trex.org

Data stored in TREXIO

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.

Dimensions are given both in row-major [] and column-major () formats. Pick the one adapted to the programming language in which you use TREXIO (Numpy is by default row-major, and Fortran is column-major). In the column-major representation, A(i,j) and A(i+1,j) are contiguous in memory. In the row-major representation, A[i,j] and A[i,j+1] are contiguous.

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	Row-major Dimensions	Column-major Dimensions	Description
code_num	dim			Number of codes used to produce the file
code	str	[metadata.code_num]	(metadata.code_num)	Names of the codes used
author_num	dim			Number of authors of the file
author	str	[metadata.author_num]	(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.

System

Nucleus (nucleus group)

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

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
num	dim			Number of nuclei
charge	float	[nucleus.num]	(nucleus.num)	Charges of the nuclei
coord	float	[nucleus.num, 3]	(3, nucleus.num)	Coordinates of the atoms
label	str	[nucleus.num]	(nucleus.num)	Atom labels
point_group	str			Symmetry point group
repulsion	float			Nuclear repulsion energy

Cell (cell group)

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

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
a	float	[3]	(3)	First real space lattice vector
b	float	[3]	(3)	Second real space lattice vector
c	float	[3]	(3)	Third real space lattice vector
g_a	float	[3]	(3)	First reciprocal space lattice vector
g_b	float	[3]	(3)	Second reciprocal space lattice vector
g_c	float	[3]	(3)	Third reciprocal space lattice vector
two_pi	int			0 or 1. If two_pi=1, $2π$ is included in the reciprocal vectors.

Periodic boundary calculations (pbc group)

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
periodic	int			1: true or 0: false
k_point_num	dim			Number of $k$-points
k_point	float	[3]	(3)	$k$-point sampling
k_point_weight	float	[pbc.k_point_num]	(pbc.k_point_num)	$k$-point weight

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 ($N_↑$) and spin down ($N_↓$). 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	Row-major Dimensions	Column-major Dimensions	Description
num	dim			Number of electrons
up_num	int			Number of ↑-spin electrons
dn_num	int			Number of ↓-spin electrons

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	Row-major Dimensions	Column-major Dimensions	Description
num	dim			Number of states (including the ground state)
id	index			Index of the current state (0 is ground state)
energy	float			Energy of the current state
current_label	str			Label of the current state
label	str	[state.num]	(state.num)	Labels of all states
file_name	str	[state.num]	(state.num)	Names of the TREXIO files linked to the current one (i.e. containing data for other states)

Basis functions

Basis set (basis group)

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 $γ_ks$ and coefficients $a_ks$: \[ R_s(\mathbf{r}) = \mathcal{N}_s |\mathbf{r}-\mathbf{R}_A|^n_s ∑_k=1^{N_{\text{prim}}} a_ks\, f_ks(γ_ks,p)\, exp \left( - γ_ks | \mathbf{r}-\mathbf{R}_A | ^p \right). \]

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.

Numerical orbitals

Trexio supports numerical atom centered orbitals. The implementation is based on the approach of FHI-aims [Blum, V. et al; Ab initio molecular simulations with numeric atom-centered orbitals; Computer Physics Communications 2009]. These orbitals are defined by the atom they are centered on, their angular momentum and a radial function $R_s$, which is of the form \[ R_s(\mathbf{r}) = \mathcal{N}_s \frac{u_i(\mathbf{r})}{r^-n_s}. \] Where $u_i(\mathbf{r})$ is numerically tabulated on a dense logarithmic grid. It is constructed to vanish for any $\mathbf{r}$ outside of the grid. The reference points are stored in nao_grid_r and nao_grid_phi. Additionaly, a separate spline for the first and second derivative of $u(\mathbf{r})$ can be stored in nao_grid_grad and nao_grid_lap. Storing them in this form allows to calculate the actual first and second derivatives easily as follows:

\[ \frac{∂ φ}{∂ x} = \frac{x}{r^2}\left( u^′\left(r\right) - \frac{u\left(r\right)}{r}\right) \] \[ \frac{∂^2 φ}{∂ x^2} = \frac{1}{r^3}\left(x^2 u^′\prime(r) + \left( 3x^2-r^2\right) \left( \frac{u(r)}{r^2} - \frac{u’(r)}{r}\right) \right) \]

The index of the first data point for each shell is stored in nao_grid_start, the number of data points per spline is stored in nao_grid_size for convenience.

