DFT-D2

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

[math]\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}) }[/math]

where the summations are over all atoms [math]\displaystyle{ N_{at} }[/math] and all translations of the unit cell [math]\displaystyle{ {L}=(l_1,l_2,l_3) }[/math]. The prime indicates that [math]\displaystyle{ i\not=j }[/math] for [math]\displaystyle{ {L}=0 }[/math], [math]\displaystyle{ C_{6ij} }[/math] denotes the dispersion coefficient for the atom pair [math]\displaystyle{ ij }[/math], [math]\displaystyle{ {r}_{ij,L} }[/math] is the distance between atom [math]\displaystyle{ i }[/math] located in the reference cell [math]\displaystyle{ L=0 }[/math] and atom [math]\displaystyle{ j }[/math] in the cell [math]\displaystyle{ L }[/math] and the term [math]\displaystyle{ f(r_{ij}) }[/math] 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 [math]\displaystyle{ E_{\mathrm{disp}} }[/math] corresponding to interactions over distances longer than a certain suitably chosen cutoff radius contribute only negligibly to [math]\displaystyle{ E_{\mathrm{disp}} }[/math] and can be ignored. Parameters [math]\displaystyle{ C_{6ij} }[/math] and [math]\displaystyle{ R_{0ij} }[/math] are computed using the following combination rules:

[math]\displaystyle{ C_{6ij} = \sqrt{C_{6ii} C_{6jj}} }[/math]

and

[math]\displaystyle{ R_{0ij} = R_{0i}+ R_{0j}. }[/math]

The values for [math]\displaystyle{ C_{6ii} }[/math] and [math]\displaystyle{ R_{0i} }[/math] are tabulated for each element and are insensitive to the particular chemical situation (for instance, [math]\displaystyle{ C_6 }[/math] 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:

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

whereby the global scaling parameter [math]\displaystyle{ s_6 }[/math] has been optimized for several different DFT functionals such as PBE ([math]\displaystyle{ s_6=0.75 }[/math]), BLYP ([math]\displaystyle{ s_6=1.2 }[/math]) or B3LYP ([math]\displaystyle{ s_6=1.05 }[/math]). The parameter [math]\displaystyle{ s_R }[/math] 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 [math]\displaystyle{ \AA }[/math]) for pair interactions
• VDW_S6=0.75 global scaling factor [math]\displaystyle{ s_6 }[/math] (available in VASP.5.3.4 and later)
• VDW_SR=1.00 scaling factor [math]\displaystyle{ s_R }[/math] (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 [math]\displaystyle{ d }[/math]
• VDW_C6=[real array] [math]\displaystyle{ C_6 }[/math] parameters ([math]\displaystyle{ \mathrm{Jnm}^{6}\mathrm{mol}^{-1} }[/math]) for each species defined in the POSCAR file
• VDW_R0=[real array] [math]\displaystyle{ R_0 }[/math] parameters ([math]\displaystyle{ \AA }[/math]) for each species defined in the POSCAR file
• LVDW_EWALD=.FALSE. decides whether lattice summation in [math]\displaystyle{ E_{disp} }[/math] 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 [math]\displaystyle{ \AA }[/math].

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

Related Tags and Sections

IVDW, IALGO, DFT-D3, Tkatchenko-Scheffler method, Tkatchenko-Scheffler method with iterative Hirshfeld partitioning, Self-consistent screening in Tkatchenko-Scheffler method, Many-body dispersion energy, dDsC dispersion correction

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