5.8 Empirical Corrections for Basis Set Superposition Error 5.8 Empirical Corrections for Basis Set Superposition Error 5.9 Double-Hybrid Density Functional Theory

5.8.1 Overview

(February 4, 2022)

This section describes DFT-C,¹²²⁷ an empirical correction for basis set superposition error (BSSE) in DFT calculations that is an adaptation of Grimme’s geometrical counterpoise (gCP) correction.⁵⁹⁶ Unlike the traditional Boys-Bernardi counterpoise correction (Section 8.9),¹²⁴ 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\})

(5.54)

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})\;.

(5.55)

The quantity

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

(5.56)

is the undamped pairwise BSSE and

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

(5.57)

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}

(5.58)

where

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

(5.59)

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

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

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5.8.1 Overview