mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-08-25 14:41:47 +02:00
2088 lines
76 KiB
HTML
2088 lines
76 KiB
HTML
<?xml version="1.0" encoding="utf-8"?>
|
||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
|
||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||
<head>
|
||
<!-- 2021-02-04 Thu 07:47 -->
|
||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||
<title>Quantum Monte Carlo</title>
|
||
<meta name="generator" content="Org mode" />
|
||
<meta name="author" content="Anthony Scemama, Claudia Filippi" />
|
||
<style type="text/css">
|
||
<!--/*--><![CDATA[/*><!--*/
|
||
.title { text-align: center;
|
||
margin-bottom: .2em; }
|
||
.subtitle { text-align: center;
|
||
font-size: medium;
|
||
font-weight: bold;
|
||
margin-top:0; }
|
||
.todo { font-family: monospace; color: red; }
|
||
.done { font-family: monospace; color: green; }
|
||
.priority { font-family: monospace; color: orange; }
|
||
.tag { background-color: #eee; font-family: monospace;
|
||
padding: 2px; font-size: 80%; font-weight: normal; }
|
||
.timestamp { color: #bebebe; }
|
||
.timestamp-kwd { color: #5f9ea0; }
|
||
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
|
||
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
|
||
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
|
||
.underline { text-decoration: underline; }
|
||
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
|
||
p.verse { margin-left: 3%; }
|
||
pre {
|
||
border: 1px solid #ccc;
|
||
box-shadow: 3px 3px 3px #eee;
|
||
padding: 8pt;
|
||
font-family: monospace;
|
||
overflow: auto;
|
||
margin: 1.2em;
|
||
}
|
||
pre.src {
|
||
position: relative;
|
||
overflow: visible;
|
||
padding-top: 1.2em;
|
||
}
|
||
pre.src:before {
|
||
display: none;
|
||
position: absolute;
|
||
background-color: white;
|
||
top: -10px;
|
||
right: 10px;
|
||
padding: 3px;
|
||
border: 1px solid black;
|
||
}
|
||
pre.src:hover:before { display: inline;}
|
||
/* Languages per Org manual */
|
||
pre.src-asymptote:before { content: 'Asymptote'; }
|
||
pre.src-awk:before { content: 'Awk'; }
|
||
pre.src-C:before { content: 'C'; }
|
||
/* pre.src-C++ doesn't work in CSS */
|
||
pre.src-clojure:before { content: 'Clojure'; }
|
||
pre.src-css:before { content: 'CSS'; }
|
||
pre.src-D:before { content: 'D'; }
|
||
pre.src-ditaa:before { content: 'ditaa'; }
|
||
pre.src-dot:before { content: 'Graphviz'; }
|
||
pre.src-calc:before { content: 'Emacs Calc'; }
|
||
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
|
||
pre.src-fortran:before { content: 'Fortran'; }
|
||
pre.src-gnuplot:before { content: 'gnuplot'; }
|
||
pre.src-haskell:before { content: 'Haskell'; }
|
||
pre.src-hledger:before { content: 'hledger'; }
|
||
pre.src-java:before { content: 'Java'; }
|
||
pre.src-js:before { content: 'Javascript'; }
|
||
pre.src-latex:before { content: 'LaTeX'; }
|
||
pre.src-ledger:before { content: 'Ledger'; }
|
||
pre.src-lisp:before { content: 'Lisp'; }
|
||
pre.src-lilypond:before { content: 'Lilypond'; }
|
||
pre.src-lua:before { content: 'Lua'; }
|
||
pre.src-matlab:before { content: 'MATLAB'; }
|
||
pre.src-mscgen:before { content: 'Mscgen'; }
|
||
pre.src-ocaml:before { content: 'Objective Caml'; }
|
||
pre.src-octave:before { content: 'Octave'; }
|
||
pre.src-org:before { content: 'Org mode'; }
|
||
pre.src-oz:before { content: 'OZ'; }
|
||
pre.src-plantuml:before { content: 'Plantuml'; }
|
||
pre.src-processing:before { content: 'Processing.js'; }
|
||
pre.src-python:before { content: 'Python'; }
|
||
pre.src-R:before { content: 'R'; }
|
||
pre.src-ruby:before { content: 'Ruby'; }
|
||
pre.src-sass:before { content: 'Sass'; }
|
||
pre.src-scheme:before { content: 'Scheme'; }
|
||
pre.src-screen:before { content: 'Gnu Screen'; }
|
||
pre.src-sed:before { content: 'Sed'; }
|
||
pre.src-sh:before { content: 'shell'; }
|
||
pre.src-sql:before { content: 'SQL'; }
|
||
pre.src-sqlite:before { content: 'SQLite'; }
|
||
/* additional languages in org.el's org-babel-load-languages alist */
|
||
pre.src-forth:before { content: 'Forth'; }
|
||
pre.src-io:before { content: 'IO'; }
|
||
pre.src-J:before { content: 'J'; }
|
||
pre.src-makefile:before { content: 'Makefile'; }
|
||
pre.src-maxima:before { content: 'Maxima'; }
|
||
pre.src-perl:before { content: 'Perl'; }
|
||
pre.src-picolisp:before { content: 'Pico Lisp'; }
|
||
pre.src-scala:before { content: 'Scala'; }
|
||
pre.src-shell:before { content: 'Shell Script'; }
|
||
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
|
||
/* additional language identifiers per "defun org-babel-execute"
|
||
in ob-*.el */
|
||
pre.src-cpp:before { content: 'C++'; }
|
||
