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<h1 class="title">Quantum Monte Carlo</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
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<li><a href="#orgc777be1">1. Introduction</a></li>
<li><a href="#orga3ef4eb">2. Numerical evaluation of the energy</a>
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<ul>
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<li><a href="#org3c36fbd">2.1. Local energy</a>
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<ul>
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<li><a href="#org434b5d3">2.1.1. Exercise 1</a></li>
<li><a href="#orga9bd823">2.1.2. Exercise 2</a></li>
<li><a href="#org19f7ef1">2.1.3. Exercise 3</a></li>
<li><a href="#orgb7ba480">2.1.4. Exercise 4</a></li>
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</ul>
</li>
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<li><a href="#orge0a15c9">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#org58344c8">2.2.1. Exercise</a></li>
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</ul>
</li>
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<li><a href="#org8216be8">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#org8e43312">2.3.1. Exercise</a></li>
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</ul>
</li>
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<li><a href="#org3c4621e">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#orgd7606c8">2.4.1. Exercise</a></li>
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</ul>
</li>
</ul>
</li>
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<li><a href="#org2082f05">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#orgbcaa16b">3.1. Computation of the statistical error</a>
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<ul>
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<li><a href="#orga919e66">3.1.1. Exercise</a></li>
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</ul>
</li>
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<li><a href="#org3f0ebe8">3.2. Uniform sampling in the box</a>
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<ul>
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<li><a href="#orgbbdcb31">3.2.1. Exercise</a></li>
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</ul>
</li>
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<li><a href="#org51cfa79">3.3. Gaussian sampling</a>
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<ul>
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<li><a href="#orgbec6ea5">3.3.1. Exercise</a></li>
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</ul>
</li>
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<li><a href="#org39b0dd3">3.4. Sampling with \(\Psi^2\)</a>
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<ul>
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<li><a href="#org174a24c">3.4.1. Importance sampling</a></li>
<li><a href="#org1a431c2">3.4.2. Metropolis algorithm</a></li>
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</ul>
</li>
</ul>
</li>
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<li><a href="#org767855f">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a>
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<ul>
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<li><a href="#org50b0f1f">4.1. Hydrogen atom</a></li>
<li><a href="#org64f2d1e">4.2. Dihydrogen</a></li>
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</ul>
</li>
</ul>
</div>
</div>
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<div id="outline-container-orgc777be1" class="outline-2">
<h2 id="orgc777be1"><span class="section-number-2">1</span> Introduction</h2>
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<div class="outline-text-2" id="text-1">
<p>
We propose different exercises to understand quantum Monte Carlo (QMC)
methods. In the first section, we propose to compute the energy of a
hydrogen atom using numerical integration. The goal of this section is
to introduce the <i>local energy</i>.
Then we introduce the variational Monte Carlo (VMC) method which
computes a statistical estimate of the expectation value of the energy
associated with a given wave function.
Finally, we introduce the diffusion Monte Carlo (DMC) method which
gives the exact energy of the \(H_2\) molecule.
</p>
<p>
Code examples will be given in Python and Fortran. Whatever language
can be chosen.
</p>
<p>
We consider the stationary solution of the Schrödinger equation, so
the wave functions considered here are real: for an \(N\) electron
system where the electrons move in the 3-dimensional space,
\(\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}\). In addition, \(\Psi\)
is defined everywhere, continuous and infinitely differentiable.
</p>
<p>
<b>Note</b>
</p>
<div class="important">
<p>
In Fortran, when you use a double precision constant, don't forget
to put d0 as a suffix (for example 2.0d0), or it will be
interpreted as a single precision value
</p>
</div>
</div>
</div>
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<div id="outline-container-orga3ef4eb" class="outline-2">
<h2 id="orga3ef4eb"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
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<div class="outline-text-2" id="text-2">
<p>
In this section we consider the Hydrogen atom with the following
wave function:
</p>
<p>
\[
\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
\]
</p>
<p>
We will first verify that \(\Psi\) is an eigenfunction of the Hamiltonian
</p>
<p>
\[
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
\]
</p>
<p>
when \(a=1\), by checking that \(\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})\) for
all \(\mathbf{r}\). We will check that the local energy, defined as
</p>
<p>
\[
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
\]
</p>
<p>
is constant. We will also see that when \(a \ne 1\) the local energy
is not constant, so \(\hat{H} \Psi \ne E \Psi\).
</p>
<p>
The probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
with respect to a probability density function \(p(x)\) is given by
</p>
<p>
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx \].
</p>
<p>
Recall that a probability density function \(p(x)\) is non-negative
and integrates to one:
</p>
<p>
\[ \int_{-\infty}^\infty p(x)\,dx = 1 \].
</p>
<p>
The electronic energy of a system is the expectation value of the
local energy \(E(\mathbf{r})\) with respect to the 3N-dimensional
electron density given by the square of the wave function:
</p>
\begin{eqnarray*}
E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
= \langle E_L \rangle_{\Psi^2}
\end{eqnarray*}
</div>
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<div id="outline-container-org3c36fbd" class="outline-3">
<h3 id="org3c36fbd"><span class="section-number-3">2.1</span> Local energy</h3>
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<div class="outline-text-3" id="text-2-1">
</div>
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<h4 id="org434b5d3"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
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<div class="outline-text-4" id="text-2-1-1">
<div class="exercise">
<p>
Write a function which computes the potential at \(\mathbf{r}\).
The function accepts a 3-dimensional vector <code>r</code> as input arguments
and returns the potential.
