MOLECULAR VIBRATIONAL-ROTATIONAL SPECTRA

Transcription

1 Studies in Physical and Theoretical Chemistry 17 MOLECULAR VIBRATIONAL-ROTATIONAL SPECTRA Theory and Applications of High Resolution Infrared, Microwave and Raman Spectroscopy of Polyatomic Molecules by D. PAPOUSEK J. Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, Prague, Czechoslovakia M. R. ALIEV Institute of Spectroscopy, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxford - New York 1982

2 CONTENTS INTRODUCTION 13 SUMMARY OF CONVENTIONS 15 I. SEMIRIGID MOLECULES - BASIC THEORY Principles and Applications of Molecular Vibrational-Rotational Spectroscopy Spectral regions of rotational and vibrational-rotational transitions Measurement of rotational and vibrational-rotational molecular spectra Physical and chemical applications of vibrational-rotational spectroscopy. 2. Basic Classical and Quantum Mechanical Treatment of Molecular Vibration and Rotation Two basic quantum mechanical approaches to molecular systems Basic approach to the vibrational-rotational problem of a polyatomic molecule Eckart conditions, kinetic energy of translation, rotation and vibration Molecule-fixed axis system. 3. Normal Coordinates of Molecular Vibrations, the GF Matrix Problem Expansion of the potential energy function in terms of the vibrational coordinates Internal coordinates of vibration The GF matrix problem. 4. Vibrational-Rotational Hamiltonian of a Semirigid Molecule Classical kinetic energy of a vibrating and rotating molecule Hamiltonian form of kinetic energy Quantum mechanical Hamiltonian of a nonlinear molecule The classical Hamiltonian of a linear molecule The quantum mechanical Hamiltonian of a linear molecule Harmonic oscillator, rigid rotor approximation Comments on the relationships between molecular parameters and experimental data on the transitions between vibrational-rotational energy levels. 5. Harmonic Oscillator : The Schrödinger equation of the harmonic oscillator The linear harmonic oscillator The two-dimensionally isotropic harmonic oscillator The threedimensionally isotropic harmonic oscillator Matrix elements of the linear vibrational momenta and coordinates. 6. Rigid Rotor Commutators of the angular momentum operators Eigenvalues of J 2, \ z and \ Matrix elements of the angular momentum operators Classification of molecules according to the relationships among the moments of inertia Rotational energy levels of symmetric top molecules Rotational energy levels of linear and spherical top molecules Symmetric top rotational wavefunctions Rotational energy levels of asymmetric top molecules.

3 II. SYMMETRY CLASSIFICATION OF MOLECULAR ENERGY LEVELS, SELECTION RULES Molecular Symmetry Permutation-inversion groups Point groups Relation between point groups and permutation-inversion groups Symmetry classification of energy levels. 8. Symmetry of Normal Vibrations Representation of a symmetry group in the basis of vibrational coordinates Symmetry coordinates Factorization of the secular equation of molecular vibrations. 9. Symmetry Classification of the Eigenstates of Symmetric Top Molecules Point group and permutation-inversion group of a rigid symmetric top molecule Symmetry classification of the vibronic states Symmetry classification of the rovibronic states Optical selection rules Selection rules for transitions to degenerate vibronic levels. 10. Symmetry Classification of the Eigenstates of Linear Molecules Extended permutation-inversion groups of linear molecules Classification of the rovibronic states of rigid linear molecules Selection rules for rovibronic transitions. 11. Symmetry Classification of the Eigenstates of Spherical Top Molecules Symmetry group for methane Classification of the rotational levels in the three-dimensional rotation group and in the T d group Classification of the vibronic states in the T d group Coupling of the vibrational and rotational wavefunctions Selection rules for rovibronic transitions. 12. Symmetry Classification of the Eigenstates of Asymmetric Top Molecules The K-group of an asymmetric top Wang transformation Symmetry species of the J K K energy levels Selection rules for rovibronic transitions. 13. Selection Rules and Line Intensities in Vibrational-Rotational Raman Spectra The scattering tensor Matrix elements of the polarizability tensor Selection rules Line strengths. 14. Statistical Weights of Molecular Energy Levels Fermi-Dirac and Bose-Einstein statististics Spin statistical weights of molecular rotational levels. III. VIBRATIONAL-ROTATIONAL INTERACTIONS IN SEMIRIGID MOLECULES Expansion of the Vibrational-Rotational Hamiltonian. The Method of Contact Transformations Expansion of the tensor ц Expansion of the vibrational-rotational Hamiltonian Orders of magnitude of various vibrational-rotational terms and the problem of convergence General symmetry requirements for the vibrational-rotational Hamiltonian Contact transformations of the vibrational-rotational Hamiltonian Rotational commutators Determination of transformation functions.

