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DFT-D2 method

In the D2 method of Grimme [127], the correction term takes the form:

$\displaystyle E_{\rm disp} = -\frac{1}{2} \sum_{i=1}^{N_{at}} \sum_{j=1}^{N_{at}} \sum_{{\bf L}}{}^\prime \frac{C_{6ij}} {r_{ij,L}^6} f_{d,6}({r}_{ij,{L}}),$ (6.87)

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 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 contributions from interactions within typical bonding distances. In practice, the terms in eq. 6.87 corresponding to interactions over distances longer than a certain suitably chosen cutoff radius contribute only negligibly to $ E_{\rm disp}$ and can be ignored. Parameters $ C_{6ij}$ and $ R_{0ij}$ are computed using the following combination rules:

$\displaystyle C_{6ij} = \sqrt{C_{6ii} C_{6jj}},$ (6.88)

$\displaystyle R_{0ij} = R_{0i}+ R_{0j},$ (6.89)

the values of $ 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 [127], Fermi-type damping function is used:

$\displaystyle f_{d,6}(r_{ij}) = \frac{s_6}{1+e^{-d(r_{ij}/(s_R R_{0ij})-1)}},$ (6.90)

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$ \vert$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 (this parameter is obsolete as of VASP.5.3.3).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 (Å) 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 ( $ Jnm^6mol^{-1}$) for each species
    defined in POSCAR
VDW_R0 = [real array] $ R_0$ parameters (Å) for each species
    defined in POSCAR
LVDW_EWALD = .FALSE.$ \vert$.TRUE. compute lattice summation in $ E_{disp}$ expression
    by means of Ewald's summation - no$ \vert$yes
    (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 J. Phys. Chem. A 114, 11814 (2010).


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