  An effective set of the Hartree-Fock (HF) equations are derived for electrons
of the muonic systems, i.e., molecules containing a positively charged muon,
conceiving the muon as a quantum oscillator, which are completely equivalent to
the usual two-component HF equations used to derive stationary states of the
muonic molecules. In these effective equations, a non-Coulombic potential is
added to the orthodox coulomb and exchange potential energy terms, which
describes the interaction of the muon and the electrons effectively and is
optimized during the self-consistent field cycles. While in the two-component
HF equations muon is treated as a quantum wave, in the effective HF equations
it is absorbed into the effective potential and practically transformed into an
effective potential field experienced by electrons. The explicit form of the
effective potential depends on the nature of muon vibrations and is derivable
from the basis set used to expand the muonic spatial orbital. The resulting
effective Hartree-Fock equations are implemented computationally and used
successfully, as a proof of concept, in a series of the muonic molecules
containing all atoms from the second and third rows of the Periodic Table. To
solve the algebraic version of the equations muon-specific gaussian basis sets
are designed for both muon and surrounding electrons and it is demonstrated
that the optimized exponents are quite distinct from those derived for the
hydrogen isotopes. The developed effective HF theory is quite general and in
principle can be used for any muonic system while it is the starting point for
a general effective electronic structure theory that incorporates various types
of quantum correlations into the muonic systems beyond the HF equations.
