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An effective set of the Hartree-Fock (HF) equations are derived for electronsof the muonic systems, i.e., molecules containing a positively charged muon,conceiving the muon as a quantum oscillator, which are completely equivalent tothe usual two-component HF equations used to derive stationary states of themuonic molecules. In these effective equations, a non-Coulombic potential isadded to the orthodox coulomb and exchange potential energy terms, whichdescribes the interaction of the muon and the electrons effectively and isoptimized during the self-consistent field cycles. While in the two-componentHF equations muon is treated as a quantum wave, in the effective HF equationsit is absorbed into the effective potential and practically transformed into aneffective potential field experienced by electrons. The explicit form of theeffective potential depends on the nature of muon vibrations and is derivablefrom the basis set used to expand the muonic spatial orbital. The resultingeffective Hartree-Fock equations are implemented computationally and usedsuccessfully, as a proof of concept, in a series of the muonic moleculescontaining all atoms from the second and third rows of the Periodic Table. Tosolve the algebraic version of the equations muon-specific gaussian basis setsare designed for both muon and surrounding electrons and it is demonstratedthat the optimized exponents are quite distinct from those derived for thehydrogen isotopes. The developed effective HF theory is quite general and inprinciple can be used for any muonic system while it is the starting point fora general effective electronic structure theory that incorporates various typesof quantum correlations into the muonic systems beyond the HF equations.

Toward a muon-specific electronic structure theory: Effective electronic Hartree-Fock equations for the muonic molecules