Contents
Preface to the Second Edition
Preface to the First Edition
1 Electric Charges and their Properties
1.1 Point Charges
1.2 Coulomb’s Law
1.3 Pair Wise Additivity
1.4 Electric Field
1.5 Work
1.6 Charge Distributions
1.7 Mutual Potential Energy, U
1.8 Relationship between Force and Mutual Potential Energy
1.9 Electric Multipoles
1.10 Electrostatic Potential
1.11 Polarization and Polarizability
1.12 Dipole Polarizability
1.13 Many-Body Forces
1.14 Problem Set
2 The Forces between Molecules
2.1 Pair Potential
2.2 Multipole Expansion
2.3 Charge–Dipole Interaction
2.4 Dipole–Dipole Interaction
2.5 Taking Account of the Temperature
2.6 Induction Energy
2.7 Dispersion Energy
2.8 Repulsive Contributions
2.9 Combination Rules
2.10 Comparison with Experiment
2.11 Improved Pair Potentials
2.12 A Numerical Potential
2.13 Site–Site Potentials
2.14 Problem Set
References
3 Balls on Springs
3.1 Vibrational Motion
3.2 The Force Law
3.3 A Simple Diatomic
3.4 Three Problems
3.5 The Morse Potential
3.6 More Advanced Potentials
References
4 Molecular Mechanics
4.1 More About Balls on Springs
4.2 Larger Systems of Balls on Springs
4.3 Force Fields
4.4 Molecular Mechanics (MM)
4.5 Modelling the Solvent
4.6 Time- and Money-Saving Tricks
4.7 Modern Force Fields
4.8 Some Commercial Force Fields
References
5 The Molecular Potential Energy Surface
5.1 Multiple Minima
5.2 Saddle Points
5.3 Characterization
5.4 Finding Minima
5.5 Multivariate Grid Search
5.6 Derivative Methods
5.7 First-Order Methods
5.8 Second-Order Methods
5.9 Choice of Method
5.10 The Z-Matrix
5.11 The End of the Z-Matrix
5.12 Redundant Internal Coordinates
References
6 Molecular Mechanics Examples
6.1 Geometry Optimization
6.2 Conformation Searches
6.3 Amino Acids
6.4 QSAR
6.5 Problem Set
References
7 Sharing Out the Energy
7.1 Games of Chance
7.2 Enumeration
7.3 Boltzmann Probability
7.4 Safety in Numbers
7.5 Partition Function
7.6 Two-Level Quantum System
7.7 Lindemann’s Theory of Melting
7.8 Problem Set
8 Introduction to Statistical Thermodynamics
8.1 The Ensemble
8.2 Internal Energy, Uth
8.3 Helmholtz Energy, A
8.4 Entropy S
8.5 Equation of State and Pressure
8.6 Phase Space
8.7 Configurational Integral
8.8 Virial of Clausius
9 Monte Carlo Simulations
9.1 An Early Paper
9.2 The First ‘Chemical’ Monte Carlo Simulation
9.3 Importance Sampling
9.4 Periodic Box
9.5 Cut-Offs
9.6 MC Simulation of Rigid Molecules
9.7 Flexible Molecules
References
10 Molecular Dynamics
10.1 Radial Distribution Function
10.2 Pair Correlation Functions
10.3 Molecular Dynamics Methodology
10.4 Algorithms for Time Dependence
10.5 Molten Salts
10.6 Liquid Water
10.7 Different Types of Molecular Dynamics
10.8 Uses in Conformational Studies
References
11 Introduction to Quantum Modelling
11.1 The Schrödinger Equation
11.2 The Time-Independent Schrödinger Equation
11.3 Particles in Potential Wells
11.4 Correspondence Principle
11.5 Two-Dimensional Infinite Well
11.6 Three-Dimensional Infinite Well
11.7 Two Noninteracting Particles
11.8 Finite Well
11.9 Unbound States
11.10 Free Particles
11.11 Vibrational Motion
12 Quantum Gases
12.1 Sharing Out the Energy
12.2 Rayleigh Counting
12.3 Maxwell–Boltzmann Distribution of Atomic Kinetic Energies
12.4 Black Body Radiation
12.5 Modelling Metals
12.6 Indistinguishability
12.7 Spin
12.8 Fermions and Bosons
