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

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?