Oscillator coordinates

The operator U shifts the qj oscillator coordinate to its equilibrium through the distance QoCj/cOj, the sign depending on the state of the TLS. All the coupling now is put into the term proportional to the tunneling matrix element and the small parameter of the theory is zIq rather... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

Fig. 7.2.1 Time evolution of a harmonic-oscillator coordinate, according to Eq. (7.20). The phase is chosen as

There are well-known temperature effects, particularly dealing with the two first moments of the spectra that evoke those of the thermal average appearing in the statistical mechanics of quantum harmonic oscillator coordinates. 

Here, (Q2) is the fluctuation of the Brownian oscillator coordinate at equilibrium, the classical and quantum statistical expressions of which are given in Table L.l Now, take the Laplace transform of both sides of Eq. (L.3),... 

Similar modelling has been performed for both of these systems, based on the cube model. Following Hand and Harris, the molecular motion was coupled to the surface oscillator via a rigid shift of the Z-coordinate in the PES, i.e. V(Z, r,. .., y) = V(Z — y, r...), where y is the oscillator coordinate. For the H2/Pd system [80], six molecular degrees-of-freedom were included in a classical treatment, while four molecular degrees-of-freedom were included in a quantum solution for the H2/Cu system [81, 82]. In the classical calculations, the surface temperature dependence was introduced by sampling the surface vibration from a Boltzmann distribution. In quantum calculations, this is not possible, and many calculations were required, each in a different initial surface oscillator state. The results... 

Although experiments are most directly expressed in terms of the oscillator coordinate q(r), the quantity of physical relevance is the instantaneous oscillator frequency deviation from the mean frequency, w(t) = oo(t) — (co). This frequency deviation is the direct consequence of time-dependent forces exerted on the oscillator by its environment. At a semiclassical level, the FID correlation function is related to the time-dependent frequency by... 

A standard model of VP, which is repeatedly used in the study of VP dynamics, corresponds to two coupled Morse (M) oscillators, one low-frequency (LF) dissociating oscillator, coordinate R, and the other high-frequency (HF) oscillator, coordinate r, in bound states. The Hamiltonian of such a Morse-Morse system is written as... 

The most important physical inputs into the stochastic model are the transition probabilities per unit time between any two vibrational levels. Naturally these transition rates will be proportional to the number Z of collisions undergone by the molecule per unit time. For each collision we assume that the transition probability between oscillator states n and m is proportional to Qmn, the absolute square of the matrix element of the oscillator coordinate q between these states, given by (cf, Eq, (2,141)) ... 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted ... 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

In the classical limit (hcoj k T), the reaction coordinate X t) in each quantum state can be described as a Gaussian stochastic process [203]. It is Gaussian because of the assumed linear response. As follows from the discussion in Section II.A, if the collective solvent polarization follows the linear response, the ET system can be effectively represented by two sets of harmonic oscillators with the same frequencies but different equilibrium positions corresponding to the initial and final electronic states [26, 203]. The reaction coordinate, defined as the energy difference between the reactant and the product states, is a linear combination of the oscillator coordinates, that is, it is a linear combination of harmonic functions and is, therefore, Gaussian. The mean value is = — , for state 1 and = , for state 2, respectively. We can represent Xi(r) and X2 t) in terms of a single Gaussian stochastic process x(t) with zero mean as follows ... 

The Landau-Teller model considers a linear collision of a structureless particle A with a harmonic oscillator BC within an approach which by now is known as a semiclassical method the relative particle-oscillator motion (coordinate R) is described classically and the vibrational motion of the oscillator (coordinate x) by quantum mechanics the interaction between incoming particle A and the nearest end B of the oscillator BC is taken to be exponential, I/(/ g) c exp(-aR g). The expression for the transition probability in the near-adiabatic limit was found [4] to have the following generic form ... 

Thus, the fifth term in eq. (5.9) arises from the expectation value of the linear term of the intermolecular potential function (which is assumed to be expanded up to second order in the oscillator coordinates) and is referred to as the V-T (vibration-translation) coupling term. The V-T term mainly depends on the linear force fk, the first term of which contains the coupling term BkF which couples the reaction path to the perpendicular vibrational mode A , and a matrix element hk s) of the second quantized creation and anhilation operators. BkF is a measure of the curvature of the reaction path and determines the amount of coupling to a given vibrational degree of freedom. 

Carry out the integration to show that the harmonic oscillator coordinate wave function in Eq. (15.4-10) is normalized. 

FIGURE 8.2 Schematic representation of the interaction of the atom with mass m with a phonon with frequency approach angle, z the distance to the surface, r the oscillator coordinate, and R = z + r. 

Finally, one more remark on the construction of terms. Although, in principle, we plot U as a function of nuclear coordinates, we are justified in using some other coordinates for complex systems of the type of a polar solvent considered in the preceding section. Indeed, if in the process under consideration some molecules are held intact, there is a one-to-one correspondence between the coordinates of nuclei and the coordinate of the molecule as a whole (oscillator coordinates), and we can go over from one coordinate system to another. 

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