Dirk Feil

X-ray and Electron Diffraction Study of MgO

Precise X-ray and high-energy transmission electron diffraction methods were used for the study of electron density and electrostatic potential in MgO crystals. The structure amplitudes were determined and their accuracy estimated using ab initio Hartree-Fock structure amplitudes as criteria. The electrostatic potential distributions, reconstructed using Fourier series from both X-ray and electron diffraction data, are in satisfactory mutual agreement and are similar to the theory. They, however, suffer from restricted experimental resolution and, therefore, the reconstruction of the electrostatic potential via an analytical structural model is preferable. The model of electron density was adjusted to X-ray experimental structure amplitudes and those calculated by the Hartree-Fock method. The electrostatic potential, deformation electron density and the Laplacian of the electron density were calculated with this model. The critical points in both experimental and theoretical model electron densities were found and compared with those for procrystals from spherical atoms and ions. A disagreement concerning the type of critical point at (,,0) in the area of low, near-uniform electron density is observed. It is noted that topological analysis of the electron density in crystals can be related with a close-packing concept.

Electron density study of urea using TDS-corrected X-ray diffraction data: quantitative comparison of experimental and theoretical results

The electron-density distribution in urea, CO(NH(2))(2), was studied by high-precision single-crystal X-ray diffraction analysis at 148 (1) K. An experimental correction for TDS was applied to the X-ray intensities. R(merge)(F(2)) = 0.015. The displacement parameters agree quite well with results from neutron diffraction. The deformation density was obtained by refinement of 145 unique low-order reflections with the Hansen & Coppens [Acta Cryst. (1978), A34, 909-921] multipole model, resulting in R = 0.008, wR = 0.011 and S = 1.09. Orbital calculations were carried out applying different potentials to account for correlation and exchange: Hartree-Fock (HF), density-functional theory/local density approximation (DFT/LDA) and density-functional theory/generalized gradient approximation (DFT/GGA). Extensive comparisons of the deformation densities and structure factors were made between the results of the various calculations and the outcome of the refinement. The agreement between the experimental and theoretical results is excellent, judged by the deformation density and the structure factors [wR(HF) = 0.023, wR(DFT) = 0.019] and fair with respect to the results of a topological analysis. Density-functional calculations seem to yield slightly better results than Hartree-Fock calculations.

Critical points in a crystal and procrystal

The critical points in the model electron density distributions of LiF, NaF, NaCl, and MgO crystals, constructed from accurate X-ray diffraction data, are determined. For LiF and MgO they are compared with those obtained from a Hartree–Fock electron density calculation. Both experiment and theory show the same type of critical points on the bond lines. The topological features in areas between structural units, where the electron density is low and near-uniform, turn out to be model dependent and cannot be established well with the data available. Topological analysis of procrystals (hypothetical systems consisting of spherical atoms or ions placed on the same sites as atoms in real crystal) show that (3, −1) critical points, usually connected with bonding interaction, are observed on interatomic lines in these nonbonded systems as well.

Extracting charge density distributions from diffraction data: a model study on urea

The quality of the extraction of electron density distributions by means of a multipole refinement method is investigated. Structure factors of the urea crystal have been obtained from an electron density distribution (EDD) resulting from a density function calculation with the CRYSTAL95 package. To account for the thermal motion of the atoms, the stockholder-partioned densities of the atoms have been convoluted with thermal smearing functions, which were obtained from a neutron diffraction experiment. A POP multipole refinement yielded a good fit, R = 0.6%. This disagreement factor is based on magnitudes only. Comparison with the original structure factors gave a disagreement of 0.8% owing to differences in magnitude and phase. The fitted EDD still showed all the characteristics of the interaction density. After random errors corresponding to the experimental situation were added to the structure factors, the refinement was repeated. The fit was R = 1.1%. This time the resulting interaction density was heavily deformed. Repetition with another set of random errors from the same distribution yielded a widely different interaction density distribution. The conclusion is that interaction densities cannot be obtained from X-ray diffraction data on non-centrosymmetric crystals.

