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:
where the contribution from permanent electrostatics is defined as the Coulomb interaction between isolated fragment charge distributions:
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
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: . This is achieved by minimizing an objective function as follows:
while interfragment orthogonality is enforced between ’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 ():
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:
It is suggested that HF is an appropriate DFXC to be used for dispersion-corrected hybrid functionals (e.g. 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:
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 term621 is also reported, which is simply defined as
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
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 () are used instead of their distorted counterparts ():