Effective oscillation

For reactants having complex intramolecular structure, some coordinates Qk describe the intramolecular degrees of freedom. For solutions in which the motion of the molecules is not described by small vibrations, the coordinates Qk describe the effective oscillators corresponding to collective excitations in the medium. Summation rules have been derived which enable us to relate the characteristics of the effective oscillators with the dielectric properties of the fi edium.5... 

This formula is easily obtained from Eq. (30) if we use the summation rules relating the parameters of the effective oscillators with the dielectric properties of the medium.5... 

The normal vibrations q and q are related to the shifts of the ions Y and X . The low-frequency part of the inertial polarization of the medium, k(cok co 9 co ), cannot follow these shifts. The high-frequency part of the inertial polarization, /(a>/ co 1, co )9 adiabatically follows the shifts of the ions Y" and X-, and the equilibrium coordinates of the effective oscillators describing this part of the polarization depend on the normal coordinates of the corresponding normal vibrations, viz. /0i(gl), (iof(q )-... 

Applying the usual steady-state treatment for consecutive first-order reactions kt at 16 torr pressure over the temperature range 597-701 °C is given by 1.8 x 1011 exp(—47,000/Kr) sec Within experimental error, reactions (1) and (2) were homogeneous processes. However, both k2 and k2 were functions of the total pressure in the system. This dependence is shown in Fig. 1. The methyl zinc decomposition is apparently in its second-order region. Therefore, assuming four effective oscillators and a mean temperature of 1050 °K, = Eohs.+i nRT... 

Fig. 9. Log-log plot of the homogeneous unimolecular rate constant vs. pressure in the decomposition of C2F40 at 126°C. The solid lines represent theoretical curves from Kassel theory with 8 and 9 effective oscillators. From Lenzi and Mele106 with permission of the American Institute of Physics.

We have shown the molecular orbital theory origin of structure - function relationships for electronic hyperpolarizability. Yet, much of the common language of nonlinear optics is phrased in terms of anharmonic oscillators. How are the molecular orbital and oscillator models reconciled with one another The potential energy function of a spring maps the distortion energy as a function of its displacement. A connection can indeed be drawn between the molecular orbitals of a molecule and its corresponding effective oscillator . 

At the classical description of the nuclear degrees of freedom (it is the classical approach that is used in the theory of electron transfer in polar media (see Chapter 2, Section 5)), the fulfillment of criterion (73) results in the following. Instead of one effective oscillator (see Chapter 2) where the equilibrium position is shifted at the transition, it should now introduce two... 

Equation (1) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. Even for linear polarization, the oscillation of the induced dipole moment may be damped (by material resonances) and thereby phase-shifted with respect to the oscillation of the external electric field. The usual way of expressing this phase shift is by considering the relationship between the Fourier components of the induced effect (oscillation of the induced dipole) and the stimulus (the electric field), with the damping and phase shift conveniently expressed by treating the terms involved as complex. Thus, the linear polarizability can be written as... 

Several attempts have been made to make s more than just a fitting parameter without losing the simplicity of the model. For example Benson suggested that the number of effective oscillators s could be estimated from the molar vibrational heat capacity, s = [15], and Troe has similarly used the energy,... 

The strong collision correction factor Fsc is a function of two further parameters that arise in Kassel theory (see Section 3). These are the number of effective oscillators, S, and B = Eg/kT, a measure of the relative magnitudes of the threshold energy and thermal energy. Troe used an energy criterion to obtain the number of effective oscillators. 

Eq. (4) and all consequences can be generalized to a set of harmonic oscillators. Then, combination bands can be observed (see below Sec. 3). Both MSAs and effective oscillator masses highlight the very nature of normal modes. 

Fitting procedures give information on wave functions via mean-square displacements (ufj for each vibration and effective oscillator masses. It transpires that proton dynamics for bending modes correspond very closely to isolated harmonic oscillators with a mass of 1 amu [Ikeda 2002], They are largely de-... 

As a conclusion for this section, INS studies of KHCOs single-crystals provide the most detailed, and hopefully the most tutorial, view of proton dynamics ever obtained. The limitation of optical techniques to establishing an unambiguous representation of proton dynamics is emphasized. Effective oscillator masses of 1 amu are determined for each normal mode. Then arises a new fundamental question which mechanisms can account for the decoupling of proton dynamics from the lattice ... 

The choice of normal coordinates is arbitrary [Wilson 1964] and definitions at variance from (11) can be found in text books, see for example [Cohen-Tan-noudji 1977], However, the under determination holds only in the classical regime. The effective oscillator mass for coupled proton oscillators is clearly 1 amu according to the scattering function (see above Sec. 3). Only normal coordinates defined in (11) have a physical reality in the quantum regime. At the present time, there is no obvious justification, apart from experiments, for this choice. 

Neutron scattering experiments shed a new light on proton dynamics in the extended arrays of hydrogen bonded centrosymmetric dimer entities of the KHCO3 crystal. Proton dynamics are decoupled from the lattice. Measurements of effective oscillator masses (namely, 1 amu) contribute to full determination of normal coordinates. 

A minor product (corresponding to about 0-5 % of the cyclopentene peak) was detected which had the same retention time as methylenecyclo-butane. At pressures below 3 mm the rate constant decreased with pressure and fell to approximately one half of the high-pressure value at 0-07 mm. The results obtained could be fitted by the Kassel equation by assuming that the reactant had eighteen effective oscillators. The data are thus consistent with the isomerization being a truly unimolecular transformation. On the basis both of the observed energy of activation and thermochemical data and of estimates of bond strengths, these... 

M provide a set of equations for the frequencies Qy and effective oscillator strength = Zifif Qy (/J( and fif depend on the external field (eq All). For example, for linear response we have / = —S i fy and fiy = —Sitiiiy. Here fy and fiy are the oscillator strength and the ground-state dipole, respectively.) ... 

The Hamiltonian of the effective oscillators of the medium for the spatially homogeneous system is of the standard form ... 

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