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#+TITLE: Data stored in TREXIO
#+STARTUP: latexpreview
#+SETUPFILE: docs/theme.setup
For simplicity, the singular form is always used for the names of
groups and attributes, and all data are stored in atomic units.
The dimensions of the arrays in the tables below are given in
column-major order (as in Fortran), and the ordering of the dimensions
is reversed in the produced ~trex.json~ configuration file as the
library is written in C.
#+begin_src python :tangle trex.json :exports none
{
#+end_src
* Metadata (metadata group)
As we expect TREXIO files to be archived in open-data repositories,
we give the possibility to the users to store some metadata inside
the files. We propose to store the list of names of the codes which
have participated to the creation of the file, a list of authors of
the file, and a textual description.
#+NAME: metadata
| Variable | Type | Dimensions (for arrays) | Description |
|-------------------+-------+-------------------------+------------------------------------------|
| ~code_num~ | ~dim~ | | Number of codes used to produce the file |
| ~code~ | ~str~ | ~(metadata.code_num)~ | Names of the codes used |
| ~author_num~ | ~dim~ | | Number of authors of the file |
| ~author~ | ~str~ | ~(metadata.author_num)~ | Names of the authors of the file |
| ~package_version~ | ~str~ | | TREXIO version used to produce the file |
| ~description~ | ~str~ | | Text describing the content of file |
| ~unsafe~ | ~int~ | | ~1~: true, ~0~: false |
**Note:** The ~unsafe~ attribute of the ~metadata~ group indicates
whether the file has been previously opened with ~'u'~ mode. It is
automatically written in the file upon the first unsafe opening. If
the user has checked that the TREXIO file is valid (e.g. using ~trexio-tools~) after unsafe operations, then the ~unsafe~ attribute
value can be manually overwritten (in unsafe mode) from ~1~ to ~0~.
#+CALL: json(data=metadata, title="metadata")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"metadata": {
"code_num" : [ "dim", [] ]
, "code" : [ "str", [ "metadata.code_num" ] ]
, "author_num" : [ "dim", [] ]
, "author" : [ "str", [ "metadata.author_num" ] ]
, "package_version" : [ "str", [] ]
, "description" : [ "str", [] ]
, "unsafe" : [ "int", [] ]
} ,
#+end_src
:end:
* System
** Nucleus (nucleus group)
The nuclei are considered as fixed point charges. Coordinates are
given in Cartesian $(x,y,z)$ format.
#+NAME: nucleus
| Variable | Type | Dimensions | Description |
|---------------+---------+-------------------+--------------------------|
| ~num~ | ~dim~ | | Number of nuclei |
| ~charge~ | ~float~ | ~(nucleus.num)~ | Charges of the nuclei |
| ~coord~ | ~float~ | ~(3,nucleus.num)~ | Coordinates of the atoms |
| ~label~ | ~str~ | ~(nucleus.num)~ | Atom labels |
| ~point_group~ | ~str~ | | Symmetry point group |
| ~repulsion~ | ~float~ | | Nuclear repulsion energy |
#+CALL: json(data=nucleus, title="nucleus")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"nucleus": {
"num" : [ "dim" , [] ]
, "charge" : [ "float", [ "nucleus.num" ] ]
, "coord" : [ "float", [ "nucleus.num", "3" ] ]
, "label" : [ "str" , [ "nucleus.num" ] ]
, "point_group" : [ "str" , [] ]
, "repulsion" : [ "float", [] ]
} ,
#+end_src
:end:
** Cell (cell group)
3 Lattice vectors to define a box containing the system, for example
used in periodic calculations.
#+NAME: cell
| Variable | Type | Dimensions | Description |
|----------+---------+------------+--------------------------------------------------------------------------|
| ~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")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"cell": {
"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:
** Periodic boundary calculations (pbc group)
A single $k$-point per TREXIO file can be stored. The $k$-point is
defined in this group.
#+NAME: pbc
| Variable | Type | Dimensions | Description |
|------------+---------+------------+-------------------------|
| ~periodic~ | ~int~ | | ~1~: true or ~0~: false |
| ~k_point~ | ~float~ | ~(3)~ | $k$-point sampling |
#+CALL: json(data=pbc, title="pbc")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"pbc": {
"periodic" : [ "int" , [] ]
, "k_point" : [ "float", [ "3" ] ]
} ,
#+end_src
:end:
** Electron (electron group)
We consider wave functions expressed in the spin-free formalism, where
the number of \uparrow and \downarrow electrons is fixed.
#+NAME:electron
| Variable | Type | Dimensions | Description |
|----------+-------+------------+-------------------------------------|
| ~num~ | ~dim~ | | Number of electrons |
| ~up_num~ | ~int~ | | Number of \uparrow-spin electrons |
| ~dn_num~ | ~int~ | | Number of \downarrow-spin electrons |
#+CALL: json(data=electron, title="electron")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"electron": {
"num" : [ "dim", [] ]
, "up_num" : [ "int", [] ]
, "dn_num" : [ "int", [] ]
} ,
#+end_src
:end:
** Ground or excited states (state group)
This group contains information about excited states. Since only a
single state can be stored in a TREXIO file, it is possible to store
in the main TREXIO file the names of auxiliary files containing the
information of the other states.
The ~file_name~ and ~label~ arrays have to be written only for the
main file, e.g. the one containing the ground state wave function
together with the basis set parameters, molecular orbitals,
integrals, etc.
The ~id~ and ~current_label~ attributes need to be specified for each file.
#+NAME: state
| Variable | Type | Dimensions | Description |
|-----------------+-------+---------------+---------------------------------------------------------------------------------------------|
| ~num~ | ~dim~ | | Number of states (including the ground state) |
| ~id~ | ~int~ | | Index of the current state (0 is ground state) |
| ~current_label~ | ~str~ | | Label of the current state |
| ~label~ | ~str~ | ~(state.num)~ | Labels of all states |
| ~file_name~ | ~str~ | ~(state.num)~ | Names of the TREXIO files linked to the current one (i.e. containing data for other states) |
#+CALL: json(data=state, title="state")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"state": {
"num" : [ "dim", [] ]
, "id" : [ "int", [] ]
, "current_label" : [ "str", [] ]
, "label" : [ "str", [ "state.num" ] ]
, "file_name" : [ "str", [ "state.num" ] ]
} ,
#+end_src
:end:
* Basis functions
** Basis set (basis group)
*** Gaussian and Slater-type orbitals
We consider here basis functions centered on nuclei. Hence, we enable
the possibility to define /dummy atoms/ to place basis functions in
random positions.
The atomic basis set is defined as a list of shells. Each shell $s$ is
centered on a center $A$, possesses a given angular momentum $l$ and a
radial function $R_s$. The radial function is a linear combination of
$N_{\text{prim}}$ /primitive/ functions that can be of type
Slater ($p=1$) or Gaussian ($p=2$),
parameterized by exponents $\gamma_{ks}$ and coefficients $a_{ks}$:
\[
R_s(\mathbf{r}) = \mathcal{N}_s \vert\mathbf{r}-\mathbf{R}_A\vert^{n_s}
\sum_{k=1}^{N_{\text{prim}}} a_{ks}\, f_{ks}(\gamma_{ks},p)\,
\exp \left( - \gamma_{ks}
\vert \mathbf{r}-\mathbf{R}_A \vert ^p \right).
\]
In the case of Gaussian functions, $n_s$ is always zero.
Different codes normalize functions at different levels. Computing
normalization factors requires the ability to compute overlap
integrals, so the normalization factors should be written in the
file to ensure that the file is self-contained and does not need the
client program to have the ability to compute such integrals.
Some codes assume that the contraction coefficients are for a linear
combination of /normalized/ primitives. This implies that a normalization
constant for the primitive $ks$ needs to be computed and stored. If
this normalization factor is not required, $f_{ks}=1$.
Some codes assume that the basis function are normalized. This
implies the computation of an extra normalization factor, $\mathcal{N}_s$.
If the the basis function is not considered normalized, $\mathcal{N}_s=1$.
All the basis set parameters are stored in one-dimensional arrays.
*** Plane waves
A plane wave is defined as
\[
\chi_j(\mathbf{r}) = \exp \left( -i \mathbf{G}_j \cdot \mathbf{r} \right)
\]
The basis set is defined as the array of $k$-points in the
reciprocal space $\mathbf{G}_j$, defined in the ~pbc~ group. The
kinetic energy cutoff ~e_cut~ is the only input data relevant to
plane waves.
*** Data definitions
#+NAME: basis
| Variable | Type | Dimensions | Description |
|-----------------+---------+---------------------+-----------------------------------------------------------------|
| ~type~ | ~str~ | | Type of basis set: "Gaussian", "Slater" or "PW" for plane waves |
| ~prim_num~ | ~dim~ | | Total number of primitives |
| ~shell_num~ | ~dim~ | | Total number of shells |
| ~nucleus_index~ | ~index~ | ~(basis.shell_num)~ | One-to-one correspondence between shells and atomic indices |
| ~shell_ang_mom~ | ~int~ | ~(basis.shell_num)~ | One-to-one correspondence between shells and angular momenta |
| ~shell_factor~ | ~float~ | ~(basis.shell_num)~ | Normalization factor of each shell ($\mathcal{N}_s$) |
| ~r_power~ | ~int~ | ~(basis.shell_num)~ | Power to which $r$ is raised ($n_s$) |
| ~shell_index~ | ~index~ | ~(basis.prim_num)~ | One-to-one correspondence between primitives and shell index |
| ~exponent~ | ~float~ | ~(basis.prim_num)~ | Exponents of the primitives ($\gamma_{ks}$) |
| ~coefficient~ | ~float~ | ~(basis.prim_num)~ | Coefficients of the primitives ($a_{ks}$) |
| ~prim_factor~ | ~float~ | ~(basis.prim_num)~ | Normalization coefficients for the primitives ($f_{ks}$) |
| ~e_cut~ | ~float~ | | Energy cut-off for plane-wave calculations |
#+CALL: json(data=basis, title="basis")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"basis": {
"type" : [ "str" , [] ]
, "prim_num" : [ "dim" , [] ]
, "shell_num" : [ "dim" , [] ]
, "nucleus_index" : [ "index", [ "basis.shell_num" ] ]
, "shell_ang_mom" : [ "int" , [ "basis.shell_num" ] ]
, "shell_factor" : [ "float", [ "basis.shell_num" ] ]
, "r_power" : [ "int" , [ "basis.shell_num" ] ]
, "shell_index" : [ "index", [ "basis.prim_num" ] ]
, "exponent" : [ "float", [ "basis.prim_num" ] ]
, "coefficient" : [ "float", [ "basis.prim_num" ] ]
, "prim_factor" : [ "float", [ "basis.prim_num" ] ]
, "e_cut" : [ "float", [] ]
} ,
#+end_src
:end:
*** Example
For example, consider H_2 with the following basis set (in GAMESS
format), where both the AOs and primitives are considered normalized:
#+BEGIN_EXAMPLE
HYDROGEN
S 5
1 3.387000E+01 6.068000E-03
2 5.095000E+00 4.530800E-02
3 1.159000E+00 2.028220E-01
4 3.258000E-01 5.039030E-01
5 1.027000E-01 3.834210E-01
S 1
1 3.258000E-01 1.000000E+00
S 1
1 1.027000E-01 1.000000E+00
P 1
1 1.407000E+00 1.000000E+00
P 1
1 3.880000E-01 1.000000E+00
D 1
1 1.057000E+00 1.000000E+00
#+END_EXAMPLE
In TREXIO representaion we have:
#+BEGIN_EXAMPLE
type = "Gaussian"
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 ]
# 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 ]
# 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,
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 ]
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,
1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01,
1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00,
7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01,
3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00,
4.3649547399719840e-01, 1.8135965626177861e+00 ]
#+END_EXAMPLE
** Effective core potentials (ecp group)
An effective core potential (ECP) $V_A^{\text{ECP}}$ replacing the
core electrons of atom $A$ can be expressed as
\[
V_A^{\text{ECP}} =
V_{A \ell_{\max}+1} +
\sum_{\ell=0}^{\ell_{\max}} V_{A \ell}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} |
\]
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.