Electron-density-based calculations of intermolecular energy: case of urea

The intermolecular interaction energy in crystalline urea has been calculated both from diffraction data and from the Hartree-Fock crystalline electron-density distribution, using a modified atom-atom approximation scheme. The electrostatic part of this energy has been calculated from the atomic multipole moments, obtained by adjustment of the multipole model to experimental X-ray and to theoretical Hartree-Fock structure amplitudes. To obtain the induction energy, multipole moments were calculated from structure amplitudes for the crystalline electron density and from those that refer to the electron density of a superposition of isolated molecules. This worked well for the calculation of the interaction energy from Hartree-Fock data (6% difference from the sublimation-energy value), but not for the interaction energy from experimental data, where the moments of the superposition have to come from Hartree-Fock calculations: the two sets of multipole moments are far too different. The uncertainty of the phases of the structure amplitudes, combined with systematic errors in the theoretical data and noise in the experimental values, may account for the discrepancies. The nature of the different contributions to intermol-ecular interactions for urea is examined.

A theoretical study of elastic X-ray scattering

Bragg X-ray scattering intensities are defined as scattering by the thermodynamic average electron-charge density. Purely elastic, kinematic X-ray scattering by a target in thermal equilibrium is always larger than Bragg scattering. At low temperatures, the elastic scattering becomes Bragg scattering. For large molecules, such as a crystal, at ordinary temperatures the elastic and Bragg scattering differ in a relative sense by O(N-1), where N is the number of vibrational degrees of freedom. For most practical cases the Bragg scattering is essentially the same as purely elastic scattering of X-rays.

Electrostatic Molecular Interaction from X-ray Diffraction Data. I. Development of the Method; Test on Pyrazine

Electrostatic interaction is often an important part of the total interaction between molecules. It depends on the electron density distribution in the participating molecules, which can, in principle, be determined by X-ray diffraction methods. A method is described to calculate the electrostatic interaction between two nonpenetrating molecules by adding the pair-wise interaction between the constituent atoms. The molecular electron density distribution is expressed in terms of the densities corresponding with spherical atoms and deformations according to Hirshfelds method. The electrostatic interaction between the various deformation densities is replaced by the interaction between the atomic multipole moments corresponding with the deformation densities. Application of the method to pyrazine, C4H4N2, showed qualitative agreement with results based on quantum-chemical calculations.

Comparison of the Hartree-Fock, Møller-Plesset, and Hartree-Fock-Slater method with respect to electrostatic properties of small molecules. Effects of electron correlation

SummaryThe results of various quantum chemical calculations, the Hartree-Fock (HF) method, the Møller-Plesset perturbation theory (MP2), and the Hartree-Fock-Slater (HFS) method are compared. Atomic charges, dipole moments, topological properties of the electron density distribution and polarizabilities, and first hyperpolarizabilities are calculated. Atomic charges obtained with the HFS method are found to be very close to those calculated with the MP2 method, from which we conclude that the HFS method describes to some extent electron correlation effects. Performing an MP2 calculation after an HF calculation improves the molecular dipole moments considerably, yielding values close to the experimental ones. HFS calculations are computationally less demanding than MP2 and yield comparable results for the electron density distributions, dipole moments and polarizabilities.

Structure and electron density distribution of the nitrate ion and urea molecule upon protonation

SummaryThe changes in the structure and electron density distribution of the nitrate ion and urea molecule upon the presence of a point charge are discussed. Analyses of the Cambridge Structural Database are performed as well as Hartree-Fock calculations on the appropriate molecules in the presence of a point charge. The Hartree-Fock calculations confirm the correlations in structural parameters found in the database. A charge analysis of the molecules explains part of the structural changes caused by the presence of the point charge. The electrostatic potential and Laplacian of the electron density distribution explain the position of the point charge relative to the molecules.

The molecular electron density distribution. Meeting place of X-ray diffraction and quantum chemistry intermediate between theory and experiment

Quantum chemistry and the concepts used daily in chemistry are increasingly growing apart. Among the concepts that are able to bridge the gap between theory and experimental practice, electron density distribution has an important place. The study of this distribution has led to new developments in theory, including Hellmann-Feynman theory and the density functional theory. The possibilities and limitations of these methods are discussed. Various ways of analysing the electron density distribution are presented and discussed. X-ray diffraction enables us to ?observe? the electron density distribution and electrostatic properties. Experimental results are compared with the results of quantum chemical calculations. It is shown that even intermolecular interaction is observable with this method. Problems in determining ionic charges are seen to be inherent in the method.