What kind of spline is used can be provided in the interpolator_kind field. For example, FHI-aims uses a cubic spline, so the interpolator_kind is "Polynomial" and the interp_coeff_cnt is $4$. In this case, the first interpolation coefficient per data point is the absolute term, the second is for the linear term etc. The interpolation coefficients for the wave function are given in the interpolator_phi array. The interpolator_grad and interpolator_lap arrays provide a spline for the gradient and Laplacian, respectively. The argument passed to the interpolants is on the logarithmic scale of the reference points: If the argument is an integer $i$, the interpolant will return the value of $u(\mathbf{r})$ at the $i$th reference point. A radius is converted to this scale by (note the zero-indexing) \[ i_log = \frac{1}{c} ⋅ log \left( \frac{r}{r_0} \right) \] where \[ c = log\left(\frac{r_1}{r_0}\right) \] For convenience, this conversion and functions to evaluate the splines are provided with trexio. Since these implementations are not adapted to a specific software architecture, a programm using these orbitals should reimplement them with consideration for its specific needs.

Plane waves

A plane wave is defined as

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

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.

Oscillating orbitals

Basis functions can be made oscillating as \[ R_s(\mathbf{r}) = \mathcal{N}_s |\mathbf{r}-\mathbf{R}_A|^n_s ∑_k=1^{N_{\text{prim}}} a_ks\, f_ks(γ_ks,p)\, exp \left( - γ_ks | \mathbf{r}-\mathbf{R}_A | ^p \right)\, cos \left( β_ks | \mathbf{r}-\mathbf{R}_A | ^q \right) \]

Oscillation kind can be:

• Cos1 for $q=1$
• Cos2 for $q=2$

Data definitions

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
type	str			Type of basis set: “Gaussian”, “Slater”, “Numerical” or “PW” for plane waves
prim_num	dim			Total number of primitives
shell_num	dim			Total number of shells
nao_grid_num	dim			Total number of grid points for numerical orbitals
interp_coeff_cnt	dim			Number of coefficients for the numerical orbital interpolator
nucleus_index	index	[basis.shell_num]	(basis.shell_num)	One-to-one correspondence between shells and atomic indices
shell_ang_mom	int	[basis.shell_num]	(basis.shell_num)	One-to-one correspondence between shells and angular momenta
shell_factor	float	[basis.shell_num]	(basis.shell_num)	Normalization factor of each shell ($\mathcal{N}_s$)
r_power	int	[basis.shell_num]	(basis.shell_num)	Power to which $r$ is raised ($n_s$)
nao_grid_start	index	[basis.shell_num]	(basis.shell_num)	Index of the first data point for a given numerical orbital
nao_grid_size	dim	[basis.shell_num]	(basis.shell_num)	Number of data points per numerical orbital
shell_index	index	[basis.prim_num]	(basis.prim_num)	One-to-one correspondence between primitives and shell index
exponent	float	[basis.prim_num]	(basis.prim_num)	Exponents of the primitives ($γks$)
exponent_im	float	[basis.prim_num]	(basis.prim_num)	Imaginary part of the exponents of the primitives ($γks$)
coefficient	float	[basis.prim_num]	(basis.prim_num)	Coefficients of the primitives ($aks$)
coefficient_im	float	[basis.prim_num]	(basis.prim_num)	Imaginary part of the coefficients of the primitives ($aks$)
oscillation_arg	float	[basis.prim_num]	(basis.prim_num)	Additional argument to have oscillating orbitals ($βks$)
oscillation_kind	str			Kind of Oscillating function:”Cos1” or “Cos2”
prim_factor	float	[basis.prim_num]	(basis.prim_num)	Normalization coefficients for the primitives ($fks$)
e_cut	float			Energy cut-off for plane-wave calculations
nao_grid_radius	float	[basis.nao_grid_num]	(basis.nao_grid_num)	Radii of grid points for numerical orbitals
nao_grid_phi	float	[basis.nao_grid_num]	(basis.nao_grid_num)	Wave function values for numerical orbitals
nao_grid_grad	float	[basis.nao_grid_num]	(basis.nao_grid_num)	Radial gradient of numerical orbitals
nao_grid_lap	float	[basis.nao_grid_num]	(basis.nao_grid_num)	Laplacian of numerical orbitals
interpolator_kind	str			Kind of spline, e.g. “Polynomial”
interpolator_phi	float	[basis.nao_grid_num, basis.interp_coeff_cnt]	(basis.interp_coeff_cnt, basis.nao_grid_num)	Coefficients for numerical orbital interpolation function
interpolator_grad	float	[basis.nao_grid_num, basis.interp_coeff_cnt]	(basis.interp_coeff_cnt, basis.nao_grid_num)	Coefficients for numerical orbital gradient interpolation function
interpolator_lap	float	[basis.nao_grid_num, basis.interp_coeff_cnt]	(basis.interp_coeff_cnt, basis.nao_grid_num)	Coefficients for numerical orbital laplacian interpolation function

