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documentation:standard_operators:coulomb_repulsion [2017/02/27 11:28] – Maurits W. Haverkortdocumentation:standard_operators:coulomb_repulsion [2017/02/27 12:43] – Maurits W. Haverkort
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 NewOperator("U", NF, IndexUp, IndexDn, SlaterIntegrals) NewOperator("U", NF, IndexUp, IndexDn, SlaterIntegrals)
 </code> </code>
-whereby SlaterIntegrals represents a list of $F^{(k)}$ with $k$ running from $0$ to $2l$ in steps of $2$.+whereby SlaterIntegrals represents a list of $F^{(k)}$ with $k$ running from $0$ to $2l$ in steps of $2$, i.e. $k$ is even.
 ### ###
  
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 ===== Two shells, shell occupation conserving ===== ===== Two shells, shell occupation conserving =====
 +
  
 ### ###
-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,
 \end{equation} \end{equation}
-with $0 \leq k \leq \mathrm{Min}[2l_1,2l_2]$ in steps of 2. +with $0 \leq k \leq \mathrm{Min}[2l_1,2l_2]$ in steps of 2, i.e. $k$ is even. 
  
 The indirect term is given by the exchange integrals: The indirect term is given by the exchange integrals:
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 G^{(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]R_1[r_j]R_2[r_i]R_2[r_j]\mathrm{d}r_i\mathrm{d}r_j, G^{(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]R_1[r_j]R_2[r_i]R_2[r_j]\mathrm{d}r_i\mathrm{d}r_j,
 \end{equation} \end{equation}
-with $|l_1-l_2| \leq k \leq |l_1+l_2|$ in steps of 2.+with $|l_1-l_2| \leq k \leq |l_1+l_2|$ in steps of 2, i.e. $k$ is even if both $l_1$ and $l_2$ are even or odd and $k$ is odd if one of the angular momenta involved is even and the other is odd.
 ### ###
  
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 OppG3pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {0,0}, {0,1}) OppG3pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {0,0}, {0,1})
 </code> </code>
 +###
 +
 +===== General case of 4 different shells =====
 +
 +###
 +{{:documentation:standard_operators:coulomb_diagram_llll.png?nolink&200 |}}
 +
 ### ###
  

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