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.
#+NAME: ecp
| Variable | Type | Dimensions | Description |
|----------------------+---------+-----------------+----------------------------------------------------------------------------------------|
| ~max_ang_mom_plus_1~ | ~int~ | ~(nucleus.num)~ | $\ell_{\max}+1$, one higher than the max angular momentum in the removed core orbitals |
| ~z_core~ | ~int~ | ~(nucleus.num)~ | Number of core electrons to remove per atom |
| ~num~ | ~dim~ | | Total number of ECP functions for all atoms and all values of $\ell$ |
| ~ang_mom~ | ~int~ | ~(ecp.num)~ | One-to-one correspondence between ECP items and the angular momentum $\ell$ |
| ~nucleus_index~ | ~index~ | ~(ecp.num)~ | One-to-one correspondence between ECP items and the atom index |
| ~exponent~ | ~float~ | ~(ecp.num)~ | $\alpha_{A q \ell}$ all ECP exponents |
| ~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
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. The 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~. The latter causes issues
when written before the ~ang_mom~ array in the TREXIO file.
*Update*: in fact, the aforementioned issue has only been observed
when using HDF5 version 1.10.4 installed via ~apt-get~. Installing
the same version from the ~conda-forge~ channel and running it in
an isolated ~conda~ environment works just fine. Thus, it seems to
be a bug in the ~apt~-provided package.
If you encounter the aforementioned issue, please report it to our
[[https://github.com/TREX-CoE/trexio/issues][issue tracker on GitHub]].
#+CALL: json(data=ecp, title="ecp")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"ecp": {
"max_ang_mom_plus_1" : [ "int" , [ "nucleus.num" ] ]
, "z_core" : [ "int" , [ "nucleus.num" ] ]
, "num" : [ "dim" , [] ]
, "ang_mom" : [ "int" , [ "ecp.num" ] ]
, "nucleus_index" : [ "index", [ "ecp.num" ] ]
, "exponent" : [ "float", [ "ecp.num" ] ]
, "coefficient" : [ "float", [ "ecp.num" ] ]
, "power" : [ "int" , [ "ecp.num" ] ]
} ,
#+end_src
:end:
*** Example
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
0.00000000000000 2 1.000000000000000
#+END_EXAMPLE
In TREXIO representation this would be:
#+BEGIN_EXAMPLE
num = 8
# lmax+1 per atom
max_ang_mom_plus_1 = [ 1, 1 ]
# number of core electrons to remove per atom
zcore = [ 0, 0 ]
# first 4 ECP elements correspond to the first H atom ; the remaining 4 elements are for the second H atom
nucleus_index = [
0, 0, 0, 0,
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
ang_mom = [
1, 1, 1, 0,
1, 1, 1, 0
]
# ECP quantities that can be attributed to atoms and/or angular momenta based on the aforementioned ecp_nucleus and ecp_ang_mom arrays
coefficient = [
1.00000000000000, 21.24359508259891, -10.85192405303825, 0.00000000000000,
1.00000000000000, 21.24359508259891, -10.85192405303825, 0.00000000000000
]
exponent = [
21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000,
21.24359508259891, 21.24359508259891, 21.77696655044365, 1.000000000000000
]
power = [
-1, 1, 0, 0,
-1, 1, 0, 0
]
#+END_EXAMPLE
** Numerical integration grid (grid group)
In some applications, such as DFT calculations, integrals have to
be computed numerically on a grid. A common choice for the angular
grid is the one proposed by Lebedev and Laikov
[Russian Academy of Sciences Doklady Mathematics, Volume 59, Number 3, 1999, pages 477-481].
For the radial grids, many approaches have been developed over the years.
The structure of this group is adapted for the [[https://github.com/dftlibs/numgrid][numgrid]] library.
Feel free to submit a PR if you find missing options/functionalities.
#+NAME: grid
| Variable | Type | Dimensions | Description |
|-----------------+---------+------------------+-------------------------------------------------------------------------|
| ~description~ | ~str~ | | Details about the used quadratures can go here |
| ~rad_precision~ | ~float~ | | Radial precision parameter (not used in some schemes like Krack-Köster) |
| ~num~ | ~dim~ | | Number of grid points |
| ~max_ang_num~ | ~int~ | | Maximum number of angular grid points (for pruning) |
| ~min_ang_num~ | ~int~ | | Minimum number of angular grid points (for pruning) |
| ~coord~ | ~float~ | ~(grid.num)~ | Discretized coordinate space |
| ~weight~ | ~float~ | ~(grid.num)~ | Grid weights according to a given partitioning (e.g. Becke) |
| ~ang_num~ | ~dim~ | | Number of angular integration points (if used) |
| ~ang_coord~ | ~float~ | ~(grid.ang_num)~ | Discretized angular space (if used) |
| ~ang_weight~ | ~float~ | ~(grid.ang_num)~ | Angular grid weights (if used) |
| ~rad_num~ | ~dim~ | | Number of radial integration points (if used) |
| ~rad_coord~ | ~float~ | ~(grid.rad_num)~ | Discretized radial space (if used) |
| ~rad_weight~ | ~float~ | ~(grid.rad_num)~ | Radial grid weights (if used) |
#+CALL: json(data=grid, title="grid")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"grid": {
"description" : [ "str" , [] ]
, "rad_precision" : [ "float", [] ]
, "num" : [ "dim" , [] ]
, "max_ang_num" : [ "int" , [] ]
, "min_ang_num" : [ "int" , [] ]
, "coord" : [ "float", [ "grid.num" ] ]
, "weight" : [ "float", [ "grid.num" ] ]
, "ang_num" : [ "dim" , [] ]
, "ang_coord" : [ "float", [ "grid.ang_num" ] ]
, "ang_weight" : [ "float", [ "grid.ang_num" ] ]
, "rad_num" : [ "dim" , [] ]
, "rad_coord" : [ "float", [ "grid.rad_num" ] ]
, "rad_weight" : [ "float", [ "grid.rad_num" ] ]
} ,
#+end_src
:end:
* Orbitals
** Atomic orbitals (ao group)
Going from the atomic basis set to AOs implies a systematic
construction of all the angular functions of each shell. We
consider two cases for the angular functions: the real-valued
spherical harmonics, and the polynomials in Cartesian coordinates.
