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.

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**

Mathematical Methods for Physical and Analytical Chemistry

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