Coupled oscillator methods

Bromo[2.2]metacyclophane-4-carbonitril (60, which had previously been correlated with many optically active 4-monosubstituted, 4,14-homo-, 4,12- and 4,14-heterodisubstituted [2.2]metacyclophanes) 77,78) was chosen as a nicely crystalline reference substance lo6) its chirality (i ) as determined by the Bijvoet-method 106) contrasts with the previously assigned chirality based on results of a kinetic resolution 77) (vide infra), but agrees with the assignment by the coupled oscillator method (to the bisester 57) 1071 and by optical comparison and correlation 84... 

The NECO method is the same as the extended coupled oscillator method if all of the chromophores are identical. 

The relative displacements of the masses in the two normal modes of this coupled oscillator are shown to the right in Fig. 2. This method the form of the normal modes is particularly useful in the analysis of molec vibrations (see Chapter 9). 

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... 

These spectral data were interpreted via a straightforward coupled oscillator model, and calculations were carried out to find a molecular conformation in < > and space around the center alanine residue which best reproduces the observed spectral data. The computational methods employed were discussed above (cf. Section 3) and detailed arguments are given in the original report [23]. A similar, yet not as detailed, effort had been published before for peptide models in non-aqueous solution [46]. 

Contributions of Aromatic Side Chains to the Far UV CD of Proteins. Numerous theoretical studies of the effects of aromatic groups on both the far and near UV CD spectra of proteins have been conducted by Hooker and co-workers [143-154], While the calculations on larger proteins were limited in scope, they do provide the only comprehensive attempt to include these chromophores into CD calculations (see below). Other researchers have attempted coupled-oscillator calculations on proteins such as insulin [155, 156], to assess the effects of tertiary structure on near UV CD spectra. More recent work by Woody and co-workers expanded the matrix method to include more elaborate descriptions of... 

The Laue method involves a stationary crystal and polychromatic ( white ) X rays. In the other camera methods, monochromatic radiation is used. In these cases the crystal may be oscillated over a small angular range (oscillation method) or rotated 360° about an axis (rotation method). The layer lines so formed may be selected individually. In the Weissenberg method, the oscillation of the crystal is coupled with a movement of the photographic film. The Buerger precession method, by a more complex motion of the instrument, produces an undistorted and magnified picture of the reciprocal lattice. 

For the description of the CD spectra of molecules with two (or more) identical or at least similar chromophores in a chiral arrangement the method of coupled oscillators (exciton chirality method) has proven particularly successful. Such systems can occur either as chiral dimers, or they can be obtained by introducing suitable chromophores into a chiral molecule. The best-known example is given by the dibenzoates derived from 1,2-cyclo-hexanediol of the general formula 4 ... 

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. 

Finally, semi-classical approaches to non-adiabatic dynamics have also been formulated and successfully applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This method is also quite accurate, except that the number of required trajectories grows with time, as in any semi-classical approach. 

Equation (3.C.1) represents the Brownian motions of coupled oscillators. Similar to the discrete case (Eqs. (3.31)-(3.33)), the standard method to solve the differential equation of the continuous Rouse model is to find the normal coordinates, each with its own independent motion. Considering the boundary conditions given by Eq. (3.C.3), we may write the Fomier expansion for Rn(t) in terms of the normal coordinates X, as... 

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