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# 11.5.4 Pairwise Fragment Energy Decomposition and Pairwise Fragment Excited-State Energy Decomposition Analysis

(July 14, 2022)

Decomposition of the interaction energy of the QM and EFP regions in the energy components and in the contributions of individual solvent molecules is available for the ground and excited states. The ground state QM/EFP energy is decomposed as:

 \displaystyle\begin{aligned} \displaystyle E_{\text{QM--EF, gr}}&\displaystyle% =E_{\text{elec}}^{(1)}+E_{\text{pol-solute}}^{(0)}+E_{\text{pol-solute}}^{(1)}% +E_{\text{pol-frag}}+E_{\text{QM--EFP}}^{\text{disp}}+E_{\text{QM--EF}}^{\text% {ex-rep}}\\ &\displaystyle=\langle\Psi_{\text{gr}}^{0}|\hat{V}^{\text{Coul}}|\Psi_{\text{% gr}}^{0}\rangle+\Big{[}\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{H}^{\text{QM}% }|\Psi_{\text{gr}}^{\text{sol}}\rangle-\langle\Psi_{\text{gr}}^{0}|\hat{H}^{% \text{QM}}|\Psi_{\text{gr}}^{0}\rangle\Big{]}\\ &\displaystyle\qquad+\Big{[}\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{V}^{% \text{Coul}}|\Psi_{\text{gr}}^{\text{sol}}\rangle-\langle\Psi_{\text{gr}}^{0}|% \hat{V}^{\text{Coul}}|\Psi_{\text{gr}}^{0}\rangle\Big{]}+\Big{[}E_{\text{QM--% EF, gr}}^{\text{pol}}+\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{V}^{\text{pol}% }|\Psi_{\text{gr}}^{\text{sol}}\rangle\Big{]}\\ &\displaystyle\qquad+E_{\text{QM-EF}}^{\text{disp}}+E_{\text{QM--EF}}^{\text{% ex-rep}}\end{aligned} (11.79)

where the terms (from left to right) mean the first-order electrostatic energy, solute polarization energy of the zero- and first orders, solvent polarization energy, and additive dispersion and exchange-repulsion terms. Superscripts “sol” and “0” denote QM wavefunction optimized in a solvent and gas phase, respectively. Each of the integrals involving $\hat{V}^{\text{Coul}}$ and $\hat{V}^{\text{pol}}$ operators can be decomposed into individual fragment contributions, e.g.,

 $E_{\text{elec}}^{(1)}=\langle\Psi_{\text{gr}}^{0}|\hat{V}^{\text{Coul}}|\Psi_{% \text{gr}}^{0}\rangle=\sum_{A}^{\text{fragments}}\langle\Psi_{\text{gr}}^{0}|% \sum_{k\in A}\hat{V}_{k}^{\text{Coul}}|\Psi_{\text{gr}}^{0}\rangle$ (11.80)

and similarly for the other terms. Polarization energy can be approximately decomposed into individual fragment contributions as:

 $E_{\text{QM--EF, gr}}^{\text{pol}}=\frac{1}{2}\sum_{A}^{\text{fragments}}\sum_% {p\in A}(-\mu^{p}F^{\text{ai,nuc,p}}+{\bar{\mu}^{p}F^{\text{ai,elec},p}})$ (11.81)

where $p$ are polarizability expansion points. Dispersion and exchange-repulsion terms are also pairwise-additive.

The only term that cannot be similarly split into fragment contributions is the zero-order solute polarization energy:

 $E_{\text{pol-solute}}^{(0)}=\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{H}^{% \text{QM}}|\Psi_{\text{gr}}^{\text{sol}}\rangle-\langle\Psi_{\text{gr}}^{0}|% \hat{H}^{\text{QM}}|\Psi_{\text{gr}}^{0}\rangle\;.$ (11.82)

This term is referred to as "non-separable term" in the output printout. From perturbation theory, this term is expected to be about twice smaller and of the opposite sign than the first-order solute polarization term:

 $E_{\text{pol-solute}}^{(1)}=\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{V}^{% \text{Coul}}|\Psi_{\text{gr}}^{\text{sol}}\rangle-\langle\Psi_{\text{gr}}^{0}|% \hat{V}^{\text{Coul}}|\Psi_{\text{gr}}^{0}\rangle\;.$ (11.83)

Application of the energy decomposition analysis to the electronically excited states is described below. The zero-order total solvatochromic shift can be represented as:

 $E_{\text{solv}}^{\text{QM/EFP}}=\sum_{A}^{\text{fragments}}\big{(}\Delta E_{% \text{ex/gr}}^{\text{elec(1),A}}+\Delta E_{\text{ex/gr}}^{\text{pol-solute(1)}% ,A}+\Delta E_{\text{ex/gr}}^{\text{pol-frag(1)},A}\big{)}+\Delta E_{\text{ex/% gr}}^{\text{pol-solute(0)},A}\;.$ (11.84)