</p>
</div>
<p>
\(\mathbf{r}=\left( \begin{array}{c} x \\ y\\ z\end{array} \right)\), so
\[
V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}}
\]
</p>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">potential</span>(r):
<span style="color: #a020f0;">return</span> -1. / np.sqrt(np.dot(r,r))
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">potential</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> r(3)</span>
potential = -1.d0 / dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orga9bd823" class="outline-4">
<h4 id="orga9bd823"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-1-2">
<div class="exercise">
<p>
Write a function which computes the wave function at \(\mathbf{r}\).
The function accepts a scalar <code>a</code> and a 3-dimensional vector <code>r</code> as
input arguments, and returns a scalar.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">psi</span>(a, r):
<span style="color: #a020f0;">return</span> np.exp(-a*np.sqrt(np.dot(r,r)))
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">psi</span><span style="color: #000000; background-color: #ffffff;">(a, r)</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
psi = dexp(-a * dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">psi</span>
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org19f7ef1" class="outline-4">
<h4 id="org19f7ef1"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-1-3">
<div class="exercise">
<p>
Write a function which computes the local kinetic energy at \(\mathbf{r}\).
The function accepts <code>a</code> and <code>r</code> as input arguments and returns the
local kinetic energy.
</p>
</div>
<p>
The local kinetic energy is defined as \[-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.\]
</p>
<p>
We differentiate \(\Psi\) with respect to \(x\):
</p>
<p>
\[\Psi(\mathbf{r}) = \exp(-a\,|\mathbf{r}|) \]
\[\frac{\partial \Psi}{\partial x}
= \frac{\partial \Psi}{\partial |\mathbf{r}|} \frac{\partial |\mathbf{r}|}{\partial x}
= - \frac{a\,x}{|\mathbf{r}|} \Psi(\mathbf{r}) \]
</p>
<p>
and we differentiate a second time:
</p>
<p>
\[
\frac{\partial^2 \Psi}{\partial x^2} =
\left( \frac{a^2\,x^2}{|\mathbf{r}|^2} -
\frac{a(y^2+z^2)}{|\mathbf{r}|^{3}} \right) \Psi(\mathbf{r}).
\]
</p>
<p>
The Laplacian operator \(\Delta = \frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\)
applied to the wave function gives:
</p>
<p>
\[
\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})
\]
</p>
<p>
So the local kinetic energy is
\[
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
\]
</p>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">kinetic</span>(a,r):
<span style="color: #a020f0;">return</span> -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">kinetic</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
kinetic = -0.5d0 * (a*a - (2.d0*a) / <span style="color: #a020f0;">&amp;</span>
dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ) )
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orgb7ba480" class="outline-4">
<h4 id="orgb7ba480"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-1-4">
<div class="exercise">
<p>
Write a function which computes the local energy at \(\mathbf{r}\).
The function accepts <code>x,y,z</code> as input arguments and returns the
local energy.
</p>
</div>
<p>
\[
E_L(\mathbf{r}) = -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) + V(\mathbf{r})
\]
</p>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">e_loc</span>(a,r):
<span style="color: #a020f0;">return</span> kinetic(a,r) + potential(r)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic, potential</span>
e_loc = kinetic(a,r) + potential(r)
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">e_loc</span>
</pre>
</div>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orge0a15c9" class="outline-3">
<h3 id="orge0a15c9"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-2-2">
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org58344c8" class="outline-4">
<h4 id="org58344c8"><span class="section-number-4">2.2.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-2-1">
<div class="exercise">
<p>
For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
local energy along the \(x\) axis.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
<span style="color: #a020f0;">import</span> matplotlib.pyplot <span style="color: #a020f0;">as</span> plt
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
<span style="color: #a0522d;">x</span>=np.linspace(-5,5)
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">make_array</span>(a):
<span style="color: #a0522d;">y</span>=np.array([ e_loc(a, np.array([t,0.,0.]) ) <span style="color: #a020f0;">for</span> t <span style="color: #a020f0;">in</span> x])
<span style="color: #a020f0;">return</span> y
plt.figure(figsize=(10,5))
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 1., 1.5, 2.]:
<span style="color: #a0522d;">y</span> = make_array(a)
plt.plot(x,y,label=f<span style="color: #8b2252;">"a={a}"</span>)
plt.tight_layout()
plt.legend()
plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</pre>
</div>
<div class="figure">
<p><img src="./plot_py.png" alt="plot_py.png" />
</p>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">plot</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), energy, dx, r(3), a(6)</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, j</span>
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
x(i) = -5.d0 + (i-1)*dx
<span style="color: #a020f0;">end do</span>
r(:) = 0.d0
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'# a='</span>, a(j)
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
r(1) = x(i)
energy = e_loc( a(j), r )
<span style="color: #a020f0;">print</span> *, x(i), energy
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">''</span>
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">''</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">plot</span>
</pre>
</div>
<p>
To compile and run:
</p>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
./plot_hydrogen &gt; data
</pre>
</div>
<p>
To plot the data using gnuplot:
</p>
<div class="org-src-container">
<pre class="src src-gnuplot">set grid
set xrange [-5:5]
set yrange [-2:1]
plot './data' index 0 using 1:2 with lines title 'a=0.1', \
'./data' index 1 using 1:2 with lines title 'a=0.2', \
'./data' index 2 using 1:2 with lines title 'a=0.5', \
'./data' index 3 using 1:2 with lines title 'a=1.0', \
'./data' index 4 using 1:2 with lines title 'a=1.5', \
'./data' index 5 using 1:2 with lines title 'a=2.0'
</pre>
</div>
<div class="figure">
<p><img src="plot.png" alt="plot.png" />
</p>
</div>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org8216be8" class="outline-3">
<h3 id="org8216be8"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-2-3">
<p>
If the space is discretized in small volume elements \(\mathbf{r}_i\)
of size \(\delta \mathbf{r}\), the expression of \(\langle E_L \rangle_{\Psi^2}\)
becomes a weighted average of the local energy, where the weights
are the values of the probability density at \(\mathbf{r}_i\)
multiplied by the volume element:
</p>
<p>
\[
\langle E \rangle_{\Psi^2} \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
w_i = \left[\Psi(\mathbf{r}_i)\right]^2 \delta \mathbf{r}
\]
</p>
<div class="note">
<p>
The energy is biased because:
</p>
<ul class="org-ul">
<li>The volume elements are not infinitely small (discretization error)</li>
<li>The energy is evaluated only inside the box (incompleteness of the space)</li>
</ul>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org8e43312" class="outline-4">
<h4 id="org8e43312"><span class="section-number-4">2.3.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-3-1">
<div class="exercise">
<p>
Compute a numerical estimate of the energy in a grid of
\(50\times50\times50\) points in the range \((-5,-5,-5) \le
\mathbf{r} \le (5,5,5)\).
