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# Data stored in TREXIO

## Table of Contents

- 1. Metadata (metadata group)
- 2. System
  - 2.1. Nucleus (nucleus group)
  - 2.2. Cell (cell group)
  - 2.3. Periodic boundary calculations (pbc group)
  - 2.4. Electron (electron group)
  - 2.5. Ground or excited states (state group)
- 3. Basis functions
  - 3.1. Basis set (basis group)
    - 3.1.1. Gaussian and Slater-type orbitals
    - 3.1.2. Plane waves
    - 3.1.3. Data definitions
    - 3.1.4. Example
  - 3.2. Effective core potentials (ecp group)
    - 3.2.1. Example
  - 3.3. Numerical integration grid (grid group)
- 4. Orbitals
  - 4.1. Atomic orbitals (ao group)
    - 4.1.1. One-electron integrals (`ao_1e_int` group)
    - 4.1.2. Two-electron integrals (`ao_2e_int` group)
  - 4.2. Molecular orbitals (mo group)
    - 4.2.1. One-electron integrals (`mo_1e_int` group)
    - 4.2.2. Two-electron integrals (`mo_2e_int` group)
- 5. Multi-determinant information
  - 5.1. Slater determinants (determinant group)
  - 5.2. Configuration state functions (csf group)
  - 5.3. Amplitudes (amplitude group)
  - 5.4. Reduced density matrices (rdm group)
- 6. Correlation factors
  - 6.1. Jastrow factor (jastrow group)
    - 6.1.1. CHAMP
    - 6.1.2. Mu
    - 6.1.3. Table of values
- 7. Quantum Monte Carlo data (qmc group)

For simplicity, the singular form is always used for the names of groups and attributes, and all data are stored in atomic units. The dimensions of the arrays in the tables below are given in column-major order (as in Fortran), and the ordering of the dimensions is reversed in the produced `trex.json` configuration file as the library is written in C.

## 1 Metadata (metadata group)

As we expect TREXIO files to be archived in open-data repositories, we give the possibility to the users to store some metadata inside the files. We propose to store the list of names of the codes which have participated to the creation of the file, a list of authors of the file, and a textual description.

Variable | Type | Dimensions (for arrays) | Description
--- | --- | --- | ---
`code_num` | `dim` |  | Number of codes used to produce the file
`code` | `str` | `(metadata.code_num)` | Names of the codes used
`author_num` | `dim` |  | Number of authors of the file
`author` | `str` | `(metadata.author_num)` | Names of the authors of the file
`package_version` | `str` |  | TREXIO version used to produce the file
`description` | `str` |  | Text describing the content of file
`unsafe` | `int` |  | `1`: true, `0`: false

Note: The `unsafe` attribute of the `metadata` group indicates whether the file has been previously opened with `'u'` mode. It is automatically written in the file upon the first unsafe opening. If the user has checked that the TREXIO file is valid (e.g. using `trexio-tools`) after unsafe operations, then the `unsafe` attribute value can be manually overwritten (in unsafe mode) from `1` to `0`.

## 2 System

### 2.1 Nucleus (nucleus group)

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

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

### 2.2 Cell (cell group)

3 Lattice vectors to define a box containing the system, for example used in periodic calculations.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`a` | `float` | `(3)` | First real space lattice vector
`b` | `float` | `(3)` | Second real space lattice vector
`c` | `float` | `(3)` | Third real space lattice vector
`G_a` | `float` | `(3)` | First reciprocal space lattice vector
`G_b` | `float` | `(3)` | Second reciprocal space lattice vector
`G_c` | `float` | `(3)` | Third reciprocal space lattice vector
`two_pi` | `int` |  | `0` or `1`. If `two_pi=1`, \(2\pi\) is included in the reciprocal vectors.

### 2.3 Periodic boundary calculations (pbc group)

A single $k$-point per TREXIO file can be stored. The $k$-point is defined in this group.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`periodic` | `int` |  | `1`: true or `0`: false
`k_point` | `float` | `(3)` | $k$-point sampling

### 2.4 Electron (electron group)

The chemical system consists of nuclei and electrons, where the nuclei are considered as fixed point charges with Cartesian coordinates. The wave function is stored in the spin-free formalism, and therefore, it is necessary for the user to explicitly store the number of electrons with spin up (\(N_\uparrow\)) and spin down (\(N_\downarrow\)). These numbers correspond to the normalization of the spin-up and spin-down single-particle reduced density matrices.

