Introductory Quantum Mechanics with MATLAB: For Atoms, Molecules, Clusters, and Nanocrystals

224 Pages·2019·5.34 MB·English

by  James R. Chelikowsky

Introductory Quantum Mechanics with MATLAB®

Introductory Quantum Mechanics with
MATLAB®

For Atoms, Molecules, Clusters, and Nanocrystals

James R. Chelikowsky

Author
Prof. James R. Chelikowsky
University of Texas at Austin
Institute for Computational Engineering
and Sciences
Center for Computational Materials
201 East 24th Street
Austin, TX 78712
United States

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10
9
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2
1

To Ellen

vii

Contents

Preface
xi

1
Introduction
1
1.1
Different Is Usually Controversial
1
1.2
The Plan: Addressing Dirac’s Challenge
2
Reference
4

2
The Hydrogen Atom
5
2.1
The Bohr Model
5
2.2
The Schrödinger Equation
8
2.3
The Electronic Structure of Atoms and the Periodic Table
15
References
18

3
Many-electron Atoms
19
3.1
The Variational Principle
19
3.1.1
Estimating the Energy of a Helium Atom
21
3.2
The Hartree Approximation
22
3.3
The Hartree–Fock Approximation
25
References
27

4
The Free Electron Gas
29
4.1
Free Electrons
29
4.2
Hartree–Fock Exchange in a Free Electron Gas
35
References
36

5
Density Functional Theory
37
5.1
Thomas–Fermi Theory
37
5.2
The Kohn–Sham Equation
40
References
43

6
Pseudopotential Theory
45
6.1
The Pseudopotential Approximation
45
6.1.1
Phillips–Kleinman Cancellation Theorem
47

viii
Contents

6.2
Pseudopotentials Within Density Functional Theory
50
References
57

7
Methods for Atoms
59
7.1
The Variational Approach
59
7.1.1
Estimating the Energy of the Helium Atom.
59
7.2
Direct Integration
63
7.2.1
Many-electron Atoms Using Density Functional Theory
67
References
69

8
Methods for Molecules, Clusters, and Nanocrystals
71
8.1
The H2 Molecule: Heitler–London Theory
71
8.2
General Basis
76
8.2.1
Plane Wave Basis
79
8.2.2
Plane Waves Applied to Localized Systems
87
8.3
Solving the Eigenvalue Problem
89
8.3.1
An Example Using the Power Method
92
References
95

9
Engineering Quantum Mechanics
97
9.1
Computational Considerations
97
9.2
Finite Difference Methods
99
9.2.1
Special Diagonalization Methods: Subspace Filtering
101
References
104

10
Atoms
107
10.1
Energy levels
107
10.2
Ionization Energies
108
10.3
Hund’s Rules
110
10.4
Excited State Energies and Optical Absorption
113
10.5
Polarizability
122
References
124

11
Molecules
125
11.1
Interacting Atoms
125
11.2
Molecular Orbitals: Simplified
125
11.3
Molecular Orbitals: Not Simplified
130
11.4
Total Energy of a Molecule from the Kohn–Sham Equations
132
11.5
Optical Excitations
137
11.5.1
Time-dependent Density Functional Theory
138
11.6
Polarizability
140
11.7
The Vibrational Stark Effect in Molecules
140
References
150

12
Atomic Clusters
153
12.1
Defining a Cluster
153

Contents
ix

12.2
The Structure of a Cluster
154
12.2.1
Using Simulated Annealing for Structural Properties
155
12.2.2
Genetic Algorithms
159
12.2.3
Other Methods for Determining Structural Properties
162
12.3
Electronic Properties of a Cluster
164
12.3.1
The Electronic Polarizability of Clusters
164
12.3.2
The Optical Properties of Clusters
166
12.4
The Role of Temperature on Excited-state Properties
167
12.4.1
Magnetic Clusters of Iron
169
References
174

13
Nanocrystals
177
13.1
Semiconductor Nanocrystals: Silicon
179
13.1.1
Intrinsic Properties
179
13.1.1.1 Electronic Properties
179
13.1.1.2 Effective Mass Theory
184
13.1.1.3 Vibrational Properties
187
13.1.1.4 Example of Vibrational Modes for Si Nanocrystals
188
13.1.2
Extrinsic Properties of Silicon Nanocrystals
190
13.1.2.1 Example of Phosphorus-Doped Silicon Nanocrystals
191
References
197

A
Units
199

B
A Working Electronic Structure Code
203
References
206

Index
207

xi

Preface

It is safe to say that nobody understands quantum mechanics.
Richard Feynman.

Those who are not shocked when they first come across quantum theory
cannot possibly have understood it.
Niels Bohr.

The more success the quantum theory has, the sillier it looks.
Albert Einstein.

