Quantum Monte-Carlo Programming E-Book

Wolfgang Schattke

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 386

Veröffentlichungsjahr: 2013

Beschreibung

Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems to solve many body systems in order to investigate the electronic structure of many-body systems. This book presents a numeric approach to determine the electronic structure of atoms, molecules and solids.
Because of the simplicity of its theoretical concept, the authors focus on the variational Quantum-Monte-Carlo (VQMC) scheme. The reader is enabled to proceed from simple examples as the hydrogen atom to advanced ones as the Lithium solid. Several intermediate steps cover the Hydrogen molecule, how to deal with a two electron systems, going over to three electrons, and expanding to an arbitrary number of electrons to finally treat the three-dimensional periodic array of Lithium atoms in a crystal.
The exmples in the field of VQMC are followed by the subject of diffusion Monte-Calro (DMC) which covers a common example, the harmonic ascillator.

The book is unique as it provides both theory and numerical programs. It includes rather practical advices to do what is usually described in a theoretical textbook, and presents in more detail the physical understanding of what the manual of a code usually promises as result. Detailed derivations can be found at the appendix, and the references are chosen with respect to their use for specifying details or getting an deeper understanding .

The authors address an introductory readership in condensed matter physics, computational phyiscs, chemistry and materials science. As the text is intended to open the reader's view towards various possibilities of choices of computing schemes connected with the method of QMC, it might also become a welcome literature for researchers who would like to know more about QMC methods.

The book is accompanied with a collection of programs, routines, and data. To download the codes, please follow http://www.wiley-vch.de/books/sample/3527408517_codes.tar.gz

Leseprobe

Contents

Preface

1 A First Monte Carlo Example

1.1 Energy of Interacting Classical Gas

2 Variational Quantum Monte Carlo for a One-Electron System

3 Two Electrons with Two Adiabatically Decoupled Nuclei: Hydrogen Molecule

3.1 Theoretical Description of the System

3.2 Numerical Results of Moderate Accuracy

3.3 Controlling the Accuracy

3.4 Details of Numerical Program

4 Three Electrons: Lithium Atom

4.1 More Electrons, More Problems: Particle and Spin Symmetry

4.2 Electron Orbitals for the Slater Determinant

4.3 Slater Determinants: Evaluation and Update

4.4 Some Important Observables in Atoms?

4.5 Statistical Accuracy

4.6. Ground State Results

4.7 Optimization?

5 Many-Electron Confined Systems

5.1 Model Systems with Few Electrons

5.2 Orthorhombic Quantum Dot

5.3 Spherical Quantum Dot

6 Many-Electron Atomic Aggregates: Lithium Cluster

6.1 Clusters and Nanophysics

6.2 Cubic BCC Arrangement of Lithium Atoms

6.3 The Cluster: Intermediate between Atom and Solid

7 Infinite Number of Electrons: Lithium Solid

7.1 Infinite Lattice

7.2 Wave Function

7.3 Jastrow Factor

7.4 Results for the 3 × 3 × 3 and 4 × 4 × 4 Superlattice Solid

8 Diffusion Quantum Monte Carlo (DQMC)

8.1 Towards a First DQMC Program

8.2 Conclusion

9 Epilogue

Appendix

A.1 The Interacting Classical Gas: High Temperature Asymptotics

A.2 Pseudorandom Number Generators

A.3 Some Generalization of the Jastrow Factor

A.4 Series Expansion

A.5 Wave Function Symmetry and Spin

A.6 Infinite Lattice: Ewald Summation

A.7 Lattice Sums: Calculation

References

Index

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The Authors

Prof. Wolfgang Schattke

Institute of Theoretical Physics and Astrophysics

Christian-Albrechts-University Kiel

Leibnizstr. 15

24118 Kiel

and

Ikerbasque Foundation/Donostia International

Physics Center

P. Manuel de Lardizabal 4

20018 Donostia – San Sebastián

Spain

Dr. Ricardo Díez Muiño

Centro de Física de Materiales CSIC-UPV/EHU

and

Donostia Intern. Physics Cente

P. Manuel de Lardizabal 4

20018 Donostia – San Sebastian

Spain

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A catalogue record for this book is available from the British Library.

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.

© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN 978-3-527-40851-1

ePDF ISBN 978-3-527-67574-6

ePub ISBN 978-3-527-67532-6

mobi ISBN 978-3-527-67531-9

Preface

The reader might be inclined not to read the preface when starting with the book, but rather at a later time when laziness or leisure leaves time for it. In the worst case, the reader might come back to the preface angered by some lack of understanding or, quite the opposite, angered by reading some undergraduate simplistic explanations. By consulting the preface, the reader is asking the authors about their goals in writing the book.

