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Table of Contents

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1 Introduction

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1 Introduction

We propose different exercises to understand quantum Monte Carlo (QMC) @@ -364,8 +364,8 @@ interpreted as a single precision value

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2 Numerical evaluation of the energy

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2 Numerical evaluation of the energy

In this section we consider the Hydrogen atom with the following @@ -439,13 +439,13 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle} \end{eqnarray*}

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2.1 Local energy

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2.1 Local energy

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2.1.1 Exercise 1

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2.1.1 Exercise 1

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2.1.2 Exercise 2

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2.1.2 Exercise 2

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2.1.3 Exercise 3

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2.1.3 Exercise 3

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2.1.4 Exercise 4

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2.1.4 Exercise 4

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2.2 Plot of the local energy along the \(x\) axis

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2.2 Plot of the local energy along the \(x\) axis

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2.2.1 Exercise

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2.2.1 Exercise

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2.3 Numerical estimation of the energy

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2.3 Numerical estimation of the energy

If the space is discretized in small volume elements \(\mathbf{r}_i\) @@ -806,8 +806,8 @@ The energy is biased because:

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2.3.1 Exercise

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2.3.1 Exercise

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2.4 Variance of the local energy

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2.4 Variance of the local energy

The variance of the local energy is a functional of \(\Psi\) @@ -940,8 +940,8 @@ energy can be used as a measure of the quality of a wave function.

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2.4.1 Exercise

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2.4.1 Exercise

@@ -1083,8 +1083,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.806881

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3 Variational Monte Carlo

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3 Variational Monte Carlo

Numerical integration with deterministic methods is very efficient @@ -1100,8 +1100,8 @@ interval.

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3.1 Computation of the statistical error

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3.1 Computation of the statistical error

To compute the statistical error, you need to perform \(M\) @@ -1141,8 +1141,8 @@ And the confidence interval is given by

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3.1.1 Exercise

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3.1.1 Exercise

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3.2 Uniform sampling in the box

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3.2 Uniform sampling in the box

We will now do our first Monte Carlo calculation to compute the @@ -1226,8 +1226,8 @@ statistical error.

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3.2.1 Exercise

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3.2.1 Exercise

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3.3 Gaussian sampling

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3.3 Gaussian sampling

We will now improve the sampling and allow to sample in the whole @@ -1434,8 +1434,8 @@ average energy can be computed as

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3.3.1 Exercise

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3.3.1 Exercise

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3.4 Sampling with \(\Psi^2\)

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3.4 Sampling with \(\Psi^2\)

We will now use the square of the wave function to make the sampling: @@ -1572,8 +1572,8 @@ the local energies, each with a weight of 1.

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3.4.1 Importance sampling

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3.4.1 Importance sampling

To generate the probability density \(\Psi^2\), we consider a @@ -1686,7 +1686,7 @@ variance \(\tau\,2D\).

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  1. Exercise 1
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  2. Exercise 1

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  3. TODO Exercise 2
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  4. TODO Exercise 2

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3.4.2 Metropolis algorithm

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3.4.2 Metropolis algorithm

Discretizing the differential equation to generate the desired @@ -1896,7 +1896,7 @@ the simulation.

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  1. Exercise
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  2. Exercise

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    4 TODO Diffusion Monte Carlo

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    4 TODO Diffusion Monte Carlo

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    4.1 Hydrogen atom

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    4.1 Hydrogen atom

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    1. Exercise
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    2. Exercise

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      4.2 Dihydrogen

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      4.2 Dihydrogen

      We will now consider the H2 molecule in a minimal basis composed of the @@ -2244,7 +2244,7 @@ the nuclei.

      Author: Anthony Scemama, Claudia Filippi

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      Created: 2021-01-13 Wed 17:25

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      Created: 2021-01-13 Wed 17:26

      Validate

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