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< div id = "org-div-home-and-up" >
< a accesskey = "h" href = "" > UP < / a >
|
< a accesskey = "H" href = "index.html" > HOME < / a >
< / div > < div id = "content" >
< h1 class = "title" > TREX Configuration file< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orga6493ca" > 1. Metadata (metadata group)< / a > < / li >
< li > < a href = "#orge2ec59e" > 2. Electron (electron group)< / a > < / li >
< li > < a href = "#org56dcc50" > 3. Nucleus (nucleus group)< / a > < / li >
< li > < a href = "#org9eb1c49" > 4. Effective core potentials (ecp group)< / a >
2021-10-21 19:40:52 +02:00
< ul >
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< li > < a href = "#orga4fab01" > 4.1. Example< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org26f5cd8" > 5. Basis set (basis group)< / a >
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< ul >
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< li > < a href = "#org2248f8a" > 5.1. Example< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgbcdac54" > 6. Atomic orbitals (ao group)< / a >
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< ul >
< li > < a href = "#ao_one_e" > 6.1. One-electron integrals (< code > ao_1e_int< / code > group)< / a > < / li >
< li > < a href = "#ao_two_e" > 6.2. Two-electron integrals (< code > ao_2e_int< / code > group)< / a > < / li >
< / ul >
< / li >
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< li > < a href = "#orgbbdf51a" > 7. Molecular orbitals (mo group)< / a >
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< ul >
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< li > < a href = "#org4be9962" > 7.1. One-electron integrals (< code > mo_1e_int< / code > group)< / a > < / li >
< li > < a href = "#orgdf049fa" > 7.2. Two-electron integrals (< code > mo_2e_int< / code > group)< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc9f42fc" > 8. < span class = "todo TODO" > TODO< / span > Slater determinants< / a > < / li >
< li > < a href = "#orge085214" > 9. < span class = "todo TODO" > TODO< / span > Reduced density matrices (rdm group)< / a > < / li >
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< / ul >
< / div >
< / div >
< p >
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 < code > trex.json< / code > configuration file,
which itself is compiled from different sections (groups) presented below.
< / p >
< p >
For more information about the automatic generation on the source code
or regarding possible modifications, please contact the TREXIO developers.
< / p >
< p >
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 < code > trex.json< / code > configuration file as the library is
written in C.
< / p >
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< p >
TREXIO currently supports < code > int< / code > , < code > float< / code > and < code > str< / code > types for both single attributes and arrays.
Note, that some attributes might have < code > dim< / code > type (e.g. < code > num< / code > of the < code > nucleus< / code > group).
This type is treated exactly the same as < code > int< / code > with the only difference that < code > dim< / code > variables
cannot be negative or zero. This additional constraint is required because < code > dim< / code > attributes
are used internally to allocate memory and to check array boundaries in the memory-safe API.
Most of the times, the < code > dim< / code > variables contain < code > num< / code > suffix.
< / p >
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< p >
In Fortran, the arrays are 1-based and in most other languages the
arrays are 0-based. Hence, we introduce the < code > index< / code > type which is an
1-based < code > int< / code > in the Fortran interface and 0-based otherwise.
< / p >
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< div id = "outline-container-orga6493ca" class = "outline-2" >
< h2 id = "orga6493ca" > < span class = "section-number-2" > 1< / span > Metadata (metadata group)< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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.
< / p >
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< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions (for arrays)< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
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< td class = "org-left" > < code > code_num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
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< td class = "org-left" > < code > code< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" > < code > (metadata.code_num)< / code > < / td >
< td class = "org-left" > Names of the codes used< / td >
< / tr >
< tr >
< td class = "org-left" > < code > author_num< / code > < / td >
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< td class = "org-left" > < code > author< / code > < / td >
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< td class = "org-left" > < code > (metadata.author_num)< / code > < / td >
< td class = "org-left" > Names of the authors of the file< / td >
< / tr >
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< tr >
< td class = "org-left" > < code > package_version< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > TREXIO version used to produce the file< / td >
< / tr >
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< tr >
< td class = "org-left" > < code > description< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > Text describing the content of file< / td >
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< / tbody >
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< div id = "outline-container-orge2ec59e" class = "outline-2" >
< h2 id = "orge2ec59e" > < span class = "section-number-2" > 2< / span > Electron (electron group)< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
We consider wave functions expressed in the spin-free formalism, where
the number of ↑ and ↓ electrons is fixed.
