7: Approximation Methods Last updated 13:44, 9 Jul 2016 Save as PDF Share Share Share Tweet Share An Physical Chemistry Libretexts Textmap organized around the textbook Physical Chemistry: A Molecular Approach by Donald A. McQuarrie and John D. Simon Chapter 1 Chapter 1: The Dawn of the Quantum Theory 1.1: Blackbody Radiation cannot Be Explained Classically 1.2: Quantum Hypothesis used for Blackbody Radiation Law 1.3: Photoelectric Effect Explained with Quantum Hypothesis 1.4: The Hydrogen Atomic Spectrum 1.5: The Rydberg Formula and the Hydrogen Atomic Spectrum 1.6: Matter Has Wavelike Properties 1.7: de Broglie Waves can be Experimentally Observed 1.8: The Bohr Theory of the Hydrogen Atom 1.9: The Heisenberg Uncertainty Principle 1.E: The Dawn of the Quantum Theory (Exercises) • Chapter 2 Chapter 2: The Classical Wave Equation 2.1: The One-Dimensional Wave Equation 2.2: The Method of Separation of Variables 2.3: Oscillatory Solutions to Differential Equations 2.4: The General Solution is a Superposition of Normal Modes 2.5: A Vibrating Membrane 2.E: The Classical Wave Equation (Exercises) • Chapter 3 Chapter 3: The Schrödinger Equation and a Particle in a Box 3.1: The Schrödinger Equation 3.2: Linear Operators in Quantum Mechanics 3.3: The Schrödinger Equation is an Eigenvalue Problem 3.4: Wavefunctions Have a Probabilistic Interpretation 3.5: The Energy of a Particle in a Box is Quantized 3.6: Wave Functions Must Be Normalized 3.7: The Average Momentum of a Particle in a Box is Zero 3.8: The Uncertainty Principle 3.9: A Particle in a Three-Dimensional Box 3.E: The Schrödinger Equation and a Particle in a Box (Exercises) • Chapter 4 Chapter 4: Postulates and Principles of Quantum Mechanics 4.1: The Wavefunction Specifies the State of a System 4.2: Quantum Operators Represent Classical Variables 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators 4.4: The Time-Dependent Schrödinger Equation 4.5: The Eigenfunctions of Operators are Orthogonal 4.6: Commuting Operators Allow Infinite Precision 4.E: Postulates and Principles of Quantum Mechanics (Exercises) • Chapter 5 Chapter 5: The Harmonic Oscillator and the Rigid Rotor 5.1: A Harmonic Oscillator Obeys Hooke's Law 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule 5.3: The Harmonic Oscillator is an Approximation 5.4: The Harmonic Oscillator Energy Levels 5.5: The Harmonic Oscillator and Infrared Spectra 5.6: The Harmonic-Oscillator Wavefunctions Involve Hermite Polynomials 5.7: Hermite Polynomials are either Even or Odd Functions 5.8: The Energy Levels of a Rigid Rotor 5.9: The Rigid Rotator is a Model for a Rotating Diatomic Molecule 5.E: The Harmonic Oscillator and the Rigid Rotor (Exercises) • Chapter 6 Chapter 6: The Hydrogen Atom 6.1: The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly 6.2: The Wavefunctions of a Rigid Rotator are Called Spherical Harmonics 6.3: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision 6.4: Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers 6.5: s Orbitals are Spherically Symmetric 6.6: Orbital Angular Momentum and the p-orbitals 6.7: The Helium Atom Cannot Be Solved Exactly 6.E: The Hydrogen Atom (Exercises) • Chapter 7 Chapter 7: Approximation Methods 7.1: The Variational Method 7.1B: Five Trial Functions applied to the Helium Atom 7.2: Linear Variational Method and the Secular Determinant 7.3: Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Parameters 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems 7.E: Approximation Methods (Exercises) • Chapter 8 Chapter 8: Multielectron Atoms 8.1: Atomic and Molecular Calculations are Expressed in Atomic Units 8.2: Pertubation Theory and the Variational Method for Helium 8.3: Hartree-Fock Equations are Solved by the Self-Consistent Field Method 8.4: An Electron Has An