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.
Mathematical Methods for Physical and Analytical Chemistry covers:
- CALCULUS—review of the basics, coordinate systems, degrees of freedom, special functions, numerical methods, complex numbers, singular points, improper integrals, Taylor series
- STATISTICS—probability theory, distribution functions, confidence intervals, propagation of error, significance of difference, ANOVA, method of least squares, calibration, model testing, fits with error in both variables, experiment design, randomization, optimization
- DIFFERENTIAL EQUATIONS—chemical reaction rate equations, Lagrangian and Hamiltonian mechanics, transport equations, the superposition principle, separation of variables, methods for exact, approximate, and numerical solutions
- LINEAR ALGEBRA—groups, Hilbert spaces, basis sets, matrices, determinants, orthogonal polynomials, spherical harmonics, Fourier series, eigenvalue equations, diagonalization, Fourier transform, spectral lineshapes, convolution, principles of quantum mechanics, Schrödinger's equation, hydrogen orbitals, hybrid orbitals, molecular orbitals
Key Features
- Modern topics such as Monte Carlo simulation, robust estimation, and discrete Fourier transform, which are otherwise available only in more specialized texts.
- Numerous figures and worked out examples and more than 200 exercises, many of which take advantage of computer algebra.
- An annotated bibliography of references for further study.
Contents
Part I. Calculus
- 1 Functions: General Properties
- 1.1 Mappings
- 1.2 Differentials and Derivatives
- 1.3 Partial Derivatives
- 1.4 Integrals
- 1.5 Critical Points
2 Functions: Examples
- 2.1 Algebraic Functions
- 2.2 Transcendental Functions
- 2.3 Functional
3 Coordinate Systems
- 3.1 Points in Space
- 3.2 Coordinate Systems for Molecules
- 3.3 Abstract Coordinates
- 3.4 Constraints
- 3.5 Differential Operators in Polar Coordinates
4 Integration
- 4.1 Change of Variables in Integrands
- 4.2 Gaussian Integrals
- 4.3 Improper Integrals
- 4.4 Dirac Delta Function
- 4.5 Line Integrals
5 Numerical Methods
- 5.1 Interpolation
- 5.2 Numerical Differentiation
- 5.3 Numerical Integration
- 5.4 Random Numbers
- 5.5 Root Finding
- 5.6 Minimization
6 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
7 Extrapolation
- 7.1 Taylor Series
- 7.2 Partial Sums
- 7.3 Applications of Taylor Series
- 7.4 Convergence
- 7.5 Summation Approximants
Part II. Statistics
8 Estimation
8 Estimation
- 8.1 Error and Estimation Ill
- 8.2 Probability Distributions
- 8.3 Outliers
- 8.4 Robust Estimation
9 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
10 Fitting
- 10.1 Method of Least Squares
- 10.2 Fitting with Error in Both Variables
- 10.3 Nonlinear Fitting
11 Quality of Fit 165
- 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 Experiment Design
- 12.1 Risk Assessment
- 12.2 Randomization
- 12.3 Multiple Comparisons
- 12.4 Optimization
Part III. Differential Equations
13 Examples of Differential Equations
13 Examples of Differential Equations
- 13.1 Chemical Reaction Rates
- 13.2 Classical Mechanics
- 13.3 Differentials in Thermodynamics
- 13.4 Transport Equations
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
15 Solving Differential Equations, II
- 15.1 Numerical Solution
- 15.2 Chemical Reaction Mechanisms
- 15.3 Approximation Methods
Part IV. Linear Algebra
16 Vector Spaces
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
17 Spaces of Functions
- 17.1 Orthogonal Polynomials
- 17.2 Function Resolution
- 17.3 Fourier Series
- 17.4 Spherical Harmonics
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
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
20 Schrödinger's Equation
- 20.1 Quantum Mechanics
- 20.2 Atoms and Molecules
- 20.3 The One-Electron Atom
- 20.4 Hybrid Orbitals
- 20.5 Antisymmetry
- 20.6 Molecular Orbitals
21 Fourier Analysis
- 21.1 The Fourier Transform
- 21.2 Spectral Line Shapes
- 21.3 Discrete Fourier Transform
- 21.4 Signal Processing
Appendices
- A Computer Programs
- A.l Robust Estimators
- A.2 FREML
- A.3 Neider-Mead Simplex Optimization
- B Answers to Selected Exercises
- C Bibliography
Index
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.
Product Details
- Hardcover: 408 pages
- Publisher: Wiley; 1 edition (2011)
- Language: English
- ISBN-10: 0470473541
- ISBN-13: 978-0470473542
- Product Dimensions: 9.3 x 6.1 x 1 inches
List Price: $99.95