# 12.7.5 ALMO-EDA with Implicit Solvent Models

(May 16, 2021)

Since the majority of chemical processes occur in the condensed phase, it is often desirable to investigate intermolecular interactions in the presence of solvents. The solvation environment can affect intermolecular interactions in a variety of ways such that the gas-phase ALMO-EDA may not be capable of revealing the physical picture of these interactions correctly. To address this gap, Mao et al. have proposed the ALMO-EDA(solv) approach, which, unlike many other EDA schemes, incorporates the solvation effect in the evaluation of all the energy components. Currently, ALMO-EDA(solv) supports two widely used implicit solvent models: PCM and SMD (see Sec. 11.2). More solvation models will be made compatible in future releases of Q-Chem.

Within the ALMO-EDA(solv) scheme, the interaction energy to be decomposed is given by the energy difference between the solvated, fully relaxed complex and the sum of the energies of individually solvated, non-interacting fragments. As in gas-phase ALMO-EDA, the total interaction energy (INT) can be partitioned into frozen (FRZ), polarization (POL), and charge transfer (CT) contributions:

 \displaystyle\begin{aligned} \displaystyle\Delta E_{\mathrm{INT}}^{(\mathrm{s}% )}&\displaystyle=E_{\mathrm{Full}}^{(\mathrm{s})}-\sum_{A}E_{A}^{(\mathrm{s})}% \\ &\displaystyle=\Delta E^{(\mathrm{s})}_{\mathrm{FRZ}}+\Delta E^{(\mathrm{s})}_% {\mathrm{POL}}+\Delta E^{(\mathrm{s})}_{\mathrm{CT}}\end{aligned} (12.15)

Here the superscript “(s)” indicates that the energetic terms are evaluated with the solvent taken into account.

The frozen interaction energy ($\Delta E_{\mathrm{FRZ}}^{(\mathrm{s})}$) is defined as the energy change upon the formation of a solvated complex from several individually solvated non-interacting fragments without relaxing their orbitals, which can be further decomposed into permanent electrostatics (ELEC), Pauli repulsion (PAULI), and dispersion (DISP) contributions:

 \displaystyle\begin{aligned} \displaystyle\Delta E_{\mathrm{FRZ}}^{(\mathrm{s}% )}&\displaystyle=E_{\mathrm{FRZ}}^{(\mathrm{s})}-\sum_{A}E_{A}^{(\mathrm{s})}% \\ &\displaystyle=(E_{\mathrm{FRZ}}^{(\mathrm{s})}-E_{\mathrm{FRZ}}^{(\mathrm{0})% })-\sum_{A}(E_{A}^{(\mathrm{s})}-E_{A}^{(0)})+E_{\mathrm{FRZ}}^{(0)}-\sum_{A}E% _{A}^{(0)}\\ &\displaystyle=\Delta E_{\mathrm{SOL}}+\Delta E^{(0)}_{\mathrm{ELEC}}+\Delta E% ^{(0)}_{\mathrm{PAULI}}+\Delta E^{(0)}_{\mathrm{DISP}}\end{aligned} (12.16)

Here we have introduced a new term,

 $\Delta E_{\mathrm{SOL}}=(E_{\mathrm{FRZ}}^{(\mathrm{s})}-E_{\mathrm{FRZ}}^{(% \mathrm{0})})-\sum_{A}(E_{A}^{(\mathrm{s})}-E_{A}^{(0)})$ (12.17)

to quantify the loss/gain of solvation energy upon the formation of the frozen complex. The other three terms in Eq. 12.16, $\Delta E^{(0)}_{\mathrm{ELEC}})$, $\Delta E^{(0)}_{\mathrm{PAULI}}$, and $\Delta E^{(0)}_{\mathrm{DISP}}$, are evaluated in the same way as in vacuum (as indicated by the superscripts “(0)") but using MOs of solvated fragments.

