Quantum number of rotational

The wavefunction which fits the equation and leads to discrete values of V is the eigenfunction. The search for such eigenfunctions and eigenvalues can be a most demanding mathematical excercise, and need not be considered here. Let us note however that the solutions of the Schrodinger equation lead to the definitions of the orbital quantum numbers n, l and m. The quantum numbers of rotational and vibrational levels are also derived from the Schrodinger equation. 

Inversion doubling has been observed in microwave spectrum of methylamine CH3NH2. This splitting depends on the quantum numbers of rotation and torsion vibrations [Shimoda et al., 1954 Lide, 1957 Tsuboi et al., 1964]. Inversion of NH2 alone leads to the eclipsed configuration corresponding to the maximum barrier for torsion. Thus, the transition between equilibrium configurations involves simultaneous NH2 inversion and internal rotation of CH3 that is, inversion appears to be strongly coupled with internal rotation. The inversion splits each rotation-vibration (n, k) level into a doublet, whose components, in turn, are split into three levels with m = 0, 1 by internal rotation of the... 

We denote by G the set of all the experimentally observable quantities (called physical observables) which must be reproduced. Such quantities are, for instance, the collision energy, the quantum numbers defining the intramolecular state (vibrations and the principal quantum number of rotation), the total angular momentum etc... However, there are other dynamical variables which have a clear meaning in Classical Mechanics but correspond to no physical observable because of the Uncertainty Principle. We call them phase variables and denote them globally by g. The phase variables must be given particular values to obtain, at given G, a particular trajectory. Such variables are, for instance, the various intramolecular normal vibrational phases, the intermolecular orientation, the secondary rotation quantum numbers, the impact parameter, etc... Thus we look for relationships of the type qo = qq (G, g) and either qo = qo (G, g) or po = Po (G, g)... 

As we did for hydrogen, we here ignore spin. Thus is the quantum number of rotation of the atoms about their center of mass, m is the projection of this angular momentum in the lab frame, and is its projection in the body frame. In this basis, the matrix elements of the Stark interaction are computed using Equation 2.10 to yield... 

V J un. quantum number of vibrational state and quantum number of rotational state for the constants in columns 3—6 unassigned to a particular state v,J values from molecular beam magnetic resonance experiments... 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

The symbols in the second column represent the electronic state in particular the first number is the total quantum number of the excited electron. We shall see later that in one case at least the symbol is probably incorrect. The third column gives the wave-number of the lowest oscillational-rotational level, the fourth the effective quantum number, the fifth and sixth the oscillational wave-number and the average intemuclear distance for the lowest oscillational-rotational level. The data for H2+ were obtained by extrapolation, except rQ, which is Burrau s theoretical value (Section Via). 

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]... 

For linear molecules or ions the symbols are usually those derived from the term symbols for the electronic states of diatomic and other linear molecules. A capital Greek letter E, n, A, O,... is used, corresponding to k — 0,1,2,3,..., where A. is the quantum number for rotation about the molecular axis. For E species a superscript + or - is added to indicate the symmetry with respect to a plane that contains the molecular axis. 

Fig. 12. Partitionings of hydrogen fragment translational energy distribution into three components. The solid line denotes the contribution from H2S — 8H(,4 "S+ ) + H which yields a resolved structure with a rovibrational state assignment on the top. The dotted line denotes the contribution of hydrogen from the SH(442 +) —> S(3P) + H reaction, which is a reflection of the solid curve but the structure is smeared out. The corresponding rotational quantum numbers of the parent molecule SI I (A 2>l 1 ) l =0 is marked on the bottom. The remaining part of the P(E) spectrum is represented by the square-like dashed curve.

Soon after Dennison had deduced from the specific-heat curve that ordinary hydrogen gas consists of a mixture of two types of molecule, the so-called ortho and para hydrogen, a similar state of affairs in the case of iodine gas was demonstrated by direct experiment by R. W. Wood and F. W. Loomis.1 In brief, these experimenters found that the iodine bands observed in fluorescence stimulated by white light differ from those in the fluorescence excited by the green mercury line X 5461, which happens to coincide with one of the iodine absorption lines. Half of the lines are missing in the latter case, only those being present which are due to transitions in which the rotational quantum number of the upper state is an even integer. In other words, in the fluorescence spectrum excited by X 5461 only those lines appear which are due to what we may provisionally call the ortho type of iodine molecule. 

We tend to give the letter J to the rotational quantum states. The rotational ground state has a rotational quantum number of J and the excited rotational quantum number is J. To be allowed (in the quantum-mechanical sense), the excitation from J to J must follow... 

The rotation-vibration interaction of the previous section can be rewritten in terms of the usual quantum numbers of linear molecules vav hbv,.JM > by making use of Eq. (4.53). By explicit evaluation, one can show that, for fixed values of va, vh,vc, the matrix elements of Eq. (4.127) have selection rules... 

Because of the large number of rotational levels in the upper and lower states, the overlap between the exciting laser line and the dopp-ler broadened absorption profile may be nonzero simultaneously for several transitions (u", / ) (v, f) with different vibrational quantum numbers v and rotational numbers J. This means, in other words, that the energy conservation law allows several upper levels to be populated by absorption of laser photons from different lower levels. 

This method is equally applicable to atoms 26) and to molecules 22). In molecules the Zeeman splitting depends on the quantum number / of the total angular momentum and therefore the fluorescence from a single rotational level (v, f) need be observed. Because of this necessarily selective excitation, these molecular level-crossing experiments can be performed much more easily with lasers than with conventional light sources and have been sucessfully performed with Naj 2 > and NaK 29). 

Application of the symmetry correlation scheme to reaction (12) is summarized in Table 4 where N is the Himd s coupling case (b) rotational quantum number for O2 and - 5 is the difference of the Hund s coupling case (a) quantum numbers of total angular momentum and electron spin (5 = 1/2) angular momentum, respectively. To consider the high-symmetry isotopomer system first, the results in Table 4 indicate that only odd / collisions - 5 = odd) with 2 can lead... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

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