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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">Gaussian Gaussian</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="#org75b028d">1. Summary</a></li>
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<li><a href="#org0603510">2. Atomic shell</a>
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<ul>
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<li><a href="#orgdf52365">2.1. Type</a></li>
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<li><a href="#orgc6225f5">2.2. Access</a></li>
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<li><a href="#org6b8dfa1">2.3. Creation</a></li>
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<li><a href="#orga9e61b8">2.4. Printers</a></li>
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</ul>
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</li>
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<li><a href="#orgd615a55">3. Atomic shell pair couple</a>
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<ul>
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<li><a href="#orgda0a937">3.1. Type</a></li>
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<li><a href="#org4991d87">3.2. Access</a></li>
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<li><a href="#orgb976b3f">3.3. Creation</a></li>
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<li><a href="#orgc9cfb8d">3.4. Printers</a></li>
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</ul>
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</li>
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<li><a href="#orgc9b4c06">4. Atomic shell pair</a>
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<ul>
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<li><a href="#org2cc8ae6">4.1. Type</a></li>
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<li><a href="#org758a97d">4.2. Access</a></li>
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<li><a href="#orgf49ab7d">4.3. Creation</a></li>
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<li><a href="#org3b5b666">4.4. Printers</a></li>
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</ul>
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</li>
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<li><a href="#orgfc88a58">5. Summary</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org75b028d" class="outline-2">
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<h2 id="org75b028d"><span class="section-number-2">1</span> Summary</h2>
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</div>
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<div id="outline-container-org0603510" class="outline-2">
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<h2 id="org0603510"><span class="section-number-2">2</span> Atomic shell</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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Set of contracted Gaussians differing only by the powers of \(x\), \(y\) and \(z\), with a
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constant <code>Angular_momentum.t</code>, all centered on the same point.
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</p>
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<p>
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In other words, it is the set of all contracted shells sharing the same center.
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</p>
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\begin{align*}
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\chi_{n_x,n_y,n_z}(r) & = f(n_x,n_y,n_z) \sum_{j=1}^{n} \sum_{i=1}^{m} \mathcal{N}_{ij}\, d_{ij}\, g_{ij\,n_x,n_y,n_z}(r) \\
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& = (x-X_A)^{n_x} (y-Y_A)^{n_y} (z-Z_A)^{n_z} f(n_x,n_y,n_z) \sum_{j=1}^{n} \sum_{i=1}^{m} \mathcal{N}_{ij}\, d_{ij}\, \exp \left( -\alpha_{ij} |r-R_A|^2 \right)
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\end{align*}
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<p>
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where:
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</p>
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<ul class="org-ul">
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<li>\(g_{ij\,n_x,n_y,n_z}(r)\) is the $i$-th <code>PrimitiveShell.t</code> of the $j$-th <code>Contracted_shell.t</code></li>
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<li>\(n_x + n_y + n_z = l\), the total angular momentum</li>
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<li>\(\alpha_{ij}\) are the exponents (tabulated) of the $j$-th <code>Contracted_shell.t</code></li>
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<li>\(d_{ij}\) are the contraction coefficients of the $j$-th <code>Contracted_shell.t</code></li>
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<li>\(\mathcal{N}_{ij}\) is the normalization coefficient of the $i$-th primitive shell
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(<code>PrimitiveShell.norm_coef</code>) of the $j$-th <code>Contracted_shell.t</code></li>
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<li>\(f(n_x,n_y,n_z)\) is a scaling factor adjusting the normalization coefficient for the
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particular powers of \(x,y,z\) (<code>PrimitiveShell.norm_coef_scale</code>)</li>
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</ul>
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</div>
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<div id="outline-container-orgdf52365" class="outline-3">
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<h3 id="orgdf52365"><span class="section-number-3">2.1</span> Type</h3>
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<div class="outline-text-3" id="text-2-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">type t
|
|
|
|
open Common
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc6225f5" class="outline-3">
|
|
<h3 id="orgc6225f5"><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">val ang_mom : t -> Angular_momentum.t
|
|
val center : t -> Coordinate.t
|
|
val coefficients : t -> float array array
|
|
val contracted_shells : t -> Contracted_shell.t array
|
|
val exponents : t -> float array array
|
|
val index : t -> int
|
|
val normalizations : t -> float array array
|
|
val norm_scales : t -> float array
|
|