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">norm</span> = 0.
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[0] = x
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[1] = y
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[2] = z
<span style="color: #a0522d;">w</span> = psi(a,r)
<span style="color: #a0522d;">w</span> = w * w * delta
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
<span style="color: #a0522d;">norm</span> += w
<span style="color: #a0522d;">E</span> = E / norm
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">energy_hydrogen</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, dx, r(3), a(6), norm</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
x(i) = -5.d0 + (i-1)*dx
<span style="color: #a020f0;">end do</span>
delta = dx**3
r(:) = 0.d0
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
energy = 0.d0
norm = 0.d0
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
r(1) = x(i)
<span style="color: #a020f0;">do</span> k=1,<span style="color: #a020f0;">size</span>(x)
r(2) = x(k)
<span style="color: #a020f0;">do</span> l=1,<span style="color: #a020f0;">size</span>(x)
r(3) = x(l)
w = psi(a(j),r)
w = w * w * delta
energy = energy + w * e_loc(a(j), r)
norm = norm + w
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
energy = energy / norm
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">energy_hydrogen</span>
</pre>
</div>
<p>
To compile the Fortran and run it:
</p>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
./energy_hydrogen
</pre>
</div>
<pre class="example">
a = 0.10000000000000001 E = -0.24518438948809140
a = 0.20000000000000001 E = -0.26966057967803236
a = 0.50000000000000000 E = -0.38563576125173815
a = 1.0000000000000000 E = -0.50000000000000000
a = 1.5000000000000000 E = -0.39242967082602065
a = 2.0000000000000000 E = -8.0869806678448772E-002
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org3c4621e" class="outline-3">
<h3 id="org3c4621e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-2-4">
<p>
The variance of the local energy is a functional of \(\Psi\)
which measures the magnitude of the fluctuations of the local
energy associated with \(\Psi\) around the average:
</p>
<p>
\[
\sigma^2(E_L) = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \left[
E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
\]
</p>
<p>
If the local energy is constant (i.e. \(\Psi\) is an eigenfunction of
\(\hat{H}\)) the variance is zero, so the variance of the local
energy can be used as a measure of the quality of a wave function.
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orgd7606c8" class="outline-4">
<h4 id="orgd7606c8"><span class="section-number-4">2.4.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-2-4-1">
<div class="exercise">
<p>
Compute a numerical estimate of the variance of the local energy
in a grid of \(50\times50\times50\) points in the range
\((-5,-5,-5)
\le \mathbf{r} \le (5,5,5)\) for different values of \(a\).
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">norm</span> = 0.
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[0] = x
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[1] = y
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[2] = z
<span style="color: #a0522d;">w</span> = psi(a, r)
<span style="color: #a0522d;">w</span> = w * w * delta
<span style="color: #a0522d;">El</span> = e_loc(a, r)
<span style="color: #a0522d;">E</span> += w * El
<span style="color: #a0522d;">norm</span> += w
<span style="color: #a0522d;">E</span> = E / norm
<span style="color: #a0522d;">s2</span> = 0.
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[0] = x
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[1] = y
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[2] = z
<span style="color: #a0522d;">w</span> = psi(a, r)
<span style="color: #a0522d;">w</span> = w * w * delta
<span style="color: #a0522d;">El</span> = e_loc(a, r)
<span style="color: #a0522d;">s2</span> += w * (El - E)**2
<span style="color: #a0522d;">s2</span> = s2 / norm
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">variance_hydrogen</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, dx, r(3), a(6), norm, s2</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
x(i) = -5.d0 + (i-1)*dx
<span style="color: #a020f0;">end do</span>
delta = dx**3
r(:) = 0.d0
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
energy = 0.d0
norm = 0.d0
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
r(1) = x(i)
<span style="color: #a020f0;">do</span> k=1,<span style="color: #a020f0;">size</span>(x)
r(2) = x(k)
<span style="color: #a020f0;">do</span> l=1,<span style="color: #a020f0;">size</span>(x)
r(3) = x(l)
w = psi(a(j),r)
w = w * w * delta
energy = energy + w * e_loc(a(j), r)
norm = norm + w
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
energy = energy / norm
s2 = 0.d0
norm = 0.d0
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
r(1) = x(i)
<span style="color: #a020f0;">do</span> k=1,<span style="color: #a020f0;">size</span>(x)
r(2) = x(k)
<span style="color: #a020f0;">do</span> l=1,<span style="color: #a020f0;">size</span>(x)
r(3) = x(l)
w = psi(a(j),r)
w = w * w * delta
s2 = s2 + w * ( e_loc(a(j), r) - energy )**2
norm = norm + w
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end do</span>
s2 = s2 / norm
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy, <span style="color: #8b2252;">' s2 = '</span>, s2
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">variance_hydrogen</span>
</pre>
</div>
<p>
To compile and run:
</p>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
./variance_hydrogen
</pre>
</div>
<pre class="example">
a = 0.10000000000000001 E = -0.24518438948809140 s2 = 2.6965218719733813E-002
a = 0.20000000000000001 E = -0.26966057967803236 s2 = 3.7197072370217653E-002
a = 0.50000000000000000 E = -0.38563576125173815 s2 = 5.3185967578488862E-002
a = 1.0000000000000000 E = -0.50000000000000000 s2 = 0.0000000000000000
a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909180444
a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270851303
</pre>
</div>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org2082f05" class="outline-2">
<h2 id="org2082f05"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-2" id="text-3">
<p>
Numerical integration with deterministic methods is very efficient
in low dimensions. When the number of dimensions becomes large,
instead of computing the average energy as a numerical integration
on a grid, it is usually more efficient to do a Monte Carlo sampling.