< / p >
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< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
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< td class = "org-left" > < code > up_num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
< td class = "org-left" > Number of ↑ -spin electrons< / td >
< / tr >
< tr >
< td class = "org-left" > < code > dn_num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
< td class = "org-left" > Number of ↓ -spin electrons< / td >
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< h2 id = "org56dcc50" > < span class = "section-number-2" > 3< / span > Nucleus (nucleus group)< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
The nuclei are considered as fixed point charges. Coordinates are
given in Cartesian \((x,y,z)\) format.
< / p >
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< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
< td class = "org-left" > Number of nuclei< / td >
< / tr >
< tr >
< td class = "org-left" > < code > charge< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (nucleus.num)< / code > < / td >
< td class = "org-left" > Charges of the nuclei< / td >
< / tr >
< tr >
< td class = "org-left" > < code > coord< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (3,nucleus.num)< / code > < / td >
< td class = "org-left" > Coordinates of the atoms< / td >
< / tr >
< tr >
< td class = "org-left" > < code > label< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" > < code > (nucleus.num)< / code > < / td >
< td class = "org-left" > Atom labels< / td >
< / tr >
< tr >
< td class = "org-left" > < code > point_group< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > Symmetry point group< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org9eb1c49" class = "outline-2" >
< h2 id = "org9eb1c49" > < span class = "section-number-2" > 4< / span > Effective core potentials (ecp group)< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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An effective core potential (ECP) \(V_A^{\text{ECP}}\) replacing the
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core electrons of atom \(A\) can be expressed as
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\[
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V_A^{\text{ECP}} =
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V_{A \ell_{\max}+1} +
\sum_{\ell=0}^{\ell_{\max}}
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\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
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V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} |
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\]
< / p >
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< p >
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.
< / p >
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< p >
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The functions \(V_{A\ell}\) are parameterized as:
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\[
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V_{A \ell}(\mathbf{r}) =
\sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
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e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
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\]
< / p >
< p >
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See < a href = "http://dx.doi.org/10.1063/1.4984046" > http://dx.doi.org/10.1063/1.4984046< / a > or < a href = "https://doi.org/10.1063/1.5121006" > https://doi.org/10.1063/1.5121006< / a > for more info.
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< / p >
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< table id = "orga649329" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
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< td class = "org-left" > < code > max_ang_mom_plus_1< / code > < / td >
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< td class = "org-left" > < code > int< / code > < / td >
< td class = "org-left" > < code > (nucleus.num)< / code > < / td >
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< td class = "org-left" > \(\ell_{\max}+1\), one higher than the max angular momentum in the removed core orbitals< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > z_core< / code > < / td >
< td class = "org-left" > < code > int< / code > < / td >
< td class = "org-left" > < code > (nucleus.num)< / code > < / td >
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< td class = "org-left" > Number of core electrons to remove per atom< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
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< td class = "org-left" > Total number of ECP functions for all atoms and all values of \(\ell\)< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > ang_mom< / code > < / td >
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< td class = "org-left" > < code > int< / code > < / td >
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< td class = "org-left" > < code > (ecp.num)< / code > < / td >
< td class = "org-left" > One-to-one correspondence between ECP items and the angular momentum \(\ell\)< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > nucleus_index< / code > < / td >
< td class = "org-left" > < code > index< / code > < / td >
< td class = "org-left" > < code > (ecp.num)< / code > < / td >
< td class = "org-left" > One-to-one correspondence between ECP items and the atom index< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > exponent< / code > < / td >
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< td class = "org-left" > < code > float< / code > < / td >
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< td class = "org-left" > < code > (ecp.num)< / code > < / td >
< td class = "org-left" > \(\alpha_{A q \ell}\) all ECP exponents< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > coefficient< / code > < / td >
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< td class = "org-left" > < code > float< / code > < / td >
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< td class = "org-left" > < code > (ecp.num)< / code > < / td >
< td class = "org-left" > \(\beta_{A q \ell}\) all ECP coefficients< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > power< / code > < / td >
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< td class = "org-left" > < code > int< / code > < / td >
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< td class = "org-left" > < code > (ecp.num)< / code > < / td >
< td class = "org-left" > \(n_{A q \ell}\) all ECP powers< / td >
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< / tr >
< / tbody >
< / table >
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< p >
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.
