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 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment 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Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 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 To Betsy Contents Preface xiii List of Examples xv Greek Alphabet xix Part I Calculus Functions: General Properties 1.1 Mappings 1.2 Differentials and Derivatives 1.3 Partial Derivatives 1.4 Integrals 1.5 Critical Points 3 14 Functions: Examples 2.1 Algebraic Functions 2.2 Transcendental Functions 2.2.1 Logarithm and Exponential 2.2.2 Circular Functions 2.2.3 Gamma and Beta Functions 2.3 Functional 19 19 21 21 24 26 31 Coordinate Systems 3.1 Points in Space 3.2 Coordinate Systems for Molecules 3.3 Abstract Coordinates 3.4 Constraints 3.4.1 Degrees of Freedom 3.4.2 Constrained Extrema* 3.5 Differential Operators in Polar Coordinates 33 33 35 37 39 39 40 43 Integration 4.1 Change of Variables in Integrands 4.1.1 Change of Variable: Examples 4.1.2 Jacobian Determinant 4.2 Gaussian Integrals 4.3 Improper Integrals 4.4 Dirac Delta Function 4.5 Line Integrals 47 47 47 49 51 53 56 57 Numerical Methods 5.1 Interpolation 5.2 Numerical Differentiation 5.3 Numerical Integration 5.4 Random Numbers 5.5 Root Finding 5.6 Minimization* 61 61 63 65 70 71 74 "This section treats an advanced topic It can be skipped without loss of continuity VII viii CONTENTS Complex Numbers 6.1 Complex Arithmetic 6.2 Fundamental Theorem of Algebra 6.3 The Argand Diagram 6.4 Functions of a Complex Variable* 6.5 Branch Cuts* Extrapolation 7.1 Taylor Series 7.2 Partial Sums 7.3 Applications of Taylor Series 7.4 Convergence 7.5 Summation Approximants* Part II 79 79 81 83 87 89 93 93 97 99 102 104 Statistics Estimation 8.1 Error and Estimation 8.2 Probability Distributions 8.2.1 Probability Distribution Functions 8.2.2 The Normal Distribution 8.2.3 The Poisson Distribution 8.2.4 The Binomial Distribution* 8.2.5 The Boltzmann Distribution* 8.3 Outliers 8.4 Robust Estimation 111 Ill 113 113 115 119 120 121 124 126 Analysis of Significance 9.1 Confidence Intervals 9.2 Propagation of Error 9.3 Monte Carlo Simulation of Error 9.4 Significance of Difference 9.5 Distribution Testing* 131 131 136 139 140 144 10 Fitting 10.1 Method of Least Squares 10.1.1 Polynomial Fitting 10.1.2 Weighted Least Squares 10.1.3 Generalizations of the Least-Squares Method* 10.2 Fitting with Error in Both Variables 10.2.1 Uncontrolled Error in ж 10.2.2 Controlled Error in ж 10.3 Nonlinear Fitting 151 151 151 154 155 157 157 160 162 ix CONTENTS 11 Quality of Fit 11.1 Confidence Intervals for Parameters 11.2 Confidence Band for a Calibration Line 11.3 Outliers and Leverage Points 11.4 Robust Fitting* 11.5 Model Testing 12 E x p e r i m e n t Design 12.1 Risk Assessment 12.2 Randomization 12.3 Multiple Comparisons 12.3.1 ANOVA* 12.3.2 Post-Hoc Tests* 12.4 Optimization* Part III ' 165 165 168 171 173 176 181 181 185 188 189 191 195 Differential Equations 13 Examples of Differential Equations 13.1 Chemical Reaction Rates 13.2 Classical Mechanics 13.2.1 Newtonian Mechanics 13.2.2 Lagrangian and Hamiltonian Mechanics 13.2.3 Angular Momentum 13.3 Differentials in Thermodynamics 13.4 Transport Equations 203 203 205 205 208 211 212 213 14 Solving Differential Equations, I 14.1 Basic Concepts 14.2 The Superposition Principle 14.3 First-Order ODE's 14.4 Higher-Order ODE's 14.5 Partial Differential Equations 217 217 220 222 225 228 15 Solving Differential Equations, II 15.1 Numerical Solution 15.1.1 Basic Algorithms 15.1.2 The Leapfrog Method* 15.1.3 Systems of Differential Equations 15.2 Chemical Reaction Mechanisms 15.3 Approximation Methods 15.3.1 Taylor Series* 15.3.2 Perturbation Theory* 231 231 231 234 235 236 239 239 242 x Part IV CONTENTS Linear Algebra 16 Vector Spaces 16.1 Cartesian Coordinate Vectors 16.2 Sets 16.3 Groups 16.4 Vector Spaces 16.5 Functions as Vectors 16.6 Hilbert Spaces 16.7 Basis Sets 247 247 248 249 251 252 253 256 17 Spaces of Functions 17.1 Orthogonal Polynomials 17.2 Function Resolution 17.3 Fourier Series 17.4 Spherical Harmonics 261 