# Rotational-vibrational spectroscopy of a molecule

I want to ask if the rotational-vibrational spectra of a molecule is related to any one mode of frequency or if it is related to the whole molecule?

Additionally, the vibrational levels that we say are composed of lower rotational energy levels. Are these vibrational energy levels of certain vibrational normal modes or of the whole molecule?

I'm not quite sure what you mean by 'vibrational levels composed of lower rotational energy levels' so I try to explain the whole thing, briefly.

In analysing the electronic, vibrational & rotational spectra of molecules the Born-Oppenheimer approximation has to be used. This is applicable because the vibrational period (a few femtoseconds) is far faster than rotation period (a few picoseconds) thus these motions do to interact appreciably with one another to a good approximation. This allows the energy levels of, rotation and vibrational motions to be added together.

A normal mode vibration is the motion of all atoms in the molecule in a fixed phase relationship with one another. For N atoms there are $$3N-6$$ normal modes ($$3N-5$$ for a linear molecule). How how far and in what direction each atom moves has to be determined by calculation, but the overall symmetry species ($$A_1, B_g, E_u$$ etc) of the total motion of all atoms is given by the Point Group of the molecule, $$C_{3v},D_{2h}, O_h$$ etc. The symmetry species determines the relative phase of each atoms motion with respect to all the others. By 'phase' is meant that as some bonds contract others extend but they always do this together.

As energy levels add each vibrational normal mode has its own stack of rotational levels. (Because the B.O. approximation is not exact a more detailed analysis in needed to add the interaction between vibration and rotation.)

The figure shows a typical spectrum (calculated) of a (perpendicular band) of a symmetric top molecule such as $$\ce{PCl3 , CHCl3} \mathrm { ~or~ } \ce{CH3Cl}$$. The total spectrum is shown on the bottom line and its separation into individual PQR rotational bands above this. There will be similar complicated features for each vibrational normal mode. Clearly the analysis of spectra such as this is non-trivial :)

(In symmetric top molecules two of its three moments of inertia are the same but different from the third. Two quantum numbers are needed J for total angular momentum and K to fix the angular momentum about the symmetry axis and this is why there are so many rotational bands for a single vibration.)

(Spectrum is taken from G. Herzberg, ' Infra-red and Raman Spectra of Polyatomic Molecules')

I want to ask if the rotational-vibrational spectra of a molecule is related to any one mode of frequency or if it is related to the whole molecule?

Both. The frequencies at which you see "lines in the spectra" are a consequence of the structure of the molecule - if you change the structure, the frequencies (and the spectra) will change as well.

Additionally, the vibrational levels that we say are composed of lower rotational energy levels.

I'm not sure that's true. There is energy transfer between rotational and vibrational modes, but vibrational and rotational levels are distinct. Rotationally resolved IR spectroscopy just happens to let you see both...