mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-12-22 20:36:15 +01:00
deploy: 31336ac6de
This commit is contained in:
scemama
parent
99539f2445
commit
133703df94
735
index.html
735
index.html
@ -3,7 +3,7 @@
|
|||||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||||
<head>
|
<head>
|
||||||
<!-- 2021-02-04 Thu 07:47 -->
|
<!-- 2021-02-04 Thu 10:25 -->
|
||||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||||
<title>Quantum Monte Carlo</title>
|
<title>Quantum Monte Carlo</title>
|
||||||
@ -329,75 +329,92 @@ for the JavaScript code in this tag.
|
|||||||
<h2>Table of Contents</h2>
|
<h2>Table of Contents</h2>
|
||||||
<div id="text-table-of-contents">
|
<div id="text-table-of-contents">
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org5ef0488">1. Introduction</a>
|
<li><a href="#org01a29e9">1. Introduction</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org900f1d4">1.1. Energy and local energy</a></li>
|
<li><a href="#orgd30e15c">1.1. Energy and local energy</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#orgeb42fdf">2. Numerical evaluation of the energy of the hydrogen atom</a>
|
<li><a href="#org08f395b">2. Numerical evaluation of the energy of the hydrogen atom</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org4a6f916">2.1. Local energy</a>
|
<li><a href="#orgb5021b6">2.1. Local energy</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#orgd3ba702">2.1.1. Exercise 1</a></li>
|
<li><a href="#org493ceaf">2.1.1. Exercise 1</a></li>
|
||||||
<li><a href="#org0939e15">2.1.2. Exercise 2</a></li>
|
<li><a href="#org32df485">2.1.2. Exercise 2</a></li>
|
||||||
<li><a href="#org8603375">2.1.3. Exercise 3</a></li>
|
<li><a href="#org4c17cf7">2.1.3. Exercise 3</a></li>
|
||||||
<li><a href="#org4d5cd01">2.1.4. Exercise 4</a></li>
|
<li><a href="#orgc43c8fc">2.1.4. Exercise 4</a></li>
|
||||||
<li><a href="#org3af2b6e">2.1.5. Exercise 5</a></li>
|
<li><a href="#org0da2733">2.1.5. Exercise 5</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org5ab9959">2.2. Plot of the local energy along the \(x\) axis</a>
|
<li><a href="#org107b849">2.2. Plot of the local energy along the \(x\) axis</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#orga7ce1d2">2.2.1. Exercise</a></li>
|
<li><a href="#orgce43b8f">2.2.1. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#orgd0d619f">2.3. Numerical estimation of the energy</a>
|
<li><a href="#orgcc5a700">2.3. Numerical estimation of the energy</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org1229a69">2.3.1. Exercise</a></li>
|
<li><a href="#org33238fa">2.3.1. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#orgad1ee8c">2.4. Variance of the local energy</a>
|
<li><a href="#org24aa290">2.4. Variance of the local energy</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org0afc4b1">2.4.1. Exercise (optional)</a></li>
|
<li><a href="#orgbd8786f">2.4.1. Exercise (optional)</a></li>
|
||||||
<li><a href="#org10f6a19">2.4.2. Exercise</a></li>
|
<li><a href="#orgdf837df">2.4.2. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org0de2b52">3. Variational Monte Carlo</a>
|
<li><a href="#orgf24e4c1">3. Variational Monte Carlo</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org4abd3b8">3.1. Computation of the statistical error</a>
|
<li><a href="#org5a7850e">3.1. Computation of the statistical error</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org5a1c95b">3.1.1. Exercise</a></li>
|
<li><a href="#org86ece31">3.1.1. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org8a01ad5">3.2. Uniform sampling in the box</a>
|
<li><a href="#orgf3538bc">3.2. Uniform sampling in the box</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#orgae0d75a">3.2.1. Exercise</a></li>
|
<li><a href="#org5ea7de2">3.2.1. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org23b5713">3.3. Metropolis sampling with \(\Psi^2\)</a>
|
<li><a href="#orgc5a1ec2">3.3. Metropolis sampling with \(\Psi^2\)</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org06c5905">3.3.1. Optimal step size</a></li>
|
<li><a href="#org5e266bf">3.3.1. Optimal step size</a></li>
|
||||||
<li><a href="#org654010b">3.3.2. Exercise</a></li>
|
<li><a href="#org7580d40">3.3.2. Exercise</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#org215f308">3.4. Generalized Metropolis algorithm</a>
|
<li><a href="#orgcaabae7">3.4. Generalized Metropolis algorithm</a>
|
||||||
<ul>
|
<ul>
|
||||||
<li><a href="#org31cb975">3.4.1. Gaussian random number generator</a></li>
|
<li><a href="#org5445700">3.4.1. Gaussian random number generator</a></li>
|
||||||
<li><a href="#org48820db">3.4.2. Exercise 1</a></li>
|
<li><a href="#org1b2244b">3.4.2. Exercise 1</a></li>
|
||||||
<li><a href="#org1ce793c">3.4.3. Exercise 2</a></li>
|
<li><a href="#org0368ce7">3.4.3. Exercise 2</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
</ul>
|
</ul>
|
||||||
</li>
|
</li>
|
||||||
<li><a href="#orgc4c9346">4. Project</a></li>
|
<li><a href="#org82635bc">4. Diffusion Monte Carlo</a>
|
||||||
<li><a href="#org79362c8">5. Acknowledgments</a></li>
|
<ul>
|
||||||
|