pre.src-abc:before { content: 'ABC'; }
|
||
pre.src-coq:before { content: 'Coq'; }
|
||
pre.src-groovy:before { content: 'Groovy'; }
|
||
/* additional language identifiers from org-babel-shell-names in
|
||
ob-shell.el: ob-shell is the only babel language using a lambda to put
|
||
the execution function name together. */
|
||
pre.src-bash:before { content: 'bash'; }
|
||
pre.src-csh:before { content: 'csh'; }
|
||
pre.src-ash:before { content: 'ash'; }
|
||
pre.src-dash:before { content: 'dash'; }
|
||
pre.src-ksh:before { content: 'ksh'; }
|
||
pre.src-mksh:before { content: 'mksh'; }
|
||
pre.src-posh:before { content: 'posh'; }
|
||
/* Additional Emacs modes also supported by the LaTeX listings package */
|
||
pre.src-ada:before { content: 'Ada'; }
|
||
pre.src-asm:before { content: 'Assembler'; }
|
||
pre.src-caml:before { content: 'Caml'; }
|
||
pre.src-delphi:before { content: 'Delphi'; }
|
||
pre.src-html:before { content: 'HTML'; }
|
||
pre.src-idl:before { content: 'IDL'; }
|
||
pre.src-mercury:before { content: 'Mercury'; }
|
||
pre.src-metapost:before { content: 'MetaPost'; }
|
||
pre.src-modula-2:before { content: 'Modula-2'; }
|
||
pre.src-pascal:before { content: 'Pascal'; }
|
||
pre.src-ps:before { content: 'PostScript'; }
|
||
pre.src-prolog:before { content: 'Prolog'; }
|
||
pre.src-simula:before { content: 'Simula'; }
|
||
pre.src-tcl:before { content: 'tcl'; }
|
||
pre.src-tex:before { content: 'TeX'; }
|
||
pre.src-plain-tex:before { content: 'Plain TeX'; }
|
||
pre.src-verilog:before { content: 'Verilog'; }
|
||
pre.src-vhdl:before { content: 'VHDL'; }
|
||
pre.src-xml:before { content: 'XML'; }
|
||
pre.src-nxml:before { content: 'XML'; }
|
||
/* add a generic configuration mode; LaTeX export needs an additional
|
||
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
|
||
pre.src-conf:before { content: 'Configuration File'; }
|
||
|
||
table { border-collapse:collapse; }
|
||
caption.t-above { caption-side: top; }
|
||
caption.t-bottom { caption-side: bottom; }
|
||
td, th { vertical-align:top; }
|
||
th.org-right { text-align: center; }
|
||
th.org-left { text-align: center; }
|
||
th.org-center { text-align: center; }
|
||
td.org-right { text-align: right; }
|
||
td.org-left { text-align: left; }
|
||
td.org-center { text-align: center; }
|
||
dt { font-weight: bold; }
|
||
.footpara { display: inline; }
|
||
.footdef { margin-bottom: 1em; }
|
||
.figure { padding: 1em; }
|
||
.figure p { text-align: center; }
|
||
.inlinetask {
|
||
padding: 10px;
|
||
border: 2px solid gray;
|
||
margin: 10px;
|
||
background: #ffffcc;
|
||
}
|
||
#org-div-home-and-up
|
||
{ text-align: right; font-size: 70%; white-space: nowrap; }
|
||
textarea { overflow-x: auto; }
|
||
.linenr { font-size: smaller }
|
||
.code-highlighted { background-color: #ffff00; }
|
||
.org-info-js_info-navigation { border-style: none; }
|
||
#org-info-js_console-label
|
||
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
|
||
.org-info-js_search-highlight
|
||
{ background-color: #ffff00; color: #000000; font-weight: bold; }
|
||
.org-svg { width: 90%; }
|
||
/*]]>*/-->
|
||
</style>
|
||
<link rel="stylesheet" title="Standard" href="worg.css" type="text/css" />
|
||
|
||
<script type="text/javascript" src="org-info.js">
|
||
/**
|
||
*
|
||
* @source: org-info.js
|
||
*
|
||
* @licstart The following is the entire license notice for the
|
||
* JavaScript code in org-info.js.
|
||
*
|
||
* Copyright (C) 2012-2019 Free Software Foundation, Inc.
|
||
*
|
||
*
|
||
* The JavaScript code in this tag is free software: you can
|
||
* redistribute it and/or modify it under the terms of the GNU
|
||
* General Public License (GNU GPL) as published by the Free Software
|
||
* Foundation, either version 3 of the License, or (at your option)
|
||
* any later version. The code is distributed WITHOUT ANY WARRANTY;
|
||
* without even the implied warranty of MERCHANTABILITY or FITNESS
|
||
* FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
|
||
*
|
||
* As additional permission under GNU GPL version 3 section 7, you
|
||
* may distribute non-source (e.g., minimized or compacted) forms of
|
||
* that code without the copy of the GNU GPL normally required by
|
||
* section 4, provided you include this license notice and a URL
|
||
* through which recipients can access the Corresponding Source.
|
||
*
|
||
* @licend The above is the entire license notice
|
||
* for the JavaScript code in org-info.js.
|
||
*
|
||
*/
|
||
</script>
|
||
|
||
<script type="text/javascript">
|
||
|
||
/*
|
||
@licstart The following is the entire license notice for the
|
||
JavaScript code in this tag.
