What Do i and j Represent in the Vibrations of Diatomic Molecules?

[jaejoon89]

What do i and j stand for here? My teacher substituted them for masses (in our example, atoms in a molecule) although I'm not sure that makes sense since when you take the Hessian force constant matrix (on the next page of the link) I believe it must have dimensions determined by the number of degrees of freedom. In other words, for a two mass system (diatomic) wouldn't you have a 6x6 matrix? Is this correct? Again, what do i and j stand for?

From Feynmann's book on Statistical Mechanics:

http://books.google.com/books?id=4Y... order to motivate the procedure that&f=false

 

[vela]

The subscript labels the coordinates, and you are correct in that you'll need a coordinate per degree of freedom. In your teacher's example, perhaps there was only one degree of freedom per molecule, so the same label for the mass could be used to specify the coordinates as well. For instance, a diatomic molecule is like two masses connected by a spring, so there's only one mode of vibration and one corresponding coordinate, namely the distance between the two atoms. The other degrees of freedom you're thinking of have to do with other types of motion, like translation and rotation.

 

[jaejoon89]

Thanks, but then what does it mean to take the mass-weighted Hessian - as Feynman does in the link - in other words, how does it make sense to say that M_i and M_j are the masses of the ith and jth degrees of freedom rather than the ith and jth atoms? And what would that be?

Again, thanks for the help.

 

[vela]

Say degrees 1, 2, and 3 belong to atom A and degrees 4, 5, and 6 belong to atom B. Then M_1=M_2=M_3=M_A and M_4=M_5=M_6=M_B.

 

[jaejoon89]

Thanks, I guess what is confusing is Feynman uses i and j for both the cartesian and mass-weighted coordinate cases.

One last question: how are the explicit values in the Hessian matrix - in this case, 6x6 - determined?

 

[vela]

I assume you're referring to the P_i's. It's not a Hessian matrix. Which case are you referring to, the classical or quantum mechanical?

 

[jaejoon89]

I'm referring to the classical case. In the stuff I've read about it, it's called the "mass-weighted Hessian matrix," ||Cij|| in pg. 15 of Feynmann's book (link to view it is in my original post). In any case, I don't know how to find the values for it for my particular example (hydroxyl radical) so that I can get the eigenvalues.

 

[vela]

That's a different matrix than what I thought you were talking about. (It would have helped if I had read the book more closely.) You need a potential energy function V that describes the interaction between the atoms. Its derivatives will give you the entries of C'_ij, and when you scale the entries by \sqrt{M_i M_j}, you get C_ij.

 

[jaejoon89]

When I solve the way my teacher did by labeling each atom as the i, j values, I get a 2x2 matrix that I solve to obtain

w = sqrt(C_HO)

That means w = sqrt(C ' _HO / sqrt(M_O M_H))

But w = sqrt(k / mew) and the above doesn't simplify to that - what am I missing?

 

[vela]

You need to describe the problem and what you did more fully. I don't really know what you're calculating.