Molecular vibrations classical mechanics analysis

Phenomenological treatments which approximate the molecular potential field (Born-Oppenheimer approximation) by a series of classical energy equations and adjustable parameters. These treatments may be called classical mechanical only in the sense that harmonic force-field expressions stemming from vibrational analysis methods are often introduced, though strictly speaking one is free to select any set of functions that reproduces the experimental data whitin chosen limits of accuracy. 

Molecular spectroscopy is now a mature field of study. It is, however, difficult to find references superior to the classic treatise written by Herzberg nearly 50 years ago (1). The origin of vibrational spectra is usually considered in terms of mechanical oscillations associated with mass of the nuclei and interconnecting springs (9). Vibrational spectroscopy considers the frequency, shape, and intensity of internuclear motions due to incident electromagnetic fields. In the harmonic approximation, the vibrational bands are associated with transitions between nearest vibrational states. When higher order transitions, resonance, and coupling between vibrational motions require analysis, quantum mechanical treatment is mandated (1). Improvements and advancements in poljuner spectroscopy are driven by the many problems of interest in the polymer community. 

Molecular mechanics calculations employ a set of superimposed potential functions (or force field) for the Bom-Oppen-heimer surface, whose mathematical form is familiar from classical mechanics, and which is partly empirical in nature. This set of potential functions, which was originally developed in vibrational analysis, contains adjustable parameters that are optimized to obtain the best fit between the calculated and actual properties of the molecules. These properties typically include geometries, conformational energies, heats of formation, vibrational spectra, etc. 

C. M. Morales and W. H. Thompson. Mixed quantum-classical molecular dynamics analysis of the molecular-level mechanisms of vibrational frequency shifts. J.Phys. Chem. A, lll(25) 5422-5433, JUN 28 2007. 

The time-dependent classical statistical mechanics of systems of simple molecules is reviewed. The Liouville equation is derived the relationship between the generalized susceptibility and time-correlation function of molecular variables is obtained and a derivation of the generalized Langevin equation from the Liouville equation is given. The G.L.E. is then simplified and/or approximated by introducing physical assumptions that are appropriate to the problem of rotational motion in a dense fluid. Finally, the well-known expressions for spectral intensity of infrared and Raman vibration-rotation bands are reformulated in terms of time correlation functions. As an illustration, a brief discussion of the application of these results to the analysis of spectral data for liquid benzene is presented. 

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