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(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.
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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
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J. Comput. Chem.

(2006),
27,
pp. 1787.
Link

${E}_{\mathrm{DISP}}$ | $=$ | $$ | (11.110) | ||

${C}_{6}^{AB}$ | $=$ | ${({C}_{6}^{A}{C}_{6}^{B})}^{1/2}$ | (11.111) | ||

${f}_{\mathrm{dmp}}({R}_{AB})$ | $=$ | ${[1+{e}^{-d({R}_{AB}/{R}_{AB}^{0}-1)}]}^{-1}$ | (11.112) |

where ${R}_{AB}$ and ${R}_{AB}^{0}$ are the internuclear separation and sum of the van der Waals radii of atoms $A$ and $B$ respectively, ${C}_{6}^{AB}$ is the dispersion coefficient for atom pair $AB$, ${s}_{6}$ is a scaling factor and ${f}_{\mathrm{dmp}}({R}_{AB})$ is the damping function. In the AIRBED approach, ${E}_{\mathrm{DISP}}$ is replaced by ${E}_{\mathrm{vdW}}$, where ${E}_{\mathrm{vdW}}$ describes the repulsion at short inter-nuclear separations in addition to the dispersion interaction through a modification of the nature of the damping function.

${E}_{\mathrm{vdW}}$ | $=$ | $-{s}_{6}{\displaystyle \sum _{A}^{{N}_{e}}}{\displaystyle \sum _{B}^{{N}_{m}}}{\displaystyle \frac{{C}_{6}^{AB}}{{R}_{AB}^{6}}}{f}_{\mathrm{dmp}}^{r+d}({R}_{AB})$ | (11.113) | ||

${f}_{\mathrm{dmp}}^{r+d}({R}_{AB})$ | $=$ | $1-{e}^{[-d({R}_{AB}/{R}_{AB}^{0}-1)+\alpha ]}.$ | (11.114) |

Here the system is partitioned into the “molecule" and “environment", with
${N}_{e}$ environment atoms and ${N}_{m}$ molecule atoms. The ${R}_{AB}^{0}$ values used
for this contribution to the energy are derived from the experimental van der
Waals radii,
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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 ${s}_{6}$ are used which
are unchanged from DFT-D2. The additional parameter $\alpha $ 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.