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Document RDM

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Anthony Scemama 2022-01-10 10:25:18 +01:00
parent 1aaca05b51
commit 4c6b84685c
2 changed files with 58 additions and 7 deletions
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24
examples.org
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@ -23,19 +23,19 @@ program print_energy
This program computes the energy as:
\[
E = E_{\text{NN}} + \sum_{ij} D_{ij}\, \langle i | h | j \rangle\,
+\, \frac{1}{2} \sum_{ijkl} \Gamma_{ijkl}\, \langle i j | k l
E = E_{\text{NN}} + \sum_{ij} \gamma_{ij}\, \langle j | h | i \rangle\,
+\, \frac{1}{2} \sum_{ijkl} \Gamma_{ijkl}\, \langle k l | i j
\rangle\; \textrm{ with } \; 0 < i,j,k,l \le n
\]
One needs to read from the TREXIO file:
- $n$ :: The number of molecular orbitals
- $E_{\text{NN}}$ :: The nuclear repulsion energy
- $D_{ij}$ :: The one-body reduced density matrix
- $\langle i |h| j \rangle$ :: The one-electron Hamiltonian integrals
- $\gamma_{ij}$ :: The one-body reduced density matrix
- $\langle j |h| i \rangle$ :: The one-electron Hamiltonian integrals
- $\Gamma_{ijkl}$ :: The two-body reduced density matrix
- $\langle i j | k l \rangle$ :: The electron repulsion integrals
- $\langle k l | i j \rangle$ :: The electron repulsion integrals
#+begin_src f90
integer :: n
@ -228,7 +228,17 @@ program print_energy
*** Compute the energy
As $(n,m)$ 2D arrays are stored in memory as $(\n times m)$ 1D
When the orbitals are real, we can use
\begin{eqnarray*}
E &=& E_{\text{NN}} + \sum_{ij} \gamma_{ij}\, \langle j | h | i \rangle\,
+\, \frac{1}{2} \sum_{ijkl} \Gamma_{ijkl}\, \langle k l | i j
\rangle \\
&=& E_{\text{NN}} + \sum_{ij} \gamma_{ij}\, \langle i | h | j \rangle\,
+\, \frac{1}{2} \sum_{ijkl} \Gamma_{ijkl}\, \langle i j | k l
\rangle \\
\end{eqnarray*}
As $(n,m)$ 2D arrays are stored in memory as $(n \times m)$ 1D
arrays, we could pass the matrices to the ~ddot~ BLAS function to
perform the summations in a single call for the 1-electron quantities.
Instead, we prefer to interleave the 1-electron (negative) and

41
trex.org
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@ -631,6 +631,45 @@ prim_factor =
* TODO Slater determinants
* Reduced density matrices (rdm group)
The reduced density matrices are defined in the basis of molecular
orbitals.
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
\end{eqnarray*}
and the spin-summed one-body density matrix is
\[
\gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij}
\]
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$
components of the two-body density matrix are given by
\begin{eqnarray*}
\Gamma_{ijkl}^{\uparrow \uparrow} &=&
\langle \Psi | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\alpha} a_{j\alpha}\, 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 \\
\Gamma_{ijkl}^{\uparrow \downarrow} &=&
+\langle \Psi | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\beta} a_{j\beta}\, 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 \\
\end{eqnarray*}
and the spin-summed one-body density matrix is
\[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}
\Gamma_{ijkl}^{\downarrow \uparrow}
\]
The total energy can be computed as:
\[
E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle +
\frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle
\]
#+NAME: rdm
| Variable | Type | Dimensions | Description |
|-----------+----------------+------------------------------------+-----------------------------------------------------------------------|
@ -641,6 +680,7 @@ prim_factor =
| ~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 |
#+CALL: json(data=rdm, title="rdm", last=1)
@ -655,6 +695,7 @@ prim_factor =
, "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" ] ]
}
#+end_src
:end: