Analytic potentials use the parameters $\alpha_i$ as arguments to analytic function templates. A well-known example for this would be the Lennard-Jones potential, which uses two parameters $\epsilon$ and $\sigma$ to define a potential:

$$V(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]$$

Using an analytic form for $V(r)$ allows to define the atomic interaction for all possible distances between atoms with very few parameters compared to tabulated potentials. The problem of local parameters (see Tabulated potentials) is also not present, parameters are usually global parameters. In certain cases it is even possible to attribute some physical meaning to individual parameters.

The analytic potential implementation in potfit has additional features which are only useful with this kind of interactions. They are described on the analytic format page.