next up previous contents index
Next: Hartree-Fock and DFT Response Up: Hartree-Fock and DFT Calculations Previous: Calculation Setup   Contents   Index


Empirical Dispersion Correction for DFT Calculations

Based on an idea that has earlier been proposed for Hartree-Fock calculations[67,68], an general empirical dispersion correction has been proposed by Stefan Grimme for density functional calculations [69]. 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

$\displaystyle E_{DFT-D}=E_{KS-DFT}+E_{disp}$ (6.8)

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

$\displaystyle E_{disp}=-s_6\sum_{i=1}^{N_{at}-1}\sum_{j=i+1}^{N_{at}} \frac{C_6^{ij}}{R_{ij}^6} f_{dmp}(R_{ij})  .$ (6.9)

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

$\displaystyle C_6^{ij}=\sqrt{C_6^i C_6^j}.$ (6.10)

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

$\displaystyle C_6^{ij}=2\cdot\frac{C_6^i \cdot C_6^j}{C_6^i + C_6^j}$ (6.11)

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

$\displaystyle f_{dmp}(R_{ij})=\frac{1}{1+e^{{-d(R_{ij}/R_{r}-1)}}}$ (6.12)

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[69] 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 DFT-D
Density Functional $ s_6$ $ s_6$ (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.[70]



Table 6.3: $ C_6$ parameters$ ^a$ (in $ J nm^6 mol^{-1}$) and van der Waals radii$ ^b$ $ R_0$ (in Å) for elements H-Xe.
element $ C_6$ $ R_0$ element $ C_6$ $ R_0$
H 0.14 1.001K 10.80$ ^c$ 1.485
He 0.08 1.012Ca 10.80$ ^c$ 1.474
Li 1.61 0.825Sc-Zn10.80$ ^c$ 1.562$ ^d$
Be 1.61 1.408Ga 16.99 1.650
B 3.13 1.485Ge 17.10 1.727
C 1.75 1.452As 16.37 1.760
N 1.23 1.397Se 12.64 1.771
O 0.70 1.342Br 12.47 1.749
F 0.75 1.287Kr 12.01 1.727
Ne 0.63 1.243Rb 24.67$ ^c$ 1.628
Na 5.71$ ^c$ 1.144Sr 24.67$ ^c$ 1.606
Mg 5.71$ ^c$ 1.364Y-Cd 24.67$ ^c$ 1.639$ ^d$
Al 10.79 1.639In 37.32 1.672
Si 9.23 1.716Sn 38.71 1.804
P 7.84 1.705Sb 38.44 1.881
S 5.57 1.683Te 31.74 1.892
Cl 5.07 1.639I 31.50 1.892
Ar 4.61 1.595Xe 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 $ C_6$ parameters$ ^a$ (in $ J nm^6 mol^{-1}$) and van der Waals radii$ ^b$ $ R_0$ (in Å) for elements H-Ne.
element $ C_6$ $ R_0$ element $ C_6$ $ R_0$
H 0.16 1.11O 0.70 1.49
C 1.65 1.61F 0.57 1.43
N 1.11 1.55Ne 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.


next up previous contents index
Next: Hartree-Fock and DFT Response Up: Hartree-Fock and DFT Calculations Previous: Calculation Setup   Contents   Index
TURBOMOLE