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More documentation in TREX JSON
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@ -1,6 +1,11 @@
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#+TITLE: TREX Configuration file
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All the quantities are saved in atomic units.
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The dimensions of the arrays in the tables below are given in
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column-major order (as in Fortran), and the ordering of the dimensions
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is reversed in the produces JSON configuration file as the library is
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written in C.
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#+begin_src python :tangle trex.json
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{
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#+end_src
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@ -8,9 +13,15 @@ All the quantities are saved in atomic units.
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* Metadata
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As we expect our files to be archived in open-data repositories, we
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need to give the possibility to the users to store some metadata
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inside the files. We propose to store the list of names of the codes
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which have participated to the creation of the file, a list of
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authors of the file, and a textual description.
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#+NAME: metadata
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| ~code_num~ | ~int~ | | Number of codes used to produce the file |
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| ~code ~ | ~str~ | ~(metadata.code_num)~ | Names of the codes used |
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| ~code~ | ~str~ | ~(metadata.code_num)~ | Names of the codes used |
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| ~author_num~ | ~int~ | | Number of authors of the file |
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| ~author~ | ~str~ | ~(metadata.author_num)~ | Names of the authors of the file |
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| ~description~ | ~str~ | | Text describing the content of file |
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@ -19,21 +30,24 @@ All the quantities are saved in atomic units.
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#+RESULTS:
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:results:
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#+begin_src python :tangle trex.json
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"metadata": {
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"code_num" : [ "int", [] ]
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, "code " : [ "str", [ "metadata.code_num" ] ]
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, "author_num" : [ "int", [] ]
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, "author" : [ "str", [ "metadata.author_num" ] ]
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, "description" : [ "str", [] ]
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} ,
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"metadata": {
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"code_num" : [ "int", [] ]
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, "code " : [ "str", [ "metadata.code_num" ] ]
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, "author_num" : [ "int", [] ]
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, "author" : [ "str", [ "metadata.author_num" ] ]
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, "description" : [ "str", [] ]
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} ,
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#+end_src
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:end:
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* Electron
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We consider wave functions expressed in the spin-free formalism, where
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the number of \uparrow and \downarrow electrons is fixed.
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#+NAME:electron
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| ~up_num~ | ~int~ | Number of $\uparrow$-spin electrons |
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| ~dn_num~ | ~int~ | Number of $\downarrow$-spin electrons |
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| ~up_num~ | ~int~ | Number of \uparrow-spin electrons |
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| ~dn_num~ | ~int~ | Number of \downarrow-spin electrons |
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#+CALL: json(data=electron, title="electron")
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#+RESULTS:
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@ -48,12 +62,15 @@ All the quantities are saved in atomic units.
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* Nucleus
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The nuclei are considered as fixed point charges. Coordinates are
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given in Cartesian $(x,y,z)$ format.
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#+NAME: nucleus
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| ~num~ | ~int~ | | Number of nuclei |
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| ~charge~ | ~float~ | ~(nucleus.num)~ | Charges of the nuclei |
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| ~coord~ | ~float~ | ~(nucleus.num, 3)~ | Coordinates of the atoms |
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| ~label~ | ~str~ | ~(nucleus.num)~ | Atom labels |
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| ~point_group~ | ~str~ | | Symmetry point group |
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| ~num~ | ~int~ | | Number of nuclei |
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| ~charge~ | ~float~ | ~(nucleus.num)~ | Charges of the nuclei |
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| ~coord~ | ~float~ | ~(3,nucleus.num)~ | Coordinates of the atoms |
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| ~label~ | ~str~ | ~(nucleus.num)~ | Atom labels |
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| ~point_group~ | ~str~ | | Symmetry point group |
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#+CALL: json(data=nucleus, title="nucleus")
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#+RESULTS:
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@ -71,47 +88,101 @@ All the quantities are saved in atomic units.
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* TODO Effective core potentials
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- $\hat{V}_\text{ecp,l} =$ : local component
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- $\hat{V}_\text{ecp,nl} =$ : non-local component
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An effective core potential (ECP) $V_A^{\text{pp}}$ replacing the
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core electrons of atom $A$ is the sum of a local component
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$V_A^{\text{l}}$ and a non-local component $V_A^{\text{nl}}$.
