Local mode potential

The local modes introduced in the previous section can interact. The interaction potential can be written as... 

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]... 

Values of the F , Fm, and Fm force constants are determined from an ab initio potential energy curve calculated as a function of displacements of the local mode coordinate from equilibrium. The potential energy curves extend from —0.2 A to... 

Figure 7.4 A typical action potential, (a) Rising phase Na+ ions are flowing into the axon from outside (b) Falling phase Na+ permeability has now dropped but permeability to K+ has increased and K+ ions are moving out of the axon (c) Positive phase this is due to maintained high K+ permeability (d) Negative after-potential local high K+ outside the axon gives net K+ influx, which delays equilibration. (Adapted from Corbett, J.R., Wright, K., and Baillie, A.C., The Biochemical Mode of Action of Pesticides, 2nd ed., Academic Press, New York, 1984. With permission.)...

A brief review and reassessment of data on the photophysics of benzene has been presented by Pereira. Evidence for the l E2g valence state has been obtained by u.v. two-photon spectroscopy.Slow electron impact excites fluorescence in thin films of benzene at 77 K as well as emission from isomers." The fluorescence yields and quenching by chloroform of alkyl-benzenes and 1-methylnaphthalene after excitation into Si, Sz, and S3 states and after photoionization have been measured. The channel-three process has been reconsidered in terms of the effects of local modes and Morse oscillator potentials. Excited-state dipole moments of some monosubstituted benzenes have been estimated from solvent effects on electronic absorption spectra, Structural imperfections influence the photochemistry of durene in crystals at low temperatures. Relaxation time studies on excited oxido-substituted p-oligophenylenes have been made by fluorescence depolarization... 

Another illustrative example of mode-specific decay is the dissociation of water. Local-mode and hyperspherical-mode resonance states exist in the same energy range and decay with significantly different rates [70,80,81]. Correlations between assignments and vibrational PSD s can also be established [51]. Often, molecules featuring mode-specific decay have low densities of states and shallow potential wells. However, the most important requirement is that the coupling between vibrational modes is weak, so that the dynamics is close to separable. Two examples, HCO and HOCl, will be discussed in Sect. 5. 

The reorganization free energy is usually split in two parts. The local mode contribution is obtained in standard routines which require local potentials (say harmonic potentials) and vibrational frequencies in the reactants and products states. The collective modes associated with the proteins and the solvent, however, pose complications. One complication arises because classical electrostatics needs modification when the spatial extension of the electric field and charge distributions are comparable with the local structure extensions of the environment. Other complications are associated with the presence of interfaces such as metal/solution, protein/solution, and metal/film/solution interfaces. These issues are only partly resolved, say by nonlocal dielectric theory and dielectric theory of anisotropic media. 

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. 

We will now develop the equations used to compute the local mode energies. After defining the local mode Hamiltonian, we will convert it into the normal coordinates that were used to define the basis sets used for the dynamical calculations. The local mode Hamiltonian, for mode /, is defined in terms of the harmonic oscillator potential (except for the initially excited stretch, as described later)... 

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. 

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