Mathematical Methods for Physical and Analytical Chemistry

Mathematical Methods for Physical and Analytical Chemistry presents mathematical and statistical methods to students of chemistry at the intermediate, post-calculus level. The content includes a review of general calculus; a review of numerical techniques often omitted from calculus courses, such as cubic splines and Newton's method; a detailed treatment of statistical methods for experimental data analysis; complex numbers; extrapolation; linear algebra; and differential equations. With numerous example problems and helpful anecdotes, this text gives chemistry students the mathematical knowledge they need to understand the analytical and physical chemistry professional literature.

Table of 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 2.3 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 3.5 Differential Operators in Polar Coordinates 43 4 Integration 47 4.1 Change of Variables in Integrands 47 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 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 8.3 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 10.2 Fitting with Error in Both Variables 157 10.3 Nonlinear Fitting 162 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 12.4 Optimization* 195 Part III. Differential Equations 13 Examples of Differential Equations 203 13.1 Chemical Reaction Rates 203 13.2 Classical Mechanics 205 13.3 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 15.2 Chemical Reaction Mechanisms 236 15.3 Approximation Methods 239 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 Schrodinger's Equation 313 20.1 Quantum Mechanics 313 20.2 Atoms and Molecules 319 20.3 The One-Electron Atom 321 20.4 Hybrid Orbitals 325 20.5 Antisymmetry* 327 20.6 Molecular Orbitals* 329 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 A Computer Programs 351 A.l Robust Estimators 351 A.2 FREML 352 A.3 Neider-Mead Simplex Optimization 352 B Answers to Selected Exercises 355 C Bibliography 367 Index 373

About the Author :
David Z. Goodson , Associate Professor of Chemistry at the University of Massachusetts Dartmouth, has a BA in chemistry from Pomona College and a PhD in chemical physics from Harvard University. An interdisciplinary scientist, he is author of numerous articles on a wide range of topics including quantum chemistry, molecular spectroscopy, reaction rate theory, atomic physics, and applied mathematics.

Review :
"Finally it can be said that this book helps to refresh and extend the knowledge about mathematical and statistical methods to be used for physico-chemical or analytical applications." ( Materials and Corrosion , 1 November 2012)