Example

For example, consider H_2 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 ]

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 \[ V_A^\text{ECP} = V_{A \ell_{max}+1} + ∑_\ell=0^\ell_{max} δ V_{A \ell}∑_m=-\ell^\ell | Y_{\ell m} \rangle \langle Y_{\ell m} | \]

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 \(δ V_{A \ell}\) and \(V_{A \ell_{max}+1}\) are parameterized as: \begin{eqnarray} δ V_{A \ell}(\mathbf{r}) &=& ∑_q=1^{N_{q \ell}} β_{A q \ell}\, |\mathbf{r}-\mathbf{R}_A|^{n_{A q \ell}}\, e^{-α_{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|}+δ 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.

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.

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
max_ang_mom_plus_1	int	[nucleus.num]	(nucleus.num)	$\ellmax+1$, one higher than the max angular momentum in the removed core orbitals
z_core	int	[nucleus.num]	(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]	(ecp.num)	One-to-one correspondence between ECP items and the angular momentum $\ell$
nucleus_index	index	[ecp.num]	(ecp.num)	One-to-one correspondence between ECP items and the atom index
exponent	float	[ecp.num]	(ecp.num)	$αA q \ell$ all ECP exponents
coefficient	float	[ecp.num]	(ecp.num)	$βA q \ell$ all ECP coefficients
power	int	[ecp.num]	(ecp.num)	$nA q \ell$ all ECP powers

Example

For example, consider H_2 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
  ]

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	Row-major Dimensions	Column-major 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]	(grid.num)	Discretized coordinate space
weight	float	[grid.num]	(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]	(grid.ang_num)	Discretized angular space (if used)
ang_weight	float	[grid.ang_num]	(grid.ang_num)	Angular grid weights (if used)
rad_num	dim			Number of radial integration points (if used)
rad_coord	float	[grid.rad_num]	(grid.rad_num)	Discretized radial space (if used)
rad_weight	float	[grid.rad_num]	(grid.rad_num)	Radial grid weights (if used)

Orbitals

Atomic orbitals (ao group)

AOs are defined as

\[ χ_i (\mathbf{r}) = \mathcal{N}_i’\, P_η(i)(\mathbf{r})\, R_s(i) (\mathbf{r}) \]

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.

$η(i)$ 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 $0, +1, -1, +2, -2, …, + m, -m$ (see Wikipedia). In the case of polynomials, the canonical (or alphabetical) ordering is used,

$p$	$p_x, p_y, p_z$
$d$	$dxx, dxy, dxz, dyy, dyz, dzz$
$f$	$fxxx, fxxy, fxxz, fxyy, fxyz$,
	$fxzz, fyyy, fyyz, fyzz, fzzz$
$\vdots$

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. $\mathcal{N}_d≡\mathcal{N}_z^2$) 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	Row-major Dimensions	Column-major Dimensions	Description
cartesian	int			1: true, 0: false
num	dim			Total number of atomic orbitals
shell	index	[ao.num]	(ao.num)	Basis set shell for each AO
normalization	float	[ao.num]	(ao.num)	Normalization factor $\mathcal{N}_i$

One-electron integrals (ao_1e_int group)

• \[ \hat{V}_\text{ne} = ∑_A=1^N_\text{nucl} ∑_i=1^N_\text{elec} \frac{-Z_A }{| \mathbf{R}_A - \mathbf{r}_i |} \] : electron-nucleus attractive potential,
• \[ \hat{T}_\text{e} = ∑_i=1^N_\text{elec} -\frac{1}{2}\hat{Δ}_i \] : 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 \[ \langle p | \hat{O} | q \rangle \], returned as a matrix over atomic orbitals.

