Energy local/normal mode model

Thus, the Morse potential is the most appropriate for estimating bond energy in a molecule [4-9], The C-X and C-Y bond stretching modes in YCX3 molecules are treated as anharmonic Morse-like diatomic, non-linear coupled oscillators. The well-known [26-37] effective Hamiltonian of the local and normal mode model is used. 

Prediction of the overtone vibrational spectra of the CX3Y methylhalides can be based on the effective local and normal mode model Hamiltonian. D parameter, differs from De parameter of the Morse potential by E0 energy of zero-point vibrations, coincides with dissociation energy of separated C-Y bond with experimental accuracy in calculations using minimal basis set, if the correlation... 

In Section 9.4.12.4 the simplest possible local mode HlqCAL, expressed in terms of four independently adjustable parameters (the Morse De and a parameters, and two 1 1 kinetic and potential energy coupling parameters, Grr and km,), is transformed to the simplest possible normal mode H )oRMAL, which is also expressed in terms of four independent parameters. However, the interrelationships between parameters, based on the 1 1 coupled local Morse oscillator model, result in only 3 independent fit parameters. This paradox is resolved when one realizes that the 4 parameter local-Morse model generates the Darling-Dennison 2 2 coupling term in the normal mode model. However, the full effects of this (A ssaa/16hc)[(at + as)2(a+ + aa)2] coupling term are not taken into account in the local mode model. 

Vibrational states can be described in terms of the normal mode (NM) [50, 51] or the local mode (LM) [37, 52, 53] model. In the former, vibrations in polyatomic molecules are treated as infinitesimal displacements of the nuclei in a harmonic potential, a picture that naturally includes the coupling among the bonds in a molecule. The general formula for the energies of the vibrational levels in a polyatomic molecule is given by [54]... 

In describing the normal modes of a protein, it is instructive to compare them conceptually with those of a simple model of a polymer, such as a chain of atoms, both periodic and aperiodic. In a harmonic periodic chain, the normal modes carry energy without resistance from one end of the ID crystal to the other. On the other hand, the vast majority of normal modes of an aperiodic chain are spatially localized [138]. Protein molecules, which are of course not periodic, can be better characterized as an aperiodic chain of atoms, and most normal modes of proteins are likewise localized in space [111,112,126-128]. If a normal mode a is exponentially localized, then the vibrational amplitude of atoms in mode a decays from the center of excitation, Ro, as... 

Experimental studies have had an enormous impact on the development of unimolecular rate theory. A set of classical thermal unimolecular dissociation reactions by Rabinovitch and co-workers [6-10], and chemical activation experiments by Rabinovitch and others [11-14], illustrated that the separability and symmetry of normal modes assumed by Slater theory is inconsistent with experiments. Eor many molecules and experimental conditions, RRKM theory is a substantially more accurate model for the unimolecular rate constant. Chemical activation experiments at high pressures [15,16] also provided information regarding the rate of vibrational energy flow within molecules. Experiments [17,18] for which molecules are vibrationally excited by overtone excitation of a local mode (e.g. C-H or O-H bond) gave results consistent with the chemical activation experiments and in overall good agreement with RRKM theory [19]. 

More subtle than the lack of ZPE in bound modes after the collision is the problem of ZPE during the collision. For instance, as a trajectory passes over a saddle point in a reactive collision, all but one of the vibrational (e.g., normal) modes are bound. Each of these bound modes is subject to quantization and should contain ZPE. In classical mechanics, however, there is no such restriction. This has been most clearly shown in model studies of reactive collisions (28,35), in which it could be seen that the classical threshold for reaction occurred at a lower energy than the quantum threshold, since the classical trajectories could pass under the quantum mechanical vibrationally adiabatic barrier to reaction. However, this problem is conspicuous only near threshold, and may even compensate somewhat for the lack of tunneling exhibited by quantum mechanics. One approach in which ZPE for local modes was added to the potential energy (44) has had some success in improving reaction threshold calculations. 

Finally, a few comments shall be made on the concept of local modes as compared to normal modes [3,33-35], The main idea of the local mode model is to treat a molecule as if it were made up of a set of equivalent diatomic oscillators, and the reason for the local mode behavior at high energy (>8000 cm ) may be understood qualitatively as follows. As the stretching vibrations are excited to high energy levels, the anharmonicity term / vq (Equation (2.9)) tends, in certain cases, to overrule the effect of interbond coupling and the vibrations become uncoupled vibrations and occur as local modes. ... 

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