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documentation:standard_operators:coulomb_repulsion [2017/02/27 12:42] – Maurits W. Haverkortdocumentation:standard_operators:coulomb_repulsion [2017/02/27 12:43] – Maurits W. Haverkort
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 ===== Two shells, shell occupation conserving ===== ===== Two shells, shell occupation conserving =====
  
-{{:documentation:standard_operators:coulomb_diagram_ll.png?nolink|}} 
  
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-The Coulomb repulsion between two shells which does not change the number of electrons is given by a direct term ($l_1=l_3$ and $l_2=l_4$) and an indirect or exchange term ($l_1=l_4$ and $l_2=l_3$). The direct term is given by the Slater integrals:+{{:documentation:standard_operators:coulomb_diagram_ll.png?nolink&200 |}}The Coulomb repulsion between two shells which does not change the number of electrons is given by a direct term ($l_1=l_3$ and $l_2=l_4$) and an indirect or exchange term ($l_1=l_4$ and $l_2=l_3$). The direct term is given by the Slater integrals:
 \begin{equation} \begin{equation}
 F^{(k)}=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]^2R_2[r_j]^2\mathrm{d}r_i\mathrm{d}r_j, F^{(k)}=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]^2R_2[r_j]^2\mathrm{d}r_i\mathrm{d}r_j,
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 ===== General case of 4 different shells ===== ===== General case of 4 different shells =====
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-{{:documentation:standard_operators:coulomb_diagram_llll.png?nolink|}} 
  
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 +{{:documentation:standard_operators:coulomb_diagram_llll.png?nolink&200 |}}
  
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