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

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`num` | `dim` |  | Number of electrons
`up_num` | `int` |  | Number of ↑-spin electrons
`dn_num` | `int` |  | Number of ↓-spin electrons

### 2.5 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.

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)

## 3 Basis functions

### 3.1 Basis set (basis group)

#### 3.1.1 Gaussian and Slater-type orbitals

We consider here basis functions centered on nuclei. Hence, it is possibile to define dummy atoms to place basis functions in arbitrary 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.

#### 3.1.2 Plane waves

A plane wave is defined as

\[ \chi_j(\mathbf{r}) = \exp \left( -i \mathbf{G}_j \cdot \mathbf{r} \right) \]

The basis set is defined as the array of $k$-points in the reciprocal space \(\mathbf{G}_j\), defined in the `pbc` group. The kinetic energy cutoff `e_cut` is the only input data relevant to plane waves.

#### 3.1.3 Data definitions

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

#### 3.1.4 Example

For example, consider H2 with the following basis set (in GAMESS format), where both the AOs and primitives are considered normalized:

    HYDROGEN
    S   5
    1         3.387000E+01           6.068000E-03
    2         5.095000E+00           4.530800E-02
    3         1.159000E+00           2.028220E-01
    4         3.258000E-01           5.039030E-01
    5         1.027000E-01           3.834210E-01
    S   1
    1         3.258000E-01           1.000000E+00
    S   1
    1         1.027000E-01           1.000000E+00
    P   1
    1         1.407000E+00           1.000000E+00
    P   1
    1         3.880000E-01           1.000000E+00
    D   1
    1         1.057000E+00           1.000000E+00

In TREXIO representaion we have:

    type  = "Gaussian"
    prim_num   = 20
    shell_num   = 12

    # 6 shells per H atom
    nucleus_index =
    [ 0, 0, 0, 0, 0, 0,
      1, 1, 1, 1, 1, 1 ]

    # 3 shells in S (l=0), 2 in P (l=1), 1 in D (l=2)
    shell_ang_mom =
    [ 0, 0, 0, 1, 1, 2,
      0, 0, 0, 1, 1, 2 ]

    # no need to renormalize shells
    shell_factor =
    [ 1., 1., 1., 1., 1., 1.,
      1., 1., 1., 1., 1., 1. ]

    # 5 primitives for the first S shell and then 1 primitive per remaining shells in each H atom
    shell_index =
    [ 0, 0, 0, 0, 0, 1, 2, 3, 4, 5,
      6, 6, 6, 6, 6, 7, 8, 9, 10, 11 ]

    # parameters of the primitives (10 per H atom)
    exponent =
    [ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057,
      33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057 ]

    coefficient =
    [ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0,
      0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0 ]

    prim_factor =
    [ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
      3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
      1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01,
      1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00,
      7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01,
      3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00,
      4.3649547399719840e-01, 1.8135965626177861e+00 ]

### 3.2 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 \[ V_A^{\text{ECP}} = V_{A \ell_{\max}+1} + \sum_{\ell=0}^{\ell_{\max}} \delta V_{A \ell}\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} | \]

The first term in this equation is attributed to the local channel, while the remaining terms correspond to non-local channel projections. \(\ell_{\max}\) refers to the maximum angular momentum in the non-local component of the ECP. The functions \(\delta V_{A \ell}\) and \(V_{A \ell_{\max}+1}\) are parameterized as:

\begin{eqnarray} \delta 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 } \nonumber\\ V_{A \ell_{\max}+1}(\mathbf{r}) &=& -\frac{Z_\text{eff}}{|\mathbf{r}-\mathbf{R}_{A}|}+\delta V_{A \ell_{\max}+1}(\mathbf{r}) \end{eqnarray}

where \(Z_\text{eff}\) is the effective nuclear charge of the center.

