2020-10-29 00:57:26 +01:00
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<title>Atomic Orbitals</title>
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2020-10-29 01:03:33 +01:00
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2020-10-29 00:57:26 +01:00
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</head>
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<body>
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<div id="content">
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<h1 class="title">Atomic Orbitals</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#sec-1">1. Polynomials</a>
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<ul>
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<li><a href="#sec-1-1">1.1. <code>qmckl_ao_powers</code></a>
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<ul>
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<li><a href="#sec-1-1-1">1.1.1. Arguments</a></li>
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<li><a href="#sec-1-1-2">1.1.2. Requirements</a></li>
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<li><a href="#sec-1-1-3">1.1.3. Header</a></li>
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<li><a href="#sec-1-1-4">1.1.4. Source</a></li>
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</ul>
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</li>
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<li><a href="#sec-1-2">1.2. <code>qmckl_ao_polynomial_vgl</code></a>
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<ul>
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<li><a href="#sec-1-2-1">1.2.1. Arguments</a></li>
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<li><a href="#sec-1-2-2">1.2.2. Requirements</a></li>
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<li><a href="#sec-1-2-3">1.2.3. Header</a></li>
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<li><a href="#sec-1-2-4">1.2.4. Source</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#sec-2">2. <span class="todo TODO">TODO</span> Gaussian basis functions</a></li>
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<li><a href="#sec-3">3. <span class="todo TODO">TODO</span> Slater basis functions</a></li>
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</ul>
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</div>
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</div>
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<p>
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This files contains all the routines for the computation of the
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values, gradients and Laplacian of the atomic basis functions.
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</p>
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<p>
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4 files are produced:
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</p>
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<ul class="org-ul">
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<li>a header file : <code>qmckl_ao.h</code>
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</li>
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<li>a source file : <code>qmckl_ao.f90</code>
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</li>
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<li>a C test file : <code>test_qmckl_ao.c</code>
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</li>
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<li>a Fortran test file : <code>test_qmckl_ao_f.f90</code>
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</li>
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</ul>
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<div id="outline-container-sec-1" class="outline-2">
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<h2 id="sec-1"><span class="section-number-2">1</span> Polynomials</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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\[
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P_l(\mathbf{r},\mathbf{R}_i) = (x-X_i)^a (y-Y_i)^b (z-Z_i)^c
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\]
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</p>
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\begin{eqnarray*}
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\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 \\
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\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 \\
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\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} \\
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\end{eqnarray*}
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\begin{eqnarray*}
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\left( \frac{\partial }{\partial x^2} +
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\frac{\partial }{\partial y^2} +
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\frac{\partial }{\partial z^2} \right) P_l
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\left(\mathbf{r},\mathbf{R}_i \right) & = &
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a(a-1) (x-X_i)^{a-2} (y-Y_i)^b (z-Z_i)^c + \\
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&& b(b-1) (x-X_i)^a (y-Y_i)^{b-1} (z-Z_i)^c + \\
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&& c(c-1) (x-X_i)^a (y-Y_i)^b (z-Z_i)^{c-1}
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\end{eqnarray*}
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</div>
