Empirical Dispersion Correction for DFT Calculations

Based on an idea that has earlier been proposed for Hartree-Fock calculations[65,66], an general empirical dispersion correction has been proposed by Stefan Grimme for density functional calculations [67]. A modified version of the approach with extension to more elements and more functionals has been published in ref. [59].

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

$$E_{DFT-D}=E_{KS-DFT}+E_{disp}$$ (4.8)

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

$$E_{disp}=-s_6\sum_{i=1}^{N_{at}-1}\sum_{j=i+1}^{N_{at}} \frac{C_6^{ij}}{R_{ij}^6} f_{dmp}(R_{ij}) .$$ (4.9)

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

$$C_6^{ij}=\sqrt{C_6^i C_6^j}.$$ (4.10)

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

$$C_6^{ij}=2\cdot\frac{C_6^i \cdot C_6^j}{C_6^i + C_6^j}$$ (4.11)

In order to avoid near-singularities for small $R$, a damping function $f_{dmp}$ must be used which is given by

$$f_{dmp}(R_{ij})=\frac{1}{1+e^{{-d(R_{ij}/R_{r}-1)}}}$$ (4.12)

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[67] 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 4.3 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 4.2: $s_6$ parameters for functionals in the old and the revised implementation of DFT-D

$$s_6 s_6$$ Density Functional

BP86 1.05 1.30

B-LYP 1.20 1.40

PBE 0.75 0.70

$$^a$$ B3-LYP 1.05

$$^a$$ TPSS 1.00

$^a$ Not available $^b$ See Ref.[68]

Table 4.3: $C_6$ parameters$^a$ (in $J nm^6 mol^{-1}$) and van der Waals radii$^b$ $R_0$ (in Å) for elements H-Xe.

$$C_6 R_0 C_6 R_0$$ element element

$$^c$$ H 0.14 1.001 K 1.485

$$^c$$ He 0.08 1.012 Ca 1.474

$$^c ^d$$ Li 1.61 0.825 Sc-Zn

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

$$^c$$ Ne 0.63 1.243 Rb 1.628

$$^c ^c$$ Na 1.144 Sr 1.606

$$^c ^c ^d$$ Mg 1.364 Y-Cd

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 4.4: old $C_6$ parameters$^a$ (in $J nm^6 mol^{-1}$) and van der Waals radii$^b$ $R_0$ (in Å) for elements H-Ne.

$$C_6 R_0 C_6 R_0$$ element element

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 $C_6$ parameter of 0.0. This results in an incomplete description of the dispersion energy.