Many-body dispersion energy: Difference between revisions

Revision as of 12:15, 20 January 2017

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

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

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 reference ^[3] for details) is as follows

$E_{\mathrm{disp}} = -\int_{\mathrm{FBZ}}\frac{d{\mathbf{k}}}{v_{\mathrm{FBZ}}} \int_0^{\infty} {\frac{d\omega}{2\pi}} \, {\mathrm{Tr}}\left \{ \mathrm{ln} \left ({\mathbf{1}}-{\mathbf{A}}^{(0)}_{LR}(\omega) {\mathbf{T}}_{LR}({\mathbf{k}}) \right ) \right \}$

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 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 ( $\beta$ ) that must be adjusted for each exchange-correlation functional. The default value of $\beta$ =0.83 corresponds to the PBE functional. If another functional is used, the value of $\beta$ 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):

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

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.

Related Tags and Sections

IVDW, IALGO, DFT-D2, DFT-D3, Tkatchenko-Scheffler method, Tkatchenko-Scheffler method with iterative Hirshfeld partitioning

References

Contents

[tkatchenko2012-1] A. Tkatchenko, R. A. Di Stasio, R. Car, and M. Scheffler, Phys. Rev. Lett. 108, 236402 (2012).

[tkatchenko2014-2] A. Ambrosetti, A. M. Reilly, R. A. DiStasio, J. Chem. Phys. 140, 018A508 (2014).

[bucko2016-3] T. Bucko, S. Lebègue, T. Gould, and J. G. Ángyán, J. Phys.: Condens. Matter 28, 045201 (2016).

[1]

[2]

[3]

@@ Line 3: / Line 3: @@
 <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.
+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
-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}
+<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>
-         {\tt VDW\_SR} & = 0.83& scaling parameter $\beta$ \\
-         {\tt LVDWEXPANSION} &=.FALSE.$|$.TRUE. & write the two- to six- body contributions to MBD\\
+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
-                       & &  dispersion energy in the output file (OUTCAR) - no$|$yes\\
+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
-         {\tt LSCSGRAD}& =.TRUE.$|$.FALSE.& compute gradients - yes$|$no\\
+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}}
-        %{\tt VDW\_alpha}, {\tt VDW\_C6}, {\bf VDW\_R0} & & atomic reference, see Sec.~\ref{sec:vdwTS}.
+(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):
- \end{tabular}
-\\
+*{{TAG|VDW_SR}}=0.83 scaling parameter <math>\beta</math>
-\hspace{5mm}
+*{{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)
-\noindent Details of implementation of the MBD@rsSCS method in VASP
+*{{TAG|VDW_alpha}}, {{TAG|VDW_C6}}, {{TAG|VDW_R0}} atomic reference (see alse {{TAG|Tkatchenko-Scheffler method}})
-are presented in
-J. Phys: Condens. Matter 28, 045201 (2016).
+Details of implementation of the MBD@rsSCS method in VASP are presented in reference <ref name="bucko2016"/>.
-\\
-\hspace{5mm}
+== IMPORTANT NOTES ==
-\\
-\noindent IMPORTANT NOTES:
+*This method requires the use of {{TAG|POTCAR}} files from the PAW dataset version 52 or later.
-\begin{itemize}
+*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.
-\item
+*The charge-density dependence of gradients is neglected.
-this method requires the use of POTCAR files from the
+*This method is incompatible with the setting {{TAG|ADDGRID}}=''.TRUE.''.
-PAW dataset version 52 or later
+*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''}).
-\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 ''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.
-the input reference
+* Analytical gradients of energy are implemented (fore details see reference <ref name="bucko2016"/>) and hence the atomic and lattice relaxations can be performed.
-data for non-interacting atoms are available only for elements
+*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.
-of the first six rows of periodic table except of lanthanides.
+*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.
-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 ==
@@ Line 99: / Line 35: @@
 {{TAG|IALGO}},
 {{TAG|DFT-D2}},
-{{TAG|DFT-D3}}
+{{TAG|DFT-D3}},
+{{TAG|Tkatchenko-Scheffler method}},
+{{TAG|Tkatchenko-Scheffler method with iterative Hirshfeld partitioning}}
 == References ==
@@ Line 105: / Line 43: @@
 <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="kerber">[http://onlinelibrary.wiley.com/doi/10.1002/jcc.21069/abstract Kerber and J. Sauer, J. Comp. Chem. 29, 2088 (2008).]</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>
 ----