Rotor, rigid

ExX = 0 and similarly for Y and Z. We have solved this before so we can write 

Note this could be the solution for a freely flying particle in the universe if we knew the dimensions of the universe. Because a, b, and c would be billions of light years squared in the denominator while the triad (n, Uy, n ) are integers, the difference between any two energy levels would be (are) so tiny that translational energy does approach a continuous variable. However, the point of this exercise is to show that often we can assume a multivariable wave function can be factored into parts, which only depend on one variable at a time. For fumre reference, note two parts of the procedure. First, we assumed the wave function could be factored and second, we assumed that the unknown energy could be made up of separate energy values for each coordinate and it worked  

We can use the reduced mass jx for diatomic molecules and this can be generalized for polyatomic cases but we will use diatomic examples here for simplicity. So far in this text we have tried to show only examples that can be worked out cleanly with calculus (and perseverance) but now we come to a situation where it is questionable whether it is worth the time and frustration to get past a big hmdle with full details. The problem is that we have to convert (x, y, z) to (r, 0, )). We will sketch out the strategy but this is the sort of thing that a graduate student in physical chemistry or physics needs to check once in their life but here we only need to know the result. 

With these relationships, we can set up chain rule derivatives and use trigonometry identities to simplify the results. We note for instance that, in general, the x derivative actually depends on all 

FIGURE 13.2 Spherical coordinate system, E the equatorial x-y plane, ( measured from the c-axis in the equatorial plane and 6 measured from the vertical z-axis. (Reprinted with permission from White, H.E., Phys. Rev., 37, 1416, 1931. Copyright 1931 by the American Physical Society.) 

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For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

It is straightforward to introduce active and adiabatic treatments of K into the widely used RRKM model which represents vibration and rotation as separable and the rotations as rigid rotors [41,42]. Eor a synnnetric top, tlie rotational energy is given by... 

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. 

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. 

Within this "rigid rotor" model, the absorption speetrum of a rigid diatomie moleeule should display a series of peaks, eaeh of whieh eorresponds to a speeifie J ==> J + 1 transition. The energies at whieh these peaks oeeur should grow linearally with J. An example of sueh a progression of rotational lines is shown in the figure below. 

Solution of the Schrddinger equation for a rigid rotor shows that the rotational energy is quantized with values... 

The first three Stokes lines in the rotational Raman spectrum of 02 are separated by 14.4 cm, 25.8 cm and 37.4 cm from the exciting radiation. Using the rigid rotor approximation obtain a value for tq. 

Figures 5-16a and 5-16b show vibration modes of a uniform shaft supported at its ends by flexible supports. Figure 5-16a shows rigid supports and a flexible rotor. Figure 5-16b shows flexible supports and rigid rotors.

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

ISO 1940/1, Mechanical Vibration—Balance Quality Requirements of Rigid Rotors—Part 1 Determination of Permissible Residual I nbal-ance, First Edition, International Organization for Standardization, Geneva, Switzerland, 1986. 

Calculating the electronic barrier with an accuracy of 0.1 kcal/mol is only possible for very simple systems. An accuracy of 1 kcal/mol is usually considered a good, but hard to get, level of accuracy. The situation is slightly better for relative energies of stable species, but a 1 kcal/mol accuracy still requires a significant computational effort. Thermodynamic corrections beyond the rigid rotor/harmonic vibrations approximation are therefore rarely performed. 

Table 55.1 Balance Quality Grades for Various Groups of Rigid Rotors...

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

Molecules in the gas phase have rotational freedom, and the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator the expression for the available energy levels is ... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

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