AN EFFICIENT LOCAL ANALYTIC REPRESENTATION OF ELECTRON DENSITY MAPS: THE SPECTRAL SPLINE APPROXIMATION. Erik Nelson and Lynn Ten Eyck, San Diego Supercomputer Center P.O. Box 85608 La Jolla, CA 92186-9784

The analysis of molecular crystals frequently requires an accurate functional representation of the electron density so that basic mathematical operations, such as differentiation or integration, can be performed analytically before resorting to complicated numerical methods. A prime example is the location of the critical points which describe the topology of the map -- others appear in phase refinement by non-crystallographic symmetry averaging, and resampling of maps in different coordinate systems.

The spectral spline approximation, first introduced by Eric Gross (Approximation Theory V. 1986), generates such a representation by a single Fast Fourier Transform of the (scaled) structure factors. The output of the FFT is a unique set of coefficients that describe the entire map as a basis expansion in terms of ``basis spline'' functions of an arbitrarily specified order k. The spline representation is a smooth, k - 1 differentiable polynomial function. The map is local in the sense that only k³ coefficients are needed to specify the density at any point in the crystal, however each coefficient contains information from the entire space of Fourier coefficients. Furthermore, the approximation does not rely on any a priori information concerning atomic or molecular structure.

This paper presents applications of the spectral spline approximation recently implemented in the XtalView software package for macromolecular crystallography.