Golden Rule approximation

Generalized relativstic effective core potentials (GRECP), ab initio calculations, P,T-odd interactions, 253-259 Gene transcription, multiparticle collisions, reactive dynamics, 108-111 Geometric transition state theory, 195-201 Gillespie s algorithm, multiparticle collisions, reactive dynamics, 109-111 Golden Rule approximation, two-pathway... 

In the golden rule approximation, the decay rates are the eigenvalues of the matrix T. The result that when the bound-state subspace is dense, T is of a rank smaller than the number of bound states means that, in the golden rule approximation, some of these bound states will remain bound. That is, they are trapped states and do not decay. Of course, the golden rule is just... 

The matter that is central to our photochemical interests is the magnitude of the three matrix elements i ° S), u = X, y, z, that connect each of the three sublevels of T witn the singlet level S, because in the Fermi golden rule approximation I I if o I 12 is proportional to the rate at which molecules interconvert between T and S. The three numbers, (T I i ° I S), Ty I if o I S), and (T iJ ° S), can be viewed as tne X, y, and z coordinates of the spin-oroit coupling vector (SOC), respectively. Under many experimental conditions, molecules equilibrate among the three T sublevels much faster than intersystem crossing to S takes place. In such a case, the intersystem crossing rate is proportional to... 

The partial photodissociation cross section to a particular final state in the golden rule approximation is proportional to the following quantity... 

A convenient method for deriving the dissociation rate is based on Fermi s Golden rule. Approximations inherent in this approach are similar to those required to separate (adiabatically via averaged interactions) the diatom and the van der Waals motions. The two states of interest are the excited complex in which the diatom has been excited to v = 1, and the final dissociated state in which the diatom is back in the ground state (v" = 0) with the energy released as translational (or rotational) energy. Within the approximation of separable motions, the system wave function in these two states are... 

In this section we derive expressions for the rates of electron transfer within the Fermi Golden Rule approximation. As we described for exciton transport in Section 9.2.4, these rates can be used to model charge transport using the density matrix formalism. There is a wide and thorough discussion of this topic in May and Kiihn (2000). 

Predissociation rates and line widths for use with the golden rule approximation and the nonadiabatic coupling (equation 4) have been calculated for the first few levels of the and 3 E+ states of ArH and ArD. In this case the repulsive part of the and 3 E+ states couple with the repulsive part of the ground state. The computed line width of... 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

This expresses the widely used El approximation to the Fermi-Wentzel golden rule. 

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,... 

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

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). 

Dick [1977] explained this behavior within the framework of a phonon-assisted tunneling mechanisms using the TLS approximation and golden rule formalism (see Sections 2.3 and 6.4). One-phonon transitions dominate the mechanism at low temperatures, resulting in a linear dependence of k with 7 this follows directly from relation (6.27) when j3/Wl. At higher temperatures, the main contribution comes from Raman processes, leading to a T4 dependence of the rate constant. This predicted T4 temperature dependence for RbBr OH- is analogous to results obtained by Silbey and Trommsdorf [1990] for two-proton transfer in benzoic acid crystals (see Section 6.4). 

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 ... 

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,... 

Other methods of calculating the O N separation dependent proton transfer rates, such as a Fermi Golden Rule approach (Siebrand et al. 1984), can also be employed. In this approach, two harmonic potential wells (e.g., O-H N and, O H-N) are considered to be coupled by an intermolecular term in the Hamiltonian. Inclusion of the van der Waals modes into this approximation involves integration of the coupling term over the proton and van der Waals mode wavefunctions for all initial and final states populated at a given temperature of the system. Such a procedure requires the reaction exothermicity and a functional form for the variation of the coupling as a function of well separation. In the present study, we employ the barrier penetration approach this approach is calculationally straightforward and leads to a clear qualitative physical picture of the proton transfer process. 

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