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

$$E_{\text{DFT-C}}=\sigma \sum _A^{\text{atoms}}{c}_A\sum _{B\ne A}^{\text{atoms}}{g}_{A{B}^{\ast }}^{\text{DFT-C}}(R_{AB})h_{A{B}^{\ast }}(\{A,B,\mathrm{\dots }\})$$

(5.47)

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

$$g_{A{B}^{\ast }}^{\text{DFT-C}}(R_{AB})=d(R_{AB})f_{A{B}^{\ast }}^{\text{DFT-C}}(R_{AB})+\left[1-d(R_{AB})\right]f_{A{B}^{\ast }}^{\text{DFT-C}}(R_{\text{cov},AB}).$$

(5.48)

The quantity

$$f_{A{B}^{\ast }}^{\text{DFT-C}}(R_{AB})=c_{AB}\exp \left(-{\alpha }_{AB}{R}_{AB}^2+{\beta }_{AB}{R}_{AB}\right)$$

(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 $h_{A{B}^{\ast }}(\{A,B,\mathrm{\dots }\})$ is a many-body correction to the two-body BSSE correction, given by

$$h_{A{B}^{\ast }}(\{A,B,\mathrm{\dots }\})={\left[1+\sum _{C\ne A,B}\frac{N_C^{\text{virt}}}{N_B^{\text{virt}}}\mathrm{terfc}(R_{AC},R_{AB})\mathrm{terfc}(R_{BC},R_{AB})\right]}^{-1}$$

(5.51)

where

$$\mathrm{terfc}(x,y)=1-\frac{1}{2}\left[\mathrm{erf}(x+y)+\mathrm{erf}(x-y)\right].$$

(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:

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

$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