trexio/trex.org

#+TITLE: TREX Configuration file
#+STARTUP: latexpreview

All the quantities are saved 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 produces JSON configuration file as the library is
written in C.

  #+begin_src python :tangle trex.json 
{
  #+end_src

* Metadata
  
  As we expect our files to be archived in open-data repositories, we
  need to 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
  | ~code_num~    | ~int~ |                         | Number of codes used to produce the file |
  | ~code~        | ~str~ | ~(metadata.code_num)~   | Names of the codes used                  |
  | ~author_num~  | ~int~ |                         | Number of authors of the file            |
  | ~author~      | ~str~ | ~(metadata.author_num)~ | Names of the authors of the file         |
  | ~description~ | ~str~ |                         | Text describing the content of file      |

  #+CALL: json(data=metadata, title="metadata")
  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "metadata": {
             "code_num" : [ "int", []                        ]
        ,        "code" : [ "str", [ "metadata.code_num" ]   ]
        ,  "author_num" : [ "int", []                        ]
        ,      "author" : [ "str", [ "metadata.author_num" ] ]
        , "description" : [ "str", []                        ]
      } ,
  #+end_src
  :end:

* Electron

   We consider wave functions expressed in the spin-free formalism, where
   the number of \uparrow and \downarrow electrons is fixed.

  #+NAME:electron
  | ~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": {
          "up_num" : [ "int", []  ]
        , "dn_num" : [ "int", []  ]
      } ,
  #+end_src
  :end:

* Nucleus

  The nuclei are considered as fixed point charges. Coordinates are
  given in Cartesian $(x,y,z)$ format.

  #+NAME: nucleus
  | ~num~         | ~int~   |                   | 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     |

  #+CALL: json(data=nucleus, title="nucleus")
  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "nucleus": {
                  "num" : [ "int"  , []                     ]
        ,      "charge" : [ "float", [ "nucleus.num" ]      ]
        ,       "coord" : [ "float", [ "nucleus.num", "3" ] ]
        ,       "label" : [ "str"  , [ "nucleus.num" ]      ]
        , "point_group" : [ "str"  , []                     ]
      } ,
  #+end_src
  :end:

* TODO Effective core potentials

  An effective core potential (ECP) $V_A^{\text{pp}}$ replacing the
  core electrons of atom $A$ is the sum of a local component
  $V_A^{\text{l}}$ and a non-local component $V_A^{\text{nl}}$.
  The local component is given by

  \[
  \hat{V}_A^{\text{l}}(r) = -\frac{Z_A^{\text{eff}}}{r} +
  \frac{Z_A^{\text{eff}}}{r}\, \exp\left( -\alpha_A\, r^2\right) +
  Z_{\text{eff}}\, \alpha_A\, r\, \exp\left( -\beta_A\, r^2\right) +
  \gamma_A \exp\left( -\delta_A\, r^2\right),
  \]

  and the component obtained after localizing the non-local operator is

  \[
  \hat{V}_A^{\text{nl}}(r) =
  \zeta_A\, \exp\left( -\eta_A\, r^2\right) |0\rangle \langle 0| +
  \mu_A \,  \exp\left( -\nu_A \, r^2\right) |1\rangle \langle 1|
  \]

  where $r=|\mathbf{r-R}_A|$ is the distance to the nucleus on which the
  potential is centered, $Z_A^{\text{eff}}$ is the effective charge
  due to the removed electrons, $|0\rangle \langle 0|$
  and $|1\rangle \langle 1|$ are projections over zero and one principal angular
  momenta, respectively (generalization to higher angular momenta is
  straightforward),  and all the parameters labeled by Greek
  letters are parameters.