In the case of real spherical harmonics, the AOs are ordered as
$0, +1, -1, +2, -2, \dots, +m, -m$ (see [[https://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics][Wikipedia]]).
In the case of polynomials we
impose the canonical (or alphabetical) ordering), i.e
\begin{eqnarray}
p & : & p_x, p_y, p_z \nonumber \\
d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\
f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\
{\rm etc.} \nonumber
\end{eqnarray}
Note that there is no exception for $p$ orbitals in spherical
coordinates: the ordering is $0,+1,-1$ which corresponds $p_z, p_x, p_y$.
AOs are defined as
\[
\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r})
\]
where $i$ is the atomic orbital index,
$P$ encodes for either the
polynomials or the spherical harmonics, $s(i)$ returns the
shell on which the AO is expanded, and $\eta(i)$ denotes which
angular function is chosen.
$\mathcal{N}_i$ is a normalization factor that enables the
possibility to have different normalization coefficients within a
shell, as in the GAMESS convention where
$\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}$ because
\[ \left[ \iiint \left(x-X_A \right)^2 R_{s(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne
\left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{s(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
In such a case, one should set the normalization of the shell (in
the [[Basis set (basis group)][Basis set]] section) to $\mathcal{N}_{z^2}$, which is the
normalization factor of the atomic orbitals in spherical coordinates.
The normalization factor of the $xy$ function which should be
introduced here should be $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
#+NAME: ao
| Variable | Type | Dimensions | Description |
|-----------------+---------+------------+--------------------------------------|
| ~cartesian~ | ~int~ | | ~1~: true, ~0~: false |
| ~num~ | ~dim~ | | Total number of atomic orbitals |
| ~shell~ | ~index~ | ~(ao.num)~ | Basis set shell for each AO |
| ~normalization~ | ~float~ | ~(ao.num)~ | Normalization factor $\mathcal{N}_i$ |
#+CALL: json(data=ao, title="ao")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"ao": {
"cartesian" : [ "int" , [] ]
, "num" : [ "dim" , [] ]
, "shell" : [ "index", [ "ao.num" ] ]
, "normalization" : [ "float", [ "ao.num" ] ]
} ,
#+end_src
:end:
*** One-electron integrals (~ao_1e_int~ group)
: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,
- \[ \hat{T}_{\text{e}} =
\sum_{i=1}^{N_\text{elec}} -\frac{1}{2}\hat{\Delta}_i \] : electronic kinetic energy
- $\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} + \hat{V}_\text{ECP}$ : core electronic Hamiltonian
The one-electron integrals for a one-electron operator $\hat{O}$ are
\[ \langle p \vert \hat{O} \vert q \rangle \], returned as a matrix
over atomic orbitals.
#+NAME: ao_1e_int
| Variable | Type | Dimensions | Description |
|-----------------------+---------+--------------------+--------------------------------------------------------------------------|
| ~overlap~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ |
| ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ |
| ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ |
| ~ecp~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle$ |
| ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ |
| ~overlap_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ (imaginary part) |
| ~kinetic_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ (imaginary part) |
| ~potential_n_e_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ (imaginary part) |
| ~ecp_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ECP}} \vert q \rangle$ (imaginary part) |
| ~core_hamiltonian_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ (imaginary part) |
#+CALL: json(data=ao_1e_int, title="ao_1e_int")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"ao_1e_int": {
"overlap" : [ "float", [ "ao.num", "ao.num" ] ]
, "kinetic" : [ "float", [ "ao.num", "ao.num" ] ]
, "potential_n_e" : [ "float", [ "ao.num", "ao.num" ] ]
, "ecp" : [ "float", [ "ao.num", "ao.num" ] ]
, "core_hamiltonian" : [ "float", [ "ao.num", "ao.num" ] ]
, "overlap_im" : [ "float", [ "ao.num", "ao.num" ] ]
, "kinetic_im" : [ "float", [ "ao.num", "ao.num" ] ]
, "potential_n_e_im" : [ "float", [ "ao.num", "ao.num" ] ]
, "ecp_im" : [ "float", [ "ao.num", "ao.num" ] ]
, "core_hamiltonian_im" : [ "float", [ "ao.num", "ao.num" ] ]
} ,
#+end_src
:end:
*** Two-electron integrals (~ao_2e_int~ group)
:PROPERTIES:
:CUSTOM_ID: ao_two_e
:END:
The two-electron integrals for a two-electron operator $\hat{O}$ are
\[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists
notation or \[ ( pr \vert \hat{O} \vert qs ) \] in chemists
notation, where $p,q,r,s$ are indices over atomic orbitals.
Functions are provided to get the indices in physicists or chemists
notation.