See http://dx.doi.org/10.1063/1.4984046 or https://doi.org/10.1063/1.5121006 for more info.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`max_ang_mom_plus_1` | `int` | `(nucleus.num)` | \(\ell_{\max}+1\), one higher than the max angular momentum in the removed core orbitals
`z_core` | `int` | `(nucleus.num)` | Number of core electrons to remove per atom
`num` | `dim` |  | Total number of ECP functions for all atoms and all values of \(\ell\)
`ang_mom` | `int` | `(ecp.num)` | One-to-one correspondence between ECP items and the angular momentum \(\ell\)
`nucleus_index` | `index` | `(ecp.num)` | One-to-one correspondence between ECP items and the atom index
`exponent` | `float` | `(ecp.num)` | \(\alpha_{A q \ell}\) all ECP exponents
`coefficient` | `float` | `(ecp.num)` | \(\beta_{A q \ell}\) all ECP coefficients
`power` | `int` | `(ecp.num)` | \(n_{A q \ell}\) all ECP powers

There might be some confusion in the meaning of the \(\ell_{\max}\). It can be attributed to the maximum angular momentum occupied in the core orbitals, which are removed by the ECP. On the other hand, it can be attributed to the maximum angular momentum of the ECP that replaces the core electrons. Note, that the latter \(\ell_{\max}\) is always higher by 1 than the former.

Note for developers: avoid having variables with similar prefix in their name. The HDF5 back end might cause issues due to the way `find_dataset` function works. For example, in the ECP group we use `max_ang_mom` and not `ang_mom_max`. The latter causes issues when written before the `ang_mom` array in the TREXIO file. Update: in fact, the aforementioned issue has only been observed when using HDF5 version 1.10.4 installed via `apt-get`. Installing the same version from the `conda-forge` channel and running it in an isolated `conda` environment works just fine. Thus, it seems to be a bug in the `apt`-provided package. If you encounter the aforementioned issue, please report it to our issue tracker on GitHub.

#### 3.2.1 Example

For example, consider H2 molecule with the following effective core potential (in GAMESS input format for the H atom):

    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

In TREXIO representation this would be:

    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
      ]

### 3.3 Numerical integration grid (grid group)

In some applications, such as DFT calculations, integrals have to be computed numerically on a grid. A common choice for the angular grid is the one proposed by Lebedev and Laikov [Russian Academy of Sciences Doklady Mathematics, Volume 59, Number 3, 1999, pages 477-481]. For the radial grids, many approaches have been developed over the years.

The structure of this group is adapted for the numgrid library. Feel free to submit a PR if you find missing options/functionalities.

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)

## 4 Orbitals

### 4.1 Atomic orbitals (ao group)

AOs are defined as

\[ \chi_i (\mathbf{r}) = \mathcal{N}_i'\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r}) \]

where \(i\) is the atomic orbital index, \(P\) refers to either polynomials or spherical harmonics, and \(s(i)\) specifies the shell on which the AO is expanded.

\(\eta(i)\) denotes the chosen angular function. The AOs can be expressed using real spherical harmonics or 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 Wikipedia). In the case of polynomials, the canonical (or alphabetical) ordering is used,

\(p\) | \(p_x, p_y, p_z\)
\(d\) | \(d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz}\)
\(f\) | \(f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}\),
| \(f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, f_{zzz}\)
\(\vdots\) |

Note that for \(p\) orbitals in spherical coordinates, the ordering is \(0,+1,-1\) which corresponds to \(p_z, p_x, p_y\).

\(\mathcal{N}_i'\) is a normalization factor that allows for different normalization coefficients within a single shell, as in the GAMESS convention where each individual function is unit-normalized. Using GAMESS convention, the normalization factor of the shell \(\mathcal{N}_d\) in the `basis` group is appropriate for instance for the \(d_z^2\) function (i.e. \(\mathcal{N}_{d}\equiv\mathcal{N}_{z^2}\)) but not for the \(d_{xy}\) AO, so the correction factor \(\mathcal{N}_i'\) for \(d_{xy}\) in the `ao` groups is the ratio \(\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}\).