When some of the scientific giants of the twentieth century express doubt about a
theory, e.g. referring to it as a “silly” or “shocking” theory, one might ask whether it
makes sense to use such a theory in practical applications. Well, it does – because
it works. Bohr, Einstein, and Feynman were reflecting on the exceptional nature
of quantum theory and how it differs so strongly from previous theories. They
did not question the success of the theory in describing the microscopic nature
of matter. Indeed, quantum theory may be the most successful and revolutionary
theory devised to date.
The theory of quantum mechanics provides a framework to accurately predict
the spatial and energetic distributions of electronic states and the concurrent
properties in atoms, molecules, clusters, nanostructures, and bulk liquids and
solids. As a specific example, consider the electrical conductivity of elemental
crystals. The ratio of the conductivities for a metal crystal such as silver to
an insulating crystal such as sulfur can exceed 24 orders of magnitude. A
comparable ratio is the size of our galaxy divided by the size of the head of a pin,
i.e. an astronomical difference! Classical physics provides no explanation for such
widely different conductivities; yet, quantum mechanics, specifically energy band
theory, does.
That is the good news. The bad news is that quantum theory can be
extraordinarily difficult to apply to real materials. There are a variety of reasons
for this. One notable characteristic of quantum theory is the wave –particle
duality of an electron, i.e. sometimes an electron behaves like a point particle,
other times it behaves like a wave. When asked if an electron is a particle
or a wave, an expert in quantum theory might respond with the seemingly
incongruous answer: yes.

xii
Preface

This is one reason why quantum theory seems so paradoxical and strange
within the framework of traditional physics. Using quantum mechanics,
we cannot ascertain the spatial characteristics of an electron by enumerat-
ing three spatial coordinates. Rather, quantum mechanics can only give us
information on where the electron is likely to be, not where it actually is.
The probabilistic nature of quantum theory adds many degrees of freedom
to the problem. We need to specify a probability for finding an electron at
every possible point in space. As a first pass, this issue alone would appear to
doom any quantitative approach to solving for the quantum behavior of an
electron.
Yet, the intellectual framework and the computational machinery have
developed in spurts over the last several decades, which makes practical
approaches to quantum mechanics feasible. Numerous textbooks often discuss
the machinery of how to apply quantum mechanics, but fail to give the reader
practical tools for accomplishing this goal. This is an odd situation whose origin
likely resides in intellectual “latency.” In the not too distant past, it would require
a huge and complex computer code, run on a large mainframe, to do quantum
mechanics for a relatively small molecule. This is no longer the case. A modest
amount of computing power, such as that available with a laptop computer or in
principle even a “smartphone,” is sufficient to allow one to implement quantum
theory for many systems of interest such as atoms, molecules, clusters, and other
nanosacle structures. The goal of this book is to illustrate how this framework
and machinery works. We will endeavor to give the reader a practical guide for
applying quantum mechanics to interesting problems.
Many people are owed thanks to this effort. Yousef Saad helped me frame many
of the electronic structure codes and gave guidance about algorithms for solv-
ing complex numerical problems. I also express a deep appreciation to a number
of mentors and friends, including Marvin Cohen, Jim Phillips, and Steve Louie.
Of course, I also thank my students and postdocs who did much of the heavy
lifting.

Austin, Texas
James R. Chelikowsky

1

1

Introduction

The great end of life is not knowledge but action.
– T. H. Huxley

1.1
Different Is Usually Controversial

Perhaps all breakthroughs in science are initially clouded with controversies.
Consider the discovery of gravity. Isaac Newton invoked the concept of “action
at a distance” when he developed his theory of gravity. Action at distance couples
the motion of objects; yet, the objects possess no clear physical connection.
Newton argued that the motion of an apple falling from a tree was similar to
the motion of the moon falling toward (and fortunately missing) the earth. The
source of the motion of both objects is consistent with an “action at a distance”
caused by the presence of the earth and its gravitational field.
We can contrast the trajectory of the moon with a simpler object such as a golf
ball. It is easy to understand that a golfer can make the ball move by striking it. A
ball struck just right will carry hundreds of yards (or meters). Residents of New-
ton’s time would be comfortable with this idea. The golf club directly contacts
the ball, albeit for a very short time. The physical connection to the ball is the
club swung by the golfer. But how can the earth change the moon’s trajectory?
The earth does not carry a big golf club to strike the moon. While the action at a
distance theory may not be apparent to a lay person, or even a good scientist in
Newton’s time, the laws of gravity predicted the behavior of astronomical bod-
ies such as the moon’s orbit incredibly well. Hardly anyone would argue that we
ignore the practical application of Newton’s theory until someone resolved this
action at distance business. For years, scientists argued the meaning of “action at
distance” and the nature of space itself. Eventually, scientists agreed that the con-
cept of Newtonian space was problematic. It was left to Einstein to straighten out
issues of space, time, and gravity. In some sense, it hardly mattered if you wanted
to predict planetary motion. A practical application of Newton’s theory accom-
plished that really well, save some relatively minor fixes from Einstein. (We are
not going to worry about issues such as worm holes or gravity waves.)

