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<a accesskey="h" href=""> UP </a>
|
<a accesskey="H" href="index.html"> HOME </a>
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<h1 class="title">Atomic Orbitals</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="#orgf1cad8d">1. Polynomial part</a>
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<ul>
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<li><a href="#org7e5edc1">1.1. Powers of \(x-X_i\)</a></li>
<li><a href="#org8d828f1">1.2. Value, Gradient and Laplacian of a polynomial</a></li>
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</ul>
</li>
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<li><a href="#org6f3397c">2. Gaussian basis functions</a></li>
<li><a href="#orged13bf2">3. <span class="todo TODO">TODO</span> Slater basis functions</a></li>
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</ul>
</div>
</div>
<p>
The atomic basis set is defined as a list of shells. Each shell \(s\) is
centered on a nucleus \(A\), possesses a given angular momentum \(l\) and a
radial function \(R_s\). The radial function is a linear combination of
\emph{primitive} functions that can be of type Slater (\(p=1\)) or
Gaussian (\(p=2\)):
</p>
<p>
\[
R_s(\mathbf{r}) = \mathcal{N}_s |\mathbf{r}-\mathbf{R}_A|^{n_s}
\sum_{k=1}^{N_{\text{prim}}} a_{ks}
\exp \left( - \gamma_{ks} | \mathbf{r}-\mathbf{R}_A | ^p \right). |
\]
</p>
<p>
In the case of Gaussian functions, \(n_s\) is always zero.
The normalization factor \(\mathcal{N}_s\) ensures that all the functions
of the shell are normalized to unity. As this normalization requires
the ability to compute overlap integrals, it should be written in the
file to ensure that the file is self-contained and does not require
the client program to have the ability to compute such integrals.
</p>
<p>
Atomic orbitals (AOs) are defined as
</p>
<p>
\[
\chi_i (\mathbf{r}) = P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r})
\]
</p>
<p>
where \(\theta(i)\) returns the shell on which the AO is expanded,
and \(\eta(i)\) denotes which angular function is chosen.
</p>
<p>
In this section we describe the kernels used to compute the values,
gradients and Laplacian of the atomic basis functions.
</p>
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<div id="outline-container-orgf1cad8d" class="outline-2">
<h2 id="orgf1cad8d"><span class="section-number-2">1</span> Polynomial part</h2>
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<div class="outline-text-2" id="text-1">
</div>
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<div id="outline-container-org7e5edc1" class="outline-3">
<h3 id="org7e5edc1"><span class="section-number-3">1.1</span> Powers of \(x-X_i\)</h3>
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<div class="outline-text-3" id="text-1-1">
<p>
The <code>qmckl_ao_power</code> function computes all the powers of the <code>n</code>
input data up to the given maximum value given in input for each of
the \(n\) points:
</p>
<p>
\[ P_{ik} = X_i^k \]
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>context</code></td>
<td class="org-left">input</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left"><code>n</code></td>
<td class="org-left">input</td>
<td class="org-left">Number of values</td>
</tr>
<tr>
<td class="org-left"><code>X(n)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the input values</td>
</tr>
<tr>
<td class="org-left"><code>LMAX(n)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the maximum power for each value</td>
</tr>
<tr>
<td class="org-left"><code>P(LDP,n)</code></td>
<td class="org-left">output</td>
<td class="org-left">Array containing all the powers of <code>X</code></td>
</tr>
<tr>
<td class="org-left"><code>LDP</code></td>
<td class="org-left">input</td>
<td class="org-left">Leading dimension of array <code>P</code></td>
</tr>
</tbody>
</table>
<p>
Requirements:
</p>
<ul class="org-ul">
<li><code>context</code> is not <code>QMCKL_NULL_CONTEXT</code></li>
<li><code>n</code> &gt; 0</li>
<li><code>X</code> is allocated with at least \(n \times 8\) bytes</li>
<li><code>LMAX</code> is allocated with at least \(n \times 4\) bytes</li>
<li><code>P</code> is allocated with at least \(n \times \max_i \text{LMAX}_i \times 8\) bytes</li>
<li><code>LDP</code> &gt;= \(\max_i\) <code>LMAX[i]</code></li>
</ul>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span>
