Nuclear statistical weights

The requirement set by the exclusion principle, given in equation (6.235), restricts the nuclear spin states which can combine with a rotational level of a given parity. They can be either symmettic or antisymmettic with respect to Pu. The number of symmetric (ortho) states is always greater than the number of antisymmetric (para) states. For equivalent nuclei with spin I,  

Consequently the rotational levels of a given parity with which they combine will have different weights (the so-called nuclear statistical weights). The consequences of this are readily observable in spectroscopy, as we will see elsewhere. 

Let us consider some specific examples whose spectra occur in this book. A very simple case is O2 inits excited1 Ag state. The predominant nucleus, 160, has / = 0 and so is a boson. IT is also zero and there is only one nuclear spin function (IT = 0, M/r = 0) which is symmetric with respect to Pn. Thus for each value of J only one A-doublet component is allowed by the exclusion principle, namely, that which has positive parity all the rotational levels of O2 in its 1 Ag state therefore have positive parity. The other A-doublet component is missing. 

A slightly more complicated example is that of H2 in its X1S + state. The hydrogen nucleus is, of course, the proton with / = 1 /2. In this case there are four possible nuclear spin wave functions, which are 

The first three of these functions are symmetric with respect to Pn and constitute the three components of a spin triplet with IT = 1 the fourth spin function is antisymmetric with respect to Pn and represents a spin singlet. Recalling that the total wave function consists of factors for electron orbital, electron spin, vibrational, rotational and nuclear 

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Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

For C22H2, the nuclear spin of C12 is zero and contributes a factor of 1 to the nuclear statistical weights. The statistical weights are therefore the same as in H2. For the ground vibronic state, the even J levels are s and have nuclear statistical weight 1, corresponding to the one possible ns the odd J levels are a and have nuclear statistical weight 3. The usual selection rule (4.138) holds for collisions as well as radiative transitions, and we have ortho and para acetylene. The two forms have not been separated. 

For nonlinear molecules with identical nuclei that are interchangeable by rotation, the derivation of the nuclear statistical weights is not as simple as for linear molecules. The problem is most efficiently dealt with using group theory. We will not attempt a complete discussion, but will only point out some features of the results.16... 

For 62v molecules where the C2 rotation interchanges two spin- nuclei (e.g., the normal isotopic species of H20 and H2C=0), the function ns involves the same spin functions as for H2. Thus the a levels of H20 have a nuclear statistical weight of 3, whereas the s levels have a nuclear weight of 1 we have ortho and para water. For molecules where the C2 rotation interchanges two spin-zero nuclei (e.g., SOj6), half the rotational levels are missing, just as for C026. 

Fermi resonance in, 278-279 IR bands, table of, 264 normal modes, 242, 248, 262, 265 nuclear statistical weights for, 287-288 vibrational constants of, 275 vibrational levels of, 253 vibrations of, and group theory, 449-451,456... 

Figure 10. Thermal rate constants for capture of N2 by an ion (SACM calculation [33] with channels generating from rotational states N = 0, 1,2, accounting for nuclear statistical weights left figure positive ion right figure negative ion).

The second part of this equation follows because the proton is a fermion. Thus for the lowest rotational level with J = 0, the nuclear wave function must be the fourth in equation (6.253) with IT = 0. In the second rotational level, on the other hand, i//ns must be symmetric and so corresponds to the triplet spin function. In general, therefore, rotational levels with even J are associated with the singlet nuclear spin state and are called para-H2 (with the lower nuclear statistical weight of one). Rotational levels with odd J have triplet nuclear spin states and are called o/v/w-Iblwith the higher nuclear statistical weight of three). 

The differences between the eigenvalues of both equations will furnish the relative locations of the torsional bands, whereas the Franck-Condon factors (120) will give the relative intensities. Notice, that the nuclear statistical weights have to be taken into account to compare the intensities of transitions belonging to different degrees of degeneracy. 

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