Refinement of X-ray and electron diffraction crystal structures using analytical Fourier transforms of Slater-type atomic wavefunctions in Olex2.

An implementation of Slater-type spherical scattering factors for X-ray and electron diffraction for elements in the range Z = 1-103 is presented within the software Olex2. Both high- and low-angle Fourier behaviour of atomic electron density and electrostatic potential can thus be addressed, in contrast to the limited flexibility of the four Gaussian plus constant descriptions which are currently the most widely used method for calculating atomic scattering factors during refinement. The implementation presented here accommodates the increasing complexity of the electronic structure of heavier elements by using complete atomic wavefunctions without any interpolation between precalculated tables or intermediate fitting functions. Atomic wavefunctions for singly charged ions are implemented and made accessible, and these show drastic changes in electron diffraction scattering factors compared with the neutral atom. A comparison between the two different spherical models of neutral atoms is presented as an example for four different kinds of X-ray and two electron diffraction structures, and comparisons of refinement results using the existing diffraction data are discussed. A systematic but slight improvement in R values and residual densities can be observed when using the new scattering factors, and this is discussed relative to effects on the atomic displacement parameters and atomic positions, which are prominent near the heavier elements in a structure.

Refinement of anomalous dispersion correction parameters in single-crystal structure determinations.

Correcting for anomalous dispersion is part of any refinement of an X-ray dif-fraction crystal structure determination. The procedure takes the inelastic scattering in the diffraction experiment into account. This X-ray absorption effect is specific to each chemical compound and is particularly sensitive to radiation energies in the region of the absorption edges of the elements in the compound. Therefore, the widely used tabulated values for these corrections can only be approximations as they are based on calculations for isolated atoms. Features of the unique spatial and electronic environment that are directly related to the anomalous dispersion are ignored, although these can be observed spectroscopically. This significantly affects the fit between the crystallographic model and the measured intensities when the excitation wavelength in an X-ray diffraction experiment is close to an element's absorption edge. Herein, we report on synchrotron multi-wavelength single-crystal X-ray diffraction, as well as X-ray absorption spectroscopy experiments which we performed on the mol-ecular compound Mo(CO)6 at energies around the molybdenum K edge. The dispersive (f') and absorptive (f'') terms of the anomalous dispersion can be refined as independent parameters in the full-matrix least-squares refinement. This procedure has been implemented as a new feature in the well-established OLEX2 software suite. These refined parameters are in good agreement with the independently recorded X-ray absorption spectrum. The resulting crystallographic models show significant improvement compared to those employing tabulated values.

Vanishing of the atomic form factor derivatives in non-spherical structural refinement - a key approximation scrutinized in the case of Hirshfeld atom refinement.

When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR2 factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

Accurate crystal structures and chemical properties from NoSpherA2.

The relationship between the structure and the properties of a drug or material is a key concept of chemistry. Knowledge of the three-dimensional structure is considered to be of such importance that almost every report of a new chemical compound is accompanied by an X-ray crystal structure - at least since the 1970s when diffraction equipment became widely available. Crystallographic software of that time was restricted to very limited computing power, and therefore drastic simplifications had to be made. It is these simplifications that make the determination of the correct structure, especially when it comes to hydrogen atoms, virtually impossible. We have devised a robust and fast system where modern chemical structure models replace the old assumptions, leading to correct structures from the model refinement against standard in-house diffraction data using no more than widely available software and desktop computing power. We call this system NoSpherA2 (Non-Spherical Atoms in Olex2). We explain the theoretical background of this technique and demonstrate the far-reaching effects that the improved structure quality that is now routinely available can have on the interpretation of chemical problems exemplified by five selected examples.

Linear Correlations between Spatial and Normal Noise in Triangle Meshes.

We study the relationship between the noise in the vertex coordinates of a triangle mesh and normal noise. First, we compute in closed form the expectation for the angle θ between the new and the old normal when uniform noise is added to a single vertex of a triangle. Next, we propose and experimentally validate an approximation and lower and upper bounds for θ when uniform noise is added to all three vertices of the triangle. In all cases, for small amounts of spatial noise that do not severely distort the mesh, there is a linear correlation between θ and simple functions of the heights of the triangles and thus, θ can be computed efficiently. The addition of uniform spatial noise to a mesh can be seen as a dithered quantization of its vertices. We use the obtained linear correlations between spatial and normal noise to compute the level of dithered quantization of the mesh vertices when a tolerance for the average normal distortion is given.