<span style="color: #0000ff;">qmckl_ao_power</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">n</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">X</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int32_t</span> *<span style="color: #a0522d;">LMAX</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">P</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">LDP</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">qmckl_ao_power_f</span><span style="color: #000000; background-color: #ffffff;">(context, n, X, LMAX, P, ldp) result(info)</span>
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> X(n)</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> LMAX(n)</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> P(ldp,n)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ldp</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> i,k</span>
info = QMCKL_SUCCESS
<span style="color: #a020f0;">if</span> (context == QMCKL_NULL_CONTEXT) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_CONTEXT
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (n &lt;= ldp) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_2
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
k = <span style="color: #a020f0;">MAXVAL</span>(LMAX)
<span style="color: #a020f0;">if</span> (LDP &lt; k) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_6
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (k &lt;= 0) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_4
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">do</span> i=1,n
P(1,i) = X(i)
<span style="color: #a020f0;">do</span> k=2,LMAX(i)
P(k,i) = P(k-1,i) * X(i)
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_ao_power_f</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int32_t</span>) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">test_qmckl_ao_power</span>(context) <span style="color: #a020f0;">bind</span>(C)
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int64_t</span>), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n, LDP </span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> LMAX(:) </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> X(:), P(:,:)</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> i,j</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> epsilon</span>
epsilon = qmckl_context_get_epsilon(context)
n = 100;
LDP = 10;
<span style="color: #a020f0;">allocate</span>(X(n), P(LDP,n), LMAX(n))
<span style="color: #a020f0;">do</span> j=1,n
X(j) = -5.d0 + 0.1d0 * <span style="color: #a020f0;">dble</span>(j)
LMAX(j) = 1 + <span style="color: #a020f0;">int</span>(<span style="color: #a020f0;">mod</span>(j, 5),4)
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_power = qmckl_ao_power(context, n, X, LMAX, P, LDP)
<span style="color: #a020f0;">if</span> (test_qmckl_ao_power /= 0) <span style="color: #a020f0;">return</span>
test_qmckl_ao_power = -1
<span style="color: #a020f0;">do</span> j=1,n
<span style="color: #a020f0;">do</span> i=1,LMAX(j)
<span style="color: #a020f0;">if</span> ( X(j)**i == 0.d0 ) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">if</span> ( P(i,j) /= 0.d0) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">else</span>
<span style="color: #a020f0;">if</span> ( dabs(1.d0 - P(i,j) / (X(j)**i)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end if</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_power = 0
<span style="color: #a020f0;">deallocate</span>(X,P,LMAX)
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">test_qmckl_ao_power</span>
</pre>
</div>
</div>
</div>
Deploying to gh-pages from @ TREX-CoE/qmckl@885f7b000e3a8c034ef5cba1a7913992aee9308a 🚀
2021-03-19 23:15:06 +01:00
<div id="outline-container-org8d828f1" class="outline-3">
<h3 id="org8d828f1"><span class="section-number-3">1.2</span> Value, Gradient and Laplacian of a polynomial</h3>
Deploying to gh-pages from @ TREX-CoE/qmckl@87ad36e342996f0a19429e5cb78933b7d5e472ae 🚀
2021-03-19 13:49:11 +01:00
<div class="outline-text-3" id="text-1-2">
<p>
A polynomial is centered on a nucleus \(\mathbf{R}_i\)
</p>
<p>
\[
P_l(\mathbf{r},\mathbf{R}_i) = (x-X_i)^a (y-Y_i)^b (z-Z_i)^c
\]
</p>
<p>
The gradients with respect to electron coordinates are
</p>
\begin{eqnarray*}
\frac{\partial }{\partial x} P_l\left(\mathbf{r},\mathbf{R}_i \right) &
= & a (x-X_i)^{a-1} (y-Y_i)^b (z-Z_i)^c \\
\frac{\partial }{\partial y} P_l\left(\mathbf{r},\mathbf{R}_i \right) &
= & b (x-X_i)^a (y-Y_i)^{b-1} (z-Z_i)^c \\
\frac{\partial }{\partial z} P_l\left(\mathbf{r},\mathbf{R}_i \right) &
= & c (x-X_i)^a (y-Y_i)^b (z-Z_i)^{c-1} \\
\end{eqnarray*}
<p>
and the Laplacian is
</p>
\begin{eqnarray*}
\left( \frac{\partial }{\partial x^2} +
\frac{\partial }{\partial y^2} +
\frac{\partial }{\partial z^2} \right) P_l
\left(\mathbf{r},\mathbf{R}_i \right) & = &
a(a-1) (x-X_i)^{a-2} (y-Y_i)^b (z-Z_i)^c + \\
&& b(b-1) (x-X_i)^a (y-Y_i)^{b-1} (z-Z_i)^c + \\
&& c(c-1) (x-X_i)^a (y-Y_i)^b (z-Z_i)^{c-1}.
