mirror of
https://github.com/TREX-CoE/trexio.git
synced 2025-04-25 18:04:44 +02:00
Typo in doc
This commit is contained in:
parent
9a6eb9a0f1
commit
77dbd5fa43
82
trex.org
82
trex.org
@ -134,7 +134,7 @@ fetched using multiple function calls to perform I/O on buffers.
|
||||
V_{A \ell_{\max}+1} +
|
||||
\sum_{\ell=0}^{\ell_{\max}}
|
||||
\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,
|
||||
@ -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}}\,
|
||||
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.
|
||||
|
||||
|
||||
#+NAME: ecp
|
||||
| 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 |
|
||||
|
||||
|
||||
There might be some confusion in the meaning of the $\ell_{\max}$.
|
||||
It can be attributed to the maximum angular momentum occupied
|
||||
There might be some confusion in the meaning of the $\ell_{\max}$.
|
||||
It can be attributed to the maximum angular momentum occupied
|
||||
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
|
||||
ECP that replaces the core electrons.
|
||||
On the other 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.
|
||||
*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.
|
||||
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.
|
||||
|
||||
|
||||
@ -197,16 +197,16 @@ The latter causes issues when written before ~ang_mom~ in the TREXIO file.
|
||||
|
||||
** Example
|
||||
|
||||
For example, consider H_2 molecule with the following
|
||||
[[https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess][effective core potential]]
|
||||
For example, consider H_2 molecule with the following
|
||||
[[https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess][effective core potential]]
|
||||
(in GAMESS input format for the H atom):
|
||||
|
||||
#+BEGIN_EXAMPLE
|
||||
H-ccECP GEN 0 1
|
||||
3
|
||||
1.00000000000000 1 21.24359508259891
|
||||
21.24359508259891 3 21.24359508259891
|
||||
-10.85192405303825 2 21.77696655044365
|
||||
1.00000000000000 1 21.24359508259891
|
||||
21.24359508259891 3 21.24359508259891
|
||||
-10.85192405303825 2 21.77696655044365
|
||||
1
|
||||
0.00000000000000 2 1.000000000000000
|
||||
#+END_EXAMPLE
|
||||
@ -228,7 +228,7 @@ nucleus_index = [
|
||||
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 = [
|
||||
1, 1, 1, 0,
|
||||
1, 1, 1, 0
|
||||
@ -240,13 +240,13 @@ coefficient = [
|
||||
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
|
||||
]
|
||||
|
||||
power = [
|
||||
-1, 1, 0, 0,
|
||||
power = [
|
||||
-1, 1, 0, 0,
|
||||
-1, 1, 0, 0
|
||||
]
|
||||
#+END_EXAMPLE
|
||||
@ -357,32 +357,32 @@ prim_num = 20
|
||||
shell_num = 12
|
||||
|
||||
# 6 shells per H atom
|
||||
nucleus_index =
|
||||
[ 0, 0, 0, 0, 0, 0,
|
||||
1, 1, 1, 1, 1, 1 ]
|
||||
nucleus_index =
|
||||
[ 0, 0, 0, 0, 0, 0,
|
||||
1, 1, 1, 1, 1, 1 ]
|
||||
|
||||
# 3 shells in S (l=0), 2 in P (l=1), 1 in D (l=2)
|
||||
shell_ang_mom =
|
||||
[ 0, 0, 0, 1, 1, 2,
|
||||
[ 0, 0, 0, 1, 1, 2,
|
||||
0, 0, 0, 1, 1, 2 ]
|
||||
|
||||
# no need to renormalize shells
|
||||
shell_factor =
|
||||
[ 1., 1., 1., 1., 1., 1.,
|
||||
1., 1., 1., 1., 1., 1. ]
|
||||
# no need to renormalize shells
|
||||
shell_factor =
|
||||
[ 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
|
||||
shell_index =
|
||||
[ 0, 0, 0, 0, 0, 1, 2, 3, 4, 5,
|
||||
shell_index =
|
||||
[ 0, 0, 0, 0, 0, 1, 2, 3, 4, 5,
|
||||
6, 6, 6, 6, 6, 7, 8, 9, 10, 11 ]
|
||||
|
||||
# parameters of the primitives (10 per H atom)
|
||||
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 ]
|
||||
|
||||
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 ]
|
||||
|
||||
prim_factor =
|
||||
@ -467,7 +467,7 @@ prim_factor =
|
||||
:PROPERTIES:
|
||||
:CUSTOM_ID: ao_one_e
|
||||
:END:
|
||||
|
||||
|
||||
- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}}
|
||||
\sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A -
|
||||
\mathbf{r}_i \vert} \] : electron-nucleus attractive potential,
|
||||
@ -637,39 +637,39 @@ prim_factor =
|
||||
The $\uparrow$-spin and $\downarrow$-spin components of the one-body
|
||||
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
|
||||
\gamma_{ij}^{\uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\
|
||||
\gamma_{ij}^{\downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\beta} \, \hat{a}_{i\beta} | \Psi \rangle
|
||||
\end{eqnarray*}
|
||||
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 $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$
|
||||
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 \\
|
||||
\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} &=&
|
||||
\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} &=&
|
||||
+\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} &=&
|
||||
\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*}
|
||||
and the spin-summed one-body density matrix is
|
||||
\[
|
||||
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
|
||||
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}
|
||||
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}
|
||||
\Gamma_{ijkl}^{\downarrow \uparrow}
|
||||
\]
|
||||
|
||||
|
||||
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 |
|
||||
|-----------+----------------+------------------------------------+-----------------------------------------------------------------------|
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user