#+TITLE: TREX Configuration file #+STARTUP: latexpreview #+SETUPFILE: docs/theme.setup This page contains information about the general structure of the TREXIO library. The source code of the library can be automatically generated based on the contents of the ~trex.json~ configuration file, which itself is compiled from different sections (groups) presented below. For more information about the automatic generation on the source code or regarding possible modifications, please contact the TREXIO developers. All quantities are saved in TREXIO file 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. TREXIO currently supports ~int~, ~float~ and ~str~ types for both single attributes and arrays. Note, that some attributes might have ~dim~ type (e.g. ~num~ of the ~nucleus~ group). This type is treated exactly the same as ~int~ with the only difference that ~dim~ variables cannot be negative. This additional constraint is required because ~dim~ attributes are used internally to allocate memory and to check array boundaries in the memory-safe API. Most of the times, the ~dim~ variables contain the ~num~ suffix. In Fortran, the arrays are 1-based and in most other languages the arrays are 0-based. Hence, we introduce the ~index~ type which is an 1-based ~int~ in the Fortran interface and 0-based otherwise. For sparse data structures such as electron replusion integrals, the data can be too large to fit in memory and the data needs to be fetched using multiple function calls to perform I/O on buffers. #+begin_src python :tangle trex.json :exports none { #+end_src * Metadata (metadata group) 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. #+NAME: metadata | 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:** ~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~. #+CALL: json(data=metadata, title="metadata") #+RESULTS: :RESULTS: #+begin_src python :tangle trex.json "metadata": { "code_num" : [ "dim", [] ] , "code" : [ "str", [ "metadata.code_num" ] ] , "author_num" : [ "dim", [] ] , "author" : [ "str", [ "metadata.author_num" ] ] , "package_version" : [ "str", [] ] , "description" : [ "str", [] ] , "unsafe" : [ "int", [] ] } , #+end_src :END: * Electron (electron group) We consider wave functions expressed in the spin-free formalism, where the number of \uparrow and \downarrow electrons is fixed. #+NAME:electron | Variable | Type | Dimensions | Description | |----------+-------+------------+-------------------------------------| | ~up_num~ | ~int~ | | Number of \uparrow-spin electrons | | ~dn_num~ | ~int~ | | Number of \downarrow-spin electrons | #+CALL: json(data=electron, title="electron") #+RESULTS: :results: #+begin_src python :tangle trex.json "electron": { "up_num" : [ "int", [] ] , "dn_num" : [ "int", [] ] } , #+end_src :end: * Nucleus (nucleus group) The nuclei are considered as fixed point charges. Coordinates are given in Cartesian $(x,y,z)$ format. #+NAME: nucleus | 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 | #+CALL: json(data=nucleus, title="nucleus") #+RESULTS: :results: #+begin_src python :tangle trex.json "nucleus": { "num" : [ "dim" , [] ] , "charge" : [ "float", [ "nucleus.num" ] ] , "coord" : [ "float", [ "nucleus.num", "3" ] ] , "label" : [ "str" , [ "nucleus.num" ] ] , "point_group" : [ "str" , [] ] , "repulsion" : [ "float", [] ] } , #+end_src :end: * 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 \[ V_A^{\text{ECP}} = V_{A \ell_{\max}+1} + \sum_{\ell=0}^{\ell_{\max}} \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[ V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} | \] 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. #+NAME: ecp | 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 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 [[https://github.com/TREX-CoE/trexio/issues][issue tracker on GitHub]]. #+CALL: json(data=ecp, title="ecp") #+RESULTS: :RESULTS: #+begin_src python :tangle trex.json "ecp": { "max_ang_mom_plus_1" : [ "int" , [ "nucleus.num" ] ] , "z_core" : [ "int" , [ "nucleus.num" ] ] , "num" : [ "dim" , [] ] , "ang_mom" : [ "int" , [ "ecp.num" ] ] , "nucleus_index" : [ "index", [ "ecp.num" ] ] , "exponent" : [ "float", [ "ecp.num" ] ] , "coefficient" : [ "float", [ "ecp.num" ] ] , "power" : [ "int" , [ "ecp.num" ] ] } , #+end_src :END: ** Example For example, consider H_2 molecule with the following [[https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess][effective core potential]] (in GAMESS input format for the H atom): #+BEGIN_EXAMPLE 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 #+END_EXAMPLE In TREXIO representation this