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Global Rietveld refinement.



Global optimisation methods of structure determination from powder diffraction Powder diffraction is a scientific technique using X-Ray or neutron diffraction on powder or microcrystalline samples for structural characterization of materials.

Ideally, every possible crystalline orientation is represented equally in a powdered sample.
 data have risen to prominence in a relatively short space of time and they now constitute a key approach in the examination of polycrystalline Adj. 1. polycrystalline - composed of aggregates of crystals; "polycrystalline metals"
crystalline - consisting of or containing or of the nature of crystals; "granite is crystalline"
 molecular organic materials. A correctly formulated global optimisation approach may be regarded as a "global Rietveld refinement Rietveld refinement is a technique devised by Hugo Rietveld for use in the characterisation of crystalline materials. The neutron and x-ray diffraction of powder samples results in a pattern characterised by peaks in intensity at certain positions. " that is capable of delivering accurate crystal structures from high-quality powder diffraction data. This paper focuses on how accuracy at all stages of a powder diffraction experiment impacts upon the overall structure solution process and particular attention is paid to assessing the degree of accuracy with which structures are returned from the global optimisation process.

Key words: global optimisation; powder diffraction; structure determination.

**********

1. Introduction

Faced with the challenge of applying traditional methods of crystal structure solution to powder diffraction data collected from molecular organic compounds, a limited number of options are currently available. One option is to seek to obtain a set of structure factors that is single-crystal-like in terms of quality and then apply conventional direct methods of structure solution. For anything other than structures containing strongly scattering atoms (e.g., S, Cl, Br) in small unit cells (up to say, a few hundred [[Angstrom angstrom (ăng`strəm), abbr. Å, unit of length equal to 10−10 meter (0.0000000001 meter); it is used to measure the wavelengths of visible light and of other forms of electromagnetic radiation, such as ultraviolet ].sup.3] in volume), this task demands experimental approaches that exploit techniques such as differential thermal expansion thermal expansion

Increase in volume of a material as its temperature is increased, usually expressed as a fractional change in dimensions per unit temperature change.
 or induced texture [1]. Alternatively, one can adapt direct methods of structure solution to take account of the inherent uncertainties in the structure factor magnitudes typically extracted from a powder diffraction pattern. Most productively, one can combine these approaches in order to maximise the chances of success.

The early 1990s saw numerous ingenious algorithmic developments in the areas of, for example, maximum entropy entropy (ĕn`trəpē), quantity specifying the amount of disorder or randomness in a system bearing energy or information. Originally defined in thermodynamics in terms of heat and temperature, entropy indicates the degree to which a given , direct methods and Patterson methods. However, by then, it was also clear that supplementing the available diffraction information with prior chemical knowledge of the compound under study was a powerful alternative (and sometimes complementary) approach [2-4]. In dealing with molecular organic compounds, one approach is particularly straightforward. Firstly, a three-dimensional representation of the compound of interest is created in Cartesian space using the known atom types and connectivity in conjunction with tabulated bond lengths, bond angles and bond torsion torsion, stress on a body when external forces tend to twist it about an axis. See strength of materials.  angles where appropriate. Those torsion angles that cannot be assigned accurate values in advance are simply assigned random values. Next, a trial crystal structure is constructed by randomly positioning and orienting this molecular model (by now, transformed to fractional coordinates) in the known unit cell, taking into account known space group information. After calculating diffraction data and comparing it against the measured diffraction data, the variable parameters of the model (typically, the molecular position, orientation and conformation con·for·ma·tion
n.
One of the spatial arrangements of atoms in a molecule that can come about through free rotation of the atoms about a single chemical bond.
) are adjusted in order to maximise the level of agreement between the observed and calculated data (i.e., minimise [chi square chi square (kī),
n a nonparametric statistic used with discrete data in the form of frequency count (nominal data) or percentages or proportions that can be reduced to frequencies.
]), at which point the structure is solved if the global minimum in [chi square] space has been located. However, despite its promise, this direct-space approach (so called, because the adjustable parameters lie in real rather than reciprocal space) was restricted in application to rigid or near-rigid molecules until only a few years ago. There were two main reasons for this. Firstly, many thousands of trial structure evaluations are typically required even when dealing with molecules with no internal degrees of freedom. As the calculation of even a single agreement factor based upon the full diffraction profile is, relatively speaking, a computationally intensive process, then the structure solution process is quite time consuming. Secondly, the search methods employed at the time (e.g. grid search) were not especially sophisticated.

Clearly, the ability to use intensities extracted from a diffraction profile offers a great opportunity for speeding up the structure evaluation process, i.e., the time-consuming peak shape and profile calculations need be performed only once, during the intensity extraction process. At the time though, it was widely believed that individual extracted intensities were too unreliable for this purpose due to the problem of reflection overlap. If, however, one uses the sums of intensities for groups of overlapping reflections [4] or the correlated integrated intensities extracted during a Pawley refinement of the diffraction data [5], this objection may be overturned. Accordingly, evaluation of each trial structure may be carried out using, for example:

[chi square] = [[SIGMA].sub.h][[SIGMA].sub.k][([I.sub.h]-c|[F.sub.h]|[.sup.2])([V.sup.-1])[.sub.hk]([I.sub.k] - c|[F.sub.k]|[.sup.2])] (1)

where [I.sub.h,k] is the extracted intensity from a Pawley refinement of the diffraction pattern diffraction pattern

