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<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>
<li><a href="#org7aba76e">1. Context</a>
<ul>
<li><a href="#orgc8ac1ab">1.1. Data structure</a></li>
<li><a href="#org3533f2a">1.2. Access functions</a></li>
<li><a href="#orgdf31bba">1.3. Initialization functions</a></li>
<li><a href="#org35d278d">1.4. <span class="todo TODO">TODO</span> Fortran interfaces</a></li>
</ul>
</li>
<li><a href="#org27990af">2. Polynomial part</a>
<ul>
<li><a href="#org5d0d848">2.1. Powers of \(x-X_i\)</a>
<ul>
<li><a href="#org6de94fb">2.1.1. Requirements</a></li>
<li><a href="#org5b02d59">2.1.2. C Header</a></li>
<li><a href="#org5fe4981">2.1.3. Source</a></li>
<li><a href="#org1137f69">2.1.4. C interface</a></li>
<li><a href="#orgfd371f6">2.1.5. Fortran interface</a></li>
<li><a href="#org6cb24b8">2.1.6. Test</a></li>
</ul>
</li>
<li><a href="#org4ad5b46">2.2. Value, Gradient and Laplacian of a polynomial</a>
<ul>
<li><a href="#org64aaa6a">2.2.1. Requirements</a></li>
<li><a href="#orgf5cd9a8">2.2.2. C Header</a></li>
<li><a href="#orgd725656">2.2.3. Source</a></li>
<li><a href="#orgbf36f63">2.2.4. C interface</a></li>
<li><a href="#org18e2e3f">2.2.5. Fortran interface</a></li>
<li><a href="#org3960946">2.2.6. Test</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org91133fb">3. Radial part</a>
<ul>
<li><a href="#orgf83e98b">3.1. Gaussian basis functions</a></li>
<li><a href="#orga60f15c">3.2. <span class="todo TODO">TODO</span> Slater basis functions</a></li>
<li><a href="#org2492e8b">3.3. <span class="todo TODO">TODO</span> Radial functions on a grid</a></li>
</ul>
</li>
<li><a href="#orgfa58d9c">4. Combining radial and polynomial parts</a></li>
</ul>
</div>
</div>
<div id="outline-container-org7aba76e" class="outline-2">
<h2 id="org7aba76e"><span class="section-number-2">1</span> Context</h2>
<div class="outline-text-2" id="text-1">
<p>
The following arrays are stored in the context:
</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>type</code></td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Gaussian (<code>'G'</code>) or Slater (<code>'S'</code>)</td>
</tr>
<tr>
<td class="org-left"><code>shell_num</code></td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Number of shells</td>
</tr>
<tr>
<td class="org-left"><code>prim_num</code></td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Total number of primitives</td>
</tr>
<tr>
<td class="org-left"><code>shell_center</code></td>
<td class="org-left"><code>[shell_num]</code></td>
<td class="org-left">Id of the nucleus on which each shell is centered</td>
</tr>
<tr>
<td class="org-left"><code>shell_ang_mom</code></td>
<td class="org-left"><code>[shell_num]</code></td>
<td class="org-left">Angular momentum of each shell</td>
</tr>
<tr>
<td class="org-left"><code>shell_prim_num</code></td>
<td class="org-left"><code>[shell_num]</code></td>
<td class="org-left">Number of primitives in each shell</td>
</tr>
<tr>
<td class="org-left"><code>shell_prim_index</code></td>
<td class="org-left"><code>[shell_num]</code></td>
<td class="org-left">Address of the first primitive of each shell in the <code>EXPONENT</code> array</td>
</tr>
<tr>
<td class="org-left"><code>shell_factor</code></td>
<td class="org-left"><code>[shell_num]</code></td>
<td class="org-left">Normalization factor for each shell</td>
</tr>
<tr>
<td class="org-left"><code>exponent</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Array of exponents</td>
</tr>
<tr>
<td class="org-left"><code>coefficient</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Array of coefficients</td>
</tr>
</tbody>
</table>
<p>
For H<sub>2</sub> with the following basis set,
</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 1.0000000
</pre>
<p>
we have:
</p>
<pre class="example">
type = 'G'
shell_num = 12
prim_num = 20
shell_center = [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2]
shell_ang_mom = ['S', 'S', 'S', 'P', 'P', 'D', 'S', 'S', 'S', 'P', 'P', 'D']
shell_factor = [ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]
shell_prim_num = [5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1]
shell_prim_index = [1, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20]
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]
</pre>
</div>
<div id="outline-container-orgc8ac1ab" class="outline-3">
<h3 id="orgc8ac1ab"><span class="section-number-3">1.1</span> Data structure</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-c"><span style="color: #a020f0;">typedef</span> <span style="color: #a020f0;">struct</span> <span style="color: #228b22;">qmckl_ao_basis_struct</span> {
<span style="color: #228b22;">int32_t</span> <span style="color: #a0522d;">uninitialized</span>;
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">shell_num</span>;
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">prim_num</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">shell_center</span>;
<span style="color: #228b22;">char</span> * <span style="color: #a0522d;">shell_ang_mom</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">shell_prim_num</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">shell_prim_index</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">shell_factor</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">exponent</span> ;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">coefficient</span> ;
<span style="color: #228b22;">bool</span> <span style="color: #a0522d;">provided</span>;
<span style="color: #228b22;">char</span> <span style="color: #a0522d;">type</span>;
} <span style="color: #228b22;">qmckl_ao_basis_struct</span>;
</pre>
</div>
<p>
The <code>uninitialized</code> integer contains one bit set to one for each
initialization function which has not bee called. It becomes equal
to zero after all initialization functions have been called. The
struct is then initialized and <code>provided == true</code>.
