 
Summary: Two interacting electrons in a spherical box: An exact diagonalization study
David C. Thompson* and Ali Alavi
University of Cambridge, Chemistry Department, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Received 24 June 2002; Revised manuscript received 18 September 2002; published 31 December 2002
We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded
by an infinite wall using exact diagonalization. We have also investigated the influence of an additional
parabolic potential of strength k) arising from a uniform background smeared throughout the sphere. The
convergence of the ground state energy of the singlet spin state of the system is investigated as a function of
sphere size essentially rs , the WignerSeitz density parameter for cases where there is no background
potential (k 0) and for when k 0. With k 0 and small rs , we observe a maximum in the ground state
density at the origin of the sphere. At rs 8 a.u., the ground state density acquires a minimum at the origin. For
this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate
the ground state density as a function of k and also the correlation hole density as a function of rs and k. We
invert the KohnSham equation for a two electron system and calculate the local effective potential and
correlation potential to within an additive constant as functions of the radial coordinate for a number of
values of rs and k.
DOI: 10.1103/PhysRevB.66.235118 PACS number s : 71.15.Mb, 73.21. b
I. INTRODUCTION
Improvements in experimental methods of confining elec
trons and in the techniques used to investigate these systems
