Coupled oscillator theory

The observed complex CD of the antibiotic rifamycin chromophore was simulated (250-190 nm) by means of coupled oscillator theory, from coupling of the long-axis polarized aromatic transition with dienone transition312. 

For many years no attempt was made to determine the absolute stereochemistry of transition metal complexes, although Kuhn (8) calculated the absolute configuration of (—)n-[Co(C204)3]3- from its anomalous optical rotatory dispersion, using classical coupled oscillator theory. He later (9) extended his theory to other tris-chelated cobalt complexes such as (+)n-[Co(en)3]3+. However, in 1955 Saito (10) showed by anomalous X-ray dispersion that Kuhn s suggested configuration for (+)n-[Co(en)3]3+ was incorrect. 

The amenability of the n-paraffins to treatment by simple coupled-oscillator theory was demonstrated by Snyder (32,33) i.e., he was able to show that the frequencies of the components of CHg rocking motion in each of the n-paraffins, n-CggH g through CggHjg, correlate with a... 

CD exciton chirality method 11 the most simple and reliable method applicable to a variety of natural products, because the exciton-coupled CD is based on the coupled oscillator theory and the mechanism of this method has already been established as will be briefly explained in the following sections. Therefore, numerical calculations using a computer are not necessary. 

De Voe calculation 2 13 a simple method based on the coupled oscillator theory, which is applicable to more complex chiral molecules composed of two or more groups. This method needs numerical calculations using a computer. Some examples are listed in the section of applications. 

The application of coupled oscillator and related theories to the investigation of nucleic acids and polynucleotides has been reviewed by Brahms. The optical activities of single nucleosides and derivatives have been calculated by a coupled oscillator theory in which the tt-tt transitions of the bases were decomposed into individual bond-bond transition moments which were then coupled by the dipolar coupling mechanism. Apparently such a treatment was necessary to account for deficiencies in the simple approach of chromophore-centered dipole transition moments. 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

In the preceding sections we have discussed the algebraic treatment of onedimensional coupled oscillators. We now present the general theory of two three-dimensional coupled rovibrators (van Roosmalen, Dieperink, and... 

Kellman, M. E. (1982), Group Theory of Coupled Oscillators Normal Modes as Symmetry Breaking, J. Chem. Phys. 76,4528. 

The chirality of [2.2]paracyclophane derivatives has been deduced as being (—)(R) on the basis of the exciton theory of coupled oscillators 67) and confirmed by experimental results (see 2.9.1 and 2.9.3). In these compounds a negative Cotton effect at 270 nm (corresponding to the p-band) seems to be specific for the (R)-chirality 54). 

Coupled oscillator models are extensions to the simple models developed for electronic circular dichroism. They are well known under the name exciton theory (see e.g. Harada and Nakanishi, 1972). These models, extended to vibrational transitions, describe the coupling of pairs of electric dipole transition moments. They predict equal amounts of positive and negative VCD intensity ... 

The first theoretical model of optical activity was proposed by Drude in 1896. It postulates that charged particles (i.e., electrons), if present in a dissymmetric environment, are constrained to move in a helical path. Optical activity was a physical consequence of the interaction between electromagnetic radiation and the helical electronic field. Early theoretical attempts to combine molecular geometric models, such as the tetrahedral carbon atom, with the physical model of Drude were based on the use of coupled oscillators and molecular polarizabilities to explain optical activity. All subsequent quantum mechanical approaches were, and still are, based on perturbation theory. Most theoretical treatments are really semiclassical because quantum theories require so many simplifications and assumptions that their practical applications are limited to the point that there is still no comprehensive theory that allows for the predetermination of the sign and magnitude of molecular optical activity. 

The factor b may be determined from physical theories of optical rotations (75). In particular, coupled oscillator (75), polarizability (76) or free-electron theories (77) of molar rotations in the transparent region for helical line models of molecules consisting of N interacting units i, j,. .. involve relations of the... 

FIGURE 15.4 Definition sketch for understanding the theory of electroviscoelasticity (a) rigid droplet (b) incident physical field, for example, electromagnetic (c) equivalent electrical circuit-antenna output circuit. Wd represents the emitter-coupled oscillator and Cd, and i d are capacitive, inductive, and resistive elements of the equivalent electrical circuit, respectively. Subscript d is related to the particular diameter of the droplet under consideration. (Courtesy of Marcel Dekker, Inc.) Spasic, A.M. Ref. 3., p. 854. 

Despite the knowledge that the two modes are coupled oscillators, little has been reported (for polymer systems) on attempting to explain the observed frequency and damping behaviour using classical theory. In classical theory the frequencies (to) of the coupled modes are related to those of the uncoupled modes (the natural frequencies), Q and Q2, by the equation... 

This simple description does not take into account the so-called photonic stop-band effect, which forbids propagation at wavelength exactly satisfying the Bragg condition. According to the coupled mode theory [66, 67], distributed feedback (DFB) lasers normally oscillate on the edge of this photonic stopband (Figure 15.14). 

To conclude this chapter we discuss the size-dependencies of the electric susceptibilities. The weak-coupling exciton theory and the oscillator sum rule indicate that transition dipole moments from the ground state to an excited state are proportional to vT, whereas interexcited state transition dipole moments are independent of size. This result indicates that is a linear function of L for long chains. 

A theory encompassing k, V3, and V4 would be a theory for exothermic binary reactions. While no such comprehensive theory exists, it has nevertheless been possible to make some useful qualitative predictions on very simple grounds. The determination of v can be related to the classical problem of unimolecular decomposition. No exact solution to this problem is possible, but approximate expressions for simplified models are well known and these give the correct qualitative dependence of the complex lifetime on the various controlling parameters. The simplest such expression, derived for a simple statistical model consisting of s coupled oscillators of the same frequency, is... 

Coming back to limit cycle oscillations shown by systems of ordinary differential equations, this simple mode of motion still seems to deserve some more attention, especially in relation to its role as a basic functional unit from which various dynamical complexities arise. This seems to occur in at least two ways. As mentioned above, one may start with a simple oscillator, increase [x, and obtain complicated behaviors this forms, in fact, a modern topic. However, another implication of this dynamical unit should not be left unnoticed. We should know that a limit cycle oscillator is also an important component system in various self-organization phenomena and also in other forms of spatio-temporal complexity such as turbulence. In this book, particular emphasis will be placed on this second aspect of oscillator systems. This naturally leads to the notion of the many-body theory of limit cycle oscillators we let many oscillators contact each other to form a field , and ask what modes of self-organiza-tion are possible or under what conditions spatio-temporal chaos arises, etc. A representative class of such many-oscillator systems in theory and practical application is that of the fields of diffusion-coupled oscillators (possibly with suitable modifications), so that this type of system will primarily be considered in this book. 

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