of the components (WO$_3$ plus the metal cation) from the ground state energy of the final product (tungsten bronze). A negative energy of formation therefore indicates that the compound formed is stable.

     Compound     Cubic(eV)   Hexagonal(eV)
  -------------- ----------- ---------------
      WO$_3$          0          -0.016
   H$_x$WO$_3$      3.059         1.047
   Li$_x$WO$_3$    -2.023        -0.560
   Na$_x$WO$_3$    -2.373        -0.813
   K$_x$WO$_3$     -2.354        -1.132
   Rb$_x$WO$_3$    -2.039        -1.203
   Cs$_x$WO$_3$    -0.637        -1.245

![\[fig:bronze\_energ\] Calculated energies of formation per WO$_3$ unit for cubic and hexagonal tungsten bronzes, relative to cubic WO$_3$.](bronze_energ.eps){width="85mm"}

Firstly, a comparison of cubic and hexagonal WO$_3$ shows that the hexagonal phase has a very similar energy to the cubic phase. Literature results indicate that the cubic phase is preferred, although the hexagonal phase is stable up to temperatures of 400-500$^o$C, indicating that the phase transition has a high activation energy [@90; @106]. In both the cubic and hexagonal systems, the hydrogen-intercalated bronze energy is positive and large, indicating that the hydrogen bronzes are not stable. In the experimental system the hydrogen bronzes are easily oxidised as the protons are highly mobile [@255; @hobbs]. This result also relates to hydrogen being the only intercalate that does not ionise in the bronze structures, as evidenced by the charge density plots earlier. For the 