Golden Rule calculations

Figure 29.14 Temperature dependence of the rate constants of double proton transfer in syn-sesquinorbornene disulfone-do, -d, and -c/2, illustrated in the insert, evaluated for R, 2=CH2. The symbols represent observed rate constants [56, 57] and the solid lines the results of two-dimensional semi-empirical Golden Rule calculations [57].

Lan Lane, I.C., Howie, W.H., Oir-Ewing, A.J. The UV absorption of CIO Part 2. Predissociation of the A IIq state studied by ab-initio and Fermi golden rule calculations, Phys. Chem. Chem. Phys. 1 (1999) 3087-3096. 

Golden Rule calculations using eq. (15.101) with the reorganisation energy given by eq. (15.100) and the parameters indicated in Figure 15.24 give the rates presented in... 

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,... 

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... 

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... 

For typical outer-sphere exchanges at ordinary temperatures, it seems probable that the original assumption of Hush and of Marcus that barrier penetration is a comparatively minor effect is correct. Moreover> it is, in a particular case, quite simple to calculate. The more general questions to which we do not yet have an answer are how adequate is the Golden Rule approach in discussing tunnelling, and, in particular, what would be expected for systems strictly remaining on one surface (electronically adiabatic) A number of fundamental issues involved here have been discussed in a recent series of papers (42-45). 

Finally, I refer back to the beginning of this paper, where the assumption of near-adiabaticity for electron transfers between ions of normal size in solution was mentioned. Almost all theoretical approaches which discuss the electron-phonon coupling in detail are, in fact, non-adiabatic, in which the perturbation Golden Rule approach to non-radiative transition is involved. What major differences will we expect from detailed calculations based on a truly adiabatic model—i.e., one in which only one potential surface is considered [Such an approach is, for example, essential for inner-sphere processes.] In work in my laboratory we have, as I have mentioned above,... 

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 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. 

The starting point for all calculations of transition probabilities is the well-known formula (22) sometimes called the Golden Rule. It expresses the transition probability per unit time A in terms of the density of final states... 

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. 

The golden rule is applicable to calculate the rate constant of electron tunneling provided that electron tunneling does not violate the equilibrium energy distribution in the initial state. Thus, for all the initial states making a considerable contribution to the sum on the right-hand side of eqn. (18), the following condition must be fulfilled... 

Thus, tunneling can be treated as a quantum transition L R. The probability of tunneling is directly connected with the value of the coefficient b and can be calculated with the use of the golden rule,... 

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... 

The next step is to treat tunneling as a perturbation. Following this idea, the transition rates TAA from the state A to the state A are calculated using the Fermi golden rule... 

For interacting electrons the calculation is a little bit more complicated. One should establish the relation between many-particle eigenstates of the system and single-particle tunneling. To do this, let us note, that the states /) and i) in the golden rule formula (83) are actually the states of the whole system, including the leads. We denote the initial and final states as... 

The perturbation H-Hq can be used in Fermi s Golden rule to calculate various transition rates. 

While the time resolved measurements (for example, see Figure 5-4) do not, in this case, mandate a serial IVR/VP mechanism, they are consistent with it in terms of the model calculations employing Fermi s Golden Rule for IVR and RRKM theory for VP. 

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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