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`cartesian` | `int` |  | `1`: true, `0`: false
`num` | `dim` |  | Total number of atomic orbitals
`shell` | `index` | `(ao.num)` | Basis set shell for each AO
`normalization` | `float` | `(ao.num)` | Normalization factor \(\mathcal{N}_i\)

#### 4.1.1 One-electron integrals (`ao_1e_int` group)

- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}} \sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A - \mathbf{r}_i \vert} \] : 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.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`overlap` | `float` | `(ao.num, ao.num)` | \(\langle p \vert q \rangle\)
`kinetic` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{T}_e \vert q \rangle\)
`potential_n_e` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle\)
`ecp` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle\)
`core_hamiltonian` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{h} \vert q \rangle\)
`overlap_im` | `float` | `(ao.num, ao.num)` | \(\langle p \vert q \rangle\) (imaginary part)
`kinetic_im` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{T}_e \vert q \rangle\) (imaginary part)
`potential_n_e_im` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle\) (imaginary part)
`ecp_im` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{V}_{\text{ECP}} \vert q \rangle\) (imaginary part)
`core_hamiltonian_im` | `float` | `(ao.num, ao.num)` | \(\langle p \vert \hat{h} \vert q \rangle\) (imaginary part)

#### 4.1.2 Two-electron integrals (`ao_2e_int` group)

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, where \(p,q,r,s\) are indices over atomic orbitals.

- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{\text{erf}(\mu\, \vert \mathbf{r}_i - \mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential

The Cholesky decomposition of the integrals can also be stored:

\[ \langle ij | kl \rangle = \sum_{\alpha} G_{ik\alpha} G_{jl\alpha} \]

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`eri` | `float sparse` | `(ao.num, ao.num, ao.num, ao.num)` | Electron repulsion integrals
`eri_lr` | `float sparse` | `(ao.num, ao.num, ao.num, ao.num)` | Long-range electron repulsion integrals
`eri_cholesky_num` | `dim` |  | Number of Cholesky vectors for ERI
`eri_cholesky` | `float sparse` | `(ao.num, ao.num, ao_2e_int.eri_cholesky_num)` | Cholesky decomposition of the ERI
`eri_lr_cholesky_num` | `dim` |  | Number of Cholesky vectors for long range ERI
`eri_lr_cholesky` | `float sparse` | `(ao.num, ao.num, ao_2e_int.eri_lr_cholesky_num)` | Cholesky decomposition of the long range ERI

### 4.2 Molecular orbitals (mo group)

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`type` | `str` |  | Free text to identify the set of MOs (HF, Natural, Local, CASSCF, etc)
`num` | `dim` |  | Number of MOs
`coefficient` | `float` | `(ao.num, mo.num)` | MO coefficients
`coefficient_im` | `float` | `(ao.num, mo.num)` | MO coefficients (imaginary part)
`class` | `str` | `(mo.num)` | Choose among: Core, Inactive, Active, Virtual, Deleted
`symmetry` | `str` | `(mo.num)` | Symmetry in the point group
`occupation` | `float` | `(mo.num)` | Occupation number
`energy` | `float` | `(mo.num)` | For canonical MOs, corresponding eigenvalue
`spin` | `int` | `(mo.num)` | For UHF wave functions, 0 is \(\alpha\) and 1 is \(\beta\)

#### 4.2.1 One-electron integrals (`mo_1e_int` group)

The operators as the same as those defined in the AO one-electron integrals section. Here, the integrals are given in the basis of molecular orbitals.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`overlap` | `float` | `(mo.num, mo.num)` | \(\langle i \vert j \rangle\)
`kinetic` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{T}_e \vert j \rangle\)
`potential_n_e` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle\)
`ecp` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle\)
`core_hamiltonian` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{h} \vert j \rangle\)
`overlap_im` | `float` | `(mo.num, mo.num)` | \(\langle i \vert j \rangle\) (imaginary part)
`kinetic_im` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{T}_e \vert j \rangle\) (imaginary part)
`potential_n_e_im` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle\) (imaginary part)
`ecp_im` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{V}_{\text{ECP}} \vert j \rangle\) (imaginary part)
`core_hamiltonian_im` | `float` | `(mo.num, mo.num)` | \(\langle i \vert \hat{h} \vert j \rangle\) (imaginary part)

#### 4.2.2 Two-electron integrals (`mo_2e_int` group)

The operators are the same as those defined in the AO two-electron integrals section. Here, the integrals are given in the basis of molecular orbitals.