</p>
</div>
</div>
<div id="outline-container-org3533f2a" class="outline-3">
<h3 id="org3533f2a"><span class="section-number-3">1.2</span> Access functions</h3>
<div class="outline-text-3" id="text-1-2">
<p>
When all the data for the AOs have been provided, the following
function returns <code>true</code>.
</p>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">bool</span> <span style="color: #0000ff;">qmckl_ao_basis_provided</span> (<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgdf31bba" class="outline-3">
<h3 id="orgdf31bba"><span class="section-number-3">1.3</span> Initialization functions</h3>
<div class="outline-text-3" id="text-1-3">
<p>
To set the basis set, all the following functions need to be
called. When
</p>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_type</span> (<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #a020f0;">const</span> <span style="color: #228b22;">char</span> <span style="color: #a0522d;">t</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_num</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;">shell_num</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_prim_num</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;">prim_num</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_prim_index</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;">shell_prim_index</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_center</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;">shell_center</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_ang_mom</span> (<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #a020f0;">const</span> <span style="color: #228b22;">char</span> * <span style="color: #a0522d;">shell_ang_mom</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_prim_num</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;">shell_prim_num</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_shell_factor</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;">shell_factor</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_exponent</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;">exponent</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_coefficient</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;">coefficient</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org35d278d" class="outline-3">
<h3 id="org35d278d"><span class="section-number-3">1.4</span> <span class="todo TODO">TODO</span> Fortran interfaces</h3>
</div>
</div>
<div id="outline-container-org27990af" class="outline-2">
<h2 id="org27990af"><span class="section-number-2">2</span> Polynomial part</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org5d0d848" class="outline-3">
<h3 id="org5d0d848"><span class="section-number-3">2.1</span> Powers of \(x-X_i\)</h3>
<div class="outline-text-3" id="text-2-1">
<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 id="org795cf69" 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>
<tbody>
<tr>
<td class="org-left">qmckl<sub>context</sub></td>
<td class="org-left">context</td>
<td class="org-left">in</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">n</td>
<td class="org-left">in</td>
<td class="org-left">Number of values</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">X[n]</td>
<td class="org-left">in</td>
<td class="org-left">Array containing the input values</td>
</tr>
<tr>
<td class="org-left">int32<sub>t</sub></td>
<td class="org-left">LMAX[n]</td>
<td class="org-left">in</td>
<td class="org-left">Array containing the maximum power for each value</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">P[n][ldp]</td>
<td class="org-left">out</td>
<td class="org-left">Array containing all the powers of <code>X</code></td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">ldp</td>
<td class="org-left">in</td>
<td class="org-left">Leading dimension of array <code>P</code></td>
</tr>
</tbody>
</table>
</div>
<div id="outline-container-org6de94fb" class="outline-4">
<h4 id="org6de94fb"><span class="section-number-4">2.1.1</span> Requirements</h4>
<div class="outline-text-4" id="text-2-1-1">
<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>
</div>
<div id="outline-container-org5b02d59" class="outline-4">
<h4 id="org5b02d59"><span class="section-number-4">2.1.2</span> C Header</h4>
<div class="outline-text-4" id="text-2-1-2">
<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: #228b22;">double</span>* <span style="color: #a020f0;">const</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>
</div>
<div id="outline-container-org5fe4981" class="outline-4">
<h4 id="org5fe4981"><span class="section-number-4">2.1.3</span> Source</h4>
<div class="outline-text-4" id="text-2-1-3">
<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>
</div>
<div id="outline-container-org1137f69" class="outline-4">
<h4 id="org1137f69"><span class="section-number-4">2.1.4</span> C interface</h4>
</div>
<div id="outline-container-orgfd371f6" class="outline-4">
<h4 id="orgfd371f6"><span class="section-number-4">2.1.5</span> Fortran interface</h4>
</div>
<div id="outline-container-org6cb24b8" class="outline-4">
<h4 id="org6cb24b8"><span class="section-number-4">2.1.6</span> Test</h4>
