  Rapidly accreting white dwarfs (RAWDs) have been proposed as contributors to
the chemical evolution of heavy elements in the Galaxy. Here, we test this
scenario for the first time and determine the contribution of RAWDs to the
solar composition of first-peak neutron-capture elements. We add the
metallicity-dependent contribution of RAWDs to the one-zone galactic chemical
evolution code OMEGA according to RAWD rates from binary stellar population
models combined with metallicity-dependent $i$-process stellar yields
calculated following the models of Denissenkov et al. (2017). With this
approach we find that the contribution of RAWDs to the evolution of heavy
elements in the Galaxy could be responsible for a significant fraction of the
solar composition of Kr, Rb, Sr, Y, Zr, Nb, and Mo ranging from $2$ to $45\%$
depending on the element, the enrichment history of the Galactic gas, and the
total mass ejected per RAWD. This contribution could explain the missing solar
Lighter Element Primary Process for some elements (e.g., Sr, Y, and Zr). We do
not overproduce any isotope relative to the solar composition, but $^{96}$Zr is
produced in a similar amount. The $i$ process produces efficiently the Mo
stable isotopes $^{95}$Mo and $^{97}$Mo. When nuclear reaction rate
uncertainties are combined with our GCE uncertainties, the upper limits for the
predicted RAWD contribution increase by a factor of $1.5-2$ for Rb, Sr, Y, and
Zr, and by 3.8 and 2.4 for Nb and Mo, respectively. We discuss the implication
of the RAWD stellar evolution properties on the single degenerate Type Ia
supernova scenario.
