TREX Configuration file
Table of Contents
All the quantities are saved in atomic units. The dimensions of the arrays in the tables below are given in column-major order (as in Fortran), and the ordering of the dimensions is reversed in the produces JSON configuration file as the library is written in C.
In Fortran, the arrays are 1-based and in most other languages the
arrays are 0-base. Hence, we introduce the index type which is an
1-based int in the Fortran interface and 0-based otherwise.
{
1 Metadata
As we expect our files to be archived in open-data repositories, we need to give the possibility to the users to store some metadata inside the files. We propose to store the list of names of the codes which have participated to the creation of the file, a list of authors of the file, and a textual description.
code_num |
int |
Number of codes used to produce the file | |
code |
str |
(metadata.code_num) |
Names of the codes used |
author_num |
int |
Number of authors of the file | |
author |
str |
(metadata.author_num) |
Names of the authors of the file |
description |
str |
Text describing the content of file |
2 Electron
We consider wave functions expressed in the spin-free formalism, where the number of ↑ and ↓ electrons is fixed.
up_num |
int |
Number of ↑-spin electrons | |
dn_num |
int |
Number of ↓-spin electrons |
3 Nucleus
The nuclei are considered as fixed point charges. Coordinates are given in Cartesian \((x,y,z)\) format.
num |
int |
Number of nuclei | |
charge |
float |
(nucleus.num) |
Charges of the nuclei |
coord |
float |
(3,nucleus.num) |
Coordinates of the atoms |
label |
str |
(nucleus.num) |
Atom labels |
point_group |
str |
Symmetry point group |
4 TODO Effective core potentials
An effective core potential (ECP) \(V_A^{\text{pp}}\) replacing the core electrons of atom \(A\) is the sum of a local component \(V_A^{\text{l}}\) and a non-local component \(V_A^{\text{nl}}\). The local component is given by
\[ \hat{V}_A^{\text{l}}(r) = -\frac{Z_A^{\text{eff}}}{r} + \frac{Z_A^{\text{eff}}}{r}\, \exp\left( -\alpha_A\, r^2\right) + Z_{\text{eff}}\, \alpha_A\, r\, \exp\left( -\beta_A\, r^2\right) + \gamma_A \exp\left( -\delta_A\, r^2\right), \]
and the component obtained after localizing the non-local operator is
\[ \hat{V}_A^{\text{nl}}(r) = \zeta_A\, \exp\left( -\eta_A\, r^2\right) |0\rangle \langle 0| + \mu_A \, \exp\left( -\nu_A \, r^2\right) |1\rangle \langle 1| \]
where \(r=|\mathbf{r-R}_A|\) is the distance to the nucleus on which the potential is centered, \(Z_A^{\text{eff}}\) is the effective charge due to the removed electrons, \(|0\rangle \langle 0|\) and \(|1\rangle \langle 1|\) are projections over zero and one principal angular momenta, respectively (generalization to higher angular momenta is straightforward), and all the parameters labeled by Greek letters are parameters.
- \(\hat{V}_\text{ecp,l} = \sum_A \hat{V}_A^{\text{l}}\) : local component
- \(\hat{V}_\text{ecp,nl} = \sum_A \hat{V}_A^{\text{nl}}\) : non-local component
lmax_plus_1 |
int |
(nucleus.num) |
\(l_{\max} + 1\) |
z_core |
float |
(nucleus.num) |
Charges to remove |
local_n |
int |
(nucleus.num) |
Number of local function |
local_num_n_max |
int |
Maximum value of local_n |
|
local_exponent |
float |
(ecp.local_num_n_max, nucleus.num) |
|
local_coef |
float |
(ecp.local_num_n_max, nucleus.num) |
|
local_power |
int |
(ecp.local_num_n_max, nucleus.num) |
|
non_local_n |
int |
(nucleus.num) |
|
non_local_num_n_max |
int |
||
non_local_exponent |
float |
(ecp.non_local_num_n_max, nucleus.num) |
|
non_local_coef |
float |
(ecp.non_local_num_n_max, nucleus.num) |
|
non_local_power |
int |
(ecp.non_local_num_n_max, nucleus.num) |
5 Basis set
We consider here basis functions centered on nuclei. Hence, we enable the possibility to define dummy atoms to place basis functions in random positions.
