Time-dependent Golden rule

The purpose of this work is to study the electronic predissociation from the bound states of the excited A and B adiabatic electronic states, using a time dependent Golden rule (TDGR) method, as previously used to study vibrational pre-dissociation[32, 33] as well as electronic predissociation[34, 35], The only difference with previous treatments[34, 35] is the use of an adiabatic representation, what requires the calculation of non-adiabatic couplings. The method used is described in section II, while the corresponding results are discussed in section III. Finally, some conclusions are extracted in section IV. 

Time-dependent Golden-rule treatment for Electronic predissociation... 

In this work the electronic predissociation from the A,B and B states has been studied using a time dependent Golden rule approach in an adiabatic representation. The PES s previously reported[31 ] to simulate the experimental spectrum[22] were used. Non-adiabatic couplings between A-X and B-X were computed using highly correlated electroiric wavefunctions using a finite difference method, with the MOLPRO package[42]. 

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. 

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 probability for a transition to occur between two states per unit time is determined by Fermi s golden rule and depends on the operator of interaction between the subsystem concerned and a thermostat. As orientational states are characterized by a low-energy spectrum, they will be substantially influenced by the... 

Recently, the electron-transfer kinetics in the DSSC, shown as a schematic diagram in Fig. 10, have been under intensive investigation. Time-resolved laser spectroscopy measurements are used to study one of the most important primary processes—electron injection from dye photosensitizers into the conduction band of semiconductors [30-47]. The electron-transfer rate from the dye photosensitizer into the semiconductor depends on the configuration of the adsorbed dye photosensitizers on the semiconductor surface and the energy gap between the LUMO level of the dye photosensitizers and the conduction-band level of the semiconductor. For example, the rate constant for electron injection, kini, is given by Fermi s golden rule expression ... 

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)... 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

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... 

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... 

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]. 

The conflicting serial/parallel models for IVR/VP are not readily distinguished until time resolved experiments can be performed on the systems of interest. Both models can relate the relative intensities of the emission features to the various model parameters, but the serial process seems more in line with a simple, conventional [Fermi s Golden Rule for IVR (Avouris et al. 1977 Beswick and Jortner 1981 Jortner et al. 1988 Lin 1980 Mukamel 1985 Mukamel and Jortner 1977) and RRKM theory for VP (Forst 1973 Gilbert and Smith 1990 Kelley and Bernstein 1986 Levine and Bernstein 1987 Pritchard 1984 Robinson and Holbrook 1972 Steinfeld et al. 1989)], few parameter approach. Time resolved measurements do distinguish the models because in a serial model the rises and decays of various vibronic states should be linked, whereas in a parallel one they are, in general, unrelated. Moreover, the time dependent studies allow one to determine how the rates of the IVR and VP processes vary with excitation energy, density of states, mode properties, and isotropic substitution. 

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 ... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

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... 

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) ... 

Spectra, P E), can be obtained directly from a time-dependent treatment as the Fourier transform of the autocorrelation function C t), assuming a direct transition from the initial to the final states within the framework of Fermi s golden rule [6,119] ... 

Q xt/n in the golden rule equation transforms (25) into the following useful form, readily utilized in a time-dependent picture... 

A word of caution is needed here. The golden-rule expression, Eq. (12.33) or (12,43), was obtained for the rate of decay of a level interacting with a continuous manifold (Section 9,1), not as a perturbation theory result but under certain conditions (in particular a dense manifold of final states) that are not usually satisfied for optical absorption, A similar expression is obtained in the weak coupling limit using time-dependent perturbation theory, in which case other conditions are not... 

We shall need this later to apply a well known result of quantum mechanics, involving time dependent perturbation theory, Fermi s Golden Rule [1]. 

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