\end{eqnarray*}
<p>
<code>qmckl_ao_polynomial_vgl</code> computes the values, gradients and
Laplacians at a given point in space, of all polynomials with an
angular momentum up to <code>lmax</code>.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>context</code></td>
<td class="org-left">input</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left"><code>X(3)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the coordinates of the points</td>
</tr>
<tr>
<td class="org-left"><code>R(3)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the x,y,z coordinates of the center</td>
</tr>
<tr>
<td class="org-left"><code>lmax</code></td>
<td class="org-left">input</td>
<td class="org-left">Maximum angular momentum</td>
</tr>
<tr>
<td class="org-left"><code>n</code></td>
<td class="org-left">output</td>
<td class="org-left">Number of computed polynomials</td>
</tr>
<tr>
<td class="org-left"><code>L(ldl,n)</code></td>
<td class="org-left">output</td>
<td class="org-left">Contains a,b,c for all <code>n</code> results</td>
</tr>
<tr>
<td class="org-left"><code>ldl</code></td>
<td class="org-left">input</td>
<td class="org-left">Leading dimension of <code>L</code></td>
</tr>
<tr>
<td class="org-left"><code>VGL(ldv,n)</code></td>
<td class="org-left">output</td>
<td class="org-left">Value, gradients and Laplacian of the polynomials</td>
</tr>
<tr>
<td class="org-left"><code>ldv</code></td>
<td class="org-left">input</td>
<td class="org-left">Leading dimension of array <code>VGL</code></td>
</tr>
</tbody>
</table>
<p>
Requirements:
</p>
<ul class="org-ul">
<li><code>context</code> is not <code>QMCKL_NULL_CONTEXT</code></li>
<li><code>n</code> &gt; 0</li>
<li><code>lmax</code> &gt;= 0</li>
<li><code>ldl</code> &gt;= 3</li>
<li><code>ldv</code> &gt;= 5</li>
<li><code>X</code> is allocated with at least \(3 \times 8\) bytes</li>
<li><code>R</code> is allocated with at least \(3 \times 8\) bytes</li>
<li><code>n</code> &gt;= <code>(lmax+1)(lmax+2)(lmax+3)/6</code></li>
<li><code>L</code> is allocated with at least \(3 \times n \times 4\) bytes</li>
<li><code>VGL</code> is allocated with at least \(5 \times n \times 8\) bytes</li>
<li>On output, <code>n</code> should be equal to <code>(lmax+1)(lmax+2)(lmax+3)/6</code></li>
<li>On output, the powers are given in the following order (l=a+b+c):
<ul class="org-ul">
<li>Increasing values of <code>l</code></li>
<li>Within a given value of <code>l</code>, alphabetical order of the
string made by a*"x" + b*"y" + c*"z" (in Python notation).