|
||
|
||
Copyright (C) 2012-2019 Free Software Foundation, Inc.
|
||
|
||
The JavaScript code in this tag is free software: you can
|
||
redistribute it and/or modify it under the terms of the GNU
|
||
General Public License (GNU GPL) as published by the Free Software
|
||
Foundation, either version 3 of the License, or (at your option)
|
||
any later version. The code is distributed WITHOUT ANY WARRANTY;
|
||
without even the implied warranty of MERCHANTABILITY or FITNESS
|
||
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
|
||
|
||
As additional permission under GNU GPL version 3 section 7, you
|
||
may distribute non-source (e.g., minimized or compacted) forms of
|
||
that code without the copy of the GNU GPL normally required by
|
||
section 4, provided you include this license notice and a URL
|
||
through which recipients can access the Corresponding Source.
|
||
|
||
|
||
@licend The above is the entire license notice
|
||
for the JavaScript code in this tag.
|
||
*/
|
||
|
||
<!--/*--><![CDATA[/*><!--*/
|
||
org_html_manager.set("TOC_DEPTH", "4");
|
||
org_html_manager.set("LINK_HOME", "");
|
||
org_html_manager.set("LINK_UP", "");
|
||
org_html_manager.set("LOCAL_TOC", "1");
|
||
org_html_manager.set("VIEW_BUTTONS", "0");
|
||
org_html_manager.set("MOUSE_HINT", "underline");
|
||
org_html_manager.set("FIXED_TOC", "0");
|
||
org_html_manager.set("TOC", "1");
|
||
org_html_manager.set("VIEW", "info");
|
||
org_html_manager.setup(); // activate after the parameters are set
|
||
/*]]>*///-->
|
||
</script>
|
||
<script type="text/javascript">
|
||
/*
|
||
@licstart The following is the entire license notice for the
|
||
JavaScript code in this tag.
|
||
|
||
Copyright (C) 2012-2019 Free Software Foundation, Inc.
|
||
|
||
The JavaScript code in this tag is free software: you can
|
||
redistribute it and/or modify it under the terms of the GNU
|
||
General Public License (GNU GPL) as published by the Free Software
|
||
Foundation, either version 3 of the License, or (at your option)
|
||
any later version. The code is distributed WITHOUT ANY WARRANTY;
|
||
without even the implied warranty of MERCHANTABILITY or FITNESS
|
||
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
|
||
|
||
As additional permission under GNU GPL version 3 section 7, you
|
||
may distribute non-source (e.g., minimized or compacted) forms of
|
||
that code without the copy of the GNU GPL normally required by
|
||
section 4, provided you include this license notice and a URL
|
||
through which recipients can access the Corresponding Source.
|
||
|
||
|
||
@licend The above is the entire license notice
|
||
for the JavaScript code in this tag.
|
||
*/
|
||
<!--/*--><![CDATA[/*><!--*/
|
||
function CodeHighlightOn(elem, id)
|
||
{
|
||
var target = document.getElementById(id);
|
||
if(null != target) {
|
||
elem.cacheClassElem = elem.className;
|
||
elem.cacheClassTarget = target.className;
|
||
target.className = "code-highlighted";
|
||
elem.className = "code-highlighted";
|
||
}
|
||
}
|
||
function CodeHighlightOff(elem, id)
|
||
{
|
||
var target = document.getElementById(id);
|
||
if(elem.cacheClassElem)
|
||
elem.className = elem.cacheClassElem;
|
||
if(elem.cacheClassTarget)
|
||
target.className = elem.cacheClassTarget;
|
||
}
|
||
/*]]>*///-->
|
||
</script>
|
||
<script type="text/x-mathjax-config">
|
||
MathJax.Hub.Config({
|
||
displayAlign: "center",
|
||
displayIndent: "0em",
|
||
|
||
"HTML-CSS": { scale: 100,
|
||
linebreaks: { automatic: "false" },
|
||
webFont: "TeX"
|
||
},
|
||
SVG: {scale: 100,
|
||
linebreaks: { automatic: "false" },
|
||
font: "TeX"},
|
||
NativeMML: {scale: 100},
|
||
TeX: { equationNumbers: {autoNumber: "AMS"},
|
||
MultLineWidth: "85%",
|
||
TagSide: "right",
|
||
TagIndent: ".8em"
|
||
}
|
||
});
|
||
</script>
|
||
<script type="text/javascript"
|
||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
|
||
</head>
|
||
<body>
|
||
<div id="content">
|
||
<h1 class="title">Quantum Monte Carlo</h1>
|
||
<div id="table-of-contents">
|
||
<h2>Table of Contents</h2>
|
||
<div id="text-table-of-contents">
|
||
<ul>
|
||
<li><a href="#org5ef0488">1. Introduction</a>
|
||
<ul>
|
||
<li><a href="#org900f1d4">1.1. Energy and local energy</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgeb42fdf">2. Numerical evaluation of the energy of the hydrogen atom</a>
|
||
<ul>
|
||
<li><a href="#org4a6f916">2.1. Local energy</a>
|
||
<ul>
|
||
<li><a href="#orgd3ba702">2.1.1. Exercise 1</a></li>
|
||
<li><a href="#org0939e15">2.1.2. Exercise 2</a></li>
|
||
<li><a href="#org8603375">2.1.3. Exercise 3</a></li>
|
||
<li><a href="#org4d5cd01">2.1.4. Exercise 4</a></li>
|
||
<li><a href="#org3af2b6e">2.1.5. Exercise 5</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org5ab9959">2.2. Plot of the local energy along the \(x\) axis</a>
|
||
<ul>
|
||
<li><a href="#orga7ce1d2">2.2.1. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgd0d619f">2.3. Numerical estimation of the energy</a>
|
||
<ul>
|
||
<li><a href="#org1229a69">2.3.1. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgad1ee8c">2.4. Variance of the local energy</a>
|
||
<ul>
|
||
<li><a href="#org0afc4b1">2.4.1. Exercise (optional)</a></li>
|
||
<li><a href="#org10f6a19">2.4.2. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org0de2b52">3. Variational Monte Carlo</a>
