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Mathematical Methods for Physical and Analytical Chemistry

ISBN: 978-0-470-47354-2
408 pages
October 2011
Mathematical Methods for Physical and Analytical Chemistry (0470473541) cover image
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.
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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 Schrödinger'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

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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.
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“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)

 

 

 

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