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# Many-body dispersion energy

The many-body dispersion energy method (MBD@rsSCS) of Tkatchenko et al.[1][2]is based on the random phase expression for the correlation energy

${\displaystyle E_{c}=\int _{{0}}^{{\infty }}{\frac {d\omega }{2\pi }}{\mathrm {Tr}}\left\{{\mathrm {ln}}(1-v\chi _{0}(i\omega ))+v\chi _{0}(i\omega )\right\}}$

whereby the response function ${\displaystyle \chi _{0}}$ is approximated by a sum of atomic contributions represented by quantum harmonic oscillators. The expression for dispersion energy used in our k-space implementation of the MBD@rsSCS method (see reference [3] for details) is as follows

${\displaystyle E_{{{\mathrm {disp}}}}=-\int _{{{\mathrm {FBZ}}}}{\frac {d{{\mathbf {k}}}}{v_{{{\mathrm {FBZ}}}}}}\int _{0}^{{\infty }}{{\frac {d\omega }{2\pi }}}\,{{\mathrm {Tr}}}\left\{{\mathrm {ln}}\left({{\mathbf {1}}}-{{\mathbf {A}}}_{{LR}}^{{(0)}}(\omega ){{\mathbf {T}}}_{{LR}}({{\mathbf {k}}})\right)\right\}}$

where ${\displaystyle {{\mathbf {A}}}_{{LR}}}$ is the frequency-dependent polarizability matrix and ${\displaystyle {\mathbf {T}}_{{LR}}}$ is the long-range interaction tensor, which describes the interaction of the screened polarizabilities embedded in the system in a given {geometrical } arrangement. The components of ${\displaystyle {\mathbf {A}}_{{LR}}}$ are obtained using an atoms-in-molecule approach as employed in the pairwise Tkatchenko-Scheffler method (see references [2][3] for details). The input reference data for non-interacting atoms can be optionally defined via the parameters VDW_ALPHA, VDW_C6, VDW_R0 (described by the Tkatchenko-Scheffler method). This method has one free parameter (${\displaystyle \beta }$) that must be adjusted for each exchange-correlation functional. The default value of ${\displaystyle \beta }$=0.83 corresponds to the PBE functional. If another functional is used, the value of ${\displaystyle \beta }$ must be specified via VDW_SR in the INCAR file. The MBD@rsSCS method is invoked by setting IVDW=202. Optionally, the following parameters can be user-defined (the given values are the default ones values):

Details of implementation of the MBD@rsSCS method in VASP are presented in reference [3].

## 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 are available only for elements of the first six rows of the periodic table except of the 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 and VDW_R0 (described by the Tkatchenko-Scheffler method) defined in the INCAR file.
• The charge-density dependence of gradients is neglected.
• This 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}).
• The method has sometimes numerical problems if highly polarizable atoms are located at short distances. In such a case the calculation terminates with an error message Error(vdw\_tsscs\_range\_separated\_k): d\_lr(pp)<=0. Note that this problem is not caused by a bug but rather it is due to a limitation of the underlying physical model.
• Analytical gradients of energy are implemented (fore details see reference [3]) and hence the atomic and lattice relaxations can be performed.
• Due to the long-range nature of dispersion interactions, the convergence of energy with respect to the number of k-points should be carefully examined.
• A default value for the free-parameter of this method (VDW_SR=0.83) is 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.