  - $\hat{V}_\text{ecp,l} = \sum_A \hat{V}_A^{\text{l}}$ : local component
  - $\hat{V}_\text{ecp,nl} = \sum_A \hat{V}_A^{\text{nl}}$ : non-local component

  #+NAME: ecp
  | ~lmax_plus_1~         | ~int~   | ~(nucleus.num)~                          | $l_{\max} + 1$             |
  | ~z_core~              | ~float~ | ~(nucleus.num)~                          | Charges to remove          |
  | ~local_n~             | ~int~   | ~(nucleus.num)~                          | Number of local function   |
  | ~local_num_n_max~     | ~int~   |                                          | Maximum value of ~local_n~ |
  | ~local_exponent~      | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~     |                            |
  | ~local_coef~          | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~     |                            |
  | ~local_power~         | ~int~   | ~(ecp.local_num_n_max, nucleus.num)~     |                            |
  | ~non_local_n~         | ~int~   | ~(nucleus.num)~                          |                            |
  | ~non_local_num_n_max~ | ~int~   |                                          |                            |
  | ~non_local_exponent~  | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ |                            |
  | ~non_local_coef~      | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ |                            |
  | ~non_local_power~     | ~int~   | ~(ecp.non_local_num_n_max, nucleus.num)~ |                            |

  #+CALL: json(data=ecp, title="ecp")

  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "ecp": {
                  "lmax_plus_1" : [ "int"  , [ "nucleus.num" ]                            ]
        ,              "z_core" : [ "float", [ "nucleus.num" ]                            ]
        ,             "local_n" : [ "int"  , [ "nucleus.num" ]                            ]
        ,     "local_num_n_max" : [ "int"  , []                                           ]
        ,      "local_exponent" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ]     ]
        ,          "local_coef" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ]     ]
        ,         "local_power" : [ "int"  , [ "nucleus.num", "ecp.local_num_n_max" ]     ]
        ,         "non_local_n" : [ "int"  , [ "nucleus.num" ]                            ]
        , "non_local_num_n_max" : [ "int"  , []                                           ]
        ,  "non_local_exponent" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
        ,      "non_local_coef" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
        ,     "non_local_power" : [ "int"  , [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
      } ,
  #+end_src
  :end:

* Basis set

  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:

  #+NAME: basis
  | ~type~              | ~str~   |                    | Type of basis set: "Gaussian" or "Slater"                |
  | ~num~               | ~int~   |                    | Total Number of shells                                   |
  | ~prim_num~          | ~int~   |                    | Total number of primitives                               |
  | ~nucleus_index~     | ~int~   | ~(nucleus.num)~    | Index of the first shell of each nucleus ($A$)           |
  | ~nucleus_shell_num~ | ~int~   | ~(nucleus.num)~    | Number of shells for each nucleus                        |
  | ~shell_ang_mom~     | ~int~   | ~(basis.num)~      | Angular momentum ~0:S, 1:P, 2:D, ...~                    |
  | ~shell_prim_num~    | ~int~   | ~(basis.num)~      | Number of primitives in the shell ($N_{\text{prim}}$)    |
  | ~shell_factor~      | ~float~ | ~(basis.num)~      | Normalization factor of the shell ($\mathcal{N}_s$)      |
  | ~shell_prim_index~  | ~int~   | ~(basis.num)~      | Index of the first primitive in the complete list        |
  | ~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}$) |
    
  #+CALL: json(data=basis, title="basis")

  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "basis": {
                       "type" : [ "str"  , []                   ]
        ,               "num" : [ "int"  , []                   ]
        ,          "prim_num" : [ "int"  , []                   ]
        ,     "nucleus_index" : [ "int"  , [ "nucleus.num" ]    ]
        , "nucleus_shell_num" : [ "int"  , [ "nucleus.num" ]    ]
        ,     "shell_ang_mom" : [ "int"  , [ "basis.num" ]      ]
        ,    "shell_prim_num" : [ "int"  , [ "basis.num" ]      ]
        ,      "shell_factor" : [ "float", [ "basis.num" ]      ]
        ,  "shell_prim_index" : [ "int"  , [ "basis.num" ]      ]
        ,          "exponent" : [ "float", [ "basis.prim_num" ] ]
        ,       "coefficient" : [ "float", [ "basis.prim_num" ] ]
        ,       "prim_factor" : [ "float", [ "basis.prim_num" ] ]
      } ,
  #+end_src
  :end:

  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           1.0000000
   #+END_EXAMPLE

   we have:

   #+BEGIN_EXAMPLE
type     = "Gaussian"
num      = 12
prim_num = 20

nucleus_index    = [0 , 6]
shell_ang_mom    = [0 , 0 , 0 , 1 , 1 , 2 , 0 , 0 , 0 , 1 , 1 , 2 ]
shell_prim_num   = [5 , 1 , 1 , 1 , 1 , 1 , 5 , 1 , 1 , 1 , 1 , 1 ]
shell_prim_index = [0 , 5 , 6 , 7 , 8 , 9 , 10, 15, 16, 17, 18, 19]
shell_factor     = [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]