12.9 Pauli Exclusion Principle
12.10 Boltzmann’s Counting Rule
References
13 One-Electron Atoms
13.1 Atomic Spectra
13.2 Correspondence Principle
13.3 Infinite Nucleus Approximation
13.4 Hartree’s Atomic Units
13.5 Schrödinger Treatment of the Hydrogen Atom
13.6 Radial Solutions
13.7 Atomic Orbitals199
13.8 The Stern–Gerlach Experiment
13.9 Electron Spin
13.10 Total Angular Momentum
13.11 Dirac Theory of the Electron
13.12 Measurement in the Quantum World
References
14 The Orbital Model
14.1 One- and Two-Electron Operators
14.2 Many-Body Problem
14.3 Orbital Model
14.4 Perturbation Theory
14.5 Variation Method
14.6 Linear Variation Method
14.7 Slater Determinants
14.8 Slater–Condon–Shortley Rules
14.9 Hartree Model
14.10 Hartree–Fock Model
14.11 Atomic Shielding Constants
14.12 Koopmans’ Theorem
References
15 Simple Molecules
15.1 Hydrogen Molecule Ion, H2+
15.2 LCAO Model
15.3 Elliptic Orbitals
15.4 Heitler–London Treatment of Dihydrogen
15.5 Dihydrogen MO Treatment
15.6 James and Coolidge Treatment
15.7 Population Analysis
References
16 The HF-LCAO Model
16.1 Roothaan’s 1951 Landmark Paper
16.2 The Ĵ and ˆK Operators
16.3 HF-LCAO Equations
16.4 Electronic Energy
16.5 Koopmans’ Theorem260
16.6 Open Shell Systems
16.7 Unrestricted Hartree–Fock (UHF) Model262
16.8 Basis Sets
16.9 Gaussian Orbitals
17 HF-LCAO Examples
17.1 Output
17.2 Visualization
17.3 Properties
17.4 Geometry Optimization
17.5 Vibrational Analysis
17.6 Thermodynamic Properties
17.7 Back to L-Phenylanine
17.8 Excited States
17.9 Consequences of the Brillouin Theorem
17.10 Electric Field Gradients
17.11 Hyperfine Interactions
17.12 Problem Set
References
18 Semiempirical Models
18.1 Hückel π-Electron Theory
18.2 Extended Hückel Theory
18.3 Pariser, Parr and Pople
18.4 Zero Differential Overlap
18.5 Which Basis Functions Are They?
18.6 All Valence Electron ZDO Models
18.7 CNDO
18.8 CNDO/2
18.9 CNDO/S
18.10 INDO
18.11 NDDO (Neglect of Diatomic Differential Overlap)
18.12 The MINDO Family
18.13 MNDO
18.14 Austin Model 1 (AM1)
18.15 PM3
18.16 SAM1
18.17 ZINDO/1 and ZINDO/S
18.18 Effective Core Potentials
18.19 Problem Set
References
19 Electron Correlation
19.1 Electron Density Functions
19.2 Configuration Interaction
19.3 Coupled Cluster Method
19.4 Møller–Plesset Perturbation Theory
19.5 Multiconfiguration SCF
References
20 Density Functional Theory and the Kohn–Sham LCAO Equations
20.1 Pauli and Thomas–Fermi Models
20.2 Hohenberg–Kohn Theorems
20.3 Kohn–Sham (KS-LCAO) Equations
20.4 Numerical Integration (Quadrature)
20.5 Practical Details
20.6 Custom and Hybrid Functionals
20.7 An Example
References
21 Accurate Thermodynamic Properties; the Gn Models
21.1 G1 Theory
21.2 G2 Theory
21.3 G3 Theory
References
22 Transition States
22.1 An Example
22.2 The Reaction Path
References
23 Dealing with the Solvent
23.1 Solvent Models
23.2 Langevin Dynamics
23.3 Continuum Solvation Models
23.4 Periodic Solvent Box
References
24 Hybrid Models; the QM/MM Approach
24.1 Link Atoms
24.2 IMOMM
24.3 IMOMO
24.4 ONIOM (Our Own N-layered Integrated Molecular Orbital and Molecular Mechanics)
References
Appendix A A Mathematical Aide-Mémoire
Appendix B Glossary
Appendix C List of Symbols
Index
This edition first published 2008
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Library of Congress Cataloging in Publication Data
Hinchliffe, Alan.
