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Physical Review A

Unexpected properties of a density functional

Jacek Karwowski, Monika Stanke

Published Feb 10, 2005• 2 citations
Publisher

Abstract

An observation on a pathological behavior of an exact density functional derived from either relativistic (Dirac) or nonrelativistic (L\'evy-Leblond) quantum-mechanical equation is reported. As expected, in the case of a one-electron atom the variational minimum of this functional is equal to the exact ground-state energy. However, apart from the correct density, this minimum is reached also by an infinite set of densities which do not correspond to the exact wave function. This paradoxical property of the functional is related to the multicomponent structure of both Dirac and L\'evy-Leblond wave functions. In particular, imposing the correct boundary conditions upon the trial densities removes only a part of the fake solutions. The results of this study demonstrate that in density-functional theories derived from models based on multicomponent wave functions, one should not expect any simple relation between the accuracy of the energy and the correctness of the corresponding density.

DOI
10.1103/physreva.71.024501

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An observation on a pathological behavior of an exact density functional derived from either relativistic (Dirac) or nonrelativistic (L'evy-Leblond) quantum-mechanical equation is reported. As expected, in the case of a one-electron atom the variational minimum of this functional is equal to the exact ground-state energy. However, apart from the correct density, this minimum is reached also by an infinite set of densities which do not correspond to the exact wave function. This paradoxical property of the functional is related to the multicomponent structure of both Dirac and L'evy-Leblond wave functions. In particular, imposing the correct boundary conditions upon the trial densities removes only a part of the fake solutions. The results of this study demonstrate that in density-functional theories derived from models based on multicomponent wave functions, one should not expect any simple relation between the accuracy of the energy and the correctness of the corresponding density.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
Abstract Relations between the extended wave functions defined by Carbó‐Dorca in connection with studies on quantum similarity problems and the wave functions of Schrödinger, Lévy‐Leblond, and Dirac equations are investigated. In particular, it is pointed out that solutions of the Lévy‐Leblond equation are equal to the appropriate extensions of the corresponding Schrödinger wave functions. The quantum mechanical continuity equation is applied to analyze the physical meaning of the extended wave functions. Finally, usefulness of the concept of the extended wave function in designing relativistic variational calculations in kinetically balanced basis sets is briefly discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 331–337, 2001
It is shown that the ground-state energy as a functional solely of the electron density is determined by the asymptotic value of the derivative of the degree-one homogeneous extension of the universal density functional F[n] at the given electron number. This has the consequence that its derivative cannot be properly determined. Carrying out the derivative of E[n[N,v]] with respect to v(r) leads to a paradox, which is resolved by the non-differentiability of E[n] if one follows traditional wisdom regarding the non-invertibility of the linear response function. However, considering the derivative of v[n[v]] through the one-electron case shows that this paradox has a more elementary origin, namely, an unaccounted restriction of the v(r) domain.