The various terms are defined as

 $\displaystyle\Delta E_{\text{ex/gr}}^{\text{elec(1)},A}$ $\displaystyle=\sum_{k\in A}\big{(}\langle\Psi_{\text{ex}}^{0}|\hat{V}_{k}^{% \text{Coul}}|\Psi_{\text{ex}}^{0}\rangle-\langle\Psi_{\text{gr}}^{0}|\hat{V}_{% k}^{\text{Coul}}|\Psi_{\text{gr}}^{0}\rangle\big{)}$ (11.85) $\displaystyle\Delta E_{\text{ex/gr}}^{\text{pol-solute(1)},A}$ $\displaystyle=\sum_{k\in A}\big{(}\langle\Psi_{\text{ex}}^{\text{sol}}|\hat{V}% _{k}^{\text{Coul}}|\Psi_{\text{ex}}^{\text{sol}}\rangle-\langle\Psi_{\text{ex}% }^{0}|\hat{V}_{k}^{\text{Coul}}|\Psi_{\text{ex}}^{0}\rangle-\langle\Psi_{\text% {gr}}^{\text{sol}}|\hat{V}_{k}^{\text{Coul}}|\Psi_{\text{gr}}^{\text{sol}}% \rangle+\langle\Psi_{\text{gr}}^{0}|\hat{V}_{k}^{\text{Coul}}|\Psi_{\text{gr}}% ^{0}\rangle\big{)}$ (11.86) $\displaystyle\Delta E_{\text{ex/gr}}^{\text{pol-frag(1)},A}$ $\displaystyle=\sum_{p\in A}\big{(}\langle\Psi_{\text{ex}}^{\text{sol}}|\hat{V}% _{p,\text{gr}}^{\text{pol}}|\Psi_{\text{ex}}^{\text{sol}}\rangle-\langle\Psi_{% \text{gr}}^{\text{sol}}|\hat{V}_{p,\text{gr}}^{\text{pol}}|\Psi_{\text{gr}}^{% \text{sol}}\rangle\big{)}$ (11.87) $\displaystyle\Delta E_{\text{ex/gr}}^{\text{pol-solute(0)},A}$ $\displaystyle=\langle\Psi_{\text{ex}}^{\text{sol}}|\hat{H}_{\text{QM}}|\Psi_{% \text{ex}}^{\text{sol}}\rangle-\langle\Psi_{\text{ex}}^{0}|\hat{H}_{\text{QM}}% |\Psi_{\text{ex}}^{0}\rangle-\langle\Psi_{\text{gr}}^{\text{sol}}|\hat{H}_{% \text{QM}}|\Psi_{\text{gr}}^{\text{sol}}\rangle+\langle\Psi_{\text{gr}}^{0}|% \hat{H}_{\text{QM}}|\Psi_{\text{gr}}^{0}\rangle\;.$ (11.88)

Fragment contribution of the perturbative polarization correction to the excited states [Eq.  (11.78)] can be obtained as follows:

 $\Delta E_{\text{pol},A}=\frac{1}{2}\sum_{p\in A}\Bigl{[}-(\mu_{\rm{ex}}^{p}-% \mu_{\rm{gr}}^{p})(F^{\text{mult},p}+F^{\text{nuc},p})+(\tilde{\mu}_{\text{ex}% }^{p}F_{\text{ex}}^{\text{ai},p}-\tilde{\mu}_{\text{gr}}^{p}F_{\text{gr}}^{% \text{ai},p})-(\mu_{\text{ex}}^{p}-\mu_{\text{gr}}^{p}+\tilde{\mu}_{\text{ex}}% ^{p}-\tilde{\mu}_{\text{gr}}^{p})F_{\text{ex}}^{\text{ai},p}\Bigr{]}$ (11.89)

where $A$ is a fragment of interest.

The energy is decomposed separately for all computed excited states. The excited state analysis is implemented for CIS/TD-DFT and EOM-CCSD methods both in ccman and ccman2. Energy decomposition analysis is activated by keyword EFP_PAIRWISE. Both ground and excited state energy decompositions are conducted in two steps, controlled by keyword EFP_ORDER. In the first step (EFP_ORDER = 1), the first-order electrostatic energy and $\langle\Psi_{\text{gr}}^{0}|\hat{H}^{\text{QM}}|\Psi_{\text{gr}}^{0}\rangle$ (or $\langle\Psi_{\text{ex}}^{0}|\hat{H}^{\text{QM}}|\Psi_{\text{ex}}^{0}\rangle$ for the excited states) part of the non-separable term are computed and printed. In the second step (EFP_ORDER = 2), the remaining terms are evaluated. Thus, for a complete analysis, the user is required to conduct two consequent simulations with EFP_ORDER set to 1 and 2, respectively. Table 11.8 shows notations used in the output to denote various terms in Eqs.  (11.79)–(11.89).