|
||
<ul>
|
||
<li><a href="#org4abd3b8">3.1. Computation of the statistical error</a>
|
||
<ul>
|
||
<li><a href="#org5a1c95b">3.1.1. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org8a01ad5">3.2. Uniform sampling in the box</a>
|
||
<ul>
|
||
<li><a href="#orgae0d75a">3.2.1. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org23b5713">3.3. Metropolis sampling with \(\Psi^2\)</a>
|
||
<ul>
|
||
<li><a href="#org06c5905">3.3.1. Optimal step size</a></li>
|
||
<li><a href="#org654010b">3.3.2. Exercise</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org215f308">3.4. Generalized Metropolis algorithm</a>
|
||
<ul>
|
||
<li><a href="#org31cb975">3.4.1. Gaussian random number generator</a></li>
|
||
<li><a href="#org48820db">3.4.2. Exercise 1</a></li>
|
||
<li><a href="#org1ce793c">3.4.3. Exercise 2</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgc4c9346">4. Project</a></li>
|
||
<li><a href="#org79362c8">5. Acknowledgments</a></li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org5ef0488" class="outline-2">
|
||
<h2 id="org5ef0488"><span class="section-number-2">1</span> Introduction</h2>
|
||
<div class="outline-text-2" id="text-1">
|
||
<p>
|
||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||
<a href="https://www.irsamc.ups-tlse.fr/lttc/Luchon">Tutorials in Theoretical Chemistry</a>.
|
||
</p>
|
||
|
||
<p>
|
||
We propose different exercises to understand quantum Monte Carlo (QMC)
|
||
methods. In the first section, we start with the computation of the energy of a
|
||
hydrogen atom using numerical integration. The goal of this section is
|
||
to familarize yourself with the concept of <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, and apply this approach to the
|
||
hydrogen atom.
|
||
Finally, we present the diffusion Monte Carlo (DMC) method which
|
||
we use here to estimate the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule,
|
||
starting from an approximate wave function.
|
||
</p>
|
||
|
||
<p>
|
||
Code examples will be given in Python3 and Fortran. You can use
|
||
whatever language you prefer to write the programs.
|
||
</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>
|
||
All the quantities are expressed in <i>atomic units</i> (energies,
|
||
coordinates, etc).
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org900f1d4" class="outline-3">
|
||
<h3 id="org900f1d4"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||
<div class="outline-text-3" id="text-1-1">
|
||
<p>
|
||
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\mathbf{r}\) denotes the 3N-dimensional electronic coordinates.
|
||
</p>
|
||
|
||
<p>
|
||
The electronic energy of a system, \(E\), can be rewritten in terms of the
|
||
local energy \(E_L(\mathbf{r})\) as
|
||
</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 |\Psi(\mathbf{r}) |^2 d\mathbf{r}} \\
|
||
& = & \frac{\int |\Psi(\mathbf{r})|^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||
= \frac{\int |\Psi(\mathbf{r})|^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
For few dimensions, one can easily compute \(E\) by evaluating the
|
||
integrals on a grid but, for a high number of dimensions, one can
|
||
resort to Monte Carlo techniques to compute \(E\).
|
||
</p>
|
||
|
||
<p>
|
||
To this aim, recall that 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>
|
||
where a probability density function \(P(x)\) is non-negative
|
||
and integrates to one:
|
||
</p>
|
||
|
||
<p>
|
||
\[ \int_{-\infty}^\infty P(x)\,dx = 1. \]
|
||
</p>
|
||
|
||
<p>
|
||
Similarly, we can view the the energy of a system, \(E\), as the expected value of the local energy with respect to
|
||
a probability density \(P(\mathbf{r})\) defined in 3\(N\) dimensions:
|
||
</p>
|
||
|
||
<p>
|
||
\[ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{P}\,, \]
|
||
</p>
|
||
|
||
<p>
|
||
where the probability density is given by the square of the wave function:
|
||
</p>
|
||
|
||
<p>
|
||
\[ P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. \]
|
||
</p>
|
||
|
||
<p>
|
||
If we can sample \(N_{\rm MC}\) configurations \(\{\mathbf{r}\}\)
|
||
distributed as \(P\), we can estimate \(E\) as the average of the local
|
||
energy computed over these configurations:
|
||
</p>
|
||
|
||
<p>
|
||
\[ E \approx \frac{1}{N_{\rm MC}} \sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i) \,. \]
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgeb42fdf" class="outline-2">
|
||
<h2 id="orgeb42fdf"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
||
<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, for a particular value of \(a\), \(\Psi\) is an
|
||
eigenfunction of the Hamiltonian
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
To do that, we will compute the local energy and check whether it is constant.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org4a6f916" class="outline-3">
|
||
<h3 id="org4a6f916"><span class="section-number-3">2.1</span> Local energy</h3>
|
||
<div class="outline-text-3" id="text-2-1">
|
||
<p>
|
||
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
|
||
</p>
|
||
|
||
<p>
|
||
Write all the functions of this section in a single file :
|
||
<code>hydrogen.py</code> if you use Python, or <code>hydrogen.f90</code> is you use
|
||
Fortran.
