Relative rotational motion

Strictly speaking, / is the reduced moment of inertia for the relative rotational motion of the system. For the case of a relatively light rotor such as CH3 it is the moment of inertia of the hindered rotor that appears in Eq. (143). 

Figure 2.4 (Left) Accelerated motion in Minkowski space. (Right) Two coordinate systems in relative rotational motion.

When the expectation value of (II.4) is taken over the product of the ground state wave functions for molecules a and b, the result is the sum of coupling energies for the permanent electric moments in the two molecules. If the moments are known, or taken as parameters, the electrostatic interaction energy is known for a chosen orientation, and can be compared for d and I species. By forming averages over angles we can make the calculation for molecules in relative rotational motion. First it is useful to examine the symmetry restrictions imposed by chiral character, and to see how the moments in one chiral enantiomer are related to those in the other. 

In opposition to the situation in static external fields — in which, as is well known, only the lowest rotationless state (through averaging over the spatial orientation) experiences a lowering of the energy — we see that the situation is more complicated here. Equally important is the relative rotational motion of the dipoles, namely in the first instance the difference in rotational quantum numbers. If l — k = A, then in the limit of large quantum numbers we obtmn for (26)... 

SHEAR FFF Shear field-flow fractionation (shear FFF) has been proposed as a technique in which shear forces are responsible for migration perpendicular to flow. An internal shear force field is induced in the annular space between two concentric cylinders that are in relative rotational motion. 

Information about the structure of a molecule can frequently be obtained from observations of its absorption spectrum. The positions of the absorption bands due to any molecule depend upon its atomic and electronic configuration. To a first approximation, the internal energy E oi a, molecule can be regarded as composed of additive contributions from the electronic motions within the molecule (Et), the vibrational motions of the constituent atoms relative to one another E ), and the rotational motion of the molecule as a whole (Ef) ... 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

The rotational operation of a CFB leads to a vortex motion in the freeboard which tends to inhibit particle loss by elutriation. Because of the relatively compact nature of the CFB and the operating flexibility provided by the rotational motion, the CFB has been proposed for a variety of applications including coal combustion, flue gas desulfurization, gas combustion, coal liquefaction and food drying. 

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. 

The tilde over operator r here and below indicates that the operator is calculated in the EFA, as was done in [185, 186], This treatment ignores the influence of rotational transitions, caused by the anisotropic part of the interaction, on relative translational motion of colliding particles. Therefore f (.K , differs slightly from the true operator r(K . What... 

In addition to Ti and T2, which reflect the rotational motion of water, NMR can also be used to measure the translational motion of water. If an additional, relatively small (compared to B0), steady magnetic field gradient is incorporated into a pulsed NMR experimental setup, a translational diffusion coefficient (D, m2/s) can be measured (called pulsed field gradient NMR). 

Y, and Z are connected by bonds of fixed length joined at fixed valence angles, that atoms W, X, and Y are confined to fixed positions in the plane of the paper, and that torsional rotation 0 occurs about the X-Y bond which allows Z to move on the circular path depicted. If the rotation 0 is "free such that the potential energy is constant for all values of 0, then all points on the circular locus are equally probable, and the mean position of Z, i.e., the terminus of , lies at point z. The mean vector would terminate at z for any potential function symmetric in 0 for any potential function at all, except one that allows absolutely no rotational motion, the vector will terminate at a point that is not on the circle. Thus, the mean position of Z as seen from W is not any one of the positions that Z can actually adopt, and, while the magnitude ll may correspond to some separation that W and Z can in fact achieve, it is incorrect to attribute the separation to any real conformation of the entity W-X-Y-Z. Mean conformations tiiat would place Z at a position z relative to the fixed positions of W, X, and Y have been called "virtual" conformations.i9,20it is clear that such conformations can never be identified with any conformation that the molecule can actually adopt... 

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