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Added reciprocal space vectors

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Anthony Scemama 2023-01-10 18:48:30 +01:00
parent f6623c1579
commit e2d8cd1972
1 changed files with 26 additions and 19 deletions
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@ -133,11 +133,15 @@ with the same name suffixed by ~_im~.
used in periodic calculations. used in periodic calculations.
#+NAME: cell #+NAME: cell
| Variable | Type | Dimensions | Description | | Variable | Type | Dimensions | Description |
|----------+---------+------------+-----------------------| |----------+---------+------------+--------------------------------------------------------------------------|
| ~a~ | ~float~ | ~(3)~ | First lattice vector | | ~a~ | ~float~ | ~(3)~ | First real space lattice vector |
| ~b~ | ~float~ | ~(3)~ | Second lattice vector | | ~b~ | ~float~ | ~(3)~ | Second real space lattice vector |
| ~c~ | ~float~ | ~(3)~ | Third lattice vector | | ~c~ | ~float~ | ~(3)~ | Third real space lattice vector |
| ~G_a~ | ~float~ | ~(3)~ | First reciprocal space lattice vector |
| ~G_b~ | ~float~ | ~(3)~ | Second reciprocal space lattice vector |
| ~G_c~ | ~float~ | ~(3)~ | Third reciprocal space lattice vector |
| ~two_pi~ | ~int~ | | ~0~ or ~1~. If ~two_pi=1~, $2\pi$ is included in the reciprocal vectors. |
#+CALL: json(data=cell, title="cell") #+CALL: json(data=cell, title="cell")
@ -145,9 +149,13 @@ with the same name suffixed by ~_im~.
:results: :results:
#+begin_src python :tangle trex.json #+begin_src python :tangle trex.json
"cell": { "cell": {
"a" : [ "float", [ "3" ] ] "a" : [ "float", [ "3" ] ]
, "b" : [ "float", [ "3" ] ] , "b" : [ "float", [ "3" ] ]
, "c" : [ "float", [ "3" ] ] , "c" : [ "float", [ "3" ] ]
, "G_a" : [ "float", [ "3" ] ]
, "G_b" : [ "float", [ "3" ] ]
, "G_c" : [ "float", [ "3" ] ]
, "two_pi" : [ "int" , [] ]
} , } ,
#+end_src #+end_src
:end: :end:
@ -282,12 +290,13 @@ with the same name suffixed by ~_im~.
A plane wave is defined as A plane wave is defined as
\[ \[
\chi_j(r) = \exp \left( -i \mathbf{k}_j \mathbf{r} \right) \chi_j(\mathbf{r}) = \exp \left( -i \mathbf{G}_j \cdot \mathbf{r} \right)
\] \]
The basis set is defined as the array of $k$-points in the The basis set is defined as the array of $k$-points in the
reciprocal space, defined in the ~pbc~ group. The kinetic energy reciprocal space $\mathbf{G}_j$, defined in the ~pbc~ group. The
cutoff ~e_cut~ is the only input data relevant to plane waves. kinetic energy cutoff ~e_cut~ is the only input data relevant to
plane waves.
*** Data definitions *** Data definitions
@ -390,7 +399,7 @@ exponent =
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 =
[ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01 [ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01, 3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
@ -408,21 +417,20 @@ prim_factor =
\[ \[
V_A^{\text{ECP}} = V_A^{\text{ECP}} =
V_{A \ell_{\max}+1} + V_{A \ell_{\max}+1} +
\sum_{\ell=0}^{\ell_{\max}} \sum_{\ell=0}^{\ell_{\max}} V_{A \ell}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} |
V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} |
\] \]
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,
while the remaining terms correspond to the non-local channel projections. while the remaining terms correspond to the non-local channel projections.
The functions $V_{A\ell}$ are parameterized as: All the functions $V_{A\ell}$ are parameterized as:
\[ \[
V_{A \ell}(\mathbf{r}) = V_{A \ell}(\mathbf{r}) =
\sum_{q=1}^{N_{q \ell}} \sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\, \beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 } e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
\] \].
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.
@ -438,7 +446,6 @@ prim_factor =
| ~coefficient~ | ~float~ | ~(ecp.num)~ | $\beta_{A q \ell}$ all ECP coefficients | | ~coefficient~ | ~float~ | ~(ecp.num)~ | $\beta_{A q \ell}$ all ECP coefficients |
| ~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 in It can be attributed to the maximum angular momentum occupied in
the core orbitals, which are removed by the ECP. On the other the core orbitals, which are removed by the ECP. On the other