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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="#org6291f3a">1. Context</a>
<ul>
<li><a href="#org6b00bd3">1.1. Data structure</a></li>
<li><a href="#orgdaac84b">1.2. Access functions</a></li>
<li><a href="#org8aa2802">1.3. Initialization functions</a></li>
<li><a href="#orgce5c158">1.4. Fortran interfaces</a></li>
</ul>
</li>
<li><a href="#org266ab2f">2. Radial part</a>
<ul>
<li><a href="#org034efbd">2.1. <span class="todo TODO">TODO</span> Helper functions to accelerate calculations</a></li>
<li><a href="#org045d8d0">2.2. General functions for Gaussian basis functions</a></li>
<li><a href="#org964fe5d">2.3. <span class="todo TODO">TODO</span> General functions for Slater basis functions</a></li>
<li><a href="#org4cf5927">2.4. <span class="todo TODO">TODO</span> General functions for Radial functions on a grid</a></li>
<li><a href="#orge486765">2.5. Computation of primitives</a>
<ul>
<li><a href="#orgfaa6f16">2.5.1. Get</a></li>
<li><a href="#org0d5aa85">2.5.2. Provide</a></li>
<li><a href="#org578d5a9">2.5.3. Compute</a></li>
<li><a href="#orgb0f3595">2.5.4. Test</a></li>
<li><a href="#orgfc27787">2.5.5. Ideas for improvement</a></li>
</ul>
</li>
<li><a href="#org1199d4b">2.6. Computation of shells</a>
<ul>
<li><a href="#org40ffc8d">2.6.1. Get</a></li>
<li><a href="#orgf994567">2.6.2. Provide</a></li>
<li><a href="#org870c35f">2.6.3. Compute</a></li>
<li><a href="#org4babc6b">2.6.4. Test</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgaa7f0f6">3. Polynomial part</a>
<ul>
<li><a href="#orgaaf9789">3.1. General functions for Powers of \(x-X_i\)</a>
<ul>
<li><a href="#orgda6ff78">3.1.1. Requirements</a></li>
<li><a href="#org8126427">3.1.2. C Header</a></li>
<li><a href="#orgcbd601a">3.1.3. Source</a></li>
<li><a href="#org4dde7d0">3.1.4. C interface</a></li>
<li><a href="#orgce2a7fb">3.1.5. Fortran interface</a></li>
<li><a href="#orga60ff87">3.1.6. Test</a></li>
</ul>
</li>
<li><a href="#orga8e899f">3.2. General functions for Value, Gradient and Laplacian of a polynomial</a>
<ul>
<li><a href="#org6b888bd">3.2.1. Requirements</a></li>
<li><a href="#orgf25409c">3.2.2. C Header</a></li>
<li><a href="#org62d78fd">3.2.3. Source</a></li>
<li><a href="#org3737147">3.2.4. C interface</a></li>
<li><a href="#org971f5de">3.2.5. Fortran interface</a></li>
<li><a href="#orgd4128f4">3.2.6. Test</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgaca2d68">4. Combining radial and polynomial parts</a>
<ul>
<li>
<ul>
<li><a href="#org2fe6578">4.0.1. Get</a></li>
<li><a href="#orgd117520">4.0.2. Provide</a></li>
<li><a href="#orgbf86fdf">4.0.3. Compute</a></li>
<li><a href="#orgcfa6211">4.0.4. Test</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org6291f3a" class="outline-2">
<h2 id="org6291f3a"><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>nucleus_index</code></td>
<td class="org-left"><code>[nucl_num]</code></td>
<td class="org-left">Index of the first shell of each nucleus</td>
</tr>
<tr>
<td class="org-left"><code>nucleus_shell_num</code></td>
<td class="org-left"><code>[nucl_num]</code></td>
<td class="org-left">Number of shells per nucleus</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>
<tr>
<td class="org-left"><code>prim_factor</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Normalization factors of the primtives</td>
</tr>
<tr>
<td class="org-left"><code>ao_num</code></td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Number of AOs</td>
</tr>
<tr>
<td class="org-left"><code>ao_cartesian</code></td>
<td class="org-left">&#xa0;</td>
<td class="org-left">If true, use polynomials. Otherwise, use spherical harmonics</td>
</tr>
<tr>
<td class="org-left"><code>ao_factor</code></td>
<td class="org-left"><code>[ao_num]</code></td>
<td class="org-left">Normalization factor of the AO</td>
</tr>
<tr>
<td class="org-left"><code>ao_shell</code></td>
<td class="org-left"><code>[ao_num]</code></td>
<td class="org-left">For each AO, specify to which shell it belongs</td>
</tr>
</tbody>
</table>
<p>
Computed data:
</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>
<thead>
<tr>
<th scope="col" class="org-left"><code>coefficient_normalized</code></th>
<th scope="col" class="org-left"><code>[prim_num]</code></th>
<th scope="col" class="org-left">Normalized primitive coefficients</th>
</tr>
<tr>
<th scope="col" class="org-left"><code>nucleus_prim_index</code></th>
<th scope="col" class="org-left"><code>[nucl_num]</code></th>
<th scope="col" class="org-left">Index of the first primitive for each nucleus</th>
</tr>
<tr>
<th scope="col" class="org-left"><code>nucleus_max_ang_mom</code></th>
<th scope="col" class="org-left"><code>[nucl_num]</code></th>
<th scope="col" class="org-left">Maximum angular momentum for each nucleus</th>
</tr>
<tr>
<th scope="col" class="org-left"><code>nucleus_range</code></th>
<th scope="col" class="org-left"><code>[nucl_num]</code></th>