In the most general cases, the solvent contribution to the frozen interaction ($\Delta E_{\mathrm{SOL}}$) includes both electrostatic ($\Delta E_{\mathrm{SOL}}^{\mathrm{el}}$) and non-electrostatic ($\Delta E_{\mathrm{SOL}}^{\text{non-el}}$) components, which can be combined with the “gas-phase” ELEC and PAULI terms, respectively. In addition, we ignore the solvent contribution to dispersion, an effect that cannot be captured by dispersion-corrected DFT that ALMO-EDA(solv) is based upon, which leads to $\Delta E^{(0)}_{\mathrm{DISP}}\approx\Delta E^{(\mathrm{s})}_{\mathrm{DISP}}$. The decomposition of the frozen energy in the solvation environment (Eq. 12.16) can thus be rewritten as

 \displaystyle\begin{aligned} \displaystyle\Delta E_{\mathrm{FRZ}}^{(\mathrm{s}% )}&\displaystyle=(\Delta E^{(0)}_{\mathrm{ELEC}}+\Delta E_{\mathrm{SOL}}^{% \mathrm{el}})+(\Delta E^{(0)}_{\mathrm{PAULI}}+\Delta E_{\mathrm{SOL}}^{\text{% non-el}})+\Delta E^{(0)}_{\mathrm{DISP}}\\ &\displaystyle=\Delta E^{(\mathrm{s})}_{\mathrm{ELEC}}+\Delta E^{(\mathrm{s})}% _{\mathrm{PAULI}}+\Delta E^{(\mathrm{s})}_{\mathrm{DISP}}\end{aligned} (12.18)

Starting from the solvated frozen complex, one can relax the fragment orbitals using the SCFMI technique in presence of solvent. The associated energy lowering is defined as the polarization energy in ALMO-EDA(solv) ($\Delta E_{\mathrm{POL}}^{(\mathrm{s})}$):

 $\Delta E_{\mathrm{POL}}^{(\mathrm{s})}=E_{\mathrm{POL}}^{(\mathrm{s})}-E_{% \mathrm{FRZ}}^{(\mathrm{s})}$ (12.19)

where $E_{\mathrm{POL}}^{(\mathrm{s})}$ is the converged SCFMI energy with solvent. Similarly, the charge-transfer term is given by

 $\Delta E_{\mathrm{CT}}^{(\mathrm{s})}=E_{\mathrm{Full}}^{(\mathrm{s})}-E_{% \mathrm{POL}}^{(\mathrm{s})}$ (12.20)

where $E_{\mathrm{Full}}^{(\mathrm{s})}$ is the full SCF energy evaluated with the presence of solvent. With that, the solvation effects are implicitly incorporated in the POL and CT terms produced by the ALMO-EDA(solv) scheme.

Example 12.17  EDA calculation for the water-Mg${}^{2+}$ complex in PCM water.

$molecule 2 1 -- 0 1 H1 H2 H1 1.55618 O1 H2 0.97619 H1 37.14891 -- 2 1 Mg1 O1 scan H2 127.14892 H1 180.0 scan = 1.91035$end

$rem JOBTYPE eda EDA2 2 METHOD wb97m-v BASIS 6-31+g(d) UNRESTRICTED false SCF_ALGORITHM diis SCF_CONVERGENCE 8 MAX_SCF_CYCLES 200 THRESH 14 SYMMETRY false SYM_IGNORE true SOLVENT_METHOD pcm EDA_CLS_DISP true$end

$PCM THEORY CPCM METHOD SWIG SOLVER INVERSION HPOINTS 302 HEAVYPOINTS 302$END

$SOLVENT DIELECTRIC 78.39$END


View output

Example 12.18  EDA calculation for the water-Mg${}^{2+}$ complex in SMD water.

$molecule 2 1 -- 0 1 H1 H2 H1 1.55618 O1 H2 0.97619 H1 37.14891 -- 2 1 Mg1 O1 scan H2 127.14892 H1 180.0 scan = 1.91035$end

$rem JOBTYPE eda EDA2 2 METHOD wb97m-v BASIS 6-31+g(d) UNRESTRICTED false SCF_ALGORITHM diis SCF_CONVERGENCE 8 MAX_SCF_CYCLES 200 THRESH 14 SYMMETRY false SYM_IGNORE true SOLVENT_METHOD smd EDA_CLS_DISP true$end

$smx solvent water$end


View output