The main goal is declared by the book’s title, nevertheless with some restrictions in mind.

The following publication is settled somewhere between a textbook and a computer code manual. Its level is perhaps too specialized for a textbook and too broad for a manual. A positive comment would be that its content includes rather practical advice on what is usually described in a theoretical textbook, as well as presenting in more detail the physical understanding of what the manual of a code promises as a result. Dangling between these two extremes the authors could not decide where to place the book exactly, so they decided to take the risk of sharing the common ground in both.

Of course, one purpose was to make it more reader-friendly than a scientific paper, or a review article. However, reviews such as that of Foulkes, Mitas, Needs, and Rajagopal for example, represent invaluable sources for extended studies [1]. The path to fulfilling the purpose of a “friendly” book was led only by the authors’ own experience and will differ from that of others. In other words, neither the authors attended courses on “How to Write Pedagogically Good Books” nor did they read such literature.

Of course, the reader does not expect a manual coming with a scientific code which reduces to “read the input file explanations and then go on.” Therefore, instead of presenting just one code that could cover the general field, the authors decided to break up the program into pieces, each of them devoted to one of a few leading examples.

A pedagogic, but time and space-consuming possibility would have been to develop the codes step by step, to let the reader run into the many traps of programming errors with exercises and solutions. That could fill many volumes. We tried not to expand the volume beyond acceptable limits, but to keep enough material so that the reader could start and develop from it his or her own specific programs. Therefore, we gave up on the textbook idea.

At the very first concept of this book we thought of presenting the code collected from the PhD theses of Eckstein and Bahnsen, who completed their work within the group of one of the authors (WS). It soon became clear that we would easily run into one of the difficulties cited above which we wanted to avoid. Therefore, we decided to program from scratch. In this way, we were also free to present our way of understanding the codes. In addition, we take on full responsibility of errors, not attributing them to any other source.

However, the number of mistakes unveiled and additionally those still hidden is embarrassing. Though some of the latter might be useful to track the path the code developed, they are not on purpose, we assure that. We present the code as it developed after testing and correcting as usual. Our main programming style, if we had any, was to render the code to be easily changed. This can be taken as an excuse for the lack of beauty and the lack of program efficiency. Both aspects and perspectives will be evaluated by the community differently with changing time, changing compilers, and changing computational facilities. To keep the work along the course of finding the pleasure in writing, we must admit deficiencies which we are now blamed for. We hope that the pleasure of eventually acquiring successful access to the quantum Monte Carlo scheme might outweigh the shortcomings from the reader’s point of view as well.

Thus, the book is not written to deliver an optimized program code. These codes exist and their development is left to another branch of science. Instead, we wanted to show some aspects of the vast and beautiful possibilities of the quantum Monte Carlo (QMC) method and to attract and maybe seduce the reader to devote his or her interest to this subject. We also want to touch on the various possibilities of choices of computing schemes connected with the method. The material presented here is by no way complete, and the general scientific development is not treated completely either. Some approaches are tentative and should be improved, some are clumsy and might be smoothed. Some parts are still under discussion.

After these atmospherical remarks, let us summarize the main topics that we included and some of those excluded from the content. We almost entirely focused on the variational quantum Monte Carlo (VQMC) scheme. The diffusion Monte Carlo (DMC) topic only covers a rather trivial example, the harmonic oscillator. There is another large branch of quantum Monte Carlo calculations for electron systems that we entirely omit here. It is based on the path integral with explicit fermion statistics. Relying on large computing resources it is used in a model-like manner for example for strong-coupling systems but rarely applied ab-initio to systems of material science.

VQMC is usually considered as the poor man’s version of QMC primarily because its theoretical concept is simple. One can refrain from the heavy complex machinery, which is hidden in the depths of quantum statistics, and calculate only the energy expectation value by a multidimensional integral and minimize the latter with respect to the parameters present in the wave function ansatz. The integral itself is computed with statistically chosen points of support, and that is the stage at which some statistics enters. In particular, there is the belief in the central limit theorem stating that the procedure guarantees the reliability of those points which are drawn from a random walk. The problem lies in an adequate choice of a parameterized wave function. If there are many parameters, then one additionally has to utilize regression methods to obtain the best choice of them.