val size_of_shell : t -> int
|
|
val size : t -> 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>ang_mom</code></td>
|
|
<td class="org-left">Total angular momentum : \(l = n_x + n_y + n_z\).</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>center</code></td>
|
|
<td class="org-left">Coordinate of the center \(\mathbf{A} = (X_A,Y_A,Z_A)\).</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>coefficients</code></td>
|
|
<td class="org-left">Array of contraction coefficients \(d_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>contracted_shells:</code></td>
|
|
<td class="org-left">Array of contracted gaussians</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>exponents</code></td>
|
|
<td class="org-left">Array of exponents \(\alpha_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>index</code></td>
|
|
<td class="org-left">Index in the basis set, represented as an array of contracted shells.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>normalizations</code></td>
|
|
<td class="org-left">Normalization coefficients \(\mathcal{N}_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>norm_scales</code></td>
|
|
<td class="org-left">Scaling factors \(f(n_x,n_y,n_z)\), given in the same order as <code>Angular_momentum.zkey_array ang_mom</code>.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>size</code></td>
|
|
<td class="org-left">Number of contracted functions, \(n\) in the definition.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>size_of_shell</code></td>
|
|
<td class="org-left">Number of contracted functions in the shell: length of <code>norm_coef_scale</code>.</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<pre class="example">
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6b8dfa1" class="outline-3">
|
|
<h3 id="org6b8dfa1"><span class="section-number-3">2.3</span> Creation</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val make : ?index:int -> Contracted_shell.t array -> t
|
|
|
|
val with_index : t -> int -> 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>make</code></td>
|
|
<td class="org-left">Creates a contracted shell from a list of coefficients and primitives.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>with_index</code></td>
|
|
<td class="org-left">Returns a copy of the contracted shell with a modified index.</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga9e61b8" class="outline-3">
|
|
<h3 id="orga9e61b8"><span class="section-number-3">2.4</span> Printers</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val pp : Format.formatter -> t -> unit
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd615a55" class="outline-2">
|
|
<h2 id="orgd615a55"><span class="section-number-2">3</span> Atomic shell pair couple</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
An atomic shell pair couple is the cartesian product between two sets of functions, one
|
|
set over electron one and one set over electron two. Both sets are atomic shell
|
|
pairs.
|
|
</p>
|
|
|
|
<p>
|
|
These are usually called <i>shell quartets</i> in the literature, but we prefer to use
|
|
<i>pair</i> for two functions with the same electron, and <i>couple</i> for two functions
|
|
acting on different electrons, since they will be coupled by a two-electron operator.
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orgda0a937" class="outline-3">
|
|
<h3 id="orgda0a937"><span class="section-number-3">3.1</span> Type</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">type t
|
|
|
|
open Common
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4991d87" class="outline-3">
|
|
<h3 id="org4991d87"><span class="section-number-3">3.2</span> Access</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val ang_mom : t -> Angular_momentum.t
|
|
val atomic_shell_a : t -> Atomic_shell.t
|
|
val atomic_shell_b : t -> Atomic_shell.t
|
|
val atomic_shell_c : t -> Atomic_shell.t
|
|
val atomic_shell_d : t -> Atomic_shell.t
|
|
val atomic_shell_pair_p : t -> Atomic_shell_pair.t
|
|
val atomic_shell_pair_q : t -> Atomic_shell_pair.t
|
|
val contracted_shell_pair_couples : t -> Contracted_shell_pair_couple.t list
|
|
val monocentric : t -> bool
|
|
val norm_scales : t -> float array
|
|
val zkey_array : t -> Zkey.t array
|
|
</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>ang_mom</code></td>
|
|
<td class="org-left">Total angular momentum of the shell pair couple: sum of the angular momenta of all the shells.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_a</code></td>
|
|
<td class="org-left">Returns the first atomic shell of the first shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_b</code></td>
|
|
<td class="org-left">Returns the second atomic shell of the first shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_c</code></td>
|
|
<td class="org-left">Returns the first atomic shell of the second shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_d</code></td>
|
|
<td class="org-left">Returns the second atomic shell of the second shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_pair_p</code></td>
|
|
<td class="org-left">Returns the first atomic shell pair that was used to build the shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_pair_q</code></td>
|
|
<td class="org-left">Returns the second atomic shell pair that was used to build the shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>contracted_shell_pair_couples</code></td>
|
|
<td class="org-left">Returns the list of significant contracted shell pair couples.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>monocentric</code></td>
|
|
<td class="org-left">True if all four atomic shells have the same center.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>norm_scales</code></td>
|
|
<td class="org-left">Scaling factors of normalization coefficients inside the shell. The ordering is the same as <code>zkey_array</code>.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>zkey_array</code></td>
|
|
<td class="org-left">Returns the array of <code>Zkey.t</code> relative to the four shells of the shell pair couple.</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb976b3f" class="outline-3">
|
|
<h3 id="orgb976b3f"><span class="section-number-3">3.3</span> Creation</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val make : ?cutoff:float -> Atomic_shell_pair.t -> Atomic_shell_pair.t -> t option
|
|
</pre>
|
|
</div>
|
|
<p>
|
|
Default cutoff is \(\epsilon\).
