From Crystals to Molecules

Six years later Sir John Herschel 13> observed that the algebraic sign of the optical rotation of quartz crystals could be correlated with their shape, and the dextro- and levorotatory crystals look like mutual mirror images. This correlation was extended from crystals to molecules by Pasteur 14>, who postulated that the spatial arrangement of atoms in the molecules is responsible for their optical activity, and the dextro-and levorotatory molecules are in a mirror image relation. 

A more subtle application of the same technique involves isotopic substitution. Because most C02 frequencies are reproducible from crystal to crystal to within 0.07 cm ", it is possible to demonstrate small, but significant shifts of v3 when certain hydrogen atoms are replaced by deuterium. Analogous orientation effects have been reported for mixed crystals of isotopically substituted benzene, where isotopic differences in adjacent molecules gave rise to splittings as large as 4.7 cm "1 [91],... 

The structures of the surfaces, the surface adsorption and the alkali-doped crystal and the atom diffusion path (cf. Section 4) were investigated by different quantum-chemical methods. We used foremost ab initio methodologies. The main computational tool utilized was the program CRYSTAL [54]. This program makes it possible to treat molecules and in particular crystalline solids and surfaces at an ab initio level of theory for surfaces and solids the periodic boundary conditions are applied in 2 or 3 dimensions [55]. The familiar Gaussian basis sets can be used for systems ranging from crystals to isolated molecules, which enables systematic comparative studies of chemical properties in different forms of matter. In our studies, split-valence basis sets were used [56]. 

Travel a path from crystals to nanoparticles to single molecules (Chapter 12). 

AS j can be computed as the difference between the entropy of an ideal gas and the entropy of the crystal at a given temperature and pressure. If the intra- and intermolecular contributions to the entropy of the crystal are considered to be decoupled, such that the change in intramolecular vibrational entropy for transfer from crystal to gas can be taken to be zero, then the sublimation entropy of rigid molecules can be approximated by AS where and are the... 

One of the benefits of the methods discussed in this chapter is that they provide a complete characterization of the thermodynamics of transfer of solute from crystal to aqueous solution. Since the solubility of a crystalline solute depends upon the properties of the undissolved crystalline precipitate as weU as the properties of the solution, the thermodynamic data provides valuable information in understanding not only which of the two molecules is more soluble but also why the selected molecule is more soluble. By contrast, QSPR models, which are statistical rather than first-principles approaches, provide only limited statistical information about the underlying physicochemical processes. Moreover, since most QSPR models predict solubility from molecular rather than crystal structure, they are not able to rationalize or predict different solubilities for different polymorphs of a molecule. Therefore, we believe that the bottom-up methods that utilize efficiently molecular-scale information about the solute and solvent structure will attract more attention in the future in terms of both practical applications and fundamental studies of solubility of druglike molecules. 

In the 60-80 ps section of Fig. 13.1 the benzene molecules are literally pushing the crystal lattice apart under the effect of the suddenly available kinetic energy. Diffusional and rotational freedom results and the crystal collapses to the liquid, which is then normally simulated at 300 K in the 80-120 ps section of the run. Since the part of the simulation where melting occurs is a non-equilibrium simulation, one cannot draw any conclusions from averages, nor can one claim to have simulated the actual solid-liquid equilibrium or to have predicted the melting temperature. Nevertheless, such dynamic runs offer a window over the evolution of the internal structure of the system as it goes from crystal to liquid an example taken from a study of acetic acid is shown in Fig. 13.2. One cannot say for sure that this picture is a representation of the tfue structural changes that occur when the acetic acid crystal melts molecular... 

A large number of ordered surface structures can be produced experimentally on single-crystal surfaces, especially with adsorbates [H]. There are also many disordered surfaces. Ordering is driven by the interactions between atoms, ions or molecules in the surface region. These forces can be of various types covalent, ionic, van der Waals, etc and there can be a mix of such types of interaction, not only within a given bond, but also from bond to bond in the same surface. A surface could, for instance, consist of a bulk material with one type of internal bonding (say, ionic). It may be covered with an overlayer of molecules with a different type of intramolecular bonding (typically covalent) and the molecules may be held to the substrate by yet another fomi of bond (e.g., van der Waals). 

Table 2 shows the results of our preliminary calculations of the pKa of the Cys403 residue, for several different models of the enzyme, based on two structures available from the PDB. In the case of the YPT structure, a crystal water molecule is close to Cys403 and was included in some of the calculations as part of the protein (i.e. it was treated with the same internal dielectric as that of the protein). Simulations denoted as -I-H2O in Table 2, include a crystallographically resolved, buried water molecule, situated 3.2lA from... 

It can be seen from Table 2 that the intrinsic values of the pK s are close to the model compound value that we use for Cys(8.3), and that interactions with surrounding titratable residues are responsible for the final apparent values of the ionization constants. It can also be seen that the best agreement with the experimental value is obtained for the YPT structure suplemented with the 27 N-terminal amino acids, although both the original YPT structure and the one with the crystal water molecule give values close to the experimentally determined one. Minimization, however, makes the agreement worse, probably because it w s done without the presence of any solvent molecules, which are important for the residues on the surface of the protein. For the YTS structure, which refers to the protein crystallized with an SO4 ion, the results with and without the ion included in the calculations, arc far from the experimental value. This may indicate that con-... 

The state of the surface is now best considered in terms of distribution of site energies, each of the minima of the kind indicated in Fig. 1.7 being regarded as an adsorption site. The distribution function is defined as the number of sites for which the interaction potential lies between and (rpo + d o)> various forms of this function have been proposed from time to time. One might expect the form ofto fio derivable from measurements of the change in the heat of adsorption with the amount adsorbed. In practice the situation is complicated by the interaction of the adsorbed molecules with each other to an extent depending on their mean distance of separation, and also by the fact that the exact proportion of the different crystal faces exposed is usually unknown. It is rarely possible, therefore, to formulate the distribution function for a given solid except very approximately. 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

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