Chapter 6 Vibrational spectroscopy,College of Physics, Qufu Normal University Zhang shu-dong 2011-4-15,6.1 Diatomic molecules,Harmonic oscillator,Energy levels:,Term values:,Force constant k:,Potantial energy:,k is not affected by isotopic substitution!,Hookes law,Units of k: N m-1, mdyn A-1, aJ A-2,Q: What is bond order?,6.1.1 Infrared spectra,Transition moment:,For homonuclear diatomic molecules:,For heteronuclear diatomic molecules:,-Selection rule,Band, fundamental band, first overtone band, hot band, ,The v=1-0 band Infrared spectrum of DCl. Band and lines Q: What does the intensity depend on?,6.1.2 Raman spectra,Transition moment:,For homonuclear and heteronuclear diatomic molecules:,-Selection rule,Boltzmann distribution law:,Q: The intensity of anti-Stokes line is usually weaker than that of Stokes line, why?,6.1.3 Anharmonicity,Morse potential,De: dissociation energy b: potential parameter, curvature,Term values:,Birge-Sponer plot:,Hint: to solve the area under the line,Worked example 6.1,6.1.4 vibration-rotation spectroscopy,Term values:,Infrared spectra,Selection rules:,-branch,The rotation transitions may be observed only in the gas phase at low pressure and usually in an absorption process.,Infrared spectra,Transitions:,Infrared spectra,Zero gap:,Infrared spectra,Combination difference:,The lower state combination differences,The upper state combination differences,Exercises 6.1,hints,Raman spectra,Selection rules:,-branch,Raman spectra,Transitions:,Raman spectra,Combination difference:,The lower state combination differences,The upper state combination differences,6.2 Polyatomic molecules,Some kinds of vibrational patterns,Common symbols for vibarion,6.2 Polyatomic molecules,Normal modes of N-atom molecules,Linear molecules: 3N-5 Non-linear molecules: 3N-6,Q: How to determine and to classify the normal modes symmetry, infrared activity?,6.2.1 Group vibrations and skeletal vibrations,Term values for ith normal modes,-harmonic approximation,For non-degenerated modes:,For degenerated modes:,Total values:,6.2.1 Group vibrations and skeletal vibrations,Fundamental, overtone, and combination tone,Selection rule:,6.2.1 Group vibrations and skeletal vibrations,Group vibrations: one group in a molecule which movement is more or less localized in a part of the molecule has a characteristic frequency which is almost independent of the rest of the molecule to which it is attached.,6.2.1 Group vibrations and skeletal vibrations,Skeletal vibrations: Many normal modes involve strong coupling between stretching or bending motions of atoms in a straight chain, a branch train or a ring. Such vibrations are called skeletal vibrations and tend to be specific to a particular molecule. The frequency is mostly less than 1300cm-1, and is called fingerprint frequency.,Some normal vibrations of benzene,6.2.2 Normal modes of vibration,1、Classical mechanical description,xi,yi,zi: Cartesian Coordinates,(xi-xie),(yi-yie),(zi-zie): Cartesian displacement Coordinates,(mi)1/2(xi-xie): Mass-weighted Cartesian displacement Coordinates,6.2.2 Normal modes of vibration,Lagrangian:,Lagranges equation,Lagranges equation,Assuming solution:,Matrix form:,Ai has non-trivial solution when :,6.2.2 Normal modes of vibration,For each eigenvalue li there is a normal mode coordinate Qi associated,or,6.2.2 Normal modes of vibration,2、Quantum mechanical description,Generalized momenta Pi:,Hamiltonian operator:,6.2.2 Normal modes of vibration,3、Internal coordiantes and symmetry coordinates,Internal coordinates:,Bond lengths and angles,Internal displacement coordinates:,Corresponding bond-stretching and bond-bending motions,rDq : for all internal coordinates have the same dimensions,6.2.2 Normal modes of vibration,symmetry coordinates,Projection operator:,A1,B2,A1,6.2.2 Normal modes of vibration,F and G matrix,-F matrix,6.2.2 Normal modes of vibration,F and G matrix,-G matrix,F G matrix solution:,6.2.2 Normal modes of vibration,4、How to determine the symmetries of normal modes,Each normal mode of vibration will form a basis for an irreducible representation of the point group of the molecule.,Streamline procedure for the normal modes of vibration:,1 Determine the number of unmoved atoms during each symmetry operation; 2 Gtotal=Gxyz the number of unmoved atoms; 3 Gvib=Gtotal-Gtrans- Grot.,z,y,3657cm-1,a1,1595cm-1,a1,3756cm-1,b2,6.2.2 Normal modes of vibration,5、How to construct the symmetry coordinates,For a normal mode of vibration, each atom moves to a different direction. How to determine such patterns ?- symmetry coordinates,Method: 1. Using projection operator 2. To separate stretching and bending vibrations,symmetry coordinates,Projection operator:,A1,B2,A1,6.2.3 vibrational selection rules,