Mathematical methods for physical and analytical chemistry

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Title:

Personal Author:

Goodson, David Z., 1958-

Publication Information:

Hoboken, NJ. : Wiley, c2011.

Physical Description:

xix, 382 p. : ill. ; 24 cm.

ISBN:

9780470473542

Abstract:

"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"-- Provided by publisher.

Subject Term:

Mathematical physics

Chemometrics

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010281234	QC20 G66 2011	Open Access Book	Book	Searching... Unknown

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Summary

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.

Author Notes

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.

Preface	p. xiii
List of Examples	p. xv
Greek Alphabet	p. xix
Part I Calculus
1 Functions: General Properties	p. 3
1.1 Mappings	p. 3
1.2 Differentials and Derivatives	p. 4
1.3 Partial Derivatives	p. 7
1.4 Integrals	p. 9
1.5 Critical Points	p. 14
2 Functions: Examples	p. 19
2.1 Algebraic Functions	p. 19
2.2 Transcendental Functions	p. 21
2.2.1 Logarithm and Exponential	p. 21
2.2.2 Circular Functions	p. 24
2.2.3 Gamma and Beta Functions	p. 26
2.3 Functionals	p. 31
3 Coordinate Systems	p. 33
3.1 Points in Space	p. 33
3.2 Coordinate Systems for Molecules	p. 35
3.3 Abstract Coordinates	p. 37
3.4 Constraints	p. 39
3.4.1 Degrees of Freedom	p. 39
3.4.2 Constrained Extrema*	p. 40
3.5 Differential Operators in Polar Coordinates	p. 43
4 Integration	p. 47
4.1 Change of Variables in Integrands	p. 47
4.1.1 Change of Variable: Examples	p. 47
4.1.2 Jacobian Determinant	p. 49
4.2 Gaussian Integrals	p. 51
4.3 Improper Integrals	p. 53
4.4 Dirac Delta Function	p. 56
4.5 Line Integrals	p. 57
5 Numerical Methods	p. 61
5.1 Interpolation	p. 61
5.2 Numerical Differentiation	p. 63
5.3 Numerical Integration	p. 65
5.4 Random Numbers	p. 70
5.5 Root Finding	p. 71
5.6 Minimization*	p. 74
6 Complex Numbers	p. 79
6.1 Complex Arithmetic	p. 79
6.2 Fundamental Theorem of Algebra	p. 81
6.3 The Argand Diagram	p. 83
6.4 Functions of a Complex Variable*	p. 87
6.5 Branch Cuts*	p. 89
7 Extrapolation	p. 93
7.1 Taylor Series	p. 93
7.2 Partial Sums	p. 97
7.3 Applications of Taylor Series	p. 99
7.4 Convergence	p. 102
7.5 Summation Approximants*	p. 104
Part II Statistics
8 Estimation	p. 111
8.1 Error and Estimation	p. 111
8.2 Probability Distributions	p. 113
8.2.1 Probability Distribution Functions	p. 113
8.2.2 The Normal Distribution	p. 115
8.2.3 The Poisson Distribution	p. 119
8.2.4 The Binomial Distribution*	p. 120
8.2.5 The Boltzmann Distribution*	p. 121
8.3 Outliers	p. 124
8.4 Robust Estimation	p. 126
9 Analysis of Significance	p. 131
9.1 Confidence Intervals	p. 131
9.2 Propagation of Error	p. 136
9.3 Monte Carlo Simulation of Error	p. 139
9.4 Significance of Difference	p. 140
9.5 Distribution Testing*	p. 144
10 Fitting	p. 151
10.1 Method of Least Squares	p. 151
10.1.1 Polynomial Fitting	p. 151
10.1.2 Weighted Least Squares	p. 154
10.1.3 Generalizations of the Least-Squares Method*	p. 155
10.2 Fitting with Error in Both Variables	p. 157
10.2.1 Uncontrolled Error in x	p. 157
