Is there such a thing as infinitesimal electric field? I am interested in calculating some response properties, namely, susceptibility and polarizability.
In principle, susceptibility should be the functional derivative of the electron density to a perturbing potential
$$ \chi(r,r') = \frac{\partial n(r)}{\partial \phi(r')} $$
Polarizability then should be the functional derivative of the polarization density to a perturbing field
$$ \alpha_{ab}(r,r') = \frac{\partial P_a(r)}{\partial E_b(r')} $$
Now assuming I can calculate the charge and the polarization density of an arbitrary system, then I could just place a Dirac delta potential at some point $r'$, calculate the charge density at $r$, and approximate $\chi(r,r')$ from finite differences.
I was thinking of doing the same thing for $\alpha_{ab}(r,r')$, but then I ran into the question of the infinitesimal field. Is there such a concept? I assume it must have a direction (of course it is a vector field...), but also be conservative. Doesn't conservativeness imply that it's defined at every point in space? Or can I take a very local potential, express the field as its gradient (making it conservative), but accept the fact that the field is defined everywhere?
Or am I thinking in a completely wrong direction, and there's a different way to evaluate functional derivatives numerically on a grid?
 A: The word infinitesimal may be misleading due to its long history as a mathematical concept. However, it is just a synonym for a quantity that eventually goes to zero in modern mathematics. In particular, when translating differentials or functional differentials into numerical calculations, the real meaning of infinitesimal is *a quantity small enough to justify the first-order approximation.
Going to a more formal discussion, let's assume that $F[\phi]$ is a functional of a function $\phi({\bf r})$ ($F$ may also depend on the position in the space).
The functional derivative of $F$, usually indicated in Physics as $\frac{\delta F}{\delta \phi({\bf r})}$ can be defined as the function of the point $ {\bf r}$ that approximates $F[\phi +\delta \phi]-F[\phi]$, with the exception of second order terms:
$$
F[\phi +\delta \phi]-F[\phi] = \int {\rm d}{\bf r} \frac{\delta F}{\delta \phi({\bf r})}  \delta \phi({\bf r}) + O\left({(\delta \phi({\bf r}))^2}\right). \tag{1}
$$
In a numerical treatment of functional derivatives, one has to approximate the space integrals in terms of finite sums over a finite grid of points:
$$
\int {\rm d}{\bf r} \frac{\delta F}{\delta \phi({\bf r})}  \delta \phi({\bf r}) \simeq  \sum_i \frac{\delta F}{\delta \phi({\bf r}_i)}  \delta \phi({\bf r}_i) \Delta V_i,
$$
where $\Delta V_i$ is a volume around the point ${\bf r}_i$, small enough to ensure that inside the volume, the integrand does not vary too much.
Therefore, the discretized form of the equation $(1)$, with a function $\delta \phi$ different from zero only at the grid point ${\bf r}_i$, can be used to obtain a numerical (controlled) approximation of the functional derivative at the same point (with such a $\delta \phi$, the sum reduces to a single term).
In practice, if we want to evaluate numerically the functional derivative at the point  ${\bf r}_i$ of $F[\phi]$, we start with a space discretization (then the function $\phi$ is represented by the values $\phi_j=\phi({\bf r}_j)$) and, depending on the geometry of the grid, there is a small volume $\Delta V_j$ associated to each grid point. The functional derivative at ${\bf r}_i$ can be numerically obtained with good precision by using a symmetric finite difference formula as
$$
\frac{\delta F}{\delta \phi({\bf r}_i)} = \frac{1}{\Delta V_i}\frac{F[\phi+\delta \phi]-F[\phi-\delta \phi]}{2\delta \phi_i}
$$
Notice that the two ingredients are necessary:

*

*a spatial grid fine enough (then a $\delta V_i$ small enough) to allow a safe substitution of the integral with a finite sum;

*a variation of the function $\phi$ small enough to ensure that the contributions of higher-order terms in $\delta \phi$ are negligible.

All the above discussion is valid independently on additional structures of the functionals. In particular, it holds in the same way if the functional is a component of a vector field at a point ${\bf r}$ and if the variable the functional depends on is, in turn, a component of a vector field. There is no constraint on the vector field. In particular, it may or may not be conservative.
