Structure solution from first principle

In the context of this book, structure solution from first principles (also referred to as the ab initio structure determination) means that all crystallographic data, including lattice parameters and symmetry, and the distribution of atoms in the unit cell, are inferred from the analysis of the scattered intensity as a function of Bragg angle, collected during a powder diffraction experiment. Additional information, such as the gravimetric density of a material, its chemical composition, basic physical and chemical properties, may be used as well, when available. 

Similar to structure solution from first principles, the ab initio indexing implies that no prior knowledge about symmetry and approximate unit cell dimensions of the crystal lattice exists. Indexing from first principles, therefore, usually means that Miller indices are assigned based strictly on the relationships between the observed Bragg angles. 

The total number of possible Bragg reflections below 20 = 98° is 1032. Therefore, before attempting a structure solution from first principles, a more thorough search of the relevant databases was performed. A search based on the unit cell dimensions and volume produced no results, but after searching the ICSD database for a matching stoichiometry of the molybdate anion, two compounds, both with the space group symmetry C2/c, were found and they are listed together with the title compound in Table 6.41. 

Much of quantum chemistry attempts to make more quantitative these aspects of chemists view of the periodic table and of atomic valence and structure. By starting from first principles and treating atomic and molecular states as solutions of a so-called Schrodinger equation, quantum chemistry seeks to determine what underlies the empirical quantum numbers, orbitals, the aufbau principle and the concept of valence used by spectroscopists and chemists, in some cases, even prior to the advent of quantum mechanics. 

Ffom a theoretical point of view, stacking fault energies in metals have been reliably calculated from first-principles with different electronic structure methods [4, 5, 6]. For random alloys, the Layer Korringa Kohn Rostoker method in combination with the coherent potential approximation [7] (LKKR-CPA), was shown to be reliable in the prediction of SFE in fcc-based solid solution [8, 9]. 

If the purpose of a calculation is to probe the inherent properties of a molecule as a thing in itself, or of a phenomenon centered on isolated molecules, then we do not want the complication of solvent. For example, a theoretically oriented study of the geometry and electronic structure of a novel hydrocarbon, e.g. pyramidane [6], or of the relative importance of diatropic and paratropic ring currents [7], properly examines unencumbered molecules. On the other hand, if we wish, say, to calculate from first principles the pZa of acids in water, we must calculate the relevant free energies in water [8]. Noteworthy too is the fact that solvation, in contrast to gas phase treatments, is somewhat akin to molecules in bulk, in crystals [9]. Here a molecule is solvated by its neighbors in a lattice, although the participants have a much more limited range of motion than in solution. Rates, equilibria, and molecular conformations are all affected by solvation. Bachrach has written a concise review of the computation of solvent effects with numerous apposite references [10]. 

An identification attempt using the Powder Diffraction File failed as no acceptable matches were found. Undoubtedly, such high quality of the powder diffraction data should be sufficient to solve the structure from first principles using either Patterson or direct methods. Yet, a structure solution is not fully automated and therefore, the ICSD database was searched in the following order ... 

It is important, however, to remember that the Rietveld method requires a model of a crystal structure and by itself offers no clue on how to create such a model from first principles. Thus, the Rietveld technique is nothing else than a powerful refinement and optimization tool, which may also be used to establish structural details (sometimes subtle) that were missed during a partial or complete ab initio structure solution process, i.e. as in the twelve examples described in Chapter 6. 

With the development of much faster computers and more reliable, more efficient computer codes for electronic structure calculations, a solution to both the above problems is to calculate both the orientation and principal values of the required interaction tensor from first principles, from an input molecular or (preferably) crystal structure. 

This is the entire formal structure of classical statistical mechanical perturbation theory. The reader will note how much simpler it is than quantum perturbation theory. But the devil lies in the details. How does one choose the unperturbed potential, y How does one evaluate the first-order perturbation It is quite difficult to compute the quantities in Equation P5 from first principles. Most progress has been made by some clever application of the law of corresponding states. It is not the aim of this chapter to follow this road to solution theory any further. 

A complete treatment of a small sample of molecular liquid from first principles is still beyond the reach of computationa] chemistry. In order to better understand what happens in a liquid or a solution at the molecular levels, one is (breed to adopt a simplified approach. One may perform a full statistical treatment of the sample, and then the representation of the molecule and the intermolecular forces have to be rather schematic. One may describe this approach as a true liquid of model molecules. Conversely, one may w ish to analyze in greater detail the structural modifications of a molecule - the solute or a particular molecule of a pure liquid - when it is placed in a liquid environment. One is then led to adopt a quantum chemical treatment of the molecule and to replace its actual surroundings by a simple medium, usually a continuum, having the averaged properties of the liquid. One may speak of a model liquid of true molecules. I he Self-( onsistent Reaction Field method belongs to this approach. 

An issue that often arises is whether ground professionals should be able to analyse simple structures from first principles or is it sufTicient that they can correctly apply published solutions, codes and standards and perform routine computer-based analyses. For example should a ground engineer be able to derive the bearing capacity factors Nc, Nq, and Ny Without deriving these, or at least seeing them derived, users have little appreciation of the assumptions required in the derivations and the limitations of the factors themselves. 

Transference numbers have formed one of the cornerstones in our understanding of electrolyte solutions. Hittorf s discovery in 1853 that transference numbers depended on the ion, the co-ion, and the solvent proved that each ion in a given solvent possesses its own individual mobility. Even today ionic mobilities must be determined by a combination of transference and conductance experiments for we still cannot predict their values accurately from first principles The importance of ionic mobilities can hardly be overemphasized since they are the only properties of individual ions that can be unambiguously measured (either directly or via trace diffusion coefficients). They therefore provide unique insight into ion-solvent interactions. Hittorf s later transference experiments also revealed the existence and composition of a variety of complex ions in solution. His approach has been followed in more recent structural investigations, for instance in studying the complex ions present in aluminium plating solutions (7A,). 

Theoretical attempts to relate dimensions of polymers to chemical structure were pioneered by Flory (2). Statistical macromolecular size in solution can be modeled from first principles by considering the number and length of bonds along with valence bond angles and conformational restrictions. Excluded volume, segmental interactions, specific intramolecular interactions, and chain solvation contribute to dimensions. 

The considerations above apply to fast dynamic processes in the sense that the amplitude of the modulation of the resonance frequency Amq (induced by modulation of the local field) multiplied with the correlation time Tc is much smaller than unity. This Redfield regime [27] is usually attained in solutions with low viscosity [2], but may also apply to small-amplitude libration in solids [28]. For slower reorientation in solutions with high viscosity or in soft matter above the glass transition temperature (slow tumbling), spectral lineshapes are directly influenced by exchange between different orientations of the molecule (Section 4.1). Relaxation times in solids outside the Redfield regime carmot be predicted from first principles except for a few crystalline systems with very simple structure and few defects [29]. In such systems, qualitative or semi-quantitative analysis of relaxation data can still provide some information on dynamics. 

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