Empirical Dispersion Correction for DFT Calculations

Based on an idea that has earlier been proposed for Hartree-Fock 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 DFT-D 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_DFT-D = E_KS-DFT + E_disp (6.9)

where E_KS-DFT is the usual self-consistent Kohn-Sham energy as obtained from the chosen functional and E_disp is an empirical dispersion correction given by

$$\sum_{{i=1}}^{{N_{at}-1}} \sum_{{j=i+1}}^{{N_{at}}} {\frac{{C_6^{ij}}}{{R_{ij}^6}}}$$ (6.10)

Here, N_at is the number of atoms in the system, C₆^ij denotes the dispersion coefficient for atom pair ij, s₆ is a global scaling factor that only depends on the DF used and R_ij is an interatomic distance. The interatomic C₆^ij term is calculated as geometric mean of the form

$$\sqrt{{C_6^i C_6^j}}$$ (6.11)

This yields much better results that the form used in the original paper:

$${\frac{{C_6^i \cdot C_6^j}}{{C_6^i + C_6^j}}}$$ (6.12)

In order to avoid near-singularities for small R, a damping function f_dmp must be used which is given by

$${\frac{{1}}{{1+e^{{-d(R_{ij}/R_{r}-1)}}}}}$$ (6.13)

where R_r is the sum of atomic vdW radii. These values are derived from the radius of the 0.01 a₀^-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 R₀ used are given in Table 6.3 together with new atomic C₆ 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: s₆ parameters for functionals in the old and the revised implementation of DFT-D

Density Functional s₆ s₆ (old)

BP86 1.05 1.30

B-LYP 1.20 1.40

PBE 0.75 0.70

B3-LYP 1.05 -^a

TPSS 1.00 -^a

^a Not available ^b See Ref.[74]

Table 6.3: C₆ parameters^a (in Jnm⁶mol^-1) and van der Waals radii^b R₀ (in Å) for elements H-Xe.

element C₆ R₀ element C₆ R₀

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 Sc-Zn 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 Y-Cd 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 UDFT-PBE0/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 C₆ parameters^a (in Jnm⁶mol^-1) and van der Waals radii^b R₀ (in Å) for elements H-Ne.

element C₆ R₀ element C₆ R₀

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 UDFT-PBE0/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 C₆ parameter of 0.0. This results in an incomplete description of the dispersion energy.