10.2.2 Controlled Error in x	p. 160
10.3 Nonlinear Fitting	p. 162
11 Quality of Fit	p. 165
11.1 Confidence Intervals for Parameters	p. 165
11.2 Confidence Band for a Calibration Line	p. 168
11.3 Outliers and Leverage Points	p. 171
11.4 Robust Fitting*	p. 173
11.5 Model Testing	p. 176
12 Experiment Design	p. 181
12.1 Risk Assessment	p. 181
12.2 Randomization	p. 185
12.3 Multiple Comparisons	p. 188
12.3.1 ANOVA*	p. 189
12.3.2 Post-Hoc Tests*	p. 191
12.4 Optimization*	p. 195
Part III Differential Equations
13 Examples of Differential Equations	p. 203
13.1 Chemical Reaction Rates	p. 203
13.2 Classical Mechanics	p. 205
13.2.1 Newtonian Mechanics	p. 205
13.2.2 Lagrangian and Hamiltonian Mechanics	p. 208
13.2.3 Angular Momentum	p. 211
13.3 Differentials in Thermodynamics	p. 212
13.4 Transport Equations	p. 213
14 Solving Differential Equations, I	p. 217
14.1 Basic Concepts	p. 217
14.2 The Superposition Principle	p. 220
14.3 First-Order ODE's	p. 222
14.4 Higher-Order ODE's	p. 225
14.5 Partial Differential Equations	p. 228
15 Solving Differential Equations, II	p. 231
15.1 Numerical Solution	p. 231
15.1.1 Basic Algorithms	p. 231
15.1.2 The Leapfrog Method*	p. 234
15.1.3 Systems of Differential Equations	p. 235
15.2 Chemical Reaction Mechanisms	p. 236
15.3 Approximation Methods	p. 239
15.3.1 Taylor Series*	p. 239
15.3.2 Perturbation Theory*	p. 242
Part IV Linear Algebra
16 Vector Spaces	p. 247
16.1 Cartesian Coordinate Vectors	p. 247
16.2 Sets	p. 248
16.3 Groups	p. 249
16.4 Vector Spaces	p. 251
16.5 Functions as Vectors	p. 252
16.6 Hilbert Spaces	p. 253
16.7 Basis Sets	p. 256
17 Spaces of Functions	p. 261
17.1 Orthogonal Polynomials	p. 261
17.2 Function Resolution	p. 267
17.3 Fourier Series	p. 270
17.4 Spherical Harmonics	p. 275
18 Matrices	p. 279
18.1 Matrix Representation of Operators	p. 279
18.2 Matrix Algebra	p. 282
18.3 Matrix Operations	p. 284
18.4 Pseudoinverse*	p. 286
18.5 Determinants	p. 288
18.6 Orthogonal and Unitary Matrices	p. 290
18.7 Simultaneous Linear Equations	p. 292
19 Eigenvalue Equations	p. 297
19.1 Matrix Eigenvalue Equations	p. 297
19.2 Matrix Diagonalization	p. 301
19.3 Differential Eigenvalue Equations	p. 305
19.4 Hermitian Operators	p. 306
19.5 The Variational Principle*	p. 309
20 Schrödinger's Equation	p. 313
20.1 Quantum Mechanics	p. 313
20.1.1 Quantum Mechanical Operators	p. 313
20.1.2 The Wavefunction	p. 316
20.1.3 The Basic Postulates*	p. 317
20.2 Atoms and Molecules	p. 319
20.3 The One-Electron Atom	p. 321
20.3.1 Orbitals	p. 321
20.3.2 The Radial Equation*	p. 323
20.4 Hybrid Orbitals	p. 325
20.5 Antisymmetry*	p. 327
20.6 Molecular Orbitals*	p. 329
21 Fourier Analysis	p. 333
21.1 The Fourier Transform	p. 333
21.2 Spectral Line Shapes*	p. 336
21.3 Discrete Fourier Transform*	p. 339
21.4 Signal Processing	p. 342
21.4.1 Noise Filtering*	p. 342
21.4.2 Convolution*	p. 345
A Computer Programs	p. 351
A.1 Robust Estimators	p. 351
A.2 FREML	p. 352
A.3 Nelder-Mead Simplex Optimization	p. 352
B Answers to Selected Exercises	p. 355
C Bibliography	p. 367
Index	p. 373

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Table of Contents