Despite the huge success and usefulness of today’s most popular EDA methods,
they still face some limitations in their capabilities. For instance, EDAs are
usually performed at complex geometries that are obtained from unconstrained
electronic structure calculations (*e.g.*, optimized equilibrium geometries). For
strongly interacting systems, close intermolecular contacts driven by POL and
particularly CT often result in largely unfavorable FRZ interaction, which
offers little physical insights besides indicating obviously substantial
intermolecular overlap. Another limitation is that the conventional EDA methods
often partitions a “single-point" interaction energy evaluated at a given
geometry. Therefore, the influence of FRZ, POL and CT on the structural and
vibrational properties of an intermolecular complex cannot be directly
characterized.

Recently Mao *et al.* reformulated the original ALMO-EDA method in an
adiabatic picture,^{Mao:2017} where the term “adiabatic" is borrowed from
spectroscopy and indicates that energy differences are evaluated at
relaxed geometry on each potential energy surface (PES). In this scheme, the
total binding energy (including monomer geometry distortions) is repartitioned
into adiabatic FRZ, POL and CT terms:

$$\mathrm{\Delta}{E}_{\mathrm{bind}}=\mathrm{\Delta}{E}_{\mathrm{frz}}^{(\mathrm{ad})}+\mathrm{\Delta}{E}_{\mathrm{pol}}^{(\mathrm{ad})}+\mathrm{\Delta}{E}_{\mathrm{ct}}^{(\mathrm{ad})}.$$ | (12.20) |

The adiabatic frozen interaction energy is given by the difference between the energy minimum of the frozen PES (on which the energy of each point is computed using the corresponding frozen wave function) and the sum of fully relaxed, non-interacting fragment energies:

$$\mathrm{\Delta}{E}_{\mathrm{frz}}^{(\mathrm{ad})}=E[{\mathbf{P}}_{\mathrm{frz}}^{(\mathrm{frz})}]-\sum _{A}{E}_{A}^{(0)}.$$ | (12.21) |

Similarly, the adiabatic POL and CT terms can be obtained by performing geometry optimizations on the polarized (SCFMI) and fully relaxed (unconstrained SCF) PESs:

$\mathrm{\Delta}{E}_{\mathrm{pol}}^{(\mathrm{ad})}$ | $=E[{\mathbf{P}}_{\mathrm{pol}}^{(\mathrm{pol})}]-E[{\mathbf{P}}_{\mathrm{frz}}^{(\mathrm{frz})}],$ | (12.22) | |||

$\mathrm{\Delta}{E}_{\mathrm{ct}}^{(\mathrm{ad})}$ | $=E[{\mathbf{P}}_{\mathrm{full}}^{(\mathrm{full})}]-E[{\mathbf{P}}_{\mathrm{pol}}^{(\mathrm{pol})}].$ | (12.23) |

With this method, the changes in monomer structures and intermolecular coordinates due to FRZ, POL and CT and the accompanied energetics are provided. Moreover, at the energy minimum (or other stationary points) on each PES, the other properties such as multipole points, vibrational frequencies and intensities can also be computed, therefore the effect of different intermolecular interaction components on them can also be characterized.

The geometry optimization on the frozen PES is facilitated by the analytical gradient of the frozen wave function energy implemented in Q-Chem. As for the geometry optimization on the polarized PES, the nuclear gradient of the SCFMI energy has the same form as that of the full SCF energy if the original ALMO model is used. These analytical gradients can also be used for finite difference calculations of harmonic frequencies by setting IDERIV = 1. We note that the analytical gradients of SCFMI calculations that use FERFs are not available yet, and SCFMI_MODE = 0 is required for computing the forces on the frozen and polarized PESs. Also, the current implementation of this method requires users to perform geometry optimization on the three PESs separately (see the example below) and evaluate the energy components by taking several Q-Chem outputs (including geometry optimizations for the monomers) together, which is probably not so convenient. We look forward to extending the functionality of this method and improving its implementation in the near future.

As we mentioned in 12.7.4, for systems containing radicals of highly symmetric geometries, the frozen wavefunction obtained from concatenating the fragment MOs might be non-unique. In those cases, we recommend the user to set EDA_ALIGN_FRGM_SPIN = 1 or 2 when performing geometry optimization on the frozen PES. The job will then go through the fragment spin alignment procedure in each optimization cycle.

