Determinantal Inequalities Lead to Triplet and Quartet Relationships

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


A necessary and sufficient condition that the electron density function in a crystal be non-negative everywhere is that the system of determinantal inequalities hold: where the (complex) normalized structure factor EH is defined by
$$ {E_H} = \left| {{E_H}} \right|\exp \left[ {i{\phi _H}} \right] = \frac{1}{{{}_{{\sigma _2}}1/2}}\sum\limits_{j = 1}^N {{Z_j}\exp \left[ { - 2\pi iH{r_j}} \right]} $$
H is an arbitrary reciprocal lattice vector, Z. is the atomic number and r. the position vector of the atom labeled j, N is the number of atoms in the unit cell, and
$$ {\sigma _2} = \sum\limits_{j = 1}^N {Z_j^2} $$


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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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