On the use of least-squares restraints for origin fixing in polar space groups

The theory of fixing the origin of the coordinate system in a polar space group by use of restraints (soft constraints or pseudo-observations) is developed for any space group in any setting. The coefficients of the optimal restraint equation are on the average proportional to the square of the atomic numbers. They are determined directly from the unrestrained singular normal-equations matrix. Application of the restraint results in a positive-definite matrix which is as nearly diagonal as possible for the atomic positional coordinates along the origin-free axes. Correlations between these coordinates are therefore minimized. A very compact completely general and easily implemented algorithm results which functions without user intervention.