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<div id="outline-container-sec-1-1" class="outline-3">
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<h3 id="sec-1-1"><span class="section-number-3">1.1</span> <code>qmckl_ao_powers</code></h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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Computes all the powers of the <code>n</code> input data up to the given
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maximum value given in input for each of the \(n\) points:
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</p>
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<p>
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\[ P_{ij} = X_j^i \]
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</p>
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</div>
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<div id="outline-container-sec-1-1-1" class="outline-4">
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<h4 id="sec-1-1-1"><span class="section-number-4">1.1.1</span> Arguments</h4>
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<div class="outline-text-4" id="text-1-1-1">
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<td class="left"><code>context</code></td>
|
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|
|
<td class="left">input</td>
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|
|
<td class="left">Global state</td>
|
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|
|
|
</tr>
|
|
|
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|
|
<tr>
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|
|
<td class="left"><code>n</code></td>
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|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Number of values</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>X(n)</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Array containing the input values</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>LMAX(n)</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Array containing the maximum power for each value</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>P(LDP,n)</code></td>
|
|
|
|
|
<td class="left">output</td>
|
|
|
|
|
<td class="left">Array containing all the powers of <code>X</code></td>
|
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|
|
|
</tr>
|
|
|
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|
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|
|
<tr>
|
|
|
|
|
<td class="left"><code>LDP</code></td>
|
|
|
|
|
<td class="left">input</td>
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|
|
|
<td class="left">Leading dimension of array <code>P</code></td>
|
|
|
|
|
</tr>
|
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|
|
|
</tbody>
|
|
|
|
|
</table>
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|
</div>
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|
|
</div>
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|
|
<div id="outline-container-sec-1-1-2" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-1-2"><span class="section-number-4">1.1.2</span> Requirements</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-1-2">
|
|
|
|
|
<ul class="org-ul">
|
|
|
|
|
<li><code>context</code> is not 0
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>n</code> > 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> >= \(\max_i\) <code>LMAX[i]</code>
|
|
|
|
|
</li>
|
|
|
|
|
</ul>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-1-3" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-1-3"><span class="section-number-4">1.1.3</span> Header</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-1-3">
|
|
|
|
|
<div class="org-src-container">
|
|
|
|
|
|
|
|
|
|
<pre class="src src-C">qmckl_exit_code qmckl_ao_powers(qmckl_context context,
|
|
|
|
|
int64_t n,
|
|
|
|
|
double *X, int32_t *LMAX,
|
|
|
|
|
double *P, int64_t LDP);
|
|
|
|
|
</pre>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-1-4" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-1-4"><span class="section-number-4">1.1.4</span> Source</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-1-4">
|
|
|
|
|
<div class="org-src-container">
|
|
|
|
|
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|
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|
|
<pre class="src src-f90">integer function qmckl_ao_powers_f(context, n, X, LMAX, P, ldp) result(info)
|
|
|
|
|
implicit none
|
|
|
|
|
integer*8 , intent(in) :: context
|
|
|
|
|
integer*8 , intent(in) :: n
|
|
|
|
|
real*8 , intent(in) :: X(n)
|
|
|
|
|
integer , intent(in) :: LMAX(n)
|
|
|
|
|
real*8 , intent(out) :: P(ldp,n)
|
|
|
|
|
integer*8 , intent(in) :: ldp
|
|
|
|
|
|
|
|
|
|
integer*8 :: i,j
|
|
|
|
|
|
|
|
|
|
info = 0
|
|
|
|
|
|
|
|
|
|
if (context == 0_8) then
|
|
|
|
|
info = -1
|
|
|
|
|
return
|
|
|
|
|
endif
|
|
|
|
|
|
|
|
|
|
if (LDP < MAXVAL(LMAX)) then
|
|
|
|
|
info = -2
|
|
|
|
|
return
|
|
|
|
|
endif
|
|
|
|
|
|
|
|
|
|
do j=1,n
|
|
|
|
|
P(1,j) = X(j)
|
|
|
|
|
do i=2,LMAX(j)
|
|
|
|
|
P(i,j) = P(i-1,j) * X(j)
|
|
|
|
|
end do
|
|
|
|
|
end do
|
|
|
|
|
|
|
|
|
|
end function qmckl_ao_powers_f
|
|
|
|
|
</pre>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-2" class="outline-3">
|
|
|
|
|
<h3 id="sec-1-2"><span class="section-number-3">1.2</span> <code>qmckl_ao_polynomial_vgl</code></h3>
|
|
|
|
|
<div class="outline-text-3" id="text-1-2">
|
|
|
|
|
<p>
|
|
|
|
|
Computes the values, gradients and Laplacians at a given point of
|
|
|
|
|
all polynomials with an angular momentum up to <code>lmax</code>.
