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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: 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}} \sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A 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 } 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 hand, it can be attributed to the maximum angular momentum of the
ECP that replaces the core electrons. ECP that replaces the core electrons.
*Note*, that the latter $\ell_{\max}$ is always higher by 1 than the former. *Note*, that the latter $\ell_{\max}$ is always higher by 1 than the former.
*Note for developers*: avoid having variables with similar prefix *Note for developers*: avoid having variables with similar prefix
in their name. The HDF5 back end might cause issues due to the way 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 ~find_dataset~ function works. For example, in the ECP group we
@ -868,7 +868,7 @@ power = [
} , } ,
#+end_src #+end_src
:end: :end:
* Multi-determinant information * Multi-determinant information
** Slater determinants (determinant group) ** Slater determinants (determinant group)
@ -966,7 +966,7 @@ power = [
on a reference wave function $\Psi$, where $\hat{T}_1$ is the single excitation operator, 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, $\hat{T}_2$ is the double excitation operator,
@ -986,7 +986,7 @@ power = [
\[ |\Phi\rangle = e^{\hat{T}}| \Psi \rangle \] \[ |\Phi\rangle = e^{\hat{T}}| \Psi \rangle \]
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. ~csf~ groups, and the amplitudes are stored using the current group.
The attributes with the ~exp~ suffix correspond to exponentialized operators. The attributes with the ~exp~ suffix correspond to exponentialized operators.
@ -1043,8 +1043,7 @@ power = [
\gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij} \gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij}
\] \]
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$ The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$ components of the two-body density matrix are given by
components of the two-body density matrix are given by
\begin{eqnarray*} \begin{eqnarray*}
\Gamma_{ijkl}^{\uparrow \uparrow} &=& \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 \\ \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 \\ \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 \\
\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*} \end{eqnarray*}
and the spin-summed one-body density matrix is and the spin-summed one-body density matrix is
\[ \[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow} + \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}.
\Gamma_{ijkl}^{\downarrow \uparrow}
\] \]
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:
\[ \[
@ -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_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_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_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_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_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 | | ~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_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_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_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") #+CALL: json(data=rdm, title="rdm")
@ -1119,7 +1119,6 @@ power = [
, "2e_upup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "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_dndn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_updn" : [ "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_num" : [ "dim" , [] ]
, "2e_cholesky" : [ "float sparse", [ "rdm.2e_cholesky_num", "mo.num", "mo.num" ] ] , "2e_cholesky" : [ "float sparse", [ "rdm.2e_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_upup_cholesky_num" : [ "dim" , [] ] , "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_dndn_cholesky" : [ "float sparse", [ "rdm.2e_dndn_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_updn_cholesky_num" : [ "dim" , [] ] , "2e_updn_cholesky_num" : [ "dim" , [] ]
, "2e_updn_cholesky" : [ "float sparse", [ "rdm.2e_updn_cholesky_num", "mo.num", "mo.num" ] ] , "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_src
:end: :end:
@ -1150,7 +1147,7 @@ power = [
following: following:
- ~CHAMP~ - ~CHAMP~
- ~Mu~ - ~Mu~
*** CHAMP *** CHAMP
The first form of Jastrow factor is the one used in The first form of Jastrow factor is the one used in
@ -1159,7 +1156,7 @@ power = [
\[ \[
J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R}) J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
\] \]
$J_{\text{eN}}$ contains electron-nucleus terms: $J_{\text{eN}}$ contains electron-nucleus terms:
@ -1212,17 +1209,17 @@ power = [
built such that the leading order in $1/r_{12}$ of the effective built such that the leading order in $1/r_{12}$ of the effective
two-electron potential reproduces the long-range interaction of the two-electron potential reproduces the long-range interaction of the
range-separated density functional theory. Its analytical range-separated density functional theory. Its analytical
expression reads expression reads
\[ \[
J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) + J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
J_{\text{eN}}(\mathbf{r}, \mathbf{R}) J_{\text{eN}}(\mathbf{r}, \mathbf{R})
\]. \].
The electron-electron cusp is incorporated in the three-body term. The electron-electron cusp is incorporated in the three-body term.
\[ \[
J_\text{eeN} (\mathbf{r}, \mathbf{R}) = J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \, \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha}) \Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\] \]
@ -1245,17 +1242,17 @@ power = [
\[ \[
E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right). E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
\] \]
In particular, if the parameters $\gamma_\alpha$ tend to zero, the In particular, if the parameters $\gamma_\alpha$ tend to zero, the
Mu-Jastrow factor becomes a two-body Jastrow factor: Mu-Jastrow factor becomes a two-body Jastrow factor:
\[ \[
J_{\text{ee}}(\mathbf{r}) = J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
\] \]
and for large $\gamma_\alpha$ it becomes zero. and for large $\gamma_\alpha$ it becomes zero.
To increase the flexibility of the Jastrow and improve the To increase the flexibility of the Jastrow and improve the
electron density the following electron-nucleus term is added 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 The parameter $\mu$ is stored in the ~ee~ array, the parameters
$\gamma_\alpha$ are stored in the ~een~ array, and the parameters $\gamma_\alpha$ are stored in the ~een~ array, and the parameters
$a_\alpha$ are stored in the ~en~ array. $a_\alpha$ are stored in the ~en~ array.
*** Table of values *** Table of values
#+name: jastrow #+name: jastrow
| Variable | Type | Dimensions | Description | | Variable | Type | Dimensions | Description |
|---------------+----------+---------------------+-----------------------------------------------------------------| |---------------+----------+---------------------+-----------------------------------------------------------------|
@ -1285,7 +1282,7 @@ power = [
| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter | | ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances | | ~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 | | ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP Jastrow for electron-nucleus distances |
#+CALL: json(data=jastrow, title="jastrow") #+CALL: json(data=jastrow, title="jastrow")
#+RESULTS: #+RESULTS:
@ -1338,7 +1335,7 @@ power = [
} }
#+end_src #+end_src
:end: :end:
* Appendix :noexport: * Appendix :noexport:
** Python script from table to json ** Python script from table to json