Two-component density functional theory for muonic molecules: Inclusion of the electron-muon correlation functional

It is well-known experimentally that the positively-charged muon and the muonium atom may bind to molecules and solids, and through muon$'$s magnetic interaction with unpaired electrons, valuable information on the local environment surrounding the muon is deduced. Theoretical understanding of the structure and properties of resulting muonic species requires accurate and efficient quantum mechanical computational methodologies. In this paper the two-component density functional theory, TC-DFT, as a first principles method, which treats electrons and the positive muon on an equal footing as quantum particles, are introduced and implemented computationally. The main ingredient of this theory, apart from the electronic exchange-correlation functional, is the electron-muon correlation functional which is foreign to the purely electronic DFT. A Wigner-type local electron-muon correlation functional, termed e$\mu$c-1, is proposed in this paper and its capability is demonstrated through its computational application to a benchmark set of organic muonic molecules. The TC-DFT equations containing e$\mu$c-1 are not only capable of predicting the muon$'$s binding site correctly but they also reproduce muon$'$s zero-point vibrational energies and the muonic densities much more accurately than the TC-DFT equations lacking e$\mu$c-1. Thus, this study set the stage for developing accurate electron-muon functionals, which can be used within the context of the TC-DFT to elucidate the intricate interaction of the positive muon with complex molecular systems.