would be: #+BEGIN_EXAMPLE 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 ] #+END_EXAMPLE * Basis set (basis group) 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: #+NAME: basis | 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 ($f_{ks}$) | #+CALL: json(data=basis, title="basis") #+RESULTS: :RESULTS: #+begin_src python :tangle trex.json "basis": { "type" : [ "str" , [] ] , "prim_num" : [ "dim" , [] ] , "shell_num" : [ "dim" , [] ] , "nucleus_index" : [ "index", [ "basis.shell_num" ] ] , "shell_ang_mom" : [ "int" , [ "basis.shell_num" ] ] , "shell_factor" : [ "float", [ "basis.shell_num" ] ] , "shell_index" : [ "index", [ "basis.prim_num" ] ] , "exponent" : [ "float", [ "basis.prim_num" ] ] , "coefficient" : [ "float", [ "basis.prim_num" ] ] , "prim_factor" : [ "float", [ "basis.prim_num" ] ] } , #+end_src :END: ** Example For example, consider H_2 with the following basis set (in GAMESS format), where both the AOs and primitives are considered normalized: #+BEGIN_EXAMPLE 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 #+END_EXAMPLE In TREXIO representaion we have: #+BEGIN_EXAMPLE 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 ] #+END_EXAMPLE * Atomic orbitals (ao group) 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 real spherical harmonics, the AOs are ordered as $0, +1, -1, +2, -2, \dots, +m, -m$ (see [[https://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics][Wikipedia]]). 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 (basis group)][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 $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$. #+NAME: ao | 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 | #+CALL: json(data=ao, title="ao") #+RESULTS: :RESULTS: #+begin_src python :tangle trex.json "ao": { "cartesian" : [ "int" , [] ] , "num" : [ "dim" , [] ] , "shell" : [ "index", [ "ao.num" ] ] , "normalization" : [ "float", [ "ao.num" ] ] } , #+end_src :END: ** One-electron integrals (~ao_1e_int~ group) :PROPERTIES: :CUSTOM_ID: ao_one_e :END: - \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}} \sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A - \mathbf{r}_i \vert} \] : 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. #+NAME: ao_1e_int | 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)~ | $\langle p \vert \hat{h} \vert q \rangle$ | #+CALL: json(data=ao_1e_int, title="ao_1e_int") #+RESULTS: :results: #+begin_src python :tangle trex.json "ao_1e_int": { "overlap" : [ "float", [ "ao.num", "ao.num" ] ] , "kinetic" : [ "float", [ "ao.num", "ao.num" ] ] , "potential_n_e" : [ "float", [ "ao.num", "ao.num" ] ] , "ecp" : [ "float", [ "ao.num", "ao.num" ] ] , "core_hamiltonian" : [ "float", [ "ao.num", "ao.num" ] ] } , #+end_src :end: ** Two-electron integrals (~ao_2e_int~ group) :PROPERTIES: :CUSTOM_ID: ao_two_e :END: 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. # TODO: Physicist / Chemist functions - \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : 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 #+NAME: ao_2e_int | 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 | #+CALL: json(data=ao_2e_int, title="ao_2e_int") #+RESULTS: :results: #+begin_src python :tangle trex.json "ao_2e_int": { "eri" : [ "float sparse", [ "ao.num", "ao.num", "ao.num", "ao.num" ] ] , "eri_lr" : [ "float sparse", [ "ao.num", "ao.num", "ao.num", "ao.num" ] ] } , #+end_src :end: * Molecular orbitals (mo group) #+NAME: mo | 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 | #+CALL: json(data=mo, title="mo") #+RESULTS: :RESULTS: #+begin_src python :tangle trex.json "mo": { "type" : [ "str" , [] ] , "num" : [ "dim" , [] ] , "coefficient" : [ "float", [ "mo.num", "ao.num" ] ] , "class" : [ "str" , [ "mo.num" ] ] , "symmetry" : [ "str" , [ "mo.num" ] ] , "occupation" : [ "float", [ "mo.num" ] ] } , #+end_src :END: ** One-electron integrals (~mo_1e_int~ group) The operators as the same as those defined in the [[#ao_one_e][AO one-electron integrals section]]. Here, the integrals are given in the basis of molecular orbitals. #+NAME: mo_1e_int | 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)~ | $\langle i \vert \hat{h} \vert j \rangle$ | #+CALL: json(data=mo_1e_int, title="mo_1e_int") #+RESULTS: :results: #+begin_src python :tangle trex.json "mo_1e_int": { "overlap" : [ "float", [ "mo.num", "mo.num" ] ] , "kinetic" : [ "float", [ "mo.num", "mo.num" ] ] , "potential_n_e" : [ "float", [ "mo.num", "mo.num" ] ] , "ecp" : [ "float", [ "mo.num", "mo.num" ] ] , "core_hamiltonian" : [ "float", [ "mo.num", "mo.num" ] ] } , #+end_src :end: ** Two-electron