Mathematical Methods for Physical and Analytical Chemistry Mathematical Methods for Physical and Analytical Chemistry David Z. Goodson Department of Chemistry & Biochemistry University of Massachusetts Dartmouth WILEY A JOHN WILEY & SONS, INC., PUBLICATION The text was typeset by the author using LaTex (copyright 1999, 2002-2008, LaTex3 Project) and the figures were created by the author using gnuplot (copyright 1986-1993, 1998, 2004, Thomas Williams and Colin Kelley). Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data is available. ISBN 978-0-470-47354-2 Printed in the United States of America. 10 9 8 7 6 5 4 3 21 To Betsy Contents Preface xiii List of Examples xv Greek Alphabet xix Part I. Calculus 1 Functions: General Properties 3 1.1 Mappings 3 1.2 Differentials and Derivatives 4 1.3 Partial Derivatives 7 1.4 Integrals 9 1.5 Critical Points 14 2 Functions: Examples 19 2.1 Algebraic Functions 19 2.2 Transcendental Functions 21 2.2.1 Logarithm and Exponential 21 2.2.2 Circular Functions 24 2.2.3 Gamma and Beta Functions 26 2.3 Functional 31 3 Coordinate Systems 33 3.1 Points in Space 33 3.2 Coordinate Systems for Molecules 35 3.3 Abstract Coordinates 37 3.4 Constraints 39 3.4.1 Degrees of Freedom 39 3.4.2 Constrained Extrema* 40 3.5 Differential Operators in Polar Coordinates 43 4 Integration 47 4.1 Change of Variables in Integrands 47 4.1.1 Change of Variable: Examples 47 4.1.2 Jacobian Determinant 49 4.2 Gaussian Integrals 51 4.3 Improper Integrals 53 4.4 Dirac Delta Function 56 4.5 Line Integrals 57 5 Numerical Methods 61 5.1 Interpolation 61 5.2 Numerical Differentiation 63 5.3 Numerical Integration 65 5.4 Random Numbers 70 5.5 Root Finding 71 5.6 Minimization* 74 "This section treats an advanced topic. It can be skipped without loss of continuity. VII viii CONTENTS 6 Complex Numbers 79 6.1 Complex Arithmetic 79 6.2 Fundamental Theorem of Algebra 81 6.3 The Argand Diagram 83 6.4 Functions of a Complex Variable* 87 6.5 Branch Cuts* 89 7 Extrapolation 93 7.1 Taylor Series 93 7.2 Partial Sums 97 7.3 Applications of Taylor Series 99 7.4 Convergence 102 7.5 Summation Approximants* 104 Part II. Statistics 8 Estimation 111 8.1 Error and Estimation Ill 8.2 Probability Distributions 113 8.2.1 Probability Distribution Functions 113 8.2.2 The Normal Distribution 115 8.2.3 The Poisson Distribution 119 8.2.4 The Binomial Distribution* 120 8.2.5 The Boltzmann Distribution* 121 8.3 Outliers 124 8.4 Robust Estimation 126 9 Analysis of Significance 131 9.1 Confidence Intervals 131 9.2 Propagation of Error 136 9.3 Monte Carlo Simulation of Error 139 9.4 Significance of Difference 140 9.5 Distribution Testing* 144 10 Fitting 151 10.1 Method of Least Squares 151 10.1.1 Polynomial Fitting 151 10.1.2 Weighted Least Squares 154 10.1.3 Generalizations of the Least-Squares Method* 155 10.2 Fitting with Error in Both Variables 157 10.2.1 Uncontrolled Error in ж 157 10.2.2 Controlled Error in ж 160 10.3 Nonlinear Fitting 162 CONTENTS ix 11 Quality of Fit 165 11.1 Confidence Intervals for Parameters 165 11.2 Confidence Band for a Calibration Line 168 11.3 Outliers and Leverage Points ' 171 11.4 Robust Fitting* 173 11.5 Model Testing 176 12 Experiment Design 181 12.1 Risk Assessment 181 12.2 Randomization 185 12.3 Multiple Comparisons 188 12.3.1 ANOVA* 189 12.3.2 Post-Hoc Tests* 191 12.4 Optimization* 195 Part III. Differential Equations 13 Examples of Differential Equations 203 13.1 Chemical Reaction Rates 203 13.2 Classical Mechanics 205 13.2.1 Newtonian Mechanics 205 13.2.2 Lagrangian and Hamiltonian Mechanics 208 13.2.3 Angular Momentum 211 13.3 Differentials in Thermodynamics 212 13.4 Transport Equations 213 14 Solving Differential Equations, I 217 14.1 Basic Concepts 217 14.2 The Superposition Principle 220 14.3 First-Order ODE's 222 14.4 Higher-Order ODE's 225 14.5 Partial Differential Equations 228 15 Solving Differential Equations, II 231 15.1 Numerical Solution 231 15.1.1 Basic Algorithms 231 15.1.2 The Leapfrog Method* 234 15.1.3 Systems of Differential Equations 235 15.2 Chemical Reaction Mechanisms 236 15.3 Approximation Methods 239 15.3.1 Taylor Series* 239 15.3.2 Perturbation Theory* 242 x CONTENTS Part IV. Linear Algebra 16 Vector Spaces 247 16.1 Cartesian Coordinate Vectors 247 16.2 Sets 248 16.3 Groups 249 16.4 Vector Spaces 251 16.5 Functions as Vectors 252 16.6 Hilbert Spaces 253 16.7 Basis Sets 256 17 Spaces of Functions 261 17.1 Orthogonal Polynomials 261 17.2 Function