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

Two-electron integrals (ao_2e_int group)

The two-electron integrals for a two-electron operator $\hat{O}$ are \[ \langle p q | \hat{O} | r s \rangle \] in physicists notation, where $p,q,r,s$ are indices over atomic orbitals.

• \[ \hat{W}_\text{ee} = ∑_i=2^N_\text{elec} ∑_j=1^i-1 \frac{1}{| \mathbf{r}_i - \mathbf{r}_j |} \] : electron-electron repulsive potential operator.
• \[ \hat{W}^lr_\text{ee} = ∑_i=2^N_\text{elec} ∑_j=1^i-1 \frac{\text{erf}(μ\, | \mathbf{r}_i - \mathbf{r}_j |)}{| \mathbf{r}_i - \mathbf{r}_j |} \] : electron-electron long range potential

The Cholesky decomposition of the integrals can also be stored:

\[ \langle ij | kl \rangle = ∑_α G_ikα G_jlα \]

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
eri	float sparse	[ao.num, ao.num, ao.num, ao.num]	(ao.num, ao.num, ao.num, ao.num)	Electron repulsion integrals
eri_lr	float sparse	[ao.num, ao.num, ao.num, ao.num]	(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_2e_int.eri_cholesky_num, ao.num, ao.num]	(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_2e_int.eri_lr_cholesky_num, ao.num, ao.num]	(ao.num, ao.num, ao_2e_int.eri_lr_cholesky_num)	Cholesky decomposition of the long range ERI

Molecular orbitals (mo group)

**Warning**: mo.num is the total number of MOs stored in the file. For periodic calculations, this group contains all the MOs belonging to all $k$ points so mo.num will be the sum of the numbers of MOs in each $k$ point. For UHF wave functions, mo.num is the number of spin-orbitals, twice the number of spatial orbitals.

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
type	str			Free text to identify the set of MOs (HF, Natural, Local, CASSCF, etc)
num	dim			Number of MOs
coefficient	float	[mo.num, ao.num]	(ao.num, mo.num)	MO coefficients
coefficient_im	float	[mo.num, ao.num]	(ao.num, mo.num)	MO coefficients (imaginary part)
class	str	[mo.num]	(mo.num)	Choose among: Core, Inactive, Active, Virtual, Deleted
symmetry	str	[mo.num]	(mo.num)	Symmetry in the point group
occupation	float	[mo.num]	(mo.num)	Occupation number
energy	float	[mo.num]	(mo.num)	For canonical MOs, corresponding eigenvalue
spin	int	[mo.num]	(mo.num)	For UHF wave functions, 0 is $α$ and 1 is $β$
k_point	index	[mo.num]	(mo.num)	For periodic calculations, the $k$ point to which each MO belongs

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	Row-major Dimensions	Column-major Dimensions	Description
overlap	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | j \rangle$
kinetic	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{T}_e | j \rangle$
potential_n_e	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{V}\text{ne} | j \rangle$
ecp	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{V}\text{ECP} | j \rangle$
core_hamiltonian	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{h} | j \rangle$
overlap_im	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | j \rangle$ (imaginary part)
kinetic_im	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{T}_e | j \rangle$ (imaginary part)
potential_n_e_im	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{V}\text{ne} | j \rangle$ (imaginary part)
ecp_im	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{V}\text{ECP} | j \rangle$ (imaginary part)
core_hamiltonian_im	float	[mo.num, mo.num]	(mo.num, mo.num)	$\langle i | \hat{h} | j \rangle$ (imaginary part)

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	Row-major Dimensions	Column-major Dimensions	Description
eri	float sparse	[mo.num, mo.num, mo.num, mo.num]	(mo.num, mo.num, mo.num, mo.num)	Electron repulsion integrals
eri_lr	float sparse	[mo.num, mo.num, mo.num, mo.num]	(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_2e_int.eri_cholesky_num, mo.num, mo.num]	(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_2e_int.eri_lr_cholesky_num, mo.num, mo.num]	(mo.num, mo.num, mo_2e_int.eri_lr_cholesky_num)	Cholesky decomposition of the long range ERI

Multi-determinant information

Slater determinants (determinant group)

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

\[ Ψ = ∑_I C_I D_I \]

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.