Introductory Quantum Mechanics with MATLAB® : For Atoms, Molecules, Clusters, and Nanocrystals,
First Edition. James R. Chelikowsky.
© 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

2
1 Introduction

Perhaps one should view the theory of quantum mechanics in the same man-
ner. The theory remains “mysterious” in some ways. Oddly, some of the central
components of the theory are understandable only because we can think about
them in classical terms. Still, quantum theory can be used to predict properties of
matter with unprecedented accuracy. Upon the invention of quantum mechanics,
the famous physicist Dirac wrote the following [1]:

The underlying physical laws necessary for the mathematical theory of a large part
of physics and the whole of chemistry are thus completely known, and the difficulty is
onlythattheexactapplicationoftheselawsleadstoequationsmuchtoocomplicated
to be soluble. It therefore becomes desirable that approximate practical methods of
applyingquantummechanicsshouldbedeveloped,whichcanleadtoanexplanation
of the main features of complex atomic systems with out much computation …

Dirac’s quote of nearly a century ago is correct and appropriate. In principle,
the underlying physical laws (of quantum mechanics) should allow one to pre-
dict “much of physics” (condensed matter physics as an example) and “the whole
of chemistry.” What stands in our way is a practical machinery to accomplish
this task. Computational physicists often refer to this task as addressing Dirac’s
Challenge.
Dirac’s warning about avoiding “too much computation” is an anachronism.
Dirac’s world of 1929 did not envision a computer capable of carrying out billions
or trillions of mathematical operations within a second. When we contemplate
the use of a modest computer, such as a laptop, for the work outlined in this book,
it is indeed “without too much computation.”

1.2
The Plan: Addressing Dirac’s Challenge

Dirac’s challenge for us is to develop “approximate practical methods of apply-
ing quantum mechanics.” The goal of this book is to address, or better start to
address, the challenge.
The book is roughly divided into three parts. The first part will focus on the
theory. We will use a minimum of theory to get to the “good part.” Our intent is
not to write a quantum mechanics textbook. Rather, our intent in this part of the
book is to review essential features. For example, we will consider the simplest
of all atoms, hydrogen, and we will start with the simplest of theories from Bohr.
We will then introduce the Schrödinger equation and briefly sketch out how to
solve the hydrogen atom problem analytically.
The hydrogen atom is one of the rarest of quantum systems – one where we can
do an analytical solution of the Schrödinger equation. The next several chapters
will involve introducing the real problem, one with more than a single electron.
A clear example of such a system is helium, where we have two electrons. Our
study of the helium atom will lead us to consider the Hartree and Hartree–Fock

1.2 The Plan: Addressing Dirac’s Challenge
3

approximations. Our next objective will be to consider a practical method for
more than one or two electrons. A practical theory for this is based on “density
functional theory,” which focuses on how electrons are distributed in space.
A logical pathway to take us from Hartree–Fock theory to density functional
theory arises from a “free electron” model. We introduce this model using
concepts removed from the physics of an isolated atom. We will “backtrack” in
our discussions to consider some solid-state physics concepts. Theories based
on electron density will provide some key approximations. In particular, we
will begin with the Thomas–Fermi approximation, which can lead to contem-
porary density functional theories. This approach will allow us to consider a
“one-electron” Schrödinger equation to solve a many-electron problem.
The last chapter of this section will center on the “pseudopotential approxima-
tion.” This key approximation will allow us to fix the length and energy scales
of the many-electron problem by considering only the chemically relevant elec-
tronic states. The pseudopotential approximation treats an element such as lead
on an equal footing with an element such as carbon. Both lead and carbon have
the same configuration for the outermost, or valence, electrons. These chemically
active states provide the chemical bond or “electronic glue” that holds atoms,
clusters, molecules, and nanocrysals together.
The next part of the book illustrates numerical methods. Numerical methods
are important as there are few atomic systems that can be solved analytically,
save the aforementioned hydrogen atom. This is also true for classical systems
where analytically only the two-body system is solvable.
We initially consider an isolated, spherically symmetric atom. We introduce
the variational method and show how approximate wave functions can be used
to obtain accurate estimates for the exact solution. We also solve the problem by
integrating a one-dimensional equation.
We will consider solutions for many-electron atoms and molecules, using
a numerical basis. This is the standard method for most quantum chemistry
approaches to molecules and atoms, although it may not be the best method
for these systems, especially for pedagogical purposes. An alternate is to solve
the problem in real space on a grid. This approach is easy to implement and
understand. With either a basis or a grid approach, we solve an “eigenvalue
problem.” Iterative methods can solve such problems and we will illustrate this.
The last part of the book demonstrates the application of quantum theory to
atoms, molecules, and clusters using a common numerical method. Physical
concepts such as pseudopotentials, density functional theory, and a real-space
grid form the underpinnings for computing a solution of the electronic structure
problem. The pseudopotential model of solids is widely used as the standard
model for describing atomistic systems. The model divides electronic states into
those that are valence states (chemically active) and those that are core states
(chemically inert). For example, systems made up of silicon atoms have valence
states derived from the atomic 3s23p2 configuration. The valence states form
bonds by promoting a 3s electron to form sp3 bond. One can omit the core states
1s22s22p6 altogether in pseudopotential theory. As such, the energy and length
scales for determining a basis are set by the valence state.