integrals (~mo_2e_int~ group) The operators as the same as those defined in the [[#ao_two_e][AO two-electron integrals section]]. Here, the integrals are given in the basis of molecular orbitals. #+NAME: mo_2e_int | 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 | #+CALL: json(data=mo_2e_int, title="mo_2e_int") #+RESULTS: :results: #+begin_src python :tangle trex.json "mo_2e_int": { "eri" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "eri_lr" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] } , #+end_src :end: * 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 ~mo.num~ occupation numbers. However, this requires a lot of extra memory and would be technically impossible for larger expansions (~millions of determinants). This is why the determinants are stored as bit fields in the TREXIO file (see ~int bitfield~ type in the table below). By default, the chemist notation is used, namely \[ | D_I > = | \alpha_1 \alpha_2 \ldots \alpha_{n\uparrow} \beta_1 \beta_2 \ldots \beta_{n\downarrow} > \ where $\alpha$ and $\beta$ denote $\uparrow$-spin and $\downarrow$-spin electrons, respectively, $n\uparrow$ and $n\downarrow$ correspond to ~electron.up_num~ and ~electron.dn_num~, respectively. Internally, bit fields for $\alpha$ and $\beta$ electrons are stored separately, which is why the ~determinant.list~ attribute has a second dimension. #+NAME: determinant | Variable | Type | Dimensions | Description | |---------------+----------------+------------------------+--------------------------------------------------------| | ~num~ | ~dim~ | | Number of determinants | | ~list~ | ~int bitfield~ | ~(determinant.num, 2)~ | List of determinants as integer bit fields | | ~coefficient~ | ~float~ | ~(determinant.num)~ | Coefficients of the determinants from the CI expansion | #+CALL: json(data=determinant, title="determinant") #+RESULTS: :results: #+begin_src python :tangle trex.json "determinant": { "num" : [ "dim" , [] ] , "list" : [ "int binary", [ "2", "determinant.num" ] ] , "coefficient" : [ "float" , [ "determinant.num" ] ] } , #+end_src :end: * Reduced density matrices (rdm group) The reduced density matrices are defined in the basis of molecular orbitals. The $\uparrow$-spin and $\downarrow$-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 $\uparrow \uparrow$, $\downarrow \downarrow$, $\uparrow \downarrow$, $\downarrow \uparrow$ 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 \\ \Gamma_{ijkl}^{\downarrow \uparrow} &=& \langle \Psi | \hat{a}^{\dagger}_{k\beta}\, \hat{a}^{\dagger}_{l\alpha} \hat{a}_{j\alpha}\, \hat{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: \[ E = E_{\text{NN}} + \sum_{ij} \gamma_{ij} \langle j|h|i \rangle + \frac{1}{2} \sum_{ijlk} \Gamma_{ijkl} \langle k l | i j \rangle \] #+NAME: rdm | Variable | Type | Dimensions | Description | |-----------+----------------+------------------------------------+-----------------------------------------------------------------------| | ~1e~ | ~float~ | ~(mo.num, mo.num)~ | One body density matrix | | ~1e_up~ | ~float~ | ~(mo.num, mo.num)~ | \uparrow-spin component of the one body density matrix | | ~1e_dn~ | ~float~ | ~(mo.num, mo.num)~ | \downarrow-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)~ | \uparrow\uparrow component of the two-body reduced density matrix | | ~2e_dndn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\downarrow component of the two-body reduced density matrix | | ~2e_updn~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \uparrow\downarrow component of the two-body reduced density matrix | | ~2e_dnup~ | ~float sparse~ | ~(mo.num, mo.num, mo.num, mo.num)~ | \downarrow\uparrow component of the two-body reduced density matrix | #+CALL: json(data=rdm, title="rdm", last=1) #+RESULTS: :results: #+begin_src python :tangle trex.json "rdm": { "1e" : [ "float" , [ "mo.num", "mo.num" ] ] , "1e_up" : [ "float" , [ "mo.num", "mo.num" ] ] , "1e_dn" : [ "float" , [ "mo.num", "mo.num" ] ] , "2e" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "2e_upup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "2e_dndn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "2e_updn" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] , "2e_dnup" : [ "float sparse", [ "mo.num", "mo.num", "mo.num", "mo.num" ] ] } #+end_src :end: * Appendix :noexport: ** Python script from table to json #+NAME: json #+begin_src python :var data=nucleus title="nucleus" last=0 :results output drawer 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""") #+end_src #+begin_src python :tangle trex.json :results output drawer :exports none } #+end_src