The interference pattern that results when a wave or a series of waves undergoes diffraction, as when passed through a diffraction grating or the lattices of a crystal.
, [V.sub.hk] is the covariance matrix In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.  from the Pawley refinement, c is the scale factor and [F.sub.h,k] is the calculated structure factor from the current trial structure. This approach is mathematically equivalent to the Rietveld method and for programs that implement it, such as DASH (1) [7] the resultant increase in the rate at which structures can be evaluated is impressive. Taking the structure solution of hydrochlorothiazide hydrochlorothiazide /hy·dro·chlo·ro·thi·a·zide/ (-klor?o-thi´ah-zid) a thiazide diuretic, used for treatment of hypertension and edema.

hy·dro·chlo·ro·thi·a·zide
n. Abbr.
 from synchrotron synchrotron: see particle accelerator.
synchrotron

Cyclic particle accelerator in which the particle is confined to its orbit by a magnetic field. The strength of the magnetic field increases as the particle's momentum increases.
 powder diffraction data as an example (23 atoms, 204 reflections up to 1.5 [Angstrom] resolution, 9726 points in the profile), DASH evaluates ca. 3500 trial structures per second running on a single processor 800 MHz (MegaHertZ) One million cycles per second. It is used to measure the transmission speed of electronic devices, including channels, buses and the computer's internal clock. A one-megahertz clock (1 MHz) means some number of bits (16, 32, 64, etc.  Intel Pentium III The successor to the Pentium II from Intel. Introduced in the spring of 1999 at 500 MHz, the Pentium III architecture was similar to the Pentium II with the addition of 70 new instructions optimized for multimedia (see SSE).  PC.

The now commonly employed optimisation methods of simulated annealing simulated annealing - A technique which can be applied to any minimisation or learning process based on successive update steps (either random or deterministic) where the update step length is proportional to an arbitrarily set parameter which can play the role of a temperature.  and evolutionary algorithms evolutionary algorithm - (EA) An algorithm which incorporates aspects of natural selection or survival of the fittest. An evolutionary algorithm maintains a population of structures (usually randomly generated initially), that evolves according to rules of selection, recombination,  (which include genetic algorithms Genetic algorithms

Search procedures based on the mechanics of natural selection and genetics. Such procedures are known also as evolution strategies, evolutionary programming, genetic programming, and evolutionary computation.
) possess distinct advantages over the Monte-Carlo [2] and grid search [3-4] methods previously employed, in that they search the relevant parameter space In generative art people talk about parameter space as the set of possible parameters for a generative system.

In statistics one can study the distribution of a random variable. Several models exist, the most common one being the normal distribution (or Gaussian distribution).
 in a much more efficient manner. It is the combination of these search algorithms In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions.  with the aforementioned speed gains that have taken direct-space methods from a niche interest to a mainstream approach that is now at least competitive with direct methods of structure solution for powders in the field of molecular organic materials [6,8,9]. That global optimisation can now be considered routine for many powder diffraction problems is emphasised by the significant increase in the number of published structures solved in this way. For a more detailed explanation of the underlying principles of the global optimisation approach to structure solution and the many variants on the basic theme, see Shankland and David in [1].

2. Accuracy With Respect to Structure Determination

The route taken in deducing a crystal structure from powder diffraction data is only as strong as its weakest link. If, for example, the unit cell dimensions of the structure cannot be determined, then the structure solution process stalls at this stage. As increasingly complex structure determinations are tackled by powder diffraction, it therefore becomes imperative to pay particular attention to accuracy at all stages of the process.

2.1 Sample Preparation

The usual considerations of sample purity apply to any structure determination from powder diffraction data (SDPD SDPD San Diego Police Department (San Diego, CA, USA)
SDPD Surveillance Data Processing and Distribution
SDPD System-Driven Physical Design
) attempt, i.e., is the sample chemically pure and, if so, is it a single structure or a mixture of polymorphic polymorphic - polymorphism  forms? Whilst the presence of chemical impurities or multiple polymorphs does not preclude structure solution (see, for example, the determination of telmisartan form B in the presence a second solvated form, [10], or the determination of two forms of (C[H.sub.3])[.sub.2]S[Br.sub.2] in a mixed phase [1]) it is nevertheless a significant complicating com·pli·cate  
tr. & intr.v. com·pli·cat·ed, com·pli·cat·ing, com·pli·cates
1. To make or become complex or perplexing.

2. To twist or become twisted together.

adj.
1.
 factor. Indexing two unknown cells from a single diffraction pattern is a non-trivial task, and the process of Pawley or Le Bail fitting the pattern for two phases introduces correlations between overlapping reflections in the two sets of structure factors, in addition to the correlations present within a single set. Whilst recognising that, in many cases, it is not possible (or straightforward) to obtain a "pure" sample to work with, it is certainly the case that having a single-phase sample removes one complication from the structure solution process.