<th scope="col" class="org-left">Distance beyond which all the AOs are zero</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><code>primitive_vgl</code></td>
<td class="org-left"><code>[5][elec_num][prim_num]</code></td>
<td class="org-left">Value, gradients, Laplacian of the primitives at electron positions</td>
</tr>
<tr>
<td class="org-left"><code>primitive_vgl_date</code></td>
<td class="org-left"><code>uint64_t</code></td>
<td class="org-left">Late modification date of Value, gradients, Laplacian of the primitives at electron positions</td>
</tr>
<tr>
<td class="org-left"><code>shell_vgl</code></td>
<td class="org-left"><code>[5][elec_num][shell_num]</code></td>
<td class="org-left">Value, gradients, Laplacian of the primitives at electron positions</td>
</tr>
<tr>
<td class="org-left"><code>shell_vgl_date</code></td>
<td class="org-left"><code>uint64_t</code></td>
<td class="org-left">Late modification date of Value, gradients, Laplacian of the AOs at electron positions</td>
</tr>
<tr>
<td class="org-left"><code>ao_vgl</code></td>
<td class="org-left"><code>[5][elec_num][ao_num]</code></td>
<td class="org-left">Value, gradients, Laplacian of the primitives at electron positions</td>
</tr>
<tr>
<td class="org-left"><code>ao_vgl_date</code></td>
<td class="org-left"><code>uint64_t</code></td>
<td class="org-left">Late modification date of Value, gradients, Laplacian of the AOs at electron positions</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><code>nucl_shell_index</code></td>
<td class="org-left"><code>[nucl_num]</code></td>
<td class="org-left">Index of the first shell for each nucleus</td>
</tr>
<tr>
<td class="org-left"><code>exponent_sorted</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Array of exponents for sorted primitives</td>
</tr>
<tr>
<td class="org-left"><code>coeff_norm_sorted</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Array of normalized coefficients for sorted primitives</td>
</tr>
<tr>
<td class="org-left"><code>prim_factor_sorted</code></td>
<td class="org-left"><code>[prim_num]</code></td>
<td class="org-left">Normalization factors of the sorted primtives</td>
</tr>
</tbody>
</table>
<p>
For H<sub>2</sub> with the following basis set,
</p>
<pre class="example" id="basis">
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" id="params">
type = 'G'
shell_num = 12
prim_num = 20
ao_num = 38
nucleus_index = [0 , 6]
shell_ang_mom = [0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 2]
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 = [0 , 5 , 6 , 7 , 8 , 9 , 10, 15, 16, 17, 18, 19]
exponent = [ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
0.388, 1.057, 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
0.388, 1.057]
coefficient = [ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
1.0, 1.0, 1.0, 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
1.0, 1.0, 1.0]
prim_factor = [ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01,
1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00,
7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01,
3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00,
4.3649547399719840e-01, 1.8135965626177861e+00 ]
</pre>
</div>
<div id="outline-container-org6b00bd3" class="outline-3">
<h3 id="org6b00bd3"><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;">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;">ao_num</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">nucleus_index</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">nucleus_shell_num</span>;
<span style="color: #228b22;">int32_t</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;">double</span> * <span style="color: #a0522d;">prim_factor</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">ao_factor</span>;
<span style="color: #228b22;">int64_t</span> * <span style="color: #a0522d;">nucleus_prim_index</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">coefficient_normalized</span>;
<span style="color: #228b22;">int32_t</span> * <span style="color: #a0522d;">nucleus_max_ang_mom</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">nucleus_range</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">primitive_vgl</span>;
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">primitive_vgl_date</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">shell_vgl</span>;
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">shell_vgl_date</span>;
<span style="color: #228b22;">double</span> * <span style="color: #a0522d;">ao_vgl</span>;
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">ao_vgl_date</span>;
<span style="color: #228b22;">int32_t</span> <span style="color: #a0522d;">uninitialized</span>;
<span style="color: #228b22;">bool</span> <span style="color: #a0522d;">provided</span>;
<span style="color: #228b22;">bool</span> <span style="color: #a0522d;">ao_cartesian</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 been called. It becomes equal
to zero after all initialization functions have been called. The
struct is then initialized and <code>provided == true</code>.
Some values are initialized by default, and are not concerned by
this mechanism.
</p>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_init_ao_basis</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_init_ao_basis</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>) {
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
<span style="color: #a020f0;">return</span> <span style="color: #008b8b;">false</span>;
}
<span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span>) context;
assert (ctx != <span style="color: #008b8b;">NULL</span>);
ctx-&gt;ao_basis.uninitialized = (1 &lt;&lt; 14) - 1;
/* <span style="color: #b22222;">Default values </span>*/
ctx-&gt;ao_basis.ao_cartesian = <span style="color: #008b8b;">true</span>;
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
}
</pre>
</div>
</div>
</div>
<div id="outline-container-orgdaac84b" class="outline-3">
<h3 id="orgdaac84b"><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-org8aa2802" class="outline-3">
<h3 id="org8aa2802"><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.
</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_ao_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;">ao_num</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_nucleus_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;">nucleus_index</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_nucleus_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;">nucleus_shell_num</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;">int32_t</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_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_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>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_prim_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;">prim_factor</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_ao_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;">ao_factor</span>);
<span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_ao_basis_cartesian</span> (<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #a020f0;">const</span> <span style="color: #228b22;">bool</span> <span style="color: #a0522d;">cartesian</span>);
</pre>
</div>
<p>
When the basis set is completely entered, other data structures are
computed to accelerate the calculations. The primitives within each
contraction are sorted in ascending order of their exponents, such
that as soon as a primitive is zero all the following functions
vanish. Also, it is possible to compute a nuclear radius beyond which
all the primitives are zero up to the numerical accuracy defined in
the context.
</p>
</div>
</div>
<div id="outline-container-orgce5c158" class="outline-3">
<h3 id="orgce5c158"><span class="section-number-3">1.4</span> Fortran interfaces</h3>
</div>
</div>
<div id="outline-container-org266ab2f" class="outline-2">
<h2 id="org266ab2f"><span class="section-number-2">2</span> Radial part</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org034efbd" class="outline-3">
<h3 id="org034efbd"><span class="section-number-3">2.1</span> <span class="todo TODO">TODO</span> Helper functions to accelerate calculations</h3>
</div>
<div id="outline-container-org045d8d0" class="outline-3">
<h3 id="org045d8d0"><span class="section-number-3">2.2</span> General functions for Gaussian basis functions</h3>
<div class="outline-text-3" id="text-2-2">
<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_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-org964fe5d" class="outline-3">
<h3 id="org964fe5d"><span class="section-number-3">2.3</span> <span class="todo TODO">TODO</span> General functions for Slater basis functions</h3>
</div>
<div id="outline-container-org4cf5927" class="outline-3">
<h3 id="org4cf5927"><span class="section-number-3">2.4</span> <span class="todo TODO">TODO</span> General functions for Radial functions on a grid</h3>
</div>
<div id="outline-container-orge486765" class="outline-3">
<h3 id="orge486765"><span class="section-number-3">2.5</span> Computation of primitives</h3>
<div class="outline-text-3" id="text-2-5">
</div>
<div id="outline-container-orgfaa6f16" class="outline-4">
<h4 id="orgfaa6f16"><span class="section-number-4">2.5.1</span> Get</h4>
<div class="outline-text-4" id="text-2-5-1">
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_get_ao_basis_primitive_vgl</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #228b22;">double</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">primitive_vgl</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org0d5aa85" class="outline-4">
<h4 id="org0d5aa85"><span class="section-number-4">2.5.2</span> Provide</h4>
</div>
<div id="outline-container-org578d5a9" class="outline-4">
<h4 id="org578d5a9"><span class="section-number-4">2.5.3</span> Compute</h4>
<div class="outline-text-4" id="text-2-5-3">
<table id="org2922c5a" 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">prim<sub>num</sub></td>
<td class="org-left">in</td>
<td class="org-left">Number of primitives</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">elec<sub>num</sub></td>
<td class="org-left">in</td>
<td class="org-left">Number of electrons</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">nucl<sub>num</sub></td>
<td class="org-left">in</td>
<td class="org-left">Number of nuclei</td>
</tr>
<tr>
<td class="org-left">int64<sub>t</sub></td>
<td class="org-left">nucleus<sub>prim</sub><sub>index</sub>[nucl<sub>num</sub>]</td>