For example, with a=0, b=2 and c=1 the string is "yyz"</li>
</ul></li>
</ul>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span>
<span style="color: #0000ff;">qmckl_ao_polynomial_vgl</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">X</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">R</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int32_t</span> <span style="color: #a0522d;">lmax</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> *<span style="color: #a0522d;">n</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int32_t</span> *<span style="color: #a0522d;">L</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">ldl</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">VGL</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">ldv</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">qmckl_ao_polynomial_vgl_f</span><span style="color: #000000; background-color: #ffffff;">(context, X, R, lmax, n, L, ldl, VGL, ldv) result(info)</span>
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> X(3), R(3)</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> lmax</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> n</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> L(ldl,(lmax+1)*(lmax+2)*(lmax+3)/6)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ldl</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> VGL(ldv,(lmax+1)*(lmax+2)*(lmax+3)/6)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ldv</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> i,j</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> a,b,c,d</span>
<span style="color: #228b22;">real</span>*8 ::<span style="color: #a0522d;"> Y(3)</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> lmax_array(3)</span>
<span style="color: #228b22;">real</span>*8 ::<span style="color: #a0522d;"> pows(-2:lmax,3)</span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> qmckl_ao_power_f</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> xy, yz, xz</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> da, db, dc, dd</span>
info = 0
<span style="color: #a020f0;">if</span> (context == QMCKL_NULL_CONTEXT) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_CONTEXT
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (lmax &lt;= 0) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_4
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (ldl &lt; 3) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_7
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (ldv &lt; 5) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_9
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">do</span> i=1,3
Y(i) = X(i) - R(i)
<span style="color: #a020f0;">end do</span>
lmax_array(1:3) = lmax
<span style="color: #a020f0;">if</span> (lmax == 0) <span style="color: #a020f0;">then</span>
VGL(1,1) = 1.d0
vgL(2:5,1) = 0.d0
l(1:3,1) = 0
n=1
<span style="color: #a020f0;">else if</span> (lmax &gt; 0) <span style="color: #a020f0;">then</span>
pows(-2:0,1:3) = 1.d0
<span style="color: #a020f0;">do</span> i=1,lmax
pows(i,1) = pows(i-1,1) * Y(1)
pows(i,2) = pows(i-1,2) * Y(2)
pows(i,3) = pows(i-1,3) * Y(3)
<span style="color: #a020f0;">end do</span>
VGL(1:5,1:4) = 0.d0
l (1:3,1:4) = 0
VGL(1 ,1 ) = 1.d0
vgl(1:5,2:4) = 0.d0
l (1,2) = 1
vgl(1,2) = pows(1,1)
vgL(2,2) = 1.d0
l (2,3) = 1
vgl(1,3) = pows(1,2)
vgL(3,3) = 1.d0
l (3,4) = 1
vgl(1,4) = pows(1,3)
vgL(4,4) = 1.d0
n=4
<span style="color: #a020f0;">endif</span>
! <span style="color: #b22222;">l&gt;=2</span>
dd = 2.d0
<span style="color: #a020f0;">do</span> d=2,lmax
da = dd
<span style="color: #a020f0;">do</span> a=d,0,-1
db = dd-da
<span style="color: #a020f0;">do</span> b=d-a,0,-1
c = d - a - b
dc = dd - da - db
n = n+1
l(1,n) = a
l(2,n) = b
l(3,n) = c
xy = pows(a,1) * pows(b,2)
yz = pows(b,2) * pows(c,3)
xz = pows(a,1) * pows(c,3)
vgl(1,n) = xy * pows(c,3)
xy = dc * xy
xz = db * xz
yz = da * yz
vgl(2,n) = pows(a-1,1) * yz
vgl(3,n) = pows(b-1,2) * xz
vgl(4,n) = pows(c-1,3) * xy
vgl(5,n) = <span style="color: #a020f0;">&amp;</span>
(da-1.d0) * pows(a-2,1) * yz + <span style="color: #a020f0;">&amp;</span>
(db-1.d0) * pows(b-2,2) * xz + <span style="color: #a020f0;">&amp;</span>
(dc-1.d0) * pows(c-2,3) * xy
db = db - 1.d0
<span style="color: #a020f0;">end do</span>
da = da - 1.d0
