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Data stored in TREXIO

Table of Contents

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 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) 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 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 gradients and Laplacian easily as follows:

\[ \grad_{x_i} \phi = \frac{x_i}{r^2}\left u\prime(r) - \frac{u(r)}{r}\right \] \[ \Delta \phi = \frac{1}{r^3}\left x^2 u''(r) + \left3x^2-r^2\right \left \frac{u(r)}{r^2} - \frac{u'(r)}{r}\right \right \]

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) \[ i_{log} = \frac{1}{c} \cdot \log{\frac{r}{r_0}} \] 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.

3.1.3 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.

3.1.4 Data definitions

Variable Type 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) 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 (\(n_s\))  
nao_grid_start index (basis.shell_num) Index of the first data point for a given numerical orbital  
nao_grid_size dim (basis.shell_num) Number of data points per numerical orbital  
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  
nao_grid_radius float (basis.nao_grid_num) Radii of grid points for numerical orbitals  
nao_grid_phi float (basis.nao_grid_num) Wave function values for numerical orbitals  
nao_grid_grad float (basis.nao_grid_num) Radial gradient of numerical orbitals  
nao_grid_lap float (basis.nao_grid_num) Laplacian of numerical orbitals  
interpolator_kind str   Kind of spline, e.g. "Polynomial"  
interpolator_phi float (basis.interp_coeff_cnt,basis.nao_grid_num) Coefficients for numerical orbital interpolation function  
interpolator_grad float (basis.interp_coeff_cnt,basis.nao_grid_num) Coefficients for numerical orbital gradient interpolation function  
interpolator_lap float (basis.interp_coeff_cnt,basis.nao_grid_num) Coefficients for numerical orbital laplacian interpolation function  

3.1.5 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-K√∂ster)
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: 2023-09-19 Tue 12:46

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