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

(November 19, 2024)

This section describes DFT-C, ¹³⁷⁹ Witte J., Neaton J. B., Head-Gordon M.
J. Chem. Phys.
(2017), 146, pp. 234105. Link an empirical correction for basis set superposition error (BSSE) in DFT calculations that is an adaptation of Grimme’s geometrical counterpoise (gCP) correction. ⁶⁸⁴ Kruse H., Grimme S.
J. Chem. Phys.
(2012), 136, pp. 154101. Link Unlike the traditional Boys-Bernardi counterpoise correction (Section 8.9), ¹³⁹ Boys S. F., Bernardi F.
Mol. Phys.
(1970), 19, pp. 553. Link 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