Computational Quantum Chemistry, Second Edition

**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. 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 = 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 "Computational Quantum Chemistry 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 Readers friendly. Deals with the mathematical techniques which will help the readers more efficient in solving problems. Explains a structured approach for mathematical derivations. A reference book for academicians and Scientific investigators"-- Provided by publisher