Category:Van der Waals functionals

The semilocal (SL) and hybrid exchange-correlation functionals do not include the London dispersion forces. Therefore, they can not be applied reliably on systems where the London dispersion forces play an important role. To account more properly for the London dispersion forces in DFT, a correlation dispersion term can be added to the semilocal or hybrid functional. This leads to the so-called van der Waals functionals:

${\displaystyle E_{\text{xc}} = E_{\text{xc}}^{\text{SL/hybrid}} + E_{\text{c,disp}}. }$

There are essentially three types of dispersion terms ${\displaystyle E_{\text{c,disp}}}$ that are available in VASP, and a brief sketch of them is given below along with links to pages that provide more detail. Note that libMBD is an external package that provides the Tkatchenko-Scheffler atom-pairwise methods and their many-body dispersion extensions.

Approximations

Atom-pairwise methods

They consist of a sum over the atoms pairs ${\displaystyle A}$-${\displaystyle B}$:

${\displaystyle E_{\text{c,disp}} = -\sum_{A<B}\sum_{n=6,8,10,\ldots}f_{n}^{\text{damp}}(R_{AB})\frac{C_{n}^{AB}}{R_{AB}^{n}}, }$

where ${\displaystyle C_{n}^{AB}}$ are the dispersion coefficients, ${\displaystyle R_{AB}}$ is the distance between atoms ${\displaystyle A}$ and ${\displaystyle B}$, and ${\displaystyle f_{n}^{\text{damp}}}$ is a damping function. Several variants of such atom-pair corrections exist and the most popular of them, listed below, are available in VASP and are selected with the IVDW tag.

• DFT-D2^[1]
• DFT-D3^[2][3]
• DFT-D4^[4] (available as of VASP.6.2 as external package)
• Tkatchenko-Scheffler method^[5]
• Tkatchenko-Scheffler method with iterative Hirshfeld partitioning^[6][7]
• Self-consistent screening in Tkatchenko-Scheffler method^[8]
• Library libMBD of many-body dispersion methods^[9][10][11]
• DDsC dispersion correction^[12][13]
• DFT-ulg^[14]

Many-body dispersion methods

These methods are based on the random-phase expression for the correlation energy, which is expressed as an integral over the frequency ${\displaystyle \omega}$ involving the frequency-dependent polarizability ${\displaystyle {\mathbf{A}}_{LR}}$:

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

These methods, listed below, are selected with the IVDW tag.

• Many-body dispersion energy^[8][15]
• Many-body dispersion energy with fractionally ionic model for polarizability^[16][17]
• Library libMBD of many-body dispersion methods^[9][10][11]

Nonlocal vdW-DF functionals

These are density functionals that require a double spatial integration and are, therefore, nonlocal:

${\displaystyle E_{\text{c,disp}} = \frac{1}{2}\int\int n(\textbf{r}) \Phi\left(\textbf{r},\textbf{r}'\right) n(\textbf{r}') d^{3}rd^{3}r', }$

where the kernel ${\displaystyle \Phi}$ depends on the electronic density ${\displaystyle n}$, its derivative ${\displaystyle \nabla n}$ as well as on the interelectronic distance ${\displaystyle \left\vert\bf{r}-\bf{r}'\right\vert}$. The nonlocal functionals are more expensive to calculate than semilocal functionals, however, they are efficiently implemented by using FFTs ^[18]. These methods are selected with the LUSE_VDW and IVDW_NL tags.

• Nonlocal vdW-DF functionals

References