Coupled oscillator interaction

Binding of related symmetrical dyes (naphthalene derivatives) to apohemoglobin resulted in different CD-Cotton effects in each case. As in the case of heme (see above), a coupled oscillator interaction with Ti-Ti transition moments of aromatic side chains of the protein was suggested, but no calculations were reported (194). The binding sites of the dyes used have not been identified and may include the heme pocket in some cases (195). The induced Cotton effects appear to be very sensitive to the local protein environment of the dye chromophores (194). 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

Let us consider a system of two classical oscillators with Kerr nonlinearity. Both oscillators interact with each other by way of a linear coupling moreover, they are pumped by external time-dependent forces. The Hamiltonian for the system is given by... 

If the interaction parameter a is switched on, the system of coupled oscillators (26)-(29) manifests a rich variety of spectacular behavior. Below, we concentrate on the most interesting ones. First, we answer the question as to how the attractors in Fig. 20 change when both oscillators interact with each other. 1... 

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

The coupled oscillator mechanism involves coupling between the transition moments of two adjacent chromophores these must have a spatial relationship in which the interacting moments are non-parallel. (If the moments are parallel, only the absorption spectrum is affected hyperchromism is shown if the chromophores are arranged in a head-to-tail mode, and hypochromism if they are stacked one over the other in the head-to-head mode 20). By knowing the optical activity, it is possible to deduce the relative configuration between the given chromophores, and vice versa. 

Optical activity arises from the coupling of given electric-allowed transitions with a chiral orientation (coupled oscillator mechanism or two-electron mechanism) or from the electric or magnetic moments of a transition being pertubed by a chiral static field (asymmetrically perturbed field mechanism or one-electron mechanism) in the given one molecule. A similar mechanism of the optical activity can be expected for molecular assemblies which are composed of chiral and achiral ones. This type of optical activity is called induced optical activity and depends on types of inter-molecular interaction modes. 

In the "degenerate extended coupled oscillator" (DECO) description of the optical activity of n interacting dipoles, the rotational strength R, and hence the VCD intensities, is given by [18] ... 

In the dimeric case, the coupled oscillator equation predicts the rotational strengths R for the symmetric +) and antisymmetric -) combination states of two interacting vibrations according to... 

Figure 13 Energy level scheme for a system of two coupled oscillators. The isolated peptide states (left side) are coupled by some weak interaction, which mixes them to generate the excitonic states (right side). Anharmonicity, which is crucial for understanding the 2D pump probe spectra, is introduced into this model by lowering the energies of the double excited monomeric site states i2) and j2) by A from their harmonic energies 2eu- This anharmonicity mixes into all coupled states, giving rise to diagonal anharmonicity (Ae ) and off-diagonal anharmonicity (mixed-mode anharmonicity, Ae i) in the basis of the normal modes discussed in the text.

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