Zero-order resonance

Table 5 summarizes the absolute values of the zero order resonance energies er w,D), in units of C, as computed from Eq. (63). Since it is unlikely that the zero-order energies are drastically modified by the small corrections to the isotropic nearest-neighbor Heisenberg Hamiltonian, the expected energy ordering would be... 

The DWBA and available complex coordinate results for -Im(E are given in Figure 3. The zero-order resonance energies are given by... 

Figure 1.13 Resonant scattering (cross sections and phase shifts). Plot of T(E)p and of the phase shift 3( ) determined from 5(E) as a function of the energy (79). (a) V = 0, (b) V = 0.2, and (c) V = 0.8. In the three cases the width of the zero-order resonance is r = 0.8 (arbitrary units).

The matrix representation of the non-dissipative part of (66) is diagonal in the basis of the zero-order resonances. V) characterizes the couplings between the resonances and the continuum. Index a in Va) (21) was suppressed because there is only one decay channel. For the sake of simplicity all these couplings are assumed to be equal ... 

To is the partial width of the zero-order resonances. The Green function [not to be confused with the Green operator (2)] was chosen in the form... 

Figure 2. Giant resonances (cross sections and phase shifts). The zero-order energies of the resonances are Ef = i — 1 for i = 1,2 6. All the zero-order resonances have the same partial width To = 2 and there is no direct coupling between these resonances. The parameters of the Green function (31) are (a) Ethr = 0 (b) jE thr = —10 (c) J thr = —100. In all cases 7] = 0.1 and Eend = 100 (arbitrary units).

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

Resonance states in the spectra, which are assignable in temis of zero-order basis will have a predominant expansion coefficient c.. Hose and Taylor [ ] have argued that for an assignable level r /,j>0.5... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Here 2tt Aft = trm is imposed to eliminate the effect by the zero-order interaction of the RF field, otherwise a term containing Ix will remain and it will distort the longitudinal magnetization, transverse magnetization, as well as the BSPS of the 13C . For the on-resonance condition, 5 = 0, all the higher-order average Hamiltonians vanish since [7f(/ ), = 0 for arbitrary l and... 

The corresponding zeroth-order quantum-mechanical results are obtainable by regarding the vector of actions I as having components which, in units of % are integers. Thus, zero-order quantum-mechanical states that are compatible with the resonance condition (i.e., two separable states n and iT such that n - n = m) are degenerate,... 

Prior knowledge allows to include fixed relations between some of the four parameters (amplitude, phase, frequency position, peak width) describing a symmetrical well-shaped resonance. Signal ratios, chemical shift difierences, linewidth relations and zero-order phase relations can be included. The reduction of the number of unknown parameters leads to a reduced calculation time, better convergence behaviour and improved results. However, the assumptions made to include the prior knowledge must be validated for each experiment. Differences between the parameter values set by the prior knowledge and the actual parameters could lead to systematic errors. 

What is a polyad A polyad is a subset of the zero-order states within a specifiable region of Evib (typically a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyad quantum numbers. For the C2H2S0 state, a procedure proposed by Kellman [9, 10] identifies the three polyad quantum numbers... 

The combination of low-resolution and spectral unzipping into noninteracting polyads enables systematic, model-free surveys of deperturbed Franck-Condon factors, deperturbed zero-order energy levels, and trends in intramolecular vibrational redistribution (IVR) rates and pathways [3]. The H[ res,/i polyad model permits extraction of the most important resonance strengths directly from fits to a few polyads [6-8]. Once these anharmonic... 

R. W. Field Each acetylene polyad contains zero-order states that are easily accessible via plausible direct or multiple-resonance A XAU — X Franck-Condon pumping schemes. Each Vib 16,000... 

As described in the main text of this section, the states of systems which undergo radiationless transitions are basically the same as the resonant scattering states described above. The terminology resonant scattering state is usually reserved for the case where a true continuum is involved. If the density of states in one of the zero-order subsystems is very large, but finite, the system is often said to be in a compound state. We show in the body of this section that the general theory of quantum mechanics leads to the conclusion that there is a set of features common to the compound states (or resonant scattering states) of a wide class of systems. In particular, the shapes of many resonances are very nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. It is the ubiquity of these features in many atomic and molecular processes that we emphasize in this review. 

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