Fichier:Kathmandu piris.pdf

Can NOFT bridge the gap between DFT and WFT?

Since the molecular Hamiltonian operator contains only one- and two-electron operators, the energy of a molecule can be determined exactly from the knowledge of the one- and two-particle reduced density matrices (1- and 2-RDMs). The most accurate electronic structure methods are based on N-particle wave functions. Given an N-particle wave function, we have the N-RDM, and by contraction the 1- and 2-RDMs can be derived. Unfortunately, the N-particle wave-function is a too complex object and its manipulation becomes cumbersome as the system grows larger.

In 1964, Hohenberg and Kohn demonstrated that the ground state energy can be expressed as a functional of the one-electron density only. The density functional theory (DFT) has become very popular thanks to its relatively low computational cost. In the exact DFT, we should reconstruct the 1- and 2-RDMs from the density, however, most practical implementations of DFT are based on the Kohn-Sham formulation, in which the kinetic energy is not constructed as a functional of the density but rather from an auxiliary Slater determinant. The contribution from a part of the kinetic energy in the correlation potential is probably the main source of problems of present-day KS functionals. Another obstacle is the construction of a functional capable of describing the N-particle system. This functional N-representability is related to the N-representability of the 2-RDM. Even though DFT energies may lie quite close to the exact ones, it is not fully guaranteed not to be below the exact ones, as required by the variational principle.

In 1974, Gilbert proved for the 1-RDMs an analogous theorem to the Hohenberg-Kohn theorem for the electron density. He suggested an alternative viewpoint regarding 1-RDM functional theory. One can employ the exact functional with an approximate 2-RDM that is built from the 1-RDM using a reconstruction functional. The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly constructed and does not require a functional. Like for the density, the ensemble N-representability conditions of the 1-RDM are well-known, but naturally, this does not overcome the N-representability problem of the energy functional.

The 1-RDM functional is called Natural Orbital Functional (NOF) when it is based upon the spectral expansion of the 1-RDM. A reconstruction of the 2-RDM has been achieved using the cumulant expansion leading to an approximate NOF known in the literature as PNOF. The PNOF is based on an explicit ansatz of the two-particle cumulant λ(Δ,Π) satisfying the D-, Q- and G-necessary positivity conditions for the 2-RDM. In this presentation, the theory behind the PNOF is outlined. Special emphasis will be put on the spin conserving NOF theory and on the recent proposed algorithm which yields the natural orbitals by an iterative diagonalization of a generalized pseudo-Fockian matrix. Some examples of strongly correlated systems, where density functionals yield pathological failures, are also presented to illustrate the potentiality of the NOF theory.