Intrinsic Spin Angular Momentum 8.5: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons 8.6: Antisymmetric Wave Functions can be Represented by Slater Determinants 8.7: Hartree-Fock Calculations Give Good Agreement with Experimental Data 8.8: A Term Symbol Gives a Detailed Description of an Electron Configuration 8.9: The Allowed Values of J - the Total Angular Momentum Quantum Number 8.10: Hund's Rules Determine the Term Symbols of the Ground Electronic States 8.11: Using Atomic Term Symbols to Describe Atomic Spectra 8.E: Multielectron Atoms (Exercises) • Chapter 9 Chapter 9: The Chemical Bond: Diatomic Molecules 9.1: The Born-Oppenheimer Approximation Simplifies the Schrodinger Equation for Molecules 9.2: The H+2H2+ Prototypical Species 9.3: The Overlap Integral 9.4: Chemical Bond Stability 9.5: Bonding and Antibonding Orbitals 9.6: A Simple Molecular-Orbital Treatment of H2H2 Places Both Electrons in a Bonding Orbital 9.7: Molecular Orbitals Can Be Ordered According to Their Energies 9.8: Molecular-Orbital Theory Does not Predict a Stable Diatomic Helium Molecule 9.9: Electrons Populate Molecular Orbitals According to the Pauli Exclusion Principle 9.10: Molecular Orbital Theory Predicts that Molecular Oxygen is Paramagnetic 9.11: Photoelectron Spectra Support the Existence of Molecular Orbitals 9.12: Molecular-Orbital Theory Also Applies to Heteronuclear Diatomic Molecules 9.13: An SCF-LCAO-MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently 9.14: Molecular Term Symbols Describe Electronic States of Molecules 9.15: Molecular Term Symbols Designate Symmetry 9.16: Most Molecules Have Excited Electronic States 9.E: The Chemical Bond: Diatomic Molecules (Exercises) • Chapter 10 Chapter 10: Bonding in Polyatomic Molecules 10.1: Hybrid Orbitals Account for Molecular Shape 10.2: Hybrid Orbitals in Water 10.3: Why is BeH2BeH2 Linear and H2OH2O Bent? 10.4: Photoelectron Spectroscopy 10.5: The ππ-Electron Approximation of Conjugation 10.6: Butadiene is Stabilized by a Delocalization Energy 10.7: Benzene and Aromaticity 10.E: Bonding in Polyatomic Molecules (Exercises) • Chapter 11 Chapter 11: Computational Quantum Chemistry 11.1: Gaussian Basis Sets 11.2: Extended Basis Sets 11.3: Orbital Polarization Terms in Basis Sets 11.4: The Ground-State Energy of H2H2 11.5: Quantum Calculations 11.E: Computational Quantum Chemistry (Exercises) Chapter 12 Chapter 12: Group Theory: The Exploitation of Symmetry 12.1: The Exploitation of Symmetry 12.2: Symmetry Elements 12.3: Symmetry Operations Define Groups 12.4: Symmetry Operations as Matrices 12.5: The C3VC3V Point Group 12.6: Character Tables 12.7: Characters of Irreducible Representations 12.8: Using Symmetry to Solve Secular Determinants 12.9: Generating Operators 12.E: Group Theory: The Exploitation of Symmetry (Exercises) • Chapter 13 Chapter 13: Molecular Spectroscopy 13.1: The Electromagnetic Spectrum 13.2: Rotations Accompany Vibrational Transitions 13.3: Unequal Spacings in Vibration-Rotation Spectra 13.4: Unequal Spacings in Pure Rotational Spectra 13.5: Vibrational Overtones 13.6: Electronic Spectra Contain Electronic, Vibrational, and Rotational Information 13.7: The Franck-Condon Principle 13.8: Rotational Spectra of Polyatomic Molecules 13.9: Normal Modes in Polyatomic Molecules 13.10: Irreducible Representation of Point Groups 13.11: Time-Dependent Perturbation Theory 13.12: The Selection Rule for the Rigid Rotator 13.13: The Harmonic Oscillator Selection Rule 13.14: Group Theory Determines Infrared Activity 13.E: Molecular Spectroscopy (Exercises) • Chapter 14 Chapter 14: Nuclear Magnetic Resonance Spectroscopy 14.1: Nuclei Have Intrinsic Spin Angular Momenta 14.2: Magnetic Moments Interact with Magnetic Fields 14.3: Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz 14.4: