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Empirical Dispersion Correction for DFT Calculations
Based on an idea that has earlier been proposed for HartreeFock calculations[70,71],
an general empirical dispersion correction has been proposed by Stefan Grimme for density functional
calculations [72]. A modified version of the approach with extension
to more elements and more functionals has been published in ref. [73].
The correction is invoked by the keyword $disp in the control file. The parameters
of the second DFTD publication are used. The older parameters are used when the
keyword $olddisp is found in the control file.
When using the dispersion correction, the total energy is given by
E_{DFTD} = E_{KSDFT} + E_{disp} 
(6.9) 
where
E_{KSDFT} is the usual selfconsistent KohnSham energy as obtained from the chosen
functional and E_{disp} is an empirical dispersion correction given by
E_{disp} =  s_{6}f_{dmp}(R_{ij}) . 
(6.10) 
Here, N_{at} is the number of atoms in the system, C_{6}^{ij} denotes the dispersion coefficient
for atom pair ij, s_{6} is a global scaling factor that only
depends on the DF used and R_{ij} is an interatomic distance.
The interatomic C_{6}^{ij} term is calculated as geometric mean of the form
C_{6}^{ij} = . 
(6.11) 
This yields much better results that the form used in the original paper:
C_{6}^{ij} = 2⋅ 
(6.12) 
In order to avoid nearsingularities for small R, a damping function f_{dmp}
must be used which is given by
f_{dmp}(R_{ij}) = 
(6.13) 
where R_{r} is the sum of atomic vdW radii. These values are derived from
the radius of the 0.01 a_{0}^{3} electron density contour from ROHF/TZV computations of the atoms
in the ground state. An earlier[72] used general scaling factor for the radii is decreased
from 1.22 to 1.10 in the second implementation. This improves computed intermolecular distances
especially for systems with heavier atoms. The atomic van der Waals radii R_{0}
used are given in Table 6.3 together with new atomic C_{6} coefficients (see below).
Compared to the original parameterization (d = 23), a smaller damping parameter of d = 20 provides
larger corrections at intermediate distances (but still negligible dispersion energies for
typical covalent bonding situations).
Table 6.2:
s_{6} parameters for functionals in the old and the revised implementation of DFTD
Density Functional  s_{6}  s_{6} (old) 
BP86  1.05  1.30 
BLYP  1.20  1.40 
PBE  0.75  0.70 
B3LYP  1.05  ^{a} 
TPSS  1.00  ^{a} 
^{a} Not available
^{b} See Ref.[74]

Table 6.3:
C_{6} parameters^{a} (in
Jnm^{6}mol^{1})
and van der Waals radii^{b} R_{0} (in Å) for elements HXe.
element  C_{6}  R_{0}  element  C_{6}  R_{0} 
H  0.14  1.001  K  10.80^{c}  1.485 
He  0.08  1.012  Ca  10.80^{c}  1.474 
Li  1.61  0.825  ScZn  10.80^{c}  1.562^{d} 
Be  1.61  1.408  Ga  16.99  1.650 
B  3.13  1.485  Ge  17.10  1.727 
C  1.75  1.452  As  16.37  1.760 
N  1.23  1.397  Se  12.64  1.771 
O  0.70  1.342  Br  12.47  1.749 
F  0.75  1.287  Kr  12.01  1.727 
Ne  0.63  1.243  Rb  24.67^{c}  1.628 
Na  5.71^{c}  1.144  Sr  24.67^{c}  1.606 
Mg  5.71^{c}  1.364  YCd  24.67^{c}  1.639^{d} 
Al  10.79  1.639  In  37.32  1.672 
Si  9.23  1.716  Sn  38.71  1.804 
P  7.84  1.705  Sb  38.44  1.881 
S  5.57  1.683  Te  31.74  1.892 
Cl  5.07  1.639  I  31.50  1.892 
Ar  4.61  1.595  Xe  29.99  1.881 
^{a} Derived from UDFTPBE0/QZVP computations.
^{b} Derived from atomic ROHF/TZV computations.
^{c} Average of preceeding group VIII and following group III element.
^{d} Average of preceeding group II and following group III element.

Table 6.4:
old C_{6} parameters^{a} (in
Jnm^{6}mol^{1})
and van der Waals radii^{b} R_{0} (in Å) for elements HNe.
element  C_{6}  R_{0}  element  C_{6}  R_{0} 
H  0.16  1.11  O  0.70  1.49 
C  1.65  1.61  F  0.57  1.43 
N  1.11  1.55  Ne  0.45  1.38 
^{a} Derived from UDFTPBE0/QZVP computations.
^{b} Derived from atomic ROHF/TZV computations.

Caution: if elements are present in the molecule for which no parameters are
defined, the calculation proceeds with an atomic C_{6} parameter of 0.0.
This results in an incomplete description of the dispersion energy.
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