An additional complicating factor in the global optimisation approach to structure solution is that, in general, the full molecular connectivity of the molecule under study must be known if the SDPD is to be successful. This means that NMR NMR: see magnetic resonance. , mass spectrometry mass spectrometry
 or mass spectroscopy

Analytic technique by which chemical substances are identified by sorting gaseous ions by mass using electric and magnetic fields.
, IR and elemental analysis Elemental analysis is a process where a sample of some material (e.g., soil, waste or drinking water, bodily fluids, minerals, chemical compounds) is analyzed for its elemental and sometimes isotopic composition.  information should ideally point to an unambiguous two-dimensional molecular formula that can then be translated into a three-dimensional model within the global optimisation program. If the input structure is incorrect, then the full crystal structure cannot be determined correctly. However, if the input structure is close to that of the correct structure, it may still be possible to interpret the resultant "incorrect" crystal structure in such a way as to lead to the correct structure. For example, during the structure determination of [([C.sub.5][H.sub.4]B(C[H.sub.3]))[.sub.2]Fe]-4,4'-bipyridine polymer from synchrotron x-ray powder data [11], initial attempts to solve the structure resulted in crystal structures in which the basic repeat unit was not long enough to cross the unit cell and complete the polymeric polymeric /poly·mer·ic/ (pol?i-mer´ik) exhibiting the characteristics of a polymer.

pol·y·mer·ic
adj.
1. Having the properties of a polymer.

2.
 structure, indicating an error in the input model. Upon checking, it was found that a one "B(C[H.sub.3])[.sub.2]" unit had accidentally been omitted from the two-dimensional sketch upon which the input model was based. Upon insertion of this group into the model, a crystal structure was obtained in which the monomer monomer (mŏn`əmər): see polymer.
monomer

Molecule of any of a class of mostly organic compounds that can react with other molecules of the same or other compounds to form very large molecules (polymers).
 unit was able to span the unit cell and form a polymeric chain, giving a satisfactory Rietveld refinement.

Other, less significant inaccuracies in the input molecular structure can sometimes be tolerated. For example, hydrogen atoms are frequently omitted from input models in order to simplify their construction or to speed up the calculation of structure factors by decreasing the number of contributing atoms that need to be considered. Such an omission is unlikely to hinder a structure determination unless it constitutes a significant part (for example, 20%) of the overall scattering power of the molecule. Often, decisions about the correctness of a particular structural feature can be deferred to the structure completion stage, or structural ambiguities handled in a multi-solution approach. For example, both the cis and trans isomers isomers (ī´sōmurz),
n.pl 1. organic compounds having the same empirical formula–i.e.
 of a molecule can be constructed and optimised independently against the diffraction data in order to determine which one is correct. The application of traditional "structure completion" methods such as Fourier recycling are still extremely valuable at the end of a global optimisation structure solution, as they can indicate the presence of features that the chemist failed to note or anticipate e.g. solvent of crystallisation.

2.2 Data Collection

It is difficult to collect accurate x-ray powder diffraction data to atomic resolution for the majority of molecular organic compounds, due to a combination of the Lorentz-polarisation factor and form-factor fall-off, compounded by a lack of strongly scattering atoms in the compounds. It has been known for several years now that employing a variable counting time (VCT VCT Voluntary Counseling and Testing
VCT Vinyl Composition Tile
VCT Saint Vincent and the Grenadines (ISO Country code)
VCT Venture Capital Trust (UK fiscal status) 
) scheme, in which the "weak" diffraction data at higher angles is collected for much longer than the "strong" data at lower angles, confers significant benefits at the stage of Rietveld refinement. By the same token, in structure determination, where accurate structure factors (or sums of structure factors for overlapping reflections) are required, a VCT strategy can greatly enhance the chances of success. A simple and effective VCT strategy is to calculate the data collection time, t, for each particular 2[theta Theta

A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option.
] value in the pattern using:

t([theta]) [proportional] (sin [theta] sin 2[theta])/[??][f.sub.av.sup.2]([theta])exp exp
abbr.
1. exponent

2. exponential
(-2 [B.sub.av] [sin.sup.2] [theta]/[[lambda].sup.2])[??] (2)

where [f.sub.av] is a representative atomic scattering factor (e.g., carbon), [B.sub.av] is an estimated overall Debye-Waller factor The Debye-Waller factor (DWF), named after Peter Debye and Ivar Waller, is used in condensed matter physics to describe the attenuation of x-ray scattering or neutron scattering caused by thermal motion or quenched disorder.  and [lambda] is the incident wavelength.

The resultant VCT scheme is shown in Fig. 1 for the case of a 6 h data collection from a sample of chlorothiazide chlorothiazide /chlo·ro·thi·a·zide/ (klor?o-thi´ah-zid) a thiazide diuretic used in the form of the base or the sodium salt to treat hypertension and edema.  at a wavelength of 1.1 [Angstrom], assuming a B value of 1 and setting the minimum count time to be 1 s. The benefits in terms of the corresponding diffraction data are clearly illustrated in Fig. 2. It is significant that 80% of the data collection time was spent in the range 40[degrees] to 60[degrees] and that 75% of the strong |E| values (|E| > 1.5) lie in this range. Use of the VCT scheme enabled a direct methods solution in which all the seventeen non-H atoms in the structure were identified from the top |E|-map [12].

The published literature contains many examples of structure solution problems where constant count times were used and it is not at all obvious why the authors did not take advantage of a VCT scheme. One suspects that the reasons for this are not particularly well-founded and that the popularity of the constant count time scheme is simply to do with the fact that it is "traditional"; in other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently
, "the way it has always been done".

