12.6 Second-Generation ALMO-EDA Method 12.6.3 Polarization Energy with a Well-Defined Basis Set Limit 12.6.5 Job Control for EDA2

12.6.4 Further Decomposition of the Frozen Energy

(September 1, 2024)

The frozen interaction energy in ALMO-EDA is defined as the energy difference between the unrelaxed frozen (Heitler-London) wave function and the isolated fragments. In other literature (e.g. Ref. 555 Hopffgarten M., Frenking G.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2012), 2, pp. 43. Link ), this interaction is often decomposed in a classical fashion:

\Delta E_{\mathrm{frz}}=\Delta E_{\mathrm{elec}}^{\mathrm{cls}}+\Delta E_{% \mathrm{Pauli}}^{\mathrm{cls}},

(12.5)

where the contribution from permanent electrostatics is defined as the Coulomb interaction between isolated fragment charge distributions:

\Delta E_{\mathrm{elec}}^{\mathrm{cls}}=\sum_{A<B}\int_{r_{1}}\int_{r_{2}}\rho% _{A}^{\mathrm{tot}}({\textbf{r}_{1}})\frac{1}{{\textbf{r}_{12}}}\rho_{B}^{% \mathrm{tot}}({\textbf{r}_{2}})\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}

(12.6)

and the remainder constitutes the Pauli (or exchange) term. Such a decomposition (referred to as the classical decomposition below) is associated with two issues: (i) the evaluation of permanent electrostatics makes use of the “promolecule" state (whose density is the simple sum of monomer densities) rather than a properly anti-symmetrized wave function; (ii) when dispersion-corrected density functionals are used, the Pauli term contains dispersion interaction and thus loses its original meaning.

Horn et al. ⁵⁵⁸ Horn P. R., Mao Y., Head-Gordon M.
J. Chem. Phys.
(2016), 144, pp. 114107. Link proposed a new scheme to further decompose the frozen term into contributions from permanent electrostatics (ELEC), Pauli repulsion (PAULI) and dispersion (DISP):

\Delta E_{\mathrm{frz}}=\Delta E_{\mathrm{elec}}+\Delta E_{\mathrm{Pauli}}+% \Delta E_{\mathrm{disp}}

(12.7)

This approach is compatible with the use of all kinds of density functionals except double hybrids, and all three components of the FRZ term are computed with the antisymmetrized frozen wave function. The key step of this method is the orthogonal decomposition of the 1PDM associated with the frozen wave function into contributions from individual fragments: $\mathbf{P}_{\mathrm{frz}}=\sum_{A}\tilde{\mathbf{P}}_{A}$ . This is achieved by minimizing an objective function as follows:

\Omega=\sum_{A}E_{A}[\tilde{\mathbf{P}}_{A}]-E_{A}[\mathbf{P}_{A}]

(12.8)

while interfragment orthogonality is enforced between $\tilde{\mathbf{P}}_{A}$ s. The readers are referred to Ref. 558 Horn P. R., Mao Y., Head-Gordon M.
J. Chem. Phys.
(2016), 144, pp. 114107. Link for more details about the orthogonal decomposition.

The ELEC term is then defined as the Coulomb interaction between distorted fragment densities ( $\tilde{\rho}_{A}(\mathbf{r})$ ):

\Delta E_{\mathrm{elec}}=\sum_{A<B}\int_{r_{1}}\int_{r_{2}}\tilde{\rho}_{A}^{% \mathrm{tot}}({\textbf{r}_{1}})\frac{1}{{\textbf{r}_{12}}}\tilde{\rho}_{B}^{% \mathrm{tot}}({\textbf{r}_{2}})\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}.

(12.9)

The DISP term is evaluated by subtracting the dispersion-free part of the total exchange-correlation (XC) interaction, where an auxiliary “dispersion-free" XC (DFXC) functional is used in company with the primary (target) XC functional:

\Delta E_{\mathrm{disp}}=\left(E_{\mathrm{xc}}[\mathbf{P}_{\mathrm{frz}}]-\sum% _{A}E_{\mathrm{xc}}[\tilde{\mathbf{P}}_{A}]\right)-\left(E_{\mathrm{xc}}^{% \mathrm{DF}}[\mathbf{P}_{\mathrm{frz}}]-\sum_{A}E_{\mathrm{xc}}^{\mathrm{DF}}[% \tilde{\mathbf{P}}_{A}]\right).

(12.10)

It has been suggested ⁸³⁸ Mao Y. et al.
Annu. Rev. Phys. Chem.
(2021), 72, pp. 641. Link that HF is an appropriate DFXC for use with dispersion-inclusive or dispersion-corrected hybrid functionals such as $\omega$ B97M-V or B3LYP-D3, while revPBE is appropriate for semi-local functionals such as B97M-V. However, more recent work suggests that for hybrid GGA functionals (e.g., PBE0-D3), the correlation energy should be included in the DFXC, which would then be HF-PBE rather than HF alone. ⁴⁴⁸ Gray M., Herbert J. M.
Annu. Rep. Comput. Chem.
(2024), 20, pp. 1. Link This avoids putting the correlation contribution to electrostatics and Pauli repulsion into the dispersion term, and affords better agreement with dispersion energies from symmetry-adapted perturbation theory.

The remainder of the frozen interaction goes into the PAULI term, which includes the net repulsive interaction given by Eq. (12.8) and the “dispersion-free" part of the XC interaction:

\Delta E_{\mathrm{Pauli}}=\sum_{A}(E_{A}[\tilde{\mathbf{P}}_{A}]-E_{A}[\mathbf% {P}_{A}])+\left(E_{\mathrm{xc}}^{\mathrm{DF}}[\mathbf{P}_{\mathrm{frz}}]-\sum_% {A}E_{\mathrm{xc}}^{\mathrm{DF}}[\tilde{\mathbf{P}}_{A}]\right).

