Asymmetric Top Molecules

For an asymmetric rotor, the analysis basically follows the same procedure; however, the details require more space than possible in the scope of this presentation. The Stark effect is usually second order, and the Stark energy contains contributions from all three dipole components

(if present). The contribution of each component has the form

where A and B are called Stark coefficients and depend on J, t , and the inertial asymmetry k . Though simple expressions cannot be given for these coefficients, they may be calculated from second-order perturbation theory and knowledge of the direction cosine matrix elements in the asymmetric rotor basis. They have been tabulated for low J. Once these quantities are specified, the second-order Stark energy may be calculated from Eq. (111).

With asymmetric rotors, the possibility of degeneracies or near degeneracies exists, and in this case the above second-order expression does not apply. Near degeneracies often occur between asymmetry doublets. For J = 2 and a near-prolate rotor, for instance, the pair of levels 2U, 2i,i or 22,i, 22,o interact via xa, and this interaction is very large if the levels are very close together. Ordinary second-order perturbation theory then fails. Thus a transition involving a level that can interact with a nearby level does not exhibit a typical second-order effect. To a good approximation, these levels may be separated from the other levels, and the problem can be treated by standard methods of quantum mechanics as a two-level system. Consider two levels that are eigenfunctions of the unperturbed Hamiltonian M0 with eigenvalues ¿0 and ¿0. The complete Hamiltonian is M = % + M%, with the perturbation. In the basis of M0, there are no off-diagonal elements from M0 and no diagonal elements for M%. The secular determinant thus has the form

with £0 > £20;the plus sign is used for E1 and the negative signfor E2. When( ¿0 - ¿0)2 > 4%2x22, the above can be expanded to give a second-order effect:

Note, however, that the second-order effect could be rather large if the energy denominator is small. If the perturbation is large, (¿0 - ¿0)2 < 4x12%2, then

which is a first-order effect. Once the matrix element is specified |12 = iKM/ J(J + 1), the above relations provide the Stark energies. Such effects are possible not only for asymmetric tops, they also can be observed for linear molecules in excited bending modes, where |12 involves the quantum number l rather than K.

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