|
|
|
|
|
</p>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-2-1" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-2-1"><span class="section-number-4">1.2.1</span> Arguments</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-2-1">
|
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<colgroup>
|
|
|
|
|
<col class="left" />
|
|
|
|
|
|
|
|
|
|
<col class="left" />
|
|
|
|
|
|
|
|
|
|
<col class="left" />
|
|
|
|
|
</colgroup>
|
|
|
|
|
<tbody>
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>context</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Global state</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>X(3)</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Array containing the coordinates of the points</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>R(3)</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Array containing the x,y,z coordinates of the center</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>lmax</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Maximum angular momentum</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>n</code></td>
|
|
|
|
|
<td class="left">output</td>
|
|
|
|
|
<td class="left">Number of computed polynomials</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>L(ldl,n)</code></td>
|
|
|
|
|
<td class="left">output</td>
|
|
|
|
|
<td class="left">Contains a,b,c for all <code>n</code> results</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>ldl</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Leading dimension of <code>L</code></td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>VGL(ldv,n)</code></td>
|
|
|
|
|
<td class="left">output</td>
|
|
|
|
|
<td class="left">Value, gradients and Laplacian of the polynomials</td>
|
|
|
|
|
</tr>
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
|
|
|
<td class="left"><code>ldv</code></td>
|
|
|
|
|
<td class="left">input</td>
|
|
|
|
|
<td class="left">Leading dimension of array <code>VGL</code></td>
|
|
|
|
|
</tr>
|
|
|
|
|
</tbody>
|
|
|
|
|
</table>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-2-2" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-2-2"><span class="section-number-4">1.2.2</span> Requirements</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-2-2">
|
|
|
|
|
<ul class="org-ul">
|
|
|
|
|
<li><code>context</code> is not 0
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>n</code> > 0
|
|
|
|
|
</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>lmax</code> >= 0
|
|
|
|
|
</li>
|
|
|
|
|
<li>On output, <code>n</code> should be equal to (=lmax=+1)(=lmax=+2)(=lmax=+3)/6
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>L</code> is allocated with at least \(3 \times n \times 4\) bytes
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>ldl</code> >= 3
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>VGL</code> is allocated with at least \(5 \times n \times 8\) bytes
|
|
|
|
|
</li>
|
|
|
|
|
<li><code>ldv</code> >= 5
|
|
|
|
|
</li>
|
|
|
|
|
</ul>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-2-3" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-2-3"><span class="section-number-4">1.2.3</span> Header</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-2-3">
|
|
|
|
|
<div class="org-src-container">
|
|
|
|
|
|
|
|
|
|
<pre class="src src-C">qmckl_exit_code qmckl_ao_polynomial_vgl(qmckl_context context,
|
|
|
|
|
double *X, double *R,
|
|
|
|
|
int32_t lmax, int64_t *n,
|
|
|
|
|
int32_t *L, int64_t ldl,
|
|
|
|
|
double *VGL, int64_t ldv);
|
|
|
|
|
</pre>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
<div id="outline-container-sec-1-2-4" class="outline-4">
|
|
|
|
|
<h4 id="sec-1-2-4"><span class="section-number-4">1.2.4</span> Source</h4>
|
|
|
|
|
<div class="outline-text-4" id="text-1-2-4">
|
|
|
|
|
<div class="org-src-container">
|
|
|
|
|
|
|
|
|
|
<pre class="src src-f90">integer function qmckl_ao_polynomial_vgl_f(context, X, R, lmax, n, L, ldl, VGL, ldv) result(info)
|
|
|
|
|
implicit none
|
|
|
|
|
integer*8 , intent(in) :: context
|
|
|
|
|
real*8 , intent(in) :: X(3), R(3)
|
|
|
|
|
integer , intent(in) :: lmax
|
|
|
|
|
integer*8 , intent(out) :: n
|
|
|
|
|
integer , intent(out) :: L(ldl,(lmax+1)*(lmax+2)*(lmax+3)/6)
|
|
|
|
|
integer*8 , intent(in) :: ldl
|
|
|
|
|
real*8 , intent(out) :: VGL(ldv,(lmax+1)*(lmax+2)*(lmax+3)/6)
|
|
|
|
|
integer*8 , intent(in) :: ldv
|
|
|
|
|
|
|
|
|
|
integer*8 :: i,j
|
|
|
|
|
integer :: a,b,c,d
|
|
|
|
|
real*8 :: Y(3)
|
|
|
|
|
integer :: lmax_array(3)
|
|
|
|
|
real*8 :: pows(-2:lmax,3)
|
|
|
|
|
integer, external :: qmckl_ao_powers_f