<td class="org-left">in</td>
<td class="org-left">Index of the 1st primitive of each nucleus</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">elec<sub>coord</sub>[3][elec<sub>num</sub>]</td>
<td class="org-left">in</td>
<td class="org-left">Electron coordinates</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">nucl<sub>coord</sub>[3][elec<sub>num</sub>]</td>
<td class="org-left">in</td>
<td class="org-left">Nuclear coordinates</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">expo[prim<sub>num</sub>]</td>
<td class="org-left">in</td>
<td class="org-left">Exponents of the primitives</td>
</tr>
<tr>
<td class="org-left">double</td>
<td class="org-left">primitive<sub>vgl</sub>[5][elec<sub>num</sub>][prim<sub>num</sub>]</td>
<td class="org-left">out</td>
<td class="org-left">Value, gradients and Laplacian of the primitives</td>
</tr>
</tbody>
</table>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span><span style="color: #a0522d;"> function qmckl_compute_ao_basis_primitive_gaussian_vgl_f(context, </span><span style="color: #a020f0;">&amp;</span>
prim_num, elec_num, nucl_num, <span style="color: #a020f0;">&amp;</span>
nucleus_prim_index, elec_coord, nucl_coord, expo, primitive_vgl) <span style="color: #a020f0;">&amp;</span>
<span style="color: #a020f0;">result</span>(info)
<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: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> prim_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_prim_index(nucl_num+1)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_coord(elec_num,3)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_coord(nucl_num,3)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> expo(prim_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> primitive_vgl(prim_num,elec_num,5)</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> inucl, iprim, ielec</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x, y, z, two_a, ar2, r2, v, cutoff</span>
info = QMCKL_SUCCESS
! <span style="color: #b22222;">Don't compute exponentials when the result will be almost zero.</span>
cutoff = -dlog(1.d-15)
<span style="color: #a020f0;">do</span> inucl=1,nucl_num
! <span style="color: #b22222;">C is zero-based, so shift bounds by one</span>
<span style="color: #a020f0;">do</span> iprim = nucleus_prim_index(inucl)+1, nucleus_prim_index(inucl+1)
<span style="color: #a020f0;">do</span> ielec = 1, elec_num
x = elec_coord(ielec,1) - nucl_coord(inucl,1)
y = elec_coord(ielec,2) - nucl_coord(inucl,2)
z = elec_coord(ielec,3) - nucl_coord(inucl,3)
r2 = x*x + y*y + z*z
ar2 = expo(iprim)*r2
<span style="color: #a020f0;">if</span> (ar2 &gt; cutoff) <span style="color: #a020f0;">cycle</span>
v = dexp(-ar2)
two_a = -2.d0 * expo(iprim) * v
primitive_vgl(iprim, ielec, 1) = v
primitive_vgl(iprim, ielec, 2) = two_a * x
primitive_vgl(iprim, ielec, 3) = two_a * y
primitive_vgl(iprim, ielec, 4) = two_a * z
primitive_vgl(iprim, ielec, 5) = two_a * (3.d0 - 2.d0*ar2)
<span style="color: #a020f0;">end do</span>
<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_compute_ao_basis_primitive_gaussian_vgl_f</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb0f3595" class="outline-4">
<h4 id="orgb0f3595"><span class="section-number-4">2.5.4</span> Test</h4>
</div>
<div id="outline-container-orgfc27787" class="outline-4">
<h4 id="orgfc27787"><span class="section-number-4">2.5.5</span> Ideas for improvement</h4>
<div class="outline-text-4" id="text-2-5-5">
<div class="org-src-container">
<pre class="src src-c">// <span style="color: #b22222;">j : electrons</span>
// <span style="color: #b22222;">l : primitives</span>
k=0;
<span style="color: #a020f0;">for</span> (j=0 ; j&lt;elec_num ; ++j) {
<span style="color: #a020f0;">for</span> (i=0 ; i&lt;nucl_num ; ++i) {
r2 = nucl_elec_dist[i][j];
<span style="color: #a020f0;">if</span> (r2 &lt; nucl_radius2[i]) {
<span style="color: #a020f0;">for</span> (l=0 ; l&lt;prim_num ; ++l) {
tmp[k].i = i;
tmp[k].j = j;
tmp[k].m = m;
tmp[k].ar2 = -expo[l] *r2;
++k;
}
}
}
}
// <span style="color: #b22222;">sort(tmp) in increasing ar2;</span>
// <span style="color: #b22222;">Identify first ar2 above numerical accuracy threshold</span>
// <span style="color: #b22222;">Compute vectorized exponentials on significant values</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org1199d4b" class="outline-3">
<h3 id="org1199d4b"><span class="section-number-3">2.6</span> Computation of shells</h3>
<div class="outline-text-3" id="text-2-6">
</div>
<div id="outline-container-org40ffc8d" class="outline-4">
<h4 id="org40ffc8d"><span class="section-number-4">2.6.1</span> Get</h4>
<div class="outline-text-4" id="text-2-6-1">
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_get_ao_basis_shell_vgl</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #228b22;">double</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">shell_vgl</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf994567" class="outline-4">
<h4 id="orgf994567"><span class="section-number-4">2.6.2</span> Provide</h4>
</div>
<div id="outline-container-org870c35f" class="outline-4">
<h4 id="org870c35f"><span class="section-number-4">2.6.3</span> Compute</h4>
<div class="outline-text-4" id="text-2-6-3">
<table id="orgb32980a" 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"><code>qmckl_context</code></td>
<td class="org-left"><code>context</code></td>
