Estimation of triplets with interatomic vectors

This paper investigates the exponential triplet probability distribution and the corresponding expectation values that incorporate structural information of an a priori known part of the electron density. The expectation values improve from the Cochran expectation values (no structural information) to the true values (all information known). Combination of the expectation values gives an estimation formula for triplets with structural information. The validity of the resulting estimated triplets is tested with mean absolute errors and correlation coefficients against the true values of the triplets. The single and double Patterson functions may serve to find the structural information ab initio. Some problems are mentioned that mean application of the formulae to standard structure-solution methods is still difficult. The development of more general formulae for other multiplets and higher Patterson functions is expected.