corrected functional

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

1

SRC1 functional

2

SRC2 functional

RECOMMENDATION:

None

OMEGA

Sets the Coulomb attenuation parameter for the short-range component.

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

Corresponding to , in units of bohr

RECOMMENDATION:

None

OMEGA2

Sets the Coulomb attenuation parameter for the long-range component.

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

Corresponding to , in units of bohr

RECOMMENDATION:

None

HF_SR

Sets the fraction of Hartree-Fock exchange at r=0.

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

Corresponding to HF_SR =

RECOMMENDATION:

None

HF_LR

Sets the fraction of Hartree-Fock exchange at r=.

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

Corresponding to HF_LR =

RECOMMENDATION:

None

WANG_ZIEGLER_KERNEL

Controls whether to use the Wang-Ziegler non-collinear exchange-correlation kernel in a SFDFT calculation.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE

Do not use non-collinear kernel

TRUE

Use non-collinear kernel

RECOMMENDATION:

None

As described in Section 11.2 (and especially Section 11.2.2), continuum solvent models such as C-PCM allow one to include solvent effect in the calculations. TDDFT/C-PCM allows excited-state modeling in solution. Q-Chem also features TDDFT coupled with C-PCM which extends TDDFT to calculations of properties of electronically-excited molecules in solution. In particular, TDDFT/C-PCM allows one to perform geometry optimization and vibrational analysis [339].

When TDDFT/C-PCM is applied to calculate vertical excitation energies, the solvent around vertically excited solute is out of equilibrium. While the solvent electron density equilibrates fast to the density of the solute (electronic response), the relaxation of nuclear degrees of freedom (e.g., orientational polarization) takes place on a slower timescale. To describe this situation, an optical dielectric constant is employed. To distinguish between equilibrium and non-equilibrium calculations, two dielectric constants are used in these calculations: a static constant (), equal to the equilibrium bulk value, and a fast constant () related to the response of the medium to high frequency perturbations. For vertical excitation energy calculations (corresponding to the unrelaxed solvent nuclear degrees of freedom), it is recommended to use the optical dielectric constant for ), whereas for the geometry optimization and vibrational frequency calculations, the static dielectric constant should be used [339].

The example below illustrates TDDFT/C-PCM calculations of vertical excitation energies. More information concerning the C-PCM and the various PCM job control options can be found in Section 11.2.

**Example 6.112** TDDFT/C-PCM low-lying vertical excitation energy

$molecule 0 1 C 0 0 0.0 O 0 0 1.21 $end $rem EXCHANGE B3lyp CIS_N_ROOTS 10 cis_singlets true cis_triplets true RPA TRUE BASIS 6-31+G* XC_GRID 1 solvent_method pcm $end $pcm Theory CPCM Method SWIG Solver Inversion Radii Bondi $end $solvent Dielectric 78.39 OpticalDielectric 1.777849 $end

To carry out vibrational frequency analysis of an excited state with TDDFT [340, 341], an optimization of the excited-state geometry is always necessary. Like the vibrational frequency analysis of the ground state, the frequency analysis of the excited state should be also performed at a stationary point on the excited state potential surface. The *$rem* variable CIS_STATE_DERIV should be set to the excited state for which an optimization and frequency analysis is needed, in addition to the *$rem* keywords used for an excitation energy calculation.

Compared to the numerical differentiation method, the analytical calculation of geometrical second derivatives of the excitation energy needs much less time but much more memory. The computational cost is mainly consumed by the steps to solve both the CPSCF equations for the derivatives of molecular orbital coefficients C and the CP-TDDFT equations for the derivatives of the transition vectors, as well as to build the Hessian matrix. The memory usages for these steps scale as , where is the number of basis functions and m is the number of atoms. For large systems, it is thus essential to solve all the coupled-perturbed equations in segments. In this case, the *$rem* variable CPSCF_NSEG is always needed.

In the calculation of the analytical TDDFT excited-state Hessian, one has to evaluate a large number of energy-functional derivatives: the first-order to fourth-order functional derivatives with respect to the density variables as well as their derivatives with respect to the nuclear coordinates. Therefore, a very fine integration grid for DFT calculation should be adapted to guarantee the accuracy of the results.

Analytical TDDFT/C-PCM Hessian has been implemented in Q-Chem. Normal mode analysis for a system in solution can be performed with the frequency calculation by TDDFT/C-PCM method. the *$rem* and *$pcm* variables for the excited state calculation with TDDFT/C-PCM included in the vertical excitation energy example above are needed. When the properties of large systems are calculated, you must pay attention to the memory limit. At present, only a few exchange correlation functionals, including Slater+VWN, BLYP, B3LYP, are available for the analytical Hessian calculation.

**Example 6.113** A B3LYP/6-31G* optimization in gas phase, followed by a frequency analysis for the first excited state of the peroxy radical

$molecule 0 2 C 1.004123 -0.180454 0.000000 O -0.246002 0.596152 0.000000 O -1.312366 -0.230256 0.000000 H 1.810765 0.567203 0.000000 H 1.036648 -0.805445 -0.904798 H 1.036648 -0.805445 0.904798 $end $rem jobtype opt exchange b3lyp cis_state_deriv 1 basis 6-31G* cis_n_roots 10 cis_singlets true cis_triplets false xc_grid 000075000302 RPA 0 $end @@@ $molecule Read $end $rem jobtype freq exchange b3lyp cis_state_deriv 1 basis 6-31G* cis_n_roots 10 cis_singlets true cis_triplets false RPA 0 xc_grid 000075000302 $end

**Example 6.114** The optimization and Hessian calculation for low-lying excited state with TDDFT/C-PCM

$comment 9-Fluorenone + 2 methanol in methanol solution $end $molecule 0 1 6 -1.987249 0.699711 0.080583 6 -1.987187 -0.699537 -0.080519 6 -0.598049 -1.148932 -0.131299 6 0.282546 0.000160 0.000137 6 -0.598139 1.149219 0.131479 6 -0.319285 -2.505397 -0.285378 6 -1.386049 -3.395376 -0.388447 6 -2.743097 -2.962480 -0.339290 6 -3.049918 -1.628487 -0.186285 6 -3.050098 1.628566 0.186246 6 -2.743409 2.962563 0.339341 6 -1.386397 3.395575 0.388596 6 -0.319531 2.505713 0.285633 8 1.560568 0.000159 0.000209 1 0.703016 -2.862338 -0.324093 1 -1.184909 -4.453877 -0.510447 1 -3.533126 -3.698795 -0.423022 1 -4.079363 -1.292006 -0.147755 1 0.702729 2.862769 0.324437 1 -1.185378 4.454097 0.510608 1 -3.533492 3.698831 0.422983 1 -4.079503 1.291985 0.147594 8 3.323150 2.119222 0.125454 1 2.669309 1.389642 0.084386 6 3.666902 2.489396 -1.208239 1 4.397551 3.298444 -1.151310 1 4.116282 1.654650 -1.759486 1 2.795088 2.849337 -1.768206 1 2.669205 -1.389382 -0.084343 8 3.322989 -2.119006 -0.125620 6 3.666412 -2.489898 1.207974 1 4.396966 -3.299023 1.150789 1 4.115800 -1.655485 1.759730 1 2.794432 -2.850001 1.767593 $end $rem jobtype OPT EXCHANGE B3lyp CIS_N_ROOTS 10 cis_singlets true cis_triplets true cis_state_deriv 1 Lowest TDDFT state RPA TRUE BASIS 6-311G** XC_GRID 000075000302 solvent_method pcm $end $pcm Theory CPCM Method SWIG Solver Inversion Radii Bondi $end $solvent Dielectric 32.613 $end @@@ $molecule read $end $rem jobtype freq EXCHANGE B3lyp CIS_N_ROOTS 10 cis_singlets true cis_triplets true RPA TRUE cis_state_deriv 1 Lowest TDDFT state BASIS 6-311G** XC_GRID 000075000302 solvent_method pcm mem_static 4000 mem_total 24000 cpscf_nseg 3 $end $pcm Theory CPCM Method SWIG Solver Inversion Radii Bondi $end $solvent Dielectric 32.613 $end

**Example 6.115** This example shows two jobs which request variants of time-dependent density functional theory calculations. The first job, using the default value of RPA = FALSE, performs TDDFT in the Tamm-Dancoff approximation (TDA). The second job, with RPA = TRUE performs a both TDA and full TDDFT calculations.

$comment methyl peroxy radical TDDFT/TDA and full TDDFT with 6-31+G* $end $molecule 0 2 C 1.00412 -0.18045 0.00000 O -0.24600 0.59615 0.00000 O -1.31237 -0.23026 0.00000 H 1.81077 0.56720 0.00000 H 1.03665 -0.80545 -0.90480 H 1.03665 -0.80545 0.90480 $end $rem EXCHANGE b CORRELATION lyp CIS_N_ROOTS 5 BASIS 6-31+G* SCF_CONVERGENCE 7 $end @@@ $molecule read $end $rem EXCHANGE b CORRELATION lyp CIS_N_ROOTS 5 RPA true BASIS 6-31+G* SCF_CONVERGENCE 7 $end

**Example 6.116** This example shows a calculation of the excited states of a formamide-water complex within a reduced excitation space of the orbitals located on formamide

$comment formamide-water TDDFT/TDA in reduced excitation space $end $molecule 0 1 H 1.13 0.49 -0.75 C 0.31 0.50 -0.03 N -0.28 -0.71 0.08 H -1.09 -0.75 0.67 H 0.23 -1.62 -0.22 O -0.21 1.51 0.47 O -2.69 1.94 -0.59 H -2.59 2.08 -1.53 H -1.83 1.63 -0.30 $end $rem EXCHANGE b3lyp CIS_N_ROOTS 10 BASIS 6-31++G** TRNSS TRUE TRTYPE 1 CUTOCC 60 CUTVIR 40 CISTR_PRINT TRUE $end $solute 1 2 3 4 5 6 $end

**Example 6.117** This example shows a calculation of the core-excited states at the oxygen -edge of CO with a short-range corrected functional.

$comment TDDFT with short-range corrected (SRC1) functional for the oxygen K-edge of CO $end $molecule 0 1 C 0.000000 0.000000 -0.648906 O 0.000000 0.000000 0.486357 $end $rem exchange gen basis 6-311(2+,2+)G** cis_n_roots 6 cis_triplets false trnss true trtype 3 n_sol 1 src_dft 1 omega 560 omega2 2450 HF_SR 500 HF_LR 170 $end $solute 1 $end $XC_Functional X HF 1.00 X B 1.00 C LYP 0.81 C VWN 0.19 $end

**Example 6.118** This example shows a calculation of the core-excited states at the phosphorus -edge with a short-range corrected functional.

$comment TDDFT with short-range corrected (SRC2) functional for the phosphorus K-edge of PH3 $end $molecule 0 1 H 1.196206 0.000000 -0.469131 P 0.000000 0.000000 0.303157 H -0.598103 -1.035945 -0.469131 H -0.598103 1.035945 -0.469131 $end $rem exchange gen basis 6-311(2+,2+)G** cis_n_roots 6 cis_triplets false trnss true trtype 3 n_sol 1 src_dft 2 omega 2200 omega2 1800 HF_SR 910 HF_LR 280 $end $solute 1 $end $XC_Functional X HF 1.00 X B 1.00 C LYP 0.81 C VWN 0.19 $end

**Example 6.119** SF-TDDFT SP calculation of the 6 lowest states of the TMM diradical using recommended 50-50 functional

$molecule 0 3 C C 1 CC1 C 1 CC2 2 A2 C 1 CC2 2 A2 3 180.0 H 2 C2H 1 C2CH 3 0.0 H 2 C2H 1 C2CH 4 0.0 H 3 C3Hu 1 C3CHu 2 0.0 H 3 C3Hd 1 C3CHd 4 0.0 H 4 C3Hu 1 C3CHu 2 0.0 H 4 C3Hd 1 C3CHd 3 0.0 CC1 = 1.35 CC2 = 1.47 C2H = 1.083 C3Hu = 1.08 C3Hd = 1.08 C2CH = 121.2 C3CHu = 120.3 C3CHd = 121.3 A2 = 121.0 $end $rem jobtype SP EXCHANGE GENERAL Exact exchange BASIS 6-31G* SCF_GUESS CORE SCF_CONVERGENCE 10 MAX_SCF_CYCLES 100 SPIN_FLIP 1 CIS_N_ROOTS 6 CIS_CONVERGENCE 10 MAX_CIS_CYCLES = 100 $end $xc_functional X HF 0.5 X S 0.08 X B 0.42 C VWN 0.19 C LYP 0.81 $end

**Example 6.120** SFDFT with non-collinear exchange-correlation functional for low-lying states of

$comment non-collinear SFDFT calculation for CH2 at 3B1 state geometry from EOM-CCSD(fT) calculation $end $molecule 0 3 C H 1 rCH H 1 rCH 2 HCH rCH = 1.0775 HCH = 133.29 $end $rem JOBTYPE SP UNRESTRICTED TRUE EXCHANGE PBE0 BASIS cc-pVTZ SPIN_FLIP 1 WANG_ZIEGLER_KERNEL TRUE SCF_CONVERGENCE 10 CIS_N_ROOTS 6 CIS_CONVERGENCE 10 $end