Figure 1

Two possible arrangements of H bonds within a 16-water unit cell of ice $\mathrm{I}h$. The possible H-bond isomers are summarized mathematically by directed graphs in which directional bonds point from H-bond donor to H-bond acceptor, as illustrated for an isomer (b).Reuse & Permissions

Figure 2

(a) Graph invariant fit to the energies of the 52 H-bond isomers of a 16-water unit cell of ice VII. (b) Calculated DFT energy of H-bond isomers of a 32-water ice VII cell plotted against energies predicted from graph invariant parameters derived from the 16-water cell. (c) Graph invariant fit to the energies of the 30 H-bond isomers of 8-water orthorhombic and 12-water hexagonal unit cells of ice $\mathrm{I}h$. (d) Calculated DFT energy of H-bond isomers of a 48-water ice-$\mathrm{I}h$ cell plotted against energies predicted from graph invariant parameters derived from the smaller cell. The DFT energies in plots (a)–(d) are calculated using the BLYP functional and the CPMD program. The atomic positions are fully optimized within unit cells whose size is fixed at the experimental value [24]. A line of slope unity is shown to indicate where points would lie for perfect agreement. (e) Average energy plotted as a function of temperature from Metropolis Monte Carlo simulations [24] for large simulation cells for ice $\mathrm{I}h/\mathrm{XI}$ (filled symbols) and ice $\mathrm{VII}/\mathrm{VIII}$ (open symbols). Data are presented for series of Metropolis Monte Carlo runs ascending (▲, △) and descending (▼, ▽) in temperature.Reuse & Permissions

Figure 3

Relative energy of H-bond isomers calculated by periodic electronic DFT methods for 16 isomers of an 8-water orthorhombic unit cell (open symbols) and 14 isomers of a 12-water hexagonal unit cell (filled symbols) listed in order of increasing fraction of trans H bonds. The bottom two graphs (dotted lines) give the fraction of trans H bonds associated with each isomer of the orthorhombic (◇) and hexagonal (◆) unit cells. The energy was calculated [24] using (●, ○) the BLYP functional, plane wave basis set and Troullier-Martins pseudopotentials with the CPMD program, (■, □) BLYP functional using numerical basis sets with the DMol program, or (▲, △) the PW91 functional using plane wave basis with the CASTEP program. Solid lines: energy of H-bond isomers before geometry optimization. Dashed lines: energies after optimization of the molecular coordinates, and for the CASTEP results cell dimensions as well. The 12 energy data sets are plotted with their average taken as the zero of energy to facilitate comparison of the relative energies of the isomers. The 12 sets are broken into four groups, according to the two unit cells and whether the geometry is optimized or unoptimized. Successive groups of three curves are shifted by $0.06\mathrm{kcal}/\mathrm{mol}$ for clarity. The $Cmc{2}_1$ isomers for the orthorhombic and hexagonal cells are indicated.Reuse & Permissions