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7.3 Time-Dependent Density Functional Theory (TDDFT)

7.3.4 TDDFT + PCM for Excitation Energies and Excited-State Properties

(December 20, 2021)

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 a dielectric continuum description. 461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
Link
The “conductor-like PCM” (C-PCM) is one such model, which can be combined with TDDFT to include the effects on solvent on electronic spectra. The TDDFT/C-PCM combination can also be used to perform excited-state geometry optimizations and vibrational frequency calculations in solution. 704 Liu J., Liang W.
J. Chem. Phys.
(2013), 138, pp. 024101.
Link

When PCMs are used in the context of vertical excitation energy calculations, the solvent around the vertically excited solute is out of equilibrium because the (implicit) solvent molecules cannot reorient in response to a vertical excitation process. 461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
Link
Whereas the solvent’s static or zero-frequency dielectric constant (ε0) describes all of the solvent response mechanisms (electronic, vibrational, and orientational), only the electronic part of the response is appropriate to include in a vertical excitation energy calculation. Polarization of the electrons (only) is encoded in the solvent’s optical or infinite-frequency dielectric constant (ε), which is equal to the square of the solvent’s index of refraction (ε=nrefr2). The difference between these two values can be stark, e.g., ε0=78 versus ε=1.8 for water. For geometry optimizations in solution, however, it is probably appropriate to use the numerical value of the static dielectric constant for ε, on the assumption that the solvent molecules have time to reorient in response to changes in the solute’s geometry, even in an electronically excited state.

The sample job provided below computes the a PCM contribution to the TDDFT linear-response matrix using a dielectric constant ε (specified using the keyword OpticalDielectric in the $pcm input section), whereas the value ε0) (specified using Dielectric in the $pcm section) is used to polarize the ground-state MOs. This corresponds to “full linear response theory” (LR-PCM), as described in Ref.  704 Liu J., Liang W.
J. Chem. Phys.
(2013), 138, pp. 024101.
Link
. A perturbative approximation to full LR-PCM is also available, along with state-specific corrections that are somewhat more theoretically rigorous, 461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
Link
and have also been implemented for TDDFT. 776 Mewes J.-M. et al.
J. Phys. Chem. A
(2015), 119, pp. 5446.
Link
, 1261 You Z.-Q. et al.
J. Chem. Phys.
(2015), 143, pp. 204104.
Link
Those approaches are described in Section 11.2.3.3 following a thorough introduction to the PCM approach. See Ref.  461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
Link
for a general overview of the theory of nonequilibrium dielectric continuum methods. Additional PCM job control options are discussed in Section 11.2.

Example 7.6  TDDFT/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

View output

TDDFT_PCM

TDDFT_PCM
       Controls LR-PCM for TDDFT, i.e., whether or not to add the 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_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, 461 Herbert J. M.
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.