Computational Quantum Chemistry, Second Edition

✍ Scribed by Ram Yatan Prasad, Pranita

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 715
• Edition: 2
• Category: Library

✦ Synopsis

Computational Quantum Chemistry, Second Edition, is an extremely useful tool for teaching and research alike. It stipulates information in an accessible manner for scientific investigators, researchers and entrepreneurs. The book supplies an overview of the field and explains the fundamental underlying principles. It also gives the knowledge of numerous comparisons of different methods. The book consists of a wider range of applications in each chapter. It also provides a number of references which will be useful for academic and industrial researchers. It includes a large number of worked-out examples and unsolved problems for enhancing the computational skill of the users.

Features

• Includes comprehensive coverage of most essential basic concepts
• Achieves greater clarity with improved planning of topics and is reader-friendly
• Deals with the mathematical techniques which will help readers to more efficient problem solving
• Explains a structured approach for mathematical derivations
• A reference book for academicians and scientific investigators

Ram Yatan Prasad, PhD, DSc (India), DSc (hc) Colombo, is a Professor of Chemistry and former Vice Chancellor of S.K.M University, Jharkhand, India.

Pranita, PhD, DSc (hc) Sri Lanka, FICS, is an Assistant Professor of Chemistry at Vinoba Bhave University, India.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Foreword
Preface
Authors
1 Quantum Theory
1.1 Black-Body Radiation
1.2 Wien’s Radiation Law
1.3 Rayleigh–Jeans Law
1.4 Planck’s Radiation Law
1.5 Quantum Theory
1.6 Photoelectric Effect
1.7 Compton Effect
1.8 Atomic Hydrogen Spectra
1.9 The Bohr Model
1.9.1 Energy of an Electron Revolving Around the Nucleus in a Permitted Orbit
1.9.2 Velocity of an Electron
1.9.3 Radius of the Orbit
1.9.4 Shortcoming of Bohr’s Model
Bibliography
Solved Problems
Questions on Concepts
2 Wave–Particle Duality
2.1 Dual Nature of Electron/de Broglie Wave
2.2 Davisson and Germer’s Experiment
2.3 Quantisation of Angular Momentum
2.4 Heisenberg’s Uncertainty Principle
2.5 Phase Velocity
2.6 Group Velocity
2.7 Uncertainty Relation Between Energy and Time
2.8 Experimental Evidence of Heisenberg’s Uncertainty Principle
2.8.1 Diffraction of Electrons Through a Slit
2.8.2 Gamma Ray Microscope Thought Experiment
2.8.3 Physical Significance of Uncertainty Principle
Bibliography
Solved Problems
Questions on Concepts
3 Mathematical Techniques
3.1 Differential Equations
3.1.1 Ordinary Differential Equation
3.1.2 Partial Differential Equation
3.1.3 Order and Degree of a Differential Equation
3.1.3.1 Order
3.1.3.2 Degree
3.1.4 Linear and Non-Linear Differential Equation
3.1.5 General Solution, Particular Solution, and Arbitrary Constants
3.1.5.1 General Solution
3.1.5.2 Particular Solution
3.1.5.3 Arbitrary Constants
3.1.6 Differential Equation of the First Order and the First Degree
3.1.6.1 Worked Out Examples
3.1.7 Linear Differential Equation
3.1.8 Equation of the Type dy/dx+ Py = Qy[sup(n)]
3.1.9 Linear Differential Equation with Constant Coefficient/ Second-Order Differential Equation with Constant Coefficient
3.1.10 Solving Differential Equations by Power Series
3.2 Matrices
3.2.1 Types of Matrices
3.2.1.1 Rectangular Matrix
3.2.1.2 Square Matrix
3.2.1.3 Non-Singular and Singular Matrices
3.2.1.4 Unit Matrix