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The local component is given by
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\[
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\hat{V}_A^{\text{l}}(r) = -\frac{Z_A^{\text{eff}}}{r} +
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\frac{Z_A^{\text{eff}}}{r}\, \exp\left( -\alpha_A\, r^2\right) +
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Z_{\text{eff}}\, \alpha_A\, r\, \exp\left( -\beta_A\, r^2\right) +
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\gamma_A \exp\left( -\delta_A\, r^2\right),
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\]
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and the component obtained after localizing the non-local operator is
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\[
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\hat{V}_A^{\text{nl}}(r) =
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\zeta_A\, \exp\left( -\eta_A\, r^2\right) |0\rangle \langle 0| +
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\mu_A \, \exp\left( -\nu_A \, r^2\right) |1\rangle \langle 1|
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\]
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where $r=|\mathbf{r-R}_A|$ is the distance to the nucleus on which the
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potential is centered, $Z_A^{\text{eff}}$ is the effective charge
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due to the removed electrons, $|0\rangle \langle 0|$
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and $|1\rangle \langle 1|$ are projections over zero and one principal angular
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momenta, respectively (generalization to higher angular momenta is
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straightforward), and all the parameters labeled by Greek
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letters are parameters.
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- $\hat{V}_\text{ecp,l} = \sum_A \hat{V}_A^{\text{l}}$ : local component
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- $\hat{V}_\text{ecp,nl} = \sum_A \hat{V}_A^{\text{nl}}$ : non-local component
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#+NAME: ecp
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| ~lmax_plus_1~ | ~int~ | ~(nucleus.num)~ | $l_{\max} + 1$ |
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| ~z_core~ | ~float~ | ~(nucleus.num)~ | Charges to remove |
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| ~local_n~ | ~int~ | ~(nucleus.num)~ | Number of local function |
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| ~local_num_n_max~ | ~int~ | | Maximum value of ~local_n~ |
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| ~local_exponent~ | ~float~ | ~(nucleus.num, ecp.local_num_n_max)~ | |
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| ~local_coef~ | ~float~ | ~(nucleus.num, ecp.local_num_n_max)~ | |
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| ~local_power~ | ~int~ | ~(nucleus.num, ecp.local_num_n_max)~ | |
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| ~local_exponent~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~local_coef~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~local_power~ | ~int~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~non_local_n~ | ~int~ | ~(nucleus.num)~ | |
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| ~non_local_num_n_max~ | ~int~ | | |
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| ~non_local_exponent~ | ~float~ | ~(nucleus.num, ecp.non_local_num_n_max)~ | |
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| ~non_local_coef~ | ~float~ | ~(nucleus.num, ecp.non_local_num_n_max)~ | |
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| ~non_local_power~ | ~int~ | ~(nucleus.num, ecp.non_local_num_n_max)~ | |
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| ~non_local_exponent~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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| ~non_local_coef~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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| ~non_local_power~ | ~int~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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#+CALL: json(data=ecp, title="ecp")
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#+RESULTS:
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:results:
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#+begin_src python :tangle trex.json
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"ecp": {
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"lmax_plus_1" : [ "int" , [ "nucleus.num" ] ]
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, "z_core" : [ "float", [ "nucleus.num" ] ]
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, "local_n" : [ "int" , [ "nucleus.num" ] ]
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, "local_num_n_max" : [ "int" , [] ]
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, "local_exponent" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "local_coef" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "local_power" : [ "int" , [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "non_local_n" : [ "int" , [ "nucleus.num" ] ]
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, "non_local_num_n_max" : [ "int" , [] ]
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, "non_local_exponent" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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, "non_local_coef" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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, "non_local_power" : [ "int" , [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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} ,
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"ecp": {
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"lmax_plus_1" : [ "int" , [ "nucleus.num" ] ]
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, "z_core" : [ "float", [ "nucleus.num" ] ]
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, "local_n" : [ "int" , [ "nucleus.num" ] ]
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, "local_num_n_max" : [ "int" , [] ]
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, "local_exponent" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "local_coef" : [ "float", [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "local_power" : [ "int" , [ "nucleus.num", "ecp.local_num_n_max" ] ]
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, "non_local_n" : [ "int" , [ "nucleus.num" ] ]
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, "non_local_num_n_max" : [ "int" , [] ]
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, "non_local_exponent" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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, "non_local_coef" : [ "float", [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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, "non_local_power" : [ "int" , [ "nucleus.num", "ecp.non_local_num_n_max" ] ]
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} ,
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#+end_src
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:end:
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* Basis set
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We consider here basis functions centered on nuclei. Hence, we enable
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the possibility to define \emph{dummy atoms} to place basis functions
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in random positions.