|
|
|
|
|
double precision :: xy, yz, xz
|
|
|
|
|
double precision :: da, db, dc, dd
|
|
|
|
|
|
|
|
|
|
info = 0
|
|
|
|
|
|
|
|
|
|
if (context == 0_8) then
|
|
|
|
|
info = -1
|
|
|
|
|
return
|
|
|
|
|
endif
|
|
|
|
|
|
|
|
|
|
n = (lmax+1)*(lmax+2)*(lmax+3)/6
|
|
|
|
|
|
|
|
|
|
if (ldl < 3) then
|
|
|
|
|
info = -2
|
|
|
|
|
return
|
|
|
|
|
endif
|
|
|
|
|
|
|
|
|
|
if (ldv < 5) then
|
|
|
|
|
info = -3
|
|
|
|
|
return
|
|
|
|
|
endif
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
do i=1,3
|
|
|
|
|
Y(i) = X(i) - R(i)
|
|
|
|
|
end do
|
|
|
|
|
pows(-2:-1,1:3) = 0.d0
|
|
|
|
|
pows(0,1:3) = 1.d0
|
|
|
|
|
lmax_array(1:3) = lmax
|
|
|
|
|
info = qmckl_ao_powers_f(context, 1_8, Y(1), (/lmax/), pows(1,1), size(pows,1,kind=8))
|
|
|
|
|
if (info /= 0) return
|
|
|
|
|
info = qmckl_ao_powers_f(context, 1_8, Y(2), (/lmax/), pows(1,2), size(pows,1,kind=8))
|
|
|
|
|
if (info /= 0) return
|
|
|
|
|
info = qmckl_ao_powers_f(context, 1_8, Y(3), (/lmax/), pows(1,3), size(pows,1,kind=8))
|
|
|
|
|
if (info /= 0) return
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n=1
|
|
|
|
|
vgl(1:5,1:n) = 0.d0
|
|
|
|
|
l(1:3,n) = 0
|
|
|
|
|
vgl(1,n) = 1.d0
|
|
|
|
|
dd = 1.d0
|
|
|
|
|
do d=1,lmax
|
|
|
|
|
da = 0.d0
|
|
|
|
|
do a=0,d
|
|
|
|
|
db = 0.d0
|
|
|
|
|
do b=0,d-a
|
|
|
|
|
c = d - a - b
|
|
|
|
|
dc = dd - da - db
|
|
|
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n = n+1
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l(1,n) = a
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l(2,n) = b
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l(3,n) = c
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xy = pows(a,1) * pows(b,2)
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yz = pows(b,2) * pows(c,3)
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xz = pows(a,1) * pows(c,3)
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vgl(1,n) = xy * pows(c,3)
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xy = dc * xy
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xz = db * xz
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yz = da * yz
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vgl(2,n) = pows(a-1,1) * yz
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vgl(3,n) = pows(b-1,2) * xz
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vgl(4,n) = pows(c-1,3) * xy
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vgl(5,n) = &
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(da-1.d0) * pows(a-2,1) * yz + &
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(db-1.d0) * pows(b-2,2) * xz + &
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(dc-1.d0) * pows(c-2,3) * xy
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db = db + 1.d0
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end do
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da = da + 1.d0
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end do
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dd = dd + 1.d0
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end do
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end function qmckl_ao_polynomial_vgl_f
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</pre>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-sec-2" class="outline-2">
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<h2 id="sec-2"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Gaussian basis functions</h2>
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</div>
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<div id="outline-container-sec-3" class="outline-2">
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<h2 id="sec-3"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Slater basis functions</h2>
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</div>
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</div>
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<div id="postamble" class="status">
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2020-10-29 01:03:33 +01:00
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<p class="date">Created: 2020-10-29 Thu 00:03</p>
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2020-10-29 00:57:26 +01:00
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<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 25.2.2 (<a href="http://orgmode.org">Org</a> mode 8.2.10)</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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</html>
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