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# Tkatchenko-Scheffler method

The expression for dispersion energy within the method of Tkatchenko and Scheffler[1] (DFT-TS) is formally identical to that of the DFT-D2 method. 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 coefficients, and atomic radii of an atom in a molecule or a solid are computed from their free-atomic values using the following relations:

${\displaystyle \alpha _{{i}}=\nu _{{i}}\,\alpha _{{i}}^{{free}},}$

${\displaystyle C_{{6ii}}=\nu _{{i}}^{{2}}\,C_{{6ii}}^{{free}},}$

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

The free-atomic quantities ${\displaystyle \alpha _{{i}}^{{free}},C_{{6ii}}^{{free}}}$ and ${\displaystyle R_{{0i}}^{{free}}}$ 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 ${\displaystyle \nu _{{i}}}$ are determined using the Hirshfeld partitioning of the all-electron density:

${\displaystyle \nu _{{i}}={\frac {\int r^{3}\,w_{i}({{\mathbf {r}}})n({{\mathbf {r}}})\,d^{3}{{\mathbf {r}}}}{\int r^{3}\,n_{{i}}^{{free}}({{\mathbf {r}}})\,d^{3}{{\mathbf {r}}}}}}$

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

${\displaystyle w_{i}({{\mathbf {r}}})={\frac {n_{{i}}^{{free}}({{\mathbf {r}}})}{\sum _{{j=1}}^{{N_{{at}}}}n_{{j}}^{{free}}({{\mathbf {r}}})}}.}$

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

${\displaystyle C_{{6ij}}={\frac {2C_{{6ii}}\,C_{{6jj}}}{[{\frac {\alpha _{{j}}}{\alpha _{{i}}}}C_{{6ii}}+{\frac {\alpha _{{i}}}{\alpha _{{j}}}}C_{{6jj}}]}}.}$

The parameter ${\displaystyle R_{{0ij}}}$ used in the damping function of the DFT-D2 method is obtained from the atom-in-molecule vdW radii as follows:

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

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

• VDW_RADIUS=50.0 cutoff radius (in Å) for pair interactions
• VDW_S6=1.00 global scaling factor ${\displaystyle s_{6}}$
• VDW_SR=0.94 scaling factor ${\displaystyle s_{R}}$
• VDW_D=20.0 damping parameter ${\displaystyle d}$
• VDW_ALPHA=[real array] free-atomic polarizabilities (atomic units) for each species defined in the POSCAR file
• VDW_C6AU=[real array] free-atomic ${\displaystyle C_{6}}$ parameters (atomic units) for each species defined in the POSCAR file
• VDW_C6=[real array] free-atomic ${\displaystyle C_{6}}$ parameters (${\displaystyle {\mathrm {Jnm}}^{{6}}{\mathrm {mol}}^{{-1}}}$) for each species defined in the POSCAR file (this parameter overrides VDW_C6AU)
• VDW_R0AU=[real array] free-atomic ${\displaystyle R_{0}}$ parameters (atomic units) for each species defined in the POSCAR file
• VDW_R0=[real array] ${\displaystyle R_{0}}$ parameters (in Å) for each species defined in the POSCAR file (this parameter overrides VDW_R0AU)
• LVDW_EWALD=.FALSE. decides whether to compute the lattice summation in ${\displaystyle E_{{disp}}}$ expression by means of Ewald's summation (LVDW_EWALD=.TRUE.) or not (tag available in VASP.5.3.4 and later)

Performance of PBE-TS method in optimization of various crystalline systems has been examined in reference [2].

## IMPORTANT NOTES

• This method requires the use of POTCAR files from the PAW dataset version 52 or later.
• The input reference data for non-interacting atoms is available only for elements of the first six rows of the periodic table except of lanthanides. If the system contains other elements, the user must provide the free-atomic parameters for all atoms in the system via VDW_ALPHA, VDW_C6, VDW_R0 defined in the INCAR file.
• The charge-density dependence of gradients is neglected.
• The DFT-TS method is incompatible with the setting ADDGRID=.TRUE..
• It is essential that a sufficiently dense FFT grid (controlled via NGFX(Y,Z)) is used in the DFT-TS calculation - we strongly recommend to use PREC=Accurate for this type of calculations (in any case, avoid using PREC=Low).
• Defaults for the 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, the value of VDW_SR must be specified in the INCAR file.
• Ewald's summation in the calculation of ${\displaystyle E_{{disp}}}$ (controlled via LVDW_EWALD) implemented according to reference [3] is available as of VASP.5.3.4.
• Parameters VDW_C6AU and VDW_R0AU are available as of VASP.5.3.4.
• Hirshfeld charges for all configurations generated in a calculation are written out in the OUTCAR file. The corresponding table is introduced by the expression Hirshfeld charges:.