Linear and Symmetric Top Molecules

The nuclear quadrupole energy for a linear molecule with a single coupling nucleus in the absence of external fields is given by

where x = eQq is the nuclear quadrupole coupling constant in frequency units with q = d2 V/dz2, the electric field gradient relative to the principal axis system, and where z is along the molecular axis and Q is considered known and is characteristic of the coupling nucleus. Also,

(3/4)C(C + 1) - I(I + 1) J(J + 1) 2(2 J - 1)(2 J + 3)I(21 - 1)

is a function of the various quantum numbers and the spin I. It has been tabulated for various I and J. It may be observed that Y(J, I, F) is undefined for I = 0 and 1/2, which is consistent with the requirement I ^ t for a nuclear quadrupole interaction. The E Q must be added to the rigid-rotor energy Er to give the total energy. Applying the selection rules A J = 0, A F = 0, ±1, one obtains the rotational frequencies including effects of quadrupole coupling:

where vr is the unperturbed rotational frequency and F' = F, F ± 1. The coupling constant can be evaluated from the splitting between any two hyperfine components. To evaluate the rotational constant, the rigid-rotor frequency vr is required, and this may be evaluated by correcting the hyperfine components v with the known x via the above frequency expression.

To understand the appearance of the hyperfine pattern, a knowledge of the relative intensities of the components is required. The explicit expressions require too much space to give here. However, we may point out that for any class of rotor when J > I, there are 21 + 1 components for F ^ F + 1,21 components for F ^ F, and 21 - 1 components for F ^ F - 1. Furthermore, the most intense components are those where A F = A J. An approximate intensity rule is that for J ^ J + 1 transitions, the intensities of the F ^ F + 1 components are proportional to F, while the F ^ F components are considerably weaker and the F ^ F - 1 components even weaker. For the J ^ J transition, the intensity of the F ^ F component is proportional to F, while the F ^ F ± 1 components are considerably weaker. In fact, the intensities of the components for F = A J decrease rapidly with increasing J. Also, the function Y(J, I, F) may be positive or negative, but for the maximum and minimum values of F, the function is positive. Moreover, the strongest component is usually not significantly displaced from the rigid-rotor position. Consider, for example, the 2 ^ 3 transition with I = 3/2. For J = 2, F = 1/2, 3/2, 5/2, 7/2 and for J = 3, F = 3/2, 5/2, 7/2, 9/2. We expect four F ^ F + 1 components (1/2 ^ 3/2, 3/2 ^ 5/2, 5/2 ^ 7/2, and 7/2 ^ 9/2 in order of increasing intensity), three weaker F ^ F components (3/2 ^ 3/2, 5/2 ^ 5/2, and 7/2 ^ 7/2), and two even weaker F ^ F - 1 components (5/2 ^ 3/2 and 7/2 ^ 5/2). These considerations enable the identification of the hyperfine components of a given rotational transition. Early applications of microwave spectroscopy to the study of hyperfine structure used the appearance of the fine structure, that is, the relative spacings and intensities of the components, to determine unknown nuclear spins. For example, the spin of 33S was found to be 3/2 by this method.

For a symmetric top, with the coupling atom on the symmetry axis, eQ = X

where x is the coupling constant with reference to the molecular axis of symmetry. From the above expression it follows that each J, K level splits into a number of sub-levels of different F. The selection rules are A J = ± 1, A K = 0, AI = 0, A F = 0, ±1. When K = 0, the hyper-fine pattern is like that for a linear molecule. For other K values, a similar pattern is obtained. However, when different K lines are separated by less than the quadrupole splitting, the individual patterns for each K overlap, and a quite complex overall structure can be obtained.

For coupling atoms off the symmetry axis we have a more complicated problem. An example would be HCCl3. The hyperfine structure for molecules with two or more coupling nuclei is more complex but has been treated theoretically and observed experimentally.

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