Correcting basis-set errors due to the absence of the electron-electron cusp using an adiabatic correction
This article proposes an analytical method to address the slow convergence of electronic structure calculations caused by the inability of finite one-particle basis sets to describe the electron-electron cusp. An equivalence is established between a calculation using a finite basis set with the physical Coulomb interaction and a calculation using a complete basis set with a model interaction (specifically, the error-function screened Coulomb potential characterized by a range-separation parameter $\mu$). By leveraging the adiabatic connection formalism, a simple, parameter-free correction formula is derived. It depends only on the on-top pair density and a locally defined range-separation parameter ($\mu$) derived from the basis set itself. This `adiabatic connection based basis-set error correction' (ABC) is derived from the asymptotic expansion of the wave function at large $\mu$ for small inter-electronic distances. Therefore it is applicable to both ground and excited states without the restriction imposed by the Hohenberg-Kohn theorem. Numerical tests illustrate that the method achieves chemical accuracy using smaller basis sets than typically required.