  The pyrovanadates $\beta$-Mn$_2$V$_2$O$_7$ and $\beta$-Cu$_2$V$_2$O$_7$ were
previously investigated as photoanode materials for water splitting. Neither of
them, however, was found to be sufficiently active. In this work we predict the
properties of solid solutions of these two structurally similar pyrovanadates
via density functional theory calculations to explore the suitability of their
band structure for water splitting and to assess their ease of synthesis. We
predict that substitution of up to 20\% Cu or Mn into $\beta$-Mn$_2$V$_2$O$_7$
and $\beta$-Mn$_2$V$_2$O$_7$ respectively leads to a narrowing of the band gap,
which in the former case is experimentally confirmed by UV-vis spectroscopy.
Calculations for solid solutions in the intermediate composition range,
however, yield nearly constant band gaps. Moreover, we predict solid-solutions
with higher substitution levels to be increasingly difficult to synthesize,
implying that solid solutions with low substitution levels are most relevant in
terms of band gaps and ease of synthesis.
