# Mathematical Methods XIA

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Physical Chemistry: An Advanced Treatise: Mathematical Methods, Volume XIA, is devoted to mathematical techniques of interest to chemists. The purpose of this treatise is to present a comprehensive treatment of physical chemistry for advanced students and investigators in a reasonably small number of volumes. An attempt has been made to include all important topics in physical chemistry together with borderline subjects which are of particular interest and importance. The book begins with discussions of elementary concepts such as linear vector spaces; generalized function theory; complex variable theory; boundary-value problems; approximating functions and their applications in numerical differentiation, integration, and the solution of differential equations; and group theory. These are followed by more advanced and specialized chapters that emphasize chemical applications rather than mathematical rigor. This book provides the student of physical chemistry with a basic understanding of those additional mathematical techniques which are important in chemistry and should enable him to read the current literature in theoretical chemistry.

## Table of Contents

List of Contributors

Foreword

Preface

Contents of Previous and Future Volumes

Chapter 1 / Linear Vector Spaces

I. Introduction: Vectors in the Physical Sciences

II. Linear Vector Spaces

III. Example: Three-Dimensional Euclidean Vectors—I

IV. Vector Transformations

V. Matrices

VI. Example: Three-Dimensional Euclidean Vectors—II

VII. Vector Spaces of Infinite Dimension

References

Chapter 2 / Generalized Functions

I. Introduction

II. Definitions

III. The Algebra of Generalized Functions

IV. The Calculus of Generalized Functions

V. Some Singular Generalized Functions

VI. Fourier Transforms

VII. Laplace Transforms

VIII. Conclusion

References

Chapter 3/Complex Variable Theory

I. Introduction

II. Complex Numbers

III. Analytic Functions of a Complex Variable

IV. Complex Integration

V. Power Series

VI. Elementary Functions

VII. Evaluation of Real Definite Integrals

VIII. Higher Transcendental Functions

IX. On Fourier Transforms

X. Quantum Chemistry Integrals

XI. A Formula of Lagrange and Nondegenerate Perturbation Theory

References

Chapter 4 / Boundary-Value Problems

I. Introduction

II. Some Typical Boundary-Value Problems

III. The D'Alembert Solution of the Wave Equation

IV. Separation of Variables

V. Eigenvalues, Eigenfunctions, and Expansion Problems

VI. Boundary-Value Problems in Cylindrical Coordinates

VII. Boundary-Value Problems in Spherical Coordinates

VIII. Green's Functions

IX. Laplace Transform Methods

X. Conformal Mapping

References

Chapter 5 / Numerical Analysis

I. Introduction

II. Approximation by Polynomial Interpolation

III. Approximation by Spline Interpolation

IV. Approximation by Least Squares

V. Numerical Differentiation

VI. Approximate Integration or Quadrature

VII. Differential Equations

VIII. Equations in a Single Unknown

IX. Systems of Linear Equations

X. Special Methods for Solving Sparse Sets of Equations

Appendix

References

Chapter 6 / Group Theory

I. Introduction

II. Definitions

III. Symmetry Operators

IV. Group Representation Theory

V. Some Applications in Molecular Quantum Mechanics

VI. The Permutation Group and Spin

VII. Continuous Groups

VIII. Group Theory and the Solid State

References

Chapter 7 / Density Matrices

I. Introduction

II. The Full Density Matrix

III. The Reduced Density Matrix

IV. The N-Representability Problem

V. The Single-Particle Reduced Density Matrix

VI. The Second-Order Reduced Density Matrix

VII. General Geminal Wave Functions

VIII. Condensation Phenomena

References

Chapter 8 / The Green's Function Method

I. Introduction

II. Double-Time Temperature-Dependent Green's Functions

III. Spectral Representations

IV. Properties of the Green's Functions

V. The Reaction of a System to an External Perturbation

VI. Calculation of the Green's Functions

VII. Charge-Transfer Spectra of Molecular Crystals

VIII. Perturbation Theory for the Green's Functions

References

Author Index

Subject Index

## Product details

- No. of pages: 588
- Language: English
- Copyright: © Academic Press 1975
- Published: January 28, 1975
- Imprint: Academic Press
- eBook ISBN: 9780323145176