An Exact Condition for the Integrand of Adiabatic Connection

In density functional theory (DFT), the exchange-correlation functional $E_{\rm XC}$ can be exactly expressed by the adiabatic connection integral [1,2]. The integrand should satisfy several exact conditions [3]. We show that for the low-density limit (as $\lambda \to \infty$), the $\lambda^{-1}$ term in the expansion of the integrand $W(\lambda)$, should vanish. We propose a simple parametric form for $W(\lambda)$, satisfying the new exact condition. We apply this interpolation form to Hooke's atom and helium atom and show that it is accurate for weakly-correlated two-electron systems. \\[3pt] [1] D.C. Langreth and J.P. Perdew, Solid State Commun. {\bf 17}, 1425 (1975). \\[0pt] [2] O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B {\bf 13}, 4274 (1976). \\[0pt] [3] M. Seidl, J.P. Perdew and M. Levy, Phys. Rev. A {\bf 59}, 51 (1999).