Novel phasing method using the origin-free modulus sum function expressed in terms of the absolute electron density

Abstract

<jats:p>The origin-free modulus sum function <jats:italic>S</jats:italic> <jats:sub>M</jats:sub> refines the set Φ of phases of the structure factors by maximizing the coincidence between the experimental origin-free modulus function and the calculated one in terms of the ρ(Φ)<jats:sup>2</jats:sup> density function [Rius, J. (1993). <jats:italic>Acta Cryst.</jats:italic> A<jats:bold>49</jats:bold>, 406–409]. Maximization is normally achieved through the recursive application of a Fourier-based algorithm. The purpose of the present study is: (i) to show that ρ(Φ)<jats:sup>2</jats:sup> can be replaced by |ρ(Φ)| in <jats:italic>S</jats:italic> <jats:sub>M</jats:sub>; (ii) to illustrate the viability of the corresponding phasing algorithm with experimental data.</jats:p>

A new density-modification procedure extending the application of the recent |ρ|-based phasing algorithm to larger crystal structures

“…The novel S M;jj phasing function is rooted in the Z R originfree modulus sum function, a nearly 30 years-old directmethods phasing function (Rius, 1993). Both mainly differ in (i) the introduction of 'Fourier transform' calculations instead of the complex manipulation of 'structure invariants' (Rius et al, 2007); (ii) the replacement of 2 ðrÞ by jðrÞj at each point r of the unit cell by using the property that 2 ðrÞ and jðrÞj are positive-definite functions with similar shape (Rius, 2020). The resulting S M;jj phase refinement function is defined by…”

“…According to Rius (2020), the values of m are 1 (for > 0), 0 (for between 0 and Àt ) and À1 (for < Àt ) with 2 being the variance of (È) and t ' 2.5. Hereafter S M;jj will be shortened to S M for simplicity.…”

“…The phasing residual (2) can be minimized with the S M algorithm (Rius, 2020), i.e. by the iterative application of the modified tangent formula…”

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The incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel preservation) into the recently published S M algorithm based on || [Rius (2020). Acta Cryst A76, 489-493] improves its phasing efficiency for larger crystal structures with atomic resolution data.

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XRD data visualization, processing and analysis with d1Dplot and d2Dplot software packages

“…Our research group has been devoted for a long time to the development of methodologies to obtain information from diffraction data. From the recent through-the-substrate methodology (tts-µXRD) [2] or the latest developments on the origin-free modulus sum function phasing method [3], to the well-established full pattern matching DAjust software [4], the Rietveld refinement program RIBOLS or the direct-space strategy TALP [5] to solve molecular structures from powder diffraction data. The data plotting capabilities never were a priority for us since there are plenty of available tools that can plot 1D and 2D XRD data as part of their data analysis capabilities, such as WinPLOTR [6], Fit2D [7], GSAS [8], EXPO [9], JPowder [10], Highscore [11] or TOPAS [12] among others.…”