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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 (this parameter is obsolete as of VASP.5.3.3).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 $\AA$  ) 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 ($\AA$  ) 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 $\AA$  .
• 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.