Crystal energy partitioning

The quantities defined by Eqs. (2)—(7) plus Vs max, Vs min, and the positive and negative areas, A and, enable detailed characterization of the electrostatic potential on a molecular surface. Over the past ten years, we have shown that subsets of these quantities can be used to represent analytically a variety of liquid-, solid-, and solution-phase properties that depend on noncovalent interactions [14-17, 84] these include boiling points and critical constants, heats of vaporization, sublimation and fusion, solubilities and solvation energies, partition coefficients, diffusion constants, viscosities, surface tensions, and liquid and crystal densities. 

Theoretical aspects of the bond valence model have been discussed by Jansen and Block (1991), Jansen et al. (1992), Burdett and Hawthorne (1993), and Urusov (1995). Recently Preiser et al. (1999) have shown that the rules of the bond valence model can be derived theoretically using the same assumptions as those made for the ionic model. The Coulomb field of an ionic crystal naturally partitions itself into localized chemical bonds whose valence is equal to the flux linking the cation to the anion (Chapter 2). The bond valence model is thus an alternative representation of the ionic model, one based on the electrostatic field rather than energy. The two descriptions are thus equivalent and complementary but, as shown in Section 2.3 and discussed further in Section 14.1.1, both apply equally well to all types of acid-base bonds, covalent as well as ionic. 

Fig. 37. Distribution of energies of water molecules about bovine pancreatic trypsin inhibitor in the crystal. Total energy ( ) partitioned into contribution from water-water (D) water-protein (O) interactions. From Hermans and Vacatello (1980).

The treatment of the defective lattice follows the customary two-region approach (Catlow and Mackrodt, 1982 this volume Chapter 7) in which the total energy of the defective system is minimized by variation of the nuclear positions (and shell displacements) around the defect. The crystal is partitioned into an inner region, immediately surrounding the defect where the relaxation is assumed to be greatest, and a less perturbed outer region. In the inner region the... 

Similar to the COOP method, this approach has been called crystal orbital Hamilton population (COHP) analysis [91]. Given a short-ranged orbital basis set, the sum of all pairwise interactions rapidly converges in real space, similar to the COOP formalism. In fact, such energy-partitioning schemes have a long... 

In addition, let us quantum-chemically analyze the electronic structure of this simple material in terms of chemical bonding. This is very easy to do because the above-mentioned TB-LMTO-ASA method operates with an extremely short-ranged basis set, such that electron and energy-partitionings are straightforward. Figure 3.3 shows the results, namely the band structure, the density-of-states, and the crystal orbital Hamilton population analysis. 

For the perfect crystal 0 reduces to Um L ) and the partition function is evaluated by expanding this potential energy in a... 

In this study, we assume that crystal structures will have the lowest possible total of intra- and inter-molecular potential energy. However, the partitioning of the potential energy between intra- and inter-molecular terms will vary among crystal structures, distorting the glucose residues away from the shape of lowest energy in a way that will reflect more-or-less random... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

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