# TODO: Physicist / Chemist functions
- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}}
\sum_{j=1}^{i-1} \frac{\text{erf}(\vert \mathbf{r}_i -
\mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential
The Cholesky decomposition of the integrals can also be stored:
\[
A_{ijkl} = \sum_{\alpha} G_{il\alpha} G_{jl\alpha}
\]
#+NAME: ao_2e_int
| Variable | Type | Dimensions | Description |
|-----------------------+----------------+---------------------------------------------------+-----------------------------------------------|
| ~eri~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Electron repulsion integrals |
| ~eri_lr~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Long-range electron repulsion integrals |
| ~eri_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for ERI |
| ~eri_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_cholesky_num)~ | Cholesky decomposition of the ERI |
| ~eri_lr_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for long range ERI |
| ~eri_lr_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_lr_cholesky_num)~ | Cholesky decomposition of the long range ERI |
#+CALL: json(data=ao_2e_int, title="ao_2e_int")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"ao_2e_int": {
"eri" : [ "float sparse", [ "ao.num", "ao.num", "ao.num", "ao.num" ] ]
, "eri_lr" : [ "float sparse", [ "ao.num", "ao.num", "ao.num", "ao.num" ] ]
, "eri_cholesky_num" : [ "dim" , [] ]
, "eri_cholesky" : [ "float sparse", [ "ao_2e_int.eri_cholesky_num", "ao.num", "ao.num" ] ]
, "eri_lr_cholesky_num" : [ "dim" , [] ]
, "eri_lr_cholesky" : [ "float sparse", [ "ao_2e_int.eri_lr_cholesky_num", "ao.num", "ao.num" ] ]
} ,
#+end_src
:end:
** Molecular orbitals (mo group)
#+NAME: mo
| Variable | Type | Dimensions | Description |
|------------------+---------+--------------------+--------------------------------------------------------------------------|
| ~type~ | ~str~ | | Free text to identify the set of MOs (HF, Natural, Local, CASSCF, /etc/) |
| ~num~ | ~dim~ | | Number of MOs |
| ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients |
| ~coefficient_im~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients (imaginary part) |
| ~class~ | ~str~ | ~(mo.num)~ | Choose among: Core, Inactive, Active, Virtual, Deleted |
| ~symmetry~ | ~str~ | ~(mo.num)~ | Symmetry in the point group |
| ~occupation~ | ~float~ | ~(mo.num)~ | Occupation number |
| ~energy~ | ~float~ | ~(mo.num)~ | For canonical MOs, corresponding eigenvalue |
| ~spin~ | ~int~ | ~(mo.num)~ | For UHF wave functions, 0 is $\alpha$ and 1 is $\beta$ |
#+CALL: json(data=mo, title="mo")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"mo": {
"type" : [ "str" , [] ]
, "num" : [ "dim" , [] ]
, "coefficient" : [ "float", [ "mo.num", "ao.num" ] ]
, "coefficient_im" : [ "float", [ "mo.num", "ao.num" ] ]
, "class" : [ "str" , [ "mo.num" ] ]
, "symmetry" : [ "str" , [ "mo.num" ] ]
, "occupation" : [ "float", [ "mo.num" ] ]
, "energy" : [ "float", [ "mo.num" ] ]
, "spin" : [ "int" , [ "mo.num" ] ]
} ,
#+end_src
:end:
*** One-electron integrals (~mo_1e_int~ group)
The operators as the same as those defined in the
[[#ao_one_e][AO one-electron integrals section]]. Here, the integrals are given in
the basis of molecular orbitals.
#+NAME: mo_1e_int
| Variable | Type | Dimensions | Description |
|-----------------------+---------+--------------------+--------------------------------------------------------------------------|
| ~overlap~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert j \rangle$ |
| ~kinetic~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{T}_e \vert j \rangle$ |
| ~potential_n_e~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle$ |
| ~ecp~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle$ |
| ~core_hamiltonian~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{h} \vert j \rangle$ |
| ~overlap_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert j \rangle$ (imaginary part) |
| ~kinetic_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{T}_e \vert j \rangle$ (imaginary part) |
| ~potential_n_e_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle$ (imaginary part) |
| ~ecp_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle$ (imaginary part) |
| ~core_hamiltonian_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{h} \vert j \rangle$ (imaginary part) |
#+CALL: json(data=mo_1e_int, title="mo_1e_int")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"mo_1e_int": {
"overlap" : [ "float", [ "mo.num", "mo.num" ] ]
, "kinetic" : [ "float", [ "mo.num", "mo.num" ] ]
, "potential_n_e" : [ "float", [ "mo.num", "mo.num" ] ]
, "ecp" : [ "float", [ "mo.num", "mo.num" ] ]
, "core_hamiltonian" : [ "float", [ "mo.num", "mo.num" ] ]
, "overlap_im" : [ "float", [ "mo.num", "mo.num" ] ]
, "kinetic_im" : [ "float", [ "mo.num", "mo.num" ] ]
, "potential_n_e_im" : [ "float", [ "mo.num", "mo.num" ] ]
, "ecp_im" : [ "float", [ "mo.num", "mo.num" ] ]
, "core_hamiltonian_im" : [ "float", [ "mo.num", "mo.num" ] ]
} ,
#+end_src
:end:
*** Two-electron integrals (~mo_2e_int~ group)
The operators are the same as those defined in the
[[#ao_two_e][AO two-electron integrals section]]. Here, the integrals are given in
the basis of molecular orbitals.
#+NAME: mo_2e_int
| Variable | Type | Dimensions | Description |
|-----------------------+----------------+---------------------------------------------------+-----------------------------------------------|
| ~eri~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | Electron repulsion integrals |
| ~eri_lr~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | Long-range electron repulsion integrals |
| ~eri_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for ERI |
| ~eri_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, mo_2e_int.eri_cholesky_num)~ | Cholesky decomposition of the ERI |
| ~eri_lr_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for long range ERI |
| ~eri_lr_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, mo_2e_int.eri_lr_cholesky_num)~ | Cholesky decomposition of the long range ERI |
#+CALL: json(data=mo_2e_int, title="mo_2e_int")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"mo_2e_int": {
"eri" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "eri_lr" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "eri_cholesky_num" : [ "dim" , [] ]
, "eri_cholesky" : [ "float sparse", [ "mo_2e_int.eri_cholesky_num", "mo.num", "mo.num" ] ]
, "eri_lr_cholesky_num" : [ "dim" , [] ]
, "eri_lr_cholesky" : [ "float sparse", [ "mo_2e_int.eri_lr_cholesky_num", "mo.num", "mo.num" ] ]
} ,
#+end_src
:end:
* Multi-determinant information
** Slater determinants (determinant group)
The configuration interaction (CI) wave function $\Psi$
can be expanded in the basis of Slater determinants $D_I$ as follows
\[
\Psi = \sum_I C_I D_I
\]
For relatively small expansions, a given determinant can be represented as a list of occupied orbitals.
However, this becomes unfeasible for larger expansions and requires more advanced data structures.
The bit field representation is used here, namely a given determinant is represented as $N_{\text{int}}$
64-bit integers where j-th bit is set to 1 if there is an electron in the j-th orbital and 0 otherwise.
