For Lennard-Jones problems, the curve is steeply exponential, almost tripling with each new atom. A cluster of more than 20 atoms will have many millions of local minima along its Lennard-Jones potential energy surface. It is physically reasonable to assume that, for any given number N of like atoms in a cluster, there is a global minimum-energy arrangement. It is the exact geometries of such arrangements that mathematicians try to find by computational means. Leary explains that there are many reliable algorithms for finding a local minimum of a potential energy function from a given starting point.
Some common ones are steepest descent, conjugate gradient, and Newton's method.
The real trick is to find ways to escape from one local minimum to find a lower one. Such algorithms operate as though the computer were the explorer in the hilly landscape. This may lead to a new local minimum in a nearby valley that is adjacent to the starting valley. This new minimum may be higher or lower in energy than the starting local minimum. The step is accepted by the algorithm if the new energy is lower than that of the starting point.