<li><a href="#org1ddc667">4.1. Schrödinger equation in imaginary time</a></li>
|
||||||
|
<li><a href="#org7695533">4.2. Relation to diffusion</a></li>
|
||||||
|
<li><a href="#orgff86247">4.3. Importance sampling</a>
|
||||||
|
<ul>
|
||||||
|
<li><a href="#orgc628347">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||||
|
</ul>
|
||||||
|
</li>
|
||||||
|
<li><a href="#org429350d">4.4. Pure Diffusion Monte Carlo</a></li>
|
||||||
|
<li><a href="#org7b3d04e">4.5. Hydrogen atom</a>
|
||||||
|
<ul>
|
||||||
|
<li><a href="#org1a74409">4.5.1. Exercise</a></li>
|
||||||
|
</ul>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</li>
|
||||||
|
<li><a href="#org60565bd">5. Project</a></li>
|
||||||
|
<li><a href="#orga85900b">6. Acknowledgments</a></li>
|
||||||
</ul>
|
</ul>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5ef0488" class="outline-2">
|
<div id="outline-container-org01a29e9" class="outline-2">
|
||||||
<h2 id="org5ef0488"><span class="section-number-2">1</span> Introduction</h2>
|
<h2 id="org01a29e9"><span class="section-number-2">1</span> Introduction</h2>
|
||||||
<div class="outline-text-2" id="text-1">
|
<div class="outline-text-2" id="text-1">
|
||||||
<p>
|
<p>
|
||||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||||||
@ -437,8 +454,8 @@ coordinates, etc).
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org900f1d4" class="outline-3">
|
<div id="outline-container-orgd30e15c" class="outline-3">
|
||||||
<h3 id="org900f1d4"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
<h3 id="orgd30e15c"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||||||
<div class="outline-text-3" id="text-1-1">
|
<div class="outline-text-3" id="text-1-1">
|
||||||
<p>
|
<p>
|
||||||
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
|
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
|
||||||
@ -521,8 +538,8 @@ energy computed over these configurations:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgeb42fdf" class="outline-2">
|
<div id="outline-container-org08f395b" class="outline-2">
|
||||||
<h2 id="orgeb42fdf"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
<h2 id="org08f395b"><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">
|
<div class="outline-text-2" id="text-2">
|
||||||
<p>
|
<p>
|
||||||
In this section, we consider the hydrogen atom with the following
|
In this section, we consider the hydrogen atom with the following
|
||||||
@ -551,8 +568,8 @@ To do that, we will compute the local energy and check whether it is constant.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4a6f916" class="outline-3">
|
<div id="outline-container-orgb5021b6" class="outline-3">
|
||||||
<h3 id="org4a6f916"><span class="section-number-3">2.1</span> Local energy</h3>
|
<h3 id="orgb5021b6"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||||
<div class="outline-text-3" id="text-2-1">
|
<div class="outline-text-3" id="text-2-1">
|
||||||
<p>
|
<p>
|
||||||
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
|
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
|
||||||
@ -579,8 +596,8 @@ to catch the error.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd3ba702" class="outline-4">
|
<div id="outline-container-org493ceaf" class="outline-4">
|
||||||
<h4 id="orgd3ba702"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
<h4 id="org493ceaf"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-1">
|
<div class="outline-text-4" id="text-2-1-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -626,8 +643,8 @@ and returns the potential.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0939e15" class="outline-4">
|
<div id="outline-container-org32df485" class="outline-4">
|
||||||
<h4 id="org0939e15"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
<h4 id="org32df485"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-2">
|
<div class="outline-text-4" id="text-2-1-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -663,8 +680,8 @@ input arguments, and returns a scalar.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org8603375" class="outline-4">
|
<div id="outline-container-org4c17cf7" class="outline-4">
|
||||||
<h4 id="org8603375"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
<h4 id="org4c17cf7"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-3">
|
<div class="outline-text-4" id="text-2-1-3">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -746,8 +763,8 @@ Therefore, the local kinetic energy is
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4d5cd01" class="outline-4">
|
<div id="outline-container-orgc43c8fc" class="outline-4">
|
||||||
<h4 id="org4d5cd01"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
<h4 id="orgc43c8fc"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-4">
|
<div class="outline-text-4" id="text-2-1-4">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -807,8 +824,8 @@ are calling is yours.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org3af2b6e" class="outline-4">
|
<div id="outline-container-org0da2733" class="outline-4">
|
||||||
<h4 id="org3af2b6e"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