4 9 16. General Form of the Vibrational-Rotational Interaction Terms Quartic interactions Quintic Coriolis interactions Sextic centrifugal distortion Reduction of the rotational Hamiltonian. 17. Vibrational-Rotational Energy of Asymmetric Top Molecules Vibrational energy Anharmonic resonances Rotational energy of asymmetric top molecules: Watson's reduced Hamiltonian X-type doubling induced by centrifugal distortion Sum rules Planarity relations Vibrational dependence of the rotational constants Coriolis interactions. 18. Vibrational-Rotational Energy of Symmetric Top Molecules Vibrational energy of symmetric top molecules. Vibrational /-type doubling Rotational energy of symmetric top molecules. K-type doubling Coriolis interactions Vibrational dependence of the rotational constants Rotational /-type doubling Transition frequencies. 19. Vibrational-Rotational Energy of Linear Molecules Vibrational energy. Fermi resonance and vibrational /-type doubling Rotational energy of linear molecules. Rotational /-type doubling. 20. Vibrational-Rotational Energy Levels of Spherical Top Molecules Irreducible tensor form of the vibrational-rotational operators Vibrational energy levels of spherical top molecules Rotational energy levels of spherical top molecules in the ground vibrational state Rotational energy levels in the first excited state of triply degenerate vibrations Rotational structure of the first excited level of the doubly degenerate vibration. 21. Forbidden Transitions Expansion of the dipole moment General symmetry requirements for the matrix elements of the dipole moment Forbidden rotational spectra in nondegenerate vibrational states Forbidden rotational transitions in degenerate vibrational states Totally symmetric vibrations of nonpolar molecules Multi-quantum transitions and the effect of resonances. 22. Coriolis Interaction and Centrifugal Distortion Constants ^-matrices in the GF matrix formulation sum rules The GF matrix formalism for the calculation of the quartic centrifugal distortion constants Isotopic relations between the т constants Relations between the т and constants Upper and lower bounds of the quartic centrifugal distortion constants. 23. Anharmonic Force Constants of Polyatomic Molecules Nonlinear coordinate transformations Transformation of the potential energy function into normal coordinates Calculation of the L coefficients. 24. Molecular Structure Equilibrium (r e ) structure Substitution (r s ) structure Mass-dependence (r~) structure Inertia defect.

5 10 IV. NONRIGID MOLECULES Molecular Inversion Nonrigid molecules general remarks Vibrational-inversional-rotational Hamiltonian of ammonia Expansion of the vibrational-inversional-rotational Hamiltonian Symmetry classification of the states and selection rules for ammonia Potential function of ammonia and the parameterization of the vibrational-inversionalrotational transition frequencies Molecular inversion in other molecules, quasilinear molecules. 26. Internal Rotation Vibrational-torsional-rotational Hamiltonian for a molecule with two coaxial symmetric tops Zeroth-order vibrational-torsional-rotational wavefunctions and energies Double-valued representations of the symmetry group of ethane-like molecules The GF matrix problem in ethane-like molecules. APPENDICES 273 Appendix A Euler Angles 273 Appendix В Sum Rules for the Vibrational-Rotational Interaction Parameters, Commutation Relations for the Angular Momentum Operators 274 B. 1. Permutation symbols. B.2. Coordinate transformation and the orthogonality relations for nonlinear molecules. B.3. Sum rules and commutation relations for nonlinear molecules. B.4. Coordinate transformation and orthogonality relations for linear molecules. B.5. Sum rules and commutation relations for linear molecules. Appendix С Adiabatic Approximations in the Treatment of Electronic, Vibrational and Rotational States of Polyatomic Molecules 280 C.l. Schrödinger equation for molecular systems. Appendix D Electronic Effects on the Moments of Inertia 283 D.I. Effective rotational Hamiltonian in the perturbed electronic state. electronic effects to the molecular g factor. D.2. Relation of Appendix E Matrix Elements of the Vibrational and Rotational Operators 286 E.l. One-dimensional harmonic oscillator. E2. Two-dimensionally isotropic harmonic oscillator. E.3. Matrix elements of the rotational operators \ I. Appendix F Representations of Finite Groups and Rotation Group 289 F.l. Representations of finite groups. F.2. Rotation groups. F.3. Representations of the two-dimensional pure rotation group. F.4. Single-valued continuous representations of the three-dimensional rotation group. F.5. Direct product of representations. F.6. Clebsch-Gordan coefficients. Appendix G Coupling of Angular Momenta and Irreducible Tensors 296 G.l. Coupling of two angular momenta. The Clebsch-Gordan coefficients and the Wigner Ъ-j symbols. G.2. 6-j and 9-j symbols. G.3. Irreducible tensor operators. G.4. Reversed angular momentum. G.5. The Wigner-Eckart theorem. G.6. Matrix elements of

6 11 irreducible tensor operators between rotational states of molecules. G.7. 3-j symbols adapted to cubic symmetry. Appendix H Tables of Characters of the Irreducible Representations of Some Point Groups 307 Appendix I Symmetry Relations Between Molecular Parameters 309 I.l. General symmetry principles. (-constants 1.2. Symmetry relations between the nonvanishing Appendix J Line Strengths for Infrared Transitions 312 SUBJECT INDEX 315