# 12.7.5 ALMO-EDA with Implicit Solvent Models

In real-life applications, it is often desired to perform EDA under a solvent environment. Q-Chem 5.2 has made EDA2 compatible with PCM and SMD models. In these EDA calculations, the interaction energy to be decomposed is the energy difference between the solvated, fully relaxed complex and the individually solvated non-interacting fragments, which can still be partitioned into contributions from frozen interaction (FRZ), polarization (POL), and charge transfer (CT):Mao:2020b

 \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)

Unlike many other EDA schemes in which the solvent contribution is treated as a correction to the gas-phase EDA results, in ALMO-EDA the effect of solvation is accounted for in the preparation of each intermediate state. The frozen interaction energy ($\Delta E_{\mathrm{FRZ}}$) is defined as the energy change upon the formation of a solvated complex from several 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{v}})-% \sum_{A}(E_{A}^{\mathrm{s}}-E_{A}^{\mathrm{v}})+E_{\mathrm{FRZ}}^{\mathrm{v}}-% \sum_{A}E_{A}^{\mathrm{v}}\\ &\displaystyle=(\Delta E_{\mathrm{SOL}}+\Delta E^{\mathrm{v}}_{\mathrm{ELEC}})% +\Delta E^{\mathrm{v}}_{\mathrm{PAULI}}+\Delta E^{\mathrm{v}}_{\mathrm{DISP}}% \\ &\displaystyle=\Delta E^{\mathrm{s}}_{\mathrm{ELEC}}+\Delta E^{\mathrm{v}}_{% \mathrm{PAULI}}+\Delta E^{\mathrm{v}}_{\mathrm{DISP}}\end{aligned} (12.16)

Note that the superscript “s" denotes intermediate state energies evaluated with the solvent contribution taken into account. We have also introduced a new term,

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

to describe the loss/gain of solvation energy upon the formation of the frozen complex. The other three terms in eq. 12.16, $\Delta E^{\mathrm{v}}_{\mathrm{ELEC}})$, $\Delta E^{\mathrm{v}}_{\mathrm{PAULI}}$, and $\Delta E^{\mathrm{v}}_{\mathrm{DISP}}$, are evaluated in the same way as in vacuum (as indicated by the superscripts “v") but with MOs of solvated fragments. As the existence of solvent mainly affects the electrostatic interaction between interacting moieties, which is exactly true at long range, we suggest that one can combine $\Delta E_{\mathrm{SOL}}$ and $\Delta E^{\mathrm{v}}_{\mathrm{ELEC}})$ (eq. 12.16) and report that as the electrostatic interaction within solvent environment ($\Delta E^{\mathrm{s}}_{\mathrm{ELEC}}$).

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

 $\Delta E_{\mathrm{POL}}^{\mathrm{s}}=E_{\mathrm{POL}}^{\mathrm{s}}-E_{\mathrm{% FRZ}}^{\mathrm{s}}$ (12.18)

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

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

where $E_{\mathrm{Full}}^{\mathrm{s}}$ is the full SCF energy evaluated with the presence of solvent.

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


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