Resolution 267 17.3 Fourier Series 270 17.4 Spherical Harmonics 275 18 Matrices 279 18.1 Matrix Representation of Operators 279 18.2 Matrix Algebra 282 18.3 Matrix Operations 284 18.4 Pseudoinverse* 286 18.5 Determinants 288 18.6 Orthogonal and Unitary Matrices 290 18.7 Simultaneous Linear Equations 292 19 Eigenvalue Equations 297 19.1 Matrix Eigenvalue Equations 297 19.2 Matrix Diagonalization 301 19.3 Differential Eigenvalue Equations 305 19.4 Hermitian Operators 306 19.5 The Variational Principle* 309 20 Schrödinger's Equation 313 20.1 Quantum Mechanics 313 20.1.1 Quantum Mechanical Operators 313 20.1.2 The Wavefunction 316 20.1.3 The Basic Postulates* 317 20.2 Atoms and Molecules 319 20.3 The One-Electron Atom 321 20.3.1 Orbitals 321 20.3.2 The Radial Equation* 323 20.4 Hybrid Orbitals 325 20.5 Antisymmetry* 327 20.6 Molecular Orbitals* 329 CONTENTS xi 21 Fourier Analysis 333 21.1 The Fourier Transform 333 21.2 Spectral Line Shapes* 336 21.3 Discrete Fourier Transform* 339 21.4 Signal Processing 342 21.4.1 Noise Filtering* 342 21.4.2 Convolution* 345 A Computer Programs 351 A.l Robust Estimators 351 A.2 FREML 352 A.3 Neider-Mead Simplex Optimization 352 В Answers to Selected Exercises 355 С Bibliography 367 Index 373 Preface This is an intermediate level post-calculus text on mathematical and statisti- cal methods, directed toward the needs of chemists. It has developed out of a course that I teach at the University of Massachusetts Dartmouth for third- year undergraduate chemistry majors and, with additional assignments, for chemistry graduate students. However, I have designed the book to also serve as a supplementary text to accompany undergraduate physical and analyti- cal chemistry courses and as a resource for individual study by students and professionals in all subfields of chemistry and in related fields such as envi- ronmental science, geochemistry, chemical engineering, and chemical physics. I expect the reader to have had one year of physics, at least one year of chemistry, and at least one year of calculus at the university level. While many of the examples are taken from topics treated in upper-level physical and analytical chemistry courses, the presentation is sufficiently self contained that almost all the material can be understood without training in chemistry beyond a first-year general chemistry course. Mathematics courses beyond calculus are no longer a standard part of the chemistry curriculum in the United States. This is despite the fact that advanced mathematical and statistical methods are steadily becoming more and more pervasive in the chemistry literature. Methods of physical chemistry, such as quantum chemistry and spectroscopy, have become routine tools in all subfields of chemistry, and developments in statistical theory have raised the level of mathematical sophistication expected for analytical chemists. This book is intended to bridge the gap from the point at which calculus courses end to the level of mathematics needed to understand the physical and analytical chemistry professional literature. Even in the old days, when a chemistry degree required more formal math- ematics training than today, there was a mismatch between the intermediate- level mathematics taught by mathematicians (in the one or two additional math courses that could be fit into the crowded undergraduate chemistry curriculum) and the kinds of mathematical methods relevant to chemists. In- deed, to cover all the topics included in this book, a student would likely have needed to take separate courses in linear algebra, differential equations, numerical methods, statistics, classical mechanics, and quantum mechanics. Condensing six semesters of courses into just one limits the depth of cov- erage, but it has the advantage of focusing attention on those ideas and tech- niques most likely to be encountered by chemists. In a work of such breadth yet of such relatively short length it is impossible to provide rigorous proofs of all results, but I have tried to provide enough explanation of the logic and underlying strategies of the methods to make them at least intuitively reasonable. An annotated bibliography is provided to assist the reader in- terested in additional detail. Throughout the book there are sections and examples marked with an asterisk (*) to indicate an advanced or specialized topic. These starred sections can be skipped without loss of continuity. xiii