As described in Section 11.2.3, polarizable continuum models (PCMs) are
a simple means of including solvation effects in quantum chemistry calculations
at the level of the dielectric continuum. In this section, the available options
in Q-Chem for PCM solvation of TDDFT excited states are discussed. For further details
on the associated keywords, see Sections 11.2.3.3
and 11.2.4. For background information on PCM solvation models,
see Ref.
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
.
TDDFT can be combined with PCM in two different formalisms: linear response (LR)-
and state-specific (SS)-PCM. The LR-PCM approach
160
J. Chem. Phys.
(1999),
110,
pp. 9877.
Link
,
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
employs the transition density
between the ground and excited state and describes the coupling between the excitation
process on the solute and the solvent environment. (For more details, see
Section 11.2.3.3.) LR-PCM entails only a minor additional
computational cost. However, the use of the transition density limits its applicability
to optically-allowed transitions, whereas the LR-PCM contribution vanishes for
charge-transfer states or singlet to triplet excitations. In Q-Chem, TDDFT/LR-PCM
is available for excitation energies, as well as excited-state energy gradients and
Hessians,
764
J. Chem. Phys.
(2013),
138,
pp. 024101.
Link
to allow for geometry optimization and vibrational frequency calculations as described in
Section 7.3.5.
The state-specific PCM formalism
159
J. Chem. Phys.
(2005),
122,
pp. 104513.
Link
,
166
J. Chem. Phys.
(2006),
124,
pp. 124520.
Link
,
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
is difference-density-based
and takes into account the instantaneous response of the solvent to the change in
the solute wave function. This contribution is not included with LR-PCM and is non-zero
even for vanishing transition densities in optically-forbidden transitions.
Q-Chem includes SS-PCM both as a perturbative correction
1368
J. Chem. Phys.
(2015),
143,
pp. 204104.
Link
,
842
J. Phys. Chem. A
(2015),
119,
pp. 5446.
Link
(ptSS-PCM) as well as in two fully iterative implementations.
Generally, the SS-PCM implementation for TDDFT follows
closely the ADC()-family of methods described in Section 7.12.10.
The ptSS-PCM approach,
1368
J. Chem. Phys.
(2015),
143,
pp. 204104.
Link
,
842
J. Phys. Chem. A
(2015),
119,
pp. 5446.
Link
which is
alternatively called corrected linear-response (cLR-PCM),
166
J. Chem. Phys.
(2006),
124,
pp. 124520.
Link
evaluates a state-specific correction based on the TDDFT excited state density
for each calculated state. This first-order correction excludes the response of
the excited state density to the state-specific reaction field, as to include this response requires
an iterative procedure. The ptSS-PCM approach can be combined with a perturbative approximation to LR-PCM,
ptLR-PCM. For more details, see Section 11.2.3.3.
To go beyond a first-order correction, one of the iterative implementations
of SS-PCM has to be used. Here, a state-specific Schrödinger equation
is solved self-consistently via iterative optimization of the reaction field
for a specific state. In Q-Chem, iterative SS-PCM is implemented either via external iteration
(EI-SS-PCM, sometimes called the Improta-Barone-Scalmani-Frisch approach),
549
J. Chem. Phys.
(2007),
127,
pp. 074504.
Link
or else via internal iteration (II-SS-PCM,
otherwise known as the vertical excitation model or VEM).
815
Chem. Sci.
(2011),
2,
pp. 2143.
Link
Whereas ptSS-PCM adds only negligible computational cost, the iterative procedures
increase it by a factor of the required number of iterations.
Excited-state solvation with TDDFT is available in two regimes:
equilibrium and nonequilibrium
solvation. Nonequilibrium solvation applies to fast processes
(e.g., during vertical excitation or emission), in which the slow nuclear degrees of
freedom of the solvent remain equilibrated with respect to the initial state.
Hence, the solute has to be
modeled for an out-of-equilibrium state of the (implicit) solvent, with only the
fast electronic polarization of the solvent relaxed using the final state’s charge density. This fast
solvent response is governed by the infinite-frequency or “optical” dielectric constant (),
often taken to be the square of the solvent’s index of refraction:
.
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
Equilibrium solvation, on the other hand,
is appropriate for excited states with a sufficiently long lifetime to allow for complete
relaxation of the solvent. Here, the zero-frequency or static dielectric constant
() is the relevant one, as it includes all polarization mechanisms including both the
fast electronic degrees of freedom and the slow vibrational and orientational degrees of freedom.
LR-PCM for TDDFT is activated by setting TDDFT_LR_PCM = TRUE
in the $pcm section. The default is a nonequilibrium
calculation employing the optical dielectric constant
for the excited state, whereas the static dielectric constant is employed for the
ground-state solvation of the used MOs.
