Adiabatic connection in the low-density limit

In density-functional theory, the exchange-correlation functional can be exactly expressed by the adiabatic connection integral. It had been noticed that as $\lambda \to \infty$, the ${\lambda }^{-1}$ term in the expansion of $W\left(\lambda \right)$ vanishes. We provide a simple derivation of this exact condition in this work. We propose a simple parametric form for the integrand, satisfying this condition, and show that it is highly accurate for weakly correlated two-electron systems.

Figure 1

Comparison of three different approximations to $W\left(\lambda \right)$ for Hooke’s atom $\left(k=1/4\right)$, plotted as $\Delta W=W^{\text{model}}-W^{\text{exact}}$. The exact curve (up to $\lambda =3$) is taken from Ref. [19].

Figure 2

Comparison of three different approximations to $W\left(\lambda \right)$ for helium atom, plotted as $\Delta W=W^{\text{model}}-W^{\text{exact}}$. The discrete values are shown, as well as fitting curves to aid the eyes. $W^{\text{exact}}$ values (up to $\lambda =1$) are taken from Ref. [22].