Chemistry Reference
In-Depth Information
(a)
(b)
0.1
0.1
0.05
0.05
0
0
0 0.05 0.1
gr. inv. prediction (kcal mol
-1
water
-1
)
0
0.05
0.1
gr. inv. prediction (kcal mol
-1
water
-1
)
(c)
0.1
0.05
0
0
0.05
0.1
gr. inv. prediction (kcal mol
-1
water
-1
)
Figure 12.
(a) Graph invariant (gr. inv.) fit to the energies of the 14 H-bond isomers of a 12-water
hexagonal(
•
) unit cell and the 16 H-bond isomers of an 8-water orthorhombic(
) unit cell of ice Ih.
(b) Calculated DFT energy of H-bond isomers of a 48-water hexagonal ice Ih unit cell plotted against
energies predicted from graph invariant parameters derived from the small unit cells. (c) Graph invariant
fit to the energies of the 63 “semirandomly” chosen H-bond isomers of a 48-water hexagonal unit cell
of ice Ih. A line of slope unity is shown to indicate where points would lie for perfect agreement.
Electronic DFT calculations for ice Ih using three different combinations of
density functionals and basis sets for two smaller unit cells of ice Ih are reported
in Fig. 8. The results of these calculations were used to fit the coefficients (the
α
's)
in Eq. (11). Three graph invariant functions plus an overall constant (the overall
constant can be regarded as a fourth invariant) were used to fit the energy of the
H-bond isomers, indicating the economy of this description.. The quality of that
fit is shown in Fig. 12a. According to the theory of Section II.C, the parameters
obtained from calculations on the smaller 8-water orthorhombic (half of Fig. 2)
and 12-water hexagonal (Fig. 13a) unit cells, can be used to predict the energies of
larger unit cells. The quality of the prediction for the larger 48-water cell shown
