Editing Rotational spectroscopy (section)

===Effect of rotation on vibrational spectra===
{{Main|Rotational–vibrational spectroscopy}}
Historically, the theory of rotational energy levels was developed to account for observations of vibration-rotation spectra of gases in [[infrared spectroscopy]], which was used before microwave spectroscopy had become practical. To a first approximation, the rotation and vibration can be treated as [[Separable partial differential equation|separable]], so the energy of rotation is added to the energy of vibration. For example, the rotational energy levels for linear molecules (in the rigid-rotor approximation) are

:<math>E_\text{rot} = hc BJ(J + 1).</math>

In this approximation, the vibration-rotation wavenumbers of transitions are

:<math>\tilde\nu = \tilde\nu_\text{vib} + B''J''(J'' + 1) - B'J'(J' + 1),</math>

where <math>B''</math> and <math>B'</math> are rotational constants for the upper and lower vibrational state respectively, while <math>J''</math> and  <math>J'</math> are the rotational quantum numbers of the upper and lower levels. In reality, this expression has to be modified for the effects of anharmonicity of the vibrations, for centrifugal distortion and for Coriolis coupling.<ref>{{harvnb|Banwell|McCash|1994|p=63}}.</ref>

For the so-called ''R'' branch of the spectrum, <math>J' = J'' + 1</math> so that there is simultaneous excitation of both vibration and rotation. For the ''P'' branch, <math>J' = J'' - 1</math> so that a quantum of rotational energy is lost while a quantum of vibrational energy is gained. The purely vibrational transition, <math>\Delta J=0</math>, gives rise to the ''Q'' branch of the spectrum. Because of the thermal population of the rotational states the ''P'' branch is slightly less intense than the ''R'' branch.

Rotational constants obtained from infrared measurements are in good accord with those obtained by microwave spectroscopy, while the latter usually offers greater precision.