I am trying to phrase this as a mathematical question, but it originates from a physical problem, which I will come to in a minute.
Suppose you have a function of $N$ variables $f(\vec{R})$, with $\vec{R} = (x_1, x_2, ..., x_N)$. $f$ is a smooth function in all variables, and has a minimum around the point $\vec{R}_0$. What you normally do is expand $f$ around $\vec{R}_0$ up to second order. Using the coefficients of this second-order expansion you can normally derive the frequency of oscillations of a system given a small perturbation around $\vec{R}_0$.
Now, suppose you want to consider instead a different point very close to $\vec{R}_0$, which we may call $\vec{R}_1 = \vec{R}_0 + \epsilon$. Here the first derivative is not zero.
The question is: say you expand $f$ around $\vec{R}_1$ using only second order (i.e. ignoring altogether the linear term). Does this still tell you something meaningful on the curvature of $f$ around $\vec{R}_1$? Can I find a justification of this practice to locally estimate the curvature of $f$?
The physical significance of this question is the following: using the Born-Oppenheimer approximation you write the energy of a physical system $E$ as a function of the ionic coordinates $E(\vec{R})$, and $\vec{R}_{0}$ is an equilibrium state of your system. Then, the expansion around $\vec{R}_0$ gives you important information on the harmonic oscillations around equilibrium. However, small ions are not exactly point-like, so the Born-Oppenheimer approximation does not hold as well for those. Small ions are in fact "smeared out" over a small region around $\vec{R}_{0}$. The idea of expanding only up to quadratic order around $\vec{R}_1 = \vec{R}_0 + \epsilon$ then is to sample the harmonic frequencies in points that might be occupied despite not being exactly the equilibrium ones.
References to papers or books where I may find some enlightenment are also a perfectly fine answer.