QUANTUM CHEMISTRY

By PATRICIA HARRIS

CONTENTS CLASSICAL WAVES AND THE TIME-INDEPENDENT SCHR¨ODINGER WAVE EQUATION INTRODUCTION WAVES THE CLASSICAL WAVE EQUATION STANDING WAVES IN A CLAMPED STRING LIGHT AS AN ELECTROMAGNETIC WAVE THE PHOTOELECTRIC EFFECT THE WAVE NATURE OF MATTER A DIFFRACTION EXPERIMENT WITH ELECTRONS SCHR¨ODINGER’S TIME-INDEPENDENT WAVE EQUATION CONDITIONS ON ? SOME INSIGHT INTO THE SCHR¨ODINGER EQUATION SUMMARY PROBLEMS MULTIPLE CHOICE QUESTIONS REFERENCE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS THE PARTICLE IN A ONE-DIMENSIONAL “BOX” DETAILED EXAMINATION OF PARTICLE-IN-A-BOX SOLUTIONS THE PARTICLE IN A ONE-DIMENSIONAL “BOX” WITH ONE FINITE WALL THE PARTICLE IN AN IN?NITE “BOX” WITH A FINITE CENTRAL BARRIER THE FREE PARTICLE IN ONE DIMENSION THE PARTICLE IN A RING OF CONSTANT POTENTIAL THE PARTICLE IN A THREE-DIMENSIONAL BOX: SEPARATION OF VARIABLES THE SCATTERING OF PARTICLES IN ONE DIMENSION REFERENCES THE ONE-DIMENSIONAL HARMONIC OSCILLATOR SOME CHARACTERISTICS OF THE CLASSICAL ONE-DIMENSIONAL HARMONIC OSCILLATOR THE QUANTUM-MECHANICAL HARMONIC OSCILLATOR SOLUTION OF THE HARMONIC OSCILLATOR SCHR¨ODINGER EQUATION QUANTUM-MECHANICAL AVERAGE VALUE OF THE POTENTIAL ENERGY VIBRATIONS OF DIATOMIC MOLECULES THE HYDROGENLIKE ION, ANGULAR MOMENTUM, AND THE RIGID ROTOR THE SCHR¨ODINGER EQUATION AND THE NATURE OF ITS SOLUTIONS SEPARATION OF VARIABLES SOLUTION OF THE R, , AND EQUATIONS ATOMIC UNITS ANGULAR MOMENTUM AND SPHERICAL HARMONICS ANGULAR MOMENTUM AND MAGNETIC MOMENT ANGULAR MOMENTUM IN MOLECULAR ROTATION—THE RIGID ROTOR MANY-ELECTRON ATOMS THE INDEPENDENT ELECTRON APPROXIMATION SIMPLE PRODUCTS AND ELECTRON EXCHANGE SYMMETRY ELECTRON SPIN AND THE EXCLUSION PRINCIPLE SLATER DETERMINANTS AND THE PAULI PRINCIPLE SINGLET AND TRIPLET STATES FOR THE 1S2S CON?GURATION OF HELIUM THE SELF-CONSISTENT FIELD, SLATER-TYPE ORBITALS, AND THE AUFBAU PRINCIPLE ELECTRON ANGULAR MOMENTUM IN ATOMS OVERVIEW POSTULATES AND THEOREMS OF QUANTUM MECHANICS THE WAVEFUNCTION POSTULATE THE POSTULATE FOR CONSTRUCTING OPERATORS THE TIME-DEPENDENT SCHR¨ODINGER EQUATION POSTULATE THE POSTULATE RELATING MEASURED VALUES TO EIGENVALUES THE POSTULATE FOR AVERAGE VALUES HERMITIAN OPERATORS PROOF THAT EIGENVALUES OF HERMITIAN OPERATORS ARE REAL PROOF THAT NONDEGENERATE EIGENFUNCTIONS OF A HERMITIAN OPERATOR FORM AN ORTHOGONAL SET DEMONSTRATION THAT ALL EIGENFUNCTIONS OF A HERMITIAN OPERATOR MAY BE EXPRESSED AS AN ORTHONORMAL SET PROOF THAT COMMUTING OPERATORS HAVE SIMULTANEOUS EIGENFUNCTIONS COMPLETENESS OF EIGENFUNCTIONS OF A HERMITIAN OPERATOR THE VARIATION PRINCIPLE THE PAULI