self-consistent approaches, namely the state specific (SS) [31] and the vertical
excitation model (VEM) [32] approaches. One of the principal differences between
the two approaches is that the former implies a modification of the GS reference
during the self-consistent process, whereas the latter does not. When the change of
polarity of the chromophore between the GS and ES is large, e.g., for chargetransfer (CT) transitions, going beyond the LR-PCM approximation is
recommended, though the most adequate model in that case remains a matter of
debate [34, 35].
We underline that, although all these approaches can be used to determine E
ES
analytically, analytic gradients (and hence efficient access to R
ES ) are only available with the LR approach [20]. Subsequently, a popular approach is to determine
E
vertÀa , E
vertÀ f , and E
adia with one of the three refined PCM approaches (cLR, SS,
and VEM) on geometries computed within the LR-PCM model. Likewise, ΔE
ZPVE
is often calculated at the LR-PCM level, so that the results of (10) are generally
obtained with mixed environmental models, the energy (geometry and vibrations)
being obtained with a refined (simpler) PCM level of theory [33].
2.3.2 0–0 Energies with Mixed DFT/Wavefunction Approaches
Besides TD-DFT, there is a wide panel of alternative and (very) accurate ab initio
methods with, on the one hand, multi-reference approaches, e.g., Complete Active
Space second-order Perturbation Theory (CAS-PT2) [36] and Multi-Reference
Configuration Interaction (MR-CI) [37], and, on the other hand, single-reference
(highly-)correlated schemes, e.g., Equation-of-Motion Coupled Cluster (EOM-CC)
[38–41], Symmetry Adapted Cluster CI (SAC-CI) [42], Algebraic Diagrammatic
Construction (ADC) [43], and Configuration Interaction singles with a perturbative
correction for double excitations [CIS(D)] [44, 45]. Despite the rapid developments
of these approaches and the implementations of efficient protocols (e.g., the resolution of identity scheme), their less favorable scalings with system size than
TD-DFT generally limit their applications to vertical calculations but for rather
small molecules. Therefore, it has been proposed to combine TD-DFT’s ES geometries and vibrations to E
vertÀa and E
vertÀ f obtained with these more advanced
approaches. In the protocol proposed by Goerigk and Grimme [13], the experimental 0–0 energies are first transformed into “experimental” vertical energies by
applying successive corrections for solvation, vibration, and geometrical reorganization effects determined with TD-DFT. Alternatively, one can determine AFCP
energies through (10) and next correct them through wavefunction (Ψ) vertical
calculations performed on the DFT GS and TD-DFT ES geometries [46, 47]. For
approaches that can only be used for gas-phase vertical transition energies, the
corrected AFCP energy simply becomes
Computational Molecular Electronic Spectroscopy with TD-DFT
353
excitation model (VEM) [32] approaches. One of the principal differences between
the two approaches is that the former implies a modification of the GS reference
during the self-consistent process, whereas the latter does not. When the change of
polarity of the chromophore between the GS and ES is large, e.g., for chargetransfer (CT) transitions, going beyond the LR-PCM approximation is
recommended, though the most adequate model in that case remains a matter of
debate [34, 35].
We underline that, although all these approaches can be used to determine E
ES
analytically, analytic gradients (and hence efficient access to R
ES ) are only available with the LR approach [20]. Subsequently, a popular approach is to determine
E
vertÀa , E
vertÀ f , and E
adia with one of the three refined PCM approaches (cLR, SS,
and VEM) on geometries computed within the LR-PCM model. Likewise, ΔE
ZPVE
is often calculated at the LR-PCM level, so that the results of (10) are generally
obtained with mixed environmental models, the energy (geometry and vibrations)
being obtained with a refined (simpler) PCM level of theory [33].
2.3.2 0–0 Energies with Mixed DFT/Wavefunction Approaches
Besides TD-DFT, there is a wide panel of alternative and (very) accurate ab initio
methods with, on the one hand, multi-reference approaches, e.g., Complete Active
Space second-order Perturbation Theory (CAS-PT2) [36] and Multi-Reference
Configuration Interaction (MR-CI) [37], and, on the other hand, single-reference
(highly-)correlated schemes, e.g., Equation-of-Motion Coupled Cluster (EOM-CC)
[38–41], Symmetry Adapted Cluster CI (SAC-CI) [42], Algebraic Diagrammatic
Construction (ADC) [43], and Configuration Interaction singles with a perturbative
correction for double excitations [CIS(D)] [44, 45]. Despite the rapid developments
of these approaches and the implementations of efficient protocols (e.g., the resolution of identity scheme), their less favorable scalings with system size than
TD-DFT generally limit their applications to vertical calculations but for rather
small molecules. Therefore, it has been proposed to combine TD-DFT’s ES geometries and vibrations to E
vertÀa and E
vertÀ f obtained with these more advanced
approaches. In the protocol proposed by Goerigk and Grimme [13], the experimental 0–0 energies are first transformed into “experimental” vertical energies by
applying successive corrections for solvation, vibration, and geometrical reorganization effects determined with TD-DFT. Alternatively, one can determine AFCP
energies through (10) and next correct them through wavefunction (Ψ) vertical
calculations performed on the DFT GS and TD-DFT ES geometries [46, 47]. For
approaches that can only be used for gas-phase vertical transition energies, the
corrected AFCP energy simply becomes
Computational Molecular Electronic Spectroscopy with TD-DFT
353