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

* Atomic orbitals

  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 spherical harmonics, the AOs are ordered in
  increasing magnetic quantum number ($-l \le m \le l$), and in the case
  of polynomials we impose the canonical ordering of the
  Libint2 library, 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}

  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 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
  | ~cartesian~     | ~int~   |            | ~1~: true, ~0~: false           |
  | ~num~           | ~int~   |            | Total number of atomic orbitals |
  | ~shell~         | ~int~   | ~(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" : [ "int"  , []           ]
        ,         "shell" : [ "int"  , [ "ao.num" ] ]
        , "normalization" : [ "float", [ "ao.num" ] ]
      } ,
  #+end_src
  :end:
    
** One-electron integrals
   :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
   - $\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} +
     \hat{V}_\text{ecp,l} + \hat{V}_\text{ecp,nl}$ : 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
   | ~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_local~        | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,l} \vert q \rangle$  |
   | ~ecp_non_local~    | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,nl} \vert q \rangle$ |
   | ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$                |

   #+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_local" : [ "float", [ "ao.num", "ao.num" ] ]
         ,    "ecp_non_local" : [ "float", [ "ao.num", "ao.num" ] ]
         , "core_hamiltonian" : [ "float", [ "ao.num", "ao.num" ] ]
       } ,
   #+end_src
   :end:

** Two-electron integrals
   :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
   | ~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

  #+NAME: mo
  | ~type~        | ~str~   |                    | String identify the set of MOs           |
  | ~num~         | ~int~   |                    | Number of MOs                            |
  | ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients                          |
  | ~class~       | ~str~   | ~(mo.num)~         | Core, Inactive, Active, Virtual, Deleted |
  | ~symmetry~    | ~str~   | ~(mo.num)~         | Symmetry in the point group              |
  | ~occupation~  | ~float~ | ~(mo.num)~         | Occupation number                        |

  #+CALL: json(data=mo, title="mo")

  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "mo": {
                 "type" : [ "str"  , []                     ]
        ,         "num" : [ "int"  , []                     ]
        , "coefficient" : [ "float", [ "mo.num", "ao.num" ] ]
        ,       "class" : [ "str"  , [ "mo.num" ]           ]
        ,    "symmetry" : [ "str"  , [ "mo.num" ]           ]
        ,  "occupation" : [ "float", [ "mo.num" ]           ]
      } ,
  #+end_src
  :end:

** One-electron integrals

   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
   | ~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_local~        | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ecp,l} \vert j \rangle$  |
   | ~ecp_non_local~    | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{V}_{\text{ecp,nl} \vert j \rangle$ |
   | ~core_hamiltonian~ | ~float~ | ~(mo.num, mo.num)~ | $\langle i \vert \hat{h} \vert j \rangle$                |

   #+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_local" : [ "float", [ "mo.num", "mo.num" ] ]
         ,    "ecp_non_local" : [ "float", [ "mo.num", "mo.num" ] ]
         , "core_hamiltonian" : [ "float", [ "mo.num", "mo.num" ] ]
       } ,
   #+end_src
   :end:

** Two-electron integrals

   The operators as 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
   | ~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 |

   #+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" ] ]
       } ,
   #+end_src
   :end:

* TODO Slater determinants
* TODO Reduced density matrices
  
  #+NAME: rdm
  | ~one_e~    | ~float~        | ~(mo.num, mo.num)~                |
  | ~one_e_up~ | ~float~        | ~(mo.num, mo.num)~                |
  | ~one_e_dn~ | ~float~        | ~(mo.num, mo.num)~                |
  | ~two_e~    | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ |

  #+CALL: json(data=rdm, title="rdm", last=1)

  #+RESULTS:
  :results:
  #+begin_src python :tangle trex.json
      "rdm": {
             "one_e" : [ "float"       , [ "mo.num", "mo.num" ]                     ]
        , "one_e_up" : [ "float"       , [ "mo.num", "mo.num" ]                     ]
        , "one_e_dn" : [ "float"       , [ "mo.num", "mo.num" ]                     ]
        ,    "two_e" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.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
}
  #+end_src