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.3 Profile Fitting

The ability to fit the shapes of individual diffraction peaks accurately has important implications not only for the final Rietveld refinement, but also for all stages of the structure determination process once the diffraction data has been collected. Many diffraction patterns can be adequately described using a Voigt (or pseudoVoigt) peak shape and the increasing use of corrections for axial axial /ax·i·al/ (ak´se-al) of or pertaining to the axis of a structure or part.

ax·i·al
adj.
1. Relating to or characterized by an axis; axile.

2.
 divergence divergence

In mathematics, a differential operator applied to a three-dimensional vector-valued function. The result is a function that describes a rate of change. The divergence of a vector v is given by
 [13] means that fitting asymmetry Asymmetry

A lack of equivalence between two things, such as the unequal tax treatment of interest expense and dividend payments.
 in peaks at low diffraction angles is no longer a problem.

2.3.1 Indexing

Accurate fitting of low angle peaks returns the accurate peak positions that are essential in obtaining a good indexing solution. On a modern synchrotron powder diffraction beamline such as BM16 at the ESRF ESRF European Synchrotron Radiation Facility (Grenoble, France)
ESRF Environmental Studies Research Funds (Canada)
ESRF Endstage Renal Failure (kidney failure) 
, the intrinsic accuracy of the diffractometer A Diffractometer (Main Entry: dif·frac·tom·e·ter Pronunciation: di-"frak-'tä-m&-t&r Function: noun) is a measuring instrument for analyzing the structure of a usually crystalline substance from the scattering pattern produced when a beam of radiation or particles (as X rays or  is such that high figures-of-merit for powder indexing solutions are the norm, even if the peak positions of the first twenty or so lines are simply estimated using a cursor (1) The symbol used to point to some element on screen. On Windows, Mac and other graphics-based screens, it is also called a "pointer," and it changes shape as it is moved with the mouse into different areas of the application. . With data as good as this, accurate peak fitting can result in unexpectedly high figures-of-merit. For example, an F(40) value >2150 was obtained for the best cell corresponding to a diffraction data set collected on station BM16 from a proprietary pharmaceutical compound.

2.3.2 Structure Factor Extraction

It is particularly important to have a good fit to the diffraction peaks during the stage of intensity extraction. Poorly fitted peaks lead to poor intensity estimates that then mislead mis·lead  
tr.v. mis·led , mis·lead·ing, mis·leads
1. To lead in the wrong direction.

2. To lead into error of thought or action, especially by intentionally deceiving. See Synonyms at deceive.
 structure solution attempts. Of particular importance here is the higher angle diffraction region, where it is difficult to distinguish the weaker Bragg diffraction The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in  features from the background "noise". Correlations between the parameters of a refineable background profile and the refineable intensities in a Pawley or a LeBail fit can lead to inaccurate intensity estimates. The chances of this happening are significantly decreased if a VCT strategy has been employed and they can be decreased still further if the pattern being fitted is first carefully background subtracted, as the need to then simultaneously refine background parameters during the extraction step is eliminated.

2.3.3 Space Group Determination

This is a stage of the structure determination process that is often considered to be straightforward but which stills relies heavily upon the intuition and ingenuity of the crystallographer crys·tal·log·ra·phy  
n.
The science of crystal structure and phenomena.



crystal·log
, plus some knowledge of the relative frequencies of occurrence of the common space groups exhibited by organic materials. Often, the choice of space group will be clear, based on the size of the unit cell, the volume of the molecule and the presence or absence of certain diagnostic low-angle reflections. However, due to the problem of peak overlap, space group choices are frequently made on the basis of the presence or absence of only one or two diffraction peaks. Higher order systematic absences occurring at higher two-theta positions in the diffraction pattern are often obscured by other Bragg peaks and so cannot easily be factored into the decision making process (Fig. 3). A less subjective approach to the problem has been outlined by Markvardsen et al. [14]. In this approach, all the data (i.e., the extracted correlated integrated intensities from a Pawley fit to the diffraction data) are consulted and the probabilities of each of the possible extinction symbols (consistent with the crystal symmetry) relative to the extinction symbol possessing no systematic absences, is calculated. Typical output from such a calculation is shown in Table 1 for the monoclinic mon·o·clin·ic  
adj.
Of or relating to three unequal crystal axes, two of which intersect obliquely and are perpendicular to the third.


monoclinic
Adjective

Crystallog
 structure decaflouroquarterphenyl (DFQP). Extinction symbol I1a1 is consistent with space groups 1a and 12/a, and in the case of DFQP, the molecule is centrosymmetric and the correct space group is 12/a. Experience has shown that a good fit to the diffraction profile is a prerequisite to the successful application of this approach.

2.4 Structure Solution

Ultimately, the accuracy of crystal structures determined by global optimisation methods is determined by the quality of the diffraction data against which the crystal structure is finally refined. In that regard, global optimisation is no different from any other structure solution method. Of more interest to those involved in the structure determination process is the question "how accurate is the answer output from the global optimisation process?" That is to say, "how close is the optimised structure to the final refined structure?" Again, the quality of the diffraction data plays a significant role, both in terms of the resolution to which it has been collected and the quality of the Bragg peaks at the highest data resolution.

