Anharmonic resonance

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. 

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 most serious obstacle to extracting useful information from the low-resolution DF spectrum is the progressively more severe spectral overlap between separate polyad features as Vjb increases. This is due to two effects the density of Franck-Condon bright ZOBSs increases (approximately as anc width of each polyad increases (owing to the i>4 scaling of the dominant anharmonic resonance experienced by the ZOBS). 

Figure 6. The entire bend Darling-Dennison stack, which contains the ZOBS, pulls away from the other Darling-Dennison stacks as V2 increases. The different stacks are connected by the 3, 245 anharmonic resonance, Although the inter-stack Hy increase as V2 increases, the Hy increase more slowly than A -, thus the mixing angle decreases slowly as vz increases.

Since the perturbations turn on at slightly lower Vj than the onset of changes in fine-structure parameters [5], it seems likely that the perturbations in (0, 32, 0) are due mostly to new oohx 4oobend anharmonic resonances and the rapid changes in fine-structure parameters at (0, 36,0) due mostly to new oohx = 3oobend Coriolis resonances. 

At higher energies, a transition around 13,000 cm 1 above the first vibrational level has been observed when the asymmetric stretching mode starts to interact with the other modes [7]. Further anharmonic resonances may be expected in this region, which are the object of current studies. 

In the energy range 0-16,000 cm-1, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [112]... 

On a single electronic surface, by accumulation of anharmonic resonances that cause the intermixing of the vibrational levels within the superpolyads. 

M. Quack In answer to the question by Prof. Kellman on exact analytical treatments of anharmonic resonance Hamiltonians, I might point out that to the best of my knowledge no fully satisfactory result beyond perturbation theory is known. Interesting efforts concern very recent perturbation theories by Sibert and co-workers and by Duncan and co-workers as well as by ourselves using as starting point internal coordinate Hamiltonians, normal coordinate Hamiltonians, and perhaps best, Fermi modes [1]. Of course, Michael Kellman himself has contributed substantial work on this question. Although all the available analytical results are still rather rough approximations, one can always... 

The dynamical interpretation is that these clusters represent the first tier of the energy flow. Time-resolved predictions can be performed by Fourier transforming the energy picture, thus simulating a coherent excitation of the whole cluster. The stronger and more numerous the anharmonic resonances within the cluster, the faster the energy flow. 

Despite the complication which resonances introduce into the analysis of a spectrum and the theoretical treatment of the hamiltonian, when they can be analysed they often give valuable information on the force field which cannot be obtained directly in the absence of a resonance. We consider briefly the two commonest types of resonance interaction, Fermi (or anharmonic) resonance and Coriolis resonance, to illustrate this point. 

For example, Q1Q2, the dimensionless coupling term for the ubiquitous 2 1 anharmonic resonance (that arises from the = Q1Q2 term in the Taylor series expansion of V(Qi, Q2)), becomes... 

As a result, the early time dynamics of acetylene could be affected by numerous near resonant anharmonic interactions. Nine such anharmonic resonances are well characterized and known to be dynamically relevant. One needs a framework to relate the Fourier amplitudes and phases of the time-dependence of any observable quantity to the expectation values of each of the resonance operators in order to establish the relative importance of each of the resonances. 

The 9 important anharmonic resonances destroy all of the 7 normal mode quantum numbers (iq, rq, v3, u4, I4, v3, I5), but three approximately conserved polyad quantum numbers remain,... 

Figure 30. Construction of the 1 2 CH stretch/CC bend anharmonic resonance in benzene.

Although we will not discuss in detail this particular aspect of anharmonic resonances, it is important to note that Darling-Dennison couplings are automatically included by the action of the Majorana operator. A practical way to convince ourselves of this inclusion is to diagonalize (either numerically or in closed form) the Hamiltonian matrix explicitly for the first two polyads of levels and then to convert, in normal-mode notation, the vibrational states obtained. As discussed in Ref. 11, the Hamiltonian (4.38) can also be written (neglecting Cj2 and Cj2 interactions) as... 

Higher order terms are often necessary and the operator for anharmonic resonances may be written as... 

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