261 267 270 275 18 Matrices 18.1 Matrix Representation of Operators 18.2 Matrix Algebra 18.3 Matrix Operations 18.4 Pseudoinverse* 18.5 Determinants 18.6 Orthogonal and Unitary Matrices 18.7 Simultaneous Linear Equations 279 279 282 284 286 288 290 292 19 Eigenvalue Equations 19.1 Matrix Eigenvalue Equations 19.2 Matrix Diagonalization 19.3 Differential Eigenvalue Equations 19.4 Hermitian Operators 19.5 The Variational Principle* 297 297 301 305 306 309 20 Schrödinger's Equation 20.1 Quantum Mechanics 20.1.1 Quantum Mechanical Operators 20.1.2 The Wavefunction 20.1.3 The Basic Postulates* 20.2 Atoms and Molecules 20.3 The One-Electron Atom 20.3.1 Orbitals 20.3.2 The Radial Equation* 20.4 Hybrid Orbitals 20.5 Antisymmetry* 20.6 Molecular Orbitals* 313 313 313 316 317 319 321 321 323 325 327 329 CONTENTS xi 21 Fourier Analysis 21.1 The Fourier Transform 21.2 Spectral Line Shapes* 21.3 Discrete Fourier Transform* 21.4 Signal Processing 21.4.1 Noise Filtering* 21.4.2 Convolution* 333 333 336 339 342 342 345 A Computer Programs A.l Robust Estimators A.2 FREML A.3 Neider-Mead Simplex Optimization 351 351 352 352 В Answers to Selected Exercises 355 С Bibliography 367 Index 373 Preface This is an intermediate level post-calculus text on mathematical and statistical methods, directed toward the needs of chemists It has developed out of a course that I teach at the University of Massachusetts Dartmouth for thirdyear 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 analytical chemistry courses and as a resource for individual study by students and professionals in all subfields of chemistry and in related fields such as environmental 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 mathematics training than today, there was a mismatch between the intermediatelevel 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 Indeed, 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 coverage, but it has the advantage of focusing attention on those ideas and techniques 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 interested 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 XIV PREFACE Part I provides a review of calculus The first four chapters provide a brief overview of elementary calculus while the next three chapters treat, in relatively more detail, topics that tend to be shortchanged in a typical introductory calculus course: numerical methods, complex numbers, and Taylor series Parts II (Statistics), III (Differential Equations), and IV (Linear Algebra) can for the most part be read in any order The only exceptions are some of the starred sections, and most of Chapter 20 (Schrödinger's Equation), which draws significantly on Part III as well as Part IV The treatment of statistics is somewhat novel for a presentation at this level in that significant use is made of Monte Carlo simulation of random error Also, an emphasis is placed on robust methods of estimation Most chemists are unaware of this relatively new development in statistical theory that allows for a more satisfactory treatment of outliers than does the more familiar Q-test Exercises are included with each chapter, and answers to many of them are provided in an appendix Many of the exercises require the use of a computer algebra system The convenience and power of modern computer algebra software systems is such that they have become an invaluable tool for physical scientists However, considering that there are various different software systems in use, each with its own distinctive syntax and its own enthusiastic corps of users, I have been reluctant to make the main body of the text too dependent on computer algebra examples Occasionally, when discussing topics such as statistical estimation, Monte Carlo simulation, or Fourier transform that particularly require the use of a computer, I have presented examples in Mathematica I apologize to users of other systems, but I trust you will be able to translate to your system of choice without too much trouble I thank my students at UMass Dartmouth who have been subjected to earlier versions of these chapters over the past several years