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 physics_chemistry:coulomb_repulsion_jh_and_slater_integrals [2017/12/11 16:33]Maurits W. Haverkort physics_chemistry:coulomb_repulsion_jh_and_slater_integrals [2018/05/03 22:02] (current)Maurits W. Haverkort Both sides previous revision Previous revision 2018/05/03 22:02 Maurits W. Haverkort 2017/12/11 16:33 Maurits W. Haverkort 2016/10/10 09:40 external edit2016/10/07 08:34 Maurits W. Haverkort created 2018/05/03 22:02 Maurits W. Haverkort 2017/12/11 16:33 Maurits W. Haverkort 2016/10/10 09:40 external edit2016/10/07 08:34 Maurits W. Haverkort created Line 23: Line 23: The expansion of $e^2/​|r_1-r_2|$ is: The expansion of $e^2/​|r_1-r_2|$ is: - \sum_{i\neq j} \frac{1}{2} \frac{e^2}{|r_i-r_j|} = \sum_{i\neq j} \frac{1}{2}\sum_{k=0}^{\infty} \sum_{m=-k}^{m=k} \frac{4 \pi}{2k+1} \frac{\mathrm{Min}[r_i,​r_j]^k}{\mathrm{Max}[r_i,​r_j]^{k+1}} Y_m^{(k)}(\theta_i,​\phi_i)Y_m^{(k)}(\theta_j,​\phi_j)^*. + \sum_{i\neq j} \frac{1}{2} \frac{e^2}{|r_i-r_j|} = \sum_{i\neq j} \frac{1}{2}\sum_{k=0}^{\infty} \sum_{m=-k}^{m=k} \frac{4 \pi e^2}{2k+1} \frac{\mathrm{Min}[r_i,​r_j]^k}{\mathrm{Max}[r_i,​r_j]^{k+1}} Y_m^{(k)}(\theta_i,​\phi_i)Y_m^{(k)}(\theta_j,​\phi_j)^*.