<td class="org-left">in</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>prim_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of primitives</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>shell_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of shells</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>elec_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of electrons</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucl_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of nuclei</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucleus_shell_num[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of shells for each nucleus</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucleus_index[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Index of the 1st shell of each nucleus</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>shell_prim_index[shell_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Index of the 1st primitive of each shell</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>shell_prim_num[shell_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of primitives per shell</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>elec_coord[3][elec_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Electron coordinates</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>nucl_coord[3][elec_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Nuclear coordinates</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>expo[prim_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Exponents of the primitives</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>coef_normalized[prim_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Coefficients of the primitives</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>shell_vgl[5][elec_num][shell_num]</code></td>
<td class="org-left">out</td>
<td class="org-left">Value, gradients and Laplacian of the shells</td>
</tr>
</tbody>
</table>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span><span style="color: #a0522d;"> function qmckl_compute_ao_basis_shell_gaussian_vgl_f(context, </span><span style="color: #a020f0;">&amp;</span>
prim_num, shell_num, elec_num, nucl_num, <span style="color: #a020f0;">&amp;</span>
nucleus_shell_num, nucleus_index, shell_prim_index, shell_prim_num, <span style="color: #a020f0;">&amp;</span>
elec_coord, nucl_coord, expo, coef_normalized, shell_vgl) <span style="color: #a020f0;">&amp;</span>
<span style="color: #a020f0;">result</span>(info)
<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: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> prim_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_shell_num(nucl_num)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_index(nucl_num)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_prim_index(shell_num)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_prim_num(shell_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_coord(elec_num,3)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_coord(nucl_num,3)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> expo(prim_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> coef_normalized(prim_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> shell_vgl(shell_num,elec_num,5)</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> inucl, iprim, ielec, ishell</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x, y, z, two_a, ar2, r2, v, cutoff</span>
info = QMCKL_SUCCESS
! <span style="color: #b22222;">Don't compute exponentials when the result will be almost zero.</span>
! <span style="color: #b22222;">TODO : Use numerical precision here</span>
cutoff = -dlog(1.d-15)
<span style="color: #a020f0;">do</span> inucl=1,nucl_num
<span style="color: #a020f0;">do</span> ielec = 1, elec_num
x = elec_coord(ielec,1) - nucl_coord(inucl,1)
y = elec_coord(ielec,2) - nucl_coord(inucl,2)
z = elec_coord(ielec,3) - nucl_coord(inucl,3)
r2 = x*x + y*y + z*z
<span style="color: #a020f0;">do</span> ishell=nucleus_index(inucl)+1, nucleus_index(inucl)+nucleus_shell_num(inucl)
! <span style="color: #b22222;">C is zero-based, so shift bounds by one</span>
shell_vgl(ishell, ielec, 1) = 0.d0
shell_vgl(ishell, ielec, 2) = 0.d0
shell_vgl(ishell, ielec, 3) = 0.d0
shell_vgl(ishell, ielec, 4) = 0.d0
shell_vgl(ishell, ielec, 5) = 0.d0
<span style="color: #a020f0;">do</span> iprim = shell_prim_index(ishell)+1, shell_prim_index(ishell)+shell_prim_num(ishell)
ar2 = expo(iprim)*r2
<span style="color: #a020f0;">if</span> (ar2 &gt; cutoff) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">cycle</span>
<span style="color: #a020f0;">end if</span>
v = coef_normalized(iprim) * dexp(-ar2)
two_a = -2.d0 * expo(iprim) * v
shell_vgl(ishell, ielec, 1) = <span style="color: #a020f0;">&amp;</span>
shell_vgl(ishell, ielec, 1) + v
shell_vgl(ishell, ielec, 2) = <span style="color: #a020f0;">&amp;</span>
shell_vgl(ishell, ielec, 2) + two_a * x
shell_vgl(ishell, ielec, 3) = <span style="color: #a020f0;">&amp;</span>
shell_vgl(ishell, ielec, 3) + two_a * y
shell_vgl(ishell, ielec, 4) = <span style="color: #a020f0;">&amp;</span>
shell_vgl(ishell, ielec, 4) + two_a * z
shell_vgl(ishell, ielec, 5) = <span style="color: #a020f0;">&amp;</span>
shell_vgl(ishell, ielec, 5) + two_a * (3.d0 - 2.d0*ar2)
