Exact exact-exchange potential of two- and one-dimensional electron gases beyond the asymptotic limit

The exchange-correlation potential experienced by an electron in the free space adjacent to a solid surface or to a low-dimensional system defines the fundamental image states and is generally important in surface and nanoscience. Here we determine the potential near the two- and one-dimensional electron gases (EG), doing this analytically at the level of the exact exchange of the density-functional theory (DFT). We find that, at r kF-1, where r is the distance from the EG and kF is the Fermi radius, the potential obeys the already known asymptotic -e2/r, while at rF-1, but still in vacuum, qualitative and quantitative deviations of the exchange potential from the asymptotic law occur. The playground of the excitations to the low-lying image states falls into the latter regime, causing significant departure from the Rydberg series. In general, our analytical exchange potentials establish benchmarks for numerical approaches in the low-dimensional science, where DFT is by far the most common tool.