12.6 Second-Generation ALMO-EDA Method 12.6.5 Job Control for EDA2 12.6.7 ALMO-EDA with Non-Aufbau Electronic Configurations

12.6.6 ALMO-EDA with Implicit Solvent Models

(September 1, 2024)

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, ⁸³⁹ Mao Y. et al.
Chem. Sci.
(2021), 12, pp. 1398. Link 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. ⁸³⁹ Mao Y. et al.
Chem. Sci.
(2021), 12, pp. 1398. Link 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 ⁵⁵⁸ Horn P. R., Mao Y., Head-Gordon M.
J. Chem. Phys.
(2016), 144, pp. 114107. Link (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 SCF-MI 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 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})}

(12.20)

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
   SOLVENT_METHOD    pcm
   EDA_CLS_DISP      true
   INTEGRAL_SYMMETRY false
   POINT_GROUP_SYMMETRY false
$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
   SOLVENT_METHOD    smd
   EDA_CLS_DISP      true
   INTEGRAL_SYMMETRY false
   POINT_GROUP_SYMMETRY false
$end

$smx
   solvent water
$end

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12.6.6 ALMO-EDA with Implicit Solvent Models