# Vibrational Partition Function

1. Apr 12, 2010

### manofphysics

I have a really conceptual question on vibrational partition function for a diatomic molecule.If we consider a diatomic molecule, we write :
Energy of simple harmonic oscillator=$$E_{i}=(n + 1/2) h\nu$$.We plug this eqn. into
$$Z_{vib}=\sum e^{-\beta\epsilon_{i}}$$.

Now , my question, is that the energy of harmonic oscillator has been derived for single mass vibrating under harmonic approx., but a diatomic molecule contains "two" atoms or masses.
Somewhere on the Net it was written about considering the center of mass as vibrating...but if the the two masses are vibrating in the normal modes,the center of mass remains constant.
In addition , can anybody tell me why the degrees of freedom for a diatomic molecule for vibration are 2? Shouldn't they be 3?
I have looked in reif and huang for this but I couldn't find anything.Is there good book which explains the vibrational and rotational partition functions for di and polyatomic molecules?

Thanks a lot,

2. Apr 12, 2010

### kanato

It's instructive to work out the classical solution to two masses attached by a spring. You will see the solution ends up having two parts: motion of the center of mass like a single particle, and "internal" vibrations of the masses in the difference coordinate x_1 - x_2. The vibrational equations end up being the same as for a single mass with a spring, but using a reduced mass $$\mu = m_1m_2/(m_1+m_2)$$.

If you want to look for a text which explains vibration and rotation in detail, I think most any advanced undergraduate physical chemistry textbook should cover it in detail.

3. Apr 12, 2010

### Mute

For a diatomic molecule vibration contributes only two degrees of freedom: one is a relative motion of the two atoms vibrating about the centre of mass (the kinetic energy term) and the other is the potential energy of the spring between them.

The centre of mass motion is the usual 3 degrees of freedom that free monatomic gases have; the two vibrational degrees of freedom add to this (plus the two rotational degrees of freedom the diatomic molecule has).