Random-walk approach to mapping nodal regions of N-body wave functions: Ground-state Hartree–Fock wave functions for Li–C

Abstract

Despite the widespread acceptance of the relevance of the nodes of one-body electronic wave functions (atomic or molecular orbitals) in determining chemical properties, relatively little is known about the corresponding nodes of many-body wave functions. As an alternative to mapping the nodal surfaces present in the ground states of many-electron systems, we have focused instead on the structural domains implied by these surfaces. In the spirit of Monte Carlo techniques, the nodal hypervolumes of a series of atomic N-body Hartree–Fock level electronic wave functions have been mapped using a random-walk simulation in 3N dimensional configuration space. The basic structural elements of the domain of atomic or molecular wave functions are identified as nodal regions (continuous volumes of the same sign) and permutational cells (identical building blocks). Our algorithm determines both the relationships among nodal regions or cells (topology) as well as the geometric properties within each structural domain. Our results indicate that ground-state Hartree–Fock wave functions generally consist of four equivalent nodal regions (two positive and two negative), each constructed from one or more permutational cells. We have developed an operational method to distinguish otherwise identical permutational cells. The limitations and most probable sources of error associated with this numerical method are discussed as are directions for future research.