Rotational Energy Levels: Equally Spaced or Not?

[broegger]

Are rotational energy levels of a molecule in general equally spaced or does the spacing increase with energy? How about a diatomic molecule; I have seen a derivation showing that the rotational levels in a diatomic molecule are equally spaced, but when drawn in an energy level diagram they clearly aren't? What is right?

And another thing; in my book the expression for the rotational and vibrational energies of a diatomic molecule is derived in terms of a classical "dumb-bell"-picture? How is this justified taking quantum mechanics into account?

 

[altered-gravity]

Are rotational energy levels of a molecule in general equally spaced or does the spacing increase with energy? How about a diatomic molecule; I have seen a derivation showing that the rotational levels in a diatomic molecule are equally spaced, but when drawn in an energy level diagram they clearly aren't? What is right?

Hi broegger,

No, even in the most simple diatomic molecule rotational levels are not equally spaced. If you look at any rotational spectrum of such molecules you will clearly appreciate it. You will see how the first lines are separated more or less a 2B distance (B is the rotational constant) but they get closer when increasing frequency. But be carefull with this, the energy levels get more separated when you increase J, but the transition frequency values: E(J+1)-E(J) get closer between them when increasing J. This is the real behaviour.

The question is what model do you choose to theoretically calculate those energy levels (and the frequency values). The most simple quantum model is the rigid rotor (exuse me if that isn´t the correct word, I´m spanish). The energy expression derived from that model is:

E(cm-1)=B J (J+1)

then the frequency expression F=2B (J+1). Frequencies are equally separated. But this is not real. In order to get a better description elastic rotor is used:

$$E(cm^-1)=BJ(J+1)-DJ^2 (J+1)^2$$
$$F=2B(J+1)-4D(J+1)^3$$

This model introduces the "D" constant to allow the variation of bond length (in fact vibration movement). The frequency values obtained get closer between them while increasing J, It´s more real but It continues being a model. Theese expressions are only for diatomic molecules.

 

[broegger]

Thank you.