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# DFT-D2

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

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

where the summations are over all atoms ${\displaystyle N_{{at}}}$ and all translations of the unit cell ${\displaystyle {L}=(l_{1},l_{2},l_{3})}$. The prime indicates that ${\displaystyle i\not =j}$ for ${\displaystyle {L}=0}$, ${\displaystyle C_{{6ij}}}$ denotes the dispersion coefficient for the atom pair ${\displaystyle ij}$, ${\displaystyle {r}_{{ij,L}}}$ is the distance between atom ${\displaystyle i}$ located in the reference cell ${\displaystyle L=0}$ and atom ${\displaystyle j}$ in the cell ${\displaystyle L}$ and the term ${\displaystyle f(r_{{ij}})}$ 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 ${\displaystyle E_{{{\mathrm {disp}}}}}$ corresponding to interactions over distances longer than a certain suitably chosen cutoff radius contribute only negligibly to ${\displaystyle E_{{{\mathrm {disp}}}}}$ and can be ignored. Parameters ${\displaystyle C_{{6ij}}}$ and ${\displaystyle R_{{0ij}}}$ are computed using the following combination rules:

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

and

${\displaystyle R_{{0ij}}=R_{{0i}}+R_{{0j}}.}$

The values for ${\displaystyle C_{{6ii}}}$ and ${\displaystyle R_{{0i}}}$ are tabulated for each element and are insensitive to the particular chemical situation (for instance, ${\displaystyle C_{6}}$ for carbon in methane takes exactly the same value as that for C in benzene within this approximation). In the original method of Grimme[1], a Fermi-type damping function is used:

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

whereby the global scaling parameter ${\displaystyle s_{6}}$ has been optimized for several different DFT functionals such as PBE (${\displaystyle s_{6}=0.75}$), BLYP (${\displaystyle s_{6}=1.2}$) and B3LYP (${\displaystyle s_{6}=1.05}$). The parameter ${\displaystyle s_{R}}$ 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 (this parameter is obsolete as of VASP.5.3.3).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 ${\displaystyle \AA }$) for pair interactions
• VDW_S6=0.75 global scaling factor ${\displaystyle s_{6}}$ (available in VASP.5.3.4 and later)
• VDW_SR=1.00 scaling factor ${\displaystyle s_{R}}$ (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 ${\displaystyle d}$
• VDW_C6=[real array] ${\displaystyle C_{6}}$ parameters (${\displaystyle {\mathrm {Jnm}}^{{6}}{\mathrm {mol}}^{{-1}}}$) for each species defined in the POSCAR file
• VDW_R0=[real array] ${\displaystyle R_{0}}$ parameters (${\displaystyle \AA }$) for each species defined in the POSCAR file
• LVDW_EWALD=.FALSE. decides whether lattice summation in ${\displaystyle E_{{disp}}}$ 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 [2].\\

## 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 ${\displaystyle \AA }$.
• Ewald's summation in the calculation of ${\displaystyle E_{{disp}}}$ calculation (controlled via LVDW_EWALD) is implemented according to reference [3] and is available as of VASP.5.3.4.