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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.
I think you have the causality backwards here. It's not as if we have the vibrations to start with and they happen to be matched by the eigenvectors of the Hessian. We choose to look at these particular vibrations precisely because they are the eigenvectors of the Hessian.
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.
The normal modes aren't any more physically correct than my previous example, but they are easier to analyze because they are decoupled. In this way, the "neatness" of the normal modes is really a fundamental mathematical property rather than a fundamental physical one.