[FIGURE 3 OMITTED]

For the purposes of this discussion, assume that the molecular structure has been parameterised in terms of variable torsion angles only, i.e., as a series of rigid units connected by bonds around which those rigid units can rotate. In principle, a correctly formulated global optimisation approach to structure determination is equivalent to a "global Rietveld refinement". The global optimisation algorithm locates, orients and folds the molecule of interest within the unit cell such that agreement between the observed and calculated diffraction data is maximised. If necessary, a semi-global optimisation algorithm (such as a simplex) or a local minimiser (such as conjugate conjugate /con·ju·gate/ (kon´jdbobr-gat)
1. paired, or equally coupled; working in unison.

2. a conjugate diameter of the pelvic inlet; used alone usually to denote the true conjugate diameter; see
 gradient) can be employed in the same data / parameter space to improve the efficiency of the final step of locating the exact best minimum. Assuming that the global minimum has indeed been located, then the structure so obtained is the one that equates to a final rigid-body Rietveld refinement in which no other parameters are varied. Further improvements in the fit to the diffraction data can then be obtained only by the introduction of additional parameters to the model; for example, by employing a traditional Rietveld refinement in which the atomic positions are refined individually, or a restrained Rietveld refinement in which atomic positions are refined individually subject to a series of restraints that help to maintain chemical sense.

Here, only structures that have been obtained directly from a global optimisation structure solution process are considered. Comparisons are then made with solutions that have been obtained either from a single crystal or from a Rietveld refinement in order to gauge the level of accuracy that can be expected.

2.4.1 Tetracycline Hydrochloride tetracycline hydrochloride

Actisite, Apo-Tetra (CA), Bristacycline, Novotetra (CA), Nu-Tetra (CA), Sumycin, Sumycin Syrup

Pharmacologic class: Tetracycline

Therapeutic class: Anti-infective



The determination of the crystal structure of the hydrochloride hydrochloride /hy·dro·chlo·ride/ (-klor´id) a salt of hydrochloric acid.

hy·dro·chlo·ride
n.
A compound resulting from the reaction of hydrochloric acid with an organic base.
 salt of the antibiotic compound tetracycline tetracycline (tĕ'trəsī`klēn), any of a group of antibiotics produced by bacteria of the genus Streptomyces. They are effective against a wide range of Gram positive and Gram negative bacteria, interfering with protein  (Fig. 4) was set as a "blind test" of SDPD in 1998 [15], largely in response to the emergence of fast global optimisation methods of structure solution. Diffraction data collected from a polycrystalline sample of tetracycline hydrochloride (capillary capillary (kăp`əlĕr'ē), microscopic blood vessel, smallest unit of the circulatory system. Capillaries form a network of tiny tubes throughout the body, connecting arterioles (smallest arteries) and venules (smallest veins). , station 9.1 Daresbury SRS SRS, SRS-A

see slow-reacting substance.
, [lambda] = 0.692 [Angstrom], image plate detector, data range 2[degrees] to 40[degrees] 2[theta]) were posted on a web site along with the chemical formula ([C.sub.22][H.sub.24][N.sub.2][O.sub.9]HCl), unit cell and space group (a = 10.981 [Angstrom], b = 12.853 [Angstrom], c = 15.733 [Angstrom], P[2.sub.1][2.sub.1][2.sub.1]) of the previously unsolved crystal structure. Participants were then invited to download the data and attempt to solve the crystal structure. Interestingly, the molecular connectivity of the molecule in question was not supplied. This is not a problem for direct or Patterson methods of solution but is a significant problem for global optimisation methods, as they rely upon knowing the molecular connectivity in advance. However, a simple search of a chemical reagent reagent /re·a·gent/ (re-a´jent) a substance used to produce a chemical reaction so as to detect, measure, produce, etc., other substances.

re·a·gent
n.
 catalogue showed that the chemical formula of the molecule concerned matched that of tetracycline hydrochloride, suggesting that this was a likely candidate for the crystal structure to be determined. Using the supplied unit cell and space group, 594 correlated integrated intensities were extracted from the diffraction data in the range 3[degrees] to 30[degrees] 2[theta] by means of a Pawley fit, achieving an [R.sub.wp] value = 2.3%. Thereafter, an internal coordinate description of the positively charged Adj. 1. positively charged - having a positive charge; "protons are positive"
electropositive, positive

charged - of a particle or body or system; having a net amount of positive or negative electric charge; "charged particles"; "a charged battery"
 tetracycline ion was constructed using the crystal structure of tetracycline as a prior source of the molecular topology topology, branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size. . In this way, the molecule was parameterised as a rigid fragment with only two optimisable torsion angles connecting the -N(C[H.sub.3])[.sub.2] group and the amide group to the fused ring system. The position, orientation and conformation of the tetracycline ion and the position of the chloride counter-ion were then optimised against the extracted correlated integrated intensities using a simulated annealing technique implemented in a computer program running on a 433 MHz DEC Alpha See Alpha.

(processor) DEC Alpha - A RISC microprocessor from DEC. In November 1995, the Alpha was purportedly the fastest non-research chip used in commonly available workstations. It is superpipelined and superscalar.
 Personal Workstation Same as personal computer or workstation. . Several structure solution runs were performed and solution times varied from 26 s to 600 s. Each run converged to give the same answer and was characterised by a rapid fall in the correlated integrated intensities [chi square] value from starting values of around 7000 to finishing values of around 300. The solution thus obtained was found to be very close to the single crystal structure that was subsequently revealed by the test organisers (Fig. 5). The average separation between the positions of the non-H atoms is only 0.191 [Angstrom], with minimum and maximum deviations of 0.041 [Angstrom] and 0.544 [Angstrom], respectively. Slack constrained con·strain  
tr.v. con·strained, con·strain·ing, con·strains
1. To compel by physical, moral, or circumstantial force; oblige: felt constrained to object. See Synonyms at force.