<span style="color: #a020f0;">end do</span>
dd = dd + 1.d0
<span style="color: #a020f0;">end do</span>
info = QMCKL_SUCCESS
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_ao_polynomial_vgl_f</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int32_t</span>) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">test_qmckl_ao_polynomial_vgl</span>(context) <span style="color: #a020f0;">bind</span>(C)
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int64_t</span>), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> lmax, d, i</span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> L(:,:)</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n, ldl, ldv, j</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(3), R(3), Y(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> VGL(:,:)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> w</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> epsilon</span>
epsilon = qmckl_context_get_epsilon(context)
X = (/ 1.1 , 2.2 , 3.3 /)
R = (/ 0.1 , 1.2 , -2.3 /)
Y(:) = X(:) - R(:)
lmax = 4;
ldl = 3;
ldv = 100;
d = (lmax+1)*(lmax+2)*(lmax+3)/6
<span style="color: #a020f0;">allocate</span> (L(ldl,d), VGL(ldv,d))
test_qmckl_ao_polynomial_vgl = <span style="color: #a020f0;">&amp;</span>
qmckl_ao_polynomial_vgl(context, X, R, lmax, n, L, ldl, VGL, ldv)
<span style="color: #a020f0;">if</span> (test_qmckl_ao_polynomial_vgl /= QMCKL_SUCCESS) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">if</span> (n /= d) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">do</span> j=1,n
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
<span style="color: #a020f0;">do</span> i=1,3
<span style="color: #a020f0;">if</span> (L(i,j) &lt; 0) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(1,j) / (<span style="color: #a020f0;">&amp;</span>
Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**L(3,j) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
<span style="color: #a020f0;">if</span> (L(1,j) &lt; 1) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">if</span> (VGL(2,j) /= 0.d0) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">else</span>
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(2,j) / (<span style="color: #a020f0;">&amp;</span>
L(1,j) * Y(1)**(L(1,j)-1) * Y(2)**L(2,j) * Y(3)**L(3,j) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end if</span>
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
<span style="color: #a020f0;">if</span> (L(2,j) &lt; 1) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">if</span> (VGL(3,j) /= 0.d0) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">else</span>
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(3,j) / (<span style="color: #a020f0;">&amp;</span>
L(2,j) * Y(1)**L(1,j) * Y(2)**(L(2,j)-1) * Y(3)**L(3,j) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end if</span>
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
<span style="color: #a020f0;">if</span> (L(3,j) &lt; 1) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">if</span> (VGL(4,j) /= 0.d0) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">else</span>
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(4,j) / (<span style="color: #a020f0;">&amp;</span>
L(3,j) * Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**(L(3,j)-1) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end if</span>
test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
w = 0.d0
<span style="color: #a020f0;">if</span> (L(1,j) &gt; 1) <span style="color: #a020f0;">then</span>
w = w + L(1,j) * (L(1,j)-1) * Y(1)**(L(1,j)-2) * Y(2)**L(2,j) * Y(3)**L(3,j)
<span style="color: #a020f0;">end if</span>
<span style="color: #a020f0;">if</span> (L(2,j) &gt; 1) <span style="color: #a020f0;">then</span>
w = w + L(2,j) * (L(2,j)-1) * Y(1)**L(1,j) * Y(2)**(L(2,j)-2) * Y(3)**L(3,j)
<span style="color: #a020f0;">end if</span>
<span style="color: #a020f0;">if</span> (L(3,j) &gt; 1) <span style="color: #a020f0;">then</span>
w = w + L(3,j) * (L(3,j)-1) * Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**(L(3,j)-2)
<span style="color: #a020f0;">end if</span>
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(5,j) / w) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_polynomial_vgl = QMCKL_SUCCESS
<span style="color: #a020f0;">deallocate</span>(L,VGL)
