On the stability of some analytically solvable maximum probability domains

Abstract

The stability of some maximum probability domains (MPDs) determined for a set of analytically solvable models is examined. MPDs are regions of the physical space that maximize the probability of finding an exact, integer number of electrons and have been shown to provide vivid images of cores, lone and bonding pairs, etc. They are obtained by numerical maximization techniques based on the use of the shape derivative, which has to vanish at any stationary point. Here, we give the first steps to use the shape Hessian to gain information about the true nature of the numerical solutions. As a general conclusion, electron correlation seems to decrease the multiplicity of solutions found at the Hartree–Fock level, the remaining MPDs remaining closer to what it is expected from chemical insight.