Vibrational modes transverse

With the boundary conditions that the chain ends are free of forces, Eq. (13) is readily solved by cos-Fourier transformation, resulting in a spectrum of normal modes. Such solutions are similar, e.g. to the transverse vibrational modes of a linear chain except that relaxation motions are involved here instead of periodic vibrations. 

Similar arguments hold for ozone however, nitrogen and oxygen are symmetric and thus are not strongly affected by radiation above 400 nm. The C02 molecule is linear but can easily be bent, leading to an induced dipole moment. A transverse vibrational mode exists for C02 at 15 pm, just -where the Earth emits most of its infrared radiation. 

A direct consequence of the existence of numerous low energy transverse vibrational modes in MOEs is that these materials exhibit extreme NTE behaviours. Among a range of cubic metal cyanide systems that display isotropic NTE, ° ° the interpenetrated diamondoid phases Zn(CN)2 and Cd(CN)2 have a = -16.9 x 10" and -20.4 x 10 respectively. The desorption of volatile guests from single diamondoid network Cd( CN)2 to achieve a 64 % porous apohost phase leads to the largest isotropic NTE yet reported for any material, with a = -33.5 x 10" K" (see Figure The thermal expansivity of this phase can be tuned... 

The three terms in have the interpretation as curvature couplings, Coriolis coupling, and centrifugal distortion, respectively. The latter two obviously involve rotation-vibration coupling—i.e., they are zero if J 0. The first (curvature coupling) term describes energy transfer between the reaction coordinate s and the transverse vibrational modes The coupling functions... 

The reaction path Hamiltonian is particularly useful for evaluating the path integral representation of the trace, Eq. (3.1 ), because the flux operator F does not involve the bath" degrees of freedom (i.e., the transverse vibrational modes Q ), and since they are harmonic oscillators the path integrals over them can be carried out analytically. All that remains to be done numerically is the path integral over only the reaction coordinate degrees of freedom itself. 

Let Xs(s) denote the IRC, Qk(s)l the transverse vibration modes and (ps(s), Pk(s) ) their conjugate momenta at the point 5 along the reaction path. In this coordinate system the molecular Hamiltonian Xs,Ps/ QklilPk)) is given by ... 

New types of coupling terms have to be considered, which describe the coupling between rotation and the motion along the path and the coupling between rotations and vibrations. The latter is described (in first order) by the cross-product of the eigenvectors of the transverse vibrational modes and by centrifugal distortion terms. ... 

Analysis of the RP curvature k(s) helps to identify those path regions with strong curvature and a coupling between translational and transverse vibrational modes. For this purpose, the curvature is investigated in terms of normal mode-curvature coupling coefficients and adiabatic internal mode-curvature coupling amplitudes At.,. 

This paper describes some new developments and applications based on the reaction path model. In section II the original form of the reaction path Hamiltonian [of. equation (1) below] is transformed to a new representation that has a more desirable structure for some applications. Section III shows how the reaction path model makes application of the unified statistical model for chemical reactions especially simple, and a generalized version of the unified statistical model is also developed there. Finally, in section IV the fact that the reaction path model consists of one special degree of freedom — i.e., the reaction coordinate — coupled to a number of harmonic oscillators is exploited to derive a generalized Langevin equation (GLE) for motion along the reaction path. This is a reduced equation of motion for only the reaction coordinate, but it experiences friction" and a "random force" because of coupling to the transverse vibrational modes. 

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