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

## 5 Multi-determinant information

### 5.1 Slater determinants (determinant group)

The configuration interaction (CI) wave function \(\Psi\) can be expanded in the basis of Slater determinants \(D_I\) as follows

\[ \Psi = \sum_I C_I D_I \]

For relatively small expansions, a given determinant can be represented as a list of occupied orbitals. However, this becomes unfeasible for larger expansions and requires more advanced data structures. The bit field representation is used here, namely a given determinant is represented as \(N_{\text{int}}\) 64-bit integers where j-th bit is set to 1 if there is an electron in the j-th orbital and 0 otherwise. This gives access to larger determinant expansions by optimising the storage of the determinant lists in the memory.

\[ D_I = \alpha_1 \alpha_2 \ldots \alpha_{n_\uparrow} \beta_1 \beta_2 \ldots \beta_{n_\downarrow} \]

where \(\alpha\) and \(\beta\) denote ↑-spin and ↓-spin electrons, respectively, \(n_\uparrow\) and \(n_\downarrow\) correspond to `electron.up_num` and `electron.dn_num`, respectively.

Note: the `special` attribute is present in the types, meaning that the source node is not produced by the code generator.

An illustration on how to read determinants is presented in the examples.

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

### 5.2 Configuration state functions (csf group)

The configuration interaction (CI) wave function \(\Psi\) can be expanded in the basis of 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.

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

### 5.3 Amplitudes (amplitude group)

The wave function may be expressed in terms of action of the cluster operator \(\hat{T}\):

\[ \hat{T} = \hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \dots \]

on a reference wave function \(\Psi\), where \(\hat{T}_1\) is the single excitation operator,

\[ \hat{T}_1 = \sum_{ia} t_{i}^{a}\, \hat{a}^\dagger_a \hat{a}_i, \]

\(\hat{T}_2\) is the double excitation operator,

\[ \hat{T}_2 = \frac{1}{4} \sum_{ijab} t_{ij}^{ab}\, \hat{a}^\dagger_a \hat{a}^\dagger_b \hat{a}_j \hat{a}_i, \]

etc. Indices \(i\), \(j\), \(a\) and \(b\) denote molecular orbital indices.

Wave functions obtained with perturbation theory or configuration interaction are of the form

\[ |\Phi\rangle = \hat{T}|\Psi\rangle \]

and coupled-cluster wave functions are of the form

\[ |\Phi\rangle = e^{\hat{T}}| \Psi \rangle \]

The reference wave function is stored using the `determinant` and/or `csf` groups, and the amplitudes are stored using the current group. The attributes with the `exp` suffix correspond to exponentialized operators.

The order of the indices is chosen such that

- `t(i,a)` = \(t_{i}^{a}\).
- `t(i,j,a,b)` = \(t_{ij}^{ab}\),
- `t(i,j,k,a,b,c)` = \(t_{ijk}^{abc}\),
- `t(i,j,k,l,a,b,c,d)` = \(t_{ijkl}^{abcd}\),
- \(\dots\)

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`single` | `float sparse` | `(mo.num,mo.num)` | Single excitation amplitudes
`single_exp` | `float sparse` | `(mo.num,mo.num)` | Exponentialized single excitation amplitudes
`double` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num)` | Double excitation amplitudes
`double_exp` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num)` | Exponentialized double excitation amplitudes
`triple` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)` | Triple excitation amplitudes
`triple_exp` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)` | Exponentialized triple excitation amplitudes
`quadruple` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)` | Quadruple excitation amplitudes
`quadruple_exp` | `float sparse` | `(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)` | Exponentialized quadruple excitation amplitudes

### 5.4 Reduced density matrices (rdm group)

The reduced density matrices are defined in the basis of molecular orbitals.