This corresponds to the “full linear response theory” of Ref.
764
J. Chem. Phys.
(2013),
138,
pp. 024101.
Link
.
The value of is specified using the keyword OpticalDielectric in the $solvent input section,
or can be chosen from a list of preset values as described in Section 11.2.4.3;
an equilibrium calculation in the excited state could be requested by setting OpticalDielectric equal to the
value of the solvent’s static dielectric constant.
A perturbative approximation to full
LR-PCM is also available, along with state-specific corrections that are somewhat more
theoretically rigorous,
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
and have also been implemented for
TDDFT.
842
J. Phys. Chem. A
(2015),
119,
pp. 5446.
Link
,
1368
J. Chem. Phys.
(2015),
143,
pp. 204104.
Link
This approach is described in further detail below
in Sections 7.3.4.2 and 11.2.3.3.
If the C-PCM model is used, then analytical excited-state energy gradients are available for TDDFT + LR-PCM
and are automatically used if LR-PCM is activated for TDDFT geometry optimization.
As excited-state geometry optimization presupposes equilibration of the solute structure
in the excited state, it is probably appropriate to employ the static dielectric constant for geometry optimization.
The LR-PCM approach considers the solvent response to the electronic transition density and provides only for optically-allowed transitions a non-vanishing contribution. Optically-forbidden transitions should be described with a state-specific PCM. This includes triplet excited states accessed from a singlet ground state and also charge-transfer states with a vanishing transition density
Example 7.6 TDDFT/LR-C-PCM low-lying vertical excitation energy
$molecule 0 1 C 0.0 0.0 0.0 O 0.0 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
TDDFT_LR_PCM
TDDFT_LR_PCM
Controls LR-PCM for TDDFT, i.e., whether or not to add the LR-PCM contributions to the TDDFT eigenvalue problem.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
FALSE
Do not do LR-PCM (0th-order solvent correction only).
TRUE
Perform full LR-PCM.
RECOMMENDATION:
Assuming that PCM solvation is turned on for the ground state (SOLVENT_METHOD = PCM in the $rem section),
then disabling LR-PCM by setting TDDFT_LR_PCM = FALSE
will afford a “0th-order” solvation correction, in which solvent-polarized MOs and energy levels are used in
what is otherwise equivalent to a gas-phase TDDFT calculation. This is the first step in more sophisticated “nonequilibrium”
TDDFT + PCM methods, which are discussed in Section 11.2.3.3. The LR-PCM correction to the
excitation energies has some peculiar properties, such as the fact that it vanishes for optically-forbidden states,
505
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021),
11,
pp. e1519.
Link
and the state-specific approaches that are discussed in Section 11.2.3.3 are likely preferable.
The state-specific PCM jobs are controlled with the StateSpecific keyword in the $pcm block, which can be employed to activate either a perturbative or one of the iterative SS-PCMs.
StateSpecific
Specifies which the state-specific PCM will be used.
INPUT SECTION: $pcm
TYPE:
Various
DEFAULT:
NONE
OPTIONS:
Perturb
Perform ptSS and ptLR for vertical excitations.
External
Perform self-consistent EI-SS-PCM to the excited state (for emission).
Internal
Perform self-consistent II-SS-PCM to the excited state (for emission or excitation).
RECOMMENDATION:
Use for vertical excitation energies ptSS-PCM or in very polar cases the nonequilibrium II-SS-PCM,
and equilibrium EI-SS-PCM for emission energies of long-lived excited states.
1. Perturbative State-Specific (ptSS) PCM
Perturbative state-specific PCM (ptSS-PCM) is activated by setting
StateSpecific = Perturb in the $pcm block. By default, the ptSS correction
is evaluated in the nonequilibrium regime governed by the optical dielectric constant
(OpticalDielectric in the $solvent block). To force an equilibrium ptSS-PCM
calculation, OpticalDielectric must be set to the static value,
. To separate the fast and slow contributions to the reaction field,
the charge-separation procedure is chosen with the ChargeSeparation keyword
(Marcus is the default, for more information, see Section 11.2.3.3).
Alternatively, setting NonEquilibrium = TRUE results in a default
nonequilibrium ptSS-PCM calculation with Marcus charge separation and
deactivated LR-PCM. The ptSS-PCM output provides for each calculated state both a ptSS
and a ptLR-correction. In the nonequilibrium case, the corrected energy corresponds to the
vertical excitation energy. Results are provided both with the unrelaxed density and the relaxed
density, if the latter has been recommended. Results based on the relaxed density are
recommended.
