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Fixed RDMs in documentation

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Anthony Scemama 2023-02-27 15:47:52 +01:00
parent d37f4d6610
commit d63bc67892
2 changed files with 8 additions and 16 deletions
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2
src/README.org
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@ -30,7 +30,7 @@ variety of platforms and has interfaces for Fortran, Python, and OCaml.
- [[./lib.html][The TREXIO library]] - [[./lib.html][The TREXIO library]]
- [[./trex.html][Data stored with TREXIO]] - [[./trex.html][Data stored with TREXIO]]
- [[./tutorial_benzene.html][Tutorial]] - [[./tutorial_benzene.html][Tutorial]]
- [[./examples.html][How-to guide]] - [[./examples.html][Examples]]
- [[./templator_front.html][Front end API]] - [[./templator_front.html][Front end API]]
- [[./templator_hdf5.html][HDF5 back end]] - [[./templator_hdf5.html][HDF5 back end]]
- [[./templator_text.html][TEXT back end]] - [[./templator_text.html][TEXT back end]]

20
trex.org
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@ -677,23 +677,21 @@ power = [
The two-electron integrals for a two-electron operator $\hat{O}$ are The two-electron integrals for a two-electron operator $\hat{O}$ are
\[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists \[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists
notation or \[ ( pr \vert \hat{O} \vert qs ) \] in chemists
notation, where $p,q,r,s$ are indices over atomic orbitals. notation, where $p,q,r,s$ are indices over atomic orbitals.
Functions are provided to get the indices in physicists or chemists
notation.
# TODO: Physicist / Chemist functions # TODO: Physicist / Chemist functions
# Functions are provided to get the indices in physicists or chemists
# notation.
- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator. - \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} - \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}}
\sum_{j=1}^{i-1} \frac{\text{erf}(\vert \mathbf{r}_i - \sum_{j=1}^{i-1} \frac{\text{erf}(\mu\, \vert \mathbf{r}_i -
\mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential \mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential
The Cholesky decomposition of the integrals can also be stored: The Cholesky decomposition of the integrals can also be stored:
\[ \[
A_{ijkl} = \sum_{\alpha} G_{il\alpha} G_{jl\alpha} \langle ij | kl \rangle = \sum_{\alpha} G_{ik\alpha} G_{jl\alpha}
\] \]
#+NAME: ao_2e_int #+NAME: ao_2e_int
@ -1009,20 +1007,14 @@ power = [
\Gamma_{ijkl}^{\downarrow \downarrow} &=& \Gamma_{ijkl}^{\downarrow \downarrow} &=&
\langle \Psi | \hat{a}^{\dagger}_{k\beta}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{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} &=& \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 \\ \langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\alpha} | \Psi \rangle
+ \langle \Psi | \hat{a}^{\dagger}_{l\alpha}\, \hat{a}^{\dagger}_{k\beta} \hat{a}_{i\beta}\, \hat{a}_{j\alpha} | \Psi \rangle \\
\end{eqnarray*} \end{eqnarray*}
and the spin-summed one-body density matrix is and the spin-summed one-body density matrix is
\[ \[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}. \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}.
\] \]
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: The total energy can be computed as:
\[ \[