|
||
</p>
|
||
|
||
<div class="note">
|
||
<ul class="org-ul">
|
||
<li>When computing a square root in \(\mathbb{R}\), <b>always</b> make sure
|
||
that the argument of the square root is non-negative.</li>
|
||
<li>When you divide, <b>always</b> make sure that you will not divide by zero</li>
|
||
</ul>
|
||
|
||
<p>
|
||
If a <i>floating-point exception</i> can occur, you should make a test
|
||
to catch the error.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd3ba702" class="outline-4">
|
||
<h4 id="orgd3ba702"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||
<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 argument
|
||
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: #b22222;">!/usr/bin/env python3</span>
|
||
<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: #b22222;">TODO</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;">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>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0939e15" class="outline-4">
|
||
<h4 id="org0939e15"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||
<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: #b22222;">TODO</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;">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>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">psi</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org8603375" class="outline-4">
|
||
<h4 id="org8603375"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||
<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 \[T_L(\mathbf{r}) = -\frac{1}{2}\frac{\Delta \Psi(\mathbf{r})}{\Psi(\mathbf{r})}.\]
|
||
</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>
|
||
Therefore, the local kinetic energy is
|
||
\[
|
||
T_L (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||
\]
|
||
</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: #b22222;">TODO</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;">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>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4d5cd01" class="outline-4">
|
||
<h4 id="org4d5cd01"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||
<div class="outline-text-4" id="text-2-1-4">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function which computes the local energy at \(\mathbf{r}\),
|
||
using the previously defined functions.
|
||
The function accepts <code>a</code> and <code>r</code> as input arguments and returns the
|
||
local kinetic 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: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
|
||
<div class="note">
|
||
<p>
|
||
When you call a function in Fortran, you need to declare its
|
||
return type.
|
||
You might by accident choose a function name which is the
|
||
same as an internal function of Fortran. So it is recommended to
|
||
<b>always</b> use the keyword <code>external</code> to make sure the function you
|
||
are calling is yours.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<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</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> potential</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">e_loc</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3af2b6e" class="outline-4">
|
||
<h4 id="org3af2b6e"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||
<div class="outline-text-4" id="text-2-1-5">
|
||
<div class="exercise">
|
||
<p>
|
||
Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(\hat{H}\).
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org5ab9959" class="outline-3">
|
||
<h3 id="org5ab9959"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||
<div class="outline-text-3" id="text-2-2">
|
||
<p>
|
||
The program you will write in this section will be written in
|
||
another file (<code>plot_hydrogen.py</code> or <code>plot_hydrogen.f90</code> for
|
||
example).
|
||
It will use the functions previously defined.
|
||
</p>
|
||
|
||
<p>
|
||
In Python, you should put at the beginning of the file
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
|
||
</pre>
|
||
</div>
|
||
<p>
|
||
to be able to use the <code>e_loc</code> function of the <code>hydrogen.py</code> file.
|
||
</p>
|
||
|
||
<p>
|
||
In Fortran, you will need to compile all the source files together:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orga7ce1d2" class="outline-4">
|
||
<h4 id="orga7ce1d2"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||
<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. In Python, you can use matplotlib
|
||
for example. In Fortran, it is convenient to write in a text file
|
||
the values of \(x\) and \(E_L(\mathbf{r})\) for each point, and use
|
||
Gnuplot to plot the files. With Gnuplot, you will need 2 blank
|
||
lines to separate the data corresponding to different values of \(a\).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The potential and the kinetic energy both diverge at \(r=0\), so we
|
||
choose a grid which does not contain the origin to avoid numerical issues.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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)
|
||
plt.figure(figsize=(10,5))
|
||
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
plt.tight_layout()
|
||
plt.legend()
|
||
plt.savefig(<span style="color: #8b2252;">"plot_py.png"</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;">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), dx</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, j</span>
|
||
|
||
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>
|
||
|
||
! <span style="color: #b22222;">TODO</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 > 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>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd0d619f" class="outline-3">
|
||
<h3 id="orgd0d619f"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||
<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 wave function square 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>
|
||
|
||
|
||
<div id="outline-container-org1229a69" class="outline-4">
|
||
<h4 id="org1229a69"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-2-3-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Compute a numerical estimate of the energy using 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: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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: #b22222;">TODO</span>
|
||
<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>
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<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>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgad1ee8c" class="outline-3">
|
||
<h3 id="orgad1ee8c"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||
<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 its average:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\sigma^2(E_L) = \frac{\int |\Psi(\mathbf{r})|^2\, \left[
|
||
E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||
\]
|
||
which can be simplified as
|
||
</p>
|
||
|
||
<p>
|
||
\[ \sigma^2(E_L) = \langle E_L^2 \rangle_{\Psi^2} - \langle E_L \rangle_{\Psi^2}^2.\]
|
||
</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>
|
||
|
||
<div id="outline-container-org0afc4b1" class="outline-4">
|
||
<h4 id="org0afc4b1"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||
<div class="outline-text-4" id="text-2-4-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Prove that :
|
||
\[\langle \left( E - \langle E \rangle_{\Psi^2} \right)^2\rangle_{\Psi^2} = \langle E^2 \rangle_{\Psi^2} - \langle E \rangle_{\Psi^2}^2 \]
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-org10f6a19" class="outline-4">
|
||
<h4 id="org10f6a19"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-2-4-2">
|
||
<div class="exercise">
|
||
<p>
|
||
Add the calculation of the variance to the previous code, and
|
||
compute a numerical estimate of the variance of the local energy using
|
||
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: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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: #b22222;">TODO</span>
|
||
|
||
<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: #a0522d;"> x(50), w, delta, energy, energy2</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> dx, r(3), a(6), norm, e_tmp, s2</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</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>
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<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;">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>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0de2b52" class="outline-2">
|
||
<h2 id="org0de2b52"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||
<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 use Monte Carlo sampling.