1368
J. Chem. Phys.
(2015),
143,
pp. 204104.
Link
NonEquilibrium
Activate nonequilibrium ptSS-PCM.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
False
OPTIONS:
TRUE
Activate nonequilibrium ptSS-PCM.
FALSE
Deactivate nonequilibrium ptSS-PCM.
RECOMMENDATION:
NonEquilibrium activates a default ptSS-PCM calculation with Marcus charge separation and no additional LR-PCM.
ChargeSeparation
Partition fast and slow charges in solvent equilibrium state
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
NONE
OPTIONS:
Marcus
Do slow/fast charge separation with the Marcus partition.
Pekar
Do slow/fast charge separation with the Pekar partition.
RECOMMENDATION:
Charge separation is used in conjunction with the StateSpecific keyword in $pcm.
Example 7.7 PCM solvation effects on the vertical excitation energies of planar DMABN using the ptSS and ptLR methods.
$molecule 0 1 C 0.000046 -0.000398 1.904953 C 1.210027 0.000379 1.186051 C 1.214640 -0.000065 -0.194515 C 0.000164 -0.000616 -0.933832 C -1.214349 -0.001557 -0.194687 C -1.209753 -0.001846 1.185775 H 2.151949 0.001377 1.722018 H 2.164371 0.000481 -0.709640 H -2.164082 -0.002008 -0.709781 H -2.151763 -0.002287 1.721615 C -0.000227 0.001061 3.325302 N -0.000475 0.002405 4.484321 N 0.000053 -0.000156 -2.297372 C -1.258656 0.001284 -3.036994 H -1.041042 0.001615 -4.102376 H -1.860897 -0.885647 -2.811117 H -1.859247 0.889133 -2.810237 C 1.258563 -0.000660 -3.037285 H 1.860651 0.886208 -2.810755 H 1.859362 -0.888604 -2.811461 H 1.040664 -0.000097 -4.102609 $end $rem EXCHANGE LRC-wPBEPBE OMEGA 260 BASIS 6-31G* CIS_N_ROOTS 10 RPA 2 CIS_SINGLETS 1 CIS_TRIPLETS 0 CIS_RELAXED_DENSITY TRUE SOLVENT_METHOD PCM $end $pcm NonEquilibrium Theory IEFPCM StateSpecific Perturb $end $solvent Dielectric 35.688000 ! Acetonitrile OpticalDielectric 1.806874 $end
2. Iterative State-Specific PCM
The control options for iterative state-specific PCM follow the same setup as for the ADC()-family of methods described in Section 7.12.10. By setting the keyword EqSolv to the intended maximum number of iterations, self-consistent SS-PCM is activated for the reference state indicated in EqState. Iterations are performed until either the maximum number of iterations is reached or the convergence criteria are satisfied. These criteria are based on the excited-state energy and the surface charges and are set using Eqs_Conv. During the iterative optimization of the reaction field, states may change their ordering (root-flipping). Automatic state-following (EqState_Follow = true) based on a combination of the electrostatic potential at the surface, the energy changes, and excited state dipole moments is available. It performs well for separated states or states of substantially different character (i.e. charge-transfer and locally excited states) but can break down for similar states close to their crossing points.
Note that the excited state energy with iterative state-specific PCMs is only physically meaningful for the reference state (marked in the output as “EqS-reference”). For all other states, only energies relative to the self-consistent reference state can be interpreted. The iterative equilibrium SS-PCM output lists the emission energy to the ground state (only EI-SS-PCM), as well as transition energies to the other excited states (both EI- and II-SS-PCM) in zeroth- and first-order (with nonequilibrium ptSS correction). For nonequilibrium II-SS-PCM, only the vertical excitation energy to the self-consistently optimized excited state is physically meaningful. For the nonequilibrium II-SS-PCM and ptSS-PCM calculations, again the ChargeSeparation keyword is employed.
EqSolv
Main switch of the self-consistent SS-PCM procedure.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
No self-consistent SS-PCM.
1
Single SS-PCM calculation (SCF + TDDFT) with the solvent field found on disk.
Do a maximum of automatic solvent-field iterations.
RECOMMENDATION:
We recommend to use 15 steps max. Typical convergence is 3–5 steps. In
difficult cases 6–12. If the solvent-field iteration do not converge in 15
steps, something is wrong. For EI-SS-PCM jobs make sure a solvent field was
written to disk by a previous job. II-SS-PCM will automatically begin by setting
up the solvent field.
EqState
Specifies the state for which the solvent field is to be optimized.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
1
energetically lowest excited state
2
2nd lowest excited state
RECOMMENDATION:
Given that only one class of excited states is calculated, the state will be
selected according to its energetic position shown in the “Exited-State
Summary” of the output file.
