The theoretical background and practical procedures for phase determination by the symbolic addition method are discussed. Phase determining formulas are presented for centrosymmetric and noncentrosymmetric crystals. A probability formula is employed to evaluate the reliability of phase determination for centrosymmetric crystals and a formula for the variance is utilized for the same purpose in noncentrosymmetric ones. These probability measures play a key role in overcoming the main problems involved in carrying out the procedure, namely the nonuniqueness of the internal consistency criterion as applied to the phase determining formulas, and questions concerning the proper circumstances for assigning symbols. The method is generally applicable to centrosymmetric crystals and has been successful in several applications to noncentrosymmetric ones. Some auxiliary phase information is probably required to make the symbolic addition procedure a general one for noncentrosymmetric crystals.

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Joint probability distributions and relevant expected values and variances are obtained for selected (but typical) non-centrosymmetric space groups belonging to the four types 1P222, 2P22, 3P12, 3P22. These lead to formulas for phase determination the analysis and interpretation of which constitute the major goal of this paper. The analysis is strongly dependent on the theory of invariants and seminvariants, and the agreement between this theory and certain consequences of the probability theory is noteworthy.

ABSTRACT:We describe a kernel energy method (KEM) for applying quantum crystallography to large molecules, with an emphasis on the calculation of the molecular energy of peptides. The computational difficulty of representing the system increases only modestly with the number of atoms. The calculations are carried out on modern parallel supercomputers. By adopting the approximation that a full biological molecule can be represented by smaller "kernels" of atoms, the calculations are greatly simplified. Moreover, collections of kernels are, from a computational point of view, well suited for parallel computation. The result is a modest increase in computational time as the number of atoms increases, while retaining the ab initio character of the calculations. We describe a test of our method, and establish its accuracy using 15 different peptides of biological interest.

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