Stretch-bend polyad

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 + +... 

From 13000 to 16500 cm (polyads 5-9). The A B symmetric stretch-bending bands are stronger than the bending overtones the intensity increases quickly. 

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...

Long progressions of feature states in the two Franck-Condon active vibrational modes (CC stretch and /rani-bend) contain information about wavepacket dynamics in a two dimensional configuration space. Each feature state actually corresponds to a polyad, which is specified by three approximately conserved vibrational quantum numbers (the polyad quantum numbers nslretch, "resonance, and /total, [ r, res,fl)> and every symmetry accessible polyad is initially illuminated by exactly one a priori known Franck-Condon bright state. 

Figure 1. Unzipped polyads (bottom) from the C2H2 A 0° DF spectrum as progressions in 04 (trans bend). Each row has constant vz (CC stretch) and each column has constant 04. Integrated intensity (deperturbed Franck-Condon factor) for each polyad in the 0° DF spectrum, arranged as progressions in 04 (top). Shading indicates the value of V2 for the progression.

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. 

The polyad model for acetylene is an example of a hybrid scheme, combining ball-and-spring motion in a two-dimensional configuration space [the two Franck-Condon active modes, the C-C stretch (Q2) and the tram-bend (Q4)] with abstract motion in a state space defined by the three approximate constants of motion (the polyad quantum numbers). This state space is four dimensional the three polyad quantum numbers reduce the accessible dimensionality of state space from the seven internal vibrational degrees of freedom of a linear four-atom molecule to 7 - 3 = 4. 

Figure 10. Density probability in the ( 3, 2) plane for the eight states of uncoupled HOBr belonging to polyad [v ,P] = [0,14], The Hamiltonian is the Dunham expansion of Eq. (24) with parameters from Table 1 of Ref. 41. (OBr stretch) ranges from to —6.5 to 6.5, and qz (bend) ranges from -5.0 to 5.0. The energy (in cm ) above the quantum mechanical ground state, as well as the good quantum numbers (vj,vy) = (v3,V2), are indicated for each state.

When more than one anharmonic interaction term couples near-degenerate, zero-order levels, a simple vector orthogonalization technique can be used to generate a complete set of the dynamically important (i.e., approximately conserved) polyad quantum numbers (Fried and Ezra, 1987 Kellman, 1990). For example, in acetylene, HC = CH, where the ratios of normal mode frequencies u2 W3 W4 W5 are approximately 5 3 5 1 1, modes 1, 2, and 3 are stretching modes (respectively symmetric CH stretch, and CC stretch, and antisymmetric CH stretch), modes 4 and 5 are bending modes (trans-bend and cis-bend), each polyad is labeled by 3 polyad quantum numbers,... 

The analogous coupling between the antisymmetric stretch and bend is forbidden in the H2O Hamiltonian because of symmetry.) The 2 1 resonance is known as a Fermi resonance after its introduction [50] in molecular spectroscopy. The 2 1 resonance is often very prominent in spectra, especially between stretch and bend modes, which often have approximate 2 1 frequency ratios. The 2 1 coupling leaves unchanged as a polyad number the sum ... 

It is, of course, possible to extend this calculation to obtain, in closed analytical form, the first excited polyad, Vj = 2. The result is shown schematically in Fig. 34. In particular, we notice the direct coupling between pairs of stretching modes in light of the selection rule (for Ub = const) A(u -I-Uj) = 0 and Au, Au = 0, 1. This means that the (initially degenerate) stretches 100), 010) now mix and split under the effect of Mj2. Due to the symmetry under bond exchange, we obtain either symmetric or antisymmetric wavefunctions, as discussed for the one-dimensional case. The difference here is the presence of the bending mode, which is also involved in the coupling scheme induced by the Majorana operator. We can see in both Fig. 34 and Eq. (4.45) that... 

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