What are the different ways of calculating dispersion constants?

Question

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 $C^{ij}_n$?

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[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.

Answer 1

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.

Answer 2

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 $j\equiv B$