The Mathematical Background to Infrared Selection Rules

In the main text we introduced the selection rules for IR spectroscopy via the transition dipole moment integral. This appendix gives a little more detail on the origin of the selection rules, with explicit formulae for the vibrational wavefunctions. This also allows a more complete explanation of the observation that absorption due to transitions involving neighbouring levels (e.g. n = 0 to n = 1) are more easily observed than overtones which involve transitions to higher levels in the ladder of vibrational states. 

The diatomic molecule is a good place to start a discussion of molecular vibrations. There are six degrees of freedom three translations of the molecule, two rotations and, so, only a single vibration, which is the bond suetch. In this appendix we will consider a diatomic molecule belonging to the Coo, using H—F as an example. It is conventional in symmetfy problems to align the principal axis with the Z-direction. However, we have decided to break with convention here and align with X, because the variable x is more often used in mathematics. 

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