Classical rotation

The rotation of a macroscopic body can be described classically in terms of angular momentum about an instantaneous rotation axis. The angular momentum P is equal to the angular velocity co multiplied by a quantity I, which is the moment of inertia about the axis of rotation. For a molecule, this depends on the molecular structure, because it is the sum of the products of atomic masses, and the squares of the displacements, r,-, of the atoms from the appropriate axis. 

A moment of inertia can be defined for rotation about any axis through the molecule, but all these possibilities can be reduced to just three orthogonal principal axes. First, the maximum possible moment of inertia (by 

Structural Methods in Molecular Inorganic Chemistry, First Edition. David W. H. Rankin, Norbert W. Mitzel and Carole A. Morrison. 2013 John Wiley Sons, Ltd. Published 2013 by John Wiley Sons, Ltd. 

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This expression for the classical rotational energy of a rigid body will now be developed in terms of Euler s angles. 

The classical rotational energy of rigid body is given in the general case by Eq. (13). This expression is thus applicable to (iv), the asymmetric rotator. The other types of rotator represent special cases. 

In the derivation of quantum mechanical expressions for the rotational g tensor e nuclei are usually treated as classical rotating point charges, Z e, located at Rk- Their contribution to the g tensor is then given as... 

Because the partition function ratio / is defined in such a way that the classical rotational and translational contributions are canceled, equations 11.40, 11.41 and 11.43 must be modified by introducing the ratio of the deviations from classical rotational behavior of heavy and light hydrogen molecules. For small values of [Pg.779]

Existence of a high degree of orientational freedom is the most characteristic feature of the plastic crystalline state. We can visualize three types of rotational motions in crystals free rotation, rotational diffusion and jump reorientation. Free rotation is possible when interactions are weak, and this situation would not be applicable to plastic crystals. In classical rotational diffusion (proposed by Debye to explain dielectric relaxation in liquids), orientational motion of molecules is expected to follow a diffusion equation described by an Einstein-type relation. This type of diffusion is not known to be applicable to plastic crystals. What would be more appropriate to consider in the case of plastic crystals is collision-interrupted molecular rotation. 

Figure 10.7 Probability density of die centrifuged oxygen gas as a function of the molecular angle and the free propagation time, that is, die time elapsed since die molecules have been released from die centrifuge. The white dashed line (around 1.5 ps) marks the calculated trajectory of a dumbbell distribution rotating widi die classical rotational frequency of an oxygen molecule with an angular momentum of 39ft. Part of Fig. 4 in Ref. 39.

A rotation of the H2 molecule through 180° creates an identical electric field. In other words, for every full rotation of a H2 molecule, the dipole induced in the collisional partner X oscillates twice through the full cycle. Quadrupole induced lines occur, therefore, at twice the (classical) rotation frequencies, or with selection rules J — J + 2, like rotational Raman lines of linear molecules. Orientational transitions (J — J AM 0) occur at zero frequency and make up the translational line. Besides multipole induction of the lowest-order multipole moments consistent with... 

Fig. 4. The classic rotation experiment (A) Fj (asj6sy) is immobilized on a NiNTA modified glass surface via polyhistidine tags introduced into the N-termini of the [1-subunits. A fluorescently labeled actin filament is attached to the y-subunit via biotin/...

Although the value of A = 17.2 keV is small (it corresponds to a mo-ment of inertia of about 90 % of the rigid rotor value) it is not unexpected, since a value of A = 18.0 keV has been deduced for the ground state band of the odd-mass neighbour A = 99. If K = 1 or K = 3 were used for the fit with the classical rotational formula then values of A 27 keV and 12 keV would result which are not compatible with the knowledge about other nuclei in this mass region. Hence, K = 2 is proposed for the band in 98Y. 

Fig. 5.6. Comparison of the quantum mechanical and the classical rotational state distribution of NO following the photodissociation of C1NO in the S absorption band. Reproduced from Schinke et al. (1990).

Nasyrov, K.A. and Shalagin, A.M. (1981). Interaction of intense radiation with classically rotating atoms or molecules, Zhumal Eksperimental noi i Teoreticheskoi Fiziki, 81, 1649-1663. [SW Phys.—JETP, 54, 877-883]. 

For those cases in which one of the colliding partners is molecular hydrogen or deuterium, the classical rotator approximation is particularly poor. One then anticipates efficient vibration-rotation exchange only when the rotational level spacings are such that the energy defects are relatively small. For such a gas at temperature T, it is also necessary to consider the population of the particular initial rotational state that can participate efficiently in the energy... 

A nonlinear molecule has three moments of inertia about three principal axes, designated I a, h and Iq- The classical rotational energy can be written... 

For the measurement of the viscosity of liquids such as molten glasses, rotational method is frequently used. In the viscosity range 10 - 10 Pa s the classical rotational devices can be applied. However, the same devices could also be applied for viscosities up to 10 Pa-s working in the so-called a-periodical mode of measurement. 

Two methods of measurement are used. The first one is the classical rotational method employing the original properties of the viscosimeter. Because the measurement of viscosity of molten glasses requires a configuration with a free spindle, the determination of the angular momentum I as the function of a and the shear stress was not taken into account. Direct calibration using the experimental relation t]la =f a) showed to be very simple and relatively accurate. Based on experimental results, the linear function has been chosen in the form... 

To obtain a classical lift of the weight, the system has to be prepared in a y(9, 0) non-stationary state. For /(0, 0) to drive classical rotation, the C (0) complex-valued coefficients and the yn(9) eigenstates selected to form the x(9,0) = Cn(O)x (0) initial wave packet have to make Eq. (3) almost... 

Observation of the Semi-Classical Rotation of a Single Molecular Rotor... 

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