< b > Note< / b > , that the latter \(\ell_{\max}\) is always higher by 1 than the former.
< / p >
< p >
< b > Note for developers< / b > : avoid having variables with similar prefix in their name.
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HDF5 back end might cause issues due to the way < code > find_dataset< / code > function works.
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For example, in the ECP group we use < code > max_ang_mom< / code > and not < code > ang_mom_max< / code > .
The latter causes issues when written before < code > ang_mom< / code > in the TREXIO file.
< / p >
< / div >
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< div id = "outline-container-orga4fab01" class = "outline-3" >
< h3 id = "orga4fab01" > < span class = "section-number-3" > 4.1< / span > Example< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
For example, consider H< sub > 2< / sub > molecule with the following
< a href = "https://pseudopotentiallibrary.org/recipes/H/ccECP/H.ccECP.gamess" > effective core potential< / a >
(in GAMESS input format for the H atom):
< / p >
< pre class = "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
< / pre >
< p >
In TREXIO representation this would be:
< / p >
< pre class = "example" >
num = 8
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# lmax+1 per atom: 0 here because no core electrons to be removed
max_ang_mom_plus_1 = [ 0, 0 ]
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# 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
]
< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org26f5cd8" class = "outline-2" >
< h2 id = "org26f5cd8" > < span class = "section-number-2" > 5< / span > Basis set (basis group)< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< p >
We consider here basis functions centered on nuclei. Hence, we enable
the possibility to define < i > dummy atoms< / i > to place basis functions in
random positions.
< / p >
< p >
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
\(N_{\text{prim}}\) < i > primitive< / i > functions that can be of type
Slater (\(p=1\)) or Gaussian (\(p=2\)),
parameterized by exponents \(\gamma_{ks}\) and coefficients \(a_{ks}\):
\[
R_s(\mathbf{r}) = \mathcal{N}_s \vert\mathbf{r}-\mathbf{R}_A\vert^{n_s}
\sum_{k=1}^{N_{\text{prim}}} a_{ks}\, f_{ks}(\gamma_{ks},p)\,
\exp \left( - \gamma_{ks}
\vert \mathbf{r}-\mathbf{R}_A \vert ^p \right).
\]
< / p >
< p >
In the case of Gaussian functions, \(n_s\) is always zero.
< / p >
< p >
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.
< / p >
< p >
Some codes assume that the contraction coefficients are for a linear
combination of < i > normalized< / i > primitives. This implies that a normalization
constant for the primitive \(ks\) needs to be computed and stored. If
this normalization factor is not required, \(f_{ks}=1\).
< / p >
< p >
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, \(\mathcal{N}_s=1\).
