Hessian Matrix is a square matrix containing the elements as the second-order partial derivatives of energy-function of a molecule; the derivative is done with respect to geometric coordinates of the molecule.
However, I couldn't get why Hessian matrix is referred to as force-constant matrix; as is done in the book I'm following viz. Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics by Errol G. Lewars and in many other sources.
If the energy function is a quadratic function which is mostly the case for a molecule following 1D PES; the function which a Hookean spring follows for small displacement, then the second derivative does indeed give the value of force constant, that is $$\begin{align}E- E_0&= k(q-q_0)^2\\ \implies \frac{\mathrm{d}^2 E}{\mathrm{d}q^2}&= 2k\;.\end{align}$$
However, for molecules following multidimensional PES, most of the time, the energy function is not simple quadratic; how can the second order derivative give the force constant? Second order derivatives vary from point to point unlike the simple case above.
So, can anyone tell why Hessian matrix is called force-constant matrix?