Beyond the golden rule

Indeed, both the exponential temperature dependence that characterize the Orbach process and the T behavior associated with the Raman type process have been observed in spin lattice relaxation  

In many cases, reliable theoretical descriptions of multi-rate processes can be obtained by using master equations in which individual rates are obtained from golden-rule type calculations (see Sections 8.3.3 and 10.4). A condition for the validity of such an approach is that individual rate processes will proceed independently. For example, after evaluating the rates ki 2 and 2 1, Eqs (12.55a) and (12.55b), a description of the overall dynamics of the coupled two-level system by the kinetic scheme 1 2 relies on the assumption that after each transition, say 

Other cases that require going beyond the golden-rule involve transitions which by their nature are of high order in the interaction. Processes studied in conjunction with nonlinear spectroscopy (see Section 18.7) are obvious examples. 

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D. G. Evans, A. Nitzan, and M. A. Ratner (1998) Photoinduced electron transfer in mixed valence compounds Beyond the golden rule regime. J. Chem. Phys. 108, p. 6387... 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

Formally, the theory can be extended beyond the Golden Rule limit by considering higher order terms in the perturbation expansion of the rate in V. Starting with the Liouville equation for the density matrix, one can derive reduced equations of motion for the state populations (generalized master equations), formally exact for arbitrary coupling [88, 99, 295, 296],... 

Mechanism and control of molecular energy flow. Theor. Chem. Acc. 109,53. Gruebele, M. and R. Bigwood (1998). Molecular vibrational energy flow beyond the Golden Rule. /ni. Rev. Phys. Chem. 17,91. 

Positive rules go a step beyond negative golden rules, requiring right conduct, even exemplary, conduct toward all. Consider, for example, the Christian golden rule, a positive golden rule ... 

Approaches beyond classical theory were clearly needed, and in the 1960s and 1970s two largely complementary quantum-mechanical approaches— the golden rule and superexchange—were developed. Modem computing power has essentially fused these two theories and greatly enhanced their reach and realism. 

The golden-rule rate expressions obtained and discussed above are very useful for many processes that involve transitions between individual levels coupled to boson fields, however there are important problems whose proper description requires going beyond this simple but powerful treatment. For example, an important attribute of this formalism is that it focuses on the rate of a given process rather than on its full time evolution. Consequently, a prerequisite for the success of this approach is that the process will indeed be dominated by a single rate. In the model of Figure 12.3, after the molecule is excited to a higher vibrational level of the electronic state 2 the relaxation back into electronic state 1 is characterized by the single rate (12.34) only provided that thermal relaxation within the vibrational subspace in electronic state 2 is faster than the 2 1 electronic transition. This is... 

Caution This is not the time to practice just the Golden Rule, that is, do unto others as you would have them do unto you. Also apply the Platinum Rule, which is do unto others as they would have done unto them. Make sure you understand what your client, owner, or customer values before you decide to give them something that is beyond the project s scope. Your personal preference is a low priority. As noted by novelist Charles Dudley Warner, The exeellence of a gift lies in its appropriateness rather than its value. And, in mentioning the Platinum Rule, I am emphasizing appropriateness. 

Yet Crowley went significantly beyond the claim to scientific validity made by his immediate predecessors in the Golden Dawn and positioned the A.A. as a kind of official sanctioning body for experimental validity and for scientific training. Crowley laid down a set of numbered rules governing experiments, proclaiming, for example ... 

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... 

If the electronic coupling is further increased beyond the range of applicability of the local Golden Rule, the reaction in the normal region becomes adiabatic. The situation in the inverted region is radically different. Here the transfer is nonadiabatic for any value of electronic coupling and... 

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