The Magnetic Field Acting upon Nuclei in Molecules Is Shielded 14.5: Chemical Shifts Depend upon the Chemical Environment of the Nucleus 14.6: Spin-Spin Coupling Can Lead to Multiplets in NMR Spectra 14.7: Spin-Spin Coupling Between Chemically Equivalent Protons Is Not Observed 14.8: The n+1 Rule Applies Only to First-Order Spectra 14.9: Second-Order Spectra Can Be Calculated Exactly Using the Variational Method 14.E: Nuclear Magnetic Resonance Spectroscopy (Exercises) • Chapter 15 Chapter 15: Lasers, Laser Spectroscopy, and Photochemistry 15.1: Electronically Excited Molecules can Relax by a Number of Processes 15.2: The Dynamics of Transitions can be Modeled by Rate Equations 15.3: A Two-Level System Cannot Achieve a Population Inversion 15.4: Population Inversion can be Achieved in a Three-Level System 15.5: What is Inside a Laser? 15.6: The Helium-Neon Laser 15.7: High-Resolution Laser Spectroscopy 15.8: Pulsed Lasers Can by Used to Measure the Dynamics of Photochemical Processes 15.E: Lasers, Laser Spectroscopy, and Photochemistry (Exercises) • Chapter 16 Chapter 16: The Properties of Gases 16.1: All Dilute Gases Behave Ideally 16.2: van der Waals and Redlich-Kwong Equations 16.3: A Cubic Equation of State 16.4: The Law of Corresponding States 16.5: The Second Virial Coefficient 16.6: The Repulsive Term in the Lennard-Jones Potential 16.7: Van der Waals Constants in Terms of Molecular Parameters 16.E: The Properties of Gases (Exercises) • Chapter 17 Chapter 17: Boltzmann Factor and Partition Functions 17.1: The Boltzmann Factor 17.2: The Thermal Boltzman Distribution 17.3: The Average Ensemble Energy 17.4: Heat Capacity at Constant Volume 17.5: Pressure in Terms of Partition Functions 17.6: Partition Functions of Distinguishable Molecules 17.7: Partition Functions of Indistinguishable Molecules 17.8: Partition Functions can be Decomposed 17.E: Boltzmann Factor and Partition Functions (Exercises) • Chapter 18 18: Partition Functions and Ideal Gases 18.1: Translational Partition Functions of Monotonic Gases 18.2: Most Atoms are in the Ground Electronic State 18.3: The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms 18.4: Most Molecules are in the Ground Vibrational State 18.5: Most Molecules are Rotationally Excited at Ordinary Temperatures 18.6: Rotational Partition Functions of Diatomic Gases 18.7: Vibrational Partition Functions of Polyatomic Molecules 18.8: Rotational Partition Functions of Polyatomic Molecules 18.9: Molar Heat Capacities 18.E: Partition Functions and Ideal Gases (Exercises) Ortho and Para Hydrogen The Equipartition Principle • Chapter 19 19: The First Law of Thermodynamics 19.0: Overview of Classical Thermodynamics 19.1: Pressure-Volume Work 19.2: Work and Heat are not State Functions, but Energy is a State Function 19.3: Energy is a State Function 19.4: An Adiabatic Process is a Process in which No Energy as Heat is Transferred 19.5: The Temperature of a Gas Decreases in a Reversible Adiabatic Expansion 19.6: Work and Heat Have a Simple Molecular Interpretation 19.7: Pressure-Volume Work 19.8: Heat Capacity is a Path Function 19.9: Relative Enthalpies Can Be Determined from Heat Capacity Data and Heats of Transition 19.10: Enthalpy Changes for Chemical Equations are Additive 19.11: Heats of Reactions Can Be Calculated from Tabulated Heats of Formation 19.12: The Temperature Dependence of ΔH 19.E: The First Law of Thermodynamics (Exercises) • Chapter 20 20: Entropy and The Second Law of Thermodynamics 20.1: Energy Does not Determine Spontaneity 20.2: Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Probability 20.3: Unlike heat, Entropy Is a State Function 20.4: The Second Law of Thermodynamics 20.5: The Famous Equation of Statistical Thermodynamics 20.6: We Must Always Devise a Reversible Process to