In contrast, the complexity of DMC is derived from the evolutionary scheme of a diffusion equation for the wave function, which should converge towards the true solution. Thus, one can dispense of an optimization procedure. Instead, one has to program the steps of the evolution, which is a combination of the separate actions of the kinetic and potential energy Hamiltonians on the actual wave function, to obtain the successive approximations. This combination as well as the generation of the random walkers, which mimic the wave function is less trivial. So we thought it important to explain this theoretical background and to show how it works with those easy going examples. To satisfy oneself with the role of being a theoretically poor man when devoting oneself to VQMC, one could imagine that DMC only replaces the optimization procedure of VQMC. One would also think that the physical insight lies in the choice of the functional shape of the many-body wave function rather than in obtaining its numerical representation as from DMC. Actually, in scientific calculations, one uses VQMC as the starting point and the rich man becomes again superior to the poor.

Presenting mainly VQMC in this volume, we proceed from simple examples such as the hydrogen atom, which has a known solution, to complicated ones such as the lithium solid. Being an infinite system, the latter presents a number of additional theoretical and numerical aspects which inflate the magnitude of the first example. Several intermediate steps are therefore inserted and explained: the hydrogen molecule, to deal with a two-electron system, going over to three electrons in the lithium atom, expanding to an arbitrary number of electrons when enclosed in a simple box potential or when assembled to an aggregate as a lithium cluster, to finally treating the three-dimensional periodic array of lithium atoms in a crystal. The two-electron system provides a first glance of particle symmetry in the wave function. The lithium atom stands for multiplicity and spin symmetry. Instead of localized orbitals, plane waves are utilized in a box, which also gives an opportunity to present the pair-correlation function. With the cluster of lithium atoms we discuss the role of a physical boundary that is important for the case of the infinite solid because of its shape-dependent energy terms, which only slowly converge with system size. The theory for the solid suffers from such terms resulting in an unacceptable slowing-down of convergence. Special remedies have to be discussed to this end, which complicate the program structure in addition to the routines already needed for a solid-state system.

The solid concludes the examples in the field of VQMC followed by the subject of DMC. Some detailed derivations are found at the end of the book in an appendix. The References cite suggestions for details and a deeper understanding of the material rather than exhaust the field or give honor to contributions for their historical importance.

One of the authors (WS) feels especially and gratefully obliged to the Donostia International Physics Center (DIPC) at the University of the Basque Country (UPV/EHU) for its long-lasting and generous hospitality. The time there rendered the development of this book an exciting experience and pleasure. In addition, these activities provided the opportunity to work for a period within the Ikerbasque community, which complemented the broad range of interests where he was embedded.

The other author (RDM) would like to thank the warm hospitality of the Christian-Albrechts-Universität of Kiel during part of the writing of this book. RDM is also extremely grateful to WS for teaching him how to maneuver in the intricate world of quantum Monte Carlo, and is definitely indebted to him for his patience and generosity.

Wolfgang SchattkeRicardo Díez Muiño

Kiel

Donostia – San Sebastian

2012

1

A First Monte Carlo Example

What will be found in this chapter:We introduce randomness in a general way and we show how to deal with it in terms of probabilities and statistics. To illustrate the concepts, we start the book with an example based on classical physics, namely classical particles moving in a box. It is an example much simpler than those that involve quantum mechanics, but that already demonstrates the power of statistical physics and the deep insight offered by averaging magnitudes over many degrees of freedom.

1.1 Energy of Interacting Classical Gas

There are an overwhelming number of places in life where one is confronted with statistics: from a random binary even/odd decision, when picking petals off a flower to learn about the chances of being loved, to the refined probability distributions of health and age that life insurance companies use to estimate the premium [2].

Of course, the knowledge of how to treat ensembles of many elements appears to be much older and already shows the first traces of statistical insight. In ancient Asia Minor, for instance, the Hittites, who were strong in book-keeping, registered with eager interest the quantity of barley for their beer. No doubt that, for this purpose, they used some measurement pot instead of counting the grains in the bucket. Masters of cuneiform writing as they were, their alphabet would have had trouble counting huge numbers to enumerate the grains.

Statistical aspects emerged even more clearly in the past in the context of cryptography. In the early Islamic centuries, Arab scientists were very skilled in identifying the originality of texts which were attributed to Muhammad. To decode a text, the occurrences of single letters in a language can be counted. Such statistical analysis of languages was crucial to develop decoding algorithms able to solve outstanding problems of cryptology. For example, roman military encoded their messages mixing the letters of the alphabet in a manner only known to the intended receiver, a procedure that resisted code-breaking for many centuries [3].