Local-mode Hamiltonian

Dennison coupling produces a pattern in the spectrum that is very distinctly different from the pattern of a pure nonnal modes Hamiltonian , without coupling, such as (Al.2,7 ). Then, when we look at the classical Hamiltonian corresponding to the Darling-Deimison quantum fitting Hamiltonian, we will subject it to the mathematical tool of bifiircation analysis [M]- From this, we will infer a dramatic birth in bifiircations of new natural motions of the molecule, i.e. local modes. This will be directly coimected with the distinctive quantum spectral pattern of the polyads. Some aspects of the pattern can be accounted for by the classical bifiircation analysis while others give evidence of intrinsically non-classical effects in the quantum dynamics. 

Local-mode Hamiltonian for linear triatomic molecules... 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983 Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-... 

For bent triatomic molecules one can easily construct a local mode Hamiltonian whose eigenvalues reproduce the spectrum ... 

This Hamiltonian is diagonal in the local mode basis [Eq. (4.43)] with eigenvalues... 

The procedure for studying tetratomic molecules is identical to that followed in the study of diatomic and triatomic molecules. One begins with a local-mode Hamiltonian... 

The local mode Hamiltonian (5.16) includes only the operator C[2 that is, interactions of the Casimir type between bonds 1 and 2. One may wish, in some cases, to include also interactions of this type between bonds 1 and 3, C13, and 2 and 3, C23. These can be included by diagonalizing the secular matrix obtained by evaluating the matrix elements of C13 and C23 in the basis (5.4). These matrix elements are given by (5.15a) and (5.15b). 

Benjamin, I., van Roosmalen, O. S., and Levine, R. D. (1984), A Model Algebraic Hamiltonian for Interacting Nonequivalent Local Modes with Application to HCCD and H12C13CD, J. Chem. Phys. 81, 3352. 

Let us consider first the decay of a strongly excited local mode due to simultaneous emission of k > 2 phonons. The Hamiltonian of the system under consideration is... 

H2O in its electronic ground state is best described by a local mode expansion (Child and Halonen 1984 Child 1985 Halonen 1989). For the purpose of this chapter it suffices to consider a simple two-dimensional model in which the bending angle is frozen at its equilibrium value 104° and the oxygen atom is assumed to be infinitely heavy. For an exact three-dimensional treatment see Bacic, Watt, and Light (1988), for example. The approximate two-dimensional Hamiltonian reads... 

Our local mode model includes the OH-stretching and SOH-bending modes in the vibrational Hamiltonian [58]. The zeroth order Hamiltonian includes the two vibrational modes as uncoupled Morse oscillators according to... 

A useful model system, representing the Interaction of two "local mode" oscillators as might occur in a light-heavy-light system, such as water, is described by fbe Hamiltonian... 

A model atom approximation is permitted if all of the stretching vibrations of the molecule are ascribed to the local-mode limit. In the normal-mode limit, using the effective Hamiltonian of the whole molecule is preferable, as was shown in the example of CH3CI and CH3F. 

An excellent example of non-accidental intramolecular resonance is provided by a polyatomic molecule with two chemically identical bonds. The spectrum and dynamics of such a molecule may equally well be described by an effective Hamiltonian expressed in basis sets corresponding to either of two opposite limiting cases normal mode (H -,RMAL) and local mode (Hlocal)-... 

It turns out that the language of normal and local modes that emerged from the bifurcation analysis of the Darling-Dennison Hamiltonian is not sufficient to describe the general Fermi resonance case, because the bifurcations are qualitatively different from the normal-to-local bifurcation in figure Al.2.10. For example, in 2 1 Fermi systems, one type of bifurcation is that in which resonant collective modes are bom [54]. The resonant collective modes are illustrated in figure A12.11 their difference from the local modes of the Darling-Dennison system is evident. Other types of bifurcations are also possible in Fermi resonance systems a detailed treatment of the 2 1 resonance can be found in [44]. 

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

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