Golden rule equation

These kinetic equations are similar to the usual golden rule equations, but are more general. 

The rate constant for the radiationless relaxation of the excited states of inorganic compounds has been derived on the basis of the Fermi s golden rule (Equation 6.69). 

An alternative approach is to calculate the coupling matrix elements from the Golden Rule equation (11) if pj (E) and the rate constant kj are known. The product v 2 may then be calculated and it can be seen whether the depopulation process in question agrees with theoretical prediction. A lack of agreement could be indicative of the participation of other, unobserved states in the decay process. This approach has been used by Parmenter and Poland (185) to demonstrate the apparent paradoxical behavior of biacetyl in the low-energy absorption region. It has since been shown that this system is actually an intermediate case with three or more states involved in the decay process (72,246). 

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

As mentioned above, the inelastic scattering function can be directly calculated from these wave functions using Fermi s Golden Rule, equation (1) above. The... 

We use the Fermi golden rule (Equation 7.39) to derive the decay rate due to Fbrster interaction and obtain... 

Note that if we identify the sum over 8-fimctions with the density of states, then equation (A1.6.88) is just Femii s Golden Rule, which we employed in section A 1.6.1. This is consistent with the interpretation of the absorption spectmm as the transition rate from state to state n. 

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,... 

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 arguments for Edwards9 cancellation theorem are rather subtle, and we do not think that it is universally true (Mott 1989). For weak scattering, the relaxation time t must be proportional tog-1, by Fermi s golden rule. The mean free path l is given by the equation... 

The magnitude of the off-diagonal Hamiltonian (i.e. the energy transfer rate) thus depends on the strengths of the electron-phonon and Coulombic couplings and also the overlap of the two exciton wavefunctions[53]. Energy transfer rates from states to state jx, are calculated via the golden rule [54] and used as inputs to a master equation calculation of the excitation transfer kinetics in PSI, in which the dynamical information is included in the matrix K. 

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). 

Usually tunneling through a potential barrier is considered on the basis of the stationary Schroedinger equation with the use of matching conditions. A different approach has been developed by Bardeen (34). Bardeen s method enables one to describe tunneling as a quantum transition and to use the Golden Rule in order to evaluate the probability of penetration through the barrier. A similar method has been used in Section III to describe vibrational predissociation. This section contains a short description of Bardeen s method (see refs. 39,82-84). 

Substitution of this equation into the golden rule expression (1.14) together with the renormalized tunneling matrix element from (5.61) gives (5.65) after thermally averaging over initial energies ,. In the case of a biased potential the expression for the forward rate constant is... 

In the case of small IV I one can take IXt) d0(t). Then equation (14) coincides with the one given by the Fermi Golden Rule for arbitrary tii. 

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... 

Fig. 4. The temperature dependence of the rate of non-radiative transitions y for some (given) values of the interaction parameter w parameters co0 and cr0 are the same as in Fig. 1(a). The sharp peaks result from the divergence of the resolvent in equation (31) for w > 10 at some positive co > o>m- F°r comparison, the golden rule result is also presented (the thick line below).

The great success of Forster theory lies on the simplicity of these expressions, which can be applied from purely spectroscopic data. However, the approximations underlying these equations are not evident at first sight. It is better to turn to the Golden Rule expression of the rate ... 

Fig. 12.2. Schematic illustration of the absorption of an IR photon by the van der Waals complex Ar H2 and the subsequent dissociation into Ar+H2(n = 0). Voo(R) and Vn(i ) are the diagonal elements of the potential coupling matrix defined in Equation (3.6) which serve to define the zero-order radial wavefunc-tions employed in the Golden Rule expression for the dissociation rate. The assignment of the bound levels is (m,n), where m and n denote the number of quanta of excitation in the dissociation mode R and the vibrational mode of H2, respectively.

The manifestation of the dipole-dipole approximation can be seen explicitly in Equation (3.134) as the R 6 dependence of the energy transfer rate. In Equation (3.134) the electronic and nuclear factors are entangled because the dipole-dipole electronic coupling is partitioned between k24>d/(td R6) and the Forster spectral overlap integral, which contains the acceptor dipole strength. Therefore, for the purposes of examining the theory it is useful to write the Fermi Golden Rule expression explicitly,... 

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 rate constants in the master equation can be derived from Fermi s golden rule, with the result that (14)... 

Equation (34) is the Fermi Golden Rule expression, where Hc, is the electronic interaction, and FC is the F ranck-Condon factor. The analytical version of Eq. (34), applicable to high temperature, is given by Eq. (35) ... 

The Fermi Golden rule describes the first-order rate constant for the electron transfer process, according to equation (11), where the summation is over all the vibrational substates of the initial state i, weighted according to their probability Pi, times the square of the electron transfer matrix element in brackets. The delta function ensures conservation of energy, in that only initial and final states of the same energy contribute to the observed rate. This treatment assumes a weak coupling between D and A, also known as the nonadiabatic limit. 

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