Fermi “golden rule

From time-dependent perturbation theory, the probability of transition from one state to another can be described by Fermi s golden rule, which 

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The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

From Wentzel-Fermi Golden Rule to the Time Domain... 

The expression for the rate R (sec ) of photon absorption due to coupling V beriveen a molecule s electronic and nuclear charges and an electromagnetic field is given through first order in perturbation theory by the well known Wentzel Fermi golden rule formula (7,8) ... 

Let us now consider how similar the expression for rates of radiationless transitions induced by non Bom-Oppenheimer couplings can be made to the expressions given above for photon absorption rates. We begin with the corresponding (6,4g) Wentzel-Fermi golden rule expression given in Eq. (10) for the transition rate between electronic states Ti,f and corresponding vibration-rotation states Xi,f appropriate to the non BO case ... 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

However, it has been pointed out 13 16> for large organic molecules ( statistical limit case) that the decay times and quantum yields can legitimately be handled by the Fermi golden rule ... 

Fermi golden rule, 268 Filipescu, N., 291 Fisch, M. H., 307 Fischer, F., 379 Flash photolysis, 80-92 of aromatic hydrocarbons, 89, 90 determination of jsc, 228-230 determination of triplet lifetime, 240-242 energy of higher triplet levels, 219-220 flash kinetic spectrophotometry, 82, 83 measurement of triplet spectra, 81,82 nanosecond flash kinetic apparatus, 89 nanosecond flash spectrographic apparatus, 88... 

In the diffuse mismatch model, the scattering destroys the correlation between the wave vector of the impinging phonon and that of the diffused one. In other words, the scattering probability is the same independent of which of the two materials the phonon comes from. This probability is proportional to the phonon state density in the material (Fermi golden rule). 

Thus, having prepared the system at the initial time t=0 in the state i f >, the probability of finding the system in the state f f > at time t is given, as usual, by Cf(t) 2. The Fermi Golden-rule expression (to first order in TDPT) has the form [47]... 

The transition probability for multiphonon, nonadiabatic ET can be formulated in terms of first-order perturbation theory, i.e., by means of the Fermi golden rule, as (2)... 

Electron transfer theories in mixed-valence and related systems have been summarized elsewhere ((5) and references therein). Conventionally, the electron transfer rate is calculated perturb tionally using the Fermi golden rule assuming that the electronic perturbation (e) is small. The most detailed... 

Energy transfer in solution occurs through a dipole-dipole interaction of the emission dipole of an excited molecule (donor) and the absorptive moment of a unexcited molecule (acceptor). Forster<40) treated the interaction quantum mechanically and derived and expression for the rate of transfer between isolated stationary, homogeneously broadened donors and acceptors. Dexter(41) formulated the transfer rate using the Fermi golden rule and extended it to include quadrupole and higher transition moments in either the donor or the acceptor. Following the scheme of Dexter, the transfer rate for a specific transition is... 

Fe Oj coatings, 40 105 FejOj/y-AljOj, MSssbauer spectra, 37 30 FCjOj-I catalyst, 37 181-183 FcjOj superacid, 37 199-201 Fermi distribution, 34 228 Fermi energy, 27 217 Fermi golden rule, 34 243 Fermi level, 27 4, 5 Fermi s Golden Rule, 35 19-20 Ferric aluminate as catalyst, 20 109-112 chemical structure and catalytic activity of, 20 111, 112... 

Finally, it may be useful to note that the Fermi golden rule and time correlation function expressions often used (see ref. 12, for example) to express the rates of electron transfer have been shown [13], for other classes of dynamical processes, to be equivalent to LZ estimates of these same rates. So, it should not be surprising that our approach, in which we focus on events with no reorganization energy requirement and we use LZ theory to evaluate the intrinsic rates, is closely related to the more common approach used to treat electron transfer in condensed media where the reorganization energy plays a central role in determining the rates but the z factor plays a second central role. 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

The transition probability of an electron from i p, to Xv in first-order perturbation theory is then given by the Fermi golden rule. 

Field ionization can also be treated as a direct electronic transition from the atomic state to a vacant state at the surface. Following the Fermi golden rule, the rate of field ionization is given by31... 

Within the framework of first-order perturbation theory, the rate constant is given by the statistically averaged Fermi golden rule formula ... 

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