</p>
<p>
Moreover, a Monte Carlo sampling will alow us to remove the bias due
to the discretization of space, and compute a statistical confidence
interval.
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orgbcaa16b" class="outline-3">
<h3 id="orgbcaa16b"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-3-1">
<p>
To compute the statistical error, you need to perform \(M\)
independent Monte Carlo calculations. You will obtain \(M\) different
estimates of the energy, which are expected to have a Gaussian
distribution by the central limit theorem.
</p>
<p>
The estimate of the energy is
</p>
<p>
\[
E = \frac{1}{M} \sum_{i=1}^M E_M
\]
</p>
<p>
The variance of the average energies can be computed as
</p>
<p>
\[
\sigma^2 = \frac{1}{M-1} \sum_{i=1}^{M} (E_M - E)^2
\]
</p>
<p>
And the confidence interval is given by
</p>
<p>
\[
E \pm \delta E, \text{ where } \delta E = \frac{\sigma}{\sqrt{M}}
\]
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orga919e66" class="outline-4">
<h4 id="orga919e66"><span class="section-number-4">3.1.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-3-1-1">
<div class="exercise">
<p>
Write a function returning the average and statistical error of an
input array.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> sqrt
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">ave_error</span>(arr):
<span style="color: #a0522d;">M</span> = <span style="color: #483d8b;">len</span>(arr)
<span style="color: #a020f0;">assert</span> (M&gt;1)
<span style="color: #a0522d;">average</span> = <span style="color: #483d8b;">sum</span>(arr)/M
<span style="color: #a0522d;">variance</span> = 1./(M-1) * <span style="color: #483d8b;">sum</span>( [ (x - average)**2 <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> arr ] )
<span style="color: #a020f0;">return</span> (average, sqrt(variance/M))
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">ave_error</span>(x,n,ave,err)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> x(n) </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> variance</span>
<span style="color: #a020f0;">if</span> (n == 1) <span style="color: #a020f0;">then</span>
ave = x(1)
err = 0.d0
<span style="color: #a020f0;">else</span>
ave = <span style="color: #a020f0;">sum</span>(x(:)) / <span style="color: #a020f0;">dble</span>(n)
variance = <span style="color: #a020f0;">sum</span>( (x(:) - ave)**2 ) / <span style="color: #a020f0;">dble</span>(n-1)
err = dsqrt(variance/<span style="color: #a020f0;">dble</span>(n))
<span style="color: #a020f0;">endif</span>
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">ave_error</span>
</pre>
</div>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org3f0ebe8" class="outline-3">
<h3 id="org3f0ebe8"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-3-2">
<p>
We will now do our first Monte Carlo calculation to compute the
energy of the hydrogen atom.
</p>
<p>
At every Monte Carlo step:
</p>
<ul class="org-ul">
<li>Draw a random point \(\mathbf{r}_i\) in the box \((-5,-5,-5) \le
(x,y,z) \le (5,5,5)\)</li>
<li>Compute \([\Psi(\mathbf{r}_i)]^2\) and accumulate the result in a
variable <code>normalization</code></li>
<li>Compute \([\Psi(\mathbf{r}_i)]^2 \times E_L(\mathbf{r}_i)\), and accumulate the
result in a variable <code>energy</code></li>
</ul>
<p>
One Monte Carlo run will consist of \(N\) Monte Carlo steps. Once all the
steps have been computed, the run returns the average energy
\(\bar{E}_k\) over the \(N\) steps of the run.
</p>
<p>
To compute the statistical error, perform \(M\) runs. The final
estimate of the energy will be the average over the \(\bar{E}_k\),
and the variance of the \(\bar{E}_k\) will be used to compute the
statistical error.
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orgbbdcb31" class="outline-4">
<h4 id="orgbbdcb31"><span class="section-number-4">3.2.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-3-2-1">
<div class="exercise">
<p>
Parameterize the wave function with \(a=0.9\). Perform 30
independent Monte Carlo runs, each with 100 000 Monte Carlo
steps. Store the final energies of each run and use this array to
compute the average energy and the associated error bar.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax):
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">N</span> = 0.
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
<span style="color: #a0522d;">r</span> = np.random.uniform(-5., 5., (3))
<span style="color: #a0522d;">w</span> = psi(a,r)
<span style="color: #a0522d;">w</span> = w*w
<span style="color: #a0522d;">N</span> += w
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
<span style="color: #a020f0;">return</span> E/N
<span style="color: #a0522d;">a</span> = 0.9
<span style="color: #a0522d;">nmax</span> = 100000
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,nmax) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="note">
<p>
When running Monte Carlo calculations, the number of steps is
usually very large. We expect <code>nmax</code> to be possibly larger than 2
billion, so we use 8-byte integers (<code>integer*8</code>) to represent it, as
well as the index of the current step.
