Exchange, idempotent density matrices and beyond

Abstract

After a brief historical introduction, emphasis is placed on the treatment of exchange using density matrices that are explicitly idempotent. This embraces both Hartree–Fock and density functional ‘exchange-only’ methods. For an atom, the exchange energy density can be evaluated asymptotically, and unlike the local density approximation it involves not just one length, ρ(r)−1/3, but two, the second being the distance from the nucleus. It is also emphasized, following March and Santamaria [N.H. March, R. Santamaria, Phys. Chem. Liquids 19 (1984) 187] that non-local generalizations of kinetic and exchange energy densities can be related via idemopotency of the Dirac, first-order, density matrix. Examples are then given, starting with the two-level Be atom and its isoelectronic ions, where both Hartree–Fock and density functional exchange-only idempotent density matrices can be expressed in terms of the diagonal density and its gradient, gradρ. This example is followed by the example of a metal surface, but now within a jellium-like-model framework. Some explicit results are again given for such models. Attention is then shifted to a formally exact expression for the exchange-correlation potential, using the differential virial theorem following Holas and March [A. Holas, N.H. March, Phys. Rev. A 51 (1995) 2040]. After a brief physical discussion of the result, in terms now of interacting (as well as non-interacting reference) density matrices; both the now non-idempotent first-order density matrix and the interacting electron pair correlation function, the separation of the Holas–March exchange potential into a sum of exchange and correlation pieces is effected, following the approach of Levy and March [M. Levy, N.H. March, Phys. Rev. A 55 (1997) 1885], in which the strength of the electron–electron interactions is scaled appropriately. Finally, the exact equation of motion for the first-order density matrix is treated, and it is stressed here that this is satisfied exactly by the Hartree–Fock idempotent density matrix. Of course, the correct interacting matrix γ satisfies γ2<γ, where γ now includes the full electron–electron interactions. An example of γ as a function of the electron density is given for He-like atomic ions in the limit of large atomic number.