Darling-Dennison coupling

Lehmann, K. K. (1983), On the Relation of Child and Lawton s Harmonically Coupled Anharmonic-Oscillator Model and Darling-Dennison Coupling, J. Chem. Phys. 79, 1098. 

Figure la illustrates some of the dynamics in the state space of the [ns = 1, nres = 11, / = 0] polyad illuminated at t = 0 via the (0, 1, 0, 8°, 0 )° ZOBS [2]. The four panels of this figure depict the survival probability of the ZOBS (upper left), die probability of transfer to the two most important first-tier states [coupled by the bend Darling-Dennison interaction upper right (0, 1, 0, 6+2, 2-2)0 and lower left (0, 1, 0, 6°, 2°), and the transfer probability to one of the dynamically most remote dark states in the polyad (1, 0, 0, 0°, 6°)° (lower right), which occurs via a minimum of four anharmonic coupling steps ( 1, 244 followed by 44,55 three times). 

In Section 9.4.12.4 the simplest possible local mode HlqCAL, expressed in terms of four independently adjustable parameters (the Morse De and a parameters, and two 1 1 kinetic and potential energy coupling parameters, Grr and km,), is transformed to the simplest possible normal mode H )oRMAL, which is also expressed in terms of four independent parameters. However, the interrelationships between parameters, based on the 1 1 coupled local Morse oscillator model, result in only 3 independent fit parameters. This paradox is resolved when one realizes that the 4 parameter local-Morse model generates the Darling-Dennison 2 2 coupling term in the normal mode model. However, the full effects of this (A ssaa/16hc)[(at + as)2(a+ + aa)2] coupling term are not taken into account in the local mode model. 

Physically, why does a term like the Darling-Dennison coupling arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian formed by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in terms of particle momenta and displacements, in the representation given by the normal coordinates. Then, in general, it may contain terms proportional to all the powers of the products of the... 

In the example of H2O, we saw that the Darling-Dennison coupling between the stretches led to a profound change in the internal dynamics the birth of local modes in a bifurcation from one of the original low-energy normal modes. The question arises of the possibility of other types of couplings, if not between two identical stretch modes, then between other kinds of modes. We have seen that, effectively, only a very small subset of possible resonance couplings between... 

Besides Fermi interactions, another class of anharmonic couplings among vibrational levels exist and they are often referred to as Darling-Dennison interaction [77]. They typically arise in conjunction with a normal-mode molecular picture, since the associated matrix elements can be written as... 

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... 

Here //normal - 12 12 1 1 normul basis Hnormal is diagonal while Cj and C2 are not, so they break the dynamic symmetry. The parameter A is directly related to the local-mode anharmonicity. Consequently, we expect to obtain strong Darling-Dennison normal-mode couplings in the presence of very anharmonic local modes. 

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. 

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