  The isotope $ {}^{99} \rm{Mo} $, the generator of $ {}^{99m} \rm{Tc} $ used
for diagnostic imaging, is supplied by extracting from fission fragments of
highly enriched uranium in reactors. However, a reactor-free production method
of $ {}^{99} \rm{Mo} $ is searched over the world from the point of view of
nuclear proliferation. Recently, $ {}^{99} \rm{Mo} $ production through a muon
capture reaction was proposed and it was found that about $ 50 \, \% $ of $
{}^{100} \rm{Mo} $ turned into $ {}^{99} \rm{Mo} $ through $ {}^{100} \rm{Mo}
\left( \mu^-, n \right) $ reaction [arXiv:1908.08166]. However, the detailed
physical process of the muon capture reaction is not completely understood. We,
therefore, study the muon capture reaction of $ ^{100} \rm{Mo} $ by a
theoretical approach. We used the proton-neutron QRPA to calculate the muon
capture rate. The muon wave function is calculated with considering the
electronic distribution of the atom and the nuclear charge distribution. The
particle evaporation process from the daughter nucleus is calculated by a
statistical model. From the model calculation, about $ 38 \, \% $ of $ {}^{100}
\rm{Mo} $ is converted to $ {}^{99} \rm{Mo} $ through the muon capture
reaction, which is in a reasonable agreement with the experimental data. It is
revealed that negative parity states, especially $ 1^- $ state, play an
important role in $ {}^{100} \rm{Mo} \left( \mu^-, n \right) {}^{99} \rm{Nb} $.
The feasibility of $ {}^{99} \rm{Mo} $ production by the muon capture reaction
is also discussed. Isotope production by the muon capture reaction strongly
depends on the nuclear structure.
