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We construct theoretical models for the diffuse Galactic gamma ray continuum emission with particular emphasis on low energies where the OSSE instrument on the Compton Gamma Ray Observatory is sensitive (about 50 keV to 10 MeV). This emission is largely the sum of the contributions of electron bremsstrahlung resulting from collisions of comparable energy electrons with interstellar gas and inverse Compton scattering of much higher energy electrons with ambient interstellar radiation fields. We extrapolate the observed high energy ($>10$ GeV) cosmic ray electron spectrum down to energies below $10$ MeV. Using models for the distributions of the ambient gas and radiation fields we calculate the low energy gamma ray continuum emissivity in the Galaxy. From this we calculate model intensity distributions and fold them through the OSSE angular response function to obtain model fluxes. These are then compared with the data to constrain the parameters of the models. Previous calculations show that if this interpretation of the diffuse low energy gamma ray emission is correct then a large power ($\sim 10^{41}$ erg/s) is supplied to the low energy cosmic ray electrons. In turn, these electrons deposit much of this power into the interstellar medium and, therefore, could be the agent responsible for energetically maintaining the warm partially ionize component of the interstellar medium.
