Natural Orbital Functional for the Many-Electron Problem

The solution of the quantum mechanical many-electron problem is one of the central problems of physics. A great number of schemes that approximate the intractable many-electron Schrodinger equation have been devised to attack this problem. Most of them map the manybody problem to a self-consistent one-particle problem. Probably the most popular method at present is density functional theory (DFT) [1] especially when employed with the generalized gradient approximation (GGA) [2,3] for the exchange-correlation energy. DFT is based on the Hohenberg-Kohn theorem [4] which asserts that the electronic charge density completely determines a manyelectron system and that, in particular, the total energy is a functional of the charge density. Attempts to construct such a functional for the total energy have not been very successful because of the strong nonlocality of the kinetic energy term. The Kohn-Sham scheme [5] where the main part of the kinetic energy, the single particle kinetic energy, is calculated by solving one-particle Schrodinger equations circumvented this problem. The difference between the one-particle kinetic energy and the many-body kinetic energy is a component of the unknown exchange-correlation functional. The exchangecorrelation functional is thus a sum of a kinetic energy contribution and a potential energy contribution, and partly for this reason it does not scale homogeneously [6] under a uniform spatial scaling of the charge density. It has been known for a long time that one can also construct a total energy functional using the firstorder reduced density matrix. Several discussions of the existence and the properties of such a functional can be found in the literature [7‐ 10]. However, no explicit functional has ever been constructed and tested on real physical systems. An important advantage of this approach is that one employs an exact expression for the many-body kinetic energy. Only the small non-HartreeFock-like part of the electronic repulsion is an unknown functional [9]. We propose in this paper an explicit form of such a functional in terms of the natural orbitals. The high accuracy of this natural orbital functional theory (NOFT) is then established by applying it to several atoms and ions. If C is an arbitrary trial wave function of an N-electron system, the first- and second-order reduced density matrices [11,12], g1 and g2, are