EXCLUSION PRINCIPLE MEASUREMENT, COMMUTATORS, AND UNCERTAINTY TIME-DEPENDENT STATES THE VARIATION METHOD THE SPIRIT OF THE METHOD NONLINEAR VARIATION: THE HYDROGEN ATOM NONLINEAR VARIATION: THE HELIUM ATOM LINEAR VARIATION: THE POLARIZABILITY OF THE HYDROGEN ATOM LINEAR COMBINATION OF ATOMIC ORBITALS: THE H+ 2 MOLECULE–ION MOLECULAR ORBITALS OF HOMONUCLEAR DIATOMIC MOLECULES BASIS SET CHOICE AND THE VARIATIONAL WAVEFUNCTION BEYOND THE ORBITAL APPROXIMATION THE SIMPLE H¨UCKEL METHOD AND APPLICATIONS THE IMPORTANCE OF SYMMETRY THE ASSUMPTION OF ?–? SEPARABILITY THE INDEPENDENT ?-ELECTRON ASSUMPTION SETTING UP THE H¨UCKEL DETERMINANT SOLVING THE HMO DETERMINANTAL EQUATION FOR ORBITAL ENERGIES SOLVING FOR THE MOLECULAR ORBITALS THE CYCLOPROPENYL SYSTEM: HANDLING DEGENERACIES CHARGE DISTRIBUTIONS FROM HMOS SOME SIMPLIFYING GENERALIZATIONS HMO CALCULATIONS ON SOME SIMPLE MOLECULES SUMMARY: THE SIMPLE HMO METHOD FOR HYDROCARBONS RELATION BETWEEN BOND ORDER AND BOND LENGTH ?-ELECTRON DENSITIES AND ELECTRON SPIN RESONANCE HYPER?NE SPLITTING CONSTANTS ORBITAL ENERGIES AND OXIDATION-REDUCTION POTENTIALS ORBITAL ENERGIES AND IONIZATION ENERGIES ?-ELECTRON ENERGY AND AROMATICITY EXTENSION TO HETEROATOMIC MOLECULES SELF-CONSISTENT VARIATIONS OF ? AND ? HMO REACTION INDICES CONCLUSIONS MATRIX FORMULATION OF THE LINEAR VARIATION METHOD MATRICES AND VECTORS SOLVING THE MATRIX EQUATION 10 THE EXTENDED H¨UCKEL METHOD THE EXTENDED H¨UCKEL METHOD MULLIKEN POPULATIONS EXTENDED H¨UCKEL ENERGIES AND MULLIKEN POPULATIONS EXTENDED H¨UCKEL ENERGIES AND EXPERIMENTAL ENERGIES 11 THE SCF-LCAO-MO METHOD AND EXTENSIONS AB INITIO CALCULATIONS THE MOLECULAR HAMILTONIAN THE FORM OF THE WAVEFUNCTION THE NATURE OF THE BASIS SET THE LCAO-MO-SCF EQUATION INTERPRETATION OF THE LCAO-MO-SCF EIGENVALUES THE SCF TOTAL ELECTRONIC ENERGY BASIS SETS THE HARTREE–FOCK LIMIT CORRELATION ENERGY KOOPMANS’ THEOREM CON?GURATION INTERACTION SIZE CONSISTENCY AND THE MØLLER–PLESSET AND COUPLED CLUSTER TREATMENTS OF CORRELATION MULTIDETERMINANT METHODS DENSITY FUNCTIONAL THEORY METHODS EXAMPLES OF AB INITIO CALCULATIONS APPROXIMATE SCF-MO METHODS 12 TIME-INDEPENDENT RAYLEIGH–SCHR¨ODINGER PERTURBATION THEORY AN INTRODUCTORY EXAMPLE FORMAL DEVELOPMENT OF THE THEORY FOR NONDEGENERATE STATES A UNIFORM ELECTROSTATIC PERTURBATION OF AN ELECTRON IN A “WIRE” THE GROUND-STATE ENERGY TO FIRST-ORDER OF HELIUMLIKE SYSTEMS PERTURBATION AT AN ATOM IN THE SIMPLE H¨UCKEL MO METHOD PERTURBATION THEORY FOR A DEGENERATE STATE POLARIZABILITY OF THE HYDROGEN ATOM IN THE N = 2 STATES DEGENERATE-LEVEL PERTURBATION THEORY BY INSPECTION INTERACTION BETWEEN