< / p >
< p >
All the basis set parameters are stored in one-dimensional arrays:
< / p >
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< table id = "orgcbe6c25" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > type< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > Type of basis set: "Gaussian" or "Slater"< / td >
< / tr >
< tr >
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< td class = "org-left" > < code > prim_num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
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< td class = "org-left" > Total number of primitives< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > shell_num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
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< td class = "org-left" > Total number of shells< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > nucleus_index< / code > < / td >
< td class = "org-left" > < code > index< / code > < / td >
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< td class = "org-left" > < code > (basis.shell_num)< / code > < / td >
< td class = "org-left" > One-to-one correspondence between shells and atomic indices< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > shell_ang_mom< / code > < / td >
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< td class = "org-left" > < code > int< / code > < / td >
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< td class = "org-left" > < code > (basis.shell_num)< / code > < / td >
< td class = "org-left" > One-to-one correspondence between shells and angular momenta< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > shell_factor< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
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< td class = "org-left" > < code > (basis.shell_num)< / code > < / td >
< td class = "org-left" > Normalization factor of each shell (\(\mathcal{N}_s\))< / td >
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< / tr >
< tr >
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< td class = "org-left" > < code > shell_index< / code > < / td >
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< td class = "org-left" > < code > index< / code > < / td >
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< td class = "org-left" > < code > (basis.prim_num)< / code > < / td >
< td class = "org-left" > One-to-one correspondence between primitives and shell index< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > exponent< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (basis.prim_num)< / code > < / td >
< td class = "org-left" > Exponents of the primitives (\(\gamma_{ks}\))< / td >
< / tr >
< tr >
< td class = "org-left" > < code > coefficient< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (basis.prim_num)< / code > < / td >
< td class = "org-left" > Coefficients of the primitives (\(a_{ks}\))< / td >
< / tr >
< tr >
< td class = "org-left" > < code > prim_factor< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (basis.prim_num)< / code > < / td >
< td class = "org-left" > Normalization coefficients for the primitives (\(f_{ks}\))< / td >
< / tr >
< / tbody >
< / table >
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< / div >
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2021-10-25 17:00:57 +02:00
< div id = "outline-container-org2248f8a" class = "outline-3" >
< h3 id = "org2248f8a" > < span class = "section-number-3" > 5.1< / span > Example< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
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< p >
For example, consider H< sub > 2< / sub > with the following basis set (in GAMESS
format), where both the AOs and primitives are considered normalized:
< / p >
< pre class = "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
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1 1.057000E+00 1.000000E+00
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< / pre >
< p >
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In TREXIO representaion we have:
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< / p >
< pre class = "example" >
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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)
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shell_ang_mom =
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[ 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)
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exponent =
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[ 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 ]
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coefficient =
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[ 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 ]
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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 ]
< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgbcdac54" class = "outline-2" >
< h2 id = "orgbcdac54" > < span class = "section-number-2" > 6< / span > Atomic orbitals (ao group)< / h2 >
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< div class = "outline-text-2" id = "text-6" >
< p >
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
< / p >
\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}
< p >
AOs are defined as
< / p >
< p >
\[
\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r})
\]
< / p >
< p >
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
\[ \left[ \iiint \left(x-X_A \right)^2 R_{\theta(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne
\left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{\theta(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
< / p >
< p >
In such a case, one should set the normalization of the shell (in
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the < a href = "#org26f5cd8" > Basis set< / a > section) to \(\mathcal{N}_{z^2}\), which is the
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normalization factor of the atomic orbitals in spherical coordinates.
The normalization factor of the \(xy\) function which should be
introduced here should be \(\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}\).