Their comments (and complaints) have significantly shaped the final result I thank various friends and colleagues who have suggested topics to include and/or have read and commented on parts of the manuscript—in particular, Dr Steven Adler-Golden, Professor Bernice Auslander, Professor Gerald Manning, and Professor Michele Mandrioli Also, I gratefully acknowledge the efforts of the anonymous reviewers of the original proposal to Wiley Their insightful and thorough critiques were extremely helpful I have followed almost all of their suggestions Finally, I thank my wife Betsy Martin for her patience and wisdom DAVID Z GOODSON Newton, Massachusetts May, 2010 List of Examples 1.1 Contrasting the concepts of function and operator 4.1 Integrals involving linear polynomials 1.2 Numerical approximation of a derivative 4.2 Integral of reciprocal of a product of linear polynomials 4.3 An integral involving a product of an exponential and an algebraic function 1.3 The derivative of x2 1.4 The chain rule 1.5 Differential of Gibbs free energy of reaction 1.6 Demonstration of the triple product rule 1.7 Integrals of x variables 1.10 The critical temperature 1.11 A saddle point 2.1 Derivation of a derivative 4.6 A divergent integral 5.1 Cubic splines formula 2.2 The cube root of - 2.3 Noninteger 4.5 Cauchy principal value 4.7 Another example of a Cauchy principal value 4.8 Quantum mechanical applications of the Dirac delta function 1.8 Integration by parts 1.9 Dummy 4.4 Integration by parts with change of variable powers 2.4 Solve φ = arctan(—1) 2.5 Integral representation of the gamma function 2.6 The kinetic molecular theory of gases algorithm 5.2 Derivatives of spectra 5.3 A simple random number generator 5.4 Monte Carlo integration 5.5 Using Brent's method to determine the bond distance of the nitrogen molecule 6.1 Real and imaginary parts 3.1 Kinetic energy in spherical polar coordinates 3.2 Center of mass of a diatomic molecule 3.3 Center of mass of a planar molecule 3.4 Coordinates for a bent triatomic molecule 3.5 The triple point 3.6 Number of degrees of freedom for a mixture of liquids 3.7 Extrema of a two-coordinate function on a circle: Using the constraint to reduce the number of degrees of freedom 3.8 Extrema of a two-coordinate function on a circle: Using the method of undetermined multipliers 6.2 Calculate (2 + Зг) 6.3 Absolute value of complex numbers 6.4 Real roots 6.5 Complex numbers of unit length 6.6 Calculating a noninteger power 6.7 Integrals involving circular functions 6.8 Calculate the logarithm of + 4г 6.9 Residues of poles 6.10 Applying the residue theorem 7.1 Taylor series related to (1 — x)~l 7.2 Taylor series of y/1 + x 7.3 Taylor series related to ex 7.4 Multiplication of Taylor series 7.5 Expanding the expansion variable 7.6 Multivariate Taylor series LIST OF EXAMPLES XVI 7.7 Laurent series 7.8 Expansion about infinity 7.9 Stirling's formula as an expansion about infinity 9.11 Histogram of a normally buted data set 9.12 Probability plots 9.13 Shapiro-Wilk distri- test 7.10 Comparison of extrapolation and interpolation 10.1 Fitting with a straight line 7.11 Simplifying a functional 10.2 Experimental determination reaction rate law form 7.12 Harmonic approximation for diatomic potential energy 7.13 Buffers 7.14 Harmonic-oscillator function partition 7.15 Exponential of the first-derivative operator 7.16 Padé approximant 8.3 An illustration of the central limit theorem 8.4 Radioactive decay probability 8.5 Computer simulation of data samples the breakdown point 8.8 Median absolute deviation estimation Linearization curve 11.1 Designing an optimal procedure for estimating an unknown concentration 11.2 Least median of squares as point estimation method 11.3 Algorithm for LMS point estimation 9.1 Solving for za/2- 12.2 Multiple comparisons 12.4 Optimization using the Nelder-Mead simplex algorithm 9.3 A 95% confidence interval 9.4 Using σ to estimate σ 12.5 Polishing the optimization local modeling 9.5 Standard error 9.6 Rules for significant figures 9.7 Monte Carlo determination fitting 11.5 Choosing between models 12.3 Contour plot of a