3.2.1.5 Null Matrix or Zero Matrix
3.2.1.6 Row Matrix
3.2.1.7 Column Matrix
3.2.1.8 Diagonal Matrix
3.2.1.9 Scalar Matrix
3.2.2 Operation of Matrices
3.2.2.1 Addition of Two Matrices
3.2.2.2 Subtraction of Two Matrices
3.2.2.3 Multiplication of Two Matrices
3.2.3 Transpose of a Matrix
3.2.4 Symmetric Matrix
3.2.5 Skew-Symmetric Matrix
3.2.6 Complex Matrix
3.2.7 Complex Conjugate of a Matrix
3.2.8 Hermitian Matrix
3.2.9 Skew-Hermitian Matrix
3.2.10 Adjoint of a Matrix
3.2.11 Inverse of a Matrix
3.2.12 Orthogonal Matrices
3.3 Determinants
3.3.1 Properties of Determinants
3.3.2 Minors and Co-Factors
3.3.3 Uses of Determinants in Quantum Chemistry
3.4 Characteristics Value Problem
3.5 Similarity Transformation
3.6 Block Diagonalisation of Matrices
Bibliography
Solved Problems
Questions on Concepts
4 Quantum Mechanical Operators
4.1 Linear Operator and Non-Linear Operator
4.2 Commutator
4.2.1 Facts About Commutation
4.3 Hermitian Operator
4.3.1 Properties of Hermitian Operator
4.3.1.1 The Eigen Values of a Hermitian Operator are Real
4.3.1.2 Non-Degenerate Eigen Functions of a Hermitian Operator Form an Orthogonal Set
4.3.1.3 If a Hermitian Operator  Commutes with an Arbitrary Operator B and Ψ[sub(k)] and Ψ[sup(1)] are Two Eigen Functions of  with Non-Degenerate Eigen Values, Then Bra-Ket Notation, Prove That <Ψ[sup(k)]| B | Ψ[sub(1)]> = 0
4.3.1.4 If Two Hermitian Operators  and B Possess a Common Eigen Function, Then They Commute
4.3.1.5 If Two Hermitian Operators  and B Commute, Then They Must Have a Common Eigen Function
4.4 Schmidt Orthogonalisation
4.5 ∇ and ∇[sup(2)] Operators
4.6 Linear Momentum Operator
4.6.1 Operators of Every Two Components of the Momentum Commute
4.6.2 Momentum Components Commute with Unlike Co-Ordinates
4.6.3 Momentum Components Do Not Commute with Their Relative Co-Ordinates
4.7 Angular Momentum Operator or Angular Momentum Vector (L)
4.7.1 Operators of the Angular Momentum Components Do Not Commute
4.7.2 Operators of the Angular Momentum Components Do Commute with the Operator of the Square of the Angular Momentum
4.7.3 Angular Momentum in Spherical Polar Co-Ordinates
4.7.4 Ladder Operators or Step-Up and Step-Down Operators for Angular Momentum
4.8 Hamiltonian Operator
4.9 Commutation Relation of Angular Momentum Operators with Hamiltonian Operators and with Each Other
4.10 Projection Operators
4.11 Parity Operator (π Operator)
Bibliography
Solved Problems
Questions on Concepts
5 Postulates of Quantum Mechanics
5.1 Postulate 1
5.2 Po Stulate 2
5.2.1 Construction of Quantum Mechanical Operator
5.3 Postulate 3
5.4 Postulate 4
5.5 Postulate 5
5.6 Postulate 6
Bibliography
Solved Problems
Questions on Concepts
6 The Schrödinger Equation
6.1 Equation of Wave Motion
6.1.1 Time-Independent Schrödinger Equation
6.1.2 Time-Dependent Schrödinger Equation
6.1.3 Interpretation of Wave Function, Ψ
6.1.4 Acceptable Wave Function
6.2 Normalisation
6.3 Orthogonality
6.3.1 Orthonormality
6.3.2 EIgen Function and Eigen Value
6.3.3 Degeneracy
6.4 Transformation of the Laplacian Into Spherical Polar Co-Ordinates
6.5 Ehrenfest’s Theorem
6.6 Matrix Representation of Wave Function
6.7 Matrix Representation of Operator
6.8 Properties of Matrix Elements
6.9 Matrix Form of the Schrödinger Equation
6.9.1 Time-Dependent Schrödinger Equation in Matrix Form
Bibliography
Solved Problems
Questions on Concepts
7 Playing with the Schrödinger Equation
7.1 Particle in a One-Dimensional Box
7.1.1 Energy Level Diagram
