Stretching vibrations Hamiltonian

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

The local-to-normal transition is governed by the same parameter of Eq. (4.30). The difference is that now the local-to-normal transition occurs simultaneously for the stretching and bending vibrations. The correlation diagram for stretching vibrations is the same as in Figure 4.3. The local-to-normal transition can also be studied for XYZ molecules, for which the Hamiltonian does not have the condition A) = A2 and is... 

The Hamiltonian operator that preserves the symmetry of the molecule can now be constructed. Since all the bonds in Figure 6.1 are equivalent, the most general lowest order Hamiltonian for C-H stretching vibrations of C6Hg is... 

C-H stretching vibrations of C6H6 are therefore characterized by five quantities AH, Ahh, A, h, Xj1//, and h- If. instead of C6H6, one wishes to study C6D6, which has the same symmetry D6h of C6H6, the same Hamiltonian (6.24) will apply except that now the five quantities AH, AHH, X jj and the... 

This Hamiltonian is identical to that of stretching vibration [Eq. (6.7)]. The only difference is that the coefficients A, in front of C, are related to the parameters of the potential, D and a, in a way that is different for Morse and Poschl-Teller potentials. The energy eigenvalues of uncoupled Poschl-Teller oscillators are, however, still given by... 

Cooper, I. L., and Levine, R. D. (1989), Construction of Triatomic Potentials from Algebraic Hamiltonians Which Represent Stretching Vibrational Overtones, J. Mol. Struct. 199,201. 

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

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... 

We start with the six conventional curvilinear vibrational coordinates for ammonia [7,14] three (N-H) bond stretches r1 and three interbond angle (H-N-H) distortions 0i, (i = 1, 2, 3). The molecular vibrational Hamiltonian and, in particular, our analytical PES are built with reference to the molecular totally symmetrical planar configuration. A basic feature of this formalism is its full ZHh symmetrization. For this purpose we define symmetrized curvilinear vibrational coordinates in a somewhat unconventional, complex form ... 

We first suppose Q is the stretching mode Q, which would be appropriate for nearly aU H-bonded systems. In the isolated X-H Y system where the X H distance is q and the X - Y distance Q, with corresponding momenta p = -ih(dldq) and = —ih(dldQ ), the vibrational Hamiltonian may be written as... 

It should be useful at this point to examine the importance of coordinate choice for SCF in the context of specific examples. Roth et al.17 compared SCF results for local and for normal coordinates for nonbending models of water, and its isotopic variants, and of C02. The model Hamiltonian used for the stretching vibrations was of the following form in local coordinates ... 

Very recently, SC-SCF in hyperspherical coordinates was tested for the coupled stretching vibrations of H20 and C02.25 The hyperspherical coordinates are highly collective and are thus expected to be very suitable for SCF. Also, for the states considered here the hyperspherical system provides a good frequency separation between the coordinates involved. For a linear, nonbending molecule A-B-C, the Hamiltonian of the stretching vibrations can be written as follows in hyperspherical coordinates (which are just plane polar coordinates in this case) ... 

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. 

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). 

I. L. Cooper and R. D. Levine, Construction of triatomic potential from algebraic Hamiltonians which represent stretching vibrational overtones,/. Mol. Struct. 191 201 (1989). 

The Hamiltonian will now be summarized for one overtone excited CH oscillator interacting with the N — 1 ring modes. Normal coordinates for the ring Q2,. . . , Q.v and for the overtone excited oscillator Q, are defined by uncoupling the overtone excited oscillator from the ring. The final form for the Hamiltonian then contains terms (both potential and kinetic) which couple the CH stretch mode to the ring modes. The derivation of the vibrational Hamiltonian was presented in Section II.C of Benzene I (103), and we will only summarize the final result here. The vibrational Hamiltonian may be partitioned into the terms... 

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