< / p >
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< table id = "orgc58a3e5" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > cartesian< / code > < / td >
< td class = "org-left" > < code > int< / code > < / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > < code > 1< / code > : true, < code > 0< / code > : false< / td >
< / tr >
< tr >
< td class = "org-left" > < code > num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
< td class = "org-left" > Total number of atomic orbitals< / td >
< / tr >
< tr >
< td class = "org-left" > < code > shell< / code > < / td >
< td class = "org-left" > < code > index< / code > < / td >
< td class = "org-left" > < code > (ao.num)< / code > < / td >
< td class = "org-left" > basis set shell for each AO< / td >
< / tr >
< tr >
< td class = "org-left" > < code > normalization< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num)< / code > < / td >
< td class = "org-left" > Normalization factors< / td >
< / tr >
< / tbody >
< / table >
< / div >
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< div id = "outline-container-org7d5fb99" class = "outline-3" >
< h3 id = "ao_one_e" > < a id = "org7d5fb99" > < / a > < span class = "section-number-3" > 6.1< / span > One-electron integrals (< code > ao_1e_int< / code > group)< / h3 >
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< div class = "outline-text-3" id = "text-ao_one_e" >
< ul class = "org-ul" >
< li > \[ \hat{V}_{\text{ne}} = \sum_{A=1}^{N_\text{nucl}}
\sum_{i=1}^{N_\text{elec}} \frac{-Z_A }{\vert \mathbf{R}_A -
\mathbf{r}_i \vert} \] : electron-nucleus attractive potential,< / li >
< li > \[ \hat{T}_{\text{e}} =
\sum_{i=1}^{N_\text{elec}} -\frac{1}{2}\hat{\Delta}_i \] : electronic kinetic energy< / li >
< li > \(\hat{h} = \hat{T}_\text{e} + \hat{V}_\text{ne} +
\hat{V}_\text{ecp,l} + \hat{V}_\text{ecp,nl}\) : core electronic Hamiltonian< / li >
< / ul >
< p >
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.
< / p >
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< table id = "org91ae93c" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > overlap< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert q \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > kinetic< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert \hat{T}_e \vert q \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > potential_n_e< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > ecp_local< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert \hat{V}_{\text{ecp,l}} \vert q \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > ecp_non_local< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert \hat{V}_{\text{ecp,nl}} \vert q \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > core_hamiltonian< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num)< / code > < / td >
< td class = "org-left" > \(\langle p \vert \hat{h} \vert q \rangle\)< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org1d2373e" class = "outline-3" >
< h3 id = "ao_two_e" > < a id = "org1d2373e" > < / a > < span class = "section-number-3" > 6.2< / span > Two-electron integrals (< code > ao_2e_int< / code > group)< / h3 >
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< div class = "outline-text-3" id = "text-ao_two_e" >
< p >
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.
< / p >
< p >
Functions are provided to get the indices in physicists or chemists
notation.
< / p >
< ul class = "org-ul" >
< li > \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.< / li >
< li > \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}}
\sum_{j=1}^{i-1} \frac{\text{erf}(\vert \mathbf{r}_i -
\mathbf{r}_j \vert)}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron long range potential< / li >
< / ul >
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< table id = "orgbc3d359" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > eri< / code > < / td >
< td class = "org-left" > < code > float sparse< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num, ao.num, ao.num)< / code > < / td >
< td class = "org-left" > Electron repulsion integrals< / td >
< / tr >
< tr >
< td class = "org-left" > < code > eri_lr< / code > < / td >
< td class = "org-left" > < code > float sparse< / code > < / td >
< td class = "org-left" > < code > (ao.num, ao.num, ao.num, ao.num)< / code > < / td >
< td class = "org-left" > Long-range Electron repulsion integrals< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbbdf51a" class = "outline-2" >
< h2 id = "orgbbdf51a" > < span class = "section-number-2" > 7< / span > Molecular orbitals (mo group)< / h2 >
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< div class = "outline-text-2" id = "text-7" >
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< table id = "orgdc6666c" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > type< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" >   < / td >
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< td class = "org-left" > Free text to identify the set of MOs (HF, Natural, Local, CASSCF, < i > etc< / i > )< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > num< / code > < / td >
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< td class = "org-left" > < code > dim< / code > < / td >
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< td class = "org-left" >   < / td >
< td class = "org-left" > Number of MOs< / td >
< / tr >
< tr >
< td class = "org-left" > < code > coefficient< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (ao.num, mo.num)< / code > < / td >
< td class = "org-left" > MO coefficients< / td >
< / tr >
< tr >
< td class = "org-left" > < code > class< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" > < code > (mo.num)< / code > < / td >
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< td class = "org-left" > Choose among: Core, Inactive, Active, Virtual, Deleted< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > symmetry< / code > < / td >
< td class = "org-left" > < code > str< / code > < / td >
< td class = "org-left" > < code > (mo.num)< / code > < / td >
< td class = "org-left" > Symmetry in the point group< / td >
< / tr >
< tr >
< td class = "org-left" > < code > occupation< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num)< / code > < / td >
< td class = "org-left" > Occupation number< / td >
< / tr >
< / tbody >
< / table >
< / div >
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< div id = "outline-container-org4be9962" class = "outline-3" >
< h3 id = "org4be9962" > < span class = "section-number-3" > 7.1< / span > One-electron integrals (< code > mo_1e_int< / code > group)< / h3 >
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< div class = "outline-text-3" id = "text-7-1" >
< p >
The operators as the same as those defined in the
< a href = "#ao_one_e" > AO one-electron integrals section< / a > . Here, the integrals are given in
the basis of molecular orbitals.
