Electrostatic Properties of Molecules from Diffraction Data

Abstract

Single crystal x-ray diffraction analysis today has become an essential methodology for scientists in a wide range of disciplines that span solid-state physics, synthetic chemistry and molecular biology. Although x-ray diffraction was first exploited by the Braggs in 1913 for crystal structure determination,¹ it has only been the last three decades that the method has become routine for establishing the atomic structure in a crystallographic unit cell. The advent of digital computers has made it possible to implement direct methods, the evaluation of relations between measured diffraction intensities and phases, for the assignment of phases to observed x-ray structure factor moduli. Accurate measurements of the diffracted intensities are now made with diffractometers that operate in an automatic mode with supporting software to select out the photon counts which are mostly due to elastic scattering. For many crystals of organic compounds, which have rather low Debye temperatures, it is routine in many labs to cool the sample to about 100 K with a cold stream of nitrogen and thereby measure Bragg diffraction intensities at these reduced temperatures to larger scattering angles than is possible at room (300 K) temperature. Thus full atomic resolution is often achieved. When the phase estimates of the structure factors are adequate for locating the atoms within ten or so picometers of their mean thermal positions, least squares programs are then used to adjust both atomic positions and mean square amplitudes of vibration by fitting a structure factor model to the “observed” structure factor moduli or their squares. In this latter stage of a crystal structure anlaysis, the observation to parameter ratio is greater than ten to one and, for the case of high resolution data, the ratio may exceed fifty to one. The very large redundancy - the larger number of observations to number of atomic parameters - in x-ray crystallography is what makes the experiment a definitive method for structure determination. Upon completion of a crystal structure determination, it is routine to report atomic positions with estimated standard deviations in the range of tenths of picometers. In some cases a dynamical analysis of the structure, such as a rigid body motion model, is included in an attempt to deduce the equilibrium structure of the atoms in the crystal. The final objective, a complete determination of the atomic arrangement with precise metrics in a crystal unit cell, has been met.