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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
Line 23: Line 23:
 The expansion of $e^2/​|r_1-r_2|$ is: The expansion of $e^2/​|r_1-r_2|$ is:
 \begin{equation} \begin{equation}
-\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)^*.
 \end{equation} \end{equation}
  
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