Angular momentum, vibration-rotation

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

J and Vrepresent the rotational angular momentum quantum number and tire velocity of tire CO2, respectively. The hot, excited CgFg donor can be produced via absorjDtion of a 248 nm excimer-laser pulse followed by rapid internal conversion of electronic energy to vibrational energy as described above. Note tliat tire result of this collision is to... 

This completes our introduction to the subject of rotational and vibrational motions of molecules (which applies equally well to ions and radicals). The information contained in this Section is used again in Section 5 where photon-induced transitions between pairs of molecular electronic, vibrational, and rotational eigenstates are examined. More advanced treatments of the subject matter of this Section can be found in the text by Wilson, Decius, and Cross, as well as in Zare s text on angular momentum. 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

The elements of S-matrices are determined in the basis of orbital angular momentum l and rotational moments jt,jf of vibrational states i,f and their projections (m,m,-,m/). Both S-matrices in Eq. (4.58) have to be calculated for the same energy Ek of colliding particles. 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

In this section, we shall look at the way these various absorptions are analysed by spectroscopists. There are four kinds of quantized energy translational, rotational, vibrational and electronic, so we anticipate four corresponding kinds of spectroscopy. When a photon is absorbed or generated, we must conserve the total angular momentum in the overall process. So we must start by looking at some of the rules that allow for intense UV-visible bands (caused by electronic motion), then look at infrared spectroscopy (which follows vibrational motion) and finally microwave spectroscopy (which looks at rotation). 

Various reactions in which the reactants are in particular vibrational and rotational states have been investigated and state-to-state kinetics have been studied. Two procedures have been used in these investigations. Brooks and coworkers first employed the molecular beam method for studying the state-to-state kinetics. The reactants molecules are put into desired vibrational and rotational states by laser excitation and identified the states by their fluorescence. In molecular beam experiments, it is possible to control the translational energy and mutual orientation of the reactants and to determine the degree of polarization of the rotational angular momentum of the product. 

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