A one-dimensional model of the Martian ionosphere is used to explore the importance of atomic and molecular hydrogen chemistry in the upper atmosphere and ionosphere. Neutral and ionized H and H-2 undergo chemical reactions that lead to the production of the hydrogenated ions: H+, H-2(+), H-3(+), OH+, HCO+, ArH+, N2H+, HCO2+, and HOC+. Simulations are conducted for the cases of photochemistry only and photochemistry coupled with transport in order to asses the separate effects of plasma diffusion in the topside ionosphere. For both of these cases, the sensitivity of the ionosphere is tested for (1) molecular hydrogen abundance and (2) reaction rate, k(1), for the charge exchange between H+ and H-2. Results are reported for midday solar minimum conditions. We find that the ionospheric composition of Mars is sensitive to H-2 abundance, but relatively insensitive to the reaction rate, k(1). Depending on the conditions simulated, the topside ionosphere can contain appreciable amounts of hydrogenated species such as H-3(+), OH+, and HCO+. Comparisons are made with Viking ion density measurements as well as with results of other published Mars ionospheric models. Future comparisons with more extensive ion composition will be available when the Mars Atmosphere and Volatile Evolution mission arrives at Mars.