Many-body dispersion energy

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

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

whereby the response function $\chi _{0}$ 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.~\cite{Bucko:16} for details) is as follows \begin{equation}\label{eq_energy_k1} E_{\text{disp}} = -\int_{\text{FBZ}}\frac{d{\mathbf{k}}}{v_{\text{FBZ}}} \int_0^{\infty} {\frac{d\omega}{2\pi}} \, {\text{Tr}}\left \{ \text{ln} \left ( {\mathbf{1}}-{\mathbf{A}}^{(0)}_{LR}(\omega) {\mathbf{T}}_{LR}({\mathbf{k}}) \right ) \right \}, \end{equation} 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.~\cite{Ambrosetti:14,Bucko:16} for details); the input reference data for non-interacting atoms can be optionally defined via parameters {\tt VDW\_alpha}, {\tt VDW\_C6}, {\tt VDW\_R0} (described in sec.~\ref{sec:vdwTS}). %The input reference %data for non-interacting atoms can be optionally defined 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 {\tt VDW\_SR} in INCAR. The MBD@rsSCS method is invoked by defining {\tt IVDW}=202. Optionally, the following parameters can be user-defined:\\

\begin{tabular}{rll}

        {\tt VDW\_SR} & = 0.83& scaling parameter $\beta$ \\
        {\tt LVDWEXPANSION} &=.FALSE.$|$.TRUE. & write the two- to six- body contributions to MBD\\
                      & &  dispersion energy in the output file (OUTCAR) - no$|$yes\\
        {\tt LSCSGRAD}& =.TRUE.$|$.FALSE.& compute gradients - yes$|$no\\
       %{\tt VDW\_alpha}, {\tt VDW\_C6}, {\bf VDW\_R0} & & atomic reference, see Sec.~\ref{sec:vdwTS}. 
\end{tabular}

\\ \hspace{5mm} \\ \noindent Details of implementation of the MBD@rsSCS method in VASP are presented in J. Phys: Condens. Matter 28, 045201 (2016). \\ \hspace{5mm} \\ \noindent IMPORTANT NOTES: \begin{itemize} \item this method requires the use of POTCAR files from the PAW dataset version 52 or later \item the input reference data for non-interacting atoms are available only for elements of the first six rows of 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 {\tt VDW\_alpha}, {\tt VDW\_C6}, {\tt VDW\_R0} (described in sec.~\ref{sec:vdwTS}) defined in the INCAR file. \item the charge-density dependence of gradients is neglected \item this method is incompatible with the setting {\tt ADDGRID=.TRUE.} \item it is essential that a sufficiently dense FFT grid (controlled via {\tt NGFX(Y,Z)}) is used in the DFT-TS - we strongly recommend to use {\tt PREC=Accurate} for this type of calculations (in any case, avoid using {\tt PREC=Low}). \item 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 ({\tt 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. \item analytical gradients of energy are implemented (fore details see Ref.~\cite{Bucko:16}) and hence the atomic and lattice relaxations can be performed \item 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 \item a default value for the free-parameter of this method ({\tt VDW\_SR}=0.83) is available only for the PBE functional. If the functional other than PBE is used, the value of {\tt VDW\_SR} must be specified in INCAR. \end{itemize}

Related Tags and Sections

IVDW, IALGO, DFT-D2, DFT-D3

References

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