Vibrational states coupling

A polyad is a group of near-degenerate vibrational states coupled by one or more low-order anharmonic terms. For example, the very common 2 1 Fermi resonance , which is due to an anharmonic coupling term like ki22QiQ2,... 

The simplest way to write down the 2 x 2 Hamiltonian for two states such that its eigenvalues coincide at trigonally symmetric points in (x,y) or (q, ( )), plane is to consider the matrices of vibrational-electronic coupling of the e Jahn-Teller problem in a diabatic electronic state representation. These have been constructed by Haiperin, and listed in Appendix TV of [157], up to the third... 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

Fig. 8. Scattering the transition state from the surface. Measured vibrational distribution of NO resulting from scattering of laser-prepared NO(v = 15) from Au (111) at incidence = 5 kJ mol-1. Only a small fraction of the laser-prepared population of v = 15 remains in the initial vibrational state. The most probable scattered vibrational level is more than 150 kJ mol-1 lower in energy than the initial state. Vibrational states below v = 5 could not be detected due to background problems. These experiments provide direct evidence that the remarkable coupling of vibrational motion to metallic electrons postulated by Luntz et al. can in fact occur. (See Refs. 44 and 59.)...

Fig. 9. Incidence energy dependence of the vibrational state population distribution resulting when NO(u = 12) is scattered from LiF(OOl) at a surface temperature of (a) 480 K, and (b) 290 K. Relaxation of large amplitude vibrational motion to phonons is weak compared to what is possible on metals. Increased relaxation at the lowest incidence energies and surface temperatures are indicators of a trapping/desorption mechanism for vibrational energy transfer. Angular and rotational population distributions support this conclusion. Estimations of the residence times suggest that coupling to phonons is significant when residence times are only as long as ps. (See Ref. 58.)...

A higher level of understanding would require a knowledge of molecular dynamics and presently represents a rather distant goal. In addition to reliable knowledge of the shapes of potential energy hypersurfaces, it would also require information such as vibronic coupling elements, densities of vibrational states, detailed mechanism of the action of the heat bath, etc. 

Fig. 6. Vibrational states corresponding to axial H-atom vibrations (y-coordinate) and perpendicular B-atom vibrations (atj, x2 — coordinates) in the absence and presence of anhar-monic coupling (see text). For state mn,n2>, the m is the H-vibrational quantum number, and the n s are the B-vibrational quantum numbers. The infrared absorption corresponding to the m = 0 to m = 1 transition is sensitive to the B-isotope, as seen in the figure (solid vertical lines). Also, the transition n = 0 to n = 2 is now weakly allowed due to the mixing with the H-mode these two-phonon transitions are indicated by dashed vertical lines. Less important vibrational states are not shown on the figure.

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

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