<h4 id="org0da2733"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-5">
|
<div class="outline-text-4" id="text-2-1-5">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -820,8 +837,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5ab9959" class="outline-3">
|
<div id="outline-container-org107b849" class="outline-3">
|
||||||
<h3 id="org5ab9959"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
<h3 id="org107b849"><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">
|
<div class="outline-text-3" id="text-2-2">
|
||||||
<p>
|
<p>
|
||||||
The program you will write in this section will be written in
|
The program you will write in this section will be written in
|
||||||
@ -852,8 +869,8 @@ In Fortran, you will need to compile all the source files together:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga7ce1d2" class="outline-4">
|
<div id="outline-container-orgce43b8f" class="outline-4">
|
||||||
<h4 id="orga7ce1d2"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
<h4 id="orgce43b8f"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-2-1">
|
<div class="outline-text-4" id="text-2-2-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -949,8 +966,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd0d619f" class="outline-3">
|
<div id="outline-container-orgcc5a700" class="outline-3">
|
||||||
<h3 id="orgd0d619f"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
<h3 id="orgcc5a700"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||||
<div class="outline-text-3" id="text-2-3">
|
<div class="outline-text-3" id="text-2-3">
|
||||||
<p>
|
<p>
|
||||||
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
||||||
@ -980,8 +997,8 @@ The energy is biased because:
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org1229a69" class="outline-4">
|
<div id="outline-container-org33238fa" class="outline-4">
|
||||||
<h4 id="org1229a69"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
<h4 id="org33238fa"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-3-1">
|
<div class="outline-text-4" id="text-2-3-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1054,8 +1071,8 @@ To compile the Fortran and run it:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgad1ee8c" class="outline-3">
|
<div id="outline-container-org24aa290" class="outline-3">
|
||||||
<h3 id="orgad1ee8c"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
<h3 id="org24aa290"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||||
<div class="outline-text-3" id="text-2-4">
|
<div class="outline-text-3" id="text-2-4">
|
||||||
<p>
|
<p>
|
||||||
The variance of the local energy is a functional of \(\Psi\)
|
The variance of the local energy is a functional of \(\Psi\)
|
||||||
@ -1082,8 +1099,8 @@ energy can be used as a measure of the quality of a wave function.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0afc4b1" class="outline-4">
|
<div id="outline-container-orgbd8786f" class="outline-4">
|
||||||
<h4 id="org0afc4b1"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
<h4 id="orgbd8786f"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||||
<div class="outline-text-4" id="text-2-4-1">
|
<div class="outline-text-4" id="text-2-4-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1094,8 +1111,8 @@ Prove that :
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-org10f6a19" class="outline-4">
|
<div id="outline-container-orgdf837df" class="outline-4">
|
||||||
<h4 id="org10f6a19"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
<h4 id="orgdf837df"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-4-2">
|
<div class="outline-text-4" id="text-2-4-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1174,8 +1191,8 @@ To compile and run:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0de2b52" class="outline-2">
|
<div id="outline-container-orgf24e4c1" class="outline-2">
|
||||||
<h2 id="org0de2b52"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
<h2 id="orgf24e4c1"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||||
<div class="outline-text-2" id="text-3">
|
<div class="outline-text-2" id="text-3">
|
||||||
<p>
|
<p>
|
||||||
Numerical integration with deterministic methods is very efficient
|
Numerical integration with deterministic methods is very efficient
|
||||||
@ -1191,8 +1208,8 @@ interval.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4abd3b8" class="outline-3">
|
<div id="outline-container-org5a7850e" class="outline-3">
|
||||||
<h3 id="org4abd3b8"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
<h3 id="org5a7850e"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||||
<div class="outline-text-3" id="text-3-1">
|
<div class="outline-text-3" id="text-3-1">
|
||||||
<p>
|
<p>
|
||||||
To compute the statistical error, you need to perform \(M\)
|
To compute the statistical error, you need to perform \(M\)
|
||||||
@ -1232,8 +1249,8 @@ And the confidence interval is given by
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5a1c95b" class="outline-4">
|
<div id="outline-container-org86ece31" class="outline-4">
|
||||||
<h4 id="org5a1c95b"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
