An advanced combination rule for the van der Waals \(C_8\) dispersion coefficient

Below you find an advanced combination rule for the higher \(C_8\) dispersion coefficient of the van der Waals interaction between two atoms or molecules A and B (g = mean dipole respectively quadrupol excitation energy):

$$\quad C^{AB}_{8} = {1 \over 2} [ { {C^{A}_{6} C^{B}_{8} (1 + g^B )} \over {{C^{A}_{6}{\alpha^B_1 \over \alpha^A_1}+g^B C^{B}_{6} {\alpha^A_1 \over \alpha^B_1}}}} + { {C^{A}_{8} C^{B}_{6} (1 + g^A )} \over {{g^B C^{A}_{6}{\alpha^B_1 \over \alpha^A_1}+C^{B}_{6} {\alpha^A_1 \over \alpha^B_1}}}} ]$$

The above rule was derived in the Appendix (VII.1) of the Dissertation of J. Schleusener (1978) and improves the standard combination rule (geometric mean)

$$C^{AB}_{8} = \sqrt {C^{AA}_{8} C^{BB}_{8}}$$

significantly (especially for large size differences of the involved atoms).

The improved expression has the great advantage of being independent of quadrupole polarizability $\alpha_{2}$, but has a dependency on the ratio g of the mean dipolar or quadrupole excitation energies. For alkalis, the one-electron oscillator model suggests the choice g = 2, whereas in the case of hydrogen and noble gases, where excitation energies and ionization energies are close together, the value g = 1 seems reasonable as a first approximation.