< / p >
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< table id = "org02a49a0" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > overlap< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert j \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > kinetic< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert \hat{T}_e \vert j \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > potential_n_e< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert \hat{V}_{\text{ne}} \vert j \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > ecp_local< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert \hat{V}_{\text{ecp,l}} \vert j \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > ecp_non_local< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert \hat{V}_{\text{ecp,nl}} \vert j \rangle\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > core_hamiltonian< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" > \(\langle i \vert \hat{h} \vert j \rangle\)< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-orgdf049fa" class = "outline-3" >
< h3 id = "orgdf049fa" > < span class = "section-number-3" > 7.2< / span > Two-electron integrals (< code > mo_2e_int< / code > group)< / h3 >
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< div class = "outline-text-3" id = "text-7-2" >
< p >
The operators as the same as those defined in the
< a href = "#ao_two_e" > AO two-electron integrals section< / a > . Here, the integrals are given in
the basis of molecular orbitals.
< / p >
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< table id = "org86aba12" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > eri< / code > < / td >
< td class = "org-left" > < code > float sparse< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num, mo.num, mo.num)< / code > < / td >
< td class = "org-left" > Electron repulsion integrals< / td >
< / tr >
< tr >
< td class = "org-left" > < code > eri_lr< / code > < / td >
< td class = "org-left" > < code > float sparse< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num, mo.num, mo.num)< / code > < / td >
< td class = "org-left" > Long-range Electron repulsion integrals< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc9f42fc" class = "outline-2" >
< h2 id = "orgc9f42fc" > < span class = "section-number-2" > 8< / span > < span class = "todo TODO" > TODO< / span > Slater determinants< / h2 >
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< / div >
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< div id = "outline-container-orge085214" class = "outline-2" >
< h2 id = "orge085214" > < span class = "section-number-2" > 9< / span > < span class = "todo TODO" > TODO< / span > Reduced density matrices (rdm group)< / h2 >
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< div class = "outline-text-2" id = "text-9" >
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< table id = "org6503f17" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Variable< / th >
< th scope = "col" class = "org-left" > Type< / th >
< th scope = "col" class = "org-left" > Dimensions< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < code > one_e< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" >   < / td >
< / tr >
< tr >
< td class = "org-left" > < code > one_e_up< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" >   < / td >
< / tr >
< tr >
< td class = "org-left" > < code > one_e_dn< / code > < / td >
< td class = "org-left" > < code > float< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num)< / code > < / td >
< td class = "org-left" >   < / td >
< / tr >
< tr >
< td class = "org-left" > < code > two_e< / code > < / td >
< td class = "org-left" > < code > float sparse< / code > < / td >
< td class = "org-left" > < code > (mo.num, mo.num, mo.num, mo.num)< / code > < / td >
< td class = "org-left" >   < / td >
< / tr >
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< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: TREX-CoE< / p >
2021-10-25 17:00:57 +02:00
< p class = "date" > Created: 2021-10-25 Mon 15:00< / p >
2021-06-25 14:16:48 +02:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
