Isolated resonance

Let T as a superscript on a matrix indicate the transposed matrix and the dagger the Hermitian conjugate, and hence, = (AT) = (A )T. Also, let I 

For multichannel scattering where there are two or more open channels, the S matrix is a true matrix with elements Sy and the cross section for the transition from channel i to channel j is proportional to 5y - Sy 2. The symmetry of collision processes with respect to the time reversal leads to the symmetric property of the S matrix, ST = S, which, in turn, leads to the principle of detailed balance between mutually reverse processes. The conservation of the flux of probability density for a real potential and a real energy requires that SSf = SfS = I, i.e., S is unitary. For a complex energy or for a complex potential, in general, the flux is not conserved and S is non-unitary. 

In the following, N0 denotes the order of the S matrix, i.e., the number of open channels of a particular symmetry under consideration. 

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The theory of isolated resonances is well understood and is discussed below. Mies and Krauss [79, ] and Rice [ ] were pioneers m treating unimolecular rate theory in temis of the decomposition of isolated Feshbach resonances. 

A microcanonical ensemble of isolated resonances decays according to... 

Excitation of an Isolated Resonance Embedded in an Uncoupled Continuum... 

Equation (51) has a clear physical interpretation. Recalling the lineshape for a single excitation route, where fragmentation takes place both directly and via an isolated resonance [68], p oc (e + q)2/( 1 + e2), we have that 8j3 is maximized at the energy where interference of the direct and resonance-mediated routes is most constructive, e = (q I c(S )j2. In the limit of a symmetric resonance, where q —> oo, Eq. (51) vanishes, in accord with Eq. (53) and indeed with physical intuition. The numerator of Eq. (51) ensures that 8]3 has the correct antisymmetry with respect to interchange of 1 and 3 and that it vanishes in the case that both direct and resonance-mediated amplitudes are equal for the one-and three-photon processes. At large detunings, e —> oo, and 8j3 of Eq. (51) approaches zero. 

The physical significance of Eq. (53) is clear. At an isolated resonance the excitation and dissociation processes decouple, all memory of the two excitation pathways is lost by the time the molecule falls apart, and the associated phase vanishes. The structure described by Eq. (53) was observed in the channel phase for the dissociation of HI in the vicinity of the (isolated) 5sg resonance. The simplest model depicting this class of problems is shown schematically in Fig. 5d, corresponding to an isolated predissociation resonance. Figures 5e and 5f extend the sketches of Figs. 5c and 5d, respectively, to account qualitatively for overlapping resonances. 

Continuum excitation, coherence spectroscopy isolated resonance ... 

Correlation matrix, linear thermodynamics, regression theorem, 17-20 Coupled cluster (CC), ab initio calculations, P,T-odd interactions, 254-259 Coupled continuum, two-pathway excitation, coherence spectroscopy isolated resonances, 168-169 structureless excitation, 167 CPT theorem ... 

Two-dimensional constant matrix, transition state trajectory, white noise, 203-207 Two-pathway excitation, coherence spectroscopy atomic systems, 170-171 channel phases, 148-149 energy domain, 178-182 extended systems and dissipative environments, 177-185 future research issues, 185-186 isolated resonance, coupled continuum, 168-169... 

Uncoupled continuum, two-pathway excitation, coherence spectroscopy isolated resonances, 167-168 structureless continuum, 166... 

The F + H2 — HF + FI reaction is one of the most studied chemical reactions in science, and interest in this reaction dates back to the discovery of the chemical laser.79 In the early 1970s, a collinear quantum scattering treatment of the reaction predicted the existence of isolated resonances.80 Subsequent theoretical investigations, using various dynamical approximations on several different potential energy surfaces (PESs), essentially all confirmed this prediction. The term resonance in this context refers to a transient metastable species produced as the reaction occurs. Transient intermediates are well known in many kinds of atomic and molecular processes, as well as in nuclear and particle physics.81 What makes reactive resonances unique is that they are not necessarily associated with trapping... 

The proposed ID TOCSY-NOESY experiment is illustrated by the assignment of NOEs from anomeric protons H-lc and H-ld of the polysaccharide 1. Because the resonances of H-lc and H-ld overlapped, this assignment was not possible from a ID NOESY spectrum as shown in fig. 3(b). Although these protons differed in their chemical shifts, it was not possible to separate them by chemical-shift-selective filtration because of the very fast spin-spin relaxation of backbone protons (20-50 ms) in this polysaccharide. Instead, a ID TOCSY-NOESY experiment was performed in which the initial TOCSY transfer from an isolated resonance of H-2c was followed by a selective NOESY transfer from H-lc. The ID TOCSY-NOESY spectrum (fig. 3(c)) clearly separated NOE signals of the H-lc proton from those originating from the H-ld proton and established the linkage Ic —> 6a. 

As shown by Fig. 8.14, in most Stark spectra above the classical ionization limit there is never one isolated resonance but, more often, an irregular jumble of them. For example, in Fig. 8.15 we show the observed22 and calculated23 Na spectra near the ionization limit in a field E = 3.59 kV/cm.22 The experimental spectrum of Fig. 8.15(a) was obtained by Luk et al.22 by exciting a Na beam with two simultaneous dye laser pulses from the 3s1/2 to 3p3/2 state and then to the ionization limit. Both lasers were polarized parallel to the field, and the ions... 

Analysis of isolated resonances with weak background... 

The derivation of the eigenphase sum for the Simonius S matrix, SSim(E), is straightforward by generalizing the procedure for an isolated resonance shown in Section 2.2.2. Compared with the Breit-Wigner S matrix, Sm(E), the matrix Sr in Eq. (34) is now replaced by SPN, or the product of matrices Sv/ each having the same apparent form as Sr but with different resonance parameters and a different projection matrix. Since the determinant of the... 

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