Overview
In quantum mechanics the basis for a spectroscopic selection rule is the value of the transition moment integral.
where and are the wave functions of the two states involved in the transition and µ is the transition moment operator. If the value of this integral is zero the transition is forbidden. In practice, the integral itself does not need to be calculated to determine a selection rule. It is sufficient to determine the symmetry of transition moment function, . If the symmetry of this function spans the totally symmetric representation of the point group to which the atom or molecule belongs then its value is not zero and the transition is allowed. Otherwise, the transition is forbidden. The idea of symmetry is important when considering the integral of odd functions (which equal zero when integrated over the whole of space).
The transition moment integral is non-zero if and only if the transition moment function, is not anti-symmetric, as when y(x) = -y(-x). The symmetry of the transition moment function is the direct product of the symmetries of its three components. The symmetry characteristics of each component can be obtained from standard character tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.
| Transition type | µ transforms as | note |
|---|---|---|
| Electric dipole | x, y, z | Optical spectra |
| Electric quadrupole | x2, y2, z2, xy, xz, yz | Constraint x2 + y2 + z2 = 0 |
| Electric polarizability | x2, y2, z2, xy, xz, yz | Raman spectra |
| Magnetic dipole | Rx, Ry, Rz | Optical spectra (weak) |
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