7.9.7 Excited States in Solution with ADC/SS-PCM

(A) Theory
Let us begin with a brief review of the theoretical and technical aspects of the calculation of absorption spectra in solution. For this purpose, one would typically employ the perturbative, state-specific approach in combination with an ADC of second or third order. The first step is a self-consistent reaction field calculation [SCRF, Hartree-Fock with a PCM, Eq. (7.78)].

$$E_0=⟨0|{\widehat{H}}^{\text{vac}}+\widehat{R}(0)|0⟩$$

(7.78)

The PCM is formally represented by the reaction- or solvent-field operator $\widehat{R}$. In practice, $\widehat{R}$ is a set of point charges placed on the molecular surface, which are optimized together with the orbitals during the SCF procedure. Note that since $\widehat{R}$ accounts for the self-induced polarization of the solute, it depends on solute’s wave function (here the ground state), which will in the following be indicated in the subscript ${\widehat{R}}_0$.

After the SCRF is converged, the final surface charges and the respective operator can, according to the Franck-Condon principle, be separated into a “slow” solvent-nuclei related ($ϵ-n^2$) and “fast”, solvent-electron related ($n^2$) component (using Marcus partitioning, eq. (7.79)), and are stored on disk.

$$\widehat{R}(0)\equiv {\widehat{R}}_0^{\text{tot}}={\widehat{R}}_0^{\text{f}}+{\widehat{R}}_0^{\text{s}}$$

(7.79)

$q^f=$

${\displaystyle \frac{n^2-1}{ϵ-1}}{q}^{\text{𝑡𝑜𝑡𝑎𝑙}}\mathit{\hspace{1em}\hspace{1em}}{q}^s=$

${\displaystyle \frac{ϵ-n^2}{ϵ-1}}{q}^{\text{𝑡𝑜𝑡𝑎𝑙}}=q-q^f$

(7.80)

The “polarized” MOs resulting from the SCRF step are subjected to ordinary MP/ADC calculations, which yield the correlation energy for the ground and excitation energies for the excited states, which are both added to the HF energy to obtain the total MP ground and ADC excited-state energies. However, since the MOs contain the interaction with the complete “frozen” solvent-polarization of the SCF ground-state density (${\widehat{R}}^{\text{tot}}$, i.e., both components), the resulting excitation energies violate the Franck-Condon principle, which requires the solvents electronic degrees of freedom (fast component of the polarization) to be relaxed. Furthermore, the solvent field has been obtained for the SCF density, which more often than not provides a poor description of the electrostatic nature of the solute and may in turn lead to systematic errors in the excitation energies.

To account for the relaxation of the solvent electrons and bring the excitation/excited-state energies in accordance with Franck-Condon, we employ the perturbative ansatz shown in eq. (7.81), in which the fast component of the ground-state solvent field ${\widehat{R}}_0^{\text{f}}$ is replaced by the respective quantity computed for the excited-state density ${\widehat{R}}_{\text{i}}^{\text{f}}$. In this framework, the energy of an excited state $i$ computed with the polarized MOs can be identified as the zeroth-order energy (eq. (7.83)), while the first-order term becomes the difference between the interaction of the zeroth-order excited-state density with the fast component of the ground- and excited-state solvent fields given in eq. (7.84).

$$E_{\text{i}}^{\text{NEq}}=⟨i|{\widehat{H}}^{\text{vac}}+{\widehat{R}}_0^{\text{tot}}+\lambda ({\widehat{R}}_{\text{i}}^{\text{f}}-{\widehat{R}}_{\text{0}}^{\text{f}})|i⟩$$

(7.81)

	$E_i^{\mathrm{NEq}(0)}=$		$E_i^{(0)}+\lambda E_i^{(1)}+\mathrm{\dots }$		(7.82)
	$E_i^{(0)}=$		$⟨i^{(0)}|{\widehat{H}}^{\text{vac}}+{\widehat{R}}_0^{\text{tot}}|i^{(0)}⟩$		(7.83)
	$E_i^{(1)}=$		$⟨i^{(0)}|{\widehat{R}}_{\text{i}}^{\text{f}}-{\widehat{R}}_0^{\text{f}}|i^{(0)}⟩$		(7.84)

After the ADC iterations have converged, ${\widehat{R}}_{\text{i}}^{\text{f}}$ is computed from the respective excited-state density and ${\widehat{R}}_0^{\text{f}}$ read from disk to form the first-order correction. Adding this so-called ptSS-term to the zeroth order energy, one arrives at the vertical energy in the non-equilibrium limit $E_{\text{i}}^{\text{NEq}}$. Since the ptSS-term accounts for the response of the electron density of the implicit solvent molecules to excitation of the solute, its always negative and thus lowers the energy of the excited state w.r.t. the ground state.