7.2 Particle in a Rectangular Three-Dimensional Box or Particle in a Three-Dimensional Box
7.2.1 Energy Levels for a Cubic Potential Box
7.2.2 The Tunnel Effect or Tunnelling
7.2.3 Importance of Tunnel Effect
7.2.4 Quantum Mechanical Explanation of Emission of α-Particles
7.3 Particle on a Ring
7.3.1 Particle on a Ring (Considering the Spherical Polar Co-Ordinates)
7.4 Particle on a Sphere
7.4.1 The Legendre Polynomials
7.4.1.7 Normalisation of the Legendre Polynomial
7.4.1.2 Orthogonality of the Legendre Polynomials
7.4.2 Associated Legendre Equation
7.4.3 Associated Legendre Functions
7.4.4 Spherical Harmonics
7.4.5 Particle on a Sphere
7.5 Rigid Rotors
7.5.1 F Equation
7.5.2 T Equation
7.5.3 Energy Levels
7.6 Hermite Polynomials
7.6.1 Orthogonal Properties of Hermite Polynomials
7.7 Simple Harmonic Oscillator
7.7.1 Classical Treatment
7.7.2 Quantum Mechanical Treatment
7.7.2.7 Asymptotic Solution
7.7.2.2 Series Solution
7.7.3 Wave Function of Linear Harmonic Oscillator
Bibliography
Solved Problems
Questions on Concepts
Numerical Problems
8 Hydrogen Atom
8.1 The Hydrogen Atom (Simple Solution of the Schrödinger Equation)
8.2 Generalised Solution of the Schrödinger Equation for Hydrogen Atom/Hydrogen-Like Species
8.3 Solution of the F Equation
8.4 Solution of the T Equation or the Polar Wave Equation
8.5 The Laguerre Differential Equation
8.5.1 Laguerre Polynomials
8.5.2 The Rodrigues Formula for the Laguerre Polynomials
8.5.3 The Laguerre Associated Equation and Its Solution
8.5.4 Associated Laguerre Polynomials
8.5.5 The Rodrigues Formula for the Associated Laguerre Polynomials
8.6 Solution of the Radial Equation
8.6.1 Normalisation of The Radial Wave Function
8.6.2 Complete Wave Function for the H Atom
8.6.3 Hydrogenic Atomic Orbital
8.6.4 Radial Wave Function
8.7 Most Probable Distance of Electron from the Nucleus of H Atom
8.7.1 Average Distance of Electron from the Nucleus of H Atom
Bibliography
Solved Problems
Questions on Concepts
9 Approximate Methods
9.1 Perturbation Theory/Method for Nondegenerate States
9.1.1 First-Order Perturbation
9.1.1.1 Correction to Energy
9.1.1.2 Correction to Wave Function
9.1.2 Second-Order Perturbation
9.1.2.1 Correction to Energy
9.1.2.2 Second-Order Correction to Wave Functions
9.2 Bra–Kept Notation or Dirac’s Notation
9.2.1 Expression for First-Order Correction to Energy for Nondegenerate State Using Dirac’s Notation
9.2.2 First-Order Correction to Wave Function for Nondegenerate State Using Dirac’s Notation
9.2.3 Second-Order Correction to the Energy Using Dirac’s Notation
9.2.4 Alternatively: Second-Order Correction to the Energy Using Dirac’s Notation
9.2.5 Second-Order Correction to Wave Function Using Dirac’s Notation
9.3 Perturbation Theory: a Degenerate Case
9.3.1 First-Order Correction to Energy
9.3.2 First-Order Correction to Wave Function
9.3.3 Alternative Way to Handle Degenerate Perturbation Theory: Twofold Degeneracy
9.4 Application of Perturbation Theory
9.4.1 Anharmonic Oscillator
9.4.2 Electronic Polarisability of Hydrogen Atom
9.4.3 Helium Atom
9.4.4 Alternatively: The Helium Atom
9.5 Variation Theorem/Method
9.5.1 Variation Method
9.5.2 Variation Theorem
9.5.3 Computation of Energy Eigen Value and Wave Function by Variation Method
9.5.4 Computation of Wave Function
9.6 Application of Variation Principle/Method
9.6.1 Estimation of Energy of the Ground State of the Simple Harmonic Oscillator Using the Trial Function Ae[sup(-ax2)]
9.6.2 Ground State of Helium Atom
