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documentation:standard_operators:coulomb_repulsion [2016/10/10 09:40] – external edit 127.0.0.1documentation:standard_operators:coulomb_repulsion [2017/05/23 16:43] (current) – Maurits W. Haverkort
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 +{{indexmenu_n>5}}
 ====== Coulomb repulsion operator (U) ====== ====== Coulomb repulsion operator (U) ======
  
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 The radial part of the operator ($\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}$) is more difficult and will be cast into parameters: The radial part of the operator ($\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}$) is more difficult and will be cast into parameters:
 \begin{equation} \begin{equation}
-R^{(k)}[\tau_1\tau_2\tau_3\tau_4]=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_2[r_j]R_3[r_i]R_4[r_j]\mathrm{d}r_i\mathrm{d}r_j.+R^{(k)}[\tau_1\tau_2\tau_3\tau_4]=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_2[r_j]R_3[r_i]R_4[r_j]r_i^2 r_j^2\mathrm{d}r_i\mathrm{d}r_j.
 \end{equation} \end{equation}
  
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 ### ###
-For the case where $n_1=n_2=n_3=n_4$ and $l_1=l_2=l_3=l_4$, i.e. Coulomb repulsion within one shell one defines:+{{:documentation:standard_operators:coulomb_diagram_lll.png?nolink&200 |}}For the case where $n_1=n_2=n_3=n_4$ and $l_1=l_2=l_3=l_4$, i.e. Coulomb repulsion within one shell one defines:
 \begin{equation} \begin{equation}
 F^{(k)} = R^{(k)}[\tau_1\tau_2\tau_3\tau_4]. F^{(k)} = R^{(k)}[\tau_1\tau_2\tau_3\tau_4].
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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&400 |}}The Coulomb repulsion between two shells which does not change the number of electrons is given by a direct term ($n_1l_1=n_3l_3$ and $n_2l_2=n_4l_4$) and an indirect or exchange term ($n_1l_1=n_4l_4$ and $n_2l_2=n_3l_3$). We here assume that $n_1l_1\neq n_2l_2$. 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 $0 \leq k \leq \mathrm{Min}[2l_1,2l_2]$. The indirect term is given by the exchange integrals:+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:
 \begin{equation} \begin{equation}
 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 $|l_1-l_2| \leq k \leq |l_1+l_2|$.+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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 <code Quanty Example.Quanty> <code Quanty Example.Quanty>
 NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, Fk, Gk) NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, Fk, Gk)
-\end{lstlisting}+</code>
 For $l_1=1$ and $l_2=2$ one could define: For $l_1=1$ and $l_2=2$ one could define:
-\begin{lstlisting}+<code Quanty Example.Quanty>
 OppF0pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {1,0}, {0,0}) OppF0pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {1,0}, {0,0})
 OppF2pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {0,1}, {0,0}) OppF2pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, {0,1}, {0,0})
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 ### ###
  
 +===== General case of 4 different shells =====
 +
 +###
 +{{:documentation:standard_operators:coulomb_diagram_llll.png?nolink&200 |}}The Coulomb repulsion in the general case allows four different principle quantum numbers and angular momenta. The radial integral is given as:
 +\begin{equation}
 +R^{(k)}[n_1l_1\:n_2l_2\:n_3l_3\:n_4l_4]=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_{n_1l_1}[r_i]R_{n_2l_2}[r_j]R_{n_3l_3}[r_i]R_{n_4l_4}[r_j]\mathrm{d}r_i\mathrm{d}r_j,
 +\end{equation}
 +with $\mathrm{Max}[|l_1-l_3|,|l_2-l_4|] \leq k \leq \mathrm{Min}[l_1+l_3,l_2+l_4]$ and $(l_1+l_3)$, $(l2_+l_4)$ either both even or both odd.
 +###
 +
 +###
 +In Quanty one can implement these operators as:
 +<code Quanty Example.Quanty>
 +NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, IndexUp_3, IndexDn_3, IndexUp_4, IndexDn_4, Rk)
 +</code>
 +For $l_1=3$, $l_2=0$, $l_3=2$ and $l_4=1$ one has $k=1$ and one could define:
 +<code Quanty Example.Quanty>
 +OppR1pd = NewOperator("U", NF, IndexUp_1, IndexDn_1, IndexUp_2, IndexDn_2, IndexUp_3, IndexDn_3, IndexUp_4, IndexDn_4, {1})
 +</code>
 +###
 +
 +###
 +Note that in the general case you need to sum over all possible permutations of $n_1l_1$, $n_2l_2$, $n_3l_3$ and $n_4l_4$. Permuting $n_1l_1$ with $n_2l_2$ and at the same time $n_3l_3$ with $n_4l_4$ will not change the value and form of the operator. If $n_1l_1$ is different from $n_2l_2$ and $n_3l_3$ is different from $n_4l_4$ one can add a factor of two in front of the operator and only add one of the permutations. If one of the $n_1l_1$ is the same as $n_2l_2$ or $n_3l_3$ is the same as $n_4l_4$ a permutation will not lead to a new configuration and the factor of two disappears. If you just sum over all possible $n_il_i$ combinations things go right automatically. 
 +###
  
 ===== Table of contents ===== ===== Table of contents =====
-{{indexmenu>.#1}}+{{indexmenu>.#1|msort nsort}}

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