text: Ro-vibrational spectroscopy concerns molecules in the gas phase. There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band.
  In the simplest cases the part of the infrared spectrum involving vibrational transitions with the same rotational quantum number (ΔJ = 0) in ground and excited states is called the Q-branch. On the high frequency side of the Q-branch the energy of rotational transitions is added to the energy of the vibrational transition. This is known as the R-branch of the spectrum for ΔJ = +1. The P-branch for ΔJ = −1 lies on the low wavenumber side of the Q branch. The appearance of the R-branch is very similar to the appearance of the pure rotation spectrum (but shifted to much higher wavenumbers), and the P-branch appears as a nearly mirror image of the R-branch. The Q branch is sometimes missing because of transitions with no change in J being forbidden.
  The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation.
  The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number , and depends on temperature. Since there are actually  states with rotational quantum number , the population with value  increases with  initially, and then decays at higher . This gives the characteristic shape of the P and R branches.
  A general convention is to label quantities that refer to the vibrational ground and excited states of a transition with double prime and single prime, respectively. For example, the rotational constant for the ground state is written as  and that of the excited state as  
  Also, these constants are expressed in the molecular spectroscopist's units of cm. so that  in this article corresponds to  in the definition of rotational constant at Rigid rotor.