|
||
</p>
|
||
|
||
<p>
|
||
Moreover, Monte Carlo sampling will allow us to remove the bias due
|
||
to the discretization of space, and compute a statistical confidence
|
||
interval.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org4abd3b8" class="outline-3">
|
||
<h3 id="org4abd3b8"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||
<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 for large \(M\), according to the <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The estimate of the energy is
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E = \frac{1}{M} \sum_{i=1}^M E_i
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The variance of the average energies can be computed as
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\sigma^2 = \frac{1}{M-1} \sum_{i=1}^{M} (E_i - E)^2
|
||
\]
|
||
</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>
|
||
|
||
<div id="outline-container-org5a1c95b" class="outline-4">
|
||
<h4 id="org5a1c95b"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||
<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: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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: #b22222;">TODO</span>
|
||
<span style="color: #a020f0;">return</span> (average, error)
|
||
</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: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">ave_error</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org8a01ad5" class="outline-3">
|
||
<h3 id="org8a01ad5"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||
<div class="outline-text-3" id="text-3-2">
|
||
<p>
|
||
We will now perform our first Monte Carlo calculation to compute the
|
||
energy of the hydrogen atom.
|
||
</p>
|
||
|
||
<p>
|
||
Consider again the expression of the energy
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
E & = & \frac{\int E_L(\mathbf{r})|\Psi(\mathbf{r})|^2\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}\,.
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
Clearly, the square of the wave function is a good choice of probability density to sample but we will start with something simpler and rewrite the energy as
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
E & = & \frac{\int E_L(\mathbf{r})\frac{|\Psi(\mathbf{r})|^2}{P(\mathbf{r})}P(\mathbf{r})\, \,d\mathbf{r}}{\int \frac{|\Psi(\mathbf{r})|^2 }{P(\mathbf{r})}P(\mathbf{r})d\mathbf{r}}\,.
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
Here, we will sample a uniform probability \(P(\mathbf{r})\) in a cube of volume \(L^3\) centered at the origin:
|
||
</p>
|
||
|
||
<p>
|
||
\[ P(\mathbf{r}) = \frac{1}{L^3}\,, \]
|
||
</p>
|
||
|
||
<p>
|
||
and zero outside the cube.
|
||
</p>
|
||
|
||
<p>
|
||
One Monte Carlo run will consist of \(N_{\rm MC}\) Monte Carlo iterations. At every Monte Carlo iteration:
|
||
</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>
|
||
Once all the iterations have been computed, the run returns the average energy
|
||
\(\bar{E}_k\) over the \(N_{\rm MC}\) iterations of the run.
|
||
</p>
|
||
|
||
<p>
|
||
To compute the statistical error, perform \(M\) independent 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>
|
||
|
||
<div id="outline-container-orgae0d75a" class="outline-4">
|
||
<h4 id="orgae0d75a"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-3-2-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Parameterize the wave function with \(a=1.2\). 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="note">
|
||
<p>
|
||
To draw a uniform random number in Python, you can use
|
||
the <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.uniform.html"><code>random.uniform</code></a> function of Numpy.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
|
||
#<span style="color: #b22222;">TODO</span>
|
||
|
||
<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>
|
||
To draw a uniform random number in Fortran, you can use
|
||
the <a href="https://gcc.gnu.org/onlinedocs/gfortran/RANDOM_005fNUMBER.html"><code>RANDOM_NUMBER</code></a> subroutine.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<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>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
<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 = 1.2d0</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: #b22222;">TODO</span>
|
||
|
||
<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>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org23b5713" class="outline-3">
|
||
<h3 id="org23b5713"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||
<div class="outline-text-3" id="text-3-3">
|
||
<p>
|
||
We will now use the square of the wave function to sample random
|
||
points distributed with the probability density
|
||
\[
|
||
P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The expression of the average energy is now simplified as the average of
|
||
the local energies, since the weights are taken care of by the
|
||
sampling:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E \approx \frac{1}{N_{\rm MC}}\sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i)\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
To sample a chosen probability density, an efficient method is the
|
||
<a href="https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm">Metropolis-Hastings sampling algorithm</a>. Starting from a random
|
||
initial position \(\mathbf{r}_0\), we will realize a random walk:
|
||
</p>
|
||
|
||
<p>
|
||
\[ \mathbf{r}_0 \rightarrow \mathbf{r}_1 \rightarrow \mathbf{r}_2 \ldots \rightarrow \mathbf{r}_{N_{\rm MC}}\,, \]
|
||
</p>
|
||
|
||
<p>
|
||
according to the following algorithm.
|
||
</p>
|
||
|
||
<p>
|
||
At every step, we propose a new move according to a transition probability \(T(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1})\) of our choice.
