Resonance condition perturbation theory

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

FIGURE 7.3 Breakdown of perturbation-theory approach for Cun(H20)6 in L-band. The spectrum of the elongated CuOs octahedron (upper trace) is simulated (lower trace) with the approximative resonance condition defined in Equation 5.18. There is no fit of the first hyper-fine line at low field (Hagen 1982a). 

This is a pretty unusual expression, and it should warn us that resonance conditions derived via perturbation theory should always be checked for their validity under actual conditions. Suppose that the zero-field intradoublet splitting, /Lni))-ii,(0)... 

If operturbation theory can be used to find the energy levels. Requiring that Awa = 1) Amj = 0, the resonance condition, to second order of perturbation theory, is readily found to be as follows ... 

From time-dependent perturbation theory of quantum-mechanics, it can be stated that a transition between two states ir) and ) is allowed provided that (Vf 77p ) 0. This takes place if v vq (ie, the resonance condition) and the alternative magnetic field Bi(t) is polarized perpendicularly to the static magnetic field Bo. Concerning a spin 7 = 1 (Fig. lb), similar calculations show that only the single-quantum transitions 0) 1> and -1) 0> (and those in the opposite directions) are allowed in the first approximation and occur at the same frequency, given by equation 3. 

The important qualitative features of the scattering intensity under resonance conditions may be identified using resonance perturbation theory,Most of the important features of the elasticscattering contribution can be obtained from a purely elastic (rigid-surface) theory of the scattering. If we assume all the surface... 

Thus the response of a spatially uniform system in thermodynamic equilibrium is always characterized by translationally invariant and temporaly stationary after-effect functions. This article is restricted to a discussion of systems which prior to an application of an external perturbation are uniform and in equilibrium. The condition expressed by Eq. (7) must be satisfied. Caution must be exercised in applying linear response theory to problems in double resonance spectroscopy where non-equilibrium initial states are prepared. Having dispensed with this caveat, we adopt Eq. (7) in the remainder of this review article. 

A useful tool to be gained from this theory is the predictive power based on the large-molecule limiting case requirement that, for an intramolecular decay pathway to be available to an "isolated" molecule, the condition pfV >> 1 must be met. When pf is small (usually for small molecules or small energy gaps between initial and final states), it is most likely that no final states are in resonance with the initial state and no mixing occurs an external perturbation is required to produce the transition and the process is observed to be collision-induced. Very small values of Pf, therefore, would indicate the possibility of small-molecule behavior. 

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