Asymptotic Structure in the Classically Forbidden Region of the Hooke's Atoms

The two-electron Hooke's atom — a quantum mechanical system with two electrons bound in a harmonic potential — is well known for its exact analytical properties at certain oscillator strengths. The Hooke's atoms with more than two electrons offer more scope for valuable practical applications. In this work, we study the asymptotic structure of these Hooke's atoms in the classically forbidden region. The leading-order term of the long-range expression for the KS exchange-correlation potential vxc(r) is shown to be −1/r. The second and third higher order terms are also exactly obtained. Various components of vxc(r) are also studied. It is shown that the leading term of O(1/r) in vxc(r) is due to the pure Pauli correlation, while the leading contribution of the Coulomb correlation is of O(1/r3). Neither of them makes contribution to the term of O(1/r2), which is shown to be solely due to the kinetic correlation effect. Results for the two-electron Hooke's atom were obtained before in the literature. Our results reduce to those of the two-electron Hooke's atom as a special case. (condensed matter: electronic structure, electrical, magnetic, and optical properties)