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<div id="org-div-home-and-up">
<a accesskey="h" href=""> UP </a>
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<a accesskey="H" href="index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Atomic Orbitals</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org10034f2">1. Summary</a></li>
<li><a href="#org5474da5">2. Gaussian basis</a>
<ul>
<li><a href="#org7b7dfc2">2.1. Type</a></li>
<li><a href="#org9115219">2.2. Access</a></li>
<li><a href="#org7028216">2.3. Computation</a></li>
<li><a href="#org1f24daa">2.4. Creation</a></li>
<li><a href="#orge8bdd89">2.5. Printers</a></li>
</ul>
</li>
<li><a href="#org6b9a069">3. Basis</a>
<ul>
<li><a href="#org68c3519">3.1. Polymorphic types</a></li>
<li><a href="#org5764372">3.2. Types</a></li>
<li><a href="#org6ea365f">3.3. Conversions</a></li>
<li><a href="#org867b16c">3.4. Access</a></li>
<li><a href="#orgf5a488a">3.5. Printers</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org10034f2" class="outline-2">
<h2 id="org10034f2"><span class="section-number-2">1</span> Summary</h2>
<div class="outline-text-2" id="text-1">
<p>
This modules contains the data structures used to characterize the
atomic basis set.
</p>
</div>
</div>
<div id="outline-container-org5474da5" class="outline-2">
<h2 id="org5474da5"><span class="section-number-2">2</span> Gaussian basis</h2>
<div class="outline-text-2" id="text-2">
<p>
Data structure for Gaussian Atomic Orbitals:
</p>
<p>
\[
\chi_i(\mathbf{r}) = P_i(\mathbf{r}) \sum_k c_k \exp\left( -\alpha_k (\mathbf{r-R_A})^2 \right)
\]
</p>
<p>
where the polynomial \(P_i\) and the Gaussian part are both centered on
nucleus \(A\).
</p>
</div>
<div id="outline-container-org7b7dfc2" class="outline-3">
<h3 id="org7b7dfc2"><span class="section-number-3">2.1</span> Type</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org8bd4ea5"><code>Gaussian_basis.t</code></a>
</p>
<div class="org-src-container">
<pre class="src src-ocaml" id="org5bf215b"><span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Common</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Particles</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Linear_algebra</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Gaussian_integrals</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Operators</span>
<span class="org-tuareg-font-lock-governing">type</span> <span class="org-type">t</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org9115219" class="outline-3">
<h3 id="org9115219"><span class="section-number-3">2.2</span> Access</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">basis</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Gaussian.Basis.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">cartesian</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> bool
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ee_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Eri.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ee_lr_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Eri_long_range.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">eN_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Electron_nucleus.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">f12_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">F12.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">f12_over_r12_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Screened_eri.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">kin_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Kinetic.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">multipole</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Multipole.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ortho</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Orthonormalization.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">overlap</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Overlap.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">size</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> int
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>basis</code></td>
<td class="org-left">One-electron basis set</td>
</tr>
<tr>
<td class="org-left"><code>cartesian</code></td>
<td class="org-left">If true, use cartesian Gaussians (6d, 10f, …)</td>
</tr>
<tr>
<td class="org-left"><code>ee_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>ee_lr_ints</code></td>
<td class="org-left">Electron-electron long-range potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>eN_ints</code></td>
<td class="org-left">Electron-nucleus potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>f12_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>f12_over_r12_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>kin_ints</code></td>
<td class="org-left">Kinetic energy integrals</td>
</tr>
<tr>
<td class="org-left"><code>multipole</code></td>
<td class="org-left">Multipole matrices</td>
</tr>
<tr>
<td class="org-left"><code>ortho</code></td>
<td class="org-left">Orthonormalization matrix of the overlap</td>
</tr>
<tr>
<td class="org-left"><code>overlap</code></td>
<td class="org-left">Overlap matrix</td>
</tr>
<tr>
<td class="org-left"><code>size</code></td>
<td class="org-left">Number of atomic orbitals</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org7028216" class="outline-3">
<h3 id="org7028216"><span class="section-number-3">2.3</span> Computation</h3>
<div class="outline-text-3" id="text-2-3">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">values</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-module">Coordinate.</span>t <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-module">Gaussian.Basis.</span>t <span class="org-tuareg-font-lock-module">Vector.</span>t
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>values</code></td>
<td class="org-left">Returns the values of all the AOs evaluated at a given point</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org1f24daa" class="outline-3">
<h3 id="org1f24daa"><span class="section-number-3">2.4</span> Creation</h3>
<div class="outline-text-3" id="text-2-4">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">make</span> <span class="org-tuareg-font-lock-operator">:</span> <span class="org-tuareg-font-lock-label">basis</span><span class="org-tuareg-font-lock-operator">:</span><span class="org-tuareg-font-lock-module">Gaussian.Basis.</span>t <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-label">?operators</span><span class="org-tuareg-font-lock-operator">:</span><span class="org-tuareg-font-lock-module">Operator.</span>t list <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-label">?cartesian</span><span class="org-tuareg-font-lock-operator">:</span>bool <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-module">Nuclei.</span>t <span class="org-tuareg-font-lock-operator">-></span>
t
</pre>
</div>
<p>
Creates the data structure for atomic orbitals from a Gaussian basis and the
molecular geometry <code>Nuclei.t</code>.