</p>
</div>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>(a,nmax,energy)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, r(3), w</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
energy = 0.d0
norm = 0.d0
<span style="color: #a020f0;">do</span> istep = 1,nmax
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(r)
r(:) = -5.d0 + 10.d0*r(:)
w = psi(a,r)
w = w*w
norm = norm + w
energy = energy + w * e_loc(a,r)
<span style="color: #a020f0;">end do</span>
energy = energy / norm
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 0.9</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #a020f0;">do</span> irun=1,nruns
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">uniform_montecarlo</span>(a,nmax,X(irun))
<span style="color: #a020f0;">enddo</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
./qmc_uniform
</pre>
</div>
<pre class="example">
E = -0.49588321986667677 +/- 7.1758863546737969E-004
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org51cfa79" class="outline-3">
<h3 id="org51cfa79"><span class="section-number-3">3.3</span> Gaussian sampling</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-3-3">
<p>
We will now improve the sampling and allow to sample in the whole
3D space, correcting the bias related to the sampling in the box.
</p>
<p>
Instead of drawing uniform random numbers, we will draw Gaussian
random numbers centered on 0 and with a variance of 1.
</p>
<p>
To obtain Gaussian-distributed random numbers, you can apply the
<a href="https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform">Box Muller transform</a> to uniform random numbers:
</p>
\begin{eqnarray*}
z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\
z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2)
\end{eqnarray*}
<p>
Here is a Fortran implementation returning a Gaussian-distributed
n-dimensional vector \(\mathbf{z}\);
</p>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">random_gauss</span>(z,n)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> z(n)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> u(n+1)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> two_pi = 2.d0*dacos(-1.d0)</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
<span style="color: #a020f0;">if</span> (<span style="color: #a020f0;">iand</span>(n,1) == 0) <span style="color: #a020f0;">then</span>
! <span style="color: #b22222;">n is even</span>
<span style="color: #a020f0;">do</span> i=1,n,2
z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) * dcos( two_pi*u(i+1) )
<span style="color: #a020f0;">end do</span>
<span style="color: #a020f0;">else</span>
! <span style="color: #b22222;">n is odd</span>
<span style="color: #a020f0;">do</span> i=1,n-1,2
z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) * dcos( two_pi*u(i+1) )
<span style="color: #a020f0;">end do</span>
z(n) = dsqrt(-2.d0*dlog(u(n)))
z(n) = z(n) * dcos( two_pi*u(n+1) )
<span style="color: #a020f0;">end if</span>
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">random_gauss</span>
</pre>
</div>
<p>
Now the sampling probability can be inserted into the equation of the energy:
</p>
<p>
\[
E = \frac{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r})\right]^2}{P(\mathbf{r})}\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r}) \right]^2}{P(\mathbf{r})} d\mathbf{r}}
\]
</p>
<p>
with the Gaussian probability
</p>
<p>
\[
P(\mathbf{r}) = \frac{1}{(2 \pi)^{3/2}}\exp\left( -\frac{\mathbf{r}^2}{2} \right).
\]
</p>
<p>
As the coordinates are drawn with probability \(P(\mathbf{r})\), the
average energy can be computed as
</p>
<p>
\[
E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
w_i = \frac{\left[\Psi(\mathbf{r}_i)\right]^2}{P(\mathbf{r}_i)} \delta \mathbf{r}
\]
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-orgbec6ea5" class="outline-4">
<h4 id="orgbec6ea5"><span class="section-number-4">3.3.1</span> Exercise</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-3-3-1">
<div class="exercise">
<p>
Modify the exercise of the previous section to sample with
Gaussian-distributed random numbers. Can you see an reduction in
the statistical error?
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
<span style="color: #a0522d;">norm_gauss</span> = 1./(2.*np.pi)**(1.5)
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">gaussian</span>(r):
<span style="color: #a020f0;">return</span> norm_gauss * np.exp(-np.dot(r,r)*0.5)
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax):
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">N</span> = 0.
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
<span style="color: #a0522d;">r</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">w</span> = psi(a,r)
<span style="color: #a0522d;">w</span> = w*w / gaussian(r)
<span style="color: #a0522d;">N</span> += w
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
<span style="color: #a020f0;">return</span> E/N
<span style="color: #a0522d;">a</span> = 0.9
<span style="color: #a0522d;">nmax</span> = 100000
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,nmax) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">gaussian</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> r(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0)</span>
gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">gaussian</span>
<span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">gaussian_montecarlo</span>(a,nmax,energy)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, r(3), w</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi, gaussian</span>
energy = 0.d0
norm = 0.d0
<span style="color: #a020f0;">do</span> istep = 1,nmax
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r,3)
w = psi(a,r)
w = w*w / gaussian(r)
norm = norm + w
energy = energy + w * e_loc(a,r)
<span style="color: #a020f0;">end do</span>
energy = energy / norm
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">gaussian_montecarlo</span>
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 0.9</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #a020f0;">do</span> irun=1,nruns
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">gaussian_montecarlo</span>(a,nmax,X(irun))
<span style="color: #a020f0;">enddo</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
./qmc_gaussian
</pre>
</div>
<pre class="example">
E = -0.49517104619091717 +/- 1.0685523607878961E-004
</pre>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org39b0dd3" class="outline-3">
<h3 id="org39b0dd3"><span class="section-number-3">3.4</span> Sampling with \(\Psi^2\)</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-3-4">
<p>
We will now use the square of the wave function to make the sampling:
</p>
<p>
\[
P(\mathbf{r}) = \left[\Psi(\mathbf{r})\right]^2
\]
</p>
<p>
The expression for the energy will be simplified to the average of
the local energies, each with a weight of 1.
