were allowed to relax in alternate cycles. The atomic positions were considered relaxed when the total energy had converged to within 10 meV between ionic steps. In the calculations of the hybrid structures the relative positions of the in-plane tungsten and oxygen atoms were fixed at the origin and half-way along each planar axis, respectively.

Cubic and hexagonal alkali bronzes
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Structure
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The alkali elements (H, Li, Na, K, Rb and Cs) were each used as the intercalated species in cubic MWO$_3$ ($x=1$ bronzes) and hexagonal M$_{0.33}$WO$_3$ (full intercalation of the hexagonal tunnels). In each case the intercalated metal atom was placed in the plane of the apical oxygen atoms, in the centre of the cavity (Figure \[fig:hex-cub-struc\]). WO$_3$ does not form a perfectly cubic cell at room temperature, as the WO$_6$ octahedra are slightly distorted in terms of W$-$O bond lengths and W$-$O$-$W bond angles, due to antiferroelectric displacement of the tungsten atoms and subsequent rotation of the WO$_6$ octahedra [@186; @90]. However this is not taken into account in the calculated system; the cell always relaxes to a cubic structure. In calculations involving supercells of tungsten trioxide the distortions are seen [@521]. Tungsten bronzes on the other 