chemical synthesis 9.2 Solving for a of 95% confidence interval of the mean 9.8 Bootstrap 10.6 Enzyme kinetics: The EadieHofstee plot 12.1 Type II error for one-way comparison with a control estimators 8.10 Breakdown of Huber on 10.5 Controlled vs uncontrolled variables 11.4 LMS straight-line 8.6 The Q-test 8.9 Huber 10.4 Effect of error assumption least-squares fit 10.8 Dose-response 8.2 Expectation value of a function 8.7 Determining 10.3 Exponential fit 10.7 8.1 Mean and median of a resampling 9.9 Testing significance of difference 9.10 Monte Carlo test of significance of difference 13.1 Empirical determination reaction rate with of a 13.2 Expressing the rate law in terms of the extent of reaction 13.3 A free particle 13.4 Lagrange 's equation in one dimension LIST OF EXAMPLES XVll 13.5 Hamilton's equations in one dimension 16.10 Vector resolution in R 13.6 Rigid-body rotation 17.1 Resolution of a polynomial 13.7 Water pollution 17.2 Chebyshev approximation discontinuous function 13.8 Groundwater flow of a 13.9 Solute transport 17.3 Fourier analysis over an arbitrary range 14.1 Solutions to the differential equation of the exponential 17.4 Solute transport boundary conditions 14.2 Constant of integration for a reaction rate law 14.3 Constants of integration for a trajectory 18.1 Rotation in xy-plane 18.2 Moment of intertia tensor 14.4 Linear superpositions of p orbitale 18.3 The product of a Ay matrix and a x matrix is x 14.5 Integrated rate laws 18.4 Transpose of a sum 14.6 Classical mechanical oscillator harmonic 14.7 Separation of variables in Fick's second law 15.2 Coupled differential equations for a reaction mechanism 15.3 Steady-state analysis of a reaction mechanism 15.4 Taylor-series integration of rate laws 15.5 Perturbation theory of harmonic oscillator with friction group 16.2 Some function spaces that qualify as vector spaces 16.3 A function space that is not a vector space 16.4 The dot product qualifies as an inner product 16.5 An inner product for 18.6 Inverse of a narrow matrix 18.7 The method of least squares as a matrix computation 18.8 15.1 Euler's method 16.1 A molecular symmetry 18.5 Inverse of a square matrix functions 16.6 Linear dependence 16.7 Bases for R 16.8 A basis for P°° 16.9 Using inner products to determine coordinates Determinants 18.9 The determinant of the twodimensional rotation matrix 18.10 Linear equations with no unique solution 19.1 A x matrix equation eigenvalue 19.2 Characteristic polynomial from a determinant 19.3 Eigenvalues of similar matrices 19.4 Rigid-body moments of inertia 19.5 Principal axes of rotation for formyl chloride 19.6 Functions of Hermitian matrices 19.7 Quantum mechanical particle on a ring 19.8 Quantum mechanical oscillator harmonic 19.9 Matrix formulation of the variational principle for a basis of dimension xviii LIST OF 20.1 Constants of motion for a free particle 20.2 Calculating an expectation value 20.3 Antisymmetry exchange of electron 20.4 Slater determinant for helium EXAMPLES 21.1 Fourier analysis of a wave packet 21.2 Discrete Fourier transform of a Lorentzian signal 21.3 Savitzky-Golay 21.4 Time-domain filtering filtering 21.5 Simultaneous noise filtering and resolution of overlapping peaks Greek Alphabet /etters Name Transliteration Letters A, a alpha a N, v B, ß beta b Г, Δ, δ E, e gamma g d e Ξ, ξ О, о Π, π z, С zeta z H, η età Θ, 6» theta e th I, iota kappa L К, к Л, λ Μ, μ delta epsilon lambda k mu m XIX Transliteration mi xi n omicron о Pi rho sigma Р r Τ, τ Τ, υ Φ, φ tau upsilon t u phi X, χ Φ, φ Ω, ω chi ph kh Ρ, Ρ Σ, σ, ς i Name psi omega X s ps Mathematical Methods for Physical and Analytical Chemistry by David Z Goodson Copyright © 2011 John Wiley & Sons, Inc Part I Calculus Functions: General Properties Functions: Examples Coordinate Systems Integration Numerical Methods Complex Numbers Extrapolation Mathematical Methods for Physical and Analytical Chemistry by David Z Goodson Copyright © 2011 John Wiley & Sons, Inc Chapter Functions: General Properties This chapter provides a brief review of some basic ideas and terminology from calculus 1.1 Mappings A function is a