The ↑-spin and ↓-spin components of the one-body density matrix are given by

\begin{eqnarray*} \gamma_{ij}^{\uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\ \gamma_{ij}^{\downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{j\beta} \, \hat{a}_{i\beta} | \Psi \rangle \end{eqnarray*}

and the spin-summed one-body density matrix is \[ \gamma_{ij} = \gamma^{\uparrow}_{ij} + \gamma^{\downarrow}_{ij} \]

The \(\uparrow \uparrow\), \(\downarrow \downarrow\), \(\uparrow \downarrow\) components of the two-body density matrix are given by

\begin{eqnarray*} \Gamma_{ijkl}^{\uparrow \uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\alpha} \hat{a}_{j\alpha}\, \hat{a}_{i\alpha} | \Psi \rangle \\ \Gamma_{ijkl}^{\downarrow \downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{k\beta}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\beta} | \Psi \rangle \\ \Gamma_{ijkl}^{\uparrow \downarrow} &=& \langle \Psi | \hat{a}^{\dagger}_{k\alpha}\, \hat{a}^{\dagger}_{l\beta} \hat{a}_{j\beta}\, \hat{a}_{i\alpha} | \Psi \rangle + \langle \Psi | \hat{a}^{\dagger}_{l\alpha}\, \hat{a}^{\dagger}_{k\beta} \hat{a}_{i\beta}\, \hat{a}_{j\alpha} | \Psi \rangle \\ \end{eqnarray*}

and the spin-summed one-body density matrix is \[ \Gamma_{ijkl} = \Gamma_{ijkl}^{\uparrow \uparrow} + \Gamma_{ijkl}^{\downarrow \downarrow} + \Gamma_{ijkl}^{\uparrow \downarrow}. \]

The 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 \(g_{ik}(\mathbf{r}_1) = \phi_i(\mathbf{r}_1) \phi_k(\mathbf{r}_1)\).

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`1e` | `float` | `(mo.num, mo.num)` | One body density matrix
`1e_up` | `float` | `(mo.num, mo.num)` | ↑-spin component of the one body density matrix
`1e_dn` | `float` | `(mo.num, mo.num)` | ↓-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)` | ↑↑ component of the two-body reduced density matrix
`2e_dndn` | `float sparse` | `(mo.num, mo.num, mo.num, mo.num)` | ↓↓ component of the two-body reduced density matrix
`2e_updn` | `float sparse` | `(mo.num, mo.num, mo.num, mo.num)` | ↑↓ 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 (↑↑)
`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 (↓↓)
`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 (↑↓)

## 6 Correlation factors

### 6.1 Jastrow factor (jastrow group)

The Jastrow factor is an $N$-electron function which multiplies the CI expansion: \(\Psi = \Phi \times \exp(J)\),

In the following, we use the notations \(r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|\) and \(R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|\), where indices \(i\) and \(j\) refer 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`

#### 6.1.1 CHAMP

The first form of Jastrow factor is the one used in the CHAMP program:

\[ J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R}) \]

\(J_{\text{eN}}\) contains electron-nucleus terms:

\[ J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}\left[ \frac{a_{1,\alpha}\, f_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\, f_\alpha(R_{i\alpha})} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [f_\alpha(R_{i\alpha})]^p - J_{\text{eN}}^\infty \right] \]

\(J_{\text{ee}}\) contains electron-electron terms:

\[ J_{\text{ee}}(\mathbf{r}) = \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \left[ \frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\, f_{\text{ee}}(r_{ij})}{1+b_2\, f_{\text{ee}}(r_{ij})} + \sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, [f_{\text{ee}}(r_{ij})]^p - J_{\text{ee},ij}^\infty \right] \]

\(\delta^{\uparrow\downarrow}_{ij}\) is zero when the electrons \(i\) and \(j\) have the same spin, and one otherwise.

\(J_{\text{eeN}}\) contains electron-electron-Nucleus terms:

\[ J_{\text{eeN}}(\mathbf{r},\mathbf{R}) = \sum_{\alpha=1}^{N_{\text{nucl}}} \sum_{i=1}^{N_{\text{elec}}} \sum_{j=1}^{i-1} \sum_{p=2}^{N_{\text{ord}}} \sum_{k=0}^{p-1} \sum_{l=0}^{p-k-2\delta_{k,0}} c_{lkp\alpha} \left[ g_{\text{ee}}({r}_{ij}) \right]^k \nonumber \\ \left[ \left[ g_\alpha({R}_{i\alpha}) \right]^l + \left[ g_\alpha({R}_{j\alpha}) \right]^l \right] \left[ g_\alpha({R}_{i\,\alpha}) \, g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2} \] \(c_{lkp\alpha}\) are non-zero only when \(p-k-l\) is even.