|
|
</p>
|
|
|
|
<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>make</code></td>
|
|
<td class="org-left">Creates an atomic shell pair couple using two atomic shell pairs.</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<pre class="example">
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc9cfb8d" class="outline-3">
|
|
<h3 id="orgc9cfb8d"><span class="section-number-3">3.4</span> Printers</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val pp : Format.formatter -> t -> unit
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc9b4c06" class="outline-2">
|
|
<h2 id="orgc9b4c06"><span class="section-number-2">4</span> Atomic shell pair</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
Data structure to represent pairs of atomic shells. The products of
|
|
functions in the shell pair are one-electron functions.
|
|
</p>
|
|
|
|
<p>
|
|
An atomic shell pair is an array of pairs of contracted shells.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org2cc8ae6" class="outline-3">
|
|
<h3 id="org2cc8ae6"><span class="section-number-3">4.1</span> Type</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">type t
|
|
|
|
open Common
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org758a97d" class="outline-3">
|
|
<h3 id="org758a97d"><span class="section-number-3">4.2</span> Access</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val atomic_shell_a : t -> Atomic_shell.t
|
|
val atomic_shell_b : t -> Atomic_shell.t
|
|
val contracted_shell_pairs : t -> Contracted_shell_pair.t list
|
|
val ang_mom : t -> Angular_momentum.t
|
|
val monocentric : t -> bool
|
|
val norm_scales : t -> float array
|
|
val a_minus_b : t -> Coordinate.t
|
|
val a_minus_b_sq : t -> float
|
|
</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>atomic_shell_a</code></td>
|
|
<td class="org-left">Returns the first <code>Atomic_shell.t</code> which was used to build the atomic shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>atomic_shell_b</code></td>
|
|
<td class="org-left">Returns the second <code>Atomic_shell.t</code> which was used to build the atomic shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>contracted_shell_pairs</code></td>
|
|
<td class="org-left">Returns an array of <code>ContractedShellPair.t</code>, containing all the pairs of contracted functions used to build the atomic shell pair.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>norm_scales</code></td>
|
|
<td class="org-left">norm<sub>coef</sub>.(i) / norm<sub>coef</sub>.(0)</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>ang_mom</code></td>
|
|
<td class="org-left">Total angular Momentum</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>monocentric</code></td>
|
|
<td class="org-left">If true, the two atomic shells have the same center.</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>a_minus_b</code></td>
|
|
<td class="org-left">Returns \(A-B\)</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left"><code>a_minus_b_sq</code></td>
|
|
<td class="org-left">Returns \(\vert A-B \vert^2\)</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf49ab7d" class="outline-3">
|
|
<h3 id="orgf49ab7d"><span class="section-number-3">4.3</span> Creation</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val make : ?cutoff:float -> Atomic_shell.t -> Atomic_shell.t -> t option
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Creates an atomic shell pair from two atomic shells.
|
|
</p>
|
|
|
|
<p>
|
|
The contracted shell pairs contains the only pairs of primitives for which
|
|
the norm is greater than <code>cutoff</code>.
|
|
</p>
|
|
|
|
<p>
|
|
If all the contracted shell pairs are not significant, the function returns
|
|
<code>None</code>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val of_atomic_shell_array : ?cutoff:float -> Atomic_shell.t array -> t option array array
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Creates all possible atomic shell pairs from an array of atomic shells.
|
|
If an atomic shell pair is not significant, sets the value to <code>None</code>.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3b5b666" class="outline-3">
|
|
<h3 id="org3b5b666"><span class="section-number-3">4.4</span> Printers</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-ocaml">val pp : Format.formatter -> t -> unit
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfc88a58" class="outline-2">
|
|
<h2 id="orgfc88a58"><span class="section-number-2">5</span> Summary</h2>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: scemama</p>
|
|
<p class="date">Created: 2023-06-17 Sat 00:26</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|