Lattice coordination number

An improvement of this method—the so-called biased sampling [55] (or inversely restrieted sampling)—suggests to look ahead at least one step in order to overcome the attrition. Consider a SAW of i steps on a -coordination number lattice. To add the / + 1st step one first checks which of the = q — neighboring sites are empty. If k qQ > k>0) sites are empty one takes one of these with equal probability 1 /A if A = 0 the walk is terminated and one starts from the beginning. This reduces the attrition dramatically. Now each A-step walk has a probability PAr( i ) = Ylf=i so that dense configurations are clearly more probable. To compensate for this bias, each chain does not count as 1 in the sample but with a weight... 

Low Coordination Number Lattice Models of Real Proteins... 

Interaction Scheme in High Coordination Number Lattice Protein Models... 

There is nothing unique about the placement of this isolated segment to distinguish it from the placement of a small molecule on a lattice filled to the same extent. The polymeric nature of the solute shows up in the placement of the second segment This must be positioned in a site adjacent to the first, since the units are covalently bonded together. No such limitation exists for independent small molecules. To handle this development we assume that each site on the lattice has z neighboring sites and we call z the coordination number of the lattice. It might appear that the need for this parameter introduces into the model a quantity which would be difficult to evaluate in any eventual test of the model. It turns out, however, that the z s cancel out of the final result for, so we need not worry about this eventuality. 

Since A Aw is the change in interaction energy per 1,2 pair, it can be expressed as some multiple of kT per pair or of RT per mole of pairs. It is also conventional to consolidate the lattice coordination number and x into a single parameter x, since z and x are not measured separately. With this change of notation Az Aw is replaced by its equivalent xRT, and Eq. (8.41) becomes... 

Here the sums over /, / run over the neighbors of /. The highest term is a product of (c+ 1) factors with c the coordination number of the lattice. 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

Betyllium, because of its small size, almost invariably has a coordination number of 4. This is important in analytical chemistry since it ensures that edta, which coordinates strongly to Mg, Ca (and Al), does not chelate Be appreciably. BeO has the wurtzite (ZnS, p. 1209) structure whilst the other Be chalcogenides adopt the zinc blende modification. BeF2 has the cristobalite (SiOi, p. 342) structure and has only a vety low electrical conductivity when fused. Be2C and Be2B have extended lattices of the antifluorite type with 4-coordinate Be and 8-coordinate C or B. Be2Si04 has the phenacite structure (p. 347) in which both Be and Si... 

Pressure is another important variable in the formation of feldspars and at sufficiently high pressures there is a tendency for A1 to increase its coordination number from 4 to 6 with consequent destruction of the feldspar lattice. For example ... 

The predominantly ionic alkali metal sulfides M2S (Li, Na, K, Rb, Cs) adopt the antifluorite structure (p. 118) in which each S atom is surrounded by a cube of 8 M and each M by a tetrahedron of S. The alkaline earth sulfides MS (Mg, Ca, Sr, Ba) adopt the NaCl-type 6 6 structure (p. 242) as do many other monosulfides of rather less basic metals (M = Pb, Mn, La, Ce, Pr, Nd, Sm, Eu, Tb, Ho, Th, U, Pu). However, many metals in the later transition element groups show substantial trends to increasing covalency leading either to lower coordination numbers or to layer-lattice structures. Thus MS (Be, Zn, Cd, Hg) adopt the 4 4 zinc blende structure (p. 1210) and ZnS, CdS and MnS also crystallize in the 4 4 wurtzite modification (p. 1210). In both of these structures both M and S are tetrahedrally coordinated, whereas PtS, which also has 4 4... 

The lowest coordination number of tantalum or niobium permitted by crystal chemistry formalism is 6, which corresponds to an octahedral configuration. X Me ratios that equal 3, 2 or 1 can, therefore, be obtained by corresponding substitutions in the cationic sub-lattice. A condition for such substitution is no doubt steric similarity between the second cation and the tantalum or niobium ion so as to enable its replacement in the octahedral polyhedron. In such cases, the structure of the compound consists of oxyfluoride octahedrons that are linked by their vertexes, sides or faces, according to the compound type, MeX3, MeX2 or MeX respectively. Table 37 lists compounds that have a coordination-type structure [259-261]. 

The chemistry of silicon in very low oxidation states is one of the most fascinating research areas, which can be located between molecular compounds of silicon and elemental (perhaps amorphous) silicon [190-194]. Most interesting results have recently been obtained by structural investigations of siliddes in Zintl phases. However, compounds of silicon with negative oxidation states and very low coordination numbers of 1, 2, and 3 are so far only known in the composite of a crystal lattice. 

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