Virial Exchange-Correlation Energy Density In Hooke'S Atom

Application of the virial theorem to the interelectronic Coulomb repulsion shows that the virial of the exchange potential yields the exchange energy. However, the virial of the correlation potential does not yield the correlation energy. We have recently constructed a "hyyercorrelated" potential whose virial is the correlation energy. We apply these ideas to a system which contains two interacting electrons in an external harmonic potential, Hooke's atom. This system can be solved analytically for a set of spring constants and numerically for any spring constant. By inverting the Kohn-Sham equations, the exact exchange and correlation potentials can be found. These exact values are compared with several popular approximate functionals, namely local spin density (LSD), Perdew, Burke, and Ernzerhof (PBE), and Becke and Lee-Yang-Parr (BLYP). We illustrate our results for two values of the spring constant. At a moderate value, the density is comparable to the He atom, while for a low spring constant, we explore extremely low densities. (C) 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 533-540, 1998.