To eliminate problems resulting from the poor description of the ground-state solvent field at the SCF/HF level of theory, we use an additive correction that is based on the MP2 ground-state density. In a nutshell, it replaces the interaction between the potential of the difference density of the excited state ${\widehat{V}}_{i^{(0)}}-{\widehat{V}}_{0^{\text{𝑀𝑃}}}$ with the SCF solvent field ${\widehat{Q}}_{0^{\text{𝐻𝐹}}}$ by the respective interaction with the MP solvent field ${\widehat{Q}}_{0^{\text{𝑀𝑃}}}$:

	$E_{\text{𝑃𝑇𝐷}}=$	$({\widehat{V}}_{i^{(0)}}-{\widehat{V}}_{0^{\text{𝑀𝑃}}})\cdot {\widehat{Q}}_{0^{\text{𝑀𝑃}}}$			(7.85)
	$-$	$({\widehat{V}}_{i^{(0)}}-{\widehat{V}}_{0^{\text{𝑀𝑃}}})\cdot {\widehat{Q}}_{0^{\text{𝐻𝐹}}}$			(7.86)

We will in the following refer to this as “perturbative, density-based” (PTD) correction and denote the respective approach as ptSS(PTD). Accordingly, the non-PTD corrected results will be denoted ptSS(PTE).

In addition to the ptSS-PCM discussed above, a perturbative variant of the so-called linear response corrections (termed ptLR presented in Ref. 635) is also available. In contrast to the ptSS approach, the ptLR corrections depend on the transition rather than the state (difference) densities. Although the ptLR corrections are always computed and printed, we discourage their use with the correlated ADC variants (2 and higher), for which the ptSS approach is better suited. The ptSS and ptLR approaches are also available for TDDFT as described in Section 12.2.2.3.

A detailed, comprehensive introduction to the theory and implementation, as well as extensive benchmark data for the non-equilibrium formalism in combination with all orders of ADC can be found in Ref. 635.

(B) Usage
The calculation of vertical transition energies with the ptSS-PCM approach is fairly straightforward. One just needs to activate the PCM (set SOLVENT_METHOD in the $rem block to PCM), enable the non-equilibrium functionality (set NONEQUILIBRIUM in the $pcm block to TRUE), and specify the solvent parameters, i.e., dielectric constant $ϵ$ (DIELECTRIC) and squared refractive index $n^2$ (DIELECTRIC_INFI) in the $solvent block.

Note:  Symmetry will be disabled for all calculations with a PCM.

In the output of a ptSS-PCM calculation with any correlated ADC variant, multiple values are given for the total and transition energies depending in the level of theory. For the correlated ADC variants these include:

• •

  Zeroth-order results, direct outcome of the ADC calculation with solvated orbitals, excited states are ordered according to this value.

• •

  First-order results, incl. only ptSS non-equilibrium corrections, termed ptSS(PTE).

• •

  Corrected first-order results, incl. also the correlation correction, termed ptSS(PTD).

• •

  Scaled and corrected first-order results, incl. an empirical scaling of the correlation correction, termed ptSS(PTD*).

We recommend to use the corrected first-order results since the PTD correction generally yields the most accurate results. The ptSS(PTD) approach is typically a very good approximation to the fully consistent, but more expensive PTED scheme, in which the solvent field is made self-consistent with the MP2 density (see next section for the PTED scheme and sample-jobs for a comparison of the two). Oscillator strengths are computed using the ptSS(PTD) energies. The PTD* approach includes an empirical scaling of the PTD correction that was developed to improve the solvatochromic-shifts of a series of nitro-aromatics with a cc-pVDZ basis.⁶³⁵ However, we later found that for most other systems, the scaling slightly worsens the agreement with the fully consistent PTED scheme. In addition to the compiled total and transition energies, all contributing terms (ptSS, PTD etc.) are given separately. An even more verbose output detailing all the integrals contributing to the 1st order corrections can be obtained by increasing PCM_PRINT to 1.

Note:  The zeroth-order results are by no means identical to the gas-phase excitation energies, and in turn the ptSS-term is not the solvatochromic shift.

