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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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@ -426,7 +426,7 @@ prim_factor =
All the functions $V_{A\ell}$ are parameterized as:
\[
V_{A \ell}(\mathbf{r}) =
V_{A \ell}(\mathbf{r}) =
\sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
@ -452,7 +452,7 @@ prim_factor =
hand, it can be attributed to the maximum angular momentum of the
ECP that replaces the core electrons.
*Note*, that the latter $\ell_{\max}$ is always higher by 1 than the former.
*Note for developers*: avoid having variables with similar prefix
in their name. The HDF5 back end might cause issues due to the way
~find_dataset~ function works. For example, in the ECP group we
@ -868,7 +868,7 @@ power = [
} ,
#+end_src
:end:
* Multi-determinant information
** Slater determinants (determinant group)
@ -966,7 +966,7 @@ power = [
on a reference wave function $\Psi$, where $\hat{T}_1$ is the single excitation operator,
\[
\hat{T}_1 = \sum_{ia} t_{i}^{a}\, \hat{a}^\dagger_a \hat{a}_i,
\hat{T}_1 = \sum_{ia} t_{i}^{a}\, \hat{a}^\dagger_a \hat{a}_i,
\]
$\hat{T}_2$ is the double excitation operator,
@ -986,7 +986,7 @@ power = [
\[ |\Phi\rangle = e^{\hat{T}}| \Psi \rangle \]
The reference wave function is stored using the ~determinant~ and/or
The reference wave function is stored using the ~determinant~ and/or
~csf~ groups, and the amplitudes are stored using the current group.
The attributes with the ~exp~ suffix correspond to exponentialized operators.
@ -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:
@ -1150,7 +1147,7 @@ power = [
following:
- ~CHAMP~
- ~Mu~
*** CHAMP
The first form of Jastrow factor is the one used in
@ -1159,7 +1156,7 @@ power = [
\[
J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
\]
$J_{\text{eN}}$ contains electron-nucleus terms:
@ -1212,17 +1209,17 @@ power = [
built such that the leading order in $1/r_{12}$ of the effective
two-electron potential reproduces the long-range interaction of the
range-separated density functional theory. Its analytical
expression reads
expression reads
\[
J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
J_{\text{eN}}(\mathbf{r}, \mathbf{R})
\].
The electron-electron cusp is incorporated in the three-body term.
\[
J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\]
@ -1245,17 +1242,17 @@ power = [
\[
E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
\]
In particular, if the parameters $\gamma_\alpha$ tend to zero, the
Mu-Jastrow factor becomes a two-body Jastrow factor:
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
\]
and for large $\gamma_\alpha$ it becomes zero.
To increase the flexibility of the Jastrow and improve the
electron density the following electron-nucleus term is added
@ -1268,9 +1265,9 @@ power = [
The parameter $\mu$ is stored in the ~ee~ array, the parameters
$\gamma_\alpha$ are stored in the ~een~ array, and the parameters
$a_\alpha$ are stored in the ~en~ array.
*** Table of values
#+name: jastrow
| Variable | Type | Dimensions | Description |
|---------------+----------+---------------------+-----------------------------------------------------------------|
@ -1285,7 +1282,7 @@ power = [
| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances |
| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP Jastrow for electron-nucleus distances |
#+CALL: json(data=jastrow, title="jastrow")
#+RESULTS:
@ -1338,7 +1335,7 @@ power = [
}
#+end_src
:end:
* Appendix :noexport:
** Python script from table to json