\[ D_I = α_1 α_2 \ldots α_{n_↑} β_1 β_2 \ldots β_{n_↓} \]

where $α$ and $β$ denote ↑-spin and ↓-spin electrons, respectively, $n_↑$ and $n_↓$ 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	Row-major Dimensions	Column-major Dimensions	Description
num	dim readonly			Number of determinants
list	int special	[determinant.num]	(determinant.num)	List of determinants as integer bit fields
coefficient	float buffered	[determinant.num]	(determinant.num)	Coefficients of the determinants from the CI expansion

Configuration state functions (csf group)

The configuration interaction (CI) wave function $Ψ$ can be expanded in the basis of configuration state functions (CSFs) $Ψ_I$ as follows

\[ Ψ = ∑_I C_I ψ_I. \]

Each CSF $ψ_I$ 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 $\langle D_I | ψ_J \rangle$ needed to project the CSFs in the basis of Slater determinants.

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
num	dim readonly			Number of CSFs
coefficient	float buffered	[csf.num]	(csf.num)	Coefficients $C_I$ of the CSF expansion
det_coefficient	float sparse	[csf.num, determinant.num]	(determinant.num, csf.num)	Projection on the determinant basis

Amplitudes (amplitude group)

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

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

on a reference wave function $Ψ$, where $\hat{T}_1$ is the single excitation operator,

\[ \hat{T}_1 = ∑_ia t_i^a\, \hat{a}^†_a \hat{a}_i, \]

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

\[ \hat{T}_2 = \frac{1}{4} ∑_ijab t_ij^ab\, \hat{a}^†_a \hat{a}^†_b \hat{a}_j \hat{a}_i, \]

etc. Indices $i$, $j$, $a$ and $b$ denote molecular orbital indices.

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

\[ |Φ\rangle = \hat{T}|Ψ\rangle \]

and coupled-cluster wave functions are of the form

\[ |Φ\rangle = e^\hat{T}| Ψ \rangle \]

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) = $t_ijkl^abcd$,
• $…$

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
single	float sparse	[mo.num,mo.num]	(mo.num,mo.num)	Single excitation amplitudes
single_exp	float sparse	[mo.num,mo.num]	(mo.num,mo.num)	Exponentialized single excitation amplitudes
double	float sparse	[mo.num,mo.num,mo.num,mo.num]	(mo.num,mo.num,mo.num,mo.num)	Double excitation amplitudes
double_exp	float sparse	[mo.num,mo.num,mo.num,mo.num]	(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]	(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]	(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]	(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]	(mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num,mo.num)	Exponentialized quadruple excitation amplitudes

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*} γ_ij^↑ &=& \langle Ψ | \hat{a}^†_jα\, \hat{a}_iα | Ψ \rangle
γ_ij^↓ &=& \langle Ψ | \hat{a}^†_jβ \, \hat{a}_iβ | Ψ \rangle \end{eqnarray*} and the spin-summed one-body density matrix is \[ γ_ij = γ^↑_ij + γ^↓_ij \]

The $↑ ↑$, $↓ ↓$, $↑ ↓$ components of the two-body density matrix are given by \begin{eqnarray*} Γ_ijkl^{↑ ↑} &=& \langle Ψ | \hat{a}^†_kα\, \hat{a}^†_lα \hat{a}_jα\, \hat{a}_iα | Ψ \rangle
Γ_ijkl^{↓ ↓} &=& \langle Ψ | \hat{a}^†_kβ\, \hat{a}^†_lβ \hat{a}_jβ\, \hat{a}_iβ | Ψ \rangle \ Γ_ijkl^{↑ ↓} &=& \langle Ψ | \hat{a}^†_kα\, \hat{a}^†_lβ \hat{a}_jβ\, \hat{a}_iα | Ψ \rangle

• \langle Ψ | \hat{a}^†_lα\, \hat{a}^†_kβ \hat{a}_iβ\, \hat{a}_jα | Ψ \rangle
  

\end{eqnarray*} and the spin-summed one-body density matrix is \[ Γ_ijkl = Γ_ijkl^{↑ ↑} + Γ_ijkl^{↓ ↓} + Γ_ijkl^{↑ ↓}. \]

The total energy can be computed as: \[ E = E_\text{NN} + ∑_ij γ_ij \langle j|h|i \rangle + \frac{1}{2} ∑_ijlk Γ_ijkl \langle k l | i j \rangle \]