(C) Tips and Tricks
To model the absorption spectrum in polar solvents, it is advisable to use a structure optimized in the presence of a PCM since the influence can be quite significant.

It should also be taken into account that a PCM, being a purely electrostatic model, lacks at the description of explicit interactions like e.g. hydrogen bonds. If fairly strong h-bond donors/acceptors are present and a protic/Lewis-basic solvent is to be modeled, consider adding a few (one or maximum two per donor/acceptor site) explicit solvent molecules (and optimize them together with the molecule in the presence of a PCM). A systematic investigation of this aspect for two representative examples can be found in Ref. 635.

If large differences between the HF and MP description of the molecule exist (PTD terms $>0.2$ eV), it is advisable to employ the iterative ptSS(PTED) scheme described in the next section. Due to the inverse nature of the systematic errors of ADC(2) and ADC(3), the best guess for the excitation energy in solution is usually the average of both values.

For the PCM, we recommend the formally exact and slightly more expensive integral-equation formalism (IEF)-PCM variant (THEORY to IEFPCM in the $pcm block) in place of the approximate C-PCM, and otherwise default parameters.

(A) Theory
To model emission/absorption of solvent-equilibrated excited states and/or to investigate their photochemical reactivity, both components of the polarization have to be relaxed with respect to the desired state. This becomes evident considering that a full solvent-field equilibration (including the nuclear component) is essentially a geometry optimization for the implicit solvent, and should thus be employed whenever the geometry of the solute is optimized for the desired excited state. The Hamiltonian for a solvent-equilibrated state $|k⟩$ simply reads

$$E_{\mathrm{k}}^{\mathrm{EqS}}=⟨k|{\widehat{H}}^{\mathrm{vac}}+{\widehat{R}}_{\mathrm{k}}|k⟩.$$

(7.87)

Since the interaction with solvent field of any state has to be introduced to the MOs in the SCF step of a calculation, a solvent-field equilibration for excited states (and correlated ground states) is an iterative procedure requiring multiple SCF+ADC calculations until convergence is achieved. This also means that a guess (typically from a ptSS calculation) for the solvent field has to generated and used in the first SCF step. The MOs resulting from this first SCF are subjected to an ADC calculation, providing a first excited-state density, for which a new solvent field is computed and employed in the SCF step of the second iteration. This procedure is repeated until the solvent field (charges) and energies are converged and ultimately provides the total energy and wave function of the solvent-equilibrated excited state $|k⟩$, as well as the out-of-equilibrium wave functions of other states. However, as the name already suggest, this state-specific approach yields a meaningful energy only for the solvent-equilibrated reference state $|k⟩$. All other states have to be treated in the non-equilibrium limit and subjected to the formalism presented in the previous section to be consistent with the Franck-Condon principle. The respective generalization of the perturbative ansatz for the Hamiltonian for the $i^{\mathrm{th}}$ out-of-equilibrium state (e.g. the ground or any other excited state) in the field of the equilibrated state $|k⟩$ reads as

$$E_{\mathrm{i}}^{\mathrm{NEq}(\mathrm{k})}=⟨i|{\widehat{H}}^{\mathrm{vac}}+{\widehat{R}}_{\mathrm{k}}^{\mathrm{tot}}+\lambda ({\widehat{R}}_{\mathrm{i}}^{\mathrm{f}}-{\widehat{R}}_{\mathrm{k}}^{\mathrm{f}})|i⟩,$$

(7.88)

which can be solved using the procedure introduced in the previous section.

While most of the technical aspects concerning the application of the model will be covered in the following, we highly recommend to read at least the formalism and implementation section of Ref. 634 before using the model.

(B) Usage
The main switch for the state-specific equilibrium solvation (SS-PCM) is the variable EQSOLV in the $pcm block. Setting it to TRUE will cause the SCF+ADC calculation to be carried out using the solvent field of the first excited state (if that is found on disk), while any integer $>1$ triggers the automatic solvent-field optimization and is interpreted as the maximum number of steps. We recommend to use EQSOLV = 15. Note that to use the SS-PCM, the PCM (SOLVENT_METHOD = PCM in $rem) and its non-equilibrium functionalities (NONEQUILIBRIUM = TRUE in $pcm ) have to be activated as well. Since the solvent-field iterations require an initial guess, a SS-PCM calculation is always the second (or third…) step of a multi-step job.

Note:  Any SS-PCM calculation requires a preceding converged ptSS-PCM calculation (i.e., EQSOLV = FALSE) for the desired state to provide a guess for the initial solvent field, or it will crash during the SCF.

