Exact Two-Structure Model

Instead of the CN we can use the number of HBs for our classification into species. The five species defined in sections 7.6 and 7.8 can also be used to construct an exact TSM. For instance, we can consider the four HB d species as one species of the TSM and refer to all the other species (with n = 0, 1, 2, 3) collectively as a second species. An approximate version of this TSM is worked out in subsection 7.9.3. 

As a second example, consider the quasicomponent distribution function based on the concept of binding energy (BE). We recall that the vector (or the function) gives the composition of the system when viewed as a mixture of molecules differing in their BE. Thus, Xb(v) dv is the mole fraction of molecules with BE between v and v dv. A possible TSM constructed from this function is, dropping subscript B in Xb(v), 

The above examples illustrate the general procedure by which we construct a TSM from any quasicomponent distribution function. From now on, we assume that we have made a classification into two components, L and H, without referring to a specific example. The arguments we use will be independent of any specific classification procedure. We will see that in order for such a TSM to be useful in interpreting the properties of water, we must assume that each component in itself behaves normally (in the sense discussed below). The anomalous properties of water are then interpreted in terms of structural changes that take place in the liquid. 

Let Az. and Nh be the equilibrium numbers of L and H molecules, respectively, and Nw be the total number of water molecules in the system, Niv= Viewing the 

The temperature dependence of the volume, along the equilibrium line with respect to the reaction L H (keeping P and Nw constant), is 

---

In the next section, we shall discuss the application of an exact two-structure model based on the MM approach. 

In this section, we formulate the general aspect of the application of the simplest mixture-model (MM) approach to water. We shall use an exact two-structure model (TSM), as introduced in Section 6.8. In the next section, we illustrate the application of a prototype of an interstitial model for water to solutions and, in Section 6.7, we discuss the application of a more general MM approach to this problem. 

In Secs. 2.3 and 2.4, we turn to a thorough discussion of the MM approach to water. We shall start with some historical notes, present an example of a successful two-structure model, then proceed to the exact MM approach and some of its applications. 

The simplest version of the MM approach and the one that has most often been applied is the two-structure model (T5M). Again, we stress that the exact T5M does not involve any model... 

Consider a two-structure model of L-cules and -cules obtained by any exact classification procedure as discussed in Chapter 6. The argument presented below is general, independent of the choice of classification, and valid for any solvent. 

In this section we present an example of the application of a two-structure model, based on the exact MM approach to the theory of liquids (section 5.13). Then we extract a particular MM which can be viewed as an approximation of the general exact MM approach. The latter, because of its simplicity and solvability, is useful in the study of some thermodynamic aspects of both pure water and aqueous solutions of simple solutes. 

These dimeric complexes involve, in their neutral state, two metal atoms in the (III) oxidation state. In the vanadium complexes such as [CpV(bdt)]2 and [CpV(tft)]2, the V—V bond length, 2.54 A in [CpV(bdt)]2, are shorter than observed in model complexes with a single V—V bond, indicating a partial double-bond character, also confirmed by a measured magnetic moment of 0.6 fiB in [CpV(tfd)]2, lower than expected if the two remaining unpaired electrons contribute to the magnetic susceptibility [20, 49]. This class of complexes most probably deserves deeper attention in order to understand their exact electronic structure. 

The exact cystal structure of quinacridone pigments has been published recently. So far, models indicated a planar arrangement of the molecules within the crystal lattice [21], In fact, it was now been proved recently by three-dimensional X-ray analysis that the pigment exists in two different crystal modifications [22] ... 

However, both Cmcm and Imma models would be expected to give rise to very similar image contrasts for thin crystals in view of the similarity of their projections along the [001] axis. Therefore, in the absence of additional experimental images projected along alternative zone axes, exact differentiation between the two structures is not possible at this resolution. Hence, the proposal that the disorder present in this system may arise from stacking faults involving a translation of c/2 cannot be confirmed. 

From these studies it is clear that damage to DNA is broader than initially expected from the two-component model since products on aU four bases and the sugar moiety have been proposed. These proposals include sugar and phosphate radicals despite early failures to detect radicals in the backbone of the DNA double helix. More work is requited in order to determine the exact identity of the radical products since structural information is difficult to obtain through the methods implemented thus far. 

In Table 1, we list the irreducible ring statistics for two crystal structures, FC-2 and BC-8, and for four models of amorphous silicon. When a bond-pair switch is introduced into the otherwise perfect FC-2 structure, the number of irreducible rings is conserved. Four 5-folds and eight 7-folds are created, but twelve 6-folds are eliminated. This conservation rule holds until the regions of bond-switching overlap, and is not grossly violated even then in the randomization process. Note that the total number of irreducible rings per atom is exactly two for the FC-2 structure and it is almost two for all the amorphous structures. 

The RADACK (contraction of RADiation-induced attACK) model, that we have developed [9,10], accounts for the experimentally determined probabilities of radiolytic damages caused by the OH radical attack in all forms of DNA (B [11], Z [12], triplex [13], quadruplex [14]), in DNA-protein complexes [15] and has the potential to predict radiolytic attack probabilities in other molecules or assemblies. Direct ionisation effects are not taken into account. The determination of relative probabilities of reaction ofthe target with the OH radicals takes into account two factors 1) the accessibility of the reactive sites of the target since it uses the exact tridimensional structure of the macromolecule or assembly as determined by NMR, crystallography or as built up by molecular modelling, and 2) the chemical reactivity of the residues (nucleotides or amino-acids). 

Thus, by using the Kronecker product, it is possible to express the structural model within these two modes in one combined mode. This is exactly what is used in the three-way Tucker3 model. The second and third modes are vectorized for all rows, leading to the model in Equation (4.12). 

---