<h4 id="org86ece31"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-1-1">
|
<div class="outline-text-4" id="text-3-1-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1275,8 +1292,8 @@ input array.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org8a01ad5" class="outline-3">
|
<div id="outline-container-orgf3538bc" class="outline-3">
|
||||||
<h3 id="org8a01ad5"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
<h3 id="orgf3538bc"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||||
<div class="outline-text-3" id="text-3-2">
|
<div class="outline-text-3" id="text-3-2">
|
||||||
<p>
|
<p>
|
||||||
We will now perform our first Monte Carlo calculation to compute the
|
We will now perform our first Monte Carlo calculation to compute the
|
||||||
@ -1337,8 +1354,8 @@ compute the statistical error.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgae0d75a" class="outline-4">
|
<div id="outline-container-org5ea7de2" class="outline-4">
|
||||||
<h4 id="orgae0d75a"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
<h4 id="org5ea7de2"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-2-1">
|
<div class="outline-text-4" id="text-3-2-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1442,8 +1459,8 @@ well as the index of the current step.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org23b5713" class="outline-3">
|
<div id="outline-container-orgc5a1ec2" class="outline-3">
|
||||||
<h3 id="org23b5713"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
<h3 id="orgc5a1ec2"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||||
<div class="outline-text-3" id="text-3-3">
|
<div class="outline-text-3" id="text-3-3">
|
||||||
<p>
|
<p>
|
||||||
We will now use the square of the wave function to sample random
|
We will now use the square of the wave function to sample random
|
||||||
@ -1562,8 +1579,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org06c5905" class="outline-4">
|
<div id="outline-container-org5e266bf" class="outline-4">
|
||||||
<h4 id="org06c5905"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
<h4 id="org5e266bf"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||||||
<div class="outline-text-4" id="text-3-3-1">
|
<div class="outline-text-4" id="text-3-3-1">
|
||||||
<p>
|
<p>
|
||||||
If the box is infinitely small, the ratio will be very close
|
If the box is infinitely small, the ratio will be very close
|
||||||
@ -1598,8 +1615,8 @@ the same variable later on to store a time step.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org654010b" class="outline-4">
|
<div id="outline-container-org7580d40" class="outline-4">
|
||||||
<h4 id="org654010b"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
<h4 id="org7580d40"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-3-2">
|
<div class="outline-text-4" id="text-3-3-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1710,8 +1727,8 @@ Can you observe a reduction in the statistical error?
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org215f308" class="outline-3">
|
<div id="outline-container-orgcaabae7" class="outline-3">
|
||||||
<h3 id="org215f308"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
<h3 id="orgcaabae7"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||||||
<div class="outline-text-3" id="text-3-4">
|
<div class="outline-text-3" id="text-3-4">
|
||||||
<p>
|
<p>
|
||||||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||||||
@ -1832,8 +1849,8 @@ The algorithm of the previous exercise is only slighlty modified as:
|
|||||||
</ol>
|
</ol>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org31cb975" class="outline-4">
|
<div id="outline-container-org5445700" class="outline-4">
|
||||||
<h4 id="org31cb975"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
<h4 id="org5445700"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-1">
|
<div class="outline-text-4" id="text-3-4-1">
|
||||||
<p>
|
<p>
|
||||||
To obtain Gaussian-distributed random numbers, you can apply the
|
To obtain Gaussian-distributed random numbers, you can apply the
|
||||||
@ -1897,8 +1914,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org48820db" class="outline-4">
|
<div id="outline-container-org1b2244b" class="outline-4">
|
||||||
<h4 id="org48820db"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
<h4 id="org1b2244b"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-2">
|
<div class="outline-text-4" id="text-3-4-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1941,8 +1958,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org1ce793c" class="outline-4">
|
<div id="outline-container-org0368ce7" class="outline-4">
|
||||||
<h4 id="org1ce793c"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
<h4 id="org0368ce7"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-3">
|
<div class="outline-text-4" id="text-3-4-3">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2041,10 +2058,542 @@ Modify the previous program to introduce the drift-diffusion scheme.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc4c9346" class="outline-2">
|