The atomic basis set is defined as a list of shells. Each shell \(s\) is centered on a center \(A\), possesses a given angular momentum \(l\) and a radial function \(R_s\). The radial function is a linear combination of \(N_{\text{prim}}\) primitive functions that can be of type Slater (\(p=1\)) or Gaussian (\(p=2\)), parameterized by exponents \(\gamma_{ks}\) and coefficients \(a_{ks}\): \[ R_s(\mathbf{r}) = \mathcal{N}_s \vert\mathbf{r}-\mathbf{R}_A\vert^{n_s} \sum_{k=1}^{N_{\text{prim}}} a_{ks}\, f_{ks}(\gamma_{ks},p)\, \exp \left( - \gamma_{ks} \vert \mathbf{r}-\mathbf{R}_A \vert ^p \right). \]
In the case of Gaussian functions, \(n_s\) is always zero.
Different codes normalize functions at different levels. Computing normalization factors requires the ability to compute overlap integrals, so the normalization factors should be written in the file to ensure that the file is self-contained and does not need the client program to have the ability to compute such integrals.
Some codes assume that the contraction coefficients are for a linear combination of normalized primitives. This implies that a normalization constant for the primitive \(ks\) needs to be computed and stored. If this normalization factor is not required, \(f_{ks}=1\).
Some codes assume that the basis function are normalized. This implies the computation of an extra normalization factor, \(\mathcal{N}_s\). If the the basis function is not considered normalized, \(\mathcal{N}_s=1\).
All the basis set parameters are stored in one-dimensional arrays:
type |
str |
Type of basis set: "Gaussian" or "Slater" | |
num |
int |
Total Number of shells | |
prim_num |
int |
Total number of primitives | |
nucleus_index |
index |
(nucleus.num) |
Index of the first shell of each nucleus (\(A\)) |
nucleus_shell_num |
int |
(nucleus.num) |
Number of shells for each nucleus |
shell_ang_mom |
int |
(basis.num) |
Angular momentum 0:S, 1:P, 2:D, ... |
shell_prim_num |
int |
(basis.num) |
Number of primitives in the shell (\(N_{\text{prim}}\)) |
shell_factor |
float |
(basis.num) |
Normalization factor of the shell (\(\mathcal{N}_s\)) |
shell_prim_index |
index |
(basis.num) |
Index of the first primitive in the complete list |
exponent |
float |
(basis.prim_num) |
Exponents of the primitives (\(\gamma_{ks}\)) |
coefficient |
float |
(basis.prim_num) |
Coefficients of the primitives (\(a_{ks}\)) |
prim_factor |
float |
(basis.prim_num) |
Normalization coefficients for the primitives (\(f_{ks}\)) |
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 1.0000000
we have:
type = "Gaussian" num = 12 prim_num = 20 nucleus_index = [0 , 6] shell_ang_mom = [0 , 0 , 0 , 1 , 1 , 2 , 0 , 0 , 0 , 1 , 1 , 2 ] shell_prim_num = [5 , 1 , 1 , 1 , 1 , 1 , 5 , 1 , 1 , 1 , 1 , 1 ] shell_prim_index = [0 , 5 , 6 , 7 , 8 , 9 , 10, 15, 16, 17, 18, 19] shell_factor = [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.] exponent = [ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057, 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407, 0.388, 1.057] coefficient = [ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0, 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0, 1.0, 1.0, 1.0] prim_factor = [ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01 3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01, 1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01, 1.8135965626177861e+00 ]
6 Atomic orbitals
Going from the atomic basis set to AOs implies a systematic construction of all the angular functions of each shell. We consider two cases for the angular functions: the real-valued spherical harmonics, and the polynomials in Cartesian coordinates. In the case of spherical harmonics, the AOs are ordered in increasing magnetic quantum number (\(-l \le m \le l\)), and in the case of polynomials we impose the canonical ordering of the Libint2 library, i.e
\begin{eqnarray} p & : & p_x, p_y, p_z \nonumber \\ d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\ f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\ {\rm etc.} \nonumber \end{eqnarray}AOs are defined as
\[ \chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r}) \]
where \(i\) is the atomic orbital index, \(P\) encodes for either the polynomials or the spherical harmonics, \(\theta(i)\) returns the shell on which the AO is expanded, and \(\eta(i)\) denotes which angular function is chosen. \(\mathcal{N}_i\) is a normalization factor that enables the possibility to have different normalization coefficients within a shell, as in the GAMESS convention where \(\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}\) because \[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)} (\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne \left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)} (\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
In such a case, one should set the normalization of the shell (in the Basis set 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 \(\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}\).
