Tetrahedral order

In a normal Hartree-Fock job, the hyperpolarizability tensor is given only in the archive entry, in the section beginning HyperPolar=. This tensor is also in lower tetrahedral order, but expressed in the input (Z-matrix) orientation. (This is also true of the polarizability tensor within the archive entry.)... 

The conditions on the phase diagram for which this anomalous behavior occurs has been termed water s structurally anomalous region. Inspection of the order map (Figure 4) reveals a dome of structural anomalies within the temperature-density plane, bounded by loci of maximum tetrahedral order (at low densities) and minimum translational order (at high densities) as shown in Figure 5. Also marked on Figure 5 are regions of diffusive anomalies,... 

Figure 4 The path traversed in structural order-metric space as liquid water (SPC/E) is compressed isothermally at two different temperatures. Filled diamonds represent T = 260 K, and open triangles represent T = 400 K. The arrows indicate the direction of increasing density. A and C are states of maximum tetrahedral order at the respective temperatures, whereas B is a state of minimum translational order. Reprinted with permission from Ref. 29. 

The most direct source of structural information, the diffraction studies, provides strong evidence for predominantly tetrahedral ordering of a molecule and its nearest neighbors, both in amorphous solid and liquid H2O. Other, weaker but still direct, structural evidence comes from the ratio of separations of nearest neighbor and next nearest neighbor 00 pairs in liquid water, the ubiquity of tetrahedral ordering in the several crystalline ices, and the statistical geometry of simulated water. 

In the Frank and Evans iceberg model, ice-like structures form around hydrophobic entities, such as methane. In this model, the hydrophobic molecules enhance the local water structure (greater tetrahedral order) compared with pure water. Ordering of the water hydration shell around hydrophobic molecules has been attributed to clathrate-like behavior, in which the water hydration shell is dominated by pentagons compared to bulk liquid water (Franks and Reid, 1973). 

Figure 9.7. A representative supermethane -based motif obtained via the tetrahedral ordering of four terminal dendrimers about a single core dendrimer by the use of rigid rod connectors.

Bell, A.M.T., Redfem, S.A.T., Henderson, C.M.B., Kohn, S.C. (1994b) Stmctnral relations and tetrahedral ordering pattern of synthetic orthorhombic CSjCdSijOij lencite a combined synchrotron X-ray powder diffraction and mnltinnclear MAS NMR study. Acta Crystallogr B50 560-566 Bertram, U.C., Heine, V., Jones, I.L., Price, G.D. (1990) Computer modelling of Al/Si ordering in sillimanite. Phys Chem Minerals 17 326-333... 

Other order parameters can be used to characterize ice/water interfaces via change in average density [17, 19] and diffusivity [17] across the interface. Local order parameters can also be defined which depend only on the position of a reference molecule. It was shown in [40] that the local tetrahedral order parameter, which reflects the change in tetrahedral environment, leads to a 10-90 width of 11 A for the basal ice-water interface which is in good agreement with the estimated value from the translational order parameter. 

It is important to mention by passing that the imperfect tetrahedral order of ice III and ice IX has no impact on our calculations because their imperfections are identical in character and cancel out any distortions influencing the results. 

II is enriched with molecules, which have tetrahedrally ordered four nearest neighbours and up to 6 molecules in the first coordination shell. Phase III seems to be an analogue of HDA. Contrary to the phases I and II, first and second coordination shells of molecules in phase III is not clearly divided. Phases I, II and III are enriched with tetrahedrally ordered molecules. There is a noticeable drop of tetrahedral order in phase IV and it consists mainly of molecules with highly anisotropic distribution of the nearest neighbours. Phase IV may be considered as analogue of VHDA. 

The separation of a one-component fluid into two liquid phases should be attributed to the existence of molecules with qualitatively different local ordering. At least two components of water should be imposed in case of a single liquid-liquid phase transition. To apply mixture model for the analysis of water properties, it is necessary to determine the local ordering, which dominates in the considered water phase. In the phase I (analogue of EDA), tetrahedrally ordered four-coordinated water molecules seems to be the natural choice of the dominating component. 

