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# 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 (this parameter is obsolete as of VASP.5.3.3).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):

• 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. decides whether lattice summation in $E_{disp}$ expression by means of Ewald's summation is computed (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 .\\

## IMPORTANT NOTES

• The defaults for VDW_C6 and 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.
• The defaults for parameters controlling the damping function (VDW_S6, VDW_SR, VDW_D) are available only for the PBE functional. If a functional other than PBE is used in DFT+D2 calculation, the value of VDW_S6 (or VDW_SCALING in versions before VASP.5.3.4) must 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 $E_{disp}$ calculation (controlled via LVDW_EWALD) is implemented according to reference  and is available as of VASP.5.3.4.