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Added reciprocal space vectors

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This commit is contained in:
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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@ -134,10 +134,14 @@ with the same name suffixed by ~_im~.
#+NAME: cell
| Variable | Type | Dimensions | Description |
|----------+---------+------------+-----------------------|
| ~a~ | ~float~ | ~(3)~ | First lattice vector |
| ~b~ | ~float~ | ~(3)~ | Second lattice vector |
| ~c~ | ~float~ | ~(3)~ | Third lattice vector |
|----------+---------+------------+--------------------------------------------------------------------------|
| ~a~ | ~float~ | ~(3)~ | First real space lattice vector |
| ~b~ | ~float~ | ~(3)~ | Second real space 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")
@ -148,6 +152,10 @@ with the same name suffixed by ~_im~.
"a" : [ "float", [ "3" ] ]
, "b" : [ "float", [ "3" ] ]
, "c" : [ "float", [ "3" ] ]
, "G_a" : [ "float", [ "3" ] ]
, "G_b" : [ "float", [ "3" ] ]
, "G_c" : [ "float", [ "3" ] ]
, "two_pi" : [ "int" , [] ]
} ,
#+end_src
:end:
@ -282,12 +290,13 @@ with the same name suffixed by ~_im~.
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
reciprocal space, defined in the ~pbc~ group. The kinetic energy
cutoff ~e_cut~ is the only input data relevant to plane waves.
reciprocal space $\mathbf{G}_j$, defined in the ~pbc~ group. The
kinetic energy cutoff ~e_cut~ is the only input data relevant to
plane waves.
*** Data definitions
@ -390,7 +399,7 @@ exponent =
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 ]
`
prim_factor =
[ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
@ -408,21 +417,20 @@ prim_factor =
\[
V_A^{\text{ECP}} =
V_{A \ell_{\max}+1} +
\sum_{\ell=0}^{\ell_{\max}}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} |
\sum_{\ell=0}^{\ell_{\max}} V_{A \ell}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} |
\]
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.
The functions $V_{A\ell}$ are parameterized as:
All the functions $V_{A\ell}$ are parameterized as:
\[
V_{A \ell}(\mathbf{r}) =
\sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
\]
\].
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 |
| ~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 in
the core orbitals, which are removed by the ECP. On the other