The terms \(J_{\text{ee},ij}^\infty\) and \(J_{\text{eN}}^\infty\) are shifts to ensure that \(J_{\text{eN}}\) and \(J_{\text{ee}}\) have an asymptotic value of zero:

\[ J_{\text{eN}}^{\infty} = \frac{a_{1,\alpha}\, \kappa_\alpha^{-1}}{1+a_{2,\alpha}\, \kappa_\alpha^{-1}} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, \kappa_\alpha^{-p} \] \[ J_{\text{ee},ij}^{\infty} = \frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\, \kappa_{\text{ee}}^{-1}}{1+b_2\, \kappa_{\text{ee}}^{-1}} + \sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, \kappa_{\text{ee}}^{-p} \]

\(f\) and \(g\) are scaling function defined as

\[ f_\alpha(r) = \frac{1-e^{-\kappa_\alpha\, r}}{\kappa_\alpha} \text{ and } g_\alpha(r) = e^{-\kappa_\alpha\, r}, \]

#### 6.1.2 Mu

Mu-Jastrow is based on a one-parameter correlation factor that has been introduced in the context of transcorrelated methods. This correlation factor imposes the electron-electron cusp, and it is built such that the leading order in \(1/r_{12}\) of the effective two-electron potential reproduces the long-range interaction of the range-separated density functional theory. Its analytical expression reads

\[ J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) + J_{\text{eN}}(\mathbf{r}, \mathbf{R}) \].

The electron-electron cusp is incorporated in the three-body term

\[ J_\text{eeN} (\mathbf{r}, \mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \, \Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha}), \]

where ww\(u\) is an electron-electron function

\[ u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right]. \]

This electron-electron term is tuned by the parameter \(\mu\) which controls the depth and the range of the Coulomb hole between electrons.

An envelope function has been introduced to cancel out the Jastrow effects between two-electrons when at least one is close to a nucleus (to perform a frozen-core calculation). The envelope function is given by

\[ E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right). \]

In particular, if the parameters \(\gamma_\alpha\) tend to zero, the Mu-Jastrow factor becomes a two-body Jastrow factor:

\[ J_{\text{ee}}(\mathbf{r}) = \sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \]

and for large \(\gamma_\alpha\) it becomes zero.

To increase the flexibility of the Jastrow and improve the electron density the following electron-nucleus term is added

\[ J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \, \left[ \exp\left( a_{\alpha} R_{i \alpha}^2 \right) - 1\right]. \]

The parameter \(\mu\) is stored in the `ee` array, the parameters \(\gamma_\alpha\) are stored in the `een` array, and the parameters \(a_\alpha\) are stored in the `en` array.

#### 6.1.3 Table of values

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`type` | `string` |  | Type of Jastrow factor: `CHAMP` or `Mu`
`en_num` | `dim` |  | Number of Electron-nucleus parameters
`ee_num` | `dim` |  | Number of Electron-electron parameters
`een_num` | `dim` |  | Number of Electron-electron-nucleus parameters
`en` | `float` | `(jastrow.en_num)` | Electron-nucleus parameters
`ee` | `float` | `(jastrow.ee_num)` | Electron-electron 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

## 7 Quantum Monte Carlo data (qmc group)

In quantum Monte Carlo calculations, the wave function is evaluated at points of the 3N-dimensional space. Some algorithms require multiple independent walkers, so it is possible to store multiple coordinates, as well as some quantities evaluated at those points.

By convention, the electron coordinates contain first all the electrons of $↑$-spin and then all the $↓$-spin.

Variable | Type | Dimensions | Description
--- | --- | --- | ---
`num` | `dim` |  | Number of 3N-dimensional points
`point` | `float` | `(3, electron.num, qmc.num)` | 3N-dimensional points
`psi` | `float` | `(qmc.num)` | Wave function evaluated at the points
`e_loc` | `float` | `(qmc.num)` | Local energy evaluated at the points

Author: TREX-CoE

Created: 2023-04-26 Wed 08:13

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