This gives access to larger determinant expansions by optimising the storage of the determinant lists
in the memory.
\[
D_I = \alpha_1 \alpha_2 \ldots \alpha_{n_\uparrow} \beta_1 \beta_2 \ldots \beta_{n_\downarrow}
\]
where $\alpha$ and $\beta$ denote \uparrow-spin and \downarrow-spin electrons, respectively,
$n_\uparrow$ and $n_\downarrow$ correspond to ~electron.up_num~ and ~electron.dn_num~, respectively.
Note: the ~special~ attribute is present in the types, meaning that the source node is not
produced by the code generator.
An illustration on how to read determinants is presented in the [[./examples.html][examples]].
#+NAME: determinant
| Variable | Type | Dimensions | Description |
|---------------+------------------+---------------------+--------------------------------------------------------|
| ~num~ | ~dim readonly~ | | Number of determinants |
| ~list~ | ~int special~ | ~(determinant.num)~ | List of determinants as integer bit fields |
| ~coefficient~ | ~float buffered~ | ~(determinant.num)~ | Coefficients of the determinants from the CI expansion |
#+CALL: json(data=determinant, title="determinant")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"determinant": {
"num" : [ "dim readonly" , [] ]
, "list" : [ "int special" , [ "determinant.num" ] ]
, "coefficient" : [ "float buffered", [ "determinant.num" ] ]
} ,
#+end_src
:end:
** Configuration state functions (csf group)
The configuration interaction (CI) wave function $\Psi$ can be
expanded in the basis of [[https://en.wikipedia.org/wiki/Configuration_state_function][configuration state functions]] (CSFs)
$\Psi_I$ as follows
\[
\Psi = \sum_I C_I \psi_I.
\]
Each CSF $\psi_I$ is a linear combination of Slater determinants. Slater
determinants are stored in the =determinant= section. In this group
we store the CI coefficients in the basis of CSFs, and the
matrix $\langle D_I | \psi_J \rangle$ needed to project the CSFs in
the basis of Slater determinants.
#+NAME: csf
| Variable | Type | Dimensions | Description |
|-------------------+------------------+-----------------------------+-----------------------------------------|
| ~num~ | ~dim readonly~ | | Number of CSFs |
| ~coefficient~ | ~float buffered~ | ~(csf.num)~ | Coefficients $C_I$ of the CSF expansion |
| ~det_coefficient~ | ~float sparse~ | ~(determinant.num,csf.num)~ | Projection on the determinant basis |
#+CALL: json(data=csf, title="csf")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"csf": {
"num" : [ "dim readonly" , [] ]
, "coefficient" : [ "float buffered", [ "csf.num" ] ]
, "det_coefficient" : [ "float sparse" , [ "csf.num", "determinant.num" ] ]
} ,
#+end_src
:end:
** Amplitudes (amplitude group)
The wave function may be expressed in terms of action of the cluster
operator $\hat{T}$:
\[
\hat{T} = \hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \dots
\]
on a reference wave function $\Psi$, where $\hat{T}_1$ is the single excitation operator,
\[
\hat{T}_1 = \sum_{ia} t_{i}^{a}\, \hat{a}^\dagger_a \hat{a}_i,
\]
$\hat{T}_2$ is the double excitation operator,
\[
\hat{T}_2 = \frac{1}{4} \sum_{ijab} t_{ij}^{ab}\, \hat{a}^\dagger_a \hat{a}^\dagger_b \hat{a}_j \hat{a}_i,
\]
/etc/. Indices $i$, $j$, $a$ and $b$ denote molecular orbital indices.
Wave functions obtained with perturbation theory or configuration
interaction are of the form
\[ |\Phi\rangle = \hat{T}|\Psi\rangle \]
and coupled-cluster wave functions are of the form
\[ |\Phi\rangle = e^{\hat{T}}| \Psi \rangle \]
The reference wave function is stored using the ~determinant~ and/or
~csf~ groups, and the amplitudes are stored using the current group.
The attributes with the ~exp~ suffix correspond to exponentialized operators.
The order of the indices is chosen such that
- ~t(i,a)~ = $t_{i}^{a}$.
- ~t(i,j,a,b)~ = $t_{ij}^{ab}$,
- ~t(i,j,k,a,b,c)~ = $t_{ijk}^{abc}$,
- ~t(i,j,k,l,a,b,c,d)~ = $t_{ijkl}^{abcd}$,
- $\dots$
#+NAME: amplitude
| Variable | Type | Dimensions | Description |
|-----------------+----------------+-------------------------------------------------------------+-------------------------------------------------|
| ~single~ | ~float sparse~ | ~(mo.num,mo.num)~ | Single excitation amplitudes |
| ~single_exp~ | ~float sparse~ | ~(mo.num,mo.num)~ | Exponentialized single excitation amplitudes |
| ~double~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num)~ | Double excitation amplitudes |
| ~double_exp~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num)~ | Exponentialized double excitation amplitudes |
| ~triple~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)~ | Triple excitation amplitudes |
| ~triple_exp~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)~ | Exponentialized triple excitation amplitudes |
| ~quadruple~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)~ | Quadruple excitation amplitudes |
| ~quadruple_exp~ | ~float sparse~ | ~(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)~ | Exponentialized quadruple excitation amplitudes |
#+CALL: json(data=amplitude, title="amplitude")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"amplitude": {
"single" : [ "float sparse", [ "mo.num", "mo.num" ] ]
, "single_exp" : [ "float sparse", [ "mo.num", "mo.num" ] ]
, "double" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "double_exp" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "triple" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "triple_exp" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "quadruple" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "quadruple_exp" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num", "mo.num" ] ]
} ,
#+end_src
:end:
** Reduced density matrices (rdm group)
The reduced density matrices are defined in the basis of molecular
orbitals.
The \uparrow-spin and \downarrow-spin components of the one-body
density matrix are given by
\begin{eqnarray*}
\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
\[
\gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij}
\]
The $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$ components of the two-body density matrix are given by
\begin{eqnarray*}
\Gamma_{ijkl}^{\uparrow \uparrow} &=&
\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 | \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 | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\alpha} | \Psi \rangle \\
\end{eqnarray*}
and the spin-summed one-body density matrix is
\[
\Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} +
\Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}.