Eqs_Conv
Controls the convergence of the solvent-field iterations by setting the
convergence criteria (a mixture of SCF energy (EI-SS-PCM) or excited state energy (II-SS-PCM)
and charge-vector). SCF energy criterion computes as .
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
SCF_CONVERGENCE
OPTIONS:
3
May be sufficient for emission energies
4
Assured converged total energies (2.7 meV)
5
Very tight
RECOMMENDATION:
Use the default.
EqState_Follow
Controls the automatic state-following based on the electrostatic potential at the surface,
excited state energy changes, and excited state dipole moments.
INPUT SECTION: $pcm
TYPE:
NONE
DEFAULT:
FALSE
OPTIONS:
TRUE
Activate automatic state-following
FALSE
Deactivate automatic state-following.
RECOMMENDATION:
State-Following works well for separated states of different character
and can become problematic for nearly degenerate states. It can be advisable
to deactivate state-following if similar states are close.
2. (A) External-Iteration SS-PCM
EI-SS-PCM, otherwise known as Improta-Barone-Scalmani-Frisch PCM,
549
J. Chem. Phys.
(2007),
127,
pp. 074504.
Link
is activated by setting StateSpecific = External in the $pcm section. EI-SS-PCM
iterates over both the SCF and TDDFT calculation by returning the excited-state reaction field
into a frozen-Reaction-Field (fRF) SCF. Prior to this, the reaction field has to be set up in a separate
job by setting StateSpecific = External without adding the keyword EqSolv.
In the following iterative job, the reference state (EqState) can then be chosen freely
among the states calculated prior, and iterations can be restarted in the same way
(e.g., to manually correct the automatic state-following).
The EI-SS-PCM convergence is judged based on the change in the surface charge vector
and the fRF-SCF energy, in order to converge all excited states to the same level.
Additionally, the self-ptSS term indicates convergence though it is not used as an
actual convergence criterion.
EI-SS-PCM is always performed as equilibrium solvation employing the full dielectric response of the solvent based on the static dielectric constant (Dielectric). Non-equilibrium emission and transition energies are calculated in first order at the ptSS-PCM level. EI-SS-PCM provides the correct solution of the state-specific Schrödinger equation without contamination from the ground state MOs, and hence, should be used whenever the excited state is to be treated on equal footing with the ground state.
Example 7.7.8 TDDFT with External Iteration SS-PCM for nitrate in acetonitrile.
$molecule -1 1 N -0.068642000000 -0.600693000000 -0.723424000000 O 0.349666000000 0.711166000000 1.187490000000 O -0.948593000000 0.200668000000 -0.956940000000 O 0.659040000000 -0.386002000000 0.402650000000 $end $rem jobtype ¯¯ sp method ¯¯ PBE0 basis ¯¯¯ 3-21g point_group_symmetry False cis_n_roots 2 ! Number of excited states cis_singlets true cis_triplets false rpa false ! Tamm-Dancoff approximation cis_moments true ! Excited state multipole moments cis_relaxed_density true ! Excited state density relaxation solvent_method ¯ pcm $end $pcm StateSpecific External ! Keyword for External Iteration SS-PCM LinearResponse false ! Switch of LR-PCM $end $solvent Dielectric 35.68800 ! Acetonitrile OpticalDielectric 1.806874 $end @@@ $molecule read $end $rem jobtype ¯¯ sp method ¯¯ PBE0 basis ¯¯¯ 3-21g point_group_symmetry False cis_n_roots 2 cis_singlets true cis_triplets false rpa false cis_moments true cis_relaxed_density true solvent_method pcm $end $pcm StateSpecific External Eqsolv 15 ! Max. Number of iterations Eqstate 1 ! Ref. State to be equilibrated Eqs_Conv 4 ! Conv. criterion for SCF energy and surface charges EqState_Follow true ! State Following to remain on the same Ref. State LinearResponse false $end $solvent Dielectric 35.68800 ! Acetonitrile OpticalDielectric 1.806874 $end
2. (B) Internal-Iteration SS-PCM
II-SS-PCM, otherwise known as the vertical excitation model (VEM),
815
Chem. Sci.
(2011),
2,
pp. 2143.
Link
is activated by setting the keyword StateSpecific = Internal.
II-SS-PCM retains the same ground state wave-function in the iterations
but exchanges the ground and excited-state reaction field during the TDDFT calculation.
This leads to slight contamination of the excited state by the ground state MOs.
II-SS-PCM is available both for equilibrium (adiabatic excitation model, AEM) and nonequilibrium (VEM), and
with either the full matrix (f) or only the diagonal elements (d) of the excited state
reaction field, as described in Ref.
815
Chem. Sci.
(2011),
2,
pp. 2143.
Link
. To choose the exact model,
the keyword InternalIteration is set to
VEM(d), VEM(f), AEM(d), or AEM(f).
In contrast to EI-SS-PCM, the II-SS-PCM approach automatically begins with a setup calculation to
produce the initial excited state reaction field for the iterations, which then cannot be restarted.
To still allow besides the automatic also for manual state following, a specific sequence of states
can be listed by appending the chosen method in InternalIteration with -state.
The sequence is terminated by a zero, after which state-following is re-engaged. For II-SS-PCM,
convergence is judged on the reference state energy and the surface charge vector.
A nonequilibrium II-SS-PCM or VEM calculation corresponds to an iterative version of nonequilibrium ptSS-PCM for vertical excitation energies. Here, also the response of the density to the fast excited state reaction field is included. Hence, the VEM result for the converged reference state (indicated by “EqS-reference” in the output) corresponds to the vertical excitation energy, while the other states have no physical meaning.
Equilibrium II-SS-PCM or AEM, on the other hand, provides the excited reference state energy and transition energies to other excited states. However, emission energies from the reference state to the ground state are adiabatic because the ground state orbitals are not affected by the excited state reaction field. To obtain vertical emission energies, a further fRF ground state calculation has to be performed by setting RF_ptSS_Save = true and loading the reaction field in a consecutive job by setting RF_ptSS_Read = true. See Section 7.9.6 for further information.
InternalIteration
Controls the used internal iteration SS-PCM model. Additionally, a sequence
of state to be followed can be appended after adding -state.
INPUT SECTION: $pcm
TYPE:
NONE
DEFAULT:
NONE
OPTIONS:
VEM(f)
Activate nonequilibrium II-SS-PCM (VEM) with the full matrix contribution.
VEM(d)
Activate nonequilibrium II-SS-PCM (VEM) with only diagonal matrix elements.
AEM(f)
Activate equilibrium II-SS-PCM (AEM) with the full matrix contribution.
AEM(d)
Activate equilibrium II-SS-PCM (AEM) with only diagonal matrix elements.
…-state x y z … 0
Specify sequence of states to be followed (terminate with 0).
RECOMMENDATION:
The nonequilibrium versions (VEM) provide vertical excitation energies, while
in equilibrium (AEM) the initial state for emission energies is prepared. By attaching
the -state ending an explicit sequence of states to be followed can be
provided. After terminating with a zero, state-following will be reengaged.
Example 7.7.9 TDDFT with Internal Iteration SS-PCM and the additional ground state calculation for vertical emission energies of nitrate in acetonitrile.
$molecule -1 1 N -0.068642000000 -0.600693000000 -0.723424000000 O 0.349666000000 0.711166000000 1.187490000000 O -0.948593000000 0.200668000000 -0.956940000000 O 0.659040000000 -0.386002000000 0.402650000000 $end $rem jobtype ¯¯ sp method ¯¯ PBE0 basis ¯¯¯ 3-21g point_group_symmetry False cis_n_roots 4 ! Number of excited states cis_singlets true cis_triplets false rpa false ! Tamm-Dancoff approximation cis_moments true ! Excited state multipole moments cis_relaxed_density true ! Excited state density relaxation solvent_method pcm $end $pcm StateSpecific Internal ! Keyword for Internal Iteration SS-PCM ChargeSeparation MARCUS ! Keyword for the charge separation InternalIteration AEM(f) ! Procedure (VEM(f), VEM(d), AEM(f), or AEM(d)) + state sequence (add e.g.: -state 1 3 1 0) Eqsolv 10 ! Max. Number of iterations Eqstate 1 ! Ref. State to be equilibrated Eqs_Conv 4 ! Conv. criterion for SCF energy and surface charges EqState_Follow true ! State Following to remain on the same Ref. State theory iefpcm ! PCM Theory (else CPCM) rf_ptss_save true ! Save the eq. reaction field of Ref. State LinearResponse false ! Switch of LR-PCM $end $solvent Dielectric 35.68800 ! Acetonitrile OpticalDielectric 1.806874 $end @@@ $molecule read $end $rem jobtype sp method ¯¯ PBE0 basis 3-21g point_group_symmetry False solvent_method pcm $end $pcm theory iefpcm rf_ptss_read true ! Flag to read the reaction field and do RF-ptSS calc $end $solvent Dielectric 35.68800 ! Acetonitrile OpticalDielectric 1.806874 $end