Hessian Matrix and Physical directions: Potential Energy Surface

How do the eigenvectors of the Hessian Matrix for potential energy surface correspond directly to the physical directions of translation, rotation and vibration?

I see that they might correspond to some physical directions, but it seems fishy that they directly correspond to these very neat directions.

• Hi @Mahathi, despite there being a good answer already, I wanted to say that this was a very good question. I would like to invite you to participate as one of the first group of users in the Materials Modeling Stack Exchange: area51.stackexchange.com/proposals/122958/…. We will soon be in the private beta phase, where only a select group of users can participate. These users help to determine the style of questions for the future. – user1271772 Feb 22 at 15:00

Consider the example of water. We know that it has it has $$9$$ degrees of freedom, because each of its $$3$$ atoms are described by $$3$$ Cartesian coordinates. However, if we are interested in the energy of a configuration, we know that some of these degrees of freedom don't matter. For example, if we shift the x coordinate of each atom by the same amount, we haven't changed the molecule at all, only its location. This leads us to seek a set of coordinates that describe the potential energy surface without considering parts that don't matter. For water, we can remove $$6$$ as shifting along or rotating about any axis doesn't change the energy of the molecule.
This leaves us with $$3$$ remaining degrees of freedom, internal motions of the molecule that do affect its potential energy. What do these look like? I could come up with all sorts of valid variations on this. Off the bat, I might think that a good representation would be individual stretches of each $$\ce{O-H}$$. While there isn't anything strictly wrong with this representation, it does have the issue that motions along these coordinates are coupled, meaning changes along a given mode affect some or all of the others.
What can I do to avoid to avoid this coupling? Well, within the harmonic approximation, all the coupling between modes is described by off-diagonal elements of the Hessian matrix. So, if I can find a representation where the Hessian is diagonal, I will have a nice set of uncoupled internal motions. This is exactly what we get from the eigenvectors. Now, instead of two coupled $$\ce{O-H}$$ bond stretches, the normal modes give two decoupled anti/symm vibrations of both $$\ce{O-H}$$ bond.