Ad Hoc Approximations Within Quantal Density Functional Theory

Abstract

In quantum mechanics, one chooses the level of approximation in terms of electron correlations by the ad hoc choice of approximate wave function. For example, in the Hartree approximation [1] (see also QDFT), one chooses a wave function that is a product of single-particle spin-orbitals. The Hartree wave function does not obey the Pauli exclusion principle, as it is not antisymmetric in an interchange of the coordinates of the electrons including its spin coordinate. Thus, electron correlations due to the Pauli principle are ignored in this choice of wave function. (In calculations performed within the Hartree approximation [2], one incorporates the Pauli exclusion principle in an ad hoc manner by ensuring that no two electrons occupy the same state.) The Hartree wave function also ignores Coulomb correlations between the electrons: there are no terms, for example, involving the inter-electronic separation in the wave function. Hence, Hartree theory is said to be an independent particle approximation. The best orbitals for this product wave function in terms of the energy, are then obtained by application of the variational principle for the energy [3] leading to the Hartree equations. The energy obtained is then a rigorous upper bound to the true nonrelativistic value. It is interesting to note that this independent particle model leads to various properties that are obtained accurately. For example, atomic shell structure throughout the Periodic Table is exhibited [4] via Hartree theory, with highly accurate core-valence separations. The Sommerfeld–Hartree model of a simple metal [5], which is an example of a noninteracting uniform electron gas, also reproduces accurately experimental properties such as the electronic specific heat at low temperatures. On the other hand, as no electron correlations are accounted for in the theory, the Sommerfeld–Hartree model does not allow for cohesion in the jellium approximation of metals.