5 Density Functional Theory

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

EDFT-C=σAatomscABAatomsgABDFT-C(RAB)hAB({A,B,}) (5.47)

where gABDFT-C is a damped, pairwise BSSE correction,

gABDFT-C(RAB)=d(RAB)fABDFT-C(RAB)+[1-d(RAB)]fABDFT-C(Rcov,AB). (5.48)

The quantity

fABDFT-C(RAB)=cABexp(-αABRAB2+βABRAB) (5.49)

is the undamped pairwise BSSE and

d(RAB)=11+k1,AB(RAB/R0,AB)-k2,AB (5.50)

is a damping function. The quantity hAB({A,B,}) is a many-body correction to the two-body BSSE correction, given by

hAB({A,B,})=[1+CA,BNCvirtNBvirtterfc(RAC,RAB)terfc(RBC,RAB)]-1 (5.51)


terfc(x,y)=1-12[erf(x+y)+erf(x-y)]. (5.52)

The parameters cA, cAB, αAB, and βAB are basis-set-dependent, and the overall scaling parameter σ 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:

       Controls whether the DFT-C empirical BSSE correction should be added.
       FALSE (or 0) Do not apply the DFT-C correction TRUE (or 1) Apply the DFT-C correction

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.

   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

   JOBTYPE            opt
   BASIS              def2-SVPD
   METHOD             b97m-v
   DFT_C              true