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The atomic basis set is defined as a list of shells. Each shell $s$ is
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centered on a center $A$, possesses a given angular momentum $l$ and a
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radial function $R_s$. The radial function is a linear combination of
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$N_{\text{prim}}$ /primitive/ functions that can be of type
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Slater ($p=1$) or Gaussian ($p=2$),
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parameterized by exponents $\gamma_{ks}$ and coefficients $a_{ks}$:
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\[
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R_s(\mathbf{r}) = \mathcal{N}_s \vert\mathbf{r}-\mathbf{R}_A\vert^{n_s}
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\sum_{k=1}^{N_{\text{prim}}} a_{ks}
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\exp \left( - \gamma_{ks}
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\vert \mathbf{r}-\mathbf{R}_A \vert ^p \right).
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\]
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In the case of Gaussian functions, $n_s$ is always zero. The
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normalization factor $\mathcal{N}_s$ ensures that all the functions
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of the shell are normalized to unity. As this normalization requires
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the ability to compute overlap integrals, the normalization factors
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should be written in the file to ensure that the file is
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self-contained and does not require the client program to have the
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ability to compute such integrals.
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#+NAME: basis
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| ~type~ | ~str~ | | Type of basis set: "Gaussian" or "Slater" |
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| ~shell_num~ | ~int~ | | Total Number of shells |
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@ -146,96 +217,138 @@ All the quantities are saved in atomic units.
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* Atomic orbitals
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Going from the atomic basis set to AOs implies a systematic
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construction of all the angular functions of each shell. We
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consider two cases for the angular functions: the real-valued
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spherical harmonics, and the polynomials in Cartesian coordinates.
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In the case of spherical harmonics, the AOs are ordered in
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increasing magnetic quantum number ($-l \le m \le l$), and in the case
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of polynomials we impose the canonical ordering of the
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Libint2 library, i.e
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\begin{eqnarray}
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p & : & p_x, p_y, p_z \nonumber \\
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d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\
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f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\
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{\rm etc.} \nonumber
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\end{eqnarray}
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AOs are defined as
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\[
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\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r})
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\]
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where $i$ is the atomic orbital index,
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$P$ encodes for either the
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polynomials or the spherical harmonics, $\theta(i)$ returns the
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shell on which the AO is expanded, and $\eta(i)$ denotes which
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angular function is chosen.
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$\mathcal{N}_i$ is a normalization factor that enables the
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possibility to have different normalization coefficients within a
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shell, as in the GAMESS convention where
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$\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}$ because
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\[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)}
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(\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne
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\left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)}
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(\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
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In such a case, one should set the normalization of the shell (in
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the [[Basis set][Basis set]] section) to $\mathcal{N}_{z^2}$, which is the
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normalization factor of the atomic orbitals in spherical coordinates.
