Polyad structure

In a smaller molecule (HCP), these diagnostically important changes in vibrational resonance structure are manifest in several ways (i) the onset of rapid changes in molecular constants, especially B values and second-order vibrational fine-structure parameters associated with a doubly degenerate bending mode (ii) the abrupt onset of anharmonic and Coriolis spectroscopic perturbations and (iii) the breakup of a persistent polyad structure 15]. 

In the present contribution the interpretation of the energy-level structure of quasi-one-dimensional quantum dots of two and three electrons is reviewed in detail by examining the polyad structure of the energy levels and the symmetry of the spatial part of the Cl wave functions due to the Pauli principle. The interpretation based on the polyad quantum number is applied to the four electron case and is shown to be applicable to general multi-electron cases. The qualitative differences in the energy-level structure between quasi-one-dimensional and quasi-ta>o-dimensional quantum dots are briefly discussed by referring to differences in the structure of their internal space. 

Numerically, this is most easily analyzed by rewriting the Dunham expansion of Eq. (24) in terms of coordinates, which are adapted to the polyad structure of the spectmm. One defines new sets (/ , / ), Jp, //>), and Jo, v[/o) of conjugate action-angle-like coordinates, according to... 

The finer structure within each feature state corresponds to the dynamics of the Franck-Condon bright state within a four-dimensional state space. This dynamics in state space is controlled by the set of all known anharmonic resonances. The state space is four dimensional because, of the seven vibrational degrees of freedom of a linear four-atom molecule, three are described by approximately conserved constants of motion (the polyad quantum numbers) thus 7-3 = 4. 

The fractionation patterns exhibited % successive members of a progression of polyads (along 02, CC stretch, or along v4, trans-bend) provide a surveyor s map of IVR. One can look at the 1VR trends and see whether the multiresonance model expressed in the H nres (1 polyads provides a qualitative or quantitative representation of the fractionation patterns. The dynamics of even a four-atom molecule is so complicated that, unless one knows what to look for, one can neither identify nor explain trends in the dynamics versus V2 or u4 or Evib- Moreover, by defining the pattern of the IVR and how this pattern should scale with V2, v4, or EVib, the H res / polyad model may make it possible to detect a disruption of the pattern. Such disruptions could be due to a change in the resonance structure of the exact H near some chemically interesting topographic feature of the V(Q), such as an isomerization saddle point. 

We know that all vibrational levels can be gathered into polyads or clusters. The generic structure of each cluster is presented in die first figure, according to Ref. 1. 

Nucleic acids can form complex structures that consist of more than two strands. Recently, the interest in structures and functions of the DNA bases polyads has significantly increased. It has been shown that guanine tetrads are vital components of many biologically important processes. 

An additional series of calculations has been carried out for the complexes of the tetrads with the metal ions. It has been concluded that the stability of such tetrads is controlled by the presence of metal ions. In this presentation, we demonstrate that such cations dominate the H-bonding patterns that govern the structures of the DNA bases polyads. 

Base Polyad Motifs in Nucleic Acids - Biological Significance, Occurrence in Three-Dimensional Experimental Structures and Computational Studies (M. Meyer J. Stthnel)... 

Figure 9.13 displays the phase space structure of local and Hnormal for the N = vs + va = vr + vl = 3 [I = (N + l)/2 = 2]f polyad of H2O. Just as HloCAL and H qRMAL provide identical quality representations of the observed spectrum, so too do Wlocal and Tinormal- The phase space structures displayed in parts (a), LOCAL, and (b), NORMAL, of Fig. 9.13 are equivalent. The appearance of qualitatively different structures in the LOCAL and NORMAL representations is largely due to the mapping of the information onto an Iz, ip (or Iz, tp) planar rectangle rather than a polyad phase sphere. As shown in parts (c) and (d) of Fig. 9.13, the structures from parts (a) and (b) differ only by a rotation of the phase sphere by 7t/2 about the y axis. 

The phase space structures for two identical coupled anharmonic oscillators are relatively simple because the trajectories lie on the surface of a 2-dimensional manifold in a 4-dimensional phase space. The phase space of two identical 2-dimensional isotropic benders is 8-dimensional, the qualitative forms of the classifying trajectories are far more complicated, and there is a much wider range of possibilities for qualitative changes in the intramolecular dynamics. The classical mechanical polyad 7feff conveys unique insights into the dynamics encoded in the spectrum as represented by the Heff fit model. 

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

Figure 9.18 Surfaces of section for the HCCH [Nj, l] = [4,0] and [8,0] polyads. The surface section for the [4,0] polyad shows that all of phase space is divided between cis-bend and trans-bend normal modes. The phase space structure for the [8, 0] polyad contains large scale chaos as well as at least two new qualitative behaviors (from Jacobson, et al., 1999).

The surface of section for the [8,0] (Ka = 2.5) polyad contains several structures that are not present for the [4,0] polyad. Most importantly, a significant fraction of the surface of section is filled with the apparently random dots symptomatic of chaos. The fraction of phase space organized by the periodic trans-and eis-bending vibrations is considerably reduced relative to that for the [4,0] polyad. New closed loops have appeared (centered at Jf, ss 1.7, ipb = 7r and Jb = 0, ipb 7r/2), and the qualitatively new trajectories represent the first appearance of localized motions in which the two ends of the molecule are... 

Figure 9.19 Overview of the phase space and configuration space dynamics associated with the HCCH [JV = 22, l = 0] polyad. The top four plots are surfaces of section for four energies within the polyad. Only simple structures are found near the bottom (local bender) and top (counter-rotators) of the polyad. Chaos dominates at energies near the middle of the polyad, but two classes of stable trajectories coexist with chaos. The bottom four plots show the coordinate space motions of the left- and right-hand H-atoms that correspond to the two different periodic orbits (from Jacobson and Field, 2000b).

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