</p>
<p>
Defaults:
</p>
<ul class="org-ul">
<li><code>operators</code> : <code>[]</code></li>
<li><code>cartesian</code> : <code>false</code></li>
</ul>
<p>
Example:
</p>
<pre class="example" id="orgb9ea620">
let b = Ao.Basis_gaussian.make ~basis nuclei ;;
val b : Ao.Basis_gaussian.t = Gaussian Basis, spherical, 15 AOs
</pre>
</div>
</div>
<div id="outline-container-orge8bdd89" class="outline-3">
<h3 id="orge8bdd89"><span class="section-number-3">2.5</span> Printers</h3>
<div class="outline-text-3" id="text-2-5">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">pp</span> <span class="org-tuareg-font-lock-operator">:</span> <span class="org-tuareg-font-lock-module">Format.</span>formatter <span class="org-tuareg-font-lock-operator">-></span> t <span class="org-tuareg-font-lock-operator">-></span> unit
</pre>
</div>
</div>
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</div>
<div id="outline-container-org6b9a069" class="outline-2">
<h2 id="org6b9a069"><span class="section-number-2">3</span> Basis</h2>
<div class="outline-text-2" id="text-3">
<p>
Data structure for Atomic Orbitals.
</p>
</div>
<div id="outline-container-org68c3519" class="outline-3">
<h3 id="org68c3519"><span class="section-number-3">3.1</span> Polymorphic types</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="org221aabc"><code>Basis.t</code></a>
</p>
<div class="org-src-container">
<pre class="src src-ocaml" id="org7f2805d"><span class="org-tuareg-font-lock-governing">type</span> <span class="org-type">t</span> <span class="org-tuareg-font-lock-operator">=</span>
<span class="org-tuareg-font-lock-operator">|</span> <span class="org-tuareg-font-lock-constructor">Unknown</span>
<span class="org-tuareg-font-lock-operator">|</span> <span class="org-tuareg-font-lock-constructor">Gaussian</span> <span class="org-keyword">of</span> <span class="org-tuareg-font-lock-module">Basis_gaussian.</span>t
</pre>
</div>
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<div id="outline-container-org5764372" class="outline-3">
<h3 id="org5764372"><span class="section-number-3">3.2</span> Types</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="orgaf9bd07"><code>Basis.t</code></a>
</p>
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">type</span> <span class="org-type">t</span>
<span class="org-tuareg-font-lock-governing">type</span> <span class="org-type">ao</span> <span class="org-tuareg-font-lock-operator">=</span> <span class="org-tuareg-font-lock-module">Ao_dim.</span>t
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Common</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Particles</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Operators</span>
<span class="org-tuareg-font-lock-governing">open </span><span class="org-tuareg-font-lock-module">Linear_algebra</span>
</pre>
</div>
</div>
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<div id="outline-container-org6ea365f" class="outline-3">
<h3 id="org6ea365f"><span class="section-number-3">3.3</span> Conversions</h3>
<div class="outline-text-3" id="text-3-3">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">of_nuclei_and_basis_filename</span> <span class="org-tuareg-font-lock-operator">:</span>
<span class="org-tuareg-font-lock-label">?kind</span><span class="org-tuareg-font-lock-operator">:[></span> <span class="org-tuareg-font-lock-constructor">`Gaussian</span> <span class="org-tuareg-font-lock-operator">]</span> <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-label">?operators</span><span class="org-tuareg-font-lock-operator">:</span><span class="org-tuareg-font-lock-module">Operator.</span>t list <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-label">?cartesian</span><span class="org-tuareg-font-lock-operator">:</span>bool <span class="org-tuareg-font-lock-operator">-></span>
<span class="org-tuareg-font-lock-label">nuclei</span><span class="org-tuareg-font-lock-operator">:</span><span class="org-tuareg-font-lock-module">Nuclei.</span>t <span class="org-tuareg-font-lock-operator">-></span>
string <span class="org-tuareg-font-lock-operator">-></span>
t
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>of_nuclei_and_basis_filename</code></td>
<td class="org-left">Creates the data structure for the atomic orbitals basis from a molecule <code>Nuclei.t</code> and the name of the basis-set file</td>
</tr>
</tbody>
</table>
<p>
Defaults:
</p>
<ul class="org-ul">
<li><code>kind</code> : <code>`Gaussian</code></li>
<li><code>operators</code> : <code>[]</code></li>
<li><code>cartesian</code> : <code>false</code></li>
</ul>
<p>
Example:
</p>
<pre class="example" id="org65a225c">
let b = Ao.Basis.of_nuclei_and_basis_filename ~nuclei filename;;
val b : Ao.Basis.t = Gaussian Basis, spherical, 15 AOs
</pre>
</div>
</div>
<div id="outline-container-org867b16c" class="outline-3">
<h3 id="org867b16c"><span class="section-number-3">3.4</span> Access</h3>
<div class="outline-text-3" id="text-3-4">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">size</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> int