mapping of some given number into another number The function f(x) = x2, for example, maps the number into the number 9, ^ + The function is a rule that indicates the destination of the mapping An operator is a mapping of a function into another function E x a m p l e 1.1 Contrasting the concepts of function maps f(x) = x2 into f'(x) = 2x, x2 ^ The first-derivative function and operator The operator -^~ 2x f'(x) = 2x applied, for example, t o the number gives -ί—► In contrast, the operator 4- applied t o the number gives A - ^ 0, as it treats "3" as a function f(x) = and "0" as a function f(x) = In principle, a mapping can have an inverse, which undoes its effect Suppose q is the inverse of / Then g(f(x))=x (i-i) For the example f(x) — x2 we have the mappings —► —> The effect of performing a mapping and then performing its inverse mapping is to map the value of x back to itself For the function x2 the inverse is the square root function, g(y) = y/y To prove this, we simply note that if we let у be the result of the mapping / (that is, у — x2), then 9(f(x)) = vx2 = x Graphs of x2 and y/y are compared in Fig 1.1 Note that the graph of yfy can be obtained by reflecting1 the graph of x2 through the diagonal line y — x x T h e reflection of a point through a line is a mapping to the point on the opposite side such that the new point is the same distance from the line as was the original point CHAPTER FUNCTIONS: GENERAL PROPERTIES Figure 1.1: Graph of у = x2 and its inverse, y/y Reflection about the dashed line (y = x) interchanges the function and its inverse Fig 1.1 illustrates an interesting fact: An inverse mapping can in some cases be multiple valued, x2 maps to 4, but it also maps —2 to The mapping / in this case is unique, in the sense that we can say with certainty what value of f{x) corresponds to any value x The inverse mapping g in this case is not unique; given у = 4, g could map this to +2 or to —2 In Fig 1.1, values of the variable у for у > each correspond to two different values of y/y This function has two branches On one branch, g(y) = \y/y\ On the other, g(y) = -\y/y\ The inverse of f(u) is designated by the symbol / _ ( u ) This can be confusing Often, the indication of the variable, "(u)," is omitted to make the notation less cumbersome Then, / _ can be the inverse, / _ ( u ) , or the reciprocal, / ( w ) _ = l / / ( u ) Usually these are not equivalent If f(u) = u2, the inverse i s / _ = / _ ( u ) = л/й while the reciprocal is / _ = / ( i t ) - = u~2 Which meaning is intended must be determined from the context 1.2 Differentials and Derivatives A function f(x) is said to be continuous at a specified point XQ if the limit x —» xo of f(x) is finite and has the same value whether it is approached from one direction or the other Calculus is the study of continuous change It was developed by Newton to describe the motions of objects in response to change in time However, as we will see in this book, its applications are much broader The basic tool of calculus is the differential, an infinitesimal change in a variable or function, indicated by prefixing a "d" to the symbol for the English alchemist, physicist, and mathematician Isaac Newton (1642-1727) Calculus was also developed, independently and almost simultaneously, by the German philosopher, mathematician, poet, lawyer, and alchemist Gottfried Wilhelm von Leibniz (1646-1716) 1.2 DIFFERENTIALS AND DERIVATIVES quantity that is changing If x is changed to x+dx, where dx is "infinitesimally small," then f(x) changes to / + df in response The formal definition of the differential of / is df= lim [f(x + Ax)-f(x)} (1.2) This is usually written Δχ—>0 df = f(x + dx)-f(x), (1.3) where f{x + dx) — f(x) is an abbreviation for the left-hand side of Eq (1.2) The basic idea of differential calculus is that the response to an infinitesimal change is linear In other words, df is proportional to dx; that is, df = f'dx, (1.4) where the proportionality factor, / ' , is called the derivative of / Solving Eq (1.4) for / ' , we obtain / ' = df /dx We now have three different notations for the derivative, ,, , ,/