The obtained distributions of the tetrahedricity measure were used for estimation of the concentration C of the four-coordinated tetrahedrally ordered water molecules. Temperature dependence of this concentration along the liquid-vapour coexistence curve is shown in the upper panel of Fig.5. There is only slight increase of C upon cooling from the liquid-vapour critical temperature to about 350 K (due to the temperature mismatch of ST2 water and real water, about 30 to 35° lower temperature should be expected for real water). The drastic increase of C is evident at lower temperatures, when approaching the liquid-liquid phase transition. At 7 = 270 K, concentrations of the tetrahedrally ordered four-coordinated water molecules in two coexisting phases was found to be about 28% and 46.5%. Such step increase of C is related to a step decrease of density from 0.97 to 0.91 g/cm ... 

Figure 5 Temperature dependence of the concentration C of the tetrahedrally ordered four-coordinated water molecules (upper panel) and of the liquid water density (lower panel) along the liquid-vapour coexistence curve. Vertical dashed line indicates the temperature of the liquid-liquid transition. Dotted lines indicate the densities and concentrations of the coexisting phases. Stars indicate percolation transition of the tetrahedrally ordered four-coordinated molecules.

Figure 84 Tetrahedral orderings in ice Ih (left diagrams) and ice Ic (right diagrams). In both bottom drawings the axis defined by O-atoms 1 and 5 is the vertical axis of the paper sheet. The two npper diagrams are obtained from the lower ones by a 90° rotation around the horizontal axis of the paper sheet. The tetrahedra defined by O-atoms 1, 2, 3, 4 and 5 are the same in all drawings.

IM term in Eq. 1 proportional to Jg-f we calculate dNjM/dT p, where Njm is the number of molecules with complete tetrahedral order. We find that the locus of maxima of dNa/dT p [Fig. 8(b)] overlaps with the locus of maxima of Ci [Fig. 9]. 

Therefore, the maximum of Cp °° occurs where the correlation length associated with the tetrahedral order is maximum, i.e. along the Widom line associated with the LL phase transition." In MF we may compare Cp calculated for the LLCP scenario J(j > 0) with Cp calculated for the SF scenario J(j = 0) [Fig. 7(b)]. We see that the sharper maximum is present only in the LLCP scenario, while the less sharp maximum occurs at the same T in both scenarios. We conclude that the sharper maximum is due to the fluctuations of the tetrahedral order, critical at the LLCP, while the less sharp maximum is due to fluctuations in bond formation. The similarity of our results with the experiments in nanopores is striking. Data in ref. [ °] show two maxima in Cp. They have been interpreted as an out-of-equilibrium dynamic effect in [ °], but more recent experiments show that they are a feature of equilibrated confined water. Therefore, our interpretation of the two maxima is of considerable interest. 

Ordering of tetrahedral cations is quite unusual in the common mica species such as muscovite-2Mi, phlogopite-lM and annite-lM (Bailey 1975, 1984c), whereas it is common in brittle micas. Margarite, bityite and anandite are examples of minerals with Si,Al (or Fe ) tetrahedral ordering (Guggenheim 1984). 

Pavese A, Ferraris G, Pischedda V, Ibberson R (1999a) Tetrahedral order in phengite-2Mi upon heating, from powder neutron diffraction, and thermodynamic consequences. Eur J Mineral 11 309-320 Pavese A, Ferraris G, Pischedda V, Mezouar M (1999b) Synchrotron powder diffraction study of phengite 3T from the Dora-Maira massif P-V-T equation of state and petrological consequences. Phys Chem Minerals 26 460-467... 

Summarizing among the five known MDO polytypes with ideal space-group, tetrahedral ordering is not possible in M and 20 polytypes. Octahedral ordering is instead possible in all five trioctahedral polytypes for dioctahedral polytypes cf phengite below. Note that there are some hints of a limited occupancy of Ml in strictly dioctahedral micas (Brigatti et al. 1998, 2001 Pavese et al. 2001). 

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