(12.11)

The PAULI term and the ELEC term can also be combined together and reported as the dispersion-free frozen (DFFRZ) term if desired.

In Q-Chem’s implementation of “EDA2", the classical frozen decomposition and the new scheme defined by Eqs. (12.9)–(12.11) are both computed by default. The classical ELEC term only depends on monomer properties and the distances between fragments, therefore, it can be particularly useful for scenarios such as force field development (as the reference for permanent electrostatics). When the DISP term calculated by the new scheme is available, a modified classical Pauli term ⁸³⁵ Mao Y. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 5422. Link is also reported, which is simply defined as

\Delta E_{\mathrm{Pauli}}^{\mathrm{mod}}=\Delta E_{\mathrm{Pauli}}^{\mathrm{% cls}}-\Delta E_{\mathrm{disp}},

(12.12)

i.e., the dispersion contribution is removed from the classical Pauli term computed using its original definition. The overall decomposition of the frozen energy with the classical scheme is given by

\Delta E_{\mathrm{frz}}=\Delta E^{\mathrm{cls}}_{\mathrm{elec}}+\Delta E^{% \mathrm{mod}}_{\mathrm{Pauli}}+\Delta E_{\mathrm{disp}}

(12.13)

Alternatively, this can also be achieved without performing the orthogonal decomposition, by setting EDA_CLS_DISP to TRUE. This also evaluates the DISP term via Eq. (12.10) except that undistorted monomer densities ( $\{\mathbf{P}_{A}\}$ ) are used instead of their distorted counterparts ( $\{\tilde{\mathbf{P}}_{A}\}$ ):

\Delta E_{\mathrm{disp}}=\left(E_{\mathrm{xc}}[\mathbf{P}_{\mathrm{frz}}]-\sum% _{A}E_{\mathrm{xc}}[\mathbf{P}_{A}]\right)-\left(E_{\mathrm{xc}}^{\mathrm{DF}}% [\mathbf{P}_{\mathrm{frz}}]-\sum_{A}E_{\mathrm{xc}}^{\mathrm{DF}}[\mathbf{P}_{% A}]\right).

(12.14)

FRZ_ORTHO_DECOMP

FRZ_ORTHO_DECOMP
       Perform the decomposition of frozen interaction energy based on the orthogonal decomposition of the 1PDM associated with the frozen wave function.
TYPE:
       BOOLEAN
DEFAULT:
       FALSE (automatically set to TRUE by EDA2 options 1–5)
OPTIONS:
       FALSE Do not perform the orthogonal decomposition. TRUE Perform the frozen energy decomposition using orthogonal fragment densities.
RECOMMENDATION:
       Use default value automatically set by “EDA2". Note that users are allowed to turn off the orthogonal decomposition by setting FRZ_ORTHO_DECOMP to $-1$ . Also, for calculations that involve ECPs, it is automatically set to FALSE since unreasonable results will be produced otherwise.

FRZ_ORTHO_DECOMP_CONV

FRZ_ORTHO_DECOMP_CONV
       Convergence criterion for the minimization problem that gives the orthogonal fragment densities.
TYPE:
       INTEGER
DEFAULT:
       6
OPTIONS:
       $n$ $10^{-n}$
RECOMMENDATION:
       Use the default unless tighter convergence is preferred.

EDA_CLS_ELEC

EDA_CLS_ELEC
       Perform the classical decomposition of the frozen term.
TYPE:
       BOOLEAN
DEFAULT:
       FALSE (automatically set to TRUE by EDA2 options 1–5)
OPTIONS:
       FALSE Do not compute the classical ELEC and PAULI terms. TRUE Perform the classical decomposition.
RECOMMENDATION:
       TRUE

EDA_CLS_DISP

EDA_CLS_DISP
       Compute the DISP contribution without performing the orthogonal decomposition, which will then be subtracted from the classical PAULI term.
TYPE:
       BOOLEAN
DEFAULT:
       FALSE
OPTIONS:
       FALSE Use the DISP term computed with orthogonal decomposition (if available). TRUE Use the DISP term computed using undistorted monomer densities.
RECOMMENDATION:
       Set it to TRUE when orthogonal decomposition is not performed.

DISP_FREE_X

DISP_FREE_X
       Specify the employed “dispersion-free" exchange functional.
TYPE:
       STRING
DEFAULT:
       HF
OPTIONS:
       Exchange functionals (e.g. revPBE) or exchange-correlation functionals (e.g. B3LYP) supported by Q-Chem.
RECOMMENDATION:
       HF is recommended for hybrid (primary) functionals (e.g. $\omega$ B97X-V) and revPBE for semi-local ones (e.g.B97M-V). Other reasonable options (e.g. B3LYP for B3LYP-D3) can also be applied.

DISP_FREE_C

DISP_FREE_C
       Specify the employed “dispersion-free" correlation functional.
TYPE:
       STRING
DEFAULT:
       NONE
OPTIONS:
       Correlation functionals supported by Q-Chem.
RECOMMENDATION:
       Put the appropriate correlation functional paired with the chosen exchange functional (e.g. put PBE if DISP_FREE_X is revPBE); put NONE if DISP_FREE_X is set to an exchange-correlation functional.

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12.6.4 Further Decomposition of the Frozen Energy