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The normalization factor of the $xy$ function which should be
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introduced here should be $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
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#+NAME: ao
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| ~num~ | ~int~ | | Number of atomic orbitals |
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| ~cartesian~ | ~int~ | | ~1~: true, ~0~: false |
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| ~shell~ | ~int~ | ~ao.num~ | ID of the basis set shell |
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| ~normalization~ | ~float~ | ~ao.num~ | Normalization factor of the AOs |
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| ~num~ | ~int~ | | Total number of atomic orbitals |
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| ~shell~ | ~int~ | ~ao.num~ | basis set shell for each AO |
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| ~normalization~ | ~float~ | ~ao.num~ | Normalization factors |
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#+CALL: json(data=ao, title="ao")
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#+RESULTS:
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:results:
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#+begin_src python :tangle trex.json
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"ao": {
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"num" : [ "int" , [] ]
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, "cartesian" : [ "int" , [] ]
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, "shell" : [ "int" , [] ]
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, "normalization" : [ "float", [] ]
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} ,
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"ao": {
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"num" : [ "int" , [] ]
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, "cartesian" : [ "int" , [] ]
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, "shell" : [ "int" , [] ]
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, "normalization" : [ "float", [] ]
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} ,
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#+end_src
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:end:
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** One-electron integrals
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- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}}
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\sum_{i=1}^{N_\text{elec}} \frac{-Q_A }{\vert \mathbf{R}_A -
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\mathbf{r}_i \vert} \] : electron-nucleus attractive potential,
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- \[ \hat{T}_{\text{e}} =
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\sum_{i=1}^{N_\text{elec}} -\frac{1}{2}\hat{\Delta}_i \] : electronic kinetic energy
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- $\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} +
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\hat{V}_\text{ecp,l} + \hat{V}_\text{ecp,nl}$ : core electronic Hamiltonian
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- \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}}
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\sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A -
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\mathbf{r}_i \vert} \] : electron-nucleus attractive potential,
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- \[ \hat{T}_{\text{e}} =
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\sum_{i=1}^{N_\text{elec}} -\frac{1}{2}\hat{\Delta}_i \] : electronic kinetic energy
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- $\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} +
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\hat{V}_\text{ecp,l} + \hat{V}_\text{ecp,nl}$ : core electronic Hamiltonian
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The one-electron integrals for a one-electron operator $\hat{O}$ are
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\[ \langle p \vert \hat{O} \vert q \rangle \], returned as a matrix
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over atomic orbitals.
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The one-electron integrals for a one-electron operator $\hat{O}$ are
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\[ \langle p \vert \hat{O} \vert q \rangle \], returned as a matrix
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over atomic orbitals.
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#+NAME: ao_1e_int
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| ~overlap~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ |
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| ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ |
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| ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ |
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| ~ecp_local~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,l} \vert q \rangle$ |
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| ~ecp_non_local~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,nl} \vert q \rangle$ |
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| ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ |
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#+NAME: ao_1e_int
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| ~overlap~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ |
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| ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ |
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| ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ |
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| ~ecp_local~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,l} \vert q \rangle$ |
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| ~ecp_non_local~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp,nl} \vert q \rangle$ |
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| ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ |
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#+CALL: json(data=ao_1e_int, title="ao_1e_int")
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#+CALL: json(data=ao_1e_int, title="ao_1e_int")
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#+RESULTS:
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:results:
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#+begin_src python :tangle trex.json
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"ao_1e_int": {
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"overlap" : [ "float", [ "ao.num", "ao.num" ] ]
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, "kinetic" : [ "float", [ "ao.num", "ao.num" ] ]
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, "potential_n_e" : [ "float", [ "ao.num", "ao.num" ] ]
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, "ecp_local" : [ "float", [ "ao.num", "ao.num" ] ]
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, "ecp_non_local" : [ "float", [ "ao.num", "ao.num" ] ]
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, "core_hamiltonian" : [ "float", [ "ao.num", "ao.num" ] ]
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} ,
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#+end_src
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:end:
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#+RESULTS:
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:results:
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#+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_local" : [ "float", [ "ao.num", "ao.num" ] ]
|
||||
, "ecp_non_local" : [ "float", [ "ao.num", "ao.num" ] ]
|
||||
, "core_hamiltonian" : [ "float", [ "ao.num", "ao.num" ] ]
|
||||
} ,
|
||||
#+end_src
|
||||
:end:
|
||||
|
||||
** 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.
|
||||
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.
|
||||
Functions are provided to get the indices in physicists or chemists
|
||||
notation.
|
||||
|
||||
# TODO: Physicist / Chemist functions
|
||||
# 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
|
||||
- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : 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
|
||||
| ~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 |
|
||||
#+NAME: ao_2e_int
|
||||
| ~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")
|
||||
#+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:
|
||||
#+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
|
||||
|
||||
@ -371,6 +484,7 @@ for line in data:
|
||||
if '(' in dims:
|
||||
dims = dims.strip()[1:-1]
|
||||
dims = [ '"'+x.strip()+'"' for x in dims.split(',') ]
|
||||
dims.reverse()
|
||||
dims = "[ " + ", ".join(dims) + " ]"
|
||||
else:
|
||||
dims = "[]"
|
||||
|
||||
Loading…
Reference in New Issue
Block a user