<div id="outline-container-org82635bc" class="outline-2">
|
||||||
<h2 id="orgc4c9346"><span class="section-number-2">4</span> Project</h2>
|
<h2 id="org82635bc"><span class="section-number-2">4</span> Diffusion Monte Carlo</h2>
|
||||||
<div class="outline-text-2" id="text-4">
|
<div class="outline-text-2" id="text-4">
|
||||||
<p>
|
<p>
|
||||||
|
As we have seen, Variational Monte Carlo is a powerful method to
|
||||||
|
compute integrals in large dimensions. It is often used in cases
|
||||||
|
where the expression of the wave function is such that the integrals
|
||||||
|
can't be evaluated (multi-centered Slater-type orbitals, correlation
|
||||||
|
factors, etc).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Diffusion Monte Carlo is different. It goes beyond the computation
|
||||||
|
of the integrals associated with an input wave function, and aims at
|
||||||
|
finding a near-exact numerical solution to the Schrödinger equation.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org1ddc667" class="outline-3">
|
||||||
|
<h3 id="org1ddc667"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||||
|
<div class="outline-text-3" id="text-4-1">
|
||||||
|
<p>
|
||||||
|
Consider the time-dependent Schrödinger equation:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
i\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = (\hat{H} -E_{\rm ref}) \Psi(\mathbf{r},t)\,.
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where we introduced a shift in the energy, \(E_{\rm ref}\), for reasons which will become apparent below.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We can expand a given starting wave function, \(\Psi(\mathbf{r},0)\), in the basis of the eigenstates
|
||||||
|
of the time-independent Hamiltonian, \(\Phi_k\), with energies \(E_k\):
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\Psi(\mathbf{r},0) = \sum_k a_k\, \Phi_k(\mathbf{r}).
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The solution of the Schrödinger equation at time \(t\) is
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\Psi(\mathbf{r},t) = \sum_k a_k \exp \left( -i\, (E_k-E_{\rm ref})\, t \right) \Phi_k(\mathbf{r}).
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Now, if we replace the time variable \(t\) by an imaginary time variable
|
||||||
|
\(\tau=i\,t\), we obtain
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \psi(\mathbf{r}, \tau)}{\partial \tau} = (\hat{H} -E_{\rm ref}) \psi(\mathbf{r}, \tau)
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where \(\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\,\tau)\)
|
||||||
|
and
|
||||||
|
</p>
|
||||||
|
|
||||||
|
\begin{eqnarray*}
|
||||||
|
\psi(\mathbf{r},\tau) &=& \sum_k a_k \exp( -(E_k-E_{\rm ref})\, \tau) \Phi_k(\mathbf{r})\\
|
||||||
|
&=& \exp(-(E_0-E_{\rm ref})\, \tau)\sum_k a_k \exp( -(E_k-E_0)\, \tau) \Phi_k(\mathbf{r})\,.
|
||||||
|
\end{eqnarray*}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
For large positive values of \(\tau\), \(\psi\) is dominated by the
|
||||||
|
\(k=0\) term, namely, the lowest eigenstate. If we adjust \(E_{\rm ref}\) to the running estimate of \(E_0\),
|
||||||
|
we can expect that simulating the differetial equation in
|
||||||
|
imaginary time will converge to the exact ground state of the
|
||||||
|
system.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org7695533" class="outline-3">
|
||||||
|
<h3 id="org7695533"><span class="section-number-3">4.2</span> Relation to diffusion</h3>
|
||||||
|
<div class="outline-text-3" id="text-4-2">
|
||||||
|
<p>
|
||||||
|
The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\frac{\partial \psi(\mathbf{r},t)}{\partial t} = D\, \Delta \psi(\mathbf{r},t)
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where \(D\) is the diffusion coefficient. When the imaginary-time
|
||||||
|
Schrödinger equation is written in terms of the kinetic energy and
|
||||||
|
potential,
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\frac{\partial \psi(\mathbf{r}, \tau)}{\partial \tau} =
|
||||||
|
\left(\frac{1}{2}\Delta - [V(\mathbf{r}) -E_{\rm ref}]\right) \psi(\mathbf{r}, \tau)\,,
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
it can be identified as the combination of:
|
||||||
|
</p>
|
||||||
|
<ul class="org-ul">
|
||||||
|
<li>a diffusion equation (Laplacian)</li>
|
||||||
|
<li>an equation whose solution is an exponential (potential)</li>
|
||||||
|
</ul>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The diffusion equation can be simulated by a Brownian motion:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \sqrt{\delta t}\, \chi \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where \(\chi\) is a Gaussian random variable, and the potential term
|
||||||
|
can be simulated by creating or destroying particles over time (a
|
||||||
|
so-called branching process) or by simply considering it as a
|
||||||
|
cumulative multiplicative weight along the diffusion trajectory
|
||||||
|
(pure Diffusion Monte Carlo):
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\prod_i \exp \left( - (V(\mathbf{r}_i) - E_{\text{ref}}) \delta t \right).