Calculate Entropy Changes 20.7: Thermodynamics Provides Insight into the Conversion of Heat into Work 20.8: Entropy Can Be Expressed in Terms of a Partition Function 20.9: The Molecular Formula S = kB in W is Analogous to the Thermodynamic Formula dS = deltaqrev 20.E: Entropy and The Second Law of Thermodynamics (Exercises) • Chapter 21 21: Entropy & the Third Law of Thermodynamics 21.1: Entropy Increases With Increasing Temperature 21.2: Absolute Entropy 21.3: Temperatures at a Phase Transition 21.4: The Third Law of Thermodynamics 21.5: Practical Absolute Entropies Can Be Determined Calorimetrically 21.6: Practical Absolute Entropies of Gases Can Be Calculated from Partition Functions 21.7: Standard Entropies Depend Upon Molecular Mass and Structure 21.8: Spectroscopic Entropies sometimes disgree with Calorimetric Entropies 21.9: Standard Entropies Can Be Used to Calculate Entropy Changes of Chemical Reactions 21.E: Entropy & the Third Law of Thermodynamics (Exercises) • Chapter 22 22: Helmholtz and Gibbs Energies 22.1: Helmholtz Energy 22.2: Gibbs Energy 22.3: The Maxwell Relations 22.4: The Enthalpy of an Ideal Gas 22.5: Thermodynamic Functions have Natural Variables 22.6: The Standard State for a Gas is Ideal Gas 22.7: The Gibbs-Helmholtz Equation 22.8: Fugacity Measures Nonideality of a Gas Homework Problems Chapter 23 23: Phase Equilibria 23.1: A Phase Diagram Summarizes the Solid-Liquid-Gas Behavior of a Substance 23.2: Gibbs Energies and Phase Diagrams 23.3: The Chemical Potentials of a Pure Substance in Two Phases in Equilibrium 23.4: The Clausius-Clapeyron Equation 23.5: Chemical Potential Can be Evaluated From a Partition Function Homework Problems • Chapter 24 24: Solutions I: Liquid-Liquid Solutions 24.1: Partial Molar Quantities in Solutions 24.2: The Gibbs-Duhem Equation 24.3: Chemical Potential of Each Component Has the Same Value in Each Phase in Which the Component Appears 24.4: Ideal Solutions obey Raoult's Law 24.5: Most Solutions are Not Ideal 24.6: Vapor Pressures of Volatile Binary Solutions 24.7: Activities of Nonideal Solutions 24.8: Activities are Calculated with Respect to Standard States 24.9: Gibbs Energy of Mixing of Binary Solutions in Terms of the Activity Coefficient 24.E: Solutions I: Liquid-Liquid Solutions (Exercises) • Chapter 25 25. Solutions II: Solid-Liquid Solutions 25.1: Raoult's and Henry's Laws Define Standard States 25.2: The Activities of Nonvolatile Solutes 25.3: Colligative Properties Depend only on Number Density 25.4: Osmotic Pressure can Determine Molecular Masses 25.5: Electrolytes Solutions are Nonideal at Low Concentrations 25.6: The Debye-Hückel Theory 25.7: Extending Debye-Hückel Theory to Higher Concentrations Homework Problems • Chapter 25 • Chapter 26 26: Chemical Equilibrium 26.1: Equilibrium Results when Gibbs Energy is Minimized 26.2: An Equilibrium Constant Is a Function of Temperature Only 26.3: Standard Gibbs Energies of Formation Can Be Used to Calculate Equilibrium Constants 26.4: A Plot of the Gibbs Energy of a Reaction Mixture Against the Extent of Reaction Is a Minimum at Equilibrium 26.5: Reaction Quotient and Equilibrium Constant Ratio Determines Reaction Direction 26.6: The Sign of ΔG and not ΔG° Determines the Direction of Reaction Spontaneity 26.7: The Van't Hoff Equation 26.8: Equilibrium Constants in Terms of Partition Functions 26.9: Molecular Partition Functions and Related Thermodynamic Data Are Extensively Tabulated 26.10: Real Gases Are Expressed in Terms of Partial Fugacities 26.11: Thermodynamic Equilibrium Constants Are Expressed in Terms of Activities 26.12: Activities are Important for Ionic Species Homework Problems • Chapter 27 27: The Kinetic Theory of Gases 27.1: The Average Translational Kinetic Energy of a Gas 27.2: The Distribution of the Components