To compress the storage, the Cholesky decomposition of the RDMs can be stored:

\[ Γ_ijkl = ∑_α G_ijα G_klα \]

Warning: as opposed to electron repulsion integrals, the decomposition is made such that the Cholesky vectors are expanded in a two-electron basis $f_ij(\mathbf{r}_1,\mathbf{r}_2) = φ_i(\mathbf{r}_1) φ_j(\mathbf{r}_2)$, whereas in electron repulsion integrals each Cholesky vector is expressed in a basis of a one-electron function $g_ik(\mathbf{r}_1) = φ_i(\mathbf{r}_1) φ_k(\mathbf{r}_1)$.

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
1e	float	[mo.num, mo.num]	(mo.num, mo.num)	One body density matrix
1e_up	float	[mo.num, mo.num]	(mo.num, mo.num)	↑-spin component of the one body density matrix
1e_dn	float	[mo.num, mo.num]	(mo.num, mo.num)	↓-spin component of the one body density matrix
1e_transition	float	[state.num, state.num, mo.num, mo.num]	(mo.num, mo.num, state.num, state.num)	One-particle transition density matrices
2e	float sparse	[mo.num, mo.num, mo.num, mo.num]	(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]	(mo.num, mo.num, mo.num, mo.num)	↑\uparrow component of the two-body reduced density matrix
2e_dndn	float sparse	[mo.num, mo.num, mo.num, mo.num]	(mo.num, mo.num, mo.num, mo.num)	↓\downarrow component of the two-body reduced density matrix
2e_updn	float sparse	[mo.num, mo.num, mo.num, mo.num]	(mo.num, mo.num, mo.num, mo.num)	↑\downarrow component of the two-body reduced density matrix
2e_transition	float sparse	[state.num, state.num, mo.num, mo.num, mo.num, mo.num]	(mo.num, mo.num, mo.num, mo.num, state.num, state.num)	Two-particle transition density matrices
2e_cholesky_num	dim			Number of Cholesky vectors
2e_cholesky	float sparse	[rdm.2e_cholesky_num, mo.num, mo.num]	(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	[rdm.2e_upup_cholesky_num, mo.num, mo.num]	(mo.num, mo.num, rdm.2e_upup_cholesky_num)	Cholesky decomposition of the two-body RDM (↑\uparrow)
2e_dndn_cholesky_num	dim			Number of Cholesky vectors
2e_dndn_cholesky	float sparse	[rdm.2e_dndn_cholesky_num, mo.num, mo.num]	(mo.num, mo.num, rdm.2e_dndn_cholesky_num)	Cholesky decomposition of the two-body RDM (↓\downarrow)
2e_updn_cholesky_num	dim			Number of Cholesky vectors
2e_updn_cholesky	float sparse	[rdm.2e_updn_cholesky_num, mo.num, mo.num]	(mo.num, mo.num, rdm.2e_updn_cholesky_num)	Cholesky decomposition of the two-body RDM (↑\downarrow)

Correlation factors

Jastrow factor (jastrow group)

The Jastrow factor is an $N$-electron function which multiplies the CI expansion: $Ψ = Φ × exp(J)$,

In the following, we use the notations $r_ij = |\mathbf{r}_i - \mathbf{r}_j|$ and $R_iα = |\mathbf{r}_i - \mathbf{R}_α|$, where indices $i$ and $j$ refer to electrons and $α$ 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 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:

\[ J_\text{eN}(\mathbf{r},\mathbf{R}) = ∑_i=1^N_\text{elec} ∑_α=1^N_\text{nucl}\left[ \frac{a_1,α\, f_α(R_iα)}{1+a_2,α\, f_α(R_iα)} + ∑_p=2^N_\text{ord^a} a_p+1,α\, [f_α(R_iα)]^p - J_\text{eN}^∞ \right] \]

$J_\text{ee}$ contains electron-electron terms:

\[ J_\text{ee}(\mathbf{r}) = ∑_i=1^N_\text{elec} ∑_j=1^i-1 \left[ \frac{\frac{1}{2}\big(1 + δ^{↑\downarrow}_ij\big)\,b_1\, f_\text{ee}(r_ij)}{1+b_2\, f_\text{ee}(r_ij)} + ∑_p=2^N_\text{ord^b} b_p+1\, [f_\text{ee}(r_ij)]^p - J_\text{ee,ij}^∞ \right] \]

$δ^{↑\downarrow}_ij$ is zero when the electrons $i$ and $j$ have the same spin, and one otherwise.

