Dynamic geometric interpretation

In the previous sections the correspondence between the Schrodinger picture and the algebraic picture was briefly reviewed for some special cases (dynamical symmetries). In general the situation is much more complex, and one needs more elaborate methods to construct the potential functions. These methods are particularly important in the case of coupled problems. This leads to the general question of what is the geometric interpretation of algebraic models. 

The information needed about the chemical kinetics of a reaction system is best determined in terms of the structure of general classes of such systems. By structure we mean quahtative and quantitative features that are common to large well-defined classes of systems. For the classes of complex reaction systems to be discussed in detail in this article, the structural approach leads to two related but independent results. First, descriptive models and analyses are developed that create a sound basis for understanding the macroscopic behavior of complex as well as simple dynamic systems. Second, these descriptive models and the procedures obtained from them lead to a new and powerful method for determining the rate parameters from experimental data. The structural analysis is best approached by a geometrical interpretation of the behavior of the reaction system. Such a description can be readily visualized. 

Theoretically, Vineyard described GIXD with a distorted-wave approximation in the kinematical theory of x-ray diffraction [4]. In terms of the ordinary dynamical theory of Ewald [5] and Lane [6], Afanas ev and Melkonyan [7] worked out a formulation for the dynamical diffraction of x-rays under specular reflection conditions and Aleksandrov, Afanas ev, and Stepanov [8] extended this formalism to include the diffraction geometry of thin surface layers. Subsequently, the properties of wave Adds constructed during specularly diffracted reflections have been discussed in more detail by Cowan [9] and Sakata and Hashizume [10]. Meanwhile, a geometrical interpretation of GIXD based on a three-dimensional dispersion surface has been proposed by Hoche, Briimmer, and Nieber [11]. 

Since the aim of the dynamic investigation is the quantification of the time scale of the motion and the geometric interpretation of the molecular process, the type and the correlation time of dynamics determine the choice of experiments and techniques. The SSNMR represents a powerful tool for the investigation of molecular dynamics due to the possibility offered by the use of different parameters to cover a wide timescale of a fluxional process from 10 ... 

In each case, we first studied the laser driven dynamics of the system in the framework of the Floquet formalism, described in Sect. 6.5 of Chap. 6, which provides a geometrical interpretation of the laser driven dynamics and its dependence on the frequency and amplitude of the laser field, through the analysis of the eigenvalues of the Floquet operator, called quasienergies. Various effective models were used for that purpose. This analysis allowed us to explain the shape of the relevant quasienergy curves as a function of the laser parameters, and to obtain the parameters of the laser field that induce the CDT. We then used the MCTDH method to solve the TDSE for the molecule in interaction with the laser field and compare these results with those obtained from the effective Hamiltonian described in Sect. 8.2.3 above. 

Two different types of dynamic test have been devised to exploit this possibility. The first and more easily interpretable, used by Gibilaro et al [62] and by Dogu and Smith [63], employs a cell geometrically similar to the Wicke-Kallenbach apparatus, with a flow of carrier gas past each face of the porous septum. A sharp pulse of tracer is injected into the carrier stream on one side, and the response of the gas stream composition on the other side is then monitored as a function of time. Interpretation is based on the first two moments of the measured response curve, and Gibilaro et al refer explicitly to a model of the medium with a blmodal pore... 

This is the beauty of this quantity which provides specifically a direct geometrical information (1 /r% ) provided that the dynamical part of Equation (16) can be inferred from appropriate experimental determinations. This cross-relaxation rate, first discovered by Overhau-ser in 1953 about proton-electron dipolar interactions,8 led to the so-called NOE in the case of nucleus-nucleus dipolar interactions, and has found tremendous applications in NMR.2 As a matter of fact, this review is purposely limited to the determination of proton-carbon-13 cross-relaxation rates in small or medium-size molecules and to their interpretation. 

In Chapter 3 we went as far as we could in the interpretation of rocking curves of epitaxial layers directly from the features in the curves themselves. At the end of the chapter we noted the limitations of this straightforward, and largely geometrical, analysis. When interlayer interference effects dominate, as in very thin layers, closely matched layers or superlattices, the simple theory is quite inadequate. We must use a method theory based on the dynamical X-ray scattering theory, which was outlined in the previous chapter. In principle that formrrlation contains all that we need, since we now have the concepts and formtrlae for Bloch wave amplitude and propagatiorr, the matching at interfaces and the interference effects. 

The basic theories of physics - classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics - support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions) molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals surface, interface, solvent and solid-state effects excited-state dynamics, reactive collisions, and chemical reactions. 

Here three constants appear Go is the equilibrium modulus of elasticity 0p is the characteristic relaxation time, and AG is the relaxation part of elastic modulus. There are six measured quantities (components of the dynamic modulus for three frequencies) for any curing time. It is essential that the relaxation characteristics are related to actual physical mechanisms the Go value reflects the existence of a three-dimensional network of permanent (chemical) bonds 0p and AG are related to the relaxation process due to the segmental flexibility of the polymer chains. According to the model, in-termolecular interactions are modelled by assuming the existence of a network of temporary bonds, which are sometimes interpreted as physical (or geometrical) long-chain entanglements. 

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