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1533 lines
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HTML
1533 lines
49 KiB
HTML
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without even the implied warranty of MERCHANTABILITY or FITNESS
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may distribute non-source (e.g., minimized or compacted) forms of
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</head>
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href=""> UP </a>
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">TREX Configuration file</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org08b10a1">1. Metadata (metadata group)</a></li>
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<li><a href="#org1af947a">2. Electron (electron group)</a></li>
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<li><a href="#org5075112">3. Nucleus (nucleus group)</a></li>
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<li><a href="#orga1079c0">4. Effective core potentials (ecp group)</a>
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<ul>
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<li><a href="#org373a100">4.1. Example</a></li>
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</ul>
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</li>
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<li><a href="#orga70d288">5. Basis set (basis group)</a>
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<ul>
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<li><a href="#org0334663">5.1. Example</a></li>
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</ul>
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</li>
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<li><a href="#org84800d2">6. Atomic orbitals (ao group)</a>
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<ul>
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<li><a href="#ao_one_e">6.1. One-electron integrals (<code>ao_1e_int</code> group)</a></li>
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<li><a href="#ao_two_e">6.2. Two-electron integrals (<code>ao_2e_int</code> group)</a></li>
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</ul>
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</li>
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<li><a href="#org633bc24">7. Molecular orbitals (mo group)</a>
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<ul>
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<li><a href="#org297b409">7.1. One-electron integrals (<code>mo_1e_int</code> group)</a></li>
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<li><a href="#orgf1768cf">7.2. Two-electron integrals (<code>mo_2e_int</code> group)</a></li>
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</ul>
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</li>
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<li><a href="#orgdd0c0bb">8. <span class="todo TODO">TODO</span> Slater determinants</a></li>
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<li><a href="#org396b22a">9. <span class="todo TODO">TODO</span> Reduced density matrices (rdm group)</a></li>
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</ul>
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</div>
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</div>
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<p>
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This page contains information about the general structure of the
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TREXIO library. The source code of the library can be automatically
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generated based on the contents of the <code>trex.json</code> configuration file,
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which itself is compiled from different sections (groups) presented below.
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</p>
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<p>
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For more information about the automatic generation on the source code
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or regarding possible modifications, please contact the TREXIO developers.
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</p>
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<p>
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All quantities are saved in TREXIO file 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 produced <code>trex.json</code> configuration file as the library is
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written in C.
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</p>
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<p>
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TREXIO currently supports <code>int</code>, <code>float</code> and <code>str</code> types for both single attributes and arrays.
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Note, that some attributes might have <code>dim</code> type (e.g. <code>num</code> of the <code>nucleus</code> group).
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This type is treated exactly the same as <code>int</code> with the only difference that <code>dim</code> variables
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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.
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Most of the times, the <code>dim</code> variables contain <code>num</code> suffix.
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</p>
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<p>
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|
In Fortran, the arrays are 1-based and in most other languages the
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arrays are 0-based. Hence, we introduce the <code>index</code> type which is an
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1-based <code>int</code> in the Fortran interface and 0-based otherwise.
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</p>
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|
<div id="outline-container-org08b10a1" class="outline-2">
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|
<h2 id="org08b10a1"><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
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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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|
</p>
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|
<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 (for arrays)</th>
|
|
<th scope="col" class="org-left">Description</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left"><code>code_num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Number of codes used to produce the file</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<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>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Number of authors of the file</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>author</code></td>
|
|
<td class="org-left"><code>str</code></td>
|
|
<td class="org-left"><code>(metadata.author_num)</code></td>
|
|
<td class="org-left">Names of the authors of the file</td>
|
|
</tr>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1af947a" class="outline-2">
|
|
<h2 id="org1af947a"><span class="section-number-2">2</span> Electron (electron group)</h2>
|
|
<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>
|
|
|
|
<table id="orgcf2f6c9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>up_num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<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>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Number of ↓-spin electrons</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5075112" class="outline-2">
|
|
<h2 id="org5075112"><span class="section-number-2">3</span> Nucleus (nucleus group)</h2>
|
|
<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>
|
|
|
|
<table id="org159e63d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<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>
|
|
|
|
<div id="outline-container-orga1079c0" class="outline-2">
|
|
<h2 id="orga1079c0"><span class="section-number-2">4</span> Effective core potentials (ecp group)</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
An effective core potential (ECP) \(V_A^{\text{ECP}}\) replacing the
|
|
core electrons of atom \(A\) can be expressed as
|
|
\[
|
|
V_A^{\text{ECP}} =
|
|
V_{A \ell_{\max}+1} +
|
|
\sum_{\ell=0}^{\ell_{\max}}
|
|
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
|
|
V_{A \ell} - V_{A \ell_{\max}+1} \right] \langle Y_{\ell m} |
|
|
\]
|
|
</p>
|
|
|
|
<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>
|
|
|
|
<p>
|
|
The functions \(V_{A\ell}\) are parameterized as:
|
|
\[
|
|
V_{A \ell}(\mathbf{r}) =
|
|
\sum_{q=1}^{N_{q \ell}}
|
|
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
|
|
e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
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.