|
||
</p>
|
||
|
||
<p>
|
||
For simplicity, we will move the electron in a 3-dimensional box of side \(2\delta L\) centered at the current position
|
||
of the electron:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta L \, \mathbf{u}
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\delta L\) is a fixed constant, and
|
||
\(\mathbf{u}\) is a uniform random number in a 3-dimensional box
|
||
\((-1,-1,-1) \le \mathbf{u} \le (1,1,1)\).
|
||
</p>
|
||
|
||
<p>
|
||
After having moved the electron, we add the
|
||
accept/reject step that guarantees that the distribution of the
|
||
\(\mathbf{r}_n\) is \(\Psi^2\). This amounts to accepting the move with
|
||
probability
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
A(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{T(\mathbf{r}_{n+1}\rightarrow\mathbf{r}_{n}) P(\mathbf{r}_{n+1})}{T(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1})P(\mathbf{r}_{n})}\right)\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
which, for our choice of transition probability, becomes
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
A(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{P(\mathbf{r}_{n+1})}{P(\mathbf{r}_{n})}\right)= \min\left(1,\frac{|\Psi(\mathbf{r}_{n+1})|^2}{|\Psi(\mathbf{r}_{n})|^2}\right)\,.
|
||
\]
|
||
</p>
|
||
|
||
<div class="exercise">
|
||
<p>
|
||
Explain why the transition probability cancels out in the
|
||
expression of \(A\).
|
||
</p>
|
||
|
||
</div>
|
||
<p>
|
||
Also note that we do not need to compute the norm of the wave function!
|
||
</p>
|
||
|
||
<p>
|
||
The algorithm is summarized as follows:
|
||
</p>
|
||
|
||
<ol class="org-ol">
|
||
<li>Evaluate the local energy at \(\mathbf{r}_n\) and accumulate it</li>
|
||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n + \delta L\, \mathbf{u}\)</li>
|
||
<li>Evaluate \(\Psi(\mathbf{r}')\) at the new position</li>
|
||
<li>Compute the ratio \(A = \frac{\left|\Psi(\mathbf{r'})\right|^2}{\left|\Psi(\mathbf{r}_{n})\right|^2}\)</li>
|
||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||
</ol>
|
||
|
||
<div class="note">
|
||
<p>
|
||
A common error is to remove the rejected samples from the
|
||
calculation of the average. <b>Don't do it!</b>
|
||
</p>
|
||
|
||
<p>
|
||
All samples should be kept, from both accepted <i>and</i> rejected moves.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org06c5905" class="outline-4">
|
||
<h4 id="org06c5905"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||
<div class="outline-text-4" id="text-3-3-1">
|
||
<p>
|
||
If the box is infinitely small, the ratio will be very close
|
||
to one and all the steps will be accepted. However, the moves will be
|
||
very correlated and you will visit the configurational space very slowly.
|
||
</p>
|
||
|
||
<p>
|
||
On the other hand, if you propose too large moves, the number of
|
||
accepted steps will decrease because the ratios might become
|
||
small. If the number of accepted steps is close to zero, then the
|
||
space is not well sampled either.
|
||
</p>
|
||
|
||
<p>
|
||
The size of the move should be adjusted so that it is as large as
|
||
possible, keeping the number of accepted steps not too small. To
|
||
achieve that, we define the acceptance rate as the number of
|
||
accepted steps over the total number of steps. Adjusting the time
|
||
step such that the acceptance rate is close to 0.5 is a good
|
||
compromise for the current problem.
|
||
</p>
|
||
|
||
<div class="note">
|
||
<p>
|
||
Below, we use the symbol \(\delta t\) to denote \(\delta L\) since we will use
|
||
the same variable later on to store a time step.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org654010b" class="outline-4">
|
||
<h4 id="org654010b"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-3-3-2">
|
||
<div class="exercise">
|
||
<p>
|
||
Modify the program of the previous section to compute the energy,
|
||
sampled with \(\Psi^2\).
|
||
</p>
|
||
|
||
<p>
|
||
Compute also the acceptance rate, so that you can adapt the time
|
||
step in order to have an acceptance rate close to 0.5.
|
||
</p>
|
||
|
||
<p>
|
||
Can you observe a 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: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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,dt):
|
||
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">return</span> energy/nmax, N_accep/nmax
|
||
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = #<span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<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>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {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;">metropolis_montecarlo</span>(a,nmax,dt,energy,accep)
|
||
<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>(in) ::<span style="color: #a0522d;"> dt </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;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> accep</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3), psi_old, psi_new</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> v, ratio</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi, gaussian</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">metropolis_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 = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = </span>! <span style="color: #b22222;">TODO</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), Y(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;">metropolis_montecarlo</span>(a,nmax,dt,X(irun),Y(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>(Y,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 qmc_metropolis.f90 -o qmc_metropolis
|
||
./qmc_metropolis
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org215f308" class="outline-3">
|
||
<h3 id="org215f308"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||
<div class="outline-text-3" id="text-3-4">
|
||
<p>
|
||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||
</p>
|
||
|
||
<p>
|
||
The Metropolis acceptance step has to be adapted accordingly to ensure that the detailed balance condition is satisfied. This means that
|
||
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>
|
||
\[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1,
|
||
\frac{T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})}
|
||
{T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})}
|
||
\right)
|
||
\]
|
||
where \(T(\mathbf{r}_n \rightarrow \mathbf{r}_{n+1})\) is the
|
||
probability of transition from \(\mathbf{r}_n\) to
|
||
\(\mathbf{r}_{n+1}\).