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We note that the ground-state wave function of a Fermionic system is
|
||||||
|
antisymmetric and changes sign. Therefore, its interpretation as a probability
|
||||||
|
distribution is somewhat problematic. In fact, mathematically, since
|
||||||
|
the Bosonic ground state is lower in energy than the Fermionic one, for
|
||||||
|
large \(\tau\), the system will evolve towards the Bosonic solution.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
For the systems you will study, this is not an issue:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<ul class="org-ul">
|
||||||
|
<li>Hydrogen atom: You only have one electron!</li>
|
||||||
|
<li>Two-electron system (\(H_2\) or He): The ground-wave function is
|
||||||
|
antisymmetric in the spin variables but symmetric in the space ones.</li>
|
||||||
|
</ul>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgff86247" class="outline-3">
|
||||||
|
<h3 id="orgff86247"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||||
|
<div class="outline-text-3" id="text-4-3">
|
||||||
|
<p>
|
||||||
|
In a molecular system, the potential is far from being constant
|
||||||
|
and, in fact, diverges at the inter-particle coalescence points. Hence,
|
||||||
|
it results in very large fluctuations of the erm weight associated with
|
||||||
|
the potental, making the calculations impossible in practice.
|
||||||
|
Fortunately, if we multiply the Schrödinger equation by a chosen
|
||||||
|
<i>trial wave function</i> \(\Psi_T(\mathbf{r})\) (Hartree-Fock, Kohn-Sham
|
||||||
|
determinant, CI wave function, <i>etc</i>), one obtains
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) =
|
||||||
|
\left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Defining \(\Pi(\mathbf{r},\tau) = \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})\), (see appendix for details)
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
|
||||||
|
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||||||
|
\right] + (E_L(\mathbf{r})-E_{\rm ref})\Pi(\mathbf{r},\tau)
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The new "kinetic energy" can be simulated by the drift-diffusion
|
||||||
|
scheme presented in the previous section (VMC).
|
||||||
|
The new "potential" is the local energy, which has smaller fluctuations
|
||||||
|
when \(\Psi_T\) gets closer to the exact wave function.
|
||||||
|
This term can be simulated by
|
||||||
|
\[
|
||||||
|
\prod_i \exp \left( - (E_L(\mathbf{r}_i) - E_{\text{ref}}) \delta t \right).
|
||||||
|
\]
|
||||||
|
where \(E_{\rm ref}\) is the constant we had introduced above, which is adjusted to
|
||||||
|
an estimate of the average energy to keep the weights close to one.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This equation generates the <i>N</i>-electron density \(\Pi\), which is the
|
||||||
|
product of the ground state solution with the trial wave
|
||||||
|
function. You may then ask: how can we compute the total energy of
|
||||||
|
the system?
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
To this aim, we use the <i>mixed estimator</i> of the energy:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
\begin{eqnarray*}
|
||||||
|
E(\tau) &=& \frac{\langle \psi(\tau) | \hat{H} | \Psi_T \rangle}{\langle \psi(\tau) | \Psi_T \rangle}\\
|
||||||
|
&=& \frac{\int \psi(\mathbf{r},\tau) \hat{H} \Psi_T(\mathbf{r}) d\mathbf{r}}
|
||||||
|
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \\
|
||||||
|
&=& \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}
|
||||||
|
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \,.
