Quantal Density Functional Theory of the Density Amplitude

Abstract

The Quantal density functional theory (Q-DFT) mapping from a system of electrons in an external electrostatic field in any state as described by Schrödinger theory to one of noninteracting bosons in their ground state but with the same density is described. The corresponding Schrödinger equation of the model bosons is for the density amplitude, with the sole eigenvalue being the negative of the ionization potential. Via the ‘Quantal Newtonian’ first law for the model system, the local potential representative of the many-body effects in this equation is the work done in a conservative effective field. The field is the sum of a component representative of electron correlations due to the Pauli Exclusion Principle and Coulomb repulsion, and another of Correlation-Kinetic effects—the difference between these effects for the interacting fermionic and noninteracting bosonic systems. The corresponding components of the total energy are expressed in integral virial form in terms of the respective fields. The traditional density functional theory definitions of these energies and potentials in terms of energy functionals of the density and their functional derivatives are given. The Levy-Perdew-Sahni definition of the local potential in terms of the wave function written as the product of a marginal and conditional probability amplitude is derived. The maps to the model systems of noninteracting bosons and fermions having the same density are related by the Pauli potential and Pauli kinetic energy. By Q–DFT, it is shown that these energies are not a consequence of the Pauli principle but rather a consequence of kinetic effects of the model systems. The Q–DFT definitions of these energies is given. Finally, the mapping to the model of noninteracting bosons is shown to be a special case of that to noninteracting fermions.