To create the input-file for a multi step job, add "@@@" at the end of the input for the first job and append the input for the second job. See also section 3.6. Note that the solvent field used in the subsequent step is stored in the basis of the molecular-surface elements and thus, the geometry of the molecule as well as parameters that affect construction and discretization of the molecular cavity must not be changed between the jobs/steps. This, however, is not enforced.

The state for which the solvent field is to be optimized is specified using the variable EQSTATE in the $pcm block. A value of 0 refers to the MP ground-state (for PTED calculations), 1 selects the energetically lowest excited state (default), 2 the second lowest, and so on. The solvent field can be optimized for any singlet, triplet or spin-flip excited state. However, only the desired class of states should be requested, i.e., either EE_SINGLETS or EE_TRIPLETS for singlet references, and either EE_STATES or SF_STATES for triplet references. To compute, e.g., the phosphorescence energy, only triplet states should be requested and EQSTATE would typically be set to 1.

Note:  Computing multiple classes of excited states during the solvent-field iterations will confuse the state-ordering logic of the program and yield the wrong results.

Convergence is controlled by EQS_CONV. Criteria are the SCF energy as well as the RMSD, MAD and largest single difference of the surface-charges. The convergence should not need to be modified. It is per default derived from the SCF convergence parameter (SCF_CONVERGENCE$-4$). The default value of 4 (since SCF_CONVERGENCE is 8 for ADC calculations) corresponds to an maximum energy change of $2.72$ meV and will yield converged total energies for all states. Excitation energies and in particular the total energy of the reference state converge somewhat faster than the SCF energy, and a value of 3 may save some time for the computation e.g. the emission energy of large solutes.

A typical ADC/SS-PCM calculation consists of three steps/subsequent jobs:

1. 1.

   Generation of an initial guess for the solvent field using a non-equilibrium calculation (EQSTATE = FALSE). To save time, this would typically be done with a smaller basis set and lower convergence criteria (e.g. ADC_DAVIDSON_CONVERGENCE 4).

2. 2.

   Converging the solvent field for the desired state. In this step it is advisable to compute as few states as possible (maybe one more than EQSTATE to be aware of looming state crossings), and more importantly, only the desired class of excited states.

3. 3.

   Computing all excited states of interest and their properties in the previously converged solvent field. If the reference state of the solvent-field equilibration is a singlet state, this is straightforward and any number of singlet/triplet states can be computed in this final step without further input. If singlet states are to be computed in the solvent field of a triplet state, the additional variable EQS_REF in $pcm has to be set to tell the program which state is to be treated as the reference state in this last step. For this purpose, EQS_REF is set to the desired triplet state plus the number of converged singlet states. Hence, assuming 2 singlet states are to be calculated along with the triplets in previously converged solvent field of $T_1$, and furthermore that both singlets converge, EQS_REF needs to be set to 3 (this can not be realized using the variable EQSTATE, because we still want to use the solvent field of the lowest triplet state stored in the previous step).

The self-consistent SS-PCM can also be used to compute a fully consistent solvent field for the MP2 ground state by setting EQSTATE to 0. This is known as ptSS(PTED) approach and can improve vertical excitation energies if there are large differences between the electrostatic properties of the SCF and MP ground states (large PTD corrections). In most cases, however, the non-iterative PTD approach is a very good approximation to the PTED approach (see the sample jobs below). The output of a PTED calculation is essentially identical to that of a ptSS calculation.

The program possesses limited intelligence in detecting the type of the calculation (PTED or EQS/SS-PCM) as well as the target state of the solvent-field equilibration and will assemble and designate the ptSS corrections, total energies and transition energies accordingly. This logic will be confused if multiple classes of states (e.g. singlet and triplets) are computed simultaneously during the iterations, and/or if the ordering of the states changes. Nevertheless, in a final job for a previously converged solvent field of a singlet reference singlet and triplet states can be computed together yielding the correct output. However, if singlet states are computed in the converged solvent field of a triplet reference the additional variable EQS_REF has to be set (and only then, see above).

In the “HF/MP2/MP3 Summary” section, zeroth (without ptSS) and first (with ptSS) order total energies of the respective ground state in the solvent field of the target state are given along with the ptSS term for a vertical transition from the equilibrated state (emission). Note that to obtain the MP3 ground state energy during an ADC(3)/EQS calculation, the ptSS term has to be added manually (it is printed in the MP2 Summary since we use the MP2 density for this purpose).

