12.7.3 Adiabatic ALMO-EDA and VFB Analysis

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, ⁸³⁶ Mao Y., Horn P. R., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2017), 19, pp. 5944. Link where the term “adiabatic" is borrowed from spectroscopy and indicates that energy differences are evaluated at relaxed geometry on each potential energy surface. In this scheme, the total binding energy (including monomer geometry distortions) is repartitioned into adiabatic FRZ, POL and CT terms:

The adiabatic frozen interaction energy is given by the difference between the energy minimum of the frozen potential surface (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:

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

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 potential surface, 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 potential surface is facilitated by the analytical gradient of the frozen wave function energy implemented in Q-Chem. As for the geometry optimization on the polarized potential surface, the nuclear gradient of the SCF-MI 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 SCF-MI calculations that use FERFs are not available yet, and SCFMI_MODE = 0 is required for computing the forces on the frozen and polarized potential surfaces. Also, the current implementation of this method requires users to perform geometry optimization separately on the three potential surfaces (see the example below) and then 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 future.

As mentioned in Section 12.6.5, 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 potential surface. 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. ⁸¹⁰ Loipersberger M., Mao Y., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1073. Link 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:

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 SCF-MI) calculation. Also, since the definition of VFB states is based on the generalized SCF-MI technique (Section 12.6.2), SCFMI_MODE = 1 is required.

In Q-Chem v. 5.4 or later, analytical gradients for the polarized and two VFB “one-way” CT states with implicit solvent models PCM and SMD are supported so that one can perform part of the adiabatic ALMO-EDA steps (POL $\to$ CTf/CTb $\to$ Full) in solvation environments. To do this, one only needs to set the $rem variable SOLVENT_METHOD to PCM or SMD, which is similar to the usage of ALMO-EDA(solv) (see Section 12.6.6). The calculation of analytical forces on the frozen surface with implicit solvents is currently unavailable, and we look forward to enabling that in future releases of Q-Chem.