From 23b6c9008a0701d644900edc54512a79eea9ee4a Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Mon, 2 Jan 2023 16:00:57 +0100 Subject: [PATCH] Added Jastrow group and reorganize data --- ChangeLog | 1 + python/install_pytrexio.sh | 2 +- trex.org | 1807 +++++++++++++++++++----------------- 3 files changed, 967 insertions(+), 843 deletions(-) diff --git a/ChangeLog b/ChangeLog index e4afa32..b42edf5 100644 --- a/ChangeLog +++ b/ChangeLog @@ -19,6 +19,7 @@ CHANGES - Added OCaml binding - Added spin and energy in MOs - Added CSF group +- Added Jastrow group - Added Cholesky-decomposed two-electron integrals - Added Cholesky-decomposed RDMs for Gammcor - Added `trexio_flush` functionality diff --git a/python/install_pytrexio.sh b/python/install_pytrexio.sh index 0441ab7..8542874 100755 --- a/python/install_pytrexio.sh +++ b/python/install_pytrexio.sh @@ -73,7 +73,7 @@ python3 -m pip install dist/trexio-*.whl --force-reinstall #python3 -m twine upload dist/trexio-*.tar.gz # Cleaning -#rm -rf build dist trexio.egg-info +rm -rf build dist trexio.egg-info # Additional information related to the installation of the TREXIO Python API diff --git a/trex.org b/trex.org index fabce8a..8f8292c 100644 --- a/trex.org +++ b/trex.org @@ -39,7 +39,7 @@ the [[./examples.html][examples]]. The ~sparse~ data representation implies the [[https://en.wikipedia.org/wiki/Sparse_matrix#Coordinate_list_(COO)][coordinate list]] representation, namely the user has to write a list of indices and values. -For the Configuration Interfaction (CI) and Configuration State Function (CSF) +For the Configuration Interaction (CI) and Configuration State Function (CSF) groups, the ~buffered~ data type is introduced, which allows similar incremental I/O as for ~sparse~ data but without the need to write indices of the sparse values. @@ -69,10 +69,12 @@ This means that the source code is not produced by the generator, but hand-writt | ~description~ | ~str~ | | Text describing the content of file | | ~unsafe~ | ~int~ | | ~1~: true, ~0~: false | - **Note:** ~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~. + **Note:** ~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: @@ -90,290 +92,296 @@ This means that the source code is not produced by the generator, but hand-writt #+end_src :end: -* Electron (electron group) +* System +** Nucleus (nucleus group) - We consider wave functions expressed in the spin-free formalism, where - the number of \uparrow and \downarrow electrons is fixed. + The nuclei are considered as fixed point charges. Coordinates are + given in Cartesian $(x,y,z)$ format. - #+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: - -* 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: - -* 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 - \[ - V_A^{\text{ECP}} = - V_{A \ell_{\max}+1} + - \sum_{\ell=0}^{\ell_{\max}} - \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ - V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} | - \] - - The first term in the equation above is sometimes attributed to the local channel, - while the remaining terms correspond to the non-local channel projections. - - 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. -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 - -* 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 - - \[ - \chi_j(r) = \exp \left( -i \mathbf{k}_j \mathbf{r} \right) - \] - - 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. - -** 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") + #+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 - "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", [] ] + "nucleus": { + "num" : [ "dim" , [] ] + , "charge" : [ "float", [ "nucleus.num" ] ] + , "coord" : [ "float", [ "nucleus.num", "3" ] ] + , "label" : [ "str" , [ "nucleus.num" ] ] + , "point_group" : [ "str" , [] ] + , "repulsion" : [ "float", [] ] } , #+end_src :end: -** Example +** Cell (cell group) - For example, consider H_2 with the following basis set (in GAMESS - format), where both the AOs and primitives are considered normalized: + 3 Lattice vectors to define a box containing the system, for example + used in periodic calculations. - #+BEGIN_EXAMPLE + #+NAME: cell + | Variable | Type | Dimensions | Description | + |----------+---------+------------+-----------------------| + | ~a~ | ~float~ | ~(3)~ | First lattice vector | + | ~b~ | ~float~ | ~(3)~ | Second lattice vector | + | ~c~ | ~float~ | ~(3)~ | Third lattice vector | + + #+CALL: json(data=cell, title="cell") + + #+RESULTS: + :results: + #+begin_src python :tangle trex.json + "cell": { + "a" : [ "float", [ "3" ] ] + , "b" : [ "float", [ "3" ] ] + , "c" : [ "float", [ "3" ] ] + } , + #+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: + +** Numerical integration grid (grid group) + + The molecular integrals have to be computed numerically on a grid in many applications. + 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: + +** 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: + +* One-electron basis +** 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 + + \[ + \chi_j(r) = \exp \left( -i \mathbf{k}_j \mathbf{r} \right) + \] + + 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. + +*** 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 @@ -391,11 +399,11 @@ P 1 1 3.880000E-01 1.000000E+00 D 1 1 1.057000E+00 1.000000E+00 - #+END_EXAMPLE + #+END_EXAMPLE - In TREXIO representaion we have: + In TREXIO representaion we have: - #+BEGIN_EXAMPLE + #+BEGIN_EXAMPLE type = "Gaussian" prim_num = 20 shell_num = 12 @@ -437,266 +445,408 @@ prim_factor = 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 + #+END_EXAMPLE -* 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 - - \[ - \chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r}) - \] - - where $i$ is the atomic orbital index, - $P$ encodes for either the - polynomials or the spherical harmonics, $\theta(i)$ 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 - \[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)} - (\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne - \left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)} - (\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \] - - 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 factors | - - #+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)~ | $\langle p \vert q \rangle$ (real part, general case) | - | ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ (real part, general case) | - | ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ (real part, general case) | - | ~ecp~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle$ (real part, general case) | - | ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ (real part, general case) | - | ~overlap_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ (imaginary part) (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 - - #+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 | - - #+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" ] ] - } , - #+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 (real part, general case) | - | ~coefficient_im~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients (imaginary part, for periodic calculations) | - | ~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)~ | $\langle i \vert j \rangle$ (real part, general case) | - | ~kinetic~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{T}_e \vert j \rangle$ (real part, general case) | - | ~potential_n_e~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle$ (real part, general case) | - | ~ecp~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle$ (real part, general case) | - | ~core_hamiltonian~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{h} \vert j \rangle$ (real part, general case) | - | ~overlap_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert j \rangle$ (imaginary part) (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. - - The Cholesky decomposition of the integrals can also be stored: +** 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 \[ - \A_{ijkl} = \sum_{\alpha} G_{il\alpha} G_{jl\alpha} + V_A^{\text{ECP}} = + V_{A \ell_{\max}+1} + + \sum_{\ell=0}^{\ell_{\max}} + \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ + V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} | \] - #+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 | + 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. - #+CALL: json(data=mo_2e_int, title="mo_2e_int") + 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. + 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 + +** 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 + + \[ + \chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r}) + \] + + where $i$ is the atomic orbital index, + $P$ encodes for either the + polynomials or the spherical harmonics, $\theta(i)$ 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 + \[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)} + (\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne + \left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)} + (\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \] + + 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 factors | + + #+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)~ | $\langle p \vert q \rangle$ (real part, general case) | + | ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ (real part, general case) | + | ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ (real part, general case) | + | ~ecp~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle$ (real part, general case) | + | ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ (real part, general case) | + | ~overlap_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ (imaginary part) (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 + + #+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 | + + #+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" ] ] + } , + #+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 (real part, general case) | + | ~coefficient_im~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients (imaginary part, for periodic calculations) | + | ~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)~ | $\langle i \vert j \rangle$ (real part, general case) | + | ~kinetic~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{T}_e \vert j \rangle$ (real part, general case) | + | ~potential_n_e~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle$ (real part, general case) | + | ~ecp~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle$ (real part, general case) | + | ~core_hamiltonian~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{h} \vert j \rangle$ (real part, general case) | + | ~overlap_im~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert j \rangle$ (imaginary part) (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. + + The Cholesky decomposition of the integrals can also be stored: + + \[ + \A_{ijkl} = \sum_{\alpha} G_{il\alpha} G_{jl\alpha} + \] + + #+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: + :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" ] ] @@ -705,335 +855,308 @@ prim_factor = , "eri_lr_cholesky_num" : [ "dim" , [] ] , "eri_lr_cholesky" : [ "float sparse", [ "mo_2e_int.eri_lr_cholesky_num", "mo.num", "mo.num" ] ] } , - #+end_src - :end: + #+end_src + : + +* N-electron basis +** Slater determinants (determinant group) -* Slater determinants (determinant group) + The configuration interaction (CI) wave function $\Psi$ + can be expanded in the basis of Slater determinants $D_I$ as follows - The configuration interaction (CI) wave function $\Psi$ - can be expanded in the basis of Slater determinants $D_I$ as follows + \[ + \Psi = \sum_I C_I D_I + \] - \[ - \Psi = \sum_I C_I D_I - \] + 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. - 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. + \[ + D_I = \alpha_1 \alpha_2 \ldots \alpha_{n_\uparrow} \beta_1 \beta_2 \ldots \beta_{n_\downarrow} + \] - \[ - D_I = \alpha_1 \alpha_2 \ldots \alpha_{n_\uparrow} \beta_1 \beta_2 \ldots \beta_{n_\downarrow} - \] + 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. - 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. - 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]]. - 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 | - #+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: - -* 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: - -* 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 $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$ - 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 \\ - \Gamma_{ijkl}^{\downarrow \uparrow} &=& - \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 - \[ - \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + - \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow} + - \Gamma_{ijkl}^{\downarrow \uparrow} - \] - - The total energy can be computed as: - \[ - E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle + - \frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle - \] - - - 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_dnup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\uparrow 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) | - | ~2e_dnup_cholesky_num~ | ~dim~ | | Number of Cholesky vectors | - | ~2e_dnup_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_dnup_cholesky_num)~ | Cholesky decomposition of the Two-body RDM (\downarrow\uparrow) | - - #+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_dnup" : [ "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" ] ] - , "2e_dnup_cholesky_num" : [ "dim" , [] ] - , "2e_dnup_cholesky" : [ "float sparse", [ "rdm.2e_dnup_cholesky_num", "mo.num", "mo.num" ] ] - } , - #+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 lattice vector | - | ~b~ | ~float~ | ~(3)~ | Second lattice vector | - | ~c~ | ~float~ | ~(3)~ | Third lattice vector | - - #+CALL: json(data=cell, title="cell") - - #+RESULTS: - :results: - #+begin_src python :tangle trex.json - "cell": { - "a" : [ "float", [ "3" ] ] - , "b" : [ "float", [ "3" ] ] - , "c" : [ "float", [ "3" ] ] - } , - #+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: - -* Numerical integration grid (grid group) - - The molecular integrals have to be computed numerically on a grid in many applications. - 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") + #+CALL: json(data=determinant, title="determinant") #+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" ] ] + "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: + +** 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 $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$ + 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 \\ + \Gamma_{ijkl}^{\downarrow \uparrow} &=& + \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 + \[ + \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + + \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow} + + \Gamma_{ijkl}^{\downarrow \uparrow} + \] + + The total energy can be computed as: + \[ + E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle + + \frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle + \] + + + 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_dnup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\uparrow 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) | + | ~2e_dnup_cholesky_num~ | ~dim~ | | Number of Cholesky vectors | + | ~2e_dnup_cholesky~ | ~float sparse~ | ~(mo.num, mo.num, rdm.2e_dnup_cholesky_num)~ | Cholesky decomposition of the Two-body RDM (\downarrow\uparrow) | + + #+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_dnup" : [ "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" ] ] + , "2e_dnup_cholesky_num" : [ "dim" , [] ] + , "2e_dnup_cholesky" : [ "float sparse", [ "rdm.2e_dnup_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: $\Psi = \Phi \times \exp(J)$, + where + + \[ + J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R}) + \] + + 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. + + $J_{\text{eN}}$ 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 + + The "mu" Jastrow factor has only a single parameter $\mu$ for the + [[https://doi.org/10.1063/5.0044683][electron-electron term]]: + + \[ + J_{\text{ee}}(\mathbf{r}) = + \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} r_{ij} + \left( 1 - \text{erf}(\mu\, r_{ij})\right) - \frac{1}{\mu\sqrt{\pi}} + e^{-(\mu\,r_{ij})^2} + \] + +# It was then updated for frozen-core calculations by introducing a +# set of electron-electron-nucleus terms with one parameter per nucleus: + +# \[ +# J_{\text{eeN}}(\mathbf{r}) = +# \] + + +*** 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~ | ~(nucleus.num)~ | Number of Electron-nucleus parameters, per nucleus | + | ~een_num~ | ~dim~ | ~(nucleus.num)~ | Number of Electron-electron-nucleus parameters, per nucleus | + | ~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 | + * Quantum Monte Carlo data (qmc group) In quantum Monte Carlo calculations, the wave function is evaluated