cartesian |
int |
1: true, 0: false |
|
num |
int |
Total number of atomic orbitals | |
shell |
index |
(ao.num) |
basis set shell for each AO |
normalization |
float |
(ao.num) |
Normalization factors |
6.1 One-electron integrals
- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}} \sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A - \mathbf{r}_i \vert} \] : electron-nucleus attractive potential,
- \[ \hat{T}_{\text{e}} = \sum_{i=1}^{N_\text{elec}} -\frac{1}{2}\hat{\Delta}_i \] : electronic kinetic energy
- \(\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} + \hat{V}_\text{ecp,l} + \hat{V}_\text{ecp,nl}\) : core electronic Hamiltonian
The one-electron integrals for a one-electron operator \(\hat{O}\) are \[ \langle p \vert \hat{O} \vert q \rangle \], returned as a matrix over atomic orbitals.
overlap |
float |
(ao.num, ao.num) |
\(\langle p \vert q \rangle\) |
kinetic |
float |
(ao.num, ao.num) |
\(\langle p \vert \hat{T}_e \vert q \rangle\) |
potential_n_e |
float |
(ao.num, ao.num) |
\(\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle\) |
ecp_local |
float |
(ao.num, ao.num) |
\(\langle p \vert \hat{V}_{\text{ecp,l}} \vert q \rangle\) |
ecp_non_local |
float |
(ao.num, ao.num) |
\(\langle p \vert \hat{V}_{\text{ecp,nl}} \vert q \rangle\) |
core_hamiltonian |
float |
(ao.num, ao.num) |
\(\langle p \vert \hat{h} \vert q \rangle\) |
6.2 Two-electron integrals
The two-electron integrals for a two-electron operator \(\hat{O}\) are \[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists notation or \[ ( pr \vert \hat{O} \vert qs ) \] in chemists notation, where \(p,q,r,s\) are indices over atomic orbitals.
Functions are provided to get the indices in physicists or chemists notation.
- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{\text{erf}(\vert \mathbf{r}_i - \mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential
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 |
7 Molecular orbitals
type |
str |
String identify the set of MOs | |
num |
int |
Number of MOs | |
coefficient |
float |
(ao.num, mo.num) |
MO coefficients |
class |
str |
(mo.num) |
Core, Inactive, Active, Virtual, Deleted |
symmetry |
str |
(mo.num) |
Symmetry in the point group |
occupation |
float |
(mo.num) |
Occupation number |
7.1 One-electron integrals
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.
overlap |
float |
(mo.num, mo.num) |
\(\langle i \vert j \rangle\) |
kinetic |
float |
(mo.num, mo.num) |
\(\langle i \vert \hat{T}_e \vert j \rangle\) |
potential_n_e |
float |
(mo.num, mo.num) |
\(\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle\) |
ecp_local |
float |
(mo.num, mo.num) |
\(\langle i \vert \hat{V}_{\text{ecp,l}} \vert j \rangle\) |
ecp_non_local |
float |
(mo.num, mo.num) |
\(\langle i \vert \hat{V}_{\text{ecp,nl}} \vert j \rangle\) |
core_hamiltonian |
float |
(mo.num, mo.num) |
\(\langle i \vert \hat{h} \vert j \rangle\) |
7.2 Two-electron integrals
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.
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 |
8 TODO Slater determinants
9 TODO Reduced density matrices
one_e |
float |
(mo.num, mo.num) |
one_e_up |
float |
(mo.num, mo.num) |
one_e_dn |
float |
(mo.num, mo.num) |
two_e |
float sparse |
(mo.num, mo.num, mo.num, mo.num) |
10 Appendix
10.1 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.reverse() dims = "[ " + ", ".join(dims) + " ]" else: if dims.strip() != "": dims = "ERROR" else: dims = "[]" buffer = fmt % (indent, name, typ.ljust(f2), dims.ljust(f3)) indent = " , " print(buffer) if last == 0: print(" } ,") else: print(" }") print("""#+end_src""")
}