diff --git a/index.html b/index.html index 8c7f63b..23e2272 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Quantum Monte Carlo @@ -329,76 +329,75 @@ for the JavaScript code in this tag.

Table of Contents

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1 Introduction

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1 Introduction

This website contains the QMC tutorial of the 2021 LTTC winter school @@ -438,8 +437,8 @@ coordinates, etc).

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1.1 Energy and local energy

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1.1 Energy and local energy

For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as @@ -522,8 +521,8 @@ energy computed over these configurations:

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2 Numerical evaluation of the energy of the hydrogen atom

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2 Numerical evaluation of the energy of the hydrogen atom

In this section, we consider the hydrogen atom with the following @@ -552,8 +551,8 @@ To do that, we will compute the local energy and check whether it is constant.

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2.1 Local energy

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2.1 Local energy

You will now program all quantities needed to compute the local energy of the H atom for the given wave function. @@ -580,8 +579,8 @@ to catch the error.

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2.1.1 Exercise 1

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2.1.1 Exercise 1

@@ -627,8 +626,8 @@ and returns the potential.

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2.1.2 Exercise 2

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2.1.2 Exercise 2

@@ -664,8 +663,8 @@ input arguments, and returns a scalar.

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2.1.3 Exercise 3

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2.1.3 Exercise 3

@@ -747,8 +746,8 @@ Therefore, the local kinetic energy is

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2.1.4 Exercise 4

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2.1.4 Exercise 4

@@ -808,8 +807,8 @@ are calling is yours.

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2.1.5 Exercise 5

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2.1.5 Exercise 5

@@ -821,8 +820,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(

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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

The program you will write in this section will be written in @@ -853,8 +852,8 @@ In Fortran, you will need to compile all the source files together:

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2.2.1 Exercise

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2.2.1 Exercise

@@ -950,8 +949,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \

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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 \(\mathbf{r}_i\) @@ -981,8 +980,8 @@ The energy is biased because:

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2.3.1 Exercise

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2.3.1 Exercise

@@ -1055,8 +1054,8 @@ To compile the Fortran and run it:

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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 \(\Psi\) @@ -1083,8 +1082,8 @@ energy can be used as a measure of the quality of a wave function.

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2.4.1 Exercise (optional)

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2.4.1 Exercise (optional)

@@ -1095,8 +1094,8 @@ Prove that :

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2.4.2 Exercise

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2.4.2 Exercise

@@ -1175,8 +1174,8 @@ To compile and run:

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3 Variational Monte Carlo

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3 Variational Monte Carlo

Numerical integration with deterministic methods is very efficient @@ -1192,8 +1191,8 @@ 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 \(M\) @@ -1233,8 +1232,8 @@ And the confidence interval is given by

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3.1.1 Exercise

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3.1.1 Exercise

@@ -1276,8 +1275,8 @@ input array.

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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 perform our first Monte Carlo calculation to compute the @@ -1338,8 +1337,8 @@ compute the statistical error.

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3.2.1 Exercise

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3.2.1 Exercise

@@ -1443,8 +1442,8 @@ well as the index of the current step.

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3.3 Metropolis sampling with \(\Psi^2\)

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3.3 Metropolis sampling with \(\Psi^2\)

We will now use the square of the wave function to sample random @@ -1563,8 +1562,8 @@ All samples should be kept, from both accepted and rejected moves.

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3.3.1 Optimal step size

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3.3.1 Optimal step size

If the box is infinitely small, the ratio will be very close @@ -1599,8 +1598,8 @@ the same variable later on to store a time step.

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3.3.2 Exercise

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3.3.2 Exercise

@@ -1711,8 +1710,8 @@ Can you observe a reduction in the statistical error?

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3.4 Generalized Metropolis algorithm

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3.4 Generalized Metropolis algorithm

One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability. @@ -1833,8 +1832,8 @@ The algorithm of the previous exercise is only slighlty modified as:

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3.4.1 Gaussian random number generator

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3.4.1 Gaussian random number generator

To obtain Gaussian-distributed random numbers, you can apply the @@ -1898,8 +1897,8 @@ In Python, you can use the -

3.4.2 Exercise 1

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3.4.2 Exercise 1

@@ -1942,8 +1941,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P

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3.4.3 Exercise 2

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3.4.3 Exercise 2

@@ -2042,8 +2041,8 @@ Modify the previous program to introduce the drift-diffusion scheme.

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4 Project

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4 Project

Change your PDMC code for one of the following: @@ -2060,88 +2059,28 @@ And compute the ground state energy.

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