The Neighborhood Principle and the Extension Concept

  • Herbert A. Hauptman
Part of the NATO ASI Series book series (NSSB, volume 274)


Three major themes are developed:
  1. (i)

    The fundamental principle of direct methods: The structure seminvariants link the observed magnitudes |E| with the desired phases ϕ of the normalized structure factors E. Specifically, for fixed enantiomorph, the observed magnitudes |E| determine, in general, unique values for all the structure seminvariants; the latter, in turn, as certain well defined linear combinations of the phases, lead to unique values for the individual phases ϕ.

  2. (ii)

    The neighborhood principle: For fixed enantiomorph, the value of any structure seminvariant T is primarily determined, in favorable cases, by the values of one or more small sets of magnitudes |E|, the neighborhoods of T, and is relatively insensitive to the great bulk of remaining magnitudes. The conditional probability distribution of T, given the magnitudes in any of its neighborhoods, yields an estimate for T which is particularly good in the favorable case that the variance of the distribution happens to be small.

  3. (iii)

    The extension concept: By embedding the structure seminvariant T and its symmetry related variants in suitable structure invariants Q, one obtains the extensions Q of the seminvariant T. In this way the probabilistic theory of the structure seminvariants is reduced to that of the structure invariants, which is well developed.



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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Herbert A. Hauptman
    • 1
  1. 1.Medical Foundation of Buffalo, Inc.BuffaloUSA

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