DFT-D2

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In the DFT-D2 method of Grimme^[1], the correction term takes the form:

E_{\mathrm {disp} }=-{\frac {1}{2}}\sum _{i=1}^{N_{at}}\sum _{j=1}^{N_{at}}\sum _{\mathbf {L} }{}^{\prime }{\frac {C_{6ij}}{r_{ij,L}^{6}}}f_{d,6}({r}_{ij,\mathbf {L} })

where the first two summations are over all $N_{at}$ atoms in the unit cell and the third summation is over all translations of the unit cell ${\mathbf {L} }=(l_{1},l_{2},l_{3})$ where the prime indicates that $i\not =j$ for ${\mathbf {L} }=0$ . $C_{6ij}$ denotes the dispersion coefficient for the atom pair $ij$ , ${r}_{ij,\mathbf {L} }$ is the distance between atom $i$ located in the reference cell $\mathbf {L} =0$ and atom $j$ in the cell $L$ and the term $f(r_{ij})$ is a damping function whose role is to scale the force field such as to minimize the contributions from interactions within typical bonding distances. In practice, the terms in the equation for $E_{\mathrm {disp} }$ corresponding to interactions over distances longer than a certain suitably chosen cutoff radius (VDW_RADIUS, see below) contribute only negligibly to $E_{\mathrm {disp} }$ and can be ignored. Parameters $C_{6ij}$ and $R_{0ij}$ are computed using the following combination rules:

C_{6ij}={\sqrt {C_{6ii}C_{6jj}}}

and

R_{0ij}=R_{0i}+R_{0j}.

The values for $C_{6ii}$ and $R_{0i}$ are tabulated for each element and are insensitive to the particular chemical situation (for instance, $C_{6}$ for carbon in methane takes exactly the same value as that for C in benzene within this approximation). In the DFT-D2 method, a Fermi-type damping function is used:

f_{d,6}(r_{ij})={\frac {s_{6}}{1+e^{-d(r_{ij}/(s_{R}\,R_{0ij})-1)}}}

whereby the global scaling parameter $s_{6}$ has been optimized for several different DFT functionals such as PBE ( $s_{6}=0.75$ ), BLYP ( $s_{6}=1.2$ ) or B3LYP ( $s_{6}=1.05$ ). The parameter $s_{R}$ is usually fixed at 1.00. The DFT-D2 method can be activated by setting IVDW=1|10 or by specifying LVDW=.TRUE. (this parameter is obsolete as of VASP.5.3.3). Optionally, the damping function and the vdW parameters can be controlled using the following flags (the given values are the default ones):

VDW_RADIUS=50.0 : cutoff radius (in $\mathrm {\AA}$ ) for pair interactions
VDW_S6=0.75 : global scaling factor $s_{6}$ (available in VASP.5.3.4 and later)
VDW_SR=1.00 : scaling factor $s_{R}$ (available in VASP.5.3.4 and later)
VDW_SCALING=0.75 : the same as VDW_S6 (obsolete as of VASP.5.3.4)
VDW_D=20.0 : damping parameter $d$
VDW_C6=[real array] : $C_{6}$ parameters ( $\mathrm {Jnm} ^{6}\mathrm {mol} ^{-1}$ ) for each species defined in the POSCAR file
VDW_R0=[real array] : $R_{0}$ parameters ( $\mathrm {\AA}$ ) for each species defined in the POSCAR file
LVDW_EWALD=.FALSE. : the lattice summation in $E_{\mathrm {disp} }$ expression is computed by means of Ewald's summation (.TRUE. ) or via a real space summation over all atomic pairs within cutoff radius VDW_RADIUS (.FALSE.). (available in VASP.5.3.4 and later)

The performance of PBE-D2 method in optimization of various crystalline systems has been tested systematically in reference ^[2].

Important: It is recommended to use the more advanced and more accurate method DFT-D3.^[3]

Mind:

The defaults for VDW_C6 and VDW_R0 are defined only for elements in the first five rows of the periodic table (i.e. H-Xe). If the system contains other elements the user has to define these parameters in INCAR.
The defaults for parameters controlling the damping function (VDW_S6, VDW_SR, VDW_D) are available for the PBE (GGA=PE), BP, revPBE, PBE0, TPSS, and B3LYP functionals. If any other functional is used in a DFT-D2 calculation, the value of VDW_S6 (or VDW_SCALING in versions before VASP.5.3.4) has to be defined in INCAR.
As of VASP.5.3.4, the default value for VDW_RADIUS has been increased from 30 to 50 $\mathrm {\AA}$ .
Ewald's summation in the calculation of $E_{\mathrm {disp} }$ calculation (controlled via LVDW_EWALD) is implemented according to reference ^[4] and is available as of VASP.5.3.4.

References

[grimme:jcc:06-1] S. Grimme, J. Comput. Chem. 27, 1787 (2006).

[bucko:jpca:10-2] T. Bučko, J. Hafner, S. Lebègue, and J. G. Ángyán, J. Phys. Chem. A 114, 11814 (2010).

[grimme:jcp:10-3] S. Grimme, J. Antony, S. Ehrlich, and S. Krieg, J. Chem. Phys. 132, 154104 (2010).

[kerber:jcc:08-4] T. Kerber and J. Sauer, J. Comput. Chem. 29, 2088 (2008).

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