Polyad number

If only zero-order states from the same polyad are conpled together, this constitutes a fantastic simplification in the Flamiltonian. Enonnons compntational economies result in fitting spectra, becanse the spectroscopic Flamiltonian is block diagonal in the polyad nnmber. That is, only zero-order states within blocks with the same polyad number are coupled the resulting small matrix diagonalization problem is vastly simpler than diagonalizing a matrix with all the zero-order states conpled to each other. 

Figure Al.2.8. Typical energy level pattern of a sequence of levels with quantum numbers nj for the number of quanta in the symmetric and antisymmetric stretch. The bend quantum number is neglected and may be taken as fixed for the sequence. The total number of quanta (n + n = 6) is the polyad number, which...

Figure Al.2.10. Birth of local modes in a bifurcation. In (a), before the bifiircation there are stable anhamionic symmetric and antisymmetric stretch modes, as in figure Al.2.6. At a critical value of the energy and polyad number, one of the modes, in this example the symmetric stretch, becomes unstable and new stable local modes are bom in a bifurcation the system is shown shortly after the bifiircation in (b), where the new modes have moved away from the unstable syimnetric stretch. In (c), the new modes clearly have taken the character of the anliamionic local modes.

We have seen that resonance couplings destroy quantum numbers as constants of the spectroscopic Hamiltonian. Widi both the Darling-Deimison stretch coupling and the Femii stretch-bend coupling in H2O, the individual quantum numbers and were destroyed, leaving the total polyad number n + +... 

The resonance vector analysis has been used to explore all of the questions raised above on the fate of the polyad numbers in larger molecules, the most thoroughly investigated case so far probably being C2FI2- This molecule has been very extensively probed by absorption as well as stimulated emission pumping and dispersed fluorescence teclmiques [, 53, 70 and 71], the experimental spectra have been analysed in... 

Another question is the nature of the changes in the classical dynamics that occur with the breakdown of the polyad number. In all likelihood there are farther bifiircations. Apart from the identification of the individual polyad-breaking resonances, the bifiircation analysis itself presents new challenges. This is partly becanse with the breakdown... 

The existence of the polyad number as a bottleneck to energy flow on short time scales is potentially important for efforts to control molecnlar reactivity rising advanced laser techniqnes, discussed below in section Al.2.20. Efforts at control seek to intervene in the molecnlar dynamics to prevent the effects of widespread vibrational energy flow, the presence of which is one of the key assumptions of Rice-Ramsperger-Kassel-Marcns (RRKM) and other theories of reaction dynamics [6]. 

For a polyatomic, there are many questions on the role of the polyad number in energy flow from the molecule to the bath. Does polyad number conservation in the isolated molecule inlhbit energy flow to the batii Is polyad number breaking a facilitator or even a prerequisite for energy flow Finally, does the energy flow to the bath increase the polyad number breaking in the molecule One can only speculate until these questions become accessible to fiiture research. 

Since the only angle dependence conies from 0 , and the actions /, L are constant. From this point onwards we concentrate on motion under the reduced Hamiltonian which depends, apart from the scaling parameter y, only on the values of scaled coupling parameter p and the scaled detuning term p. In other words, we investigate the monodromy only in a fixed J (or polyad number N = 2J) section of the three-dimensional quanmm number space. 

Figure 14. Contour plots of the wave functions for DCO in the ground bending states 03 = 0. The numbers on the right-hand side denote the polyad number N = vi + V2 + 03/2. The quantum numbers above each column, (vi, V2, 113), indicate the assignment if the substantial mixing between the modes were not present. For more details see the text and the original publication [17], For ease of the visualization the relevant potential cut is shown in the upper left comer. (Reprinted, with permission of IOP Publishing, from Ref. 8.)...

Let us first neglect the Fermi resonance and analyze the dynamics of the uncoupled systems described by the Dunham expansion alone [Eq. (24)]. Because of the resonance condition co/ 2c0s, quantum states are organized in clumps, or polyads. Each polyad is defined by two quantum numbers, namely the number v of quanta in the uncoupled degree of freedom and the so-called polyad number P ... 

Of course it is possible to excite coherently several bright states that belong to different polyads. To obtain motion of Qi and Pi associated with mode 1, one needs to excite members of polyads belonging to polyad numbers, N = 2v +V2, that differ in steps of 2 in N, whereas to get motion of Q2 and P2 one needs to excite polyads that differ in steps of 1 in N. This offers an interesting possibility for achieving motion of Qi and Pi but not Q2 and P2 or vice versa. 

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

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