From 77dbd5fa43744c150c9c9c6f9fd668355d73ff58 Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Mon, 10 Jan 2022 10:34:27 +0100 Subject: [PATCH] Typo in doc --- trex.org | 82 ++++++++++++++++++++++++++++---------------------------- 1 file changed, 41 insertions(+), 41 deletions(-) diff --git a/trex.org b/trex.org index a4ad0c1..44b95e9 100644 --- a/trex.org +++ b/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 | |-----------+----------------+------------------------------------+-----------------------------------------------------------------------|