Subsystem functionals in density-functional theory: Investigating the exchange energy per particle

A viable way of extending the successful use of density-functional theory into studies of even more complex systems than are addressed today has been suggested by Kohn and Mattsson [W. Kohn and A. E. Mattsson, Phys. Rev. Lett. 81, 3487 (1998); A. E. Mattsson and W. Kohn, J. Chem. Phys. 115, 3441 (2001)], and is further developed in this work. The scheme consists of dividing a system into subsystems and applying different approximations for the unknown (but general) exchange-correlation energy functional to the different subsystems. We discuss a basic requirement on approximative functionals used in this scheme; they must all adhere to a single explicit choice of the exchange-correlation energy per particle. From a numerical study of a model system with a cosine effective potential, the Mathieu gas, and one of its limiting cases, the harmonic oscillator model, we show that the conventional definition of the exchange energy per particle cannot be described by an analytical series expansion in the limit of slowly varying densities. This indicates that the conventional definition is not suitable in the context of subsystem functionals. We suggest alternative definitions and approaches to subsystem functionals for slowly varying densities and discuss the implications of our findings on the future of functional development.