11.5.3 Excited-State Calculations with EFP

Interface of EFP with EOM-CCSD (both via CCMAN and CCMAN2), CIS, CIS(D), and TDDFT has been developed.^{1046, 588} In the EOM-CCSD/EFP calculations, the reference-state CCSD equations for the $T$ cluster amplitudes are solved with the HF Hamiltonian modified by the electrostatic and polarization contributions due to the effective fragments, Eq. (11.63). In the coupled-cluster calculation, the induced dipoles of the fragments are frozen at their HF values.

The transformed Hamiltonian $\overline{H}$ effectively includes Coulomb and polarization contributions from the EFP part. As $\overline{H}$ is diagonalized in an EOM calculation, the induced dipoles of the effective fragments are frozen at their reference state value, i.e., the EOM equations are solved with a constant response of the EFP environment. To account for solvent response to electron rearrangement in the EOM target states (i.e., excitation or ionization), a perturbative non-iterative correction is computed for each EOM root as follows. The one-electron density of the target EOM state (excited or ionized) is calculated and used to re-polarize the environment, i.e., to recalculate the induced dipoles of the EFP part in the field of an EOM state. These dipoles are used to compute the polarization energy corresponding to this state.

The total energy of the excited state with inclusion of the perturbative response of the EFP polarization is:

where $E_{\mathrm{EOM}}$ is the energy found from EOM-CCSD procedure and $\mathrm{\Delta }{E}_{\mathrm{pol}}$ has the following form:

where $F_{\mathrm{gr}}^{\mathrm{ai}}$ and $F_{\mathrm{ex}}^{\mathrm{ai}}$ are the fields due to the reference (HF) state and excited-state electronic densities, respectively. ${\mu }_{\mathrm{gr}}^k$ and ${\tilde{\mu }}_{\mathrm{gr}}^k$ are the induced dipole and conjugated induced dipole at the distributed polarizability point $k$ consistent with the reference-state density, while ${\mu }_{\mathrm{ex}}^k$ and ${\tilde{\mu }}_{\mathrm{ex}}^k$ are the induced dipoles corresponding to the excited state density.

The first two terms in Eq. (11.81) provide a difference of the polarization energy of the QM/EFP system in the excited and ground electronic states; the last term is the leading correction to the interaction of the ground-state-optimized induced dipoles with the wave function of the excited state.

The EOM states have both direct and indirect polarization contributions. The indirect term comes from the orbital relaxation of the solute in the field due to induced dipoles of the solvent. The direct term given by Eq. (11.81) is the response of the polarizable environment to the change in solute’s electronic density upon excitation. Note that the direct polarization contribution can be very large (tenths of eV) in EOM-IP/EFP since the electronic densities of the neutral and the ionized species are very different.

An important advantage of the perturbative EOM/EFP scheme is that it does not compromise multi-state nature of EOM and that the electronic wave functions of the target states remain orthogonal to each other since they are obtained with the same (reference-state) field of the polarizable environment. For example, transition properties between these states can be calculated.

EOM-CC/EFP scheme works with any type of the EOM excitation operator $R_k$ currently supported in Q-Chem, i.e., spin-flipping (SF), excitation energies (EE), ionization potential (IP), electron affinity (EA) (see Section 7.10.18 for details). However, direct polarization correction requires calculation of one-electron density of the excited state, and will be computed only for the methods with implemented one-electron properties.

Implementation of CIS/EFP, CIS(D)/EFP, and TDDFT/EFP methods is similar to the implementation of EOM/EFP. Polarization correction as in Eq. 11.81 is calculated and added to the CIS or TDDFT excitation energies.