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Tkatchenko-Scheffler method

The expression for dispersion energy within the method of Tkatchenko and Scheffler [131] (DFT-TS) is formally identical to that of DFT-D2 method (see eq. 6.87), the important difference is, however, that the dispersion coefficients and damping function are charge-density dependent. The DFT-TS method is therefore able to take into account variations in vdW contributions of atoms due to their local chemical environment. In this method, polarizability, dispersion coeficients, and atomic radii of an atom in molecule or solid are computed from their free-atomic values using the following relations:

$\displaystyle \alpha_{i} = \nu_{i}  \alpha_{i}^{free},$ (6.94)

$\displaystyle C_{6ii} = \nu_{i}^{2} C_{6ii}^{free},$ (6.95)

$\displaystyle R_{0i} = \left(\frac{\alpha_{i}}{\alpha_{i}^{free}} \right)^{\frac{1}{3}} R_{0i}^{free}.$ (6.96)

The free-atomic quantities $ \alpha_{i}^{free}$, $ C_{6ii}^{free}$, and $ R_{0i}^{free}$ are tabulated for all elements from the first six rows of the periodic table except of lanthanides. If a DFT-TS calculation is performed for the system containing the unsupported elements, the user must define corresponding values using the tags VDW_ALPHA, VDW_C6, and VDW_R0, see below. The effective atomic volumes $ \nu_{i}$ are determined using the Hirshfeld partitioning of the all-electron density:

$\displaystyle \nu_{i} = \frac{\int r^3  w_i({\mathbf{r}}) n({\mathbf{r}})  d^3{\mathbf{r}}}{\int r^3  n_{i}^{free}({\mathbf{r}}) d^3{\mathbf{r}}},$ (6.97)

where $ n({\mathbf{r}})$ is the total electron density, and $ n_{i}^{free}({\mathbf{r}})$ is the spherically averaged electron density of the neutral free atomic species $ i$. The Hirshfeld weight $ w_i({\mathbf{r}})$ is defined by free atomic densities as follows:

$\displaystyle w_i({\mathbf{r}}) = \frac{n_{i}^{free}({\mathbf{r}})}{\sum_{j=1}^{N_{at}} n_{j}^{free}({\mathbf{r}})}.$ (6.98)

The combination rule to define the strength of the dipole-dipole dispersion interaction between unlike species is:

$\displaystyle C_{6ij} = \frac{2C_{6ii} C_{6jj}}{[\frac{\alpha_{j}} {\alpha_{i}}C_{6ii}+\frac{\alpha_{i}}{\alpha_{j}}C_{6jj}]}.$ (6.99)

The parameter $ R_{0ij}$ used in damping function (see eq. 6.90) is obtained from the atom-in-molecule vdW radii as follows:

$\displaystyle R_{0ij} = R_{0i} + R_{0j}.$ (6.100)

The DFT-TS calculation is invoked by setting IVDW=2$ \vert$20. The following parameters can be optionally defined in INCAR:

VDW_RADIUS = 50.0 cutoff radius (Å) for pair interactions
VDW_S6 = 1.00 global scaling factor $ s_6$
VDW_SR = 0.94 scaling factor $ s_R$
VDW_D = 20.0 damping parameter $ d$
VDW_ALPHA = [real array] free-atomic polarizabilities (atomic units) for each species
    defined in POSCAR
VDW_C6AU = [real array] free-atomic $ C_6$ parameters (atomic units) for each species
    defined in POSCAR
VDW_C6 = [real array] free-atomic $ C_6$ parameters ( $ Jnm^6mol^{-1}$) for each species
    defined in POSCAR (this parameter overrides VDW_C6AU)
VDW_R0AU = [real array] free-atomic $ R_0$ parameters (atomic units) for each species
    defined in POSCAR
VDW_R0 = [real array] $ R_0$ parameters (Å) for each species
    defined in POSCAR (this parameter overrides VDW_R0AU)
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)

Performance of PBE-TS method in optimization of various crystalline systems has been examined in Phys. Rev. B. 87, 064110 (2013).


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