5.8 Empirical Corrections for Basis Set Superposition Error

This section describes DFT-C,1057 an empirical correction for basis set superposition error (BSSE) in DFT calculations that is an adaptation of Grimme’s geometrical counterpoise (gCP) correction.505 Unlike the traditional Boys-Bernardi counterpoise correction (Section 8.9),95 the cost of the DFT-C correction is essentially zero (on the scale of a DFT calculation), and the latter provides an estimate of both inter- and intramolecular BSSE. The form of this correction is

 $E_{\text{DFT-C}}=\sigma\sum^{\text{atoms}}_{A}c_{A}\sum^{\text{atoms}}_{B\neq A% }g_{AB^{\ast}}^{\text{DFT-C}}(R_{AB})\;h_{AB^{\ast}}(\{A,B,\ldots\})$ (5.47)

where $g_{AB^{\ast}}^{\text{DFT-C}}$ is a damped, pairwise BSSE correction,

 $g_{AB^{\ast}}^{\text{DFT-C}}(R_{AB})=d(R_{AB})\;f_{AB^{\ast}}^{\text{DFT-C}}(R% _{AB})+\bigl{[}1-d(R_{AB})\bigr{]}f_{AB^{\ast}}^{\text{DFT-C}}(R_{\text{cov},% AB})\;.$ (5.48)

The quantity

 $f_{AB^{\ast}}^{\text{DFT-C}}(R_{AB})=c_{AB}\exp\bigl{(}-\alpha_{AB}R_{AB}^{2}+% \beta_{AB}R_{AB}\bigr{)}$ (5.49)

is the undamped pairwise BSSE and

 $d(R_{AB})=\frac{1}{1+k_{1,AB}(R_{AB}/R_{0,AB})^{-k_{2,AB}}}$ (5.50)

is a damping function. The quantity $h_{AB^{\ast}}(\{A,B,...\})$ is a many-body correction to the two-body BSSE correction, given by

 $h_{AB^{\ast}}(\{A,B,...\})=\left[1+\sum_{C\neq A,B}\frac{N_{C}^{\text{virt}}}{% N_{B}^{\text{virt}}}\operatorname{terfc}\left(R_{AC},R_{AB}\right)% \operatorname{terfc}\left(R_{BC},R_{AB}\right)\right]^{-1}$ (5.51)

where

 $\operatorname{terfc}(x,y)=1-\frac{1}{2}\big{[}\mathrm{erf}(x+y)+\mathrm{erf}(x% -y)\bigr{]}\;.$ (5.52)

The parameters $c_{A}$, $c_{AB}$, $\alpha_{AB}$, and $\beta_{AB}$ are basis-set-dependent, and the overall scaling parameter $\sigma$ is loosely method-dependent. All of these parameters are set internally based on the method and basis $rem specifications. Note: Currently, only the def2-SVPD basis set is supported for use with DFT-C. The DFT-C correction is governed by the following$rem variable:

DFT_C
Controls whether the DFT-C empirical BSSE correction should be added.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Do not apply the DFT-C correction TRUE (or 1) Apply the DFT-C correction
RECOMMENDATION:
NONE

The DFT-C method can be applied to any local, GGA, or meta-GGA density functional, as in the following example.

Example 5.15  Geometry optimization of the methane dimer using B97M-V-C/def2-SVPD, i.e., the B97M-V functional with the DFT-C BSSE correction in the def2-SVPD basis set.

$molecule 0 1 C 0.000000 -0.000140 1.859161 H -0.888551 0.513060 1.494685 H 0.888551 0.513060 1.494685 H 0.000000 -1.026339 1.494868 H 0.000000 0.000089 2.948284 C 0.000000 0.000140 -1.859161 H 0.000000 -0.000089 -2.948284 H -0.888551 -0.513060 -1.494685 H 0.888551 -0.513060 -1.494685 H 0.000000 1.026339 -1.494868$end

$rem JOBTYPE opt BASIS def2-SVPD METHOD b97m-v DFT_C true$end