2.
 Rietveld refinement against the full diffraction data range resulted in an improved fit to the data as a result of some changes in the previously fixed molecular topology of the fused ring system (Fig. 6). A final [R.sub.wp] value of 2.9% was obtained for the Rietveld fit.

[FIGURE 4 OMITTED]

2.4.2 Capsaicin capsaicin /cap·sa·i·cin/ (kap-sa´i-sin) an alkaloid irritating to the skin and mucous membranes, the active ingredient of capsicum; used as a topical counterirritant and analgesic.

cap·sa·i·cin
n.


The molecular crystal structure of capsaicin, the hot component of chilli peppers, was solved directly from synchrotron powder diffraction data alone, using the same simulated annealing procedure outlined in the previous section [6]. The internal coordinate description of the molecule was constructed using standard bond lengths, bond angles and bond torsions where appropriate. The level of agreement between the crystal structure obtained directly from the simulated annealing and the subsequently determined single crystal structure is excellent (Fig. 7) though a small degree of preferred orientation (discovered during subsequent restrained Rietveld refinement) precluded even better agreement.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

2.4.3 Promazine Hydrochloride

The molecular crystal structure of the tranquilliser promazine hydrochloride was solved directly from synchrotron powder diffraction data alone, using the same procedure outlined for capsaicin. It is clear from Fig. 8 there is little difference between the model-independent Pawley fit to the data and the fit to the data given by a scale-factor-only refinement of the crystal structure output from the simulated annealing procedure. The level of agreement between observed and calculated data, particularly at higher angles, confirms that the crystal structure has been determined with a good degree of accuracy.

[FIGURE 8 OMITTED]

2.4.4 Uridine uridine /uri·dine/ (ur´i-den) a pyrimidine nucleoside containing uracil and ribose; it is a component of nucleic acid and its nucleosides are involved in the biosynthesis of polysaccharides. Symbol U.

Another example of an excellent fit to the data at high angle obtained from the structure output direct from the simulated annealing procedure is shown in Fig. 9. The crystal structure in question is that of uridine, which contains two molecules in the asymmetric A difference between two opposing modes. It typically refers to a speed disparity. For example, in asymmetric operations, it takes longer to compress and encrypt data than to decompress and decrypt it. Contrast with symmetric. See asymmetric compression and public key cryptography.  unit.

2.4.6 Famotidine Form B

The crystal structure of famotidine form B was first solved from synchrotron x-ray powder diffraction data in 1998 using simulated annealing. Yet again, the correct solution (Fig. 10) is in excellent agreement with a subsequently determined single crystal structure. As a moderately complex organic crystal structure that can be solved relatively easily, it has since served as an excellent structure for evaluating the effects of varying algorithmic, chemical and crystallographic crys·tal·log·ra·phy  
n.
The science of crystal structure and phenomena.



crystal·log
 variables in the simulated annealing procedure. For a full discussion of the variables investigated, see Shankland et al. [16]. One of the most important findings of this work is the effect of diffraction data resolution upon the accuracy of the structure determination. Eighty simulated annealing solutions were obtained from each of four data sets truncated truncated adjective Shortened  to the following resolutions: 1.5 [Angstrom], 2.0 [Angstrom], 2.5 [Angstrom], and 3.0 [Angstrom]. The crystal structures from the successful runs ("success" meaning that each run reached a particular pre-set [chi square] value) were then analysed and distributions of each of the optimisable torsion angles within the famotidine molecule calculated. Fig. 11 shows this distribution for one such torsion angle, whose value in the single crystal structure is 62.7[degrees]. At data resolutions as low as 2.5 [Angstrom], the distribution of torsion angles is always centred on the correct value, albeit with increasing spread as the resolution is decreased from 1.5 [Angstrom]. At 3.0 [Angstrom], the distribution is bimodal bi·mod·al  
adj.
1. Having or exhibiting two contrasting modes or forms: "American supermarket shopping shows bimodal behavior
. The results indicate that for a molecule of the complexity of famotidine, data should be collected to at least 2.5 [Angstrom] resolution for reliable structure determination.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

2.5 Accuracy of Input Structures

In sec. 2.4, only examples of structure determination in which the molecules under study had been parameterised in terms of a series of connected rigid bodies Rigid body

