Models rotational motion

Rotational motion is an important topic in chemical systems as it will be used, in the chapters to follow, to describe the rotational motion of gas phase molecules and electronic motion in atoms and molecules. The model problems presented in this chapter will be the basis for modeling rotational motion throughout the remainder of the text. 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

The examples examined earlier in this Chapter and those given in the Exereises and Problems serve as useful models for ehemieally important phenomena eleetronie motion in polyenes, in solids, and in atoms as well as vibrational and rotational motions. Their study thus far has served two purposes it allowed the reader to gain some familiarity with applieations of quantum meehanies and it introdueed models that play eentral roles in mueh of ehemistry. Their study now is designed to illustrate how the above seven rules of quantum meehanies relate to experimental reality. 

Thus far, exaetly soluble model problems that represent one or more aspeets of an atom or moleeule s quantum-state strueture have been introdueed and solved. For example, eleetronie motion in polyenes was modeled by a partiele-in-a-box. The harmonie oseillator and rigid rotor were introdueed to model vibrational and rotational motion of a diatomie moleeule. 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

In experimental measurements, sueh sharp 5-funetion peaks are, of eourse, not observed. Even when very narrow band width laser light sourees are used (i.e., for whieh g(co) is an extremely narrowly peaked funetion), speetral lines are found to possess finite widths. Let us now diseuss several sourees of line broadening, some of whieh will relate to deviations from the "unhindered" rotational motion model introdueed above. 

To inelude the effeets of eollisions on the rotational motion part of any of the above C(t) funetions, one must introduee a model for how sueh eollisions ehange the dipole-related veetors that enter into C(t). The most elementary model used to address eollisions applies to gaseous samples whieh are assumed to undergo unhindered rotational motion until stmek by another moleeule at whieh time a randomizing "kiek" is applied to the dipole veetor and after whieh the moleeule returns to its unhindered rotational movement. 

For reactions between atoms, the computation needs to model only the translational energy of impact. For molecular reactions, there are internal energies to be included in the calculation. These internal energies are vibrational and rotational motions, which have quantized energy levels. Even with these corrections included, rate constant calculations tend to lose accuracy as the complexity of the molecular system and reaction mechanism increases. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

This brief discussion of the physical meaning and mutual correspondence of different models and theories of rotational motion is intended as a guide for those who do not intend to examine the book systematically. Setting forth the material consistently, one cannot avoid certain formalisms peculiar to angular momentum theories. We hope, however, that a detailed commentary will enable readers to form a clear notion of the most important assumptions and results without referring to proofs. 

In this study, particle fluidization behaviors in a RFB were numerically analyzed by using a DEM-CFD coupling model [3]. The particle motion was calculated by DEM, which calculates the motion of each particle by integrating the Newton s equations for individual particle step by step, allowing for the external forces acting on a particle. Equations of transitional and rotational motions for individual particles are as follows ... 

Numerical examples are shown in Figs. 7-9. The model system used is a 2D model of H2O in a continuous wave (CW) laser field of wavelength 515nm and intensity lO W/cm. The ground electronic state X and the first excited state A are considered. The bending and rotational motions are neglected for... 

Model 4 is also a plate kinematic model. The retreat of a fore arc plate forms a back-arc basin. This model seems attractive. Jackson et al. (1975) found the periodicities of rotational motions of the Pacific plate. When the direction of the Pacific plate changed and obliquely subducted, the compressional force of oceanic plate to continental plate decreases. That means that the retreat of fore arc plate occurs. 

Li and coworkers49 reported a molecular motion of /1-carotene and a carotenopor-phyrin dyad (composed of a porphyrin, a trimethylene bridge and a carotenoid polyene) in solution. Internal rotational motions in carotenoid polyenes and porphyrins are of interest because they can mediate energy and electron transfer between these two moieties when the pigments are joined by covalent bonds. Such internal motions can affect the performance of synthetic model systems which mimic photosynthetic antenna function,... 

Figure 41 shows the absorption spectrum for the 24-mode model of pyrazine. As was done by Raab et al. [277], we have included a phenomenological dephasing time of T2 = 150 fs to model the experimental broadening due to hnite resolution and rotational motion. It can be seen that the inclusion of all 24 normal modes of the pyrazine molecule leads to a shape of the spectrum which is in good agreement with the experimental result (Fig. 38b). The semiclassical result is seen to be in fairly good agreement with the quantum result. The spurious structure in the semiclassical spectrum is presumably due to the statistical error.

The literature offers numerous calculation models for mass transfer in single liquid particles. However, they provide only a rough approximation to reality in industrial columns, since the processes in droplet swarms are much more complicated, especially when pulsing and rotating motion are superimposed. For estimation, the following relationships are sufficient ... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

A possible step in this direction can be made through use of earlier relaxation studies on other systems. Hunt and Powles,— when studying the proton relaxation in liquids and glasses, found the relaxation best described by a "defect-diffusion" model, in which a non-exponential correlation function corresponding to diffusion is Included together with the usual exponential function corresponding to rotational motion. The correlation function is taken as the product of the two independent reorientation pro-... 

The lattice models provide useful interpretations of spin relaxation in dissolved polymers and rubbery or amorphous bulk polymers. Very large data bases are required to distinguish the interpretive ability of lattice models from other models, but as yet no important distinction between the lattice models is apparent. In solution, the spectral density at several frequencies can be determined by observing both carbon-13 and proton relaxation processes. However, all the frequencies are rather hl unless T2 data are also included which then involves the prospect of systematic errors. It should be mentioned that only effective rotational motions of either very local or very long range nature are required to account for solution observations. The local... 

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