given in Table \[table:bronze\_cells\]. These are generally lower than the calculated values we have obtained, although most discrepancies are less than 0.5$\%$.

       Compound        Experimental a ([Å]{})   Calculated a ([Å]{})   Experimental c ([Å]{})   Calculated c ([Å]{})
  ------------------- ------------------------ ---------------------- ------------------------ ----------------------
        WO$_3$                 7.298                   7.4103                  3.899                   3.8144
   H$_{0.33}$WO$_3$             7.38                   7.4173                   3.78                   3.8111
   Li$_{0.33}$WO$_3$           7.405                   7.4007                  3.777                   3.8219
   Na$_{0.33}$WO$_3$            7.38                   7.4034                  3.775                   3.8248
   K$_{0.33}$WO$_3$             7.37                   7.4010                   3.77                   3.8282
   Rb$_{0.33}$WO$_3$            7.38                   7.4163                   3.78                   3.8289
   Cs$_{0.33}$WO$_3$            7.38                   7.4507                  3.785                   3.8342

For the hexagonal system, the changes in the lattice parameters are much less pronounced than in the cubic system, as shown in Table \[table:cub-hex-comp\]. This is due to the hexagonal tunnel spaces being much larger than the cavities in the cubic system and so the interactions between the inserted ion and the WO$_3$ lattice are smaller.

     Compound     Cubic ([Å]{}$^3$)   Hexagonal ([Å]{}$^3$)
  -------------- ------------------- -----------------------
      WO$_3$           55.4977               60.4648
   H$_x$WO$_3$         55.8946               60.5278
   Li$_x$WO$_3$        58.0127               60.4270
   Na$_x$WO$_3$        58.9116               60.5158
   K$_x$WO$_3$         61.0457               60.6785
   Rb$_x$WO$_3$        63.1294               60.7928
   Cs$_x$WO$_3$        65.9809               61.4445

Charge density
--------------

Charge density plots of the cubic system taken in a plane through the centre of the cell where the intercalated atom sits, reveal that hydrogen behaves differently from the other intercalates. This is shown in Figure \[fig:cub\_hex\_chg\]. 