Direct methods - from birth to maturity

The award of the Nobel Prize for Chemistry to two pioneers of direct methods, H. Hauptman and J. Karle, has been a recognition of the importance that these methods have attained, in crystallography in particular and in science generally. The development of direct methods is traced from its first beginnings with Harker-Kasper inequalities and the Karle & Hauptman determinantal inequalities. The range of application of these methods to centrosymmetric structures was much increased by the introduction of the sign relationship by Sayre, Cochran and Zachariasen and an ACA Monograph by Hauptman & Karle to which the origins of the application of direct methods may be traced. Sayre's paper, in 1952, developed an exact equation which applied to both centrosymmetric and non-centrosymmetric structures as did the Karle & Hauptman determinantal inequalities. However, it was the derivation of the probability distribution for an individual three-phase relationship by Cochran in 1955 and the tangent formula by Karle & Hauptman in 1956 which provided the weaponry to tackle non-centrosymmetric structures, but nothing decisive was done until, in 1964, I. L. Karle & J. Karle solved the first non-centrosymmetric structure with direct methods using their symbolic addition procedure. The advent of the computer allowed automatic multisolution procedures, such as MULTAN, SHELX and SIMPEL, to take over as the main tools for the solution of small structures. Various developments since about 1970 have slowly introduced direct-methods concepts into the field of solving macromolecular structures, starting with the maximum-determinant method of Tsoucaris and progressing to the still experimental maximum-entropy method. Various predictions are made about the progress of direct methods in the next few years. A new tangent formula, the Sayre tangent formula, which gives most of the benefit of the maximum-entropy method but is much easier to apply, is suggested as a possible new source of progress. Other progress may be made in the association of direct methods with physical methods, such as isomorphous replacement and anomalous scattering, following work which has already been done over the past few years.