Enumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensities

Abstract

A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N ≤ 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N - 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (~4(N)/N(3/2) for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.

Determination of small crystal structures from a minimum set of diffraction intensities by homotopy continuation

“…To our knowledge, we are the first to apply one of such methods, polynomial homotopy continuation, to this problem. The exponentially increasing ambiguity of crystal structure determination directly from the minimum of diffraction intensities (Al-Asadi et al, 2012, 2014 precludes brute-force root computation even for relatively small structures. Experimental uncertainty exacerbates this problem further.…”

“…a crystal structure. As N increases, the size and the degree of system (2) rapidly increase, resulting in the exponentially increasing number of its solutions (Al-Asadi et al, 2012, 2014. To resolve this ambiguity, higher-order intensity data are needed, overdetermining the system.…”

“…It was realized early on that the problem of determining an N-atom structure from the algebraic minimum of 3(N À 1) error-free intensities has multiple solutions (Hauptman & Karle, 1951). Recently, we rigorously enumerated this ambiguity and demonstrated that the number of crystal structures that yield the same minimum set of intensities increases exponentially with increasing N (Al- Asadi et al, 2012Asadi et al, , 2014. The solutions themselves can be determined by methods of elementary algebra only for theoretically relevant one-dimensional crystals of small numbers of atoms (N < 5) (Shkel et al, 2011), whereas such methods do not work for larger structures or for higher dimensionality.…”

Ambiguity of structure determination from a minimum of diffraction intensities

“…and the equations for I 0k0 and I 00l have the same functional form as system (2). System (2) is equivalent to the onedimensional crystal structure determination problem (Al-Asadi et al, 2012;Shkel et al, 2011). The number of all solutions of this system was recently obtained by our group (Al-Asadi et al, 2012) as…”

“…Nevertheless, later they demonstrated that algebraic structure determination can be achieved for small structures of one-dimensional crystals, with added bond-length constraints and noted a large structure ambiguity (Cervellino & Ciccariello, 1999). Recently, by applying Bernstein's theorem to the problem of determining the structure of an idealized one-dimensional crystal from the algebraic minimum of intensities, we obtained the structure ambiguity for this oversimplified case of a onedimensional crystal of identical point atoms as a function of the number of atoms in the unit cell (Al-Asadi et al, 2012). Here, we apply Bernstein's theorem to establish this ambiguity for the realistic cases of three-and two-dimensional crystals.…”