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

An extract from the thesis (1978) of J. Schleusener: Derivation of an against the standard method (110) improved combination rule for the \(C_8\) dispersion coefficient of the van der Waals interaction between two atoms or molecules A and B (g = mean dipole respectively quadrupol excitation energy):

$$C^{AB}_{6} = \sqrt {C^{AA}_{6} C^{BB}_{6}} \space\space\space\space\space(109)$$

$$C^{AB}_{8} = \sqrt {C^{AA}_{8} C^{BB}_{8}} \space\space\space\space\space(110)$$

$$\alpha_l(i\omega) = \sum_{n} { f^l_n \over {(E^l_n)^2 -\omega^2}} \space\space\space\space\space(111)$$

$$f^l_n = {8\pi \over (2l+1)} E^l_n |<0|\sum_{i}r^l_iP_l(cos\theta_j)|n>|^2 \space\space\space\space\space(112)$$

$$C^{AB}(l,L) = {2l+2L \over {4(2l)!(2L)!}} {2 \over \pi} \int\limits_0^\infty{\alpha^A_l}(i\omega){\alpha^B_L}(i\omega)d\omega \space\space\space\space\space(113)$$

$$C_{6} = C^{AB}(1,1) \space\space\space\space\space(114)$$

$$C_{8} = C^{AB}(2,1)+C^{AB}(1,2) \space\space\space\space\space(115)$$

$$\alpha_l(i\omega) = {8\pi \over {2l+1}} {E_l \over {E_l^2-\omega^2}} \sum_{n}|<0|r^lP_l(cos\theta)|n>|^2 \space\space\space\space\space(116)$$

$$\alpha_l(i\omega) = {2 \over {2l+1}} {E_l \over {E_l^2-\omega^2}} <{r^{2l}}> \space\space\space\space\space(117)$$

$$C^{AB}(l,L) = {(2l+2L)! \over {(2l+1)!(2L+1)!}} {{<{r^{2l}}>^A <{r^{2L}}>^B} \over {E^A_l+E^B_L}} \space\space\space\space\space(118)$$

$$C^{AB}_{8} = {1 \over 2} [ { {C^{A}_{6} C^{B}_{8} (1 + g^B )} \over {{C^{A}_{6}{\alpha^B_1 \over \alpha^A_1}+g^B C^{B}_{6} {\alpha^A_1 \over \alpha^B_1}}}} + { {C^{A}_{8} C^{B}_{6} (1 + g^A )} \over {{g^B C^{A}_{6}{\alpha^B_1 \over \alpha^A_1}+C^{B}_{6} {\alpha^A_1 \over \alpha^B_1}}}} ] \space\space\space\space\space(126)$$