P. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION

Transcription

1 P. W. Atkins and R. S. Friedman Molecular Quantum Mechanics THIRD EDITION Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997

2 Introduction and orientation 1 Black-body radiation 1 Heat capacities 2 The photoelectric and Compton effects 3 Atomic spectra 4 The duality of matter 6 Problems 7 Further reading 8 1 The foundations of quantum mechanics 9 Operators in quantum mechanics Eigenfunctions and eigenvalues Representations Commutation and non-commutation The construction of Operators Linear Operators Integrals over Operators 14 The postulates of quantum mechanics States and wavefunctions The fundamental prescription The outcome of measurements The interpretation of the wavefunction The equation for the wavefunction The Separation of the Schrödinger equation 19 Hermitian Operators Definitions Dirac bracket notation The properties of hermitian Operators 23 The specification of states: complementarity 24 The uncertainty principle Formal derivation of the principle Consequences of the uncertainty principle The uncertainty in energy and time 28 Time-evolution and conservation laws 29 Matrices in quantum mechanics Matrix elements The diagonalization of the hamiltonian 32

3 viii Contents The plausibility of the Schrödinger equation The propagation oflight The propagation ofparticles The transition to quantum mechanics 37 Problems 39 Further reading 41 2 Linear motion and the harmonic osciuator 42 The characteristics of acceptable wavefunctions 42 Some general remarks on the Schrödinger equation The curvature of the wavefunction Qualitative Solutions The emergence of quantization Penetration into nonclassical regions 45 Translational motion Energy and momentum The significance of the coefncients The flux density Wavepackets 49 Penetration into and through barriers An infinitely thick potential wall A barrier of finite width The Eckart potential barrier 54 Particle in a box The Solutions Features of the Solutions The two-dimensional Square well Degeneracy 58 The harmonic osciuator The Solutions Properties of the Solutions The classical limit 64 Translation revisited: the scattering matrix 65 Problems 67 Further reading 70 3 Rotational motion and the hydrogen atom 71 Particle on a ring 71

4 IX 3.1 The hamiltonian and the Schrödinger equation The angular momentum The shapes of the wavefunctions The classical limit 76 Particle on a sphere The Schrödinger equation and its Solution The angular momentum of the particle Properties of the Solutions The rigid rotor 81 Motion in a Coulombic field The Schrödinger equation for hydrogenic atoms The Separation of the relative coordinates The radial Schrödinger equation Probabilities and the radial distribution function Atomic orbitals The degeneracy of hydrogenic atoms 94 Problems 95 Further reading 97 4 Angular momentum 98 The angular momentum Operators The Operators and their commutation relations Angular momentum observables The shift Operators 101 The definition of the states The effect of the shift Operators The eigenvalues of the angular momentum The matrix elements of the angular momentum The angular momentum eigenfunctions Spin 109 The angular momenta of composite Systems The specification of coupled states The permitted values of the total angular momentum The vector model of coupled angular momenta The relation between schemes The coupling of several angular momenta 119 Problems 119 Further reading 121

5 5 Group theory 122 The symmetries of objects, Symmetry Operations and elements The Classification of molecules 124 The calculus of symmetry The definition of a group Group multiplication tables Matrix representations The properties of matrix representations The characters of representations Characters and classes Irreducible representations The great and little orthogonality theorems 139 Reduced representations The reduction of representations Symmetry-adapted bases 145 The symmetry properties of functions The transformation ofp-orbitals The decomposition of direct-product bases Direct-product groups Vanishing integrals Symmetry and degeneracy 156 The füll rotation group The generators of rotations The representation of the füll rotation group Coupled angular momenta 161 Applications 161 Problems 162 Further reading Techniques of approximation 164 Time-independent perturbation theory Perturbation of a two-level system Many-level Systems The first-order correction to the energy The first-order correction to the wavefunction The second-order correction to the energy Comments on the perturbation expressions The closure approximation Perturbation theory for degenerate states 175

6 xi Variation theory The Rayleigh ratio The Rayleigh-Ritz method 180 The Hellmann-Feynman theorem 182 Time-dependent perturbation theory The time-dependent behaviour of a two-level system The Rabi formula Many-level Systems: the Variation of constants The effect of a slowly switched constant perturbation The effect of an oscillating perturbation Transition rates to continuum states The Einstein transition probabilities Lifetime and energy uncertainty 198 Problems 199 Further reading Atomic spectra and atomic structure 202 The spectrum of atomic hydrogen The energies of the transitions Selection rules Orbital and spin magnetic moments Spin-orbit coupling The fine-structure of spectra Term symbols and spectral details The detailed spectrum of hydrogen 211 The structure of helium The helium atom Excited states of helium The spectrum of helium The Pauli principle 218 Many-electron atoms Penetration and shielding Periodicity Slater atomic orbitals Self-consistent fields Term symbols and transitions of many-electron atoms Hund's rules and the relative energies of terms Alternative coupling schemes 232 Atoms in external fields The normal Zeeman effect 7.20 The anomalous Zeeman effect