</p>
<p>
\[
E \approx \frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r}_i)
\]
</p>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org174a24c" class="outline-4">
<h4 id="org174a24c"><span class="section-number-4">3.4.1</span> Importance sampling</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-3-4-1">
<p>
To generate the probability density \(\Psi^2\), we consider a
diffusion process characterized by a time-dependent density
\([\Psi(\mathbf{r},t)]^2\), which obeys the Fokker-Planck equation:
</p>
<p>
\[
\frac{\partial \Psi^2}{\partial t} = \sum_i D
\frac{\partial}{\partial \mathbf{r}_i} \left(
\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right)
[\Psi(\mathbf{r},t)]^2.
\]
</p>
<p>
\(D\) is the diffusion constant and \(F_i\) is the i-th component of a
drift velocity caused by an external potential. For a stationary
density, \( \frac{\partial \Psi^2}{\partial t} = 0 \), so
</p>
\begin{eqnarray*}
0 & = & \sum_i D
\frac{\partial}{\partial \mathbf{r}_i} \left(
\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right)
[\Psi(\mathbf{r})]^2 \\
0 & = & \sum_i D
\frac{\partial}{\partial \mathbf{r}_i} \left(
\frac{\partial [\Psi(\mathbf{r})]^2}{\partial \mathbf{r}_i} -
F_i(\mathbf{r})\,[\Psi(\mathbf{r})]^2 \right) \\
0 & = &
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} -
\frac{\partial F_i }{\partial \mathbf{r}_i}[\Psi(\mathbf{r})]^2 -
\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
\end{eqnarray*}
<p>
we search for a drift function which satisfies
</p>
<p>
\[
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} =
[\Psi(\mathbf{r})]^2 \frac{\partial F_i }{\partial \mathbf{r}_i} +
\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
\]
</p>
<p>
to obtain a second derivative on the left, we need the drift to be
of the form
\[
F_i(\mathbf{r}) = g(\mathbf{r}) \frac{\partial \Psi^2}{\partial \mathbf{r}_i}
\]
</p>
<p>
\[
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} =
[\Psi(\mathbf{r})]^2 \frac{\partial
g(\mathbf{r})}{\partial \mathbf{r}_i}\frac{\partial \Psi^2}{\partial \mathbf{r}_i} +
[\Psi(\mathbf{r})]^2 g(\mathbf{r}) \frac{\partial^2
\Psi^2}{\partial \mathbf{r}_i^2} +
\frac{\partial \Psi^2}{\partial \mathbf{r}_i}
g(\mathbf{r}) \frac{\partial \Psi^2}{\partial \mathbf{r}_i}
\]
</p>
<p>
\(g = 1 / \Psi^2\) satisfies this equation, so
</p>
<p>
\[
F(\mathbf{r}) = \frac{\nabla [\Psi(\mathbf{r})]^2}{[\Psi(\mathbf{r})]^2} = 2 \frac{\nabla
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} = 2 \nabla \left( \log \Psi(\mathbf{r}) \right)
\]
</p>
<p>
In statistical mechanics, Fokker-Planck trajectories are generated
by a Langevin equation:
</p>
<p>
\[
\frac{\partial \mathbf{r}(t)}{\partial t} = 2D \frac{\nabla
\Psi(\mathbf{r}(t))}{\Psi} + \eta
\]
</p>
<p>
where \(\eta\) is a normally-distributed fluctuating random force.
</p>
<p>
Discretizing this differential equation gives the following drifted
diffusion scheme:
</p>
<p>
\[
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau\, 2D \frac{\nabla
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi
\]
where \(\chi\) is a Gaussian random variable with zero mean and
variance \(\tau\,2D\).
</p>
</div>
<ol class="org-ol">
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<li><a id="orgfcbecfe"></a>Exercise 1<br />
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-5" id="text-3-4-1-1">
<div class="exercise">
<p>
Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\).
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">drift</span>(a,r):
<span style="color: #a0522d;">ar_inv</span> = -a/np.sqrt(np.dot(r,r))
<span style="color: #a020f0;">return</span> r * ar_inv
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">drift</span>(a,r,b)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> b(3)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ar_inv</span>
ar_inv = -a / dsqrt(r(1)*r(1) + r(2)*r(2) + r(3)*r(3))
b(:) = r(:) * ar_inv
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">drift</span>
</pre>
</div>
</div>
</li>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<li><a id="orge541cc5"></a><span class="todo TODO">TODO</span> Exercise 2<br />
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-5" id="text-3-4-1-2">
<div class="exercise">
<p>
Sample \(\Psi^2\) approximately using the drifted diffusion scheme,
with a diffusion constant \(D=1/2\). You can use a time step of
0.001 a.u.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,tau,nmax):
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
# <span style="color: #b22222;">Initialization</span>
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">N</span> = 0.
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">r_new</span> = r_old + tau * d_old + chi*sq_tau
<span style="color: #a0522d;">N</span> += 1.
<span style="color: #a0522d;">E</span> += e_loc(a,r_new)
<span style="color: #a0522d;">r_old</span> = r_new
<span style="color: #a020f0;">return</span> E/N
<span style="color: #a0522d;">a</span> = 0.9
<span style="color: #a0522d;">nmax</span> = 100000
<span style="color: #a0522d;">tau</span> = 0.2
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,nmax) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,energy)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, tau</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, r_old(3), r_new(3), d_old(3), sq_tau, chi(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc</span>
sq_tau = dsqrt(tau)
! <span style="color: #b22222;">Initialization</span>
energy = 0.d0
norm = 0.d0
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
<span style="color: #a020f0;">do</span> istep = 1,nmax
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
norm = norm + 1.d0
energy = energy + e_loc(a,r_new)
r_old(:) = r_new(:)
<span style="color: #a020f0;">end do</span>
energy = energy / norm
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 0.9</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 0.2</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #a020f0;">do</span> irun=1,nruns
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,X(irun))
<span style="color: #a020f0;">enddo</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
./vmc
</pre>
</div>
<pre class="example">
E = -0.48584030499187431 +/- 1.0411743995438257E-004
</pre>
</div>
</li>
</ol>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org1a431c2" class="outline-4">
<h4 id="org1a431c2"><span class="section-number-4">3.4.2</span> Metropolis algorithm</h4>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-4" id="text-3-4-2">
<p>
Discretizing the differential equation to generate the desired
probability density will suffer from a discretization error
leading to biases in the averages. The <a href="https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm">Metropolis-Hastings
sampling algorithm</a> removes exactly the discretization errors, so
large time steps can be employed.
