An integer minimal principle and triplet sieve method for phasing centrosymmetric structures

Abstract

In this paper, a new integer minimal principle model for centrosymmetric structures is presented; one which fully accounts for reciprocal-space phase shifts present in non-symmorphic space groups. Additionally, characterization of false minima of the model is done in terms of even and odd triplets. Based on this characterization, a triplet sieve method is proposed. First, Gaussian elimination using only a subset of reliable triplets is employed for phasing. Triplet subsets are generated using a progressively smaller set of the strongest reflections. Several phase solution sets are generated by enumerating the degrees of freedom present. To facilitate computational evaluation of the quality of these phase solutions, these phase sets are passed into the crystallographic software SnB, which expands the reflection set in two cycles. The final solution is identified via statistics of two crystallographic figures of merit. Computational results are presented for a variety of structures.

SnBversion 2.3: triplet sieve phasing for centrosymmetric structures

“…These parameters are the number of sieve phases (NSP), the reduction in the number of phases (SPR) considered in each successive sieving step and the minimum number of phases to be used for sieving (NSP min ). A subroutine implementing the procedure described by Smith et al (2007) was added to the SnB program, and the additional steps introduced by triplet sieving are indicated by a gray background in the flow chart (Fig. 1).…”

“…) N A 0 . 5 N † The initial set of sieve reflections is chosen utilizing a binary search for the first instance of non-origin degrees of freedom, as described in x5 of Smith et al (2007).…”

“…For example, the phasing program SHELXD (Schneider & Sheldrick, 2002), which is also based on the Shake-and-Bake algorithm, uses Patterson minimum functions (Buerger, 1959;Nordman, 1966) to derive sets of starting atoms that are, in some way, consistent with the Patterson function. Alternatively, the triplet sieve method (Smith et al, 2007) uses an integer minimal principle to provide a subset of perfect, or nearly perfect, initial phases that can be expanded using standard Shake-and-Bake refinement.…”

Optimization techniques in molecular structure and function elucidation

“…In other words, solutions of MP do not guarantee that no two atoms will coincide. Nonetheless, recent progress on the phase problem suggests that the modeling of phases with integer variables can facilitate the use of linear and integer programming approaches to resolve the multiple minima difficulty (Vaia and Sahinidis, 2003, 2005; Smith et al, 2007; Smith, 2008), while the addition of constraints that enforce atomicity requirements via limiting electron density over a grid provides a complete formulation that leads to correct structures (Smith, 2008). Models and algorithms have been developed for centrosymmetric and non-centrosymmetric structures.…”