\]
The density matrices are normalized such that
\begin{eqnarray*}
\sum_{ij} \Gamma_{ijij}^{\uparrow \uparrow} & = & N_\uparrow\, (N_\uparrow-1)/2 \\
\sum_{ij} \Gamma_{ijij}^{\downarrow \downarrow} & = & N_\downarrow\, (N_\downarrow-1)/2 \\
\sum_{ij} \Gamma_{ijij}^{\uparrow \downarrow} & = & N_\uparrow\, N_\downarrow \\
\sum_{ij} \Gamma_{ijij} & = & (N_\uparrow+N_\downarrow)\, (N_\uparrow+N_\downarrow-1)/2.
\end{eqnarray*}
The total energy can be computed as:
\[
E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle +
\frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle
\]
To compress the storage, the Cholesky decomposition of the RDMs can
be stored:
\[
\Gamma_{ijkl} = \sum_{\alpha} G_{ij\alpha} G_{kl\alpha}
\]
Warning: as opposed to electron repulsion integrals, the
decomposition is made such that the Cholesky vectors are expanded
in a two-electron basis
$f_{ij}(\mathbf{r}_1,\mathbf{r}_2) = \phi_i(\mathbf{r}_1) \phi_j(\mathbf{r}_2)$,
whereas in electron repulsion integrals each Cholesky vector is
expressed in a basis of a one-electron function
$g_{ik}(\mathbf{r}_1) = \phi_i(\mathbf{r}_1) \phi_k(\mathbf{r}_1)$.
#+NAME: rdm
| Variable | Type | Dimensions | Description |
|------------------------+----------------+----------------------------------------------+-----------------------------------------------------------------------|
| ~1e~ | ~float~ | ~(mo.num, mo.num)~ | One body density matrix |
| ~1e_up~ | ~float~ | ~(mo.num, mo.num)~ | \uparrow-spin component of the one body density matrix |
| ~1e_dn~ | ~float~ | ~(mo.num, mo.num)~ | \downarrow-spin component of the one body density matrix |
| ~2e~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | Two-body reduced density matrix (spin trace) |
| ~2e_upup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \uparrow\uparrow component of the two-body reduced density matrix |
| ~2e_dndn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\downarrow component of the two-body reduced density matrix |
| ~2e_updn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \uparrow\downarrow component of the two-body reduced density matrix |
| ~2e_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_cholesky_num)~ | Cholesky decomposition of the two-body RDM (spin trace) |
| ~2e_upup_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_upup_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_upup_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\uparrow\uparrow) |
| ~2e_dndn_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_dndn_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_dndn_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\downarrow\downarrow) |
| ~2e_updn_cholesky_num~ | ~dim~ | | Number of Cholesky vectors |
| ~2e_updn_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_updn_cholesky_num)~ | Cholesky decomposition of the two-body RDM (\uparrow\downarrow) |
#+CALL: json(data=rdm, title="rdm")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"rdm": {
"1e" : [ "float" , [ "mo.num", "mo.num" ] ]
, "1e_up" : [ "float" , [ "mo.num", "mo.num" ] ]
, "1e_dn" : [ "float" , [ "mo.num", "mo.num" ] ]
, "2e" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_upup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_dndn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_updn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ]
, "2e_cholesky_num" : [ "dim" , [] ]
, "2e_cholesky" : [ "float sparse", [ "rdm.2e_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_upup_cholesky_num" : [ "dim" , [] ]
, "2e_upup_cholesky" : [ "float sparse", [ "rdm.2e_upup_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_dndn_cholesky_num" : [ "dim" , [] ]
, "2e_dndn_cholesky" : [ "float sparse", [ "rdm.2e_dndn_cholesky_num", "mo.num", "mo.num" ] ]
, "2e_updn_cholesky_num" : [ "dim" , [] ]
, "2e_updn_cholesky" : [ "float sparse", [ "rdm.2e_updn_cholesky_num", "mo.num", "mo.num" ] ]
} ,
#+end_src
:end:
* Correlation factors
** Jastrow factor (jastrow group)
The Jastrow factor is an $N$-electron function to which the CI
expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
In the following, we use the notations $r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|$ and
$R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|$, where indices
$i$ and $j$ correspond to electrons and $\alpha$ to nuclei.
Parameters for multiple forms of Jastrow factors can be saved in
TREXIO files, and are described in the following sections. These
are identified by the ~type~ attribute. The type can be one of the
following:
- ~CHAMP~
- ~Mu~
*** CHAMP
The first form of Jastrow factor is the one used in
the [[https://trex-coe.eu/trex-quantum-chemistry-codes/champ][CHAMP]] program:
\[
J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
\]
$J_{\text{eN}}$ contains electron-nucleus terms:
\[
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}
\frac{a_{1,\alpha}\, g_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\, g_\alpha(R_{i\alpha})} +
\sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [g_\alpha(R_{i\alpha})]^p - J_{eN}^\infty
\]
$J_{\text{ee}}$ contains electron-electron terms:
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1}
\frac{b_1\, f(r_{ij})}{1+b_2\, f(r_{ij})} +
\sum_{p=2}^{N_\text{ord}^b} a_{p+1}\, [f(r_{ij})]^p - J_{ee}^\infty
\]
and $J_{\text{eeN}}$ contains electron-electron-Nucleus terms:
\[
J_{\text{eeN}}(\mathbf{r},\mathbf{R}) =
\sum_{\alpha=1}^{N_{\text{nucl}}}
\sum_{i=1}^{N_{\text{elec}}}
\sum_{j=1}^{i-1}
\sum_{p=2}^{N_{\text{ord}}}
\sum_{k=0}^{p-1}
\sum_{l=0}^{p-k-2\delta_{k,0}}
c_{lkp\alpha} \left[ f({r}_{ij}) \right]^k
\left[ \left[ g_\alpha({R}_{i\alpha}) \right]^l + \left[ g_\alpha({R}_{j\alpha}) \right]^l \right]
\left[ g_\alpha({R}_{i\,\alpha}) \, g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2}
\]
$c_{lkp\alpha}$ are non-zero only when $p-k-l$ is even.
