View source for Many-body dispersion energy

The many-body dispersion energy method (MBD@rsSCS) of Tkatchenko et al.<ref name="tkatchenko2012"/><ref name="tkatchenko2014"/>is based on the random phase expression for the correlation energy

<math> 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\} </math>

whereby the response function <math>\chi_0</math> 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 <ref name="bucko2016"/> for details) is as follows

<math>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}}^{(0)}_{LR}(\omega) {\mathbf{T}}_{LR}({\mathbf{k}}) \right ) \right \} </math>

where <math>{\mathbf{A}}_{LR}</math> is the frequency-dependent polarizability matrix and <math>\mathbf{T}_{LR}</math> 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 <math>\mathbf{A}_{LR}</math> are obtained using an atoms-in-molecule approach as employed in the pairwise Tkatchenko-Scheffler method (see
references <ref name="tkatchenko2014"/><ref name="bucko2016"/> for details). The input reference data for non-interacting atoms can be optionally defined via the parameters {{TAG|VDW_alpha}}, {{TAG|VDW_C6}}, {{TAG|VDW_R0}}
(described by the {{TAG|Tkatchenko-Scheffler method}}). This method has one free parameter (<math>\beta</math>) that must be adjusted for each exchange-correlation functional. The default value of <math>\beta</math>=0.83 corresponds to the PBE functional. If another functional is used, the value of <math>\beta</math> must be specified via {{TAG|VDW_SR}} in the {{TAG|INCAR}} file. The MBD@rsSCS method is invoked by setting {{TAG|IVDW}}=202. Optionally, the following parameters can be user-defined (the given values are the default ones values):

*{{TAG|VDW_SR}}=0.83 scaling parameter <math>\beta</math>
*{{TAG|LVDWEXPANSION}}=.FALSE. writes the two- to six- body contributions tothe  MBD dispersion energy in the {{TAG|OUTCAR}} ({{TAG|LVDWEXPANSION}}=.TRUE.)
*{{TAG|LSCSGRAD}}=.TRUE. compute gradients (or not)
*{{TAG|VDW_alpha}}, {{TAG|VDW_C6}}, {{TAG|VDW_R0}} atomic reference (see alse {{TAG|Tkatchenko-Scheffler method}}) 

Details of implementation of the MBD@rsSCS method in VASP are presented in reference <ref name="bucko2016"/>.

== IMPORTANT NOTES ==

*This method requires the use of {{TAG|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 {{TAG|VDW_alpha}}, {{TAG|VDW_C6}} and {{TAG|VDW_R0}} (described by the {{TAG|Tkatchenko-Scheffler method}}) defined in the {{TAG|INCAR}} file.
*The charge-density dependence of gradients is neglected.
*This method is incompatible with the setting {{TAG|ADDGRID}}=''.TRUE.''.
*It is essential that a sufficiently dense FFT grid (controlled via {{TAG|NGFX(Y,Z)}}) is used in the DFT-TS calculation. We strongly recommend to use {{TAG|PREC}}=''Accurate'' for this type of calculations (in any case, avoid using {{TAG|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 <ref name="bucko2016"/>) 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 ({{TAG|VDW_SR}}=0.83) is available only for the PBE functional. If a functional other than PBE is used, the value of {{TAG|VDW_SR}} must be specified in the {{TAG|INCAR}} file.

== Related Tags and Sections ==
{{TAG|IVDW}},
{{TAG|IALGO}},
{{TAG|DFT-D2}},
{{TAG|DFT-D3}},
{{TAG|Tkatchenko-Scheffler method}},
{{TAG|Tkatchenko-Scheffler method with iterative Hirshfeld partitioning}}

== References ==
<references>
<ref name="tkatchenko2012">[http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.236402 A. Tkatchenko, R. A. Di Stasio, R. Car, and M. Scheffler, Phys. Rev. Lett. 108, 236402 (2012).]</ref>
<ref name="tkatchenko2014">[http://aip.scitation.org/doi/full/10.1063/1.4865104 A. Ambrosetti, A. M. Reilly, R. A. DiStasio, J. Chem. Phys. 140, 018A508 (2014).]</ref>
<ref name="bucko2016">[http://iopscience.iop.org/article/10.1088/0953-8984/28/4/045201/meta T. Bucko, S. Lebègue, T. Gould, and J. G. Ángyán, J. Phys.: Condens. Matter 28, 045201 (2016).]</ref>
</references>
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