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

\[ J_\text{eeN}(\mathbf{r},\mathbf{R}) = ∑_α=1^{N_{\text{nucl}}} ∑_i=1^{N_{\text{elec}}} ∑_j=1^i-1 ∑_p=2^N_{\text{ord}} ∑_k=0^p-1 ∑_l=0^p-k-2δ_{k,0} c_lkpα \left[ g_\text{ee}({r}_ij) \right]^k \nonumber
\left[ \left[ g_α({R}_iα) \right]^l + \left[ g_α({R}_jα) \right]^l \right] \left[ g_α({R}_i\,α) \, g_α({R}_jα) \right]^(p-k-l)/2 \] $c_lkpα$ are non-zero only when $p-k-l$ is even.

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

\[ J_\text{eN}^∞ = \frac{a_1,α\, κ_α^-1}{1+a_2,α\, κ_α^-1} + ∑_p=2^N_\text{ord^a} a_p+1,α\, κ_α^-p \] \[ J_\text{ee,ij}^∞ = \frac{\frac{1}{2}\big(1 + δ^{↑\downarrow}_ij\big)\,b_1\, κ_\text{ee}^-1}{1+b_2\, κ_\text{ee}^-1} + ∑_p=2^N_\text{ord^b} b_p+1\, κ_\text{ee}^-p \]

$f$ and $g$ are scaling function defined as

\[ f_α(r) = \frac{1-e^{-κ_α\, r}}{κ_α} \text{ and } g_α(r) = e^{-κ_α\, r}, \]

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 $1/r₁₂$ 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

\[ J_\text{eeN} (\mathbf{r}, \mathbf{R}) = ∑_i=1^N_\text{elec} ∑_j=1^i-1 \, u\left(μ, r_ij\right) \, Π_α=1^{N_{\text{nucl}}} \, E_α({R}_iα) \, E_α({R}_jα), \]

where ww$u$ is an electron-electron function

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

This electron-electron term is tuned by the parameter $μ$ 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

\[ E_α(R) = 1 - exp\left( - γ_α \, R^2 \right). \]

In particular, if the parameters $γ_α$ tend to zero, the Mu-Jastrow factor becomes a two-body Jastrow factor:

\[ J_\text{ee}(\mathbf{r}) = ∑_i=1^N_\text{elec} ∑_j=1^i-1 \, u\left(μ, r_ij\right) \]

and for large $γ_α$ it becomes zero.

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

\[ J_\text{eN}(\mathbf{r},\mathbf{R}) = ∑_i=1^N_\text{elec} ∑_α=1^N_\text{nucl} \, \left[ exp\left( a_α R_{i α}^2 \right) - 1\right]. \]

The parameter $μ$ is stored in the ee array, the parameters $γ_α$ are stored in the een array, and the parameters $a_α$ are stored in the en array.

Table of values

Variable	Type	Row-major Dimensions	Column-major Dimensions	Description
type	str			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]	(jastrow.en_num)	Electron-nucleus parameters
ee	float	[jastrow.ee_num]	(jastrow.ee_num)	Electron-electron parameters
een	float	[jastrow.een_num]	(jastrow.een_num)	Electron-electron-nucleus parameters
en_nucleus	index	[jastrow.en_num]	(jastrow.en_num)	Nucleus relative to the eN parameter
een_nucleus	index	[jastrow.een_num]	(jastrow.een_num)	Nucleus relative to the eeN parameter
ee_scaling	float			$κ$ value in CHAMP Jastrow for electron-electron distances
en_scaling	float	[nucleus.num]	(nucleus.num)	$κ$ value in CHAMP Jastrow for electron-nucleus distances

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	Row-major Dimensions	Column-major Dimensions	Description
num	dim			Number of 3N-dimensional points
point	float	[qmc.num, electron.num, 3]	(3, electron.num, qmc.num)	3N-dimensional points
psi	float	[qmc.num]	(qmc.num)	Wave function evaluated at the points
e_loc	float	[qmc.num]	(qmc.num)	Local energy evaluated at the points

Appendix

Python script from table to json

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 = "[ " + ", ".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""")