  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, is introduced and implemented computationally. The main ingredient
of this theory, apart from the electronic exchange-correlation functional, is
the electron-positive muon correlation functional which is foreign to the
purely electronic DFT. A Wigner-type local electron-positive 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 muonic
organic 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-positive 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.
