Direct calculation of atomic coordinates from diffraction intensities: space group P1.

A method is presented that can extend the range of convergence for the least-squares minimization technique in nonlinear systems. The problem of determining directly the positions of atomic coordinates in a crystal from the measured intensities of x-ray scattering is used to illustrate the method. The minimization is facilitated by altering the minimization function in a way that reduces the number of false minima and, in addition, by altering the character of the false minima from time to time by changing the particular deck of data that is used and the values of a variety of parameters that may occur in the defining equations. With a structure consisting of 30 equal atoms in space group P1 and the use of exact data with a Cray XMP/216 computer, convergence to a global minimum was generally obtained, from a random positioning of the 90 coordinates, in a few minutes. Convergence within an hour for a 40-atom problem was much more sporadic, showing a considerable decrease in success with the increase in complexity. The possibility of extending the method to other mathematical systems is apparent.