Asymptotic behavior of the exchange-correlation potentials from the linear-response Sham–Schlüter equation

The linear-response Sham–Schlüter equation can be used to calculate an exchange-correlation potential starting from a given approximation for the self-energy. The asymptotic behavior of these potentials is, however, much debated, a recent work suggesting that they could blow up in finite systems. Here we investigate the asymptotic behavior of the linear-response Sham–Schlüter potentials in the GW and second-order approximations for the self-energy. We show that these potentials do not diverge, and that the correlation potential itself has a −α/(2r4) tail (under appropriate conditions), where α depends on the self-energy. We also provide further justification for the quasiparticle approximation to the linear-response Sham–Schlüter equation, that is much simpler to solve while likely being of comparable accuracy. Calculations for real molecules or solids using this approximation should be within the reach of present computers.