|
||||||
|
\end{eqnarray*}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
For large \(\tau\), we have that
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\Pi(\mathbf{r},\tau) =\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \rightarrow \Phi_0(\mathbf{r}) \Psi_T(\mathbf{r})\,,
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
and, using that \(\hat{H}\) is Hermitian and that \(\Phi_0\) is an
|
||||||
|
eigenstate of the Hamiltonian, we obtain for large \(\tau\)
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
E(\tau) = \frac{\langle \psi_\tau | \hat{H} | \Psi_T \rangle}
|
||||||
|
{\langle \psi_\tau | \Psi_T \rangle}
|
||||||
|
= \frac{\langle \Psi_T | \hat{H} | \psi_\tau \rangle}
|
||||||
|
{\langle \Psi_T | \psi_\tau \rangle}
|
||||||
|
\rightarrow E_0 \frac{\langle \Psi_T | \Phi_0 \rangle}
|
||||||
|
{\langle \Psi_T | \Phi_0 \rangle}
|
||||||
|
= E_0
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Therefore, we can compute the energy within DMC by generating the
|
||||||
|
density \(\Pi\) with random walks, and simply averaging the local
|
||||||
|
energies computed with the trial wave function.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgc628347" class="outline-4">
|
||||||
|
<h4 id="orgc628347"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||||
|
<div class="outline-text-4" id="text-4-3-1">
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) =
|
||||||
|
\left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Big( \Delta \big[
|
||||||
|
\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] -
|
||||||
|
\psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) - 2
|
||||||
|
\nabla \psi(\mathbf{r},\tau) \nabla \Psi_T(\mathbf{r}) \Big) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||||||
|
\frac{1}{2} \psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) +
|
||||||
|
\Psi_T(\mathbf{r})\nabla \psi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})} + V(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||||||
|
\psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) +
|
||||||
|
\Psi_T(\mathbf{r})\nabla \psi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})} + E_L(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
\[
|
||||||
|
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||||||
|
\nabla \left[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||||||
|
\frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||||||
|
\right] + E_L(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||||||
|
\Psi_T(\mathbf{r})\),
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
|
||||||
|
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
|
||||||
|
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||||||
|
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org429350d" class="outline-3">
|
||||||
|
<h3 id="org429350d"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo</h3>
|
||||||
|
<div class="outline-text-3" id="text-4-4">
|
||||||
|
<p>
|
||||||
|
Instead of having a variable number of particles to simulate the
|
||||||
|
branching process as in the <i>Diffusion Monte Carlo</i> (DMC) algorithm, we
|
||||||
|
use variant called <i>pure Diffusion Monte Carlo</i> (PDMC) where
|
||||||
|
the potential term is considered as a cumulative product of weights:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
\begin{eqnarray*}
|
||||||
|
W(\mathbf{r}_n, \tau) = \prod_{i=1}^{n} \exp \left( -\delta t\,
|
||||||
|
(E_L(\mathbf{r}_i) - E_{\text{ref}}) \right) =
|
||||||
|
\prod_{i=1}^{n} w(\mathbf{r}_i)
|
||||||
|
\end{eqnarray*}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where \(\mathbf{r}_i\) are the coordinates along the trajectory and
|
||||||
|
we introduced a <i>time-step variable</i> \(\delta t\) to discretize the
|
||||||
|
integral.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The PDMC algorithm is less stable than the DMC algorithm: it
|
||||||
|
requires to have a value of \(E_\text{ref}\) which is close to the
|
||||||
|
fixed-node energy, and a good trial wave function. Moreover, we
|
||||||
|
can't let \(\tau\) become too large because the weight whether
|
||||||
|
explode or vanish: we need to have a fixed value of \(\tau\)
|
||||||
|
(projection time).
|
||||||
|
The big advantage of PDMC is that it is rather simple to implement
|
||||||
|
starting from a VMC code:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<ol class="org-ol">
|
||||||
|
<li>Start with \(W(\mathbf{r}_0)=1, \tau_0 = 0\)</li>
|
||||||
|
<li>Evaluate the local energy at \(\mathbf{r}_{n}\)</li>
|
||||||
|
<li>Compute the contribution to the weight \(w(\mathbf{r}_n) =
|
||||||
|
\exp(-\delta t(E_L(\mathbf{r}_n)-E_\text{ref}))\)</li>
|
||||||
|
<li>Update \(W(\mathbf{r}_{n}) = W(\mathbf{r}_{n-1}) \times w(\mathbf{r}_n)\)</li>
|
||||||
|
<li>Accumulate the weighted energy \(W(\mathbf{r}_n) \times
|
||||||
|
E_L(\mathbf{r}_n)\),
|
||||||
|
and the weight \(W(\mathbf{r}_n)\) for the normalization</li>
|
||||||
|
<li>Update \(\tau_n = \tau_{n-1} + \delta t\)</li>
|
||||||
|
<li>If \(\tau_{n} > \tau_\text{max}\), the long projection time has
|
||||||
|
been reached and we can start an new trajectory from the current
|
||||||
|
position: reset \(W(r_n) = 1\) and \(\tau_n
|
||||||
|
= 0\)</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>
|
||||||
|
</ol>
|
||||||
|
<ol class="org-ol">
|
||||||
|
<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>
|
||||||
|
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Some comments are needed:
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<ul class="org-ul">
|
||||||
|
<li><p>
|
||||||
|
You estimate the energy as
|
||||||
|
</p>
|
||||||
|
|
||||||
|
\begin{eqnarray*}
|
||||||
|
E = \frac{\sum_{k=1}^{N_{\rm MC}} E_L(\mathbf{r}_k) W(\mathbf{r}_k, k\delta t)}{\sum_{k=1}^{N_{\rm MC}} W(\mathbf{r}_k, k\delta t)}
|
||||||
|
\end{eqnarray*}</li>
|
||||||
|
|
||||||
|
<li><p>
|
||||||
|
The result will be affected by a time-step error
|
||||||
|
(the finite size of \(\delta t\)) due to the discretization of the
|
||||||
|
integral, and one has in principle to extrapolate to the limit
|
||||||
|
\(\delta t \rightarrow 0\). This amounts to fitting the energy
|
||||||
|
computed for multiple values of \(\delta t\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Here, you will be using a small enough time-step and you should not worry about the extrapolation.