Molecular modelling for beginners / Alan Hinchliffe. — 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-51313-2 (cloth) — ISBN 978-0-470-51314-9 (pbk. : alk. paper)
1. Molecules—Mathematical models. 2. Molecules—Computer simulation.
I. Title.
QD480.H58 2008
541′.22015118—dc22
2008024099
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
CLOTH: 978-0-470-51313-2
PAPER: 978-0-470-51314-9
Preface to the Second Edition
It is five years since the first edition was published, and many things have moved on sufficiently to justify this second edition.
Some things never change; I have left the elementary chapters alone and I still believe that Appendix A on relevant mathematical methods is the correct place for you to start your studies.
Some topics have matured in the last five years. Density functional theory (and especially the B3LYP choice of functionals) has become the workhorse of modern computational chemistry. I have reworked all the problems and expanded the text as appropriate.
I have also said ‘goodbye’ to a few of the older topics. For example, everyone can now do chemical drawing, so I do not need to teach it. Thankfully that bane of our lives the Z-matrix has all but disappeared; I still have fond memories of struggling to get cyclic structures symmetrical and so it still gets a page of discussion.
I have completely rewritten the chapters dealing with Monte Carlo and molecular dynamics, the Gn models, transition states and solvent models. I have also added a completely new chapter called ‘Sharing Out the Energy’, and I hope you will enjoy reading it.
It is fashionable to have an associated website with any new teaching text, and I have therefore added a website at
http://www.wileyeurope.com/college/hinchliffe
where you will find a number of problem sets and their solutions. Feel free to use them any way you like. I used them in my own teaching. Perhaps you have a corresponding set that you would like to share with the rest of us? Let me know.
I did all the illustrative calculations using either Gaussian 03 or HyperChem; these were done either on a beautiful Sony Vaio laptop or on the University of Manchester’s High Performance Computing parallel computer, a Bull Itanium2 system.
As always, I welcome comments and can be reached at: Alan.Hinchliffe@manchester. ac.uk.
Alan Hinchliffe
Manchester, UK
Preface to the First Edition
There is nothing radically new about the techniques we use in modern molecular modelling. Classical mechanics hasn’t changed since the time of Newton, Hamilton and Lagrange, the great ideas of statistical mechanics and thermodynamics were discovered by Ludwig Boltzmann and J. Willard Gibbs amongst others and the basic concepts of quantum mechanics appeared in the 1920s, by which time J.C. Maxwell’s famous electromagnetic equations had long since been published.
The chemically inspired idea that molecules can profitably be treated as a collection of balls joined together with springs can be traced back to the work of D.H. Andrews in 1930. The first serious molecular Monte Carlo simulation appeared in 1953, closely followed by B.J. Alder and T.E. Wainwright’s classic molecular dynamics study of hard discs in 1957.
The Hartrees’ 1927 work on atomic structure is the concrete foundation of our everyday concept of atomic orbitals, whilst C.C.J. Roothaan’s 1951 formulation of the HF-LCAO model arguably gave us the basis for much of modern molecular quantum theory.
If we move on a little, most of my colleagues would agree that the two recent major advances in molecular quantum theory have been density functional theory, and the elegant treatment of solvents using ONIOM. Ancient civilizations believed in the cyclic nature of time and they might have had a point for, as usual, nothing is new. Workers in solid-state physics and biology actually proposed these models many years ago. It took the chemists a while to catch up.
