Electron Density Studies of Molecular Crystals

Small molecules consisting of light-, few-electron atoms were the first species beyond atoms to yield to quantum-mechanical methods. Similarly, crystals of small light-atom molecules have served as most useful test cases of charge density mapping. The small number of core electrons in first-row atoms enhances the relative contribution of valence electron scattering to the diffraction pattern. Early studies, done just after automated diffractometers became widely available, were concerned with molecular crystals such as uracil (Stewart and Jensen 1967), s-triazine (Coppens 1967), oxalic acid dihydrate (Coppens et al. 1969), decaborane (Dietrich and Scheringer 1978), fumaramic acid (Hirshfeld 1971), glycine (Almlof et al. 1973), and tetraphenylbutatriene (Berkovitch-Yellin and Leiserowitz 1976). While thermal motion is often pronounced in molecular crystals, advances in low-temperature data collection have done much to alleviate this disadvantage. In recent years, subliquid-nitrogen cooling techniques have been increasingly applied. Among the most interesting aspects of molecular crystals are the influence of intermolecular interactions on the electronic structure. Physically meaningful Coulombic parameters pertinent to a molecule in a condensed environment can be obtained from the diffraction analysis, and can be used in the modeling of macromolecules. The enhancement of the electrostatic moments relative to those of the isolated species has been noted in chapter 7. But, beyond these considerations, molecular crystals are important in their own right. For example, crystals of aromatic molecules substituted with π-electron donor and acceptor groups are among the most strongly nonlinear optical solids known, considerably exceeding the nonlinearity of inorganic crystals such as potassium titanyl phosphate (KTP); while mixed-valence organic components of low-dimensional solids can become superconducting at low temperatures. The relation between such properties of molecular crystals and their charge distribution provides a continuing impetus for further study. The suitability of light-atom crystals for charge density analysis can be understood in terms of the relative importance of core electron scattering. As the perturbation of the core electrons by the chemical environment is beyond the reach of practically all experimental studies, the frozen-core approximation is routinely used. It assumes the intensity of the core electron scattering to be invariable, while the valence scattering is affected by the chemical environment, as discussed in chapter 3.

Charge Density Studies of Transition Metal Compounds

The electron density in transition metal complexes is of unusual interest. The chemistry of transition metal compounds is of relevance for catalysis, for solid-state properties, and for a large number of key biological processes. The importance of transition-metal-based materials needs no further mention after the discovery of the high-Tc superconducting cuprates, the properties of which depend critically on the electronic structure in the CuO2 planes. The results of theoretical calculations of systems with a large number of electrons can be ambiguous because of the approximations involved and the frequent occurrence of low-lying excited states. The X-ray charge densities provide independent evidence from a technique with very different strengths and weaknesses, and thus can make significant contributions to our understanding of the properties of transition-metal-containing molecules and solids. In inorganic and organometallic solids, the average electron concentration tends to be high. This means that absorption and extinction effects can be severe, and that the use of hard radiation and very small crystals is frequently essential. Needless to say that the advent of synchrotron radiation has been most helpful in this respect. The weaker contribution of valence electrons compared with the scattering of first-row-atom-only solids implies that great care must be taken during data collection in order to obtain reliable information on the valence electron distribution. When the field exerted by the atomic environment is not spherically symmetric, as is the case in any crystal, the degeneracy of the d-electron orbitals is lifted. In the electrostatic crystal field theory, originally developed by Bethe (1929) and Van Vleck (1932), all interactions between the transition metal atom and its ligands are treated electrostatically, and covalent bonding is neglected. Since the ligands are almost always negatively charged, electrons in orbitals pointing towards the ligands are repelled more strongly, and the corresponding orbitals will be higher in energy. The discussion is the simplest for the one d-electron case, in which d-d electron repulsions are absent.