|
||||||
|
</p></li>
|
||||||
|
<li>The accept/reject step (steps 9-12 in the algorithm) is in principle not needed for the correctness of
|
||||||
|
the DMC algorithm. However, its use reduces significantly the time-step error.</li>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="outline-container-org7b3d04e" class="outline-3">
|
||||||
|
<h3 id="org7b3d04e"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||||
|
<div class="outline-text-3" id="text-4-5">
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org1a74409" class="outline-4">
|
||||||
|
<h4 id="org1a74409"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||||
|
<div class="outline-text-4" id="text-4-5-1">
|
||||||
|
<div class="exercise">
|
||||||
|
<p>
|
||||||
|
Modify the Metropolis VMC program into a PDMC program.
|
||||||
|
In the limit \(\delta t \rightarrow 0\), you should recover the exact
|
||||||
|
energy of H for any value of \(a\), as long as the simulation is stable.
|
||||||
|
We choose here a time step of 0.05 a.u. and a fixed projection
|
||||||
|
time \(\tau\) =100 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, nmax, dt, Eref):
|
||||||
|
# <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> = 0.05
|
||||||
|
<span style="color: #a0522d;">tau</span> = 100.
|
||||||
|
<span style="color: #a0522d;">E_ref</span> = -0.5
|
||||||
|
|
||||||
|
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a, nmax, dt, E_ref) <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;">pdmc</span>(a, dt, nmax, energy, accep, tau, E_ref)
|
||||||
|
<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, 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</span>
|
||||||
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> E_ref</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;">pdmc</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 = 0.05d0</span>
|
||||||
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> E_ref = -0.5d0</span>
|
||||||
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 100.d0</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;">pdmc</span>(a, dt, nmax, X(irun), accep(irun), tau, E_ref)
|
||||||
|
<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 pdmc.f90 -o pdmc
|
||||||
|
./pdmc
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<div id="outline-container-org60565bd" class="outline-2">
|
||||||
|
<h2 id="org60565bd"><span class="section-number-2">5</span> Project</h2>
|
||||||
|
<div class="outline-text-2" id="text-5">
|
||||||
|
<p>
|
||||||
Change your PDMC code for one of the following:
|
Change your PDMC code for one of the following:
|
||||||
</p>
|
</p>
|
||||||
<ul class="org-ul">
|
<ul class="org-ul">
|
||||||
@ -2059,9 +2608,9 @@ And compute the ground state energy.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org79362c8" class="outline-2">
|
<div id="outline-container-orga85900b" class="outline-2">
|
||||||
<h2 id="org79362c8"><span class="section-number-2">5</span> Acknowledgments</h2>
|
<h2 id="orga85900b"><span class="section-number-2">6</span> Acknowledgments</h2>
|
||||||
<div class="outline-text-2" id="text-5">
|
<div class="outline-text-2" id="text-6">
|
||||||
|
|
||||||
<div class="figure">
|
<div class="figure">
|
||||||
<p><img src="https://trex-coe.eu/sites/default/files/inline-images/euflag.jpg" alt="euflag.jpg" />
|
<p><img src="https://trex-coe.eu/sites/default/files/inline-images/euflag.jpg" alt="euflag.jpg" />
|
||||||
@ -2080,7 +2629,7 @@ Union is not responsible for any use that might be made of such content.
|
|||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||||||
<p class="date">Created: 2021-02-04 Thu 07:47</p>
|
<p class="date">Created: 2021-02-04 Thu 10:25</p>
|
||||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user