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17.5 Empirical damped dispersion correction

Empirical damped dispersion corrections can be calculated in addition to Kohn-Sham calculations. This is particularly important in cases where long-range correlation effects become dominant. The dispersion correction uses the van-der-Waals radii and atomic dispersion coefficients from Refs. [1-2] and its functional form is given by:

$$E_\mathrm{disp}=-s_6\sum_{i,j<i}^{N_\mathrm{atoms}}f_\mathrm{damp}(R_{ij}) \frac{C_6^{ij}}{R_{ij}^6}~.$$ (39)

In the above equation, $N_\mathrm{atoms}$ is the total number of atoms, $R_{ij}$ is the interatomic distance of atoms $i$ and $j$, $s_6$ is a global scaling parameter depending on the choice of the functional used (see below), and the $C_6^{ij}$ values are calculated from atomic dispersion coefficients $C_6^i$ and $C_6^j$ in the following way:

$$C_6^{ij} = 2\frac{C_6^i C_6^j}{C_6^i + C_6^j} ~~~~~\mathrm{(Ref.\ [1])}$$ (40)

$$C_6^{ij} = \sqrt{C_6^i C_6^j} ~~~~~~~~~\mathrm{(Ref.\ [2])}$$ (41)

The function $f_\mathrm{damp}$ damps the dispersion correction for shorter interatomic distances and is given by:

$$f_\mathrm{damp}(R_{ij})=\frac{1}{1+\exp{[-\alpha(R_{ij}/ (R_\mathrm{vdW}^i+R_\mathrm{vdW}^j)-1)]}}$$ (42)

whith $R_\mathrm{vdW}^i$ being the van-der-Waals radius for atom $i$ and $\alpha$ is a parameter that is usually set to 23 (Ref. [1]) or 20 (Ref. [2]).

Currently the following functionals can be used in conjunction with the empirical dispersion correction:

Table: Optimised scaling parameters $s_6$ (see Eq. (41)) for density functionals

functional	Ref. [1]	Ref. [2]
BLYP	1.40	1.20
PBE	0.70	0.75
BP86	1.30	1.05
B3LYP	-	1.05

The dispersion correction can be calculated using dispcorr in the Molpro input and it can be added to DFT energies by using the following template:

 ks,<func>
 eks=energy
 dispcorr
 eks_plus_disp=eks+edisp

i.e., the dispersion corrections (Eq. (41)) are stored in variables edisp. Note that dispcorr notices which type of functional has previously been used by reading in the internal Molpro variable DFTNAME.

The dispcorr program can have the following options:

MODE

Adjusts the parametrisation used: MODE=1 uses parameters from Ref. [1] and MODE=2 uses parameters from Ref. [2] (default: MODE=1)

SCALE

Overall scaling parameter $s_6$ (see Eq. (41) and table 9 for optimised values).

ALPHA

Damping function parameter (see Eq. (44)). Smaller values lead to larger corrections for intermediate distances.

The third dispersion correction from Grimme et al. [3] takes into account also eith-order dispersion coefficients and uses different damping functions. It can be invoked in the same way as shown above, but by replacing dispcorr with dispcorr3 in the template.

The dispcorr3 program can have the following options:

FUNC

Functional name (default: FUNC='pbe').

VERSION

Can have values 2 and 3 according to parametrisations from Refs. [2] and [3] (default: VERSION=3)

ANAL

Performs a detailed analysis of pair contributions.

GRAD

Cartesian gradients are computed. (note that geometry optimisations with DFT+dispcorr3 are currently not yet possible).

TZ

Use special parameters for calculations with triple-zeta basis sets. Preliminary results in the SI of Ref. [3] indicate that results are slightly worse than with the default parameters and QZVP type basis sets. This option should be carfully tested for future use in very large computations.

(see also http://toc.uni-muenster.de/DFTD3/index.html for further documentation).

References:

$[1]$ S. Grimme, J. Comp. Chem. 25, 1463 (2004).
$[2]$ S. Grimme, J. Comp. Chem. 27, 1787 (2006).
$[3]$ S. Grimme, J. Antony, S. Ehrlich and H. Krieg, J. Chem. Phys. 132, 154104 (2010)

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