Landau-Teller

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

This conceptual link between the solvent vibrational friction and vibrational energy relaxation is actually mirrored by an important practical connection. Within the rather accurate Landau-Teller approximation, (29,33,34), the rate of vibrational energy relaxation for a diatomic with frequency coq and reduced mass /i is given by... 

At all events, the role of Coulombic forces for VET in solution was first examined in a molecular dynamics simulation for the 680 cm-1 C-Cl vibration of the CH3CI molecule (modeled as a diatomic) in water solvent by Whitnell et al. (19). The (classical) relaxation time Ti was determined both by nonequilibrium simulations and by use of the classical Landau-Teller (LT) formula (1,3,19,20). 

It is now interesting to consider the classical (h 0) limits of these two expressions. Actually, in the first instance it is inappropriate to take this limit, since just the opposite limit (/fftkuio 1) is invoked in truncating the state space to two levels. However, in the second instance, one can smoothly take the classical limit, and one finds that in this case Ti is given by the usual classical Landau-Teller result ... 

Here is a fascinating result if the harmonic scheme of Equation (18) is applied to Equation (14), using Equation (12) for Ti recovers exactly the classical Landau-Teller result, as recently shown by Bader and Berne (16). Thus, if the oscillator and bath are both harmonic and are bilinearly coupled, the exact quantum result and the exact classical result are identical This provides some justification for using the purely classical result of Equation (13) even in situations where one clearly is not in the classical limit. 

Note that for /3fi[Pg.691]

Relatively few measurements have been reported of vibrational energy transfer at low temperatures. Aside from the NO studies of Billingsley and Callear, already discussed, Miller and Millikan [190] have reported relaxation times for mixtures of CO with helium and hydrogen down to 100°K. A very clear deviation from the Landau-Teller linear extrapolation [191] was observed below 300°K for both mixtures. Shin [192] has examined the CO + He case in more detail, using the WKB approach and a Morse interaction potential. At low temperatures, he concludes, there are three important considerations ... 

Shock-tube investigations of vibrational relaxation times in hydrogen, deuterium, and their mixtures with argon and krypton have been reported by Kiefer and Lutz [129] and by Moreno [196]. By fitting their measurements to a Landau-Teller temperature dependence, Kiefer and Lutz give, for D2-D2 collisions (1100-3000°K),... 

These relaxation times, for both forms of Ha at low temperatures, are much shorter than the Landau-Teller extrapolation from higher temperatures would predict. 

E. 1. Dashevskaya, J. A. Kune, E. E. Nikitin, and I. Oref, Two-ehannel vibrational relaxation of H2 by He a bridge between the Landau-Teller and Bethe-Wigner limits, J. Chem. Phys. 118,3141 (2003)... 

Abstract A generalization of the Landau-Teller model for vibrational relaxation of diatoms in collisions with atoms at very low energies is presented. The extrapolation of the semiclassical Landau-Teller approach to the zero-energy Bethe-Wigner limit is based on the quasiclassical Landau method for calculation of transition probabilities, and the recovery of the Landau exponent from the classical collision time. The quantum suppression-enhancement probabilities are calculated for a general potential well, which supports several bound states, and for a Morse potential with arbitrary number of states. The model is applied to interpretation of quantum scattering calculations for the vibrational relaxation of H2 in collisions with He. 

Landau-Teller expression oc exp(—2 zA / Ctv) for a head-on collision. Here... 

This asymptotic form is plotted in Fig. 5. A feature of BBM(d>) is that it decreases asymptotically with frequency to zero. If the atom B is involved in vibrational motion at frequency oo (Oq, the coupling with the bath through binary collisions is small and the slow dissipation is the stochastic manifestation of slow vibrational relaxation. The most significant feature of Eq. (3.17) is that the dependence in the exponent of Eq. (3.17) is equivalent to an exponent This is just the form of the Landau-Teller theory of vibration-translation (V-T) energy transfer in atom-diatom collisions, and this form is almost universally used to fit vibrational relaxation rates in such systems. This will be dealt with in more detail in Section V C. The utility of BBM(d>) is that it pertains to atom-atom collisions in which the atom B is bonded to the other atoms by arbitrary potentials. No assumptions have been made about the intramolecular motions, although the use of BBM(d)) implies linear coupling to the displacements of atom B. Grote et al. have alluded to the form of Eq. (3.20) for di = 0 in a footnote. 

For a comparison of the assumptions underlying our theory of liquid phase VER and those of the Landau-Teller theory see sec. 23 of ref. 19. 

E.E.Nikitin, The Landau-Teller model in the theory of collisional excitation of molecules, Proceedings of second USSR summer school on physics of electronic and atomic collisions. Leningrad Physico-Technical Institute, 1973, p.l4... 

E.E.Nikitin, Generalized Landau-Zener and Landau-Teller models in theory of molecular collisions. In Chemistry Reviews, Soviet Scientific Reviews, Section B 9, ed. M. E. VoFpin, New York, Harwood Acadentic 1987, p.l87... 

Landau-Teller expression P °c exp(-2M /ft v) for a head-on collision. Here V is the collision velocity and a is the logarithmic derivative of the repulsive exponential potential [3],... 

Krylov subspace 220ff, 266ff Landau-Teller 413ff... 

After taking the coefficients as the Landau-Teller transition probabilities for the harmonic osciUator,t Montroll and Shuler... 

As long as repulsive interaction is dominant, there is no ftmdamental difference in the theoretical description between the quenching of ions and the quenching of neutral molecules. This problem was solved first by Landau and Teller (1936), leading to the famous Landau-Teller equation for the vibrational quenching rate constant. 

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