<div class="outline-text-4" id="text-2-1-6">
<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>(qmckl_context), <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_get_numprec_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 /= QMCKL_SUCCESS) <span style="color: #a020f0;">return</span>
test_qmckl_ao_power = QMCKL_FAILURE
<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 = QMCKL_SUCCESS
<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>
</div>
<div id="outline-container-org4ad5b46" class="outline-3">
<h3 id="org4ad5b46"><span class="section-number-3">2.2</span> Value, Gradient and Laplacian of a polynomial</h3>
<div class="outline-text-3" id="text-2-2">
<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 id="orgafebda5" 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>
<tbody>
<tr>
<td class="org-left">qmckl<sub>context</sub></td>
<td class="org-left">context</td>
<td class="org-left">in</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">X[3]</td>
<td class="org-left">in</td>
<td class="org-left">Array containing the coordinates of the points</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">R[3]</td>
<td class="org-left">in</td>
<td class="org-left">Array containing the x,y,z coordinates of the center</td>
</tr>
<tr>
<td class="org-left">int32<sub>t</sub></td>
<td class="org-left">lmax</td>
<td class="org-left">in</td>
<td class="org-left">Maximum angular momentum</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">n</td>
<td class="org-left">inout</td>
<td class="org-left">Number of computed polynomials</td>
</tr>
<tr>
<td class="org-left">int32<sub>t</sub></td>
<td class="org-left">L[n][ldl]</td>
<td class="org-left">out</td>
<td class="org-left">Contains a,b,c for all <code>n</code> results</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">ldl</td>
<td class="org-left">in</td>
<td class="org-left">Leading dimension of <code>L</code></td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">VGL[n][ldv]</td>
<td class="org-left">out</td>
<td class="org-left">Value, gradients and Laplacian of the polynomials</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">ldv</td>
<td class="org-left">in</td>
<td class="org-left">Leading dimension of array <code>VGL</code></td>
</tr>
</tbody>
</table>
</div>
<div id="outline-container-org64aaa6a" class="outline-4">
<h4 id="org64aaa6a"><span class="section-number-4">2.2.1</span> Requirements</h4>
<div class="outline-text-4" id="text-2-2-1">
<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>
</div>
<div id="outline-container-orgf5cd9a8" class="outline-4">
<h4 id="orgf5cd9a8"><span class="section-number-4">2.2.2</span> C Header</h4>
<div class="outline-text-4" id="text-2-2-2">
<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: #228b22;">int64_t</span>* <span style="color: #a0522d;">n</span>,
<span style="color: #228b22;">int32_t</span>* <span style="color: #a020f0;">const</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: #228b22;">double</span>* <span style="color: #a020f0;">const</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>
</div>
<div id="outline-container-orgd725656" class="outline-4">
<h4 id="orgd725656"><span class="section-number-4">2.2.3</span> Source</h4>
<div class="outline-text-4" id="text-2-2-3">
<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>
</div>
<div id="outline-container-orgbf36f63" class="outline-4">
<h4 id="orgbf36f63"><span class="section-number-4">2.2.4</span> C interface</h4>
</div>
<div id="outline-container-org18e2e3f" class="outline-4">
<h4 id="org18e2e3f"><span class="section-number-4">2.2.5</span> Fortran interface</h4>
</div>
<div id="outline-container-org3960946" class="outline-4">
<h4 id="org3960946"><span class="section-number-4">2.2.6</span> Test</h4>
<div class="outline-text-4" id="text-2-2-6">
<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_get_numprec_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>
</div>
<div id="outline-container-org91133fb" class="outline-2">
<h2 id="org91133fb"><span class="section-number-2">3</span> Radial part</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgf83e98b" class="outline-3">
<h3 id="orgf83e98b"><span class="section-number-3">3.1</span> Gaussian basis functions</h3>
<div class="outline-text-3" id="text-3-1">
<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_get_numprec_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>
<div id="outline-container-orga60f15c" class="outline-3">
<h3 id="orga60f15c"><span class="section-number-3">3.2</span> <span class="todo TODO">TODO</span> Slater basis functions</h3>
</div>
<div id="outline-container-org2492e8b" class="outline-3">
<h3 id="org2492e8b"><span class="section-number-3">3.3</span> <span class="todo TODO">TODO</span> Radial functions on a grid</h3>
</div>
</div>
<div id="outline-container-orgfa58d9c" class="outline-2">
<h2 id="orgfa58d9c"><span class="section-number-2">4</span> Combining radial and polynomial parts</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: TREX CoE</p>
<p class="date">Created: 2021-04-29 Thu 23:56</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
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