  Mechanical stresses and strains in the microstructure of cathode materials
evolving during charge/discharge cycles can reduce the long-term stability of
intercalation-type alkali-metal-ion batteries. In this context, crystalline
compounds exhibiting zero-strain (ZS) behavior are of particular interest.
Near-ZS sodiation was experimentally measured in the \textit{tetragonal
tungsten bronze} (TTB) type compound Na$_x$FeF$\mathrm{_3}$. Using a
first-principles method based on density functional theory, we investigate the
potential of iron-based fluoride compounds with tungsten bronze (TB) structures
as ZS cathode materials. Simulations were conducted to study the intercalation
of the alkali metal ions Li$\mathrm{^+}$, Na$\mathrm{^+}$, and K$\mathrm{^+}$
into the TTB and two related TB structures of the cubic perovskite (PTB) and
hexagonal (HTB) types. We describe compensating local volume effects that can
explain the experimentally measured low volume change of
Na$_x$FeF$\mathrm{_3}$. We discuss the structural and chemical prerequisites of
the host lattice for ZS insertion mechanism for alkali ions in TB structures
and present a qualitative descriptor to predict the local volume change, that
provides a way for faster screening and discovery of novel ZS battery
materials.
