Hamiltonian anharmonic resonance

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... 

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]... 

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... 

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. 

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... 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

As a consequence of the above equations, the full Hamiltonian describing the fast mode coupled to the H-bond bridge (via the strong anharmonic coupling theory) and to the bending mode (via the Fermi resonance process) may be written within the tensorial basis (222) according to [24] ... 

We shall assume that the conditions for Fermi resonance are satisfied in a free molecule, i.e. that there are two (for the sake of simplicity) nondegenerate vibrations with the frequencies fli and ily, for which, for instance, 2il m In this case, when taking the intramolecular anharmonicity (with the constant T) into account, it is necessary to add to the Hamiltonian (6.11) the sum of two terms Hq(C) and Hp(B,C), where... 

Consider a many Fermi resonance system of coupled anharmonic oscillators (10). In an analytical treatment of the energy flow, it is convenient to use an algebraic representation for the Hamiltonian. The Hamiltonian H = // 4- V consists of an unperturbed portion,... 

Here we describe exact quantum calculations on a model many-dimensional Fermi resonance system, using the methods explained in Chapters 1 and 3 (11). As in the previous section of this chapter, the Hamiltonian H = H0 + V consists of an unperturbed portion and a coupling term. As a simple means to model 5 = 6 anharmonic oscillators the normal... 

These matrix elements are equivalent to those of Eq. (3.124), apart from anharmonic contributions of the order of v/N. So we see that the extended Majorana operator has the required effect on the states involved in the resonance mechanism. At the same time > SB does not preserve the coupled 65 (2) symmetry in other words, + Vg is not conserved anymore. Consequently, the block-diagonal structure of the Hamiltonian operator is destroyed and the numerical diagonalization of... 

IVR results from nondiagonal couplings among the anharmonic modes of the effective Hamiltonian The effects of these couplings are most significant when resonances among the modes take place. The locations where nonlinear resonances can occur are given by... 

The vibration-rotation spectra and/or the rotational spectra in excited vibrational states provide the af constants and, when all the a/ constants are determined, the equilibrium rotational constants can be obtained by extrapolation. This method has often been hampered by anharmonic or harmonic resonance interactions in excited vibrational states, such as Fermi resonances arising from cubic and higher anharmonic force constants in the vibrational potential, or by Coriolis resonances. Equihbrium rotational constants have so far been determined only for a limited number of simple molecules. To be even more precise, one has further to consider the contributions of electrons to the moments of inertia, and to correct for the small effects of centrifugal distortion which arise from transformation of the original Hamiltonian to eliminate indeterminacy terms [11]. Higher-order time-independent effects such as the breakdown of the Bom-Oppenheimer separation between the electronic and nuclear motions have been discussed so far only for diatomic molecules [12]. 

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