9.6.3 Ground State of Hydrogen Atom
Bibliography
Solved Problems
Based on Variation Theory
Questions on Concepts
10 Diatomic Molecules
10.1 Born–Oppenheìmer Approximation
10.2 Hydrogen Molecule Ion
10.2.1 Evaluation of Overlap Integral
10.2.2 Evaluation of the Coulomb Integral
10.2.3 Evaluation of Resonance Integral or Exchange Integral
10.3 Evaluation of Ψ and Ψ[sup(2)] (Probability)
10.4 Hydrogen Molecule (Spin Independent)
10.5 Linear Combination of Atomic Orbitals
10.6 Molecular Orbital Theory
10.7 Valence Bond Treatment of H[sub(2)] Molecule
10.8 Configuration Interaction
10.9 Comparison of the Molecular Orbital and Valence Bond Theories
10.10 Symmetric and Antisymmetric Wave Functions
10.11 Pauli’s Exclusion Principle
10.12 Antisymmetric Wave Function and Slater Determinant
10.13 Bonding and Antibonding Orbitals
10.14 Electron Density in Molecular Hydrogen
10.15 Excited State of H[sub(2)] Molecule
10.16 Electronic Transition in Hydrogen Molecule
10.17 Homopolar Diatomic or Homonuclear Diatomic Molecules
10.17.1 Molecules with S and P valence Atomic Orbitals
10.17.2 Electronic Configuration of Homonuclear Diatomic Molecules
10.18 Heteropolar Diatomic or Heteronuclear Diatomic Molecules
Bibliography
Solved Problems
Questions on Concepts
Numerical Problems
11 Multielectronic Systems
11.1 Energy of the Many-Electron System
11.2 Fock Equation and Hartree Equation
11.2.1 Application in Two-Electron Systems – for Getting Hartree Equation and Energy of Two-Electron System
11.3 Hartree and Hartree–Fock Self-Consistent Field Methods
11.4 Excited State of Helium
11.5 Lithium in the Ground State
11.6 Atomic Magnets and Magnetic Quantum Numbers
11.6.1 Atomic Magnets
11.6.2 Magnetic Quantum Number
11.6.2.1 The Fourth Quantum Number
11.6.2.2 Electron Spin
11.6.3 Atoms Having Two or More Than Two Electrons
11.7 The Gyromagnetic Ratio and the Landé Splitting Factor
11.7.1 Landé ‘g’ Factor or Splitting Factor
11.7.2 Landé Interval Rule
11.7.3 Zeeman Effect
11.7.3.1 Origin of the Zeeman Effect
11.7.3.2 The Normal Zeeman Effect
11.7.3.3 The Anomalous/Complex Zeeman Effect
11.8 Stark Effect
11.9 Coupling of Orbital Angular Momentum
11.10 Coupling of Spin Momenta
11.11 Coupling of Orbital and Spin Angular Momenta
11.11.1 L-S or the Russell–Saunders Coupling Scheme
11.11.2 jj-Coupling Scheme
11.12 Multiplicity and Atomic States
11.13 Hund’s Rule
11.14 Atomic Terms and Symbols
11.14.1 Terms of Non-equivalent Electrons
11.14.2 Terms of Equivalent Electrons
11.14.3 Use of jj Coupling
11.15 Slater Rules
11.16 Slater-Type Orbitals
11.17 Gaussian-Type Orbitals
11.17.1 Gaussian Basis Set
11.18 Condon–Slater Rules: Evaluation of Matrix Elements
11.19 Koopman’s Theorem
11.20 Brillouin’s Theorem
11.21 Roothaan’s Equations: The Matrix Solution of the Hartree–Fock Equation
Bibliography
Solved Problems
Questions on Concepts
12 Polyatomic Molecules
12.1 Matrix Form of Roothaan’s Equations
12.2 Fock Matrix Elements
12.3 Roothaan’s Method in One Dimension
12.4 Electronic Energy
12.5 Solution of Roothaan’s Equation for he Atom
12.6 Hybridisation
12.6.1 Sp[sup(3)] hybridisation
12.6.2 Sp[sup(2)] hybridisation
12.6.3 Sp Hybridisation
12.6.4 Hybridisation in H[sub(2)]O
12.7 Semi-Empirical Methods
12.7.1 Valence Electrons
12.7.2 Zero Differential Overlap
12.7.3 π[sub(i)]-Electron Evaluation
I 12.7.4 Invariance Under Transformation
12.7.5 Complete Neglect of Differential Overlap
12.7.6 Parametrisation
12.7.7 Intermediate Neglect of Differential Overlap