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<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_compute_ao_basis_shell_gaussian_vgl_f</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_compute_ao_basis_shell_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;">int64_t</span> <span style="color: #a0522d;">prim_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">shell_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">elec_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">nucl_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span>* <span style="color: #a0522d;">nucleus_shell_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span>* <span style="color: #a0522d;">nucleus_index</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: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span>* <span style="color: #a0522d;">shell_prim_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">elec_coord</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">nucl_coord</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">expo</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">coef_normalized</span>,
<span style="color: #228b22;">double</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">shell_vgl</span> );
</pre>
</div>
</div>
</div>
<div id="outline-container-org4babc6b" class="outline-4">
<h4 id="org4babc6b"><span class="section-number-4">2.6.4</span> Test</h4>
</div>
</div>
</div>
<div id="outline-container-orgaa7f0f6" class="outline-2">
<h2 id="orgaa7f0f6"><span class="section-number-2">3</span> Polynomial part</h2>
<div class="outline-text-2" id="text-3">
<p>
Going from the atomic basis set to AOs implies a systematic
construction of all the angular functions of each shell. We
consider two cases for the angular functions: the real-valued
spherical harmonics, and the polynomials in Cartesian coordinates.
In the case of spherical harmonics, the AOs are ordered in
increasing magnetic quantum number (\(-l \le m \le l\)), and in the
case of polynomials we choose the canonical ordering, i.e
</p>
\begin{eqnarray}
p & : & p_x, p_y, p_z \nonumber \\
d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\
f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, f_{zzz} \nonumber \\
{\rm etc.} \nonumber
\end{eqnarray}
</div>
<div id="outline-container-orgaaf9789" class="outline-3">
<h3 id="orgaaf9789"><span class="section-number-3">3.1</span> General functions for Powers of \(x-X_i\)</h3>
<div class="outline-text-3" id="text-3-1">
<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="org2a09331" 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-orgda6ff78" class="outline-4">
<h4 id="orgda6ff78"><span class="section-number-4">3.1.1</span> Requirements</h4>
<div class="outline-text-4" id="text-3-1-1">
<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-org8126427" class="outline-4">
<h4 id="org8126427"><span class="section-number-4">3.1.2</span> C Header</h4>
<div class="outline-text-4" id="text-3-1-2">
<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-orgcbd601a" class="outline-4">
<h4 id="orgcbd601a"><span class="section-number-4">3.1.3</span> Source</h4>
<div class="outline-text-4" id="text-3-1-3">
<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-org4dde7d0" class="outline-4">
<h4 id="org4dde7d0"><span class="section-number-4">3.1.4</span> C interface</h4>
</div>
<div id="outline-container-orgce2a7fb" class="outline-4">
<h4 id="orgce2a7fb"><span class="section-number-4">3.1.5</span> Fortran interface</h4>
</div>
<div id="outline-container-orga60ff87" class="outline-4">
<h4 id="orga60ff87"><span class="section-number-4">3.1.6</span> Test</h4>
<div class="outline-text-4" id="text-3-1-6">
<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-orga8e899f" class="outline-3">
<h3 id="orga8e899f"><span class="section-number-3">3.2</span> General functions for Value, Gradient and Laplacian of a polynomial</h3>
<div class="outline-text-3" id="text-3-2">
<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="org7163235" 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-org6b888bd" class="outline-4">
<h4 id="org6b888bd"><span class="section-number-4">3.2.1</span> Requirements</h4>
<div class="outline-text-4" id="text-3-2-1">
<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-orgf25409c" class="outline-4">
<h4 id="orgf25409c"><span class="section-number-4">3.2.2</span> C Header</h4>
<div class="outline-text-4" id="text-3-2-2">
<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-org62d78fd" class="outline-4">
<h4 id="org62d78fd"><span class="section-number-4">3.2.3</span> Source</h4>
<div class="outline-text-4" id="text-3-2-3">
<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;">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-org3737147" class="outline-4">
<h4 id="org3737147"><span class="section-number-4">3.2.4</span> C interface</h4>
</div>
<div id="outline-container-org971f5de" class="outline-4">
<h4 id="org971f5de"><span class="section-number-4">3.2.5</span> Fortran interface</h4>
</div>
<div id="outline-container-orgd4128f4" class="outline-4">
<h4 id="orgd4128f4"><span class="section-number-4">3.2.6</span> Test</h4>
<div class="outline-text-4" id="text-3-2-6">
<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>);
assert(0 == test_qmckl_ao_polynomial_vgl(context));
</pre>
</div>
</div>
</div>
</div>
</div>
<div id="outline-container-orgaca2d68" class="outline-2">
<h2 id="orgaca2d68"><span class="section-number-2">4</span> Combining radial and polynomial parts</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org2fe6578" class="outline-4">
