A SIMPLIFIED PERTURBATION TREATMENT FOR SAS PHASING. D. Y. Guo, Robert H. Blessing and David A. Langs, Hauptman-Woodward Medical Research Institute, 73 High St. Buffalo, NY 14203,USA.

Via a simple perturbation approximation based on the probabilistic theory for Friedel-pair SAS two-phase structure invariants, it is shown that the SAS three-phase structure invariants tend to predictable positive values, and an SAS perturbed tangent formula is derived. The theoretical probabilistic results are verified by empirical statistical analyses of model-calculated phases and experimentally measured magnitudes of structure factors for a small-molecule and a protein structure.

The SAS perturbed tangent formula is

Where

W_HK = E_K E_{-H-K ,}

and [[xi]]_[[Eta]] is the probabilistic estimate of the two-phase structure invariant [[phi]]_H +[[phi]]_-H (Hauptman, Acta Cryst. A38 (1982), 623-641).

The test case for the tangent formula was the single-site K₂Pt(NO₂)₄ derivative of macromomycin [P2₁, Z=2C₄₆₁H₇₃₅N_l27O₁₆₀S₄ * 99.5H₂O (located) * 3C₆ H₁₄O₂. Ca_0.6. 3347 Friedel pairs measured to 2.5 Å resolution by Cu K[[alpha]] x-rays (Van Roey & Beerman, Proc. Natl. Acad. Sci. USA. 86(1989),6587-6591)]. Ambiguous phases for 3028 Friedel-pairs were determined by Harker constructions based on the known Pt position. The tangent formula was used to resolve the two-fold ambiguities. After one or two cycles of tangent refinement starting from random choices of the ambiguous Harker phases, the phase errors were minimized. The average phase error for all 3028 phases, including very small |E|s, is about 48deg.. For a selected set of 1428 Friedel-pairs with larger |E| values, the phase error is about 37deg..

Support from NIH grant GM-46733 is gratefully acknowledged.