7 J 7.21 The Stark effect 236 Problems 237 Further reading An introduction to molecular structure 240 The Born-Oppenheimer approximation The formulation of the approximation An application: the hydrogen molecule-ion 242 Molecular orbital theory Linear combinations of atomic orbitals The hydrogen molecule Configuration interaction Diatomic molecules Heteronuclear diatomic molecules 254 Molecular orbital theory of polyatomic molecules Symmetry-adapted linear combinations Conjugated rc-systems Ligandfield theory Further aspects of ligand field theory 265 The band theory of solids The tight-binding approximation The Kronig-Penney model Brillouin zones 271 Problems 273 Further reading The calculation of electronic structure 276 The Hartree-Fock self-consistent field method The formulation of the approach The Hartree-Fock approach Restricted and unrestricted Hartree-Fock calculations The Roothaan equations The selection of basis sets Calculational accuracy and the basis set 290 Electron correlation Configuration State functions Configuration interaction CI calculations Multiconfiguration and multireference methods 297

8 XIII 9.11 Moller-Plesset many-body perturbation theory 298 Density functional theory 301 Gradient methods and molecular properties 304 Semiempirical methods Conjugated rc-electron Systems Neglect of differential overlap 311 Software packages for electronic structure calculations 314 Problems 316 Further reading Molecular rotations and vibrations 320 Spectroscopic transitions Absorption and emission Raman processes 322 Molecular rotation Rotational energy levels Centrifugal distortion Pure rotational selection rules Rotational Raman selection rules Nuclear statistics 331 Molecular Vibration The vibrational energy levels of diatomic molecules Anharmonic oscillation Vibrational selection rules Vibration-rotation spectra of diatomic molecules Vibrational Raman transitions of diatomic molecules 341 The vibrations of polyatomic molecules Normal modes Vibrational selection rules for polyatomic molecules Group theory and molecular vibrations The effects of anharmonicity Coriolis forces Inversion doubling 354 Problems 355 Further reading Molecular electronic transitions 358 The states of diatomic molecules 358

9 11.1 The Hund coupling cases Decoupling and /t-doubling Selection rules 361 Vibronic transitions The Franck-Condon principle The structure of vibronic transitions 364 The electronic spectra of polyatomic molecules Symmetry considerations Chromophores Vibronically allowed transitions Singlet-triplet transitions 370 The fates of excited species Non-radiative decay Radiative decay The conservation of orbital symmetry Electrocyclic reactions Cycloaddition reactions Photochemically induced electrocyclic reactions Photochemically induced cycloaddition reactions 379 Problems 380 Further reading The electric properties of molecules 382 The response to electric fields Molecular response parameters The static electric polarizability Polarizability and molecular properties Polarizabilities and molecular spectroscopy Polarizabilities and dispersion forces Retardation effects 392 Bulk electrical properties The relative permittivity and the electric susceptibility Polar molecules Refractive index 397 Optical activity Circular birefringence and optical rotation Magnetically induced polarization Rotational strength 405 Problems 409 Further reading 410

10 xv 13 The magnetic properties of molecules 411 The description of magnetic fields The magnetic susceptibility Paramagnetism Vector functions Derivatives of vector functions The vector potential 416 Magnetic perturbations The perturbation hamiltonian The magnetic susceptibility The current density The diamagnetic current density The paramagnetic current density 426 Magnetic resonance parameters Shielding constants ' The diamagnetic contribution to shielding The paramagnetic contribution to shielding The g-value Spin-spin coupling Hyperfine interactions Nuclear spin-spin coupling 442 Problems 445 Further reading Scattering theory 448 The formulation of scattering events The scattering cross-section Stationary scattering states The integral scattering equation The Born approximation 455 Partial-wave stationary scattering states Partial waves The partial-wave equation Free-particle radial wavefunctions The scattering phase shift Phase shifts and scattering cross-sections Scattering by a spherical Square well Backgrounds and resonances The Breit-Wigner formula The scattering matrix element 470

11 xvi Contents Multichannel scattering Channels for scattering Multichannel stationary scattering states Inelastic collisions Reactive scattering The S matrix and multichannel resonances 479 Problems 480 Further reading 483 Further Information 484 Classical mechanics Action The canonical momentum The virial theorem Reduced mass 488 Solutions of the Schrödinger equation The motion of wavepackets The harmonic oscillator: Solution by factorization The harmonic oscillator: the Standard Solution The radial wave equation The angular wavefunction Molecular integrals The Hartree-Fock equations Green's functions The unitarity of the S matrix 504 Group theory and angular momentum The orthogonality of basis functions Vector coupling coefficients 506 Spectroscopic properties Electric dipole transitions Oscillator strength Sum rules Normal modes: an example 512 The electromagnetic field The Maxwell equations The dipolar vector potential 516 Mathematical relations Vector properties Matrices 519

12 xvii Appendix Character tables and direct products Vector coupling coefficients 527 Answers to selected problems 528 Index 533