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There are many different dispersion corrections out there. The most famous is D3 [1] (and the new D4 [2]), but there's probably other approaches too. The dispersion energy can be written as:

$$ E_{\textrm{disp}} = -\sum_{ij}\sum_{6,8,10,\cdot \cdot }\frac{C_n^{ij}f^{(n)}_{\textrm{damp}}(r)}{r^n}, $$

where the $C^{ij}_n$ coefficients are calculated differently depending on whether using D3 or D4, and the damping function $f$ can take on various forms.

What are the other ways of calculating $C^{ij}_n$?


[1] J. Chem. Phys. 132, 154104 (2010)
[2] J. Chem. Phys. 147, 034112 (2017)

P.S. For further discussion and overview, see Grimme's and Tkatchenko's review papers.

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    $\begingroup$ This answer by Geoff Hutchison might be helpful to you: materials.stackexchange.com/a/1120/5 $\endgroup$ – Nike Dattani May 30 '20 at 21:06
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    $\begingroup$ Considering that this was one of the first questions on the site, and one of the highest voted, but still remains un-answered 40 days later, I tried to reach out to Geoff last week to answer this since he answered a similar question very very well already. With still no answer, I looked at the review article Geoff suggested. It addresses your question about what all the D corrections are. No one can possibly write all methods, and compare them, and then also answer your 3rd question question about the validity of D corrections in general, without basically re-writing that paper here. $\endgroup$ – Nike Dattani Jun 7 '20 at 3:39
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    $\begingroup$ As for comparing the different methods, you can look at Tabs. 4 & 9, or Fig 15 of this review which Geoff mentioned. It's also the review Stefan Grimme sent me by email yesterday, so his group doesn't have anything more recent. I did find a review by a different group, which was published 1 yr later, & Tab 1 gives more comparison. This addresses your question about "comparisons". As for validity: you are right: none of these methods perfectly account for dispersion. $\endgroup$ – Nike Dattani Jun 7 '20 at 3:45
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    $\begingroup$ Excellent @NikeDattani! Thanks for all the comments and answer. It seems now that the original question was too broad indeed. $\endgroup$ – Felipe S. S. Schneider Jun 8 '20 at 10:56
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    $\begingroup$ @NikeDattani By the way, both reviews are excellent! $\endgroup$ – Felipe S. S. Schneider Jun 8 '20 at 11:14
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2007 (Becke & Johnson): XDM

XDM stands for "exchange-hole dipole moment" which is a model introduced by Becke and Johnson in 2007 for calculating dispersion constants. The formulas are as follows:

$$\begin{align} \!\!\!\!\!\!\!\!C_6 &= \frac{\alpha_i\alpha_j}{\mathcal{M}_i\alpha_j + \mathcal{M}_j\alpha_i} \mathcal{M}_i\mathcal{M}_j \tag{1}\label{eq1} \\ \!\!\!\!\!\!\!\!C_8 &= \frac{\alpha_i\alpha_j}{\mathcal{M}_i\alpha_j + \mathcal{M}_j\alpha_i}\left(\frac{3}{2}\mathcal{M}_{1i}\mathcal{M}_{2j} + \frac{3}{2}\mathcal{M}_{2i}\mathcal{M}_{1j}\right) \tag{2}\label{eq2}\\ \!\!\!\!\!\!\!\!C_{10} &= \frac{\alpha_i\alpha_j}{\mathcal{M}_i\alpha_j + \mathcal{M}_j\alpha_i}\left(\frac{10}{5}\mathcal{M}_{1j}\mathcal{M}_{3j} + \frac{10}{5}\mathcal{M}_{3i} \mathcal{M}_{1j} + \frac{21}{5}\mathcal{M}_{2i} \mathcal{M}_{2j} \right), \tag{3}\label{eq3} \end{align}$$

where for the system $x$: the dipole polarizability is $\alpha_{x}$ and the $l^{\textrm{th}}$ multi-pole moment is $\mathcal{M_{lx}}$, and $l=1,2,3$ correspond to the dipole, quadrupole, and octupole moments respectively.

