  A main source of capacity fading in lithium-ion batteries is the degradation
of the active cathode materials caused by the series of volume changes during
charge and discharge cycles. The quaternary colquiriite-type fluorides
Li$_x$CaFeF$\mathrm{_6}$ and Li$_x$CaCoF$\mathrm{_6}$ were reported to have
negligible volume changes in specific Li concentration ranges, making the
underlying colquiriite structure a promising candidate for so-called
zero-strain behavior. Using first-principles electronic structure calculations
based on density functional theory with a Hubbard-$U$ correlation correction on
the transition-metal ions, we systematically investigate the equilibrium
volumes of the colquiriite-type fluorides Li$_x$CaMF$\mathrm{_6}$ with M =Ti,
V, Cr, Mn, Fe, Co, and Ni at the Li concentrations $x$=0, 1, and 2. We
elucidate the connection between the total volume of the structures and the
local volumes of fluorine coordinated octahedra around the cations, and we find
trends along the series of the 3d transition-metal elements. In the lithiation
step from $x$=1 to $x$=2 we find volume changes of about 10 %, and we discuss
the discrepancy to the experimentally reported smaller value for
Li$_x$CaFeF$\mathrm{_6}$. The suitability as cathode material was furhter
investigated by calculating the theoretical voltages and capacities. From $x$=0
to $x$=1 we describe the compensating structural mechanisms that lead to an
exceptionally small volume change of Li$_x$CaMnF$\mathrm{_6}$, which posseses a
high theoretical voltage and moderate capacity. This compound is therefore a
particularly promising zero-strain cathode material.