The Electrostatic Moments of a Charge Distribution

The moments of a charge distribution provide a concise summary of the nature of that distribution. They are suitable for quantitative comparison of experimental charge densities with theoretical results. As many of the moments can be obtained by spectroscopic and dielectric methods, the comparison between techniques can serve as a calibration of experimental and theoretical charge densities. Conversely, since the full charge density is not accessible by the other experimental methods, the comparison provides an interpretation of the results of the complementary physical techniques. The electrostatic moments are of practical importance, as they occur in the expressions for intermolecular interactions and the lattice energies of crystals. The first electrostatic moment from X-rays was obtained by Stewart (1970), who calculated the dipole moment of uracil from the least-squares valence-shell populations of each of the constituent atoms of the molecule. Stewart’s value of 4.0 ± 1.3 D had a large experimental uncertainty, but is nevertheless close to the later result of 4.16 ± 0.4 D (Kulakowska et al. 1974), obtained from capacitance measurements of a solution in dioxane. The diffraction method has the advantage that it gives not only the magnitude but also the direction of the dipole moment. Gas-phase microwave measurements are also capable of providing all three components of the dipole moment, but only the magnitude is obtained from dielectric solution measurements. We will use an example as illustration. The dipole moment vector for formamide has been determined both by diffraction and microwave spectroscopy. As the diffraction experiment measures a continuous charge distribution, the moments derived are defined in terms of the method used for space partitioning, and are not necessarily equal. Nevertheless, the results from different techniques agree quite well. A comprehensive review on molecular electric moments from X-ray diffraction data has been published by Spackman (1992). Spackman points out that despite a large number of determinations of molecular dipole moments and a few determinations of molecular quadrupole moments, it is not yet widely accepted that diffraction methods lead to valid experimental values of the electrostatic moments.

The Charge Density in Extended Solids

Extended solids encompass all solids in which no well-defined molecular entities can be distinguished. This is the case for metals and alloys, covalently bonded solids like diamond and silicon, and ionic crystals of which the alkali halides are prototypes. Intermediate cases are common, such as crystals consisting of a charged covalent network with counterbalancing cations or anions. Silicates and their analogues are a prime example of often charged networks with partially covalent bonding. An increasing number of solids are known in which both an extended framework and molecular entities exist, with the molecules being embedded in the extended framework. Graphite intercalation compounds and a variety of host/guest complexes are examples of this class. The bonding features in the charge density are pronounced in crystals with extended covalent networks. The availability of perfect silicon crystals has allowed the measurement of uncommonly accurate structure factors, of millielectron accuracy. The data have served as a test of experimental formalisms for charge density analysis, and at the same time have provided a stringent criterion for quantum-mechanical methods. We will start the discussion in this chapter with silicon and its analogues, diamond and germanium, and proceed with the treatment of silicates, and metallic and ionic crystals. In the face-centered cubic structure of silicon, atoms are located at 1/8 1/8 1/8 and at the center-of-symmetry related position of −1/8 −1/8 −1/8.

X-ray Diffraction and the Electrostatic Potential

The distribution of positive and negative charge in a crystal fully defines physical properties like the electrostatic potential and its derivatives, the electric field, and the gradient of the electric field. The electrostatic potential at a point in space, defined as the energy required to bring a positive unit of charge from infinite distance to that point, is an important function in the study of chemical reactivity. As electrostatic forces are relatively long-range forces, they determine the path along which an approaching reactant will travel towards a molecule. A nucleophilic reagent will first be attracted to the regions where the potential is positive, while an electrophilic reagent will approach the negative regions of the molecule. As the electrostatic potential is of importance in the study of intermolecular interactions, it has received considerable attention during the past two decades (see, e.g., articles on the molecular potential of biomolecules in Politzer and Truhlar 1981). It plays a key role in the process of molecular recognition, including drug-receptor interactions, and is an important function in the evaluation of the lattice energy, not only of ionic crystals. This chapter deals with the evaluation of the electrostatic potential and its derivatives by X-ray diffraction. This may be achieved either directly from the structure factors, or indirectly from the experimental electron density as described by the multipole formalism. The former method evaluates the properties in the crystal as a whole, while the latter gives the values for a molecule or fragment “lifted” out of the crystal. Like other properties derived from the charge distribution, the experimental electrostatic potential will be affected by the finite resolution of the experimental data set. But as the contribution of a structure factor F(H) to the potential is proportional to H−2, as shown below, convergence is readily achieved. A summary of the dependence of electrostatic properties of the magnitude of the scattering vector H is given in Table 8.1, which shows that the electrostatic potential is among the most accessible of the properties listed.

Space Partitioning and Topological Analysis of the Total Charge Density

In partitioning space in the analysis of a continuous charge distribution, the requirement of locality, formulated by Kurki-Suonio (Kurki-Suonio 1968, 1971; Kurki-Suonio and Salmo 1971), should be preserved. It states that density at a point should be assigned to a center in the proximity of that point. In discrete boundary partitioning schemes, the density at each point is assigned to a specific basin, while in fuzzy boundary partitioning, the density at the point may be assigned to overlapping functions centered at different locations. The least-squares formalisms described in chapter 3 implicitly define a space partitioning scheme, based on the density functions used in the refinement that are each centered on a specific nucleus. Since the density functions are continuous, they overlap, so the fragments interpenetrate rather than meet at a discrete boundary. Such fuzzy boundaries correspond to smoothly varying functions, both in real and reciprocal space, and therefore to well-behaved fragment scattering factors, and reasonable fragment electrostatic moments. The interpenetratingfragment partitioning schemes are related to the Mulliken and Løwdin population analyses of theoretical chemistry. The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. The topological analysis of the density leads to a powerful classification of bonding based on the electron density. It is discussed in the final sections of this chapter. The stockholder partitioning concept is one of the important contributions to charge density analysis made by Hirshfeld (1977b). It defines a continuous sampling function wi(r), which assigns the density among the constituent atoms. The sampling function is based on the spherical-atom promolecule density—the sum of the spherically averaged ground-state atom densities.