12.7.8 Neglect of Diatomic Differential Overlap
12.7.9 The Pariser–Parr–Pople Method
12.7.9.1 Evaluation of Integrals of Pariser–Parr–Pople Method
Bibliography
Solved Problems
Questions on Concepts
13 Hückel Molecular Orbital Theory/Method
13.1 Application of the Hückel Molecular Orbital Method to π Systems
13.1.1 Ethylene
13.1.2 Determination of the Hückel Molecular Orbital Coefficients and Molecular Orbitals of Ethylene
13.1.2.1 Graphical Representation: Plots of ψ[sub(1)] and ψ[sub(2)] vs Distance
13.1.2.2 Three-Dimensional Representation
13.1.3 Allyl System
13.1.4 Delocalisation Energy of Allyl System
13.1.5 Determination of the Hückel Molecular Orbital Coefficients and Molecular Orbitals of Allyl System
13.1.5.1 Graphical Representation
13.1.5.2 Three-Dimensional Representation: Plots of ψ[sub(1)], ψ[sub(2)] and ψ[sub()3] vs Directions
13.1.6 Butadiene
13.1.7 Delocalisation Energy of Butadiene
13.1.8 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.1.8.1 Graphical Representation
13.1.8.2 Three-Dimensional Representation
13.2 Application of the Hückel Method to Some Cyclic Polyenes
13.2.1 Cyclopropenyl System
13.2.2 Delocalisation of Cyclopropenyl System
13.2.3 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.2.4 Cyclobutadiene
13.2.5 Delocalisation Energy of Cyclobutadiene
13.2.6 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.2.7 Cyclopentadienyl System
13.2.8 Delocalisation Energy of Cyclopentadienyl Systems
13.2.9 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.2.10 Benzene
13.2.11 Delocalisation Energy of Benzene
13.2.12 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.2.13 Graphical Representation of Molecular Orbitals in Benzene
13.3 Electron Density
13.3.1 Ethylene
13.3.2 Butadiene
13.3.3 Benzene
13.4 Bond Order
13.4.1 Ethylene
13.4.2 Butadiene
13.4.3 Benzene
13.5 Free Valence
13.5.1 Ethylene
13.5.2 Butadiene
13.5.3 Benzene
13.6 Generalised Treatment of the Hückel Molecular Orbital Theory to Open-Chain Conjugated System
13.6.1 Ethylene
13.6.2 Butadiene
13.7 Generalised Treatment of the Hückel Molecular Orbital Theory to Cyclic Polyenes
13.7.1 Cyclopropenyl Radical
13.7.2 Cyclobutadiene
13.7.3 Cyclopentadienyl Radical
13.7.4 Benzene
13.8 Extended Hückel Theory
13.8.1 Hetero Atom Substitutions
13.8.2 General Improvement
13.8.3 Extended Hückel Theory Applied to Pyrrole
13.8.4 Delocalisation Energy of Pyrrole
13.8.5 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.8.6 Pyridine
13.8.7 Hückel Molecular Orbital Coefficients and Molecular Orbitals
13.8.8 Electron Density
13.8.9 Bond Order
13.8.10 HMO Treatment to Naphthalene
13.8.11 Hückel Molecular Orbital Coefficients and Molecular Orbitals
References
Bibliography
Solved Problems
Questions on Concepts
14 Density Functional Theory
14.1 Function
14.2 Functional
14.3 Hohenberg–Kohn Theorem
14.3.1 Theorem 1
14.3.2 Theorem 2
14.3.3 Alternative Proof of Hohenberg–Kohn Theorems
14.3.3.1 Theorem 1
14.3.3.2 Theorem 2
14.4 Kohn–Sham Energy
14.5 Kohn–Sham Equations
14.5.1 Comments
14.6 Local Density Approximation
14.6.1 Comments on LDA
14.6.2 Application of the LDA
14.6.3 Electron Gas
14.6.4 The Local Spin Density Approximation
14.6.5 Generalised Gradient Approximation or Gradient Correlated Functional
14.6.6 Meta-Generalised Gradient Approximation
14.6.7 Hybrid Functionals
14.6.8 Time-Dependent DFT
14.6.9 Application of Density Functional Theory
Bibliography
Questions on Concepts
Appendix I
Appendix II
Appendix III
Model Question Papers
Glossary
Index