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2009 (Tkatchenko−Scheffler) TS

The Tkatchenko−Scheffler model for van der Waals interactions (vdW) defines the $C_6^{AB}$ parameters in an ab-initio fashion. In TS model the vdW energy $E_{vdw}$ is defined as,

\begin{equation} E_{\text{vdW}} = -\frac{1}{2}\sum_{A,B}f_{\text{damp}}\left(R_{AB},R^{0}_{A},R^{0}_{B}\right)C_{6}^{AB}R^{-6}_{AB} \tag{1} \end{equation}

where $R^0_{A}$ and $R^0_{B}$ are the vdW radii. The $C_6^{AB}$ parameter can defined by the Casimir-Polder integral exactly:

$$ C_6^{AB}=\frac{3}{\pi}\int_{0}^{\infty}\alpha_{{A}}(i\omega)\alpha_{{B}}(i\omega)d\omega \tag{2} \label{eq:eq2} $$

where $\alpha_{A/B}(i\omega)$ is the frequency-dependent polarizability of $A$ and $B$ evaluated at imaginary frequencies. The $\alpha_{A/B}(i\omega)$ can be replaced by an approximate $\alpha^1_{A/B}(i\omega)$, where $\alpha^1_{A}(i\omega)=\alpha^{0}_{A}/[1-(\omega/\eta_{A})^2]$. $\alpha^{0}_{A}$ is the static polarizability of $A$ and $\eta_{A}$ is an effective frequency. Simplifying \eqref{eq:eq2}, we get:

$$ C_6^{AB}=\frac{3}{2}[\eta_{A}\eta_{B}/(\eta_{A}+\eta_{B})]\alpha_{A}^0\alpha_{B}^0\tag{3}\label{eq:eq3} $$

which after further simplification results in:

$$ C_6^{AB}=\frac{2C_6^{AA}C_6^{BB}}{[\frac{\alpha_{B}^0}{\alpha_{A}^0}C_6^{AA}+\frac{\alpha_{A}^0}{\alpha_{B}^0}C_6^{BB}]}\tag{4} $$

$C_6^{AA}$ and $\alpha_{A}^0$ can be determined from highly accurate benchmark calculations.

Note: Here $C_6^{ij}\equiv C_6^{AB}$, $i\equiv A$ and $j\equiv B$

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    $\begingroup$ Hello, can help someone help me edit the equations here? The preview shows the equation format to be ok. $\endgroup$ – mykd Jul 30 '20 at 19:12
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    $\begingroup$ Would it be possible to create a community page about "Dispersion methods in density functional theory" ? My field is condensed phase dynamics, and the consensus between molecular electronic structure calculations and bulk/interfacial simulations is actually not great, for example in the works of Angelos Michaelides: aip.scitation.org/doi/10.1063/1.4944633 & aip.scitation.org/doi/10.1063/1.4754130 . I think a lot of users would benefit from such a page. $\endgroup$ – mykd Jul 30 '20 at 20:11
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    $\begingroup$ @NikeDattani , I meant a "community wiki page" meta.stackexchange.com/questions/11740/… . BTW, I edited the G16 post. $\endgroup$ – mykd Jul 30 '20 at 20:24
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    $\begingroup$ Thank you for removing your answer on the G16 post. As for community wiki, we don't have any here yet, but we could start. Mind you, it doesn't need to be a community wiki, neither was this. $\endgroup$ – Nike Dattani Jul 30 '20 at 20:34
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    $\begingroup$ @mykd Nike and I seem to have fixed the issue. Apparently, equation labels persist for the whole page and you happened to have the same equation labels as Nike for your first three expressions. $\endgroup$ – Tyberius Jul 30 '20 at 21:01

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