</p>
<p>
After the new position \(\mathbf{r}_{n+1}\) has been computed, an
additional accept/reject step is performed. The acceptance
probability \(A\) is chosen so that it is consistent with the
probability of leaving \(\mathbf{r}_n\) and the probability of
entering \(\mathbf{r}_{n+1}\):
</p>
<p>
\[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1,
\frac{g(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})}
{g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})}
\right)
\]
</p>
<p>
In our Hydrogen atom example, \(P\) is \(\Psi^2\) and \(g\) is a
solution of the discretized Fokker-Planck equation:
</p>
\begin{eqnarray*}
P(r_{n}) &=& \Psi^2(\mathbf{r}_n) \\
g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = &
\frac{1}{(4\pi\,D\,\tau)^{3/2}} \exp \left[ - \frac{\left(
\mathbf{r}_{n+1} - \mathbf{r}_{n} - 2D \frac{\nabla
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{4D\,\tau} \right]
\end{eqnarray*}
<p>
The accept/reject step is the following:
</p>
<ul class="org-ul">
<li>Compute \(A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\).</li>
<li>Draw a uniform random number \(u\)</li>
<li>if \(u \le A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\), accept
the move</li>
<li>if \(u>A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\), reject
the move: set \(\mathbf{r}_{n+1} = \mathbf{r}_{n}\), but <b>don't
remove the sample from the average!</b></li>
</ul>
<p>
The <i>acceptance rate</i> is the ratio of the number of accepted step
over the total number of steps. The time step should be adapted so
that the acceptance rate is around 0.5 for a good efficiency of
the simulation.
</p>
</div>
<ol class="org-ol">
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<li><a id="orgdec27ad"></a>Exercise<br />
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-5" id="text-3-4-2-1">
<div class="exercise">
<p>
Modify the previous program to introduce the accept/reject step.
You should recover the unbiased result.
Adjust the time-step so that the acceptance rate is 0.5.
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,tau,nmax):
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">N</span> = 0.
<span style="color: #a0522d;">accep_rate</span> = 0.
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
<span style="color: #a0522d;">d2_old</span> = np.dot(d_old,d_old)
<span style="color: #a0522d;">psi_old</span> = psi(a,r_old)
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">r_new</span> = r_old + tau * d_old + sq_tau * chi
<span style="color: #a0522d;">d_new</span> = drift(a,r_new)
<span style="color: #a0522d;">d2_new</span> = np.dot(d_new,d_new)
<span style="color: #a0522d;">psi_new</span> = psi(a,r_new)
# <span style="color: #b22222;">Metropolis</span>
<span style="color: #a0522d;">prod</span> = np.dot((d_new + d_old), (r_new - r_old))
<span style="color: #a0522d;">argexpo</span> = 0.5 * (d2_new - d2_old)*tau + prod
<span style="color: #a0522d;">q</span> = psi_new / psi_old
<span style="color: #a0522d;">q</span> = np.exp(-argexpo) * q*q
<span style="color: #a020f0;">if</span> np.random.uniform() &lt; q:
<span style="color: #a0522d;">accep_rate</span> += 1.
<span style="color: #a0522d;">r_old</span> = r_new
<span style="color: #a0522d;">d_old</span> = d_new
<span style="color: #a0522d;">d2_old</span> = d2_new
<span style="color: #a0522d;">psi_old</span> = psi_new
<span style="color: #a0522d;">N</span> += 1.
<span style="color: #a0522d;">E</span> += e_loc(a,r_old)
<span style="color: #a020f0;">return</span> E/N, accep_rate/N
<span style="color: #a0522d;">a</span> = 0.9
<span style="color: #a0522d;">nmax</span> = 100000
<span style="color: #a0522d;">tau</span> = 1.0
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,nmax) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error([x[0] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error([x[1] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,energy,accep_rate)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, tau</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep_rate</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, sq_tau, chi(3), d2_old, prod, u</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new, d2_new, argexpo, q</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
sq_tau = dsqrt(tau)
! <span style="color: #b22222;">Initialization</span>
energy = 0.d0
norm = 0.d0
accep_rate = 0.d0
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
psi_old = psi(a,r_old)
<span style="color: #a020f0;">do</span> istep = 1,nmax
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_new,d_new)
d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
psi_new = psi(a,r_new)
! <span style="color: #b22222;">Metropolis</span>
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + <span style="color: #a020f0;">&amp;</span>
(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + <span style="color: #a020f0;">&amp;</span>
(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
q = psi_new / psi_old
q = dexp(-argexpo) * q*q
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
<span style="color: #a020f0;">if</span> (u&lt;q) <span style="color: #a020f0;">then</span>
accep_rate = accep_rate + 1.d0
r_old(:) = r_new(:)
d_old(:) = d_new(:)
d2_old = d2_new
psi_old = psi_new
<span style="color: #a020f0;">end if</span>
norm = norm + 1.d0
energy = energy + e_loc(a,r_old)
<span style="color: #a020f0;">end do</span>
energy = energy / norm
accep_rate = accep_rate / norm
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 0.9</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 1.0</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #a020f0;">do</span> irun=1,nruns
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,X(irun),accep(irun))
<span style="color: #a020f0;">enddo</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis
</pre>
</div>
<pre class="example">
E = -0.49499990423528023 +/- 1.5958250761863871E-004
A = 0.78861366666666655 +/- 3.5096729498002445E-004
</pre>
</div>
</li>
</ol>
</div>
</div>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org767855f" class="outline-2">
<h2 id="org767855f"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-2" id="text-4">
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org50b0f1f" class="outline-3">
<h3 id="org50b0f1f"><span class="section-number-3">4.1</span> Hydrogen atom</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-4-1">
</div>
<ol class="org-ol">
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<li><a id="org6dfc9ee"></a>Exercise<br />
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-5" id="text-4-1-0-1">
<div class="exercise">
<p>
Modify the Metropolis VMC program to introduce the PDMC weight.
