# Mathematics for Chemistry & Physics

**By**

- George Turrell, University of Sciences and Technology of Lille, France

Chemistry and physics share a common mathematical foundation. From elementary calculus to vector analysis and group theory,

View full description**Mathematics for Chemistry and Physics**aims to provide a comprehensive reference for students and researchers pursuing these scientific fields. The book is based on the authors many classroom experience. Designed as a reference text,**Mathematics for Chemistry and Physics**will prove beneficial for students at all university levels in chemistry, physics, applied mathematics, and theoretical biology. Although this book is not computer-based, many references to current applications are included, providing the background to what goes on "behind the screen" in computer experiments.### Audience

Undergraduate and graduate students in chemistry, chemical physics, materials science, and physics. Researchers in chemistry/physics and theoretical biologists.

### Book information

- Published: December 2001
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-705051-5

### Table of Contents

Preface

1 Variables and Functions

1.1 Introduction

1.2 Functions

1.3 Classification and Properties of Functions

1.4 Exponential and Logarithmic Functions

1.5 Applications of Exponential and Logarithmic Functions

1.6 Complex Numbers

1.7 Circular Trigonometric Functions

1.8 Hyperbolic Functions

Problems

2 Limits, Derivatives and Series

2.1 Definition of a Limit

2.2 Continuity

2.3 The Derivative

2.4 Higher Derivatives

2.5 Implicit and Parametric Relations

2.6 The Extrema of a Function and Its Critical Points

2.7 The Differential

2.8 The Mean-Value Theorem and lâ€™Hospitalâ€™s Rule

2.9 Taylor's Series

2.10 Binomial Expansion

2.11 Tests of Series Convergence

2.12 Functions of Several Variables

2.13 Exact Differentials

Problems

3 Integration

3.1 The Indefinite Integral

3.2 Integration Formulas

3.3 Methods of Integration

3.3.1 Integration by Substitution

3.3.2 Integration by Parts

3.3.3 Integration of Partial Fractions

3.4 Definite Integrals

3.4.1 Definition

3.4.2 Plane Area

3.4.3 Line Integrals

3.4.4 Fido and his Master

3.4.5 The Gaussian and Its Moments

3.5 Integrating Factors

3.6 Tables of Integrals

Problems

4 Vector Analysis

4.1 Introduction

4.2 Vector Addition

4.3 Scalar Product

4.4 Vector Product

4.5 Triple Products

4.6 Reciprocal Bases

4.7 Differentiation of Vectors

4.8 Scalar and Vector Fields

4.9 The Gradient

4.10 The Divergence

4.11 The Curl or Rotation

4.12 The Laplacian

4.13 Maxwell's Equations

4.14 Line Integrals

4.15 Curvilinear Coordinates

Problems

5 Ordinary Differential Equations

5.1 First-Order Differential Equations

5.2 Second-Order Differential Equations

5.2.1 Series Solution

5.2.2 The Classical Harmonic Oscillator

5.2.3 The Damped Oscillator

5.3 The Differential Operator

5.3.1 Harmonic Oscillator

5.3.2 Inhomogeneous Equations

5.3.3 Forced Vibrations

5.4 Applications in Quantum Mechanics

5.4.1 The Particle in a Box

5.4.2 Symmetric Box

5.4.3 Rectangular Barrier: The Tunnel Effect

5.4.4 The Harmonic Oscillator in Quantum Mechanics

5.5 Special Functions

5.5.1 Hermite Polynomials

5.5.2 Associated Legendre Polynomials

5.5.3 The Associated Laguerre Polynomials

5.5.4 The Gamma Function

5.5.5 Bessel Functions

5.5.6 Mathieu Functions

5.5.7 The Hypergeometric Functions

Problems

6 Partial Differential Equations

6.1 The Vibrating String

6.1.1 The Wave Equation

6.1.2 Separation of Variables

6.1.3 Boundary Conditions

6.1.4 Initial Conditions

6.2 The Three-Dimensional Harmonic Oscillator

6.2.1 Quantum-Mechanical Applications

6.2.2 Degeneracy

6.3 The Two-Body Problem

6.3.1 Classical Mechanics

6.3.2 Quantum Mechanics

6.4 Central Forces

6.4.1 Spherical Coordinates

6.4.2 Spherical Harmonics

6.5 The Diatomic Molecule

6.5.1 The Rigid Rotator

6.5.2 The Vibrating Rotator

6.5.3 Centrifugal Forces

6.6 The Hydrogen Atom

6.6.1 Energy

6.6.2 Wavefunctions and The Probability Density

6.7 Binary Collisions

6.7.1 Conservation of Angular Momentum

6.7.2 Conservation of Energy

6.7.3 Interaction Potential: LJ (6-12)

6.7.4 Angle of Deflection

6.7.5 Quantum Mechanical Description: The Phase Shift

Problems

7 Operators and Matrices

7.1 The Algebra of Operators

7.2 Hermitian Operators and Their Eigenvalues

7.3 Matrices

7.4 The Determinant

7.5 Properties of Determinants

7.6 Jacobians

7.7 Vectors and Matrices

7.8 Linear Equations

7.9 Partitioning of Matrices

7.10 Matrix Formulation of the Eigenvalue Problem

7.11 Coupled Oscillators

7.12 Geometric Operations

7.13 The Matrix Method in Quantum Mechanics

7.14 The Harmonic Oscillator

Problems

8 Group Theory

8.1 Definition of a Group

8.2 Examples

8.3 Permutations

8.4 Conjugate Elements and Classes

8.5 Molecular Symmetry

8.6 The Character

8.7 Irreducible Representations

8.8 Character Tables

8.9 Reduction of a Representation: The â€śMagic Formulaâ€ť

8.10 The Direct Product Representation

8.11 Symmetry-Adapted Functions: Projection Operators

8.12 Hybridization of Atomic Orbitals

8.13 Crystal Symmetry

Problems

9 Molecular Mechanics

9.1 Kinetic Energy

9.2 Molecular Rotation

9.2.1 Euler's Angles

9.2.2 Classification of Rotators

9.2.3 Angular Momenta

9.2.4 The Symmetric Top in Quantum Mechanics

9.3 Vibrational Energy

9.3.1 Kinetic Energy

9.3.2 Internal Coordinates: The G Matrix

9.3.3 Potential Energy

9.3.4 Normal Coordinates

9.3.5 Secular Determinant

9.3.6 An Example: The Water Molecule

9.3.7 Symmetry Coordinates

9.3.8 Application to Molecular Vibrations

9.3.9 Form of Normal Modes

9.4 Nonrigid Molecules

9.4.1 Molecular Inversion

9.4.2 Internal Rotation

9.4.3 Molecular Conformation: The Molecular Mechanics Method

Problems

10 Probability and Statistics

10.1 Permutations

10.2 Combinations

10.3 Probability

10.4 Stirling's Approximation

10.5 Statistical Mechanics

10.6 The Lagrange Multipliers

10.7 The Partition Function

10.8 Molecular Energies

10.8.1 Translation

10.8.2 Rotation

10.8.3 Vibration

10.9 Quantum Statistics

10.9.1 The Indistinguishability of Identical Particles

10.9.2 The Exclusion Principle

10.9.3 Fermi-Dirac Statistics

10.9.4 Bose-Einstein Statistics

10.10 Ortho- and Para-Hydrogen

Problems

11 Integral Transforms

11.1 The Fourier Transform

11.1.1 Convolution

11.1.2 Fourier Transform Pairs

11.2 The Laplace Transform

11.2.1 Examples of Simple Laplace Transforms

11.2.2 The Transform of Derivatives

11.2.3 Solution of Differential Equations

11.2.4 Laplace Transforms: Convolution and Inversion

11.2.5 Green's Functions

Problems

12 Approximation Methods in Quantum Mechanics

12.1 The Born-Oppenheimer Approximation

12.2 Perturbation Theory: Stationary States

12.2.1 Nondegenerate Systems

12.2.2 First-Order Approximation

12.2.3 Second-Order Approximation

12.2.4 The Anharmonic Oscillator

12.2.5 Degenerate Systems

12.2.6 The Stark Effect of the Hydrogen Atom

12.3 Time-Dependent Perturbations

12.3.1 The SchrÂ¨Odinger Equation

12.3.2 Interaction of Light and Matter

12.3.3 Spectroscopic Selection Rules

12.4 The Variation Method

12.4.1 The Variation Theorem

12.4.2 An Example: The Particle in a Box

12.4.3 Linear Variation Functions

12.4.4 Linear Combinations of Atomic Orbitals (LCAO)

12.4.5 The HÂ¨Uckel Approximation

Problems

13 Numerical Analysis

13.1 Errors

13.1.1 The Gaussian Distribution

13.1.2 The Poisson Distribution

13.2 The Method of Least Squares

13.3 Polynomial Interpolation and Smoothing

13.4 The Fourier Transform

13.4.1 The Discrete Fourier Transform (DFT)

13.4.2 The Fast Fourier Transform (FFT)

13.4.3 An Application: Interpolation and Smoothing

13.5 Numerical Integration

13.5.1 The Trapezoid Rule

13.5.2 Simpson's Rule

13.5.3 The Method of Romberg

13.6 Zeros of Functions

13.6.1 Newton's Method

13.6.2 The Bisection Method

13.6.3 The Roots: An Example

Problems

Appendices

I The Greek Alphabet

II Dimensions and Units

III Atomic Orbitals

IV Radial Wavefunctions for Hydrogenlike Species

V The Laplacian Operator in Spherical Coordinates

VI The Divergence Theorem

VII Determination of the Molecular Symmetry Group

VIII Character Tables for Some of the More Common Point Groups

IX Matrix Elements for the Harmonic Oscillator

X Further Reading

Applied Mathematics

Chemical Physics

Author Index

Subject Index