|
|
</p>
|
|
|
|
<table id="org0334a70" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>max_ang_mom_plus_1</code></td>
|
|
<td class="org-left"><code>int</code></td>
|
|
<td class="org-left"><code>(nucleus.num)</code></td>
|
|
<td class="org-left">\(\ell_{\max}+1\), one higher than the max angular momentum in the removed core orbitals</td>
|
|
</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>
|
|
<td class="org-left">Number of core electrons to remove per atom</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Total number of ECP functions for all atoms and all values of \(\ell\)</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>ang_mom</code></td>
|
|
<td class="org-left"><code>int</code></td>
|
|
<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>
|
|
</tr>
|
|
|
|
<tr>
|
|
<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>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>exponent</code></td>
|
|
<td class="org-left"><code>float</code></td>
|
|
<td class="org-left"><code>(ecp.num)</code></td>
|
|
<td class="org-left">\(\alpha_{A q \ell}\) all ECP exponents</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>coefficient</code></td>
|
|
<td class="org-left"><code>float</code></td>
|
|
<td class="org-left"><code>(ecp.num)</code></td>
|
|
<td class="org-left">\(\beta_{A q \ell}\) all ECP coefficients</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>power</code></td>
|
|
<td class="org-left"><code>int</code></td>
|
|
<td class="org-left"><code>(ecp.num)</code></td>
|
|
<td class="org-left">\(n_{A q \ell}\) all ECP powers</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
|
|
<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.
|
|
HDF5 back end might cause issues due to the way <code>find_dataset</code> function works.
|
|
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>
|
|
|
|
<div id="outline-container-org373a100" class="outline-3">
|
|
<h3 id="org373a100"><span class="section-number-3">4.1</span> Example</h3>
|
|
<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
|
|
|
|
# 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
|
|
]
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga70d288" class="outline-2">
|
|
<h2 id="orga70d288"><span class="section-number-2">5</span> Basis set (basis group)</h2>
|
|
<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>
|
|
|
|
<table id="orgfe5143f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
<td class="org-left"><code>prim_num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Total number of primitives</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>shell_num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<td class="org-left"> </td>
|
|
<td class="org-left">Total number of shells</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>nucleus_index</code></td>
|
|
<td class="org-left"><code>index</code></td>
|
|
<td class="org-left"><code>(basis.shell_num)</code></td>
|
|
<td class="org-left">One-to-one correspondence between shells and atomic indices</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>shell_ang_mom</code></td>
|
|
<td class="org-left"><code>int</code></td>
|
|
<td class="org-left"><code>(basis.shell_num)</code></td>
|
|
<td class="org-left">One-to-one correspondence between shells and angular momenta</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>shell_factor</code></td>
|
|
<td class="org-left"><code>float</code></td>
|
|
<td class="org-left"><code>(basis.shell_num)</code></td>
|
|
<td class="org-left">Normalization factor of each shell (\(\mathcal{N}_s\))</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>shell_index</code></td>
|
|
<td class="org-left"><code>index</code></td>
|
|
<td class="org-left"><code>(basis.prim_num)</code></td>
|
|
<td class="org-left">One-to-one correspondence between primitives and shell index</td>
|
|
</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>
|
|
</div>
|
|
|
|
<div id="outline-container-org0334663" class="outline-3">
|
|
<h3 id="org0334663"><span class="section-number-3">5.1</span> Example</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<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
|
|
1 1.057000E+00 1.000000E+00
|
|
</pre>
|
|
|
|
<p>
|
|
In TREXIO representaion we have:
|
|
</p>
|
|
|
|
<pre class="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 ]
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org84800d2" class="outline-2">
|
|
<h2 id="org84800d2"><span class="section-number-2">6</span> Atomic orbitals (ao group)</h2>
|
|
<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
|
|
the <a href="#orga70d288">Basis set</a> 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 \(\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}\).
|
|
</p>
|
|
|
|
<table id="orgbb99340" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<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>
|
|
|
|
<div id="outline-container-org4492a48" class="outline-3">
|
|
<h3 id="ao_one_e"><a id="org4492a48"></a><span class="section-number-3">6.1</span> One-electron integrals (<code>ao_1e_int</code> group)</h3>
|
|
<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>
|
|
|
|
<table id="org5e0c567" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
|
|
<div id="outline-container-org06296f8" class="outline-3">
|
|
<h3 id="ao_two_e"><a id="org06296f8"></a><span class="section-number-3">6.2</span> Two-electron integrals (<code>ao_2e_int</code> group)</h3>
|
|
<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>
|
|
|
|
<table id="org49d54f7" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
|
|
<div id="outline-container-org633bc24" class="outline-2">
|
|
<h2 id="org633bc24"><span class="section-number-2">7</span> Molecular orbitals (mo group)</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<table id="org009b23e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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">Free text to identify the set of MOs (HF, Natural, Local, CASSCF, <i>etc</i>)</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>num</code></td>
|
|
<td class="org-left"><code>dim</code></td>
|
|
<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>
|
|
<td class="org-left">Choose among: Core, Inactive, Active, Virtual, Deleted</td>
|
|
</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>
|
|
|
|
<div id="outline-container-org297b409" class="outline-3">
|
|
<h3 id="org297b409"><span class="section-number-3">7.1</span> One-electron integrals (<code>mo_1e_int</code> group)</h3>
|
|
<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>
|
|
|
|
<table id="org8eed2d1" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
|
|
<div id="outline-container-orgf1768cf" class="outline-3">
|
|
<h3 id="orgf1768cf"><span class="section-number-3">7.2</span> Two-electron integrals (<code>mo_2e_int</code> group)</h3>
|
|
<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>
|
|
|
|
<table id="orgd66afed" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
|
|
<div id="outline-container-orgdd0c0bb" class="outline-2">
|
|
<h2 id="orgdd0c0bb"><span class="section-number-2">8</span> <span class="todo TODO">TODO</span> Slater determinants</h2>
|
|
</div>
|
|
<div id="outline-container-org396b22a" class="outline-2">
|
|
<h2 id="org396b22a"><span class="section-number-2">9</span> <span class="todo TODO">TODO</span> Reduced density matrices (rdm group)</h2>
|
|
<div class="outline-text-2" id="text-9">
|
|
<table id="org046e357" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<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>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: TREX-CoE</p>
|
|
<p class="date">Created: 2021-11-17 Wed 13:23</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|