Eigenvalues of Hessian matrix when the molecule is not at its equilibrium configuration

Question

Given the potential energy surface (PES) of a system, we can calculate the Hessian matrix $H$ of each point $x$ on the PES. $$H_{ij}(x) = \frac{\partial^2 E}{\partial R_i \partial R_j}|_{R = x}$$

When the structure of a molecule is optimized, the eigenvectors of $H$ are displacement vectors, the corresponding eigenvalues are the 'force constants' and they are all positive. On the other hand, when the structure corresponds to a transition state, some of the eigenvalues will be negative.

When a structure is neither the optimized one nor a transition state, the first derivate of the PES is not zero. However, we can still calculate the Hessian matrix for this structure. Moreover, as long as the PES has continuous second partial derivatives, the Hessian matrix is symmetric and thus diagonalizable (the proof). Can we interpret the eigenvalues here?

Answer 1

I am quite sure a much more thorough answer can be provided, but I will give a brief answer.

The answer is both yes and no. The answer is no in the sense that a harmonic analysis should always be performed at the true minimum on the potential energy surface (PES) because the eigenvalues have a physical interpretation as the force constant, and this must be a positive value. This is true because if one wishes to calculate a quadratic approximation to the PES, one will Taylor expand the potential about the minimum. In this case, the leading term is quadratic in the displacement and the coefficient is the second derivative with respect to the potential. This form of the potential is identical to the usual one for a harmonic oscillator, so the coefficient is a force constant and it is positive because the PES is strictly concave up at the minimum.

On the other hand, the answer is yes both approximately and exactly. First, I have seen people sometimes perform a harmonic analysis away from the minimum and simply trust the frequencies which they get out. This can be justified by noting that if the displacement is mostly along an eigenvector of the Hessian, then one can expect to basically still be at the minimum with respect to all other coordinates because these eigenvectors are mutually orthogonal. You will see this if you do a harmonic analysis slightly away from the minimum. That is, the modes whose eigenvectors do not describe a motion similar to the displacement you performed will have basically the same frequency as at the true minimum. However, if you make many displacements (i.e. randomly jiggle the atom positions) then you have no guarantees the answer you get out means anything.

On the other hand, these frequencies can be made rigorous by projecting out the eigenvectors which correspond to negative eigenvalues and re-diagonalizing the new Hessian. This guarantees all positive frequencies and can be interpreted physically as the instantaneous frequency when the system reaches a particular point on the PES. I think people have tried to use this type of procedure to determine linewidths of vibrational transitions. I am not sure how effective of a method it is. This kind of thing can at least give you some insight into how sensitive a particular frequency is to motion in other vibrational coordinates.