TREX Configuration file

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

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

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

Variable	Type	Dimensions	Description
up_num	int		Number of ↑-spin electrons
dn_num	int		Number of ↓-spin electrons

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

An effective core potential (ECP) \(V_A^{\text{ECP}}\) replacing the core electrons of atom \(A\) can be expressed as \[ V_A^{\text{ECP}} = V_{A \ell_{\max}+1} + \sum_{\ell=0}^{\ell_{\max}} \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} | \]

The first term in the equation above is sometimes attributed to the local channel, while the remaining terms correspond to the non-local channel projections.

The functions \(V_{A\ell}\) are parameterized as: \[ V_{A \ell}(\mathbf{r}) = \sum_{q=1}^{N_{q \ell}} \beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\, e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 } \]

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

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

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

Note for developers: avoid having variables with similar prefix in their name. HDF5 back end might cause issues due to the way find_dataset function works. For example, in the ECP group we use max_ang_mom and not ang_mom_max. The latter causes issues when written before ang_mom in the TREXIO file.

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

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
  ]

We consider here basis functions centered on nuclei. Hence, we enable the possibility to define dummy atoms to place basis functions in random positions.

The atomic basis set is defined as a list of shells. Each shell \(s\) is centered on a center \(A\), possesses a given angular momentum \(l\) and a radial function \(R_s\). The radial function is a linear combination of \(N_{\text{prim}}\) primitive functions that can be of type Slater (\(p=1\)) or Gaussian (\(p=2\)), parameterized by exponents \(\gamma_{ks}\) and coefficients \(a_{ks}\): \[ R_s(\mathbf{r}) = \mathcal{N}_s \vert\mathbf{r}-\mathbf{R}_A\vert^{n_s} \sum_{k=1}^{N_{\text{prim}}} a_{ks}\, f_{ks}(\gamma_{ks},p)\, \exp \left( - \gamma_{ks} \vert \mathbf{r}-\mathbf{R}_A \vert ^p \right). \]

In the case of Gaussian functions, \(n_s\) is always zero.

Different codes normalize functions at different levels. Computing normalization factors requires the ability to compute overlap integrals, so the normalization factors should be written in the file to ensure that the file is self-contained and does not need the client program to have the ability to compute such integrals.

Some codes assume that the contraction coefficients are for a linear combination of normalized primitives. This implies that a normalization constant for the primitive \(ks\) needs to be computed and stored. If this normalization factor is not required, \(f_{ks}=1\).

Some codes assume that the basis function are normalized. This implies the computation of an extra normalization factor, \(\mathcal{N}_s\). If the the basis function is not considered normalized, \(\mathcal{N}_s=1\).

All the basis set parameters are stored in one-dimensional arrays:

Variable	Type	Dimensions	Description
type	str		Type of basis set: "Gaussian" or "Slater"
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 (\(\mathcal{N}_s\))
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 (\(f_{ks}\))

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

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 ]

Going from the atomic basis set to AOs implies a systematic construction of all the angular functions of each shell. We consider two cases for the angular functions: the real-valued spherical harmonics, and the polynomials in Cartesian coordinates. In the case of spherical harmonics, the AOs are ordered as \(0, +1, -1, +2, -2, \dots, +m, -m\) and in the case of polynomials we impose the canonical (or alphabetical) ordering), i.e

\begin{eqnarray} p & : & p_x, p_y, p_z \nonumber \\ d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\ f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\ {\rm etc.} \nonumber \end{eqnarray}

Note that there is no exception for \(p\) orbitals in spherical coordinates: the ordering is \(0,+1,-1\) which corresponds \(p_z, p_x, p_y\).

AOs are defined as

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

where \(i\) is the atomic orbital index, \(P\) encodes for either the polynomials or the spherical harmonics, \(\theta(i)\) returns the shell on which the AO is expanded, and \(\eta(i)\) denotes which angular function is chosen. \(\mathcal{N}_i\) is a normalization factor that enables the possibility to have different normalization coefficients within a shell, as in the GAMESS convention where \(\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}\) because \[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)} (\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne \left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)} (\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]

In such a case, one should set the normalization of the shell (in the Basis set section) to \(\mathcal{N}_{z^2}\), which is the normalization factor of the atomic orbitals in spherical coordinates. The normalization factor of the \(xy\) function which should be introduced here should be \(\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}\).

Variable	Type	Dimensions	Description
cartesian	int		1: true, 0: false
num	dim		Total number of atomic orbitals
shell	index	(ao.num)	basis set shell for each AO
normalization	float	(ao.num)	Normalization factors

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
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

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

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

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

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

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

The total energy can be computed as: \[ E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle + \frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle \]

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_dnup	float sparse	(mo.num, mo.num, mo.num, mo.num)	↓↑ component of the two-body reduced density matrix