In the limit \(\tau \rightarrow 0\), you should recover the exact
energy of H for any value of \(a\).
</p>
</div>
<p>
<b>Python</b>
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,tau,nmax,Eref):
<span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">N</span> = 0.
<span style="color: #a0522d;">accep_rate</span> = 0.
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
<span style="color: #a0522d;">d2_old</span> = np.dot(d_old,d_old)
<span style="color: #a0522d;">psi_old</span> = psi(a,r_old)
<span style="color: #a0522d;">w</span> = 1.0
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
<span style="color: #a0522d;">el</span> = e_loc(a,r_old)
<span style="color: #a0522d;">w</span> *= np.exp(-tau*(el - Eref))
<span style="color: #a0522d;">N</span> += w
<span style="color: #a0522d;">E</span> += w * el
<span style="color: #a0522d;">r_new</span> = r_old + tau * d_old + sq_tau * chi
<span style="color: #a0522d;">d_new</span> = drift(a,r_new)
<span style="color: #a0522d;">d2_new</span> = np.dot(d_new,d_new)
<span style="color: #a0522d;">psi_new</span> = psi(a,r_new)
# <span style="color: #b22222;">Metropolis</span>
<span style="color: #a0522d;">prod</span> = np.dot((d_new + d_old), (r_new - r_old))
<span style="color: #a0522d;">argexpo</span> = 0.5 * (d2_new - d2_old)*tau + prod
<span style="color: #a0522d;">q</span> = psi_new / psi_old
<span style="color: #a0522d;">q</span> = np.exp(-argexpo) * q*q
# <span style="color: #b22222;">PDMC weight</span>
<span style="color: #a020f0;">if</span> np.random.uniform() &lt; q:
<span style="color: #a0522d;">accep_rate</span> += w
<span style="color: #a0522d;">r_old</span> = r_new
<span style="color: #a0522d;">d_old</span> = d_new
<span style="color: #a0522d;">d2_old</span> = d2_new
<span style="color: #a0522d;">psi_old</span> = psi_new
<span style="color: #a020f0;">return</span> E/N, accep_rate/N
<span style="color: #a0522d;">a</span> = 0.9
<span style="color: #a0522d;">nmax</span> = 10000
<span style="color: #a0522d;">tau</span> = .1
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,nmax,-0.5) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error([x[0] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error([x[1] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}"</span>)
</pre>
</div>
<p>
<b>Fortran</b>
</p>
<div class="org-src-container">
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,energy,accep_rate)
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, tau</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep_rate</span>
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, sq_tau, chi(3), d2_old, prod, u</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new, d2_new, argexpo, q</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
sq_tau = dsqrt(tau)
! <span style="color: #b22222;">Initialization</span>
energy = 0.d0
norm = 0.d0
accep_rate = 0.d0
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
psi_old = psi(a,r_old)
<span style="color: #a020f0;">do</span> istep = 1,nmax
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_new,d_new)
d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
psi_new = psi(a,r_new)
! <span style="color: #b22222;">Metropolis</span>
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + <span style="color: #a020f0;">&amp;</span>
(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + <span style="color: #a020f0;">&amp;</span>
(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
q = psi_new / psi_old
q = dexp(-argexpo) * q*q
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
<span style="color: #a020f0;">if</span> (u&lt;q) <span style="color: #a020f0;">then</span>
accep_rate = accep_rate + 1.d0
r_old(:) = r_new(:)
d_old(:) = d_new(:)
d2_old = d2_new
psi_old = psi_new
<span style="color: #a020f0;">end if</span>
norm = norm + 1.d0
energy = energy + e_loc(a,r_old)
<span style="color: #a020f0;">end do</span>
energy = energy / norm
accep_rate = accep_rate / norm
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 0.9</span>
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 1.0</span>
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
<span style="color: #a020f0;">do</span> irun=1,nruns
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,tau,nmax,X(irun),accep(irun))
<span style="color: #a020f0;">enddo</span>
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis
</pre>
</div>
<pre class="example">
E = -0.49499990423528023 +/- 1.5958250761863871E-004
A = 0.78861366666666655 +/- 3.5096729498002445E-004
</pre>
</div>
</li>
</ol>
</div>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<div id="outline-container-org64f2d1e" class="outline-3">
<h3 id="org64f2d1e"><span class="section-number-3">4.2</span> Dihydrogen</h3>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<div class="outline-text-3" id="text-4-2">
<p>
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
\(1s\) orbitals of the hydrogen atoms:
</p>
<p>
\[
\Psi(\mathbf{r}_1, \mathbf{r}_2) =
\exp(-(\mathbf{r}_1 - \mathbf{R}_A)) +
\]
where \(\mathbf{r}_1\) and \(\mathbf{r}_2\) denote the electron
coordinates and \(\mathbf{R}_A\) and \(\mathbf{R}_B\) the coordinates of
the nuclei.
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:07:39 +01:00
<p class="date">Created: 2021-01-13 Wed 17:07</p>
deploy: 50dc730ea5889a23b685a0c7e4824cf3ce959379
2021-01-13 18:02:56 +01:00
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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