of Molecular Speeds are Described by a Gaussian Distribution 27.3: The Distribution of Molecular Speeds Is Given by the Maxwell-Boltzmann Distribution 27.4: The Frequency of Collisions 27.5: The Maxwell-Boltzmann Distribution Has Been Verified Experimentally 27.6: The Mean Free Path 27.7: The Rate of a Gas-Phase Chemical Reactions 27.E: The Kinetic Theory of Gases (Exercises) • Chapter 28 28: Chemical Kinetics I - Rate Laws 28.1: The Time Dependence of a Chemical Reaction Is Described by a Rate Law 28.2: Rate Laws Must Be Determined Experimentally 28.3: First-Order Reactions Show an Exponential Decay of Reactant Concentration with Time 28.4: Different Rate Laws Predict Different Kinetics 28.5: Reactions Can Also Be Reversible 28.6: The Rate Constants of a Reversible Reaction Can Be Determined Using Relaxation Techniques 28.7: Rate Constants Are Usually Strongly Temperature Dependent 28.8: Transition-State Theory Can Be Used to Estimate Reaction Rate Constants 28.E: Chemical Kinetics I : Rate Laws (Exercises) • Chapter 29 29: Chemical Kinetics II: Reaction Mechanisms 29.1: A Mechanism is a Sequence of Elementary Reactions 29.2: The Principle of Detailed Balance 29.3: Multiple Mechanisms are Often Indistinguishable 29.4: The Steady-State Approximation 29.5: Rate Laws Do Not Imply Unique Mechanism 29.6: The Lindemann Mechanism 29.7: Some Reaction Mechanisms Involve Chain Reactions 29.8: A Catalyst Affects the Mechanism and Activation Energy 29.9: The Michaelis-Menten Mechanism for Enzyme Catalysis 29.E: Chemical Kinetics II: Reaction Mechanisms (Exercises) • Chapter 30 30: Gas-Phase Reaction Dynamics 30.1: The Rate of Bimolecular Gas-Phase Reaction Can Be Calculated Using Hard-Sphere Collision Theory and an Energy-Dependent Reaction Cross Section 30.2: A Reaction Cross Section Depends Upon the Impact Parameter 30.3: The Rate Constant for a Gas-Phase Chemical Reaction May Depend on the Orientations of the Colliding Molecules 30.4: The Internal Energy of the Reactants Can Affect the Cross Section of a Reaction 30.5: A Reactive Collision Can Be Described in a Center-of-Mass Coordinate System 30.6: Reactive Collisions Can be Studied Using Crossed Molecular Beam Machines 30.7: Reactions Can Produce Vibrationally Excited Molecules 30.8: The Velocity and Angular Distribution of the Products of a Reactive Collision Provide a Molecular Picture of the Chemical Reaction 30.9: Not All Gas-Phase Chemical Reactions Are Rebound Reactions 30.10: The Potential-Energy Surface Can Be Calculated Using Quantum Mechanics 30.E: Gas-Phase Reaction Dynamics (Exercises) • Chapter 33 31: Solids and Surface Chemistry 31.1: The Unit Cell Is the Fundamental Building Block of a Crystal 31.2: The Orientation of a Lattice Plane Is Described by its Miller Indices 31.3: The Spacing Between Lattice Planes Can Be Determined from X-Ray Diffraction Measurements 31.4: The Total Scattering Intensity Is Related to the Periodic Structure of the Electron Density in the Crystal 31.5: The Structure Factor and the Electron Density Are Related by a Fourier Transform 31.6: A Gas Molecule can Physisorb or Chemisorb to a Solid Surface 31.7: Isotherms are Plots of Surface Coverage as a Function of Gas Pressure at Constant Temperature 31.8: The Langmuir Isotherm Can Be Used to Derive Rate Laws for Surface-Catalyzed Gas-Phase Reactions 31.9: The Structure of a Surface is Different from that of a Bulk Solid 31.10: The Reaction Between H2(g) and N 2(g) to Make NH3 (g) Can Be Surface Catalyzed 31.E: Homework Problems • Appendices MathChapters A: Complex Numbers B: Probability and Statistics C: Vectors D: Spherical Coordinates E: Determinants F: Matrices G: Numerical Methods H: Partial Differentiation I: Series and Limits J: The Binomial Distribution and Stirling's Appromixation Topic hierarchy