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">test_qmckl_ao_polynomial_vgl</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">int</span> <span style="color: #0000ff;">test_qmckl_ao_polynomial_vgl</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>);
munit_assert_int(0, ==, test_qmckl_ao_polynomial_vgl(context));
</pre>
</div>
</div>
</div>
</div>
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<div id="outline-container-org6f3397c" class="outline-2">
<h2 id="org6f3397c"><span class="section-number-2">2</span> Gaussian basis functions</h2>
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<div class="outline-text-2" id="text-2">
<p>
<code>qmckl_ao_gaussian_vgl</code> computes the values, gradients and
Laplacians at a given point of <code>n</code> Gaussian functions centered at
the same point:
</p>
<p>
\[ v_i = \exp(-a_i |X-R|^2) \]
\[ \nabla_x v_i = -2 a_i (X_x - R_x) v_i \]
\[ \nabla_y v_i = -2 a_i (X_y - R_y) v_i \]
\[ \nabla_z v_i = -2 a_i (X_z - R_z) v_i \]
\[ \Delta v_i = a_i (4 |X-R|^2 a_i - 6) v_i \]
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<tbody>
<tr>
<td class="org-left"><code>context</code></td>
<td class="org-left">input</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left"><code>X(3)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the coordinates of the points</td>
</tr>
<tr>
<td class="org-left"><code>R(3)</code></td>
<td class="org-left">input</td>
<td class="org-left">Array containing the x,y,z coordinates of the center</td>
</tr>
<tr>
<td class="org-left"><code>n</code></td>
<td class="org-left">input</td>
<td class="org-left">Number of computed gaussians</td>
</tr>
<tr>
<td class="org-left"><code>A(n)</code></td>
<td class="org-left">input</td>
<td class="org-left">Exponents of the Gaussians</td>
</tr>
<tr>
<td class="org-left"><code>VGL(ldv,5)</code></td>
<td class="org-left">output</td>
<td class="org-left">Value, gradients and Laplacian of the Gaussians</td>
</tr>
<tr>
<td class="org-left"><code>ldv</code></td>
<td class="org-left">input</td>
<td class="org-left">Leading dimension of array <code>VGL</code></td>
</tr>
</tbody>
</table>
<p>
Requirements :
</p>
<ul class="org-ul">
<li><code>context</code> is not 0</li>
<li><code>n</code> &gt; 0</li>
<li><code>ldv</code> &gt;= 5</li>
<li><code>A(i)</code> &gt; 0 for all <code>i</code></li>
<li><code>X</code> is allocated with at least \(3 \times 8\) bytes</li>
<li><code>R</code> is allocated with at least \(3 \times 8\) bytes</li>
<li><code>A</code> is allocated with at least \(n \times 8\) bytes</li>
<li><code>VGL</code> is allocated with at least \(n \times 5 \times 8\) bytes</li>
</ul>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span>
<span style="color: #0000ff;">qmckl_ao_gaussian_vgl</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">X</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">R</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> *<span style="color: #a0522d;">n</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> *<span style="color: #a0522d;">A</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> *<span style="color: #a0522d;">VGL</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">ldv</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">qmckl_ao_gaussian_vgl_f</span><span style="color: #000000; background-color: #ffffff;">(context, X, R, n, A, VGL, ldv) result(info)</span>
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> X(3), R(3)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> A(n)</span>
<span style="color: #228b22;">real</span>*8 , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> VGL(ldv,5)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ldv</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> i,j</span>
<span style="color: #228b22;">real</span>*8 ::<span style="color: #a0522d;"> Y(3), r2, t, u, v</span>
info = QMCKL_SUCCESS
<span style="color: #a020f0;">if</span> (context == QMCKL_NULL_CONTEXT) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_CONTEXT
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (n &lt;= 0) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_4
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">if</span> (ldv &lt; n) <span style="color: #a020f0;">then</span>