|
||
</p>
|
||
|
||
<p>
|
||
In the previous example, we were using uniform sampling in a box centered
|
||
at the current position. Hence, the transition probability was symmetric
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n})
|
||
= \text{constant}\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
so the expression of \(A\) was simplified to the ratios of the squared
|
||
wave functions.
|
||
</p>
|
||
|
||
<p>
|
||
Now, if instead of drawing uniform random numbers, we
|
||
choose to draw Gaussian random numbers with zero mean and variance
|
||
\(\delta t\), the transition probability becomes:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||
\frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
|
||
\mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\delta t} \right]\,.
|
||
\]
|
||
</p>
|
||
|
||
|
||
<p>
|
||
Furthermore, to sample the density even better, we can "push" the electrons
|
||
into in the regions of high probability, and "pull" them away from
|
||
the low-probability regions. This will increase the
|
||
acceptance ratios and improve the sampling.
|
||
</p>
|
||
|
||
<p>
|
||
To do this, we can use the gradient of the probability density
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\frac{\nabla [ \Psi^2 ]}{\Psi^2} = 2 \frac{\nabla \Psi}{\Psi}\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
and add the so-called drift vector, \(\frac{\nabla \Psi}{\Psi}\), so that the numerical scheme becomes a
|
||
drifted diffusion with transition probability:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||
\frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
|
||
\mathbf{r}_{n+1} - \mathbf{r}_{n} - \delta t\frac{\nabla
|
||
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\delta t} \right]\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The corresponding move is proposed as
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \frac{\nabla
|
||
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\chi\) is a Gaussian random variable with zero mean and
|
||
variance \(\delta t\).
|
||
</p>
|
||
|
||
|
||
|
||
<p>
|
||
The algorithm of the previous exercise is only slighlty modified as:
|
||
</p>
|
||
|
||
<ol class="org-ol">
|
||
<li>Evaluate the local energy at \(\mathbf{r}_{n}\) and accumulate it</li>
|
||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)</li>
|
||
<li>Evaluate \(\Psi(\mathbf{r}')\) and \(\frac{\nabla \Psi(\mathbf{r'})}{\Psi(\mathbf{r'})}\) at the new position</li>
|
||
<li>Compute the ratio \(A = \frac{T(\mathbf{r}' \rightarrow \mathbf{r}_{n}) P(\mathbf{r}')}{T(\mathbf{r}_{n} \rightarrow \mathbf{r}') P(\mathbf{r}_{n})}\)</li>
|
||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<div id="outline-container-org31cb975" class="outline-4">
|
||
<h4 id="org31cb975"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||
<div class="outline-text-4" id="text-3-4-1">
|
||
<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>
|
||
Below is a Fortran implementation returning a Gaussian-distributed
|
||
n-dimensional vector \(\mathbf{z}\). This will be useful for the
|
||
following sections.
|
||
</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>
|
||
In Python, you can use the <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html"><code>random.normal</code></a> function of Numpy.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org48820db" class="outline-4">
|
||
<h4 id="org48820db"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||
<div class="outline-text-4" id="text-3-4-2">
|
||
<div class="exercise">
|
||
<p>
|
||
If you use Fortran, copy/paste the <code>random_gauss</code> function in
|
||
a Fortran file.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<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: #b22222;">TODO</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;">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: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">drift</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1ce793c" class="outline-4">
|
||
<h4 id="org1ce793c"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||
<div class="outline-text-4" id="text-3-4-3">
|
||
<div class="exercise">
|
||
<p>
|
||
Modify the previous program to introduce the drift-diffusion scheme.
|
||
(This is a necessary step for the next section on diffusion Monte Carlo).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<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,dt):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = # <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<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>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {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,dt,nmax,energy,accep)
|
||
<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, dt</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</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> sq_dt, chi(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new</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>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<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 = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = </span>! <span style="color: #b22222;">TODO</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,dt,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>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgc4c9346" class="outline-2">
|
||
<h2 id="orgc4c9346"><span class="section-number-2">4</span> Project</h2>
|
||
<div class="outline-text-2" id="text-4">
|
||
<p>
|
||
Change your PDMC code for one of the following:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>the Helium atom, or</li>
|
||
<li>the H<sub>2</sub> molecule at \(R\) =1.401 bohr.</li>
|
||
</ul>
|
||
|
||
<p>
|
||
And compute the ground state energy.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org79362c8" class="outline-2">
|
||
<h2 id="org79362c8"><span class="section-number-2">5</span> Acknowledgments</h2>
|
||
<div class="outline-text-2" id="text-5">
|
||
|
||
<div class="figure">
|
||
<p><img src="https://trex-coe.eu/sites/default/files/inline-images/euflag.jpg" alt="euflag.jpg" />
|
||
</p>
|
||
</div>
|
||
|
||
<p>
|
||
<a href="https://trex-coe.eu">TREX</a> : Targeting Real Chemical Accuracy at the Exascale project
|
||
has received funding from the European Union’s Horizon 2020 - Research and
|
||
Innovation program - under grant agreement no. 952165. The content of this
|
||
document does not represent the opinion of the European Union, and the European
|
||
Union is not responsible for any use that might be made of such content.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||
<p class="date">Created: 2021-02-04 Thu 07:47</p>
|
||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||
</div>
|
||
</body>
|
||
</html>
|