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Typo in doc

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Anthony Scemama 2022-01-10 10:34:27 +01:00
parent 9a6eb9a0f1
commit 77dbd5fa43
1 changed files with 41 additions and 41 deletions
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@ -134,7 +134,7 @@ fetched using multiple function calls to perform I/O on buffers.
V_{A \ell_{\max}+1} + V_{A \ell_{\max}+1} +
\sum_{\ell=0}^{\ell_{\max}} \sum_{\ell=0}^{\ell_{\max}}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} | V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} |
\] \]
The first term in the equation above is sometimes attributed to the local channel, The first term in the equation above is sometimes attributed to the local channel,
@ -147,9 +147,9 @@ fetched using multiple function calls to perform I/O on buffers.
\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 }
\] \]
See http://dx.doi.org/10.1063/1.4984046 or https://doi.org/10.1063/1.5121006 for more info. See http://dx.doi.org/10.1063/1.4984046 or https://doi.org/10.1063/1.5121006 for more info.
#+NAME: ecp #+NAME: ecp
| Variable | Type | Dimensions | Description | | Variable | Type | Dimensions | Description |
|----------------------+---------+-----------------+----------------------------------------------------------------------------------------| |----------------------+---------+-----------------+----------------------------------------------------------------------------------------|
@ -163,17 +163,17 @@ fetched using multiple function calls to perform I/O on buffers.
| ~power~ | ~int~ | ~(ecp.num)~ | $n_{A q \ell}$ all ECP powers | | ~power~ | ~int~ | ~(ecp.num)~ | $n_{A q \ell}$ all ECP powers |
There might be some confusion in the meaning of the $\ell_{\max}$. There might be some confusion in the meaning of the $\ell_{\max}$.
It can be attributed to the maximum angular momentum occupied It can be attributed to the maximum angular momentum occupied
in the core orbitals, which are removed by the ECP. in the core orbitals, which are removed by the ECP.
On the other hand, it can be attributed to the maximum angular momentum of the On the other 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 in their name. *Note for developers*: avoid having variables with similar prefix in their name.
HDF5 back end might cause issues due to the way ~find_dataset~ function works. HDF5 back end might cause issues due to the way ~find_dataset~ function works.
For example, in the ECP group we use ~max_ang_mom~ and not ~ang_mom_max~. For example, in the ECP group we use ~max_ang_mom~ and not ~ang_mom_max~.
The latter causes issues when written before ~ang_mom~ in the TREXIO file. The latter causes issues when written before ~ang_mom~ in the TREXIO file.
@ -197,16 +197,16 @@ The latter causes issues when written before ~ang_mom~ in the TREXIO file.
** Example ** Example
For example, consider H_2 molecule with the following For example, consider H_2 molecule with the following
[[https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess][effective core potential]] [[https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess][effective core potential]]
(in GAMESS input format for the H atom): (in GAMESS input format for the H atom):
#+BEGIN_EXAMPLE #+BEGIN_EXAMPLE
H-ccECP GEN 0 1 H-ccECP GEN 0 1
3 3
1.00000000000000 1 21.24359508259891 1.00000000000000 1 21.24359508259891
21.24359508259891 3 21.24359508259891 21.24359508259891 3 21.24359508259891
-10.85192405303825 2 21.77696655044365 -10.85192405303825 2 21.77696655044365
1 1
0.00000000000000 2 1.000000000000000 0.00000000000000 2 1.000000000000000
#+END_EXAMPLE #+END_EXAMPLE
@ -228,7 +228,7 @@ nucleus_index = [
1, 1, 1, 1 1, 1, 1, 1
] ]
# 3 first ECP elements correspond to potential of the P orbital (l=1), then 1 element for the S orbital (l=0) ; similar for the second H atom # 3 first ECP elements correspond to potential of the P orbital (l=1), then 1 element for the S orbital (l=0) ; similar for the second H atom
ang_mom = [ ang_mom = [
1, 1, 1, 0, 1, 1, 1, 0,
1, 1, 1, 0 1, 1, 1, 0
@ -240,13 +240,13 @@ coefficient = [
1.00000000000000, 21.24359508259891, -10.85192405303825, 0.00000000000000 1.00000000000000, 21.24359508259891, -10.85192405303825, 0.00000000000000
] ]
exponent = [ exponent = [
21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000, 21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000,
21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000 21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000
] ]
power = [ power = [
-1, 1, 0, 0, -1, 1, 0, 0,
-1, 1, 0, 0 -1, 1, 0, 0
] ]
#+END_EXAMPLE #+END_EXAMPLE
