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Removed RDM down-up which was redundant with up-down

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Anthony Scemama 2023-02-14 15:27:25 +01:00
parent fb711e1bed
commit 87035ba5c8
1 changed files with 28 additions and 31 deletions
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@ -1043,8 +1043,7 @@ power = [
\gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij}
\]
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$
components of the two-body density matrix are given by
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$ components of the two-body density matrix are given by
\begin{eqnarray*}
\Gamma_{ijkl}^{\uparrow \uparrow} &=&
\langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\alpha} \hat{a}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\
@ -1052,15 +1051,19 @@ power = [
\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 | \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 | \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
\[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow} +
\Gamma_{ijkl}^{\downarrow \uparrow}
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}.
\]
The density matrices are normalized such that
\begin{eqnarray*}
\sum_{ij} \Gamma_{ijij}^{\uparrow \uparrow} & = & N_\uparrow\, (N_\uparrow-1)/2 \\
\sum_{ij} \Gamma_{ijij}^{\downarrow \downarrow} & = & N_\downarrow\, (N_\downarrow-1)/2 \\
\sum_{ij} \Gamma_{ijij}^{\uparrow \downarrow} & = & N_\uparrow\, N_\downarrow \\
\sum_{ij} \Gamma_{ijij} & = & (N_\uparrow+N_\downarrow)\, (N_\uparrow+N_\downarrow-1)/2.
\end{eqnarray*}
The total energy can be computed as:
\[
@ -1094,7 +1097,6 @@ power = [
| ~2e_upup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \uparrow\uparrow component of the two-body reduced density matrix |
| ~2e_dndn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\downarrow component of the two-body reduced density matrix |
| ~2e_updn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \uparrow\downarrow component of the two-body reduced density matrix |
| ~2e_dnup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\uparrow component of the two-body reduced density matrix |
| ~2e_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_cholesky_num)~ | Cholesky decomposition of the two-body RDM (spin trace) |
| ~2e_upup_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
@ -1103,8 +1105,6 @@ power = [
| ~2e_dndn_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_dndn_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\downarrow\downarrow) |
| ~2e_updn_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_updn_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_updn_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\uparrow\downarrow) |
| ~2e_dnup_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_dnup_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_dnup_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\downarrow\uparrow) |
#+CALL: json(data=rdm, title="rdm")
@ -1119,7 +1119,6 @@ power = [
, "2e_upup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_dndn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_updn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_dnup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_cholesky_num" : [ "dim" , [] ]
, "2e_cholesky" : [ "float sparse", [ "rdm.2e_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_upup_cholesky_num" : [ "dim" , [] ]
@ -1128,8 +1127,6 @@ power = [
, "2e_dndn_cholesky" : [ "float sparse", [ "rdm.2e_dndn_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_updn_cholesky_num" : [ "dim" , [] ]
, "2e_updn_cholesky" : [ "float sparse", [ "rdm.2e_updn_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_dnup_cholesky_num" : [ "dim" , [] ]
, "2e_dnup_cholesky" : [ "float sparse", [ "rdm.2e_dnup_cholesky_num", "mo.num", "mo.num" ] ]
} ,
#+end_src
:end: