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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. 396), this
interaction is often decomposed in a classical fashion:

$$\mathrm{\Delta}{E}_{\mathrm{frz}}=\mathrm{\Delta}{E}_{\mathrm{elec}}^{\mathrm{cls}}+\mathrm{\Delta}{E}_{\mathrm{pauli}}^{\mathrm{cls}},$$ | (12.5) |

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

$$ | (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.* proposed a new scheme to further decompose the frozen term into
contributions from permanent electrostatics (ELEC), Pauli repulsion (PAULI) and
dispersion (DISP):^{399}

$$\mathrm{\Delta}{E}_{\mathrm{frz}}=\mathrm{\Delta}{E}_{\mathrm{elec}}+\mathrm{\Delta}{E}_{\mathrm{pauli}}+\mathrm{\Delta}{E}_{\mathrm{disp}}$$ | (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: ${\mathbf{P}}_{\mathrm{frz}}={\sum}_{A}{\stackrel{~}{\mathbf{P}}}_{A}$. This is achieved by minimizing an objective function as follows:

$$\mathrm{\Omega}=\sum _{A}{E}_{A}[{\stackrel{~}{\mathbf{P}}}_{A}]-{E}_{A}[{\mathbf{P}}_{A}]$$ | (12.8) |

while interfragment orthogonality is enforced between ${\stackrel{~}{\mathbf{P}}}_{A}$’s. The readers are referred to Ref. 399 for more details about the orthogonal decomposition.

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

$$ | (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" (DF) XC functional is used in company with the primary XC functional:

$$\mathrm{\Delta}{E}_{\mathrm{disp}}=\left({E}_{\mathrm{xc}}[{\mathbf{P}}_{\mathrm{frz}}]-\sum _{A}{E}_{\mathrm{xc}}[{\stackrel{~}{\mathbf{P}}}_{A}]\right)-\left({E}_{\mathrm{xc}}^{\mathrm{DF}}[{\mathbf{P}}_{\mathrm{frz}}]-\sum _{A}{E}_{\mathrm{xc}}^{\mathrm{DF}}[{\stackrel{~}{\mathbf{P}}}_{A}]\right).$$ | (12.10) |

It is suggested that HF is an appropriate DFXC to be used for
dispersion-corrected hybrid functionals (*e.g.* $\omega $B97M-V, B3LYP-D3), while
revPBE is appropriate for semi-local functionals (*e.g.* B97M-V).

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:

$$\mathrm{\Delta}{E}_{\mathrm{pauli}}=\sum _{A}({E}_{A}[{\stackrel{~}{\mathbf{P}}}_{A}]-{E}_{A}[{\mathbf{P}}_{A}])+\left({E}_{\mathrm{xc}}^{\mathrm{DF}}[{\mathbf{P}}_{\mathrm{frz}}]-\sum _{A}{E}_{\mathrm{xc}}^{\mathrm{DF}}[{\stackrel{~}{\mathbf{P}}}_{A}]\right).$$ | (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^{621} is
also reported, which is simply defined as

$$\mathrm{\Delta}{E}_{\mathrm{pauli}}^{\mathrm{mod}}=\mathrm{\Delta}{E}_{\mathrm{pauli}}^{\mathrm{cls}}-\mathrm{\Delta}{E}_{\mathrm{disp}},$$ | (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

$$\mathrm{\Delta}{E}_{\mathrm{frz}}=\mathrm{\Delta}{E}_{\mathrm{elec}}^{\mathrm{cls}}+\mathrm{\Delta}{E}_{\mathrm{pauli}}^{\mathrm{mod}}+\mathrm{\Delta}{E}_{\mathrm{disp}}$$ | (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 ($\{{\stackrel{~}{\mathbf{P}}}_{A}\}$):

$$\mathrm{\Delta}{E}_{\mathrm{disp}}=\left({E}_{\mathrm{xc}}[{\mathbf{P}}_{\mathrm{frz}}]-\sum _{A}{E}_{\mathrm{xc}}[{\mathbf{P}}_{A}]\right)-\left({E}_{\mathrm{xc}}^{\mathrm{DF}}[{\mathbf{P}}_{\mathrm{frz}}]-\sum _{A}{E}_{\mathrm{xc}}^{\mathrm{DF}}[{\mathbf{P}}_{A}]\right).$$ | (12.14) |

FRZ_ORTHO_DECOMP

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

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

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

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

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

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.