Fourier Methods and Maximum Entropy Enhancement

Image formation in diffraction is no different from image formation in other branches of optics, and it obeys the same mathematical equations. However, the nonexistence of lenses for X-ray beams makes it necessary to use computational methods to achieve the Fourier transform of the diffraction pattern into the image. The phase information required for this process is, in general, not available from the diffraction experiment, even though progress has been made in deriving phases from multiple-beam effects. This is the phase problem, the paramount issue in crystal structure analysis, which also affects charge density analysis of noncentrosymmetric structures. For centrosymmetric space groups, the independent-atom model is a sufficiently close approximation to allow calculation of the signs for all but a few very weak reflections. Images of the charge density are indispensable for qualitative understanding of chemical bonding, and play a central role in charge density analysis. In this chapter, we will discuss methods for imaging the experimental charge density, and define the functions used in the imaging procedure. According to Eq. (1.22), the structure factor F(H) is the Fourier transform of the electron density ρ(r) in the crystallographic unit cell. The electron density p(r) is then obtained by the inverse Fourier transformation, or . . . ρ(r)=∫F(H) exp (−2πi H ·r) dH (5.1) . . . in which F(H) are the (complex) structure factors corrected for the anomalous scattering discussed in chapter 1.

Least-Squares Methods and Their Use in Charge Density Analysis

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. In addition to the positional and thermal parameters of the atoms, least-squares procedures are used to determine the scale of the data, and parameters such as mosaic spread or particle size, which influence the intensities through multiple-beam effects (Becker and Coppens 1974a, b, 1975). It is not an exaggeration to say that modern crystallography is, to a large extent, made possible by the use of least-squares methods. Similarly, least-squares techniques play a central role in the charge density analysis with the scattering formalisms described in the previous chapter. The following description follows closely the treatment given by Hamilton (1964).

The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams

The atoms in a crystal are vibrating with amplitudes determined by the force constants of the crystal’s normal modes. This motion can never be frozen out because of the persistence of zero-point motion, and it has important consequences for the scattering intensities. Since X-ray scattering (and, to a lesser extent, neutron scattering) is a very fast process, taking place on a time scale of 10−18 s, the photon-matter interaction time is much shorter than the period of a lattice vibration, which is of the order Thus, the recorded X-ray scattering pattern is the sum over the scattering of a large number of 1/v, or ≈10−13s. instantaneous states of the crystal. To an extremely good approximation, the scattering averaged over the instantaneous distributions is equivalent to the scattering of the time-averaged distribution of the scattering matter (Stewart and Feil 1980). The structure factor expression for coherent elastic Bragg scattering of X-rays may therefore be written in terms 〈ρ(r)〉, of the thermally averaged electron density: . . . F(H)=∫unit cell〈ρ(r)〉 exp (2πi H ·r) dr (2.1) . . . The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward |S| = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. The reduction in intensity due to thermal motion is accompanied by an increase in the incoherent elastic scattering, ensuring conservation of energy. In this respect, thermal motion is much like disorder, with the Bragg intensities representing the average distribution, and the deviations from the average appearing as a continuous, though not uniform, background, generally referred to as thermal diffuse scattering or TDS. A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal.

Scattering of X-rays and Neutrons

This chapter starts with a discussion of the classical treatment of X-ray scattering, followed by a brief overview of the quantum-mechanical theory in the first Born approximation. The scattering of a periodic arrangement is derived by considering the crystal as a convolution of the unit cell contents and a periodic lattice. The atomic description of the charge density, which is the basis for structure analysis, is introduced. The origin of resonance anomalous scattering is discussed. While its effect must be accounted for before charge densities can be derived from the X-ray scattering amplitudes, resonance scattering itself can give invaluable information on the electronic states of the resonating atoms. The final section of this chapter deals with the scattering of neutrons by atomic nuclei. Nuclear neutron scattering is independent of the distribution of the electrons, and can provide atomic positions and thermal amplitudes unbiased by the bonding effects which are the subject of this book. In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations.