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Typo in doc

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Anthony Scemama 2022-01-10 10:34:27 +01:00
parent 9a6eb9a0f1
commit 77dbd5fa43
1 changed files with 41 additions and 41 deletions
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@ -637,8 +637,8 @@ prim_factor =
The $\uparrow$-spin and $\downarrow$-spin components of the one-body
density matrix are given by
\begin{eqnarray*}
\gamma_{ij}^{\uparrow} &=& \langle \Psi | a^{\dagger}_{j\alpha}\, a_{i\alpha} | \Psi \rangle \\
\gamma_{ij}^{\downarrow} &=& \langle \Psi | a^{\dagger}_{j\beta} \, a_{i\beta} | \Psi \rangle
\gamma_{ij}^{\uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\
\gamma_{ij}^{\downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\beta} \, \hat{a}_{i\beta} | \Psi \rangle
\end{eqnarray*}
and the spin-summed one-body density matrix is
\[
@ -649,13 +649,13 @@ prim_factor =
components of the two-body density matrix are given by
\begin{eqnarray*}
\Gamma_{ijkl}^{\uparrow \uparrow} &=&
\langle \Psi | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\alpha} a_{j\alpha}\, a_{i\alpha} | \Psi \rangle \\
\langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\alpha} \hat{a}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\
\Gamma_{ijkl}^{\downarrow \downarrow} &=&
\langle \Psi | a^{\dagger}_{k\beta}\, a^{\dagger}_{l\beta} a_{j\beta}\, a_{i\beta} | \Psi \rangle \\
\langle \Psi | \hat{a}^{\dagger}_{k\beta}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\beta} | \Psi \rangle \\
\Gamma_{ijkl}^{\uparrow \downarrow} &=&
+\langle \Psi | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\beta} a_{j\beta}\, a_{i\alpha} | \Psi \rangle \\
\langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\alpha} | \Psi \rangle \\
\Gamma_{ijkl}^{\downarrow \uparrow} &=&
\langle \Psi | a^{\dagger}_{k\beta}\, a^{\dagger}_{l\alpha} a_{j\alpha}\, a_{i\beta} | \Psi \rangle \\
\langle \Psi | \hat{a}^{\dagger}_{k\beta}\, \hat{a}^{\dagger}_{l\alpha} \hat{a}_{j\alpha}\, \hat{a}_{i\beta} | \Psi \rangle \\
\end{eqnarray*}
and the spin-summed one-body density matrix is
\[