To further understand the charge-transfer effects in dative complexes, in Q-Chem
5.2.2 and after, one is allowed to separate the overall CT into contributions from
forward and backward donations using the variational forward-backward (VFB)
approach.^{Loipersberger:2020} Such a decomposition is achieved by introducing
two additional constrained intermediate states in which only one direction of CT is
permitted. These two “one-way” CT states are variationally relaxed such that the
associated nuclear forces can be readily obtained. This allows for a facile integration
into the adiabatic ALMO-EDA scheme introduced above:

$\mathrm{\Delta}{E}_{\mathrm{ctf}}^{(\mathrm{ad})}$ | $=E[{\mathbf{P}}_{\mathrm{ctf}}^{(\mathrm{ctf})}]-E[{\mathbf{P}}_{\mathrm{pol}}^{(\mathrm{pol})}],$ | (12.24) | |||

$\mathrm{\Delta}{E}_{\mathrm{ctb}}^{(\mathrm{ad})}$ | $=E[{\mathbf{P}}_{\mathrm{ctb}}^{(\mathrm{ctb})}]-E[{\mathbf{P}}_{\mathrm{pol}}^{(\mathrm{pol})}],$ | (12.25) |

and thus the molecular property changes arising from forward and backward donations can be separately assigned. Note that in its Q-Chem implementation, the evaluation of a VFB state always follows a polarization (standard SCFMI) calculation. Also, since the definition of VFB states is based on the generalized SCFMI technique (Sec. 12.7.1), SCFMI_MODE = 1 is required.

FRZ_GEOM

Compute forces on the frozen PES.

TYPE:

BOOLEAN

DEFAULT:

FALSE

OPTIONS:

FALSE
Do not compute forces on the frozen PES.
TRUE
Compute forces on the frozen PES.

RECOMMENDATION:

Set it to TRUE when optimized geometry or vibrational frequencies on
the frozen PES are desired.

POL_GEOM

Compute forces on the polarized (converged SCFMI) PES.

TYPE:

BOOLEAN

DEFAULT:

FALSE

OPTIONS:

FALSE
Do not compute forces on the polarized PES.
TRUE
Compute forces on the polarized PES.

RECOMMENDATION:

Set it to TRUE when optimized geometry or vibrational frequencies on
the polarized PES are desired.

VFB_CTA

Use the Variational Forward-Backward (VFB) approach to obtain “one-way” CT PESs.

TYPE:

STRING

DEFAULT:

NONE

OPTIONS:

FORWARD
Allow 1$\to $2 CT only (1 and 2 are two fragments).
BACKWARD
Allow 2$\to $1 CT only.

RECOMMENDATION:

None

$molecule 0 1 -- 0 1 H 0.000000 0.000000 0.000000 H 0.000000 0.000000 1.629090 H 1.417687 0.000000 0.814543 N 0.473683 -0.370067 0.814542 -- 0 1 N 3.494032 -1.531250 0.814538 H 3.967715 -1.901317 -0.000008 H 2.550028 -1.901319 0.814537 H 3.967715 -1.901317 1.629083 $end $rem JOBTYPE opt !optimization on the fully relaxed PES GEN_SCFMAN true METHOD wb97x-d BASIS 6-31+g* XC_GRID 1 THRESH 14 SCF_CONVERGENCE 9 SCF_GUESS fragmo SYMMETRY false SYM_IGNORE true $end @@@ $molecule read $end $rem JOBTYPE opt POL_GEOM true !optimization on the polarized PES GEN_SCFMAN true METHOD wb97x-d BASIS 6-31+g* XC_GRID 1 THRESH 14 SCF_CONVERGENCE 9 SYMMETRY false SYM_IGNORE true SCFMI_MODE 0 $end @@@ $molecule read $end $rem JOBTYPE opt FRZ_GEOM true !optimization on the frozen PES GEN_SCFMAN true METHOD wb97x-d BASIS 6-31+g* XC_GRID 1 THRESH 14 SCF_CONVERGENCE 9 SYMMETRY false SYM_IGNORE true SCFMI_MODE 0 $end

$molecule 1 1 -- 0 1 C 0.0000000000 0.0000000000 1.3792049588 O 0.0000000000 0.0000000000 2.4988670685 -- 1 1 Cu 0.0000000000 0.0000000000 -0.9778656750 $end $rem JOBTYPE opt FRZ_GEOM true METHOD b3lyp BASIS def2-svp UNRESTRICTED false SYMMETRY false SYM_IGNORE false IDERIV 1 FD_MAT_VEC_PROD false $end @@@ $molecule read $end $rem JOBTYPE freq FRZ_GEOM true METHOD b3lyp BASIS def2-svp UNRESTRICTED false SYMMETRY false SYM_IGNORE false IDERIV 1 FD_MAT_VEC_PROD false $end

$molecule 0 1 -- 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 -- 0 1 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561 $end $rem JOBTYPE opt VFB_CTA backward METHOD wb97x-d BASIS 6-31g THRESH 14 SCF_CONVERGENCE 9 SYMMETRY FALSE SCF_ALGORITHM DIIS IDERIV 1 SCFMI_MODE 1 $end