DFT-D2

In the D2 method of Grimme, the correction term takes the form:

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

where the summations are over all atoms $N_{at}$ and all translations of the unit cell ${L}=(l_1,l_2,l_3)$. The prime indicates that $i\not=j$ for ${L}=0$, $C_{6ij}$ denotes the dispersion coefficient for the atom pair $ij$, ${r}_{ij,L}$ is the distance between atom $i$ located in the reference cell $L=0$ 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 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 original method of Grimme, 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 ($s_6=1.2$) and 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 (this parameter is obsolete as of VASP.5.3.3).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 default values are listed):

\begin{tabular}{rll} {\tt VDW\_RADIUS} &= 50.0 & cutoff radius ({\AA}) for pair interactions\\ {\tt VDW\_S6} &= 0.75 & global scaling factor $s_6$\\

                & & (available in VASP.5.3.4 and later)\\

{\tt VDW\_SR} &= 1.00 & scaling factor $s_R$\\ & & (available in VASP.5.3.4 and later)\\ {\tt VDW\_SCALING} & =0.75 & the same as {\tt VDW\_S6}\\

  & & (obsolete as of VASP.5.3.4)\\

{\tt VDW\_D} &= 20.0 & damping parameter $d$\\ {\tt VDW\_C6} &= [real array] & $C_6$ parameters ($Jnm^6mol^{-1}$) for each species\\

            & &  defined in POSCAR\\

{\tt VDW\_R0} &= [real array] & $R_0$ parameters ({\AA}) for each species \\

          & & defined in POSCAR\\

{\tt LVDW\_EWALD} &= .FALSE.$|$.TRUE. & compute lattice summation in $E_{disp}$ expression\\

& & by means of Ewald's summation - no$|$yes\\
& & (available in VASP.5.3.4 and later)\\

\end{tabular} \\ \\ \noindent The performance of PBE-D2 method in optimization of various crystalline systems has been tested systematically in J. Phys. Chem. A 114, 11814 (2010).\\ \vspace{5mm} \\ \noindent IMPORTANT NOTES: \begin{itemize} \item the defaults for {\tt VDW\_C6} and {\tt VDW\_R0} are defined only for elements in the first five rows of periodic table (i.e. H-Xe) - if the system contains other elements the user must define these parameters in INCAR. \item the defaults for parameters controlling damping function ({\tt VDW\_S6}, {\tt VDW\_SR}, {\tt VDW\_D}) are available only for the PBE functional. If functional other than PBE is used in DFT+D2 calculation, the value of {\tt VDW\_S6} (or {\tt VDW\_SCALING} in versions before VASP.5.3.4) must be defined in INCAR. \item as of VASP.5.3.4, the default value for {\tt VDW\_RADIUS} has been increased from 30 to 50 {\AA}. \item Ewald's summation in $E_{disp}$ calculation (controlled via {\tt LVDW\_EWALD}) implemented according to Ref.~\cite{Kerber:08} is available as of VASP.5.3.4 \end{itemize}

References

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