info = QMCKL_INVALID_ARG_7
<span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">do</span> i=1,3
Y(i) = X(i) - R(i)
<span style="color: #a020f0;">end do</span>
r2 = Y(1)*Y(1) + Y(2)*Y(2) + Y(3)*Y(3)
<span style="color: #a020f0;">do</span> i=1,n
VGL(i,1) = dexp(-A(i) * r2)
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">do</span> i=1,n
VGL(i,5) = A(i) * VGL(i,1)
<span style="color: #a020f0;">end do</span>
t = -2.d0 * ( X(1) - R(1) )
u = -2.d0 * ( X(2) - R(2) )
v = -2.d0 * ( X(3) - R(3) )
<span style="color: #a020f0;">do</span> i=1,n
VGL(i,2) = t * VGL(i,5)
VGL(i,3) = u * VGL(i,5)
VGL(i,4) = v * VGL(i,5)
<span style="color: #a020f0;">end do</span>
t = 4.d0 * r2
<span style="color: #a020f0;">do</span> i=1,n
VGL(i,5) = (t * A(i) - 6.d0) * VGL(i,5)
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_ao_gaussian_vgl_f</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int32_t</span>) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">test_qmckl_ao_gaussian_vgl</span>(context) <span style="color: #a020f0;">bind</span>(C)
<span style="color: #a020f0;">use</span> <span style="color: #0000ff;">qmckl</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>(<span style="color: #008b8b;">c_int64_t</span>), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n, ldv, j, i</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(3), R(3), Y(3), r2</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> VGL(:,:), A(:)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> epsilon</span>
epsilon = qmckl_context_get_epsilon(context)
X = (/ 1.1 , 2.2 , 3.3 /)
R = (/ 0.1 , 1.2 , -2.3 /)
Y(:) = X(:) - R(:)
r2 = Y(1)**2 + Y(2)**2 + Y(3)**2
n = 10;
ldv = 100;
<span style="color: #a020f0;">allocate</span> (A(n), VGL(ldv,5))
<span style="color: #a020f0;">do</span> i=1,n
A(i) = 0.0013 * <span style="color: #a020f0;">dble</span>(<span style="color: #a020f0;">ishft</span>(1,i))
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_gaussian_vgl = <span style="color: #a020f0;">&amp;</span>
qmckl_ao_gaussian_vgl(context, X, R, n, A, VGL, ldv)
<span style="color: #a020f0;">if</span> (test_qmckl_ao_gaussian_vgl /= 0) <span style="color: #a020f0;">return</span>
test_qmckl_ao_gaussian_vgl = -1
<span style="color: #a020f0;">do</span> i=1,n
test_qmckl_ao_gaussian_vgl = -11
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(i,1) / (<span style="color: #a020f0;">&amp;</span>
dexp(-A(i) * r2) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
test_qmckl_ao_gaussian_vgl = -12
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(i,2) / (<span style="color: #a020f0;">&amp;</span>
-2.d0 * A(i) * Y(1) * dexp(-A(i) * r2) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
test_qmckl_ao_gaussian_vgl = -13
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(i,3) / (<span style="color: #a020f0;">&amp;</span>
-2.d0 * A(i) * Y(2) * dexp(-A(i) * r2) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
test_qmckl_ao_gaussian_vgl = -14
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(i,4) / (<span style="color: #a020f0;">&amp;</span>
-2.d0 * A(i) * Y(3) * dexp(-A(i) * r2) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
test_qmckl_ao_gaussian_vgl = -15
<span style="color: #a020f0;">if</span> (dabs(1.d0 - VGL(i,5) / (<span style="color: #a020f0;">&amp;</span>
A(i) * (4.d0*r2*A(i) - 6.d0) * dexp(-A(i) * r2) <span style="color: #a020f0;">&amp;</span>
)) &gt; epsilon ) <span style="color: #a020f0;">return</span>
<span style="color: #a020f0;">end do</span>
test_qmckl_ao_gaussian_vgl = 0
<span style="color: #a020f0;">deallocate</span>(VGL)
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">test_qmckl_ao_gaussian_vgl</span>
</pre>
</div>
</div>
</div>
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<div id="outline-container-orged13bf2" class="outline-2">
<h2 id="orged13bf2"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Slater basis functions</h2>
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</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: TREX CoE</p>
Deploying to gh-pages from @ TREX-CoE/qmckl@885f7b000e3a8c034ef5cba1a7913992aee9308a 🚀
2021-03-19 23:15:06 +01:00
<p class="date">Created: 2021-03-19 Fri 22:14</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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