@ -357,32 +357,32 @@ prim_num = 20
shell_num = 12 shell_num = 12
# 6 shells per H atom # 6 shells per H atom
nucleus_index = nucleus_index =
[ 0, 0, 0, 0, 0, 0, [ 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1 ] 1, 1, 1, 1, 1, 1 ]
# 3 shells in S (l=0), 2 in P (l=1), 1 in D (l=2) # 3 shells in S (l=0), 2 in P (l=1), 1 in D (l=2)
shell_ang_mom = shell_ang_mom =
[ 0, 0, 0, 1, 1, 2, [ 0, 0, 0, 1, 1, 2,
0, 0, 0, 1, 1, 2 ] 0, 0, 0, 1, 1, 2 ]
# no need to renormalize shells # no need to renormalize shells
shell_factor = shell_factor =
[ 1., 1., 1., 1., 1., 1., [ 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1. ] 1., 1., 1., 1., 1., 1. ]
# 5 primitives for the first S shell and then 1 primitive per remaining shells in each H atom # 5 primitives for the first S shell and then 1 primitive per remaining shells in each H atom
shell_index = shell_index =
[ 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, [ 0, 0, 0, 0, 0, 1, 2, 3, 4, 5,
6, 6, 6, 6, 6, 7, 8, 9, 10, 11 ] 6, 6, 6, 6, 6, 7, 8, 9, 10, 11 ]
# parameters of the primitives (10 per H atom) # parameters of the primitives (10 per H atom)
exponent = exponent =
[ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057, [ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057,
33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057 ] 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057 ]
coefficient = coefficient =
[ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0, [ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0,
0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0 ] 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0 ]
prim_factor = prim_factor =
@ -467,7 +467,7 @@ prim_factor =
:PROPERTIES: :PROPERTIES:
:CUSTOM_ID: ao_one_e :CUSTOM_ID: ao_one_e
:END: :END:
- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}} - \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}}
\sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A - \sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A -
\mathbf{r}_i \vert} \] : electron-nucleus attractive potential, \mathbf{r}_i \vert} \] : electron-nucleus attractive potential,
@ -637,39 +637,39 @@ prim_factor =
The $\uparrow$-spin and $\downarrow$-spin components of the one-body The $\uparrow$-spin and $\downarrow$-spin components of the one-body
density matrix are given by density matrix are given by
\begin{eqnarray*} \begin{eqnarray*}
\gamma_{ij}^{\uparrow} &=& \langle \Psi | a^{\dagger}_{j\alpha}\, a_{i\alpha} | \Psi \rangle \\ \gamma_{ij}^{\uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\
\gamma_{ij}^{\downarrow} &=& \langle \Psi | a^{\dagger}_{j\beta} \, a_{i\beta} | \Psi \rangle \gamma_{ij}^{\downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\beta} \, \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
\[ \[
\gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij} \gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij}
\] \]
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$ The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$
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 | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\alpha} a_{j\alpha}\, 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 \\
\Gamma_{ijkl}^{\downarrow \downarrow} &=& \Gamma_{ijkl}^{\downarrow \downarrow} &=&
\langle \Psi | a^{\dagger}_{k\beta}\, a^{\dagger}_{l\beta} a_{j\beta}\, 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 | a^{\dagger}_{k\alpha}\, a^{\dagger}_{l\beta} a_{j\beta}\, 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} &=& \Gamma_{ijkl}^{\downarrow \uparrow} &=&
\langle \Psi | a^{\dagger}_{k\beta}\, a^{\dagger}_{l\alpha} a_{j\alpha}\, a_{i\beta} | \Psi \rangle \\ \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
\[ \[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow} \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}
\Gamma_{ijkl}^{\downarrow \uparrow} \Gamma_{ijkl}^{\downarrow \uparrow}
\] \]
The total energy can be computed as: The total energy can be computed as:
\[ \[
E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle + E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle +
\frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle \frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle
\] \]
#+NAME: rdm #+NAME: rdm
| Variable | Type | Dimensions | Description | | Variable | Type | Dimensions | Description |
|-----------+----------------+------------------------------------+-----------------------------------------------------------------------| |-----------+----------------+------------------------------------+-----------------------------------------------------------------------|