<h4 id="org2fe6578"><span class="section-number-4">4.0.1</span> Get</h4>
<div class="outline-text-4" id="text-4-0-1">
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_get_ao_vgl</span>(<span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #228b22;">double</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ao_vgl</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd117520" class="outline-4">
<h4 id="orgd117520"><span class="section-number-4">4.0.2</span> Provide</h4>
</div>
<div id="outline-container-orgbf86fdf" class="outline-4">
<h4 id="orgbf86fdf"><span class="section-number-4">4.0.3</span> Compute</h4>
<div class="outline-text-4" id="text-4-0-3">
<table id="org87db496" 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"><code>qmckl_context</code></td>
<td class="org-left"><code>context</code></td>
<td class="org-left">in</td>
<td class="org-left">Global state</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>ao_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of AOs</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>shell_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of shells</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>elec_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of electrons</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucl_num</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of nuclei</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>elec_coord[3][elec_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Electron coordinates</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>nucl_coord[3][nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Nuclear coordinates</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucleus_index[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Index of the 1st shell of each nucleus</td>
</tr>
<tr>
<td class="org-left"><code>int64_t</code></td>
<td class="org-left"><code>nucleus_shell_num[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Number of shells per nucleus</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>nucleus_range[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Range beyond which all is zero</td>
</tr>
<tr>
<td class="org-left"><code>int32_t</code></td>
<td class="org-left"><code>nucleus_max_ang_mom[nucl_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Maximum angular momentum per nucleus</td>
</tr>
<tr>
<td class="org-left"><code>int32_t</code></td>
<td class="org-left"><code>shell_ang_mom[shell_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Angular momentum of each shell</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>ao_factor[ao_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Normalization factor of the AOs</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>shell_vgl[5][elec_num][shell_num]</code></td>
<td class="org-left">in</td>
<td class="org-left">Value, gradients and Laplacian of the shells</td>
</tr>
<tr>
<td class="org-left"><code>double</code></td>
<td class="org-left"><code>ao_vgl[5][elec_num][ao_num]</code></td>
<td class="org-left">out</td>
<td class="org-left">Value, gradients and Laplacian of the AOs</td>
</tr>
</tbody>
</table>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">integer</span><span style="color: #a0522d;"> function qmckl_compute_ao_vgl_f(context, </span><span style="color: #a020f0;">&amp;</span>
ao_num, shell_num, elec_num, nucl_num, <span style="color: #a020f0;">&amp;</span>
elec_coord, nucl_coord, nucleus_index, nucleus_shell_num, <span style="color: #a020f0;">&amp;</span>
nucleus_range, nucleus_max_ang_mom, shell_ang_mom, <span style="color: #a020f0;">&amp;</span>
ao_factor, shell_vgl, ao_vgl) <span style="color: #a020f0;">&amp;</span>
<span style="color: #a020f0;">result</span>(info)
<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: #a0522d;"> context</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ao_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_num</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_num</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> elec_coord(elec_num,3)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucl_coord(nucl_num,3)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_index(nucl_num)</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_shell_num(nucl_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_range(nucl_num)</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nucleus_max_ang_mom(nucl_num)</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_ang_mom(shell_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> ao_factor(ao_num)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> shell_vgl(shell_num,elec_num,5)</span>
<span style="color: #228b22;">double precision</span> , <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ao_vgl(ao_num,elec_num,5)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> e_coord(3), n_coord(3)</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_poly</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> l, il, k</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> ielec, inucl, ishell</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> lstart(0:20)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x, y, z, r2</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> cutoff</span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> qmckl_ao_polynomial_vgl_f</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> poly_vgl(:,:)</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">allocatable</span> ::<span style="color: #a0522d;"> powers(:,:)</span>
