11 Molecules in Complex Environments: Solvent Models, QM/MM and QM/EFP Features, Density Embedding

11.9 Atomic Interactions Represented By Empirical Dispersion (AIRBED)

(May 16, 2021)

The properties of a molecule can be influenced by its physical environment, this can occur as a direct consequence of the molecule-environment interaction, or indirectly through the geometrical constraints imposed by the environment modifying the molecular structure. Even when there is no chemical bonding between a molecule and its environment, and the interaction is dominated by relatively weak intermolecular forces the effects of this interaction can be significant.

Recently the Atomic Interactions Represented By Empirical Dispersion (AIRBED) approach was introduced. 736 Mason S.E., Beton P.H., Besley N.A.
J. Chem. Theory Comput.
(2018), 15, pp. 5628.
Link
Within this approach, the empirical dispersion correction commonly used in DFT calculations was modified to capture the repulsion at short inter-nuclear distances, in addition to the attractive dispersion interaction with point charges included to account for electrostatic effects. This allows the important components of the interaction between the molecule and environment to be described without the electronic structure of the environment atoms being included with the DFT calculation, and can be viewed as a quantum mechanics/molecular mechanics approach integrated within the DFT calculation that will provide a more consistent treatment of the non-bonded interactions.

The AIRBED approach is based upon the DFT-D2 method 390 Grimme S.
J. Comput. Chem.
(2006), 27, pp. 1787.
Link

EDISP = -s6ANB<ANC6ABRAB6fdmp(RAB) (11.110)
C6AB = (C6AC6B)1/2 (11.111)
fdmp(RAB) = [1+e-d(RAB/RAB0-1)]-1 (11.112)

where RAB and RAB0 are the internuclear separation and sum of the van der Waals radii of atoms A and B respectively, C6AB is the dispersion coefficient for atom pair AB, s6 is a scaling factor and fdmp(RAB) is the damping function. In the AIRBED approach, EDISP is replaced by EvdW, where EvdW describes the repulsion at short inter-nuclear separations in addition to the dispersion interaction through a modification of the nature of the damping function.

EvdW = -s6ANeBNmC6ABRAB6fdmpr+d(RAB) (11.113)
fdmpr+d(RAB) = 1-e[-d(RAB/RAB0-1)+α]. (11.114)

Here the system is partitioned into the “molecule" and “environment", with Ne environment atoms and Nm molecule atoms. The RAB0 values used for this contribution to the energy are derived from the experimental van der Waals radii, 119 Bondi A.
J. Phys. Chem.
(1964), 68, pp. 441.
Link
which tend to be larger than the values used in the standard DFT-D2 corrections. Values of d=20.0 and s6 are used which are unchanged from DFT-D2. The additional parameter α is introduced to allow some additional flexibility to tune the environment-molecule interaction. Note that this modified interaction is only applied for the interaction between the atoms of the environment and the molecule, and the original, unmodified dispersion correction is used for the interaction between the atoms of the molecule. Furthermore, the atoms of the molecular environment can be assigned an arbitrary charge. The atoms of the environment specified in the $airbed block and charges can be assigned using the $external_charges block (Section C.1.7). Gradients and second-derivatives have been implemented for this model allowing the optimisation of structures and the calculation of harmonic vibrational frequencies.