Quantum vibrational

In the case where they represent quantum vibrational modes, this leads to the appearance of a small tunnel factor in the transmission coefficient k. ... 

Schmickler W. 1976. The effect of quantum vibrations on electrochemical outer sphere redox reactions. Electrochim Acta 21 161-168. 

Even more remarkable, vibrational relaxation of NO(r = 15) on Au(lll) is characterized by profound multi-quantum vibrational relaxation. Specifically, the most probable vibrational scattering channel releases more than 150kJmol-1. Vibrational relaxation events exchanging as many as 10 vibrational quanta are observed. It appears likely that even more vibrational quanta can be exchanged with significant efficiency, but background problems prevented the observation of these channels. Thus the reported... 

Below we will restrict ourselves to the Born-Oppenheimer approximation and, unlike Refs. 62, 64, and 65, we will take into account the contribution from the excited vibrational states of the tunneling particle and consider the role played by the transverse quantum vibrations of the tunneling particle itself in the preparation of the potential barrier.48... 

Computational Study of Many-Dimensional Quantum Vibrational Energy Redistribution. I. Statistics of the Survival Probability. 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

V. S. Letokhov In the case of very intense IR fields we should take into consideration the effects of both (1) rearrangement of quantum vibrational levels (e.g., Rabi splittings) and (2) distortion of potential molecular curves. But with such strong IR femtosecond pulses we perhaps cannot ignore the excited electronic states. I believe that a simple molecule can be excited to some higher vibrational levels and subse-quently can be involved in a multiphoton jump to excited electronic states. As a result, it will be difficult to observe the effect of distortion of molecular potential curves. 

This was attributed to the stronger coupling required for the double quantum vibrational transition. 

Basically there are two main nonequilibrium effects the electronic spectrum modification and excitation of vibrons (quantum vibrations). In the weak electron-vibron coupling case the spectrum modification is usually small (which is dependent, however, on the vibron dissipation rate, temperature, etc.) and the main possible nonequilibrium effect is the excitation of vibrons at finite voltages. We have developed an analytical theory for this case [124]. This theory is based on the self-consistent Born approximation (SCBA), which allows to take easily into account and calculate nonequilibrium distribution functions of electrons and vibrons. 

Note that in Eq. 8-24 we have used the typical approximation to factorize the partition function [28,32] and hence Qv,s, Qv are the (quantum) vibrational partition functions of the solvent and solute molecule, respectively, and U = + AUvfi with < > the system potential energy (i.e., electronic ground state energy surface) and NUv,o the system vibrational ground state energy shift from a reference value [28,32], typically negligible. 

In order to deal with a chemical reaction it is convenient to express the energy U by the perturbed Hamiltonian matrix as a function of the reaction coordinates t). Expressing the nuclear coordinates of the quantum center (we consider it as the solute or a part of the solute) as r = xq, t, f where xq are the internal quantum vibrational coordinates, t) the reaction coordinates (belonging to the solute classical internal coordinates Xjn) and the remaining classical coordinates. Defining with all the solute classical internal coordinates except t, i.e., Xjn =, tt, we have that the free-energy change for a chemical transition defined by %, is... 

If both the forward (absorption) and backward (emission) optical transitions are available, their first spectral moments determine the reorganization energies of quantum vibrations, Xy, and of the classical nuclear motions of the donor-acceptor skeleton and the solvent, Xj-i ... 

Figure 3 Schematic quantum representation of (a) IR absorption, (b) normal Raman scattering, and (c) RR scattering (see text). The parabolas represent harmonic potentials in the ground electronic states. The two energy levels shown in the harmonic groimd-state (GS) potentials are quantum vibrational states u = 0 is the zero-point vibrational energy, and u = 1 is the first excited vibrational state...

Summarizing, we note that the single-quantum vibrational predissoeiation of a weak van-der-Waals bond in a triatomic molecule and the classical dissociation of the same system do not represent proeesses which can be eompared in the eorrespondenee prineiple limit. Our study explains why the classieal and quantum rates sometimes are elose to each other and show the same trends with a change of the interaction parameters. The underlying logic in comparing of classical and quantum and VP rates is as follows ... 

However, the classical approach should not be used for a quantitative interpretation of the experimental results for single-quantum vibrational predissociation. The classical process, which should show the correspondence with the quantiun one, is the cleavage of a bond under a condition when the quantum of energy transferred to this bond from a diatomic fragment, hQ, is much smaller than the dissociation energy, Ej, of the bond, i.e. h l . This condition does not contradict the classical limit,... 

Previous work on the thermodynamic properties of clusters used a number of schemes to evaluate the partition function required in Eq. (3.14). In the normal-mode method, " described in the Introduction, the partition function is constructed from the standard partitioning of a polyatomic gas into classical translational and rotational terms and quantum vibrational contributions. In Monte Carlo studies it is usual to employ a state integration technique.In the state integration method Eq. (3.5) is integrated with respect to temperature to obtain... 

In a polyatomic molecule, several types of vibrational motion are possible (Fig. 20.9). Each has a different frequency, and each gives rise to a series of allowed quantum vibrational states. Infrared absorption spectra provide useful information about vibrational frequencies and force constants in molecules. For... 

For larger hydrogen-bonded systems, rigorous calculations are far more difficult to carry out, both from the point of view of obtaining full-dimensional potentials and the subsequent quantum vibrational calculations. Reduced dimensionality approaches are therefore often necessary and several chapters in this volume illustrate this approach. With increasing computational power, coupled with some new approaches, it is possible to treat modest sized H-bonded systems in full dimensionality. We have already briefly reviewed the approach we have developed for potentials for the vibrations we have primarily used the code Multimode (MM). The methods used in MM have been reviewed recently [24 and references therein, 25], and so we only give a very brief overview of the method here. 

Pang X-F. Quantum vibrational energy spectra of molecular chains in crystalline acetanilide. J Phys Chem Solids 2001 62 793-6. 

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