<span style="color: #a020f0;">allocate</span>(poly_vgl(5,ao_num), powers(3,ao_num))
! <span style="color: #b22222;">Pre-computed data</span>
<span style="color: #a020f0;">do</span> l=0,20
lstart(l) = l*(l+1)*(l+2)/6 +1
<span style="color: #a020f0;">end do</span>
info = QMCKL_SUCCESS
! <span style="color: #b22222;">Don't compute polynomials when the radial part is zero.</span>
! <span style="color: #b22222;">TODO : Use numerical precision here</span>
cutoff = -dlog(1.d-15)
<span style="color: #a020f0;">do</span> ielec = 1, elec_num
e_coord(1) = elec_coord(ielec,1)
e_coord(2) = elec_coord(ielec,2)
e_coord(3) = elec_coord(ielec,3)
k=1
<span style="color: #a020f0;">do</span> inucl=1,nucl_num
n_coord(1) = nucl_coord(inucl,1)
n_coord(2) = nucl_coord(inucl,2)
n_coord(3) = nucl_coord(inucl,3)
! <span style="color: #b22222;">Test if the electron is in the range of the nucleus</span>
x = e_coord(1) - n_coord(1)
y = e_coord(2) - n_coord(2)
z = e_coord(3) - n_coord(3)
r2 = x*x + z*z + z*z
<span style="color: #a020f0;">if</span> (r2 &gt; cutoff*nucleus_range(inucl)) <span style="color: #a020f0;">then</span>
<span style="color: #a020f0;">cycle</span>
<span style="color: #a020f0;">end if</span>
! <span style="color: #b22222;">Compute polynomials</span>
info = qmckl_ao_polynomial_vgl_f(context, e_coord, n_coord, <span style="color: #a020f0;">&amp;</span>
nucleus_max_ang_mom(inucl), n_poly, powers, 3_8, <span style="color: #a020f0;">&amp;</span>
poly_vgl, 5_8)
! <span style="color: #b22222;">Loop over shells</span>
<span style="color: #a020f0;">do</span> ishell = nucleus_index(inucl)+1, nucleus_index(inucl)+nucleus_shell_num(inucl)
l = shell_ang_mom(ishell)
<span style="color: #a020f0;">do</span> il = lstart(l), lstart(l+1)-1
! <span style="color: #b22222;">Value</span>
ao_vgl(k,ielec,1) = <span style="color: #a020f0;">&amp;</span>
poly_vgl(1,il) * shell_vgl(ishell,ielec,1) * ao_factor(k)
! <span style="color: #b22222;">Grad_x</span>
ao_vgl(k,ielec,2) = ( <span style="color: #a020f0;">&amp;</span>
poly_vgl(2,il) * shell_vgl(ishell,ielec,1) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(1,il) * shell_vgl(ishell,ielec,2) <span style="color: #a020f0;">&amp;</span>
) * ao_factor(k)
! <span style="color: #b22222;">Grad_y</span>
ao_vgl(k,ielec,3) = ( <span style="color: #a020f0;">&amp;</span>
poly_vgl(3,il) * shell_vgl(ishell,ielec,1) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(1,il) * shell_vgl(ishell,ielec,3) <span style="color: #a020f0;">&amp;</span>
) * ao_factor(k)
! <span style="color: #b22222;">Grad_z</span>
ao_vgl(k,ielec,4) = ( <span style="color: #a020f0;">&amp;</span>
poly_vgl(4,il) * shell_vgl(ishell,ielec,1) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(1,il) * shell_vgl(ishell,ielec,4) <span style="color: #a020f0;">&amp;</span>
) * ao_factor(k)
! <span style="color: #b22222;">Lapl_z</span>
ao_vgl(k,ielec,5) = ( <span style="color: #a020f0;">&amp;</span>
poly_vgl(5,il) * shell_vgl(ishell,ielec,1) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(1,il) * shell_vgl(ishell,ielec,5) + <span style="color: #a020f0;">&amp;</span>
2.d0 * ( <span style="color: #a020f0;">&amp;</span>
poly_vgl(2,il) * shell_vgl(ishell,ielec,2) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(3,il) * shell_vgl(ishell,ielec,3) + <span style="color: #a020f0;">&amp;</span>
poly_vgl(4,il) * shell_vgl(ishell,ielec,4) ) <span style="color: #a020f0;">&amp;</span>
) * ao_factor(k)
k = k+1
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">deallocate</span>(poly_vgl, powers)
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_compute_ao_vgl_f</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_compute_ao_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;">int64_t</span> <span style="color: #a0522d;">ao_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">shell_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">elec_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">nucl_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">elec_coord</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">nucl_coord</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span>* <span style="color: #a0522d;">nucleus_index</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int64_t</span>* <span style="color: #a0522d;">nucleus_shell_num</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">nucleus_range</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int32_t</span>* <span style="color: #a0522d;">nucleus_max_ang_mom</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int32_t</span>* <span style="color: #a0522d;">shell_ang_mom</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">ao_factor</span>,
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span>* <span style="color: #a0522d;">shell_vgl</span>,
<span style="color: #228b22;">double</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ao_vgl</span> );
</pre>
</div>
</div>
</div>
<div id="outline-container-orgcfa6211" class="outline-4">
<h4 id="orgcfa6211"><span class="section-number-4">4.0.4</span> Test</h4>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: TREX CoE</p>
<p class="date">Created: 2021-10-11 Mon 20:44</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
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