The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly: rather, its existence is inferred from the various experiments whose outcome is most rationally explained using a wave function interpretation of quantum mechanics. Further, the N-particle wave function is a rather complicated construction, depending on 3N spatial coordinates as well as N spin coordinates, correlated in a manner that almost defies description. By contrast, the electron density of an N-electron system is a much simpler quantity, described by three spatial coordinates and even accessible to experiment. In terms of the wave function, the electron density is expressed as . . . ρ(r) = N ∫ Ψ* (r1,r2,...,rN)Ψ (r1,r2,...,rN)dr2dr3 ...drN (14.1) . . . where the sum over spin coordinates is implicit. It might be much more convenient to have a theory based on the electron density rather than the wave function. The description would be much simpler, and with a greatly reduced (and constant) number of variables, the calculation of the electron density would hopefully be faster and less demanding. We also note that given the correct ground state density, we should be able to calculate any observable quantity of a stationary system. The answer to these hopes is density functional theory, or DFT. Over the past decade, DFT has become one of the most widely used tools of the computational chemist, and in particular for systems of some size. This success has come despite complaints about arbitrary parametrization of potentials, and laments about the absence of a universal principle (other than comparison with experiment) that can guide improvements in the way the variational principle has led the development of wave-function-based methods. We do not intend to pursue that particular discussion, but we note as a historical fact that many important early contributions to relativistic quantum chemistry were made using DFT-like methods. Furthermore, there is every reason to try to extend the success of nonrelativistic DFT methods to the relativistic domain. We suspect that their potential for conquering a sizable part of this field is at least as large as it has been in the nonrelativistic domain.
Lieb-Wu Solution, Gutzwiller-Wave-Function, and Gutzwiller-Ansatz Approximations with Adjustable Single-Particle Wave Function for the Hubbard Chain