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ao_basis</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Basis_poly.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">overlap</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-operator">(</span>ao<span class="org-tuareg-font-lock-operator">,</span>ao<span class="org-tuareg-font-lock-operator">)</span> <span class="org-tuareg-font-lock-module">Matrix.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">multipole</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> string <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-operator">(</span>ao<span class="org-tuareg-font-lock-operator">,</span>ao<span class="org-tuareg-font-lock-operator">)</span> <span class="org-tuareg-font-lock-module">Matrix.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ortho</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-operator">(</span>ao<span class="org-tuareg-font-lock-operator">,</span>'a<span class="org-tuareg-font-lock-operator">)</span> <span class="org-tuareg-font-lock-module">Matrix.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">eN_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-operator">(</span>ao<span class="org-tuareg-font-lock-operator">,</span>ao<span class="org-tuareg-font-lock-operator">)</span> <span class="org-tuareg-font-lock-module">Matrix.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">kin_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-operator">(</span>ao<span class="org-tuareg-font-lock-operator">,</span>ao<span class="org-tuareg-font-lock-operator">)</span> <span class="org-tuareg-font-lock-module">Matrix.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ee_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> ao <span class="org-tuareg-font-lock-module">Four_idx_storage.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">ee_lr_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> ao <span class="org-tuareg-font-lock-module">Four_idx_storage.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">f12_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> ao <span class="org-tuareg-font-lock-module">Four_idx_storage.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">f12_over_r12_ints</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> ao <span class="org-tuareg-font-lock-module">Four_idx_storage.</span>t
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">cartesian</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> bool
<span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">values</span> <span class="org-tuareg-font-lock-operator">:</span> t <span class="org-tuareg-font-lock-operator">-></span> <span class="org-tuareg-font-lock-module">Coordinate.</span>t <span class="org-tuareg-font-lock-operator">-></span> ao <span class="org-tuareg-font-lock-module">Vector.</span>t
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>size</code></td>
<td class="org-left">Number of atomic orbitals in the AO basis set</td>
</tr>
<tr>
<td class="org-left"><code>ao_basis</code></td>
<td class="org-left">One-electron basis set</td>
</tr>
<tr>
<td class="org-left"><code>overlap</code></td>
<td class="org-left">Overlap matrix</td>
</tr>
<tr>
<td class="org-left"><code>multipole</code></td>
<td class="org-left">Multipole matrices</td>
</tr>
<tr>
<td class="org-left"><code>ortho</code></td>
<td class="org-left">Orthonormalization matrix of the overlap</td>
</tr>
<tr>
<td class="org-left"><code>eN_ints</code></td>
<td class="org-left">Electron-nucleus potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>kin_ints</code></td>
<td class="org-left">Kinetic energy integrals</td>
</tr>
<tr>
<td class="org-left"><code>ee_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>ee_lr_ints</code></td>
<td class="org-left">Electron-electron long-range potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>f12_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>f12_over_r12_ints</code></td>
<td class="org-left">Electron-electron potential integrals</td>
</tr>
<tr>
<td class="org-left"><code>cartesian</code></td>
<td class="org-left">If true, use cartesian Gaussians (6d, 10f, …)</td>
</tr>
<tr>
<td class="org-left"><code>values</code></td>
<td class="org-left">Values of the AOs evaluated at a given point</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgf5a488a" class="outline-3">
<h3 id="orgf5a488a"><span class="section-number-3">3.5</span> Printers</h3>
<div class="outline-text-3" id="text-3-5">
<div class="org-src-container">
<pre class="src src-ocaml"><span class="org-tuareg-font-lock-governing">val</span> <span class="org-function-name">pp</span> <span class="org-tuareg-font-lock-operator">:</span> <span class="org-tuareg-font-lock-module">Format.</span>formatter <span class="org-tuareg-font-lock-operator">-></span> t <span class="org-tuareg-font-lock-operator">-></span> unit
</pre>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Anthony Scemama</p>
<p class="date">Created: 2021-01-28 Thu 00:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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</html>