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Many-body dispersion energy method

The many-body dispersion energy method (MBD@rsSCS) of Tkatchenko et al. [130,137] is based on the random phase expression of correlation energy

$\displaystyle E_c = \int_{0}^{\infty} \frac{d\omega}{2\pi}$   Tr$\displaystyle \left\{ \text{ln} (1-v\chi_0(i\omega))+v\chi_0(i\omega) \right\},$ (6.108)

whereby the response function $ \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 Ref. [136] for details) is as follows

$\displaystyle E_{\text{disp}} = -\int_{\text{FBZ}}\frac{d{\mathbf{k}}}{v_{\text...
...thbf{A}}^{(0)}_{LR}(\omega) {\mathbf{T}}_{LR}({\mathbf{k}}) \right ) \right \},$ (6.109)

where $ {\mathbf{A}}_{LR}$ is the frequency-dependent polarizability matrix and $ {\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 $ {\mathbf{A}}_{LR}$ are obtained using an atoms-in-molecule approach as employed in the pairwise Tkatchenko-Scheffler method (see Ref. [137,136] for details); the input reference data for non-interacting atoms can be optionally defined via parameters VDW_alpha, VDW_C6, VDW_R0 (described in sec. 6.77.3).This method has one free parameter ($ \beta$) that must be adjusted for each exchange-correlation functional. The default value of $ \beta$ (0.83) corresponds to PBE functional; if other functional is used, the value of $ \beta$ must be specified via VDW_SR in INCAR. The MBD@rsSCS method is invoked by defining IVDW=202. Optionally, the following parameters can be user-defined:

VDW_SR = 0.83 scaling parameter $ \beta$
LVDWEXPANSION =.FALSE.$ \vert$.TRUE. write the two- to six- body contributions to MBD
    dispersion energy in the output file (OUTCAR) - no$ \vert$yes
LSCSGRAD =.TRUE.$ \vert$.FALSE. compute gradients - yes$ \vert$no

Details of implementation of the MBD@rsSCS method in VASP are presented in J. Phys: Condens. Matter 28, 045201 (2016).


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