An idealized extended solid whose size and shape are definitely fixed and remain unaltered when forces are applied. Treatment of the motion of a rigid body in terms of Newton's laws of motion leads to an understanding of certain important
 were shown. The justification for this approach is exemplified in Fig. 12, which shows the effective "fluctuations" in the correlated integrated intensities [chi square] values seen as a result of varying certain key structural parameters during the structure solution of promazine hydrochloride. It is clear that it is the position, orientation and conformation of strongly scattering fragments that have the greatest impact upon the structure solution process, when parameters are varied within chemically sensible bounds. That is not to say that optimisation of bond lengths and bond angles may not be important in circumstances where their values are not known with sufficient accuracy in advance but, in the vast majority of circumstances, a "connected rigid body" parameterisation is likely to be effective. Furthermore, the accuracy achieved with such an approach will, in all likelihood, exceed that justified by the powder diffraction data alone. Nevertheless, a very accurate starting structure can help the structure solution process, if high quality diffraction data are available. This is illustrated in Table 2, which shows values of [chi square] obtained from repeated DASH crystal structure solutions of cimetidine cimetidine /ci·met·i·dine/ (si-met´i-den) a histamine H2 receptor antagonist, which inhibits gastric acid secretion; used as the base or the monohydrochloride salt in the treatment and prophylaxis of gastric or duodenal ulcers,  (Fig. 13) using a series of increasingly accurate input models. Successful structure determinations are indicated by the "*". The energy-minimised model yields significantly better [chi square] values than the model constructed simply using standard bond lengths and bond angles, although the structure solution success rate was unchanged in this small set of repeat runs.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

2.6 Conclusions

It is certainly true to say that global optimisation methods of structure determination from powder diffraction data are now competitive with direct methods, at least for the case of molecular organic compounds. Their effectiveness reflects their ability to incorporate prior chemical knowledge in the form of the known connectivity of the molecule under investigation. They find particular utility in cases where direct methods of structure solution currently have difficulty, i.e., with low-resolution data, poor quality data, and "equalatom" structures.

Ironically, it is now often easier to solve a moderately complex organic crystal structure to a chemically sensible answer than it is to refine it to "publication quality". Powder diffraction is entering an era in which, the "acceptance criteria" for publication need to be carefully reconsidered, if useful crystal structures are to find their way into the public domain.

SDPD can be regarded as routine in general, although there are many specific cases within the current bounds of structural complexity where structure solution remains frustratingly difficult. If the range of applicability of SDPD is to be extended beyond the current bounds, attention now needs to be focused on the basic information content of the powder diffraction pattern and on how it can be enhanced by experimental and algorithmic developments. As an added complication, the ability to solve large crystal structures has taken us to a region where large unit cells are the norm and powder indexing often becomes the limiting step. That said, the fact that very large unit cells can be indexed from powder diffraction data when the diffraction features are close to instrumental resolution [17] provides a basis for optimism. Of one thing we can be sure--the many remaining problems associated with the accurate determination of organic structures from powders will serve as a spur to some exciting developments in the near future.
Table 1. Log--likelihood values

Extinction symbol  Log-likelihood value

  I1a1                   153.2
  I1-1                   129.1
  P 1 21/n 1              60.2
  P 1 21/c 1              59.6
  P 1 n 1                 57.2
  P 1 c 1                 56.6
  P 1 21/a 1              56.2
  P 1 a 1                 53.2
  P 1 21 1                 3.0
  P 1-1                    0.0
  A 1 n 1              -3619.2
  A 1-1                -3641.0
  C 1 c 1              -4218.8
  C 1-1                -4248.0

Table 2. [chi square] values from DASH runs

Cimetidine input model  [chi square] values from DASH runs

Not energy minimised    112*, 127*, 130*, 148, 226
Energy minimised         80*, 86*, 87*, 210, 214
Single crystal           64*, 73*, 82*, 87*, 90*


Acknowledgments

The author gratefully acknowledges a long-standing and fruitful collaboration with Prof. Bill David on the topic of SDPD. Many of the examples cited in this work have arisen from work performed in collaboration with others, but particular thanks go to Norman Shankland, Alastair Florence, Lorraine McBride, Alan Kennedy Alan Phillip Kennedy (born 31 August 1954) was a footballer who played for Liverpool during their halcyon days in the late 1970s and early 1980s who had a knack of scoring in major cup finals. , Gerry Steele, Tony Csoka and Anders Markvardsen, all of whom I have worked with closely on many problems. Thanks also due to Andy Fitch and the staff of BM16 at the ESRF for their assistance in collecting many powder data sets and to Robert Dinnebier for bringing many interesting structural problems to our attention. Finally, I would like to thank the referee for his detailed reading of the submitted manuscript and his very helpful amendments.

Accepted: April 11, 2003

Available online: http://www.nist.gov/jres

(1) Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology National Institute of Standards and Technology, governmental agency within the U.S. Dept. of Commerce with the mission of "working with industry to develop and apply technology, measurements, and standards" in the national interest. , nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

3. References

[1] W. I. F. David, K. Shankland, L. B. McCusker, and Ch. Baerlocher, eds., Structure Determination from Powder Diffraction Data, Oxford University Press, Oxford (2002).

[2] K. D. M. Harris, M. Tremayne, P. Lightfoot, and P. G. Bruce, Crystal-structure determination from powder diffraction data by Monte-Carlo methods, J. Am. Chem. Soc. 116, 3543-3547 (1994).

[3] G. Reck, R. G. Kretschmer, L. Kutschabsky, and W. Pritzkow, Posit: a method for structure determination of small partially known molecules from powder diffraction data--structure of 6-methyl-1,2,3,4-tetrahydropyrimidine-2,4-dione (6-methyluracil) Acta Crystallogr. A44, 417-421 (1988).

[4] N. Masciocchi, R. Bianchi, P. Cairati, G. Mezza, T. Pilati, and A. Sironi, P-RISCON--a real-space scavenger for crystalstructure determination from powder diffraction data, J. Appl. Crystallogr. 27, 426-429 (1994).

