Vibrational, generally modes

Mention should be made here of recent attempts by Piepho, Schatz and Krausz (46) to give a general interpretation of intervalence bandshapes in terms of a Hamiltonian equivalent to that of eq 6. They use vibronic eigenfunctions (following the method of solution of Merrifield (47)) rather than adiabatic Born-Oppenheimer (ABO) functions. Thus, the aim is to interpret an observed spectrum in terms of one vibrational coupling mode, which is antisymmetric. Their analysis of the spectrum of the Creutz-Taube ion yields a value of 0 of 1.215, i.e., a rather weakly localized ground state. Using their assumed unperturbed... 

As a generalization of these observations it follows that vibrations in a central field i.e. around a special central point) are of two types, radial modes and angular modes. Laplace s equation separates into angular and radial components, of which the angular part accounts in full for the normal angular modes of vibration. Radial modes are better described by the related radial function that separates out from a Helmholtz equation. It is noted that the one-dimensional oscillator has no angular modes. 

Let a molecule have two vibrational modes with frequencies uia and uii,. If the second-order resonance condition 2uia uif, is fulfilled, then the huif, transition in the infrared spectrum can split if the interaction is allowed by symmetry of molecule into two lines of comparable intensity. The second line cannot be explained as a result of the interaction of light with the vibrational a mode because the transition with excitation of two hwa quanta is forbidden due to the well-known n — n 1 selection rule for a harmonic oscillator. Fermi explained (7) this experimental observation as a result of a nonlinear resonance interaction of two vibrational modes with each other. Since that time the notion of Fermi resonance has been generalized to processes with participation of different types of quanta (e.g. + iv2 u>3, lo + iv2 — UJ3 — 0J4, and so on) and to elec-... 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

The good news from this chapter is that all of the techniques we have used in this book so far are applicable in two and three dimensions. The bad news is that things are generally more complex in higher dimensions. Meshes are the most general way to model the spatial aspects of vibrations, and modes are the most general way to handle the pure spectral behavior. Banded waveguides are applicable in some systems, especially where clear, closed wavepaths can be identified. The next chapter will depart entirely from these types of models, and look instead at the statistical behavior of random type sounds. 

---