17 September 2001 Physics Letters A 288 (2001) 101–104 www.elsevier.com/locate/pla Pair function at coincidence and ground-state energy for interacting systems of two fermions with isotropic harmonic confinement and antiparallel spins N.H. March a,b,c , I.A. Howard a,∗ , I. Nagy b,d , A. Rubio b a Department of Physics, University of Antwerp (RUCA), Groenenborgerlaan 171, B-2020 Antwerp, Belgium b Departamento de Física de Materialas, Facultad de Ciencias Químicas, Universidad del Pais Vasco, Donostia International Physics Center (DIPC) and Centro Mixto CSIC-UPV/EHU, 20018 San Sebastían/Donostia, Basque Country, Spain c Oxford University, Oxford, UK d Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, H-1521, Budapest, Hungary Received 10 July 2001; accepted 17 July 2001 Communicated by V.M. Agranovich Abstract Harmonically confined isotropic two-dimensional fermion systems with different interparticle interactions are compared and contrasted. For a harmonic oscillator interparticle force, the pair correlation function n 2 (r, r ′ ) at coincidence is determined analytically and is shown to uniquely fix the ground-state energy. The two-electron Hookean atom is cited for comparison.  2001 Elsevier Science B.V. All rights reserved. Keywords: 23.23.+x; 56.65.Dy Density functional theory [1] has had considerable successes since its initiation in the early days of quan- tum mechanics by Thomas [2] and Fermi [3]. Un- fortunately, to date, the functional required to cal- culate the ground-state energy E from the (assumed known) fermion density remains unknown. It is there- fore of continuing interest to study further simple solu- ble models with inhomogeneous particle densities and with interparticle interactions. The focus of this Letter is on two-fermion systems with antiparallel spins, in both of which the fermions are harmonically confined. * Corresponding author. E-mail address: howard@ruca.ua.ac.be (I.A. Howard). Then different interparticle interactions are ‘switched on’. As the first case, denoted (i) below, the study of Moshinsky [4] will be utilized. In addition to harmonic confinement of the two fermions, fermion–fermion in- teractions are introduced through a harmonic oscillator force. Then the Hamiltonian H takes the form [4] (1) H = 1 2 ( p 2 1 + r 2 1 ) + 1 2 ( p 2 2 + r 2 2 ) + 1 2 K |r 1 - r 2 | 2 , where r 1 and r 2 denote the positions of fermions 1 and 2, respectively, with p 2 1 /2 and p 2 2 /2 the cor- responding kinetic energy operators. The ground- state space wave function Ψ(r 1 , r 2 ) was obtained by Moshinsky [4], and since it will be seen below to be symmetric under interchange of the two fermions, it 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0375-9601(01)00479-0