Perturbation theory golden rule

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

A simple method for predicting electronic state crossing transitions is Fermi s golden rule. It is based on the electromagnetic interaction between states and is derived from perturbation theory. Fermi s golden rule states that the reaction rate can be computed from the first-order transition matrix and the density of states at the transition frequency p as follows ... 

The golden rule is a reasonable prediction of state-crossing transition rates when those rates are slow. Crossings with fast rates are predicted poorly due to the breakdown of the perturbation theory assumption of a small interaction. 

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... 

In the time-dependent perturbation theory [Landau and Lifshitz 1981] the transition probability from the state 1 to 2 is related with the perturbation by the golden rule,... 

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) ... 

This is an application of Fermi s golden rule. The first term is the square of the matrix element of the perturbation, which appears in all versions of perturbation theory. In the second term 8(x) denotes the Dirac delta function. For a full treatment of this function we refer to the literature [2]. Here we note that S(x) is defined such that S(x) = 0 for x 7 0 at the origin S(x) is singular such that / ( r) dx — 1. The term 8 (Ef — Ei) ensures energy conservation since it vanishes unless... 

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... 

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. 

Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

From the definition of 6k it should be noted that Bk and k are Hermitian conjugates. According to the Golden Rule of time-dependent perturbation theory the probability per unit time, Wt f k, co), that the field (k, to) induces a transition in the system from the initial state f> to the final state / ) is given by... 

Berry (14) used the Golden Rule to evaluate the energy distribution of photofragments. In his development, polyatomic photodissociation is treated as a nonstationary phenomenon so that the probability of a transition i + f is given from first-order perturbation theory by... 

Note that in a sudden transition, eq. 3 is a more general relation than the Golden role (see, e.g., reference 15). In that circumstance the Golden rule appears as a special case of the FC factor, corresponding to small V and the applicability of perturbation theory. In order to evaluate the FC factor, Berry used the dressed oscillator model which, in principle, coincides with the previously described quasi-diatomic method. 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

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

If W, is small then Dfit) d(u>it) and equation (27) coincides with the Fermi golden rule (see equation (23)). To get beyond the perturbation theory, one should find D/it) more accurately. To this end, let us multiply the left- and the right-hand sides of equation (22) by e, and sum over i we obtain an equation for q/it). Then, applying the latter equation once more (this time for qtit ) under the integral in equation (22)), one obtains... 

Because of the time dependence of the vector potential A(rJ( t), the photon-atom interaction also depends on time. Hence, time-dependent perturbation theory has to be applied. The golden rule (so called by Fermi [Fer50], see also [Dir47, Sch55, LLi58]) for the transition rate w then yields for the change from an initial atomic state i> to a final atomic state f>... 

The quantum mechanical approach is based on time-dependent perturbation theory and is derived from Fermi s Golden Rule for non-radiative decay processes [1]. 

Much more is becoming known about the rates of the physical processes in competition with proton exchange reactions in excited states. (For an excellent review see Henry and Siebrand, 1973.) The factors which determine the rate constants (k) for internal conversion and intersystem crossing are neatly summarized in the Golden Rule of time-dependent perturbation theory ... 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

The method proposed by Fermi (1934) for calculating the / decay of a nucleus is based on the time-dependent perturbation theory. The small value of the weak-interaction constant makes it possible to restrict oneself to the first order in perturbation theory and to use the so-called Fermi Golden Rule... 

According to the Fermi Golden Rule, the non-adiabatic ET rate constant is strongly dependent on electronic coupling between the donor state D and acceptor state A connected by a bridge (VAb) which is given by an expression derived from the weak perturbation theory... 

The transition probability per unit time given by the time-dependent perturbation theory, that Fermi named Golden Rule in view of its prevalence in radiationless transitions, has the form... 

This expression is the exact form of Fermi s Golden Rule, familiar in time-dependent perturbation theory where F[, 0)) is approximated by o) (Merzbacher, 1970). p( ,) is the density of final states. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

This is the most direct experimental manifestation of the existence of an electronic interaction. It can occur spontaneously in mixed-valence complexes, but also in bimetallic systems after a photochemical excitation (photoinduced electron transfer). The general theory considers electron transfer as a special case of radiationless transition, with a perturbative treatment based on Fermi s Golden Rule [42]. In the nonadiabatic case, the rate constant can be written as [43] ... 

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