In the “Excited-State Summary” section the reference state is distinguished from the out-of-equilibrium states. Respective total and transition energies are given along with the non-equilibrium corrections, transition moments and some remarks. Note that for emission, in contrast to absorption, the ptSS term increases the transition energy as it lowers the energy of the out-of-equilibrium ground state. The so-called "self-ptSS term" is a perturbative estimate of how much the solvent field of this state is off from its equilibrium. Although the line in the output changes depending on the value (from "not converged" to "reasonably converged" to "converged") it is not used in the actual convergence check. Note that the self-ptSS term is computed with $n^2$ (dielectric_infi) and not $ϵ$, as it probably should be. The self-ptSS term may be used to judge how well a solvent field computed with a different methodology (basis, ADC order/variant) fits. In such a case, values  eV signal a reasonable agreement.

To calculate inter-state transition moments for excited-state absorption, the variable ADC_PROP_ES2ES has to be set to TRUE. Unfortunately, transition energies have to be computed manually from the (first-order) total energies given in the excited-state summary, since the transition energies given along with the state-to-state transition moments following the excite-state summary are incorrect (missing non-equilibrium terms).

The progress of the solvent-field iteration and their convergence is reported following the “Excited-State Summary”.

(C) Tips and Tricks
To compute fluorescence and phosphorescence energies, solute geometry AND solvent field should both be optimized for the suspected emitting state. Since hardly any programs can perform excited-state optimizations with the SS-PCM solvent models, you will probably have to use gas-phase geometries. In our experience, the errors introduced by this approximation are small to negligible (typically  eV) in non-polar solvents, but can become significant in polar solvents, in particular for polar charge-transfer states.

Concerning the predicted emission energies, we found that ADC(2)/SS-PCM typically over-stabilizes CT states, yielding emission energies that are too low. SOS-ADC(2) can improve this error, but does not eliminate it. In general, while emission energies are more accurate with (SOS-)ADC(2)/SS-PCM than with ADC(3)/SS-PCM, the latter affords better relative state energies (see Ref. 634).

Keep in mind that the solute-solvent interaction of polar solvents with polar (e.g. charge-transfer) states can become quite large (multiple eV), and may thus affect the ordering and nature of the excited states. This is quite typical for charge-transfer states even in remotely polar solvents. If they are not the lowest state in the non-equilibrium calculation, but say $S_2$, it is typically necessary to do one solvent-field iteration for the CT state (EQSOLV = 1 and EQSTATE = 2), which brings the CT down to become $S_1$, and then continue the iterations with EQSOLV = 15 and EQSTATE = 1. It is in general advisable to carefully check if the character and/or energetic ordering of the states changes during the procedure, in particular for any equilibration of the solvent field for all but the lowest excited state (e.g. $S_2$ or $S_3$). But even the solvent-field equilibration for a weakly polar $S_1$ in a polar solvent can cause a more polar state with the same dipole-vector to become the lowest state.

If the excited-states swap during the procedure, find out in which step the swap occurred and do only so many iterations (i.e., set EQSOLV accordingly). In a following job, adjust the variable EQSTATE and continue the iterations. If states start to mix when they get close, it might help to first set an artificially large dielectric constant to induce the change in ordering, and then continue with the desired dielectric constant in a following job.

If performance is critical, the calculations may be accelerated by lowering the ADC convergence during the solvent-field iterations (set ADC_DAVIDSON_CONVERGENCE = 5). The number of iterations may be reduced by first converging the solvent field with a smaller basis/at a lower level of ADC followed by another job with the full basis/level of ADC. However, in our experience ADC(2) and ADC(3) solvent fields for the same state differ quite significantly and the approach probably does not help much. In contrast, the solvent field computed with a smaller basis (e.g. 3-21G, SVP) is often a good approximation for that computed with a larger basis (e.g. def2-TZVP, see examples), such this may actually help. It is in general advisable to compute just as many states as is necessary during the solvent-field iterations and include higher lying excited states and triplets in the final job. In ADC(2) calculations for large systems, one should always employ the resolution-of-the-identity approximation.

To save time in PTED jobs, it is suggested to disable the computation of excited states during the solvent-field iterations of a PTED job by setting EE_SINGLETS (and/or EE_TRIPLETS/ EE_STATES) to 0 and compute the excited-states in a final job once the reaction field is converged.

For more tips and examples see the sample jobs.