Hamiltonian modes coupling

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. 

Let us perform the harmonic approximation on the H-bond bridge. Then, the total Hamiltonian (12) of the system formed by the high frequency mode coupled to the H-bond bridge takes the form ... 

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] ... 

The conversion of muonium (y+e ) to its antiatom antimuonium (y e+) would be an example of a muon number violating process,2 and like neutrinoless double beta decay would involve ALe=2. The M-M system also bears some relation to the K°-K7r system, since the neutral atoms M and M are degenerate in the absence of an interaction which couples them. In Table III a four-Fermion Hamiltonian term coupling M and M is postulated, and the probability that M formed at time t=0 will decay from the M mode is given. Present experimental limits22 23 for the coupling constant G are indicated and are larger than the Fermi constant Gp. 

The Hamiltonian contains coupling among the normal modes and to obtain eigenvalues and eigenfunctions requires either an approach based on PT or VT. 

Recently, Carter and Handy have addressed this very effectively by incorporating the approach taken in MULTIMODE into the Reaction Path Hamiltonian (RPH) [60]. In this approach one special, large amplitude mode is singled out and the u-mode coupling idea is applied to the normal modes orthogonal to this mode. The kinetic energy operator is somewhat complex and is given elsewhere [60]. This version of MULTIMODE is denoted MULTIMODE-RPH or abbreviated as MM-RPH. 

As the energy is increased, mode couplings come into play, and the spectrum may lose its simple regular structure (Nordholm and Rice, 1974). If the couplings are extensive, a zero-order Hamiltonian and basis cannot be found to represent the for these states. The spectrum for these states is irregular without patterns. For the most... 

The vibrational adiabatic approximation is hardly justified because the reaction channel is curved. This means that motion along s couples with some vibrational modes, and also the vibrational modes couple among themselves. Therefore, we have to use the non-adiabatic theory, and this means that we need coupling coefficients B (p. 906). The Miller-Handy-Adams reaction path Hamiltonian theory gives the following expression for the Bkk (Fig. 14.7) ... 

A problem that sometimes occurs in reaction-path Hamiltonians, especially for bend potentia1s, is the bifurcation of the reaction path. This occurs when a harmonic frequency becomes imaginary, and for the Raff surface this occurs for bends on both sides of the saddle point. initio calculations can be helpful in determining if the bifurcation is an artifact of the form of the analytic potential function or if it is present in the actual system. When the MEP bifurcates it is probably best to base the RPH on a reference path centered on the ridge between two equivalent MEP s. l This requires extra effort when computing vibrational energy levels since the vibrational potential becomes a double-minimum one, but it probably reduces mode-mode coupling, which (see Sect. 2) is hard to treat accurately. 

Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts. 

Spectra that can be assigned approximate quantum numbers are called regular spectra. As discussed above, when the energy is increased mode couplings become important and the spectrum may lose its patterns and progressions associated with a regular structure. If a zero-order Hamiltonian and basis... 

Now the physical picture of multidimensional tunneling obtained from the above analysis is explained by taking a symmetric mode coupling model potential [30]. The Hamiltonian is given by... 

Then the scaled Hamiltonian with the symmetric mode coupling (SMC) potential is given by... 

The scaled Hamiltonian with the antisymmetric mode coupling (ASMC) potential is defined by... 

In the new coordinates, the bath Hamiltonian takes a hierarchical form The effective modes Xg couple directly to the electronic subsystem, while the remaining (residual) Nb —Ngs bath modes couple in turn to the effective modes. The new bath Hamiltonian Hb of Eq. 15.8 can thus be split as follows ... 

The analogous coupling between the antisyimnetric stretch and bend is forbidden in the H2O Hamiltonian because of syimnetry.) The 2 1 resonance is known as a Femii resonance after its introduction [ ] 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 couplmg leaves unchanged as a poly ad number the sum ... 

There has been a great deal of work [62, 63] investigating how one can use perturbation theory to obtain an effective Hamiltonian like tlie spectroscopic Hamiltonian, starting from a given PES. It is found that one can readily obtain an effective Hamiltonian in temis of nomial mode quantum numbers and coupling. 

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