[5] K. Shankland, W. I. F. David, and T. Csoka, Crystal structure determination from powder diffraction data by the application of a genetic algorithm genetic algorithm - (GA) An evolutionary algorithm which generates each individual from some encoded form known as a "chromosome" or "genome". Chromosomes are combined or mutated to breed new individuals. , Z. Kristallogr. 212, 550-552 (1997).

[6] W. I. F. David, K. Shankland, and N. Shankland, Routine determination of molecular crystal structures from powder diffraction data, J. Chem. Soc. Chem. Commun, 931-932 (1998).

[7] W. I. F. David, K. Shankland, J. Cole, S. Maginn, W. D. S. Motherwell, and R. Taylor, DASH User Manual, Cambridge Crystallographic Data Centre The Cambridge Crystallographic Data Centre (CCDC) is a crystallographic organisation based in Cambridge, England. It is a non-profit organisation whose primary role is the compilation and maintenance of the Cambridge Structural Database, a database of small molecule crystal , Cambridge, UK (2001).

[8] S. Pagola, P. W. Stephens, D. S. Bohle, A. D. Kosar, and S. K. Madsen, The structure of malaria malaria, infectious parasitic disease that can be either acute or chronic and is frequently recurrent. Malaria is common in Africa, Central and South America, the Mediterranean countries, Asia, and many of the Pacific islands.  pigment pigment, substance that imparts color to other materials. In paint, the pigment is a powdered substance which, when mixed in the liquid vehicle, imparts color to a painted surface.  beta-haematin, Nature 404, 307-310 (2000).

[9] G. E. Engel, S. Wilke, O. Konig, K. D. M. Harris, and F. J. J. Leusen, PowderSolve--a complete package for crystal structure solution from powder diffraction patterns, J. Appl. Crystallogr. 32, 1169-1179 (1999).

[10] R. E. Dinnebier, P. Sieger, H. Nar, K. Shankland, and W. I. F. David, Structural characterisation of three crystalline Like a crystal. It implies a uniform structure of molecules in all dimensions. For example, phase change technology, widely used for rewritable optical discs, uses crystalline spots (bits) to reflect the laser beam. Amorphous, non-crystalline bits do not reflect light.  modifications of telmisartan by single crystal and high-resolution X-ray powder diffraction, J. Pharmaceut. Sci. 89 (11), 1465-1479 (2000).

[11] R. E. Dinnebier, M. Wagner, F. Peters, K. Shankland, and W. I. F. David, Crystal structure of the [([C.sub.5][H.sub.4]B[Me.sub.2])[.sub.2]Fe]-4,4 '-bipyridine polymer from high resolution X-ray powder diffraction, Z. Anorgan. Allgeme. Chem. 626, 1400-1405 (2000).

[12] K. Shankland, W. I. F. David, and D. S. Sivia, Routine ab initio [Latin, From the beginning; from the first act; from the inception.] An agreement is said to be "void ab initio" if it has at no time had any legal validity.  structure determination of chlorothiazide by X-ray powder diffraction using optimised data collection and analysis strategies, J. Mater. Chem. 7, 569-572 (1997).

[13] M. M. Eddy, A. K. Cheetham, and W. I. F. David, Powder neutron-diffraction study of Zeolite zeolite

Any member of a family of hydrated aluminosilicate minerals that have a framework structure enclosing interconnected cavities occupied by large metal cations (positively charged ions)—generally sodium, potassium, magnesium, calcium, and barium—and water
 NA-ZK-4--An application of new functions for peak shape and asymmetry, Zeolites 6, 449-454 (1986).

[14] A. J. Markvardsen, W. I. F. David, J. C. Johnston, and K. Shankland, A probabilistic (probability) probabilistic - Relating to, or governed by, probability. The behaviour of a probabilistic system cannot be predicted exactly but the probability of certain behaviours is known. Such systems may be simulated using pseudorandom numbers.  approach to space-group determination from powder diffraction data, Acta Crystallogr. A57, 47-54 (2001).

[15] http://pcb4122.univ-lemans.fr/SDPDRR/.

[16] K. Shankland, L. McBride, W. I. F. David, N. Shankland, and G. Steele, Molecular, crystallographic and algorithmic factors in structure determination from powder diffraction data by simulated annealing, J. Appl. Crystallogr., submitted (2002).

[17] R. B. Von Dreele, P. W. Stephens, G. D. Smith, and R. H. Blessing, The first protein crystal structure determined from X-ray powder diffraction data: a variant of [T.sub.3][R.sub.3] human insulin human insulin
n.
A protein that has the normal structure of insulin produced by the human pancreas but that is prepared by recombinant DNA techniques and by semisynthetic processes.
 zinc complex produced by grinding, Acta Crystallogr. D56, 1549-53 (2000).

Kenneth Shankland

ISIS Facility, Rutherford Appleton Laboratory The Rutherford Appleton Laboratory (RAL) at the Chilton/Harwell Science Campus is a UK scientific research laboratory near Didcot in Oxfordshire. It has a staff of around 1,200 who support the work of over 10,000 scientists and engineers, mainly from the university research , Oxfordshire OX11 0QX, United Kingdom

About the author: Kenneth Shankland is the Acting Group Leader of the Data Analysis and Visualisation Group at the ISIS Facility of the Rutherford Appleton Laboratory in Oxfordshire, U.K.
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