Scientists and engineers first got their hands on computers in the late 1960s. We have passed the point on the computer history curve where every ten years gave us an order of magnitude increase in computer power, but it is no coincidence the that growth in our understanding and application of molecular modelling has run in parallel with growth in computer power. Perhaps the two greatest driving forces in recent years have been the PC and the graphical user interface. I am humbled by the fact that my lowly 1.2 GHz AMD Athlon office PC is far more powerful than the world-beating mainframes that I used as a graduate student all those years ago, and that I can build a molecule on screen and run a B3LYP/6-311++G(3d,2p) calculation before my eyes (of which more in Chapter 20).
We have also reached a stage where tremendously powerful molecular modelling computer packages are commercially available, and the subject is routinely taught as part of undergraduate science degrees. I have made use of several such packages to produce the screenshots; obviously they look better in colour than the greyscale of this text.
There are a number of classic (and hard) texts in the field; if I’m stuck with a basic molecular quantum mechanics problem, I usually reach for Eyring, Walter and Kimball’s Quantum Chemistry but the going is rarely easy.
Equally there are a number of beautifully produced elementary texts and software reference manuals that can apparently transform you into an expert overnight. It’s a two-edged sword, and we are victims of our own success. One often meets self-appointed experts in the field who have picked up much of the jargon with little of the deep understanding. It’s no use (in my humble opinion) trying to hold a conversation about gradients, hessians and density functional theory with a colleague who has just run a molecule through one package or another but hasn’t the slightest clue what the phrases or the output mean.
It therefore seemed to me (and to the reviewers who read my new book proposal) that the time was right for a middle course. I assume that you are a ‘Beginner’ in the sense of Chambers dictionary, someone who begins; a person who is in the early stages of learning or doing anything…, and I want to tell you how we go about modern molecular modelling, why we do it, and most important of all, explain much of the basic theory behind the mouse clicks. This involves mathematics and physics, and the book neither pulls punches nor aims at instant enlightenment. Many of the concepts and ideas are difficult ones, and you will have to think long and hard about them; if it’s any consolation, so did the pioneers in our subject. I have given many of the derivations in full, and tried to avoid the dreaded phrase ‘it can be shown that’.
There are various strands to our studies, all of which eventually intertwine. We start off with molecular mechanics, a classical treatment widely used to predict molecular geometries. In Chapter 8, I give a quick guide to statistical thermodynamics (if such a thing is possible), because we need to make use of the concepts when trying to model arrays of particles at nonzero temperatures. Armed with this knowledge, we are ready for an assault on Monte Carlo and molecular dynamics.
Just as we have to bite the bullet of statistical mechanics, so we have to bite the equally difficult one of quantum mechanics, which occupies Chapters 11 and 12. We then turn to the quantum treatment of atoms, where many of the sums can be done on a postcard if armed with knowledge of angular momentum.
The Hartree–Fock and HF-LCAO models dominate much of the next few chapters, as they should. The Hartree–Fock model is great for predicting many molecular properties, but it can’t usually cope with bond breaking and bond making. Chapter 19 treats electron correlation and Chapter 20 deals with the very topical density functional theory (DFT). You won’t be taken seriously if you have not done a DFT calculation on your molecule. Quantum mechanics, statistical mechanics and electromagnetism all have a certain well-deserved reputation amongst science students; they are hard subjects. Unfortunately all three all feature in this new text. In electromagnetism it is mostly a matter of getting to grips with the mathematical notation (although I have spared you Maxwell’s beautiful equations), whilst in the other two subjects it is more a question of mastering hard concepts. In the case of quantum mechanics, the concepts are often in direct contradiction to everyday experience and common sense. I expect from you a certain level of mathematical competence; I have made extensive use of vectors and matrices not because I am perverse, but because such mathematical notation brings out the inherent simplicity and beauty of many of the equations. I have tried to help by giving a mathematical appendix, which should also make the text self-contained.
I have tried to put the text into historical perspective, and in particular I have quoted directly from a number of what I call keynote papers. It is interesting to read at first hand how the pioneers put their ideas across, and in any case they do it far better than me. For example, I am not the only author to quote Paul Dirac’s famous statement
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 only that exact application of these laws leads to equations much too complicated to be soluble.
I hope you have a profitable time in your studies, and at the very least begin to appreciate what all those options mean next time you run a modelling package!
Alan Hinchliffe
Manchester, UK