The terms $J_{\text{ee}}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
$J_{\text{ee}}$ and $J_{\text{eN}}$ have an asymptotic value of zero.
$f$ and $g$ are scaling function defined as
\[
f(r) = \frac{1-e^{-\kappa\, r}}{\kappa} \text{ and }
g_\alpha(r) = e^{-\kappa_\alpha\, r}.
\]
*** Mu
[[https://aip.scitation.org/doi/10.1063/5.0044683][Mu-Jastrow]] is based on a one-parameter correlation factor that has
been introduced in the context of transcorrelated methods. This
correlation factor imposes the electron-electron cusp and it is
built such that the leading order in $1/r_{12}$ of the effective
two-electron potential reproduces the long-range interaction of the
range-separated density functional theory. Its analytical
expression reads
\[
J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
J_{\text{eN}}(\mathbf{r}, \mathbf{R})
\].
The electron-electron cusp is incorporated in the three-body term.
\[
J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\]
$u$ is an electron-electron function given by the symmetric function
\[
u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
\]
This electron-electron term is tuned by the parameter $\mu$ which
controls the depth and the range of the Coulomb hole between
electrons.
An envelope function has been introduced to cancel out the Jastrow
effects between two-electrons when they are both close to a nucleus
(to perform a frozen-core calculation). The envelope function is
given by
\[
E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
\]
In particular, if the parameters $\gamma_\alpha$ tend to zero, the
Mu-Jastrow factor becomes a two-body Jastrow factor:
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
\]
and for large $\gamma_\alpha$ it becomes zero.
To increase the flexibility of the Jastrow and improve the
electron density the following electron-nucleus term is added
\[
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \,
\left[ \exp\left( a_{\alpha} R_{i \alpha}^2 \right) - 1\right].
\]
The parameter $\mu$ is stored in the ~ee~ array, the parameters
$\gamma_\alpha$ are stored in the ~een~ array, and the parameters
$a_\alpha$ are stored in the ~en~ array.
*** Table of values
#+name: jastrow
| Variable | Type | Dimensions | Description |
|---------------+----------+---------------------+-----------------------------------------------------------------|
| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
| ~en_nucleus~ | ~index~ | ~(jastrow.en_num)~ | Nucleus relative to the eN parameter |
| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances |
| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP Jastrow for electron-nucleus distances |
#+CALL: json(data=jastrow, title="jastrow")
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"jastrow": {
"type" : [ "string", [] ]
, "ee_num" : [ "dim" , [] ]
, "en_num" : [ "dim" , [] ]
, "een_num" : [ "dim" , [] ]
, "ee" : [ "float" , [ "jastrow.ee_num" ] ]
, "en" : [ "float" , [ "jastrow.en_num" ] ]
, "een" : [ "float" , [ "jastrow.een_num" ] ]
, "en_nucleus" : [ "index" , [ "jastrow.en_num" ] ]
, "een_nucleus" : [ "index" , [ "jastrow.een_num" ] ]
, "ee_scaling" : [ "float" , [] ]
, "en_scaling" : [ "float" , [ "nucleus.num" ] ]
} ,
#+end_src
:end:
* Quantum Monte Carlo data (qmc group)
In quantum Monte Carlo calculations, the wave function is evaluated
at points of the 3N-dimensional space. Some algorithms require multiple
independent /walkers/, so it is possible to store multiple coordinates,
as well as some quantities evaluated at those points.
By convention, the electron coordinates contain first all the electrons
of $\uparrow$-spin and then all the $\downarrow$-spin.
#+name: qmc
| Variable | Type | Dimensions | Description |
|----------+---------+------------------------------+---------------------------------------|
| ~num~ | ~dim~ | | Number of 3N-dimensional points |
| ~point~ | ~float~ | ~(3, electron.num, qmc.num)~ | 3N-dimensional points |
| ~psi~ | ~float~ | ~(qmc.num)~ | Wave function evaluated at the points |
| ~e_loc~ | ~float~ | ~(qmc.num)~ | Local energy evaluated at the points |
#+CALL: json(data=qmc, title="qmc", last=1)
#+RESULTS:
:results:
#+begin_src python :tangle trex.json
"qmc": {
"num" : [ "dim" , [] ]
, "point" : [ "float", [ "qmc.num", "electron.num", "3" ] ]
, "psi" : [ "float", [ "qmc.num" ] ]
, "e_loc" : [ "float", [ "qmc.num" ] ]
}
#+end_src
:end:
* Appendix :noexport:
** Python script from table to json
#+NAME: json
#+begin_src python :var data=nucleus title="nucleus" last=0 :results output drawer
print("""#+begin_src python :tangle trex.json""")
print(""" "%s": {"""%(title))
indent = " "
f1 = 0 ; f2 = 0 ; f3 = 0
for line in data:
line = [ x.replace("~","") for x in line ]
name = '"'+line[0]+'"'
typ = '"'+line[1]+'"'
dims = line[2]
if '(' in dims:
dims = dims.strip()[1:-1]
dims = [ '"'+x.strip()+'"' for x in dims.split(',') ]
dims = "[ " + ", ".join(dims) + " ]"
else:
dims = "[ ]"
f1 = max(f1, len(name))
f2 = max(f2, len(typ))
f3 = max(f3, len(dims))
fmt = "%%s%%%ds : [ %%%ds, %%%ds ]" % (f1, f2, f3)
for line in data:
line = [ x.replace("~","") for x in line ]
name = '"'+line[0]+'"'
typ = '"'+line[1]+'"'
dims = line[2]
if '(' in dims:
dims = dims.strip()[1:-1]
dims = [ '"'+x.strip()+'"' for x in dims.split(',') ]
dims.reverse()
dims = "[ " + ", ".join(dims) + " ]"
else:
if dims.strip() != "":
dims = "ERROR"
else:
dims = "[]"
buffer = fmt % (indent, name, typ.ljust(f2), dims.ljust(f3))
indent = " , "
print(buffer)
if last == 0:
print(" } ,")
else:
print(" }")
print("""#+end_src""")
#+end_src
#+begin_src python :tangle trex.json :results output drawer :exports none
}
#+end_src
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