The probability distribution function of structure factors with non-integral indices. III. The joint probability distribution in the P1; case.

The joint probability distribution function method has been developed in P1; for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F(p1), F(p2)) are found to be useful for phasing purposes, where p1 and p2 are any pair of vectorial indices. A variety of conditional distributions like P(|F(p1)| | |F(p2)|), P(|F(p1)| |F(p2)), P(varphi(p1)| |F(p1)|, F(p2)) are derived, which are able to estimate the modulus and phase of F(p1) given the modulus and/or phase of F(p2). The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F(1), F(2),ellipsis, F(n+1)), P(|F(1)| |F(2),ellipsis, F(n+1)) and P(varphi(1)| |F|(1), F(2),ellipsis, F(n+1)) have been obtained. Some practical tests prove the efficiency of the method.

The joint probability distribution function of structure factors with rational indices. VI. Cases with reduced dimensionality.

The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

SIR99, a program for the automatic solution of small and large crystal structures.

The moduli and flow diagram of the program SIR99 are described. New phasing algorithms are proposed working both in direct and in reciprocal space. Their cooperative work is able to solve the structures of both small and large molecules. In particular, small proteins can be solved ab initio without any use of prior information and any user intervention. The efficiency of the various algorithms employed by SIR99 has been tested, and the role of the tangent formula clarified. The user is also provided with some practical information concerning the computer power needed.