TWO ORBITALS: AN IMPORTANT CHEMICAL MODEL CONNECTION BETWEEN TIME-INDEPENDENT PERTURBATION THEORY AND SPECTROSCOPIC SELECTION RULES 13 GROUP THEORY AN ELEMENTARY EXAMPLE SYMMETRY POINT GROUPS THE CONCEPT OF CLASS SYMMETRY ELEMENTS AND THEIR NOTATION IDENTIFYING THE POINT GROUP OF A MOLECULE REPRESENTATIONS FOR GROUPS GENERATING REPRESENTATIONS FROM BASIS FUNCTIONS LABELS FOR REPRESENTATIONS SOME CONNECTIONS BETWEEN THE REPRESENTATION TABLE AND MOLECULAR ORBITALS REPRESENTATIONS FOR CYCLIC AND RELATED GROUPS ORTHOGONALITY IN IRREDUCIBLE INEQUIVALENT REPRESENTATIONS CHARACTERS AND CHARACTER TABLES USING CHARACTERS TO RESOLVE REDUCIBLE REPRESENTATIONS IDENTIFYING MOLECULAR ORBITAL SYMMETRIES DETERMINING IN WHICH MOLECULAR ORBITAL AN ATOMIC ORBITAL WILL APPEAR GENERATING SYMMETRY ORBITALS HYBRID ORBITALS AND LOCALIZED ORBITALS SYMMETRY AND INTEGRATION 14 QUALITATIVE MOLECULAR ORBITAL THEORY THE NEED FOR A QUALITATIVE THEORY HIERARCHY IN MOLECULAR STRUCTURE AND IN MOLECULAR ORBITALS H+ 2 REVISITED H2: COMPARISONS WITH H+ RULES FOR QUALITATIVE MOLECULAR ORBITAL THEORY APPLICATION OF QMOT RULES TO HOMONUCLEAR DIATOMIC MOLECULES SHAPES OF POLYATOMIC MOLECULES: WALSH DIAGRAMS FRONTIER ORBITALS QUALITATIVE MOLECULAR ORBITAL THEORY OF REACTIONS 15 MOLECULAR ORBITAL THEORY OF PERIODIC SYSTEMS THE PARTICLE IN A RING BENZENE GENERAL FORM OF ONE-ELECTRON ORBITALS IN PERIODIC POTENTIALS— BLOCH’S THEOREM A RETROSPECTIVE PAUSE AN EXAMPLE: POLYACETYLENE WITH UNIFORM BOND LENGTHS ELECTRICAL CONDUCTIVITY POLYACETYLENE WITH ALTERNATING BOND LENGTHS—PEIERLS’ DISTORTION ELECTRONIC STRUCTURE OF ALL-TRANS POLYACETYLENE COMPARISON OF EHMO AND SCF RESULTS ON POLYACETYLENE EFFECTS OF CHEMICAL SUBSTITUTION ON THE ? BANDS POLY-PARAPHENYLENE—A RING POLYMER ENERGY CALCULATIONS TWO-DIMENSIONAL PERIODICITY AND VECTORS IN RECIPROCAL SPACE PERIODICITY IN THREE DIMENSIONS—GRAPHITE CHAPTER WAVES 1-2 A TRAVELING WAVES A VERY SIMPLE EXAMPLE OF A TRAVELING WAVE IS PROVIDED BY CRACKING A WHIP A PULSE OF ENERGY IS IMPARTED TO THE WHIPCORD BY A SINGLE OSCILLATION OF THE HANDLE THIS RESULTS IN A WAVE WHICH TRAVELS DOWN THE CORD, TRANSFERRING THE ENERGY TO THE POPPER AT THE END OF THE WHIP IN FIG 1-1, AN IDEALIZATION OF THE PROCESS IS SKETCHED THE SHAPE OF THE DISTURBANCE IN THE WHIP IS CALLED THE WAVE PRO?LE AND IS USUALLY SYMBOLIZED ?(X) THE WAVE PRO?LE FOR THE TRAVELING WAVE IN FIG 1-1 SHOWS WHERE THE ENERGY IS LOCATED AT A GIVEN INSTANT IT ALSO CONTAINS THE INFORMATION NEEDED TO TELL HOW MUCH ENERGY IS BEING TRANSMITTED, BECAUSE THE HEIGHT AND SHAPE OF THE WAVE RE?ECT THE VIGOR WITH WHICH THE HANDLE WAS OSCILLATED