Now that I'm retired, I have discovered there's more to life than research in x-ray crystallography. However, there are still some obscure corners of the subject that continue to attract my attention and I present some of them here.
The discrete Fourier transform can be expressed as a matrix (or more than one matrix for a multi-dimensional transform) operating on an array of structure factors, where the elements of the matrix are complex exponentials. The eigenvectors of the matrix (or matrices) form a complete set of orthogonal functions, each of which is its own Fourier transform. Both the electron density and the structure factors are linear combinations of these functions using the same admixture coefficients. A review of the properties of these functions and how to calculate them will be available here in due course.
The constraint that the electron density is non-negative puts constraints on the structure factors such that each Karle-Hauptman matrix that can be generated must be positive semi-definite. Goedkoop matrices are Karle-Hauptman matrices that make full use of space group symmetry. They are in block-diagonal form, thus speeding up all calculations of the eigenvalues, eigenvectors and the determinant. Examples of Goedkoop matrices will be given here when I have had time to prepare them.
For a one-dimensional problem, the Karle-Hauptman matrix is in Toeplitz form. It follows therefore that a three-dimensional problem requires a six-dimensional Toeplitz matrix. The matrix eigenvectors of interest are three-dimensional orthogonal functions whose Fourier transforms form components of the electron density. The information content of each electron density component is measured by the corresponding eigenvalue. The surprise is that these calculations can be expressed in normal matrix form (two-dimensional arrays that are Karle-Hauptman matrices) so standard matrix methods can be used to perform the calculations. Theoretically, these are the best Karle-Hauptman matrices that can be generated. Work is in progress to develop efficient techniques for performing the calculations, to make use of space group symmetry and to explore the properties of these matrices. A report on progress will appear here when I get round to writing it.
There are obvious difficulties in solving and refining structures containing pseudosymmetry, i.e. those small molecule structures with more than one molecule in the asymmetric unit where the molecules are approximately related by translation(s). This gives rise to some very strong structure factors in the diffraction pattern, the rest being comparatively weak. Solving the structure using the strong reflections only gives the average of the different molecules (in a smaller unit cell and possibly a different space group), but it remains to determine their differences. I am developing a method of determining the complete structure once the average structure is known that shows good promise. It solves with ease the less complicated of the pseudosymmetric structures and, with further development, should become a reliable method for the more complicated structures as well. A progress report will be available here in due course.