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Research Collection
Doctoral Thesis
Structure envelopes and their application in structuredetermination from powder diffraction data
Author(s): Brenner, Simon
Publication Date: 1999
Permanent Link: https://doi.org/10.3929/ethz-a-003839470
Rights / License: In Copyright - Non-Commercial Use Permitted
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ETH Library
Dissertation ETH Nr. 13280
Structure Envelopes and their Application in Structure
Determination from Powder Diffraction Data
Dissertation
submitted to the
Swiss Federal Institute of Technology
Zurich
for the degree of Doctor of Natural Sciences
Presented by
Simon Brenner
Dipl. Kristallograph (University of Leipzig)
born July 1, 1965 in Germany
Accepted on the recommendation of
Prof. Dr. W. Steurer
Dr. L.B. McCusker
Dr. Jordi Rius
1999
I
Abstract
In this study, the possibility of using periodic nodal surfaces (PNS) to facilitate structure
determination from powder diffraction data was investigated. PNS are three-dimensional
curved surfaces that describe the nodes of a density distribution and they have been used to
describe physical and chemical features of various compounds with known crystal structures.
It was hoped that such surfaces could be derived for materials with unknown crystal structures
using just the data in the measured powder diffraction pattern, and that they could then be used
in the structure determination process.
The reflection intensities in a powder pattern are dependent upon the electron density
distribution within the unit cell. It was found that by assigning the correct phases to the
structure factors of a few (1-5) strong reflections m the low-angle (high «-/-spacing, low-
resolution) region of the powder pattern, a density distribution similar to a low-resolution
electron density map could be calculated. The PNS for that density distribution separates the
regions of high and low electron densities. That is, the PNS envelops the crystal structure. For
this special PNS, the term "structure envelope" has been adopted.
The reflections needed to generate a structure envelope can be selected by following some
simple rules, and this has been demonstrated for a number of zeolite structures. In most cases,
no more than five reflections are required to calculate an envelope that describes the coarse
features of the crystal structure. To calculate the structure envelope, the phases of the
reflections used must be known. In some cases, only the origin-defining reflections (odr),
whose structure factors can phased arbitrarily, are needed, but usually a few additional
reflections arc required and their phases must be determined
A computer program (SayPerm) to estimate the phases of the structure factors needed for the
structure envelope generation has been written. It combines the pseudo-atom approach, the
application of the Sayre equation and phase permutation. Pseudo-atoms replace building units
in a structure (e.g. Si04 tetrahedron in a silicate framework) to simulate atomic resolution at
high d-spacing. SayPerm is a multisolution approach in which a certain number of phase sets
are generated using error correcting codes (ecc's). These codes sample phase space in a very
efficient manner (e.g. for 14 centrosymmetric reflections only 256 of the possible 2J4 = 16384
permutations are examined and one of the phase sets will have at most two incorrect phases).
From each phase set, a phase extension using the Sayre equation is performed. The results are
then ranked by a figure-of-merit that is calculated from the validity of the Sayre equation. II
n-
could be shown that for moderate-sized centrosymmetric structures (< 10 building units
replaced by pseudo-atoms per asymmetric unit), which can be well approximated by an equal
pseudo-atom structure (e.g. tetrahedral and octahedral structures), the phases for the strongest
structure factors can be estimated reliably. SayPerm was tested on two zeolite structures
(RUB-3 and 1TQ-1), and on 0-AlF3, which has octahedral building units. It was then used to
generate quite an informative envelope for a previously unknown tri-ß-peptidc structure
(sa322) with the chemical formula C^NßOgH^.
An alternative to the Sayre-equation approach is to use permutation synthesis. Here, phases of
only seven strong structure factors are permuted. This is done using the Hamming [7,4,3 j error
correcting code. From each of the 16 structure factor sets obtained, a Fourier map is generated.
Using chemical knowledge (e.g. size and shape of an organic molecule) there is a good chance
that the map that corresponds to the correct phase set can be recognized by eye. The method
was demonstrated on the organic structure Cimetidine (CjQNgSjHjg). For the tri-ß-peptide
structure, the results of the envelope generation mentioned above could be confirmed.
For methods of structure solution that work in direct space, a structure envelope is quite useful,
because it limits the volume m a unit cell where the atoms of the crystal structure are likely to
be located. To test its effectiveness, structure envelope masks were implemented in the
program FOCUS, which was written specifically for the determination of zeolite structures
from powder data. Exhaustive grid searches for four zeolite topologies (APD, SGT, RTE, and
MWW) using structure envelopes were performed. For comparison, these tests were repeated
without the structure-envelope mask. The amount of computer time required for the search for
the APD and SGT topologies was reduced by as much as two orders of magnitude when the
envelope mask was used. To find the RTE and MWW topologies, the envelope proved to be
essential. Without the envelope mask, those grid searches were not successful even after two
days of computing time, whereas with the mask they were found in 25 min and 27 min,
respectively.
The structure envelope for the tri-ß-peptide sa322 calculated using SayPerm was used in a
direct-space search for the structure. A simulated-annealing computer program (SAFE) was
developed for this purpose. It combines simulated annealing with a structure-envelope mask.
The molecule is moved within the unit cell via a simulated annealing algorithm to find the
conformation and orientation of the molecule that produces the lowest weighted profile R-
factor. Simultaneously, the molecule is encouraged to he within the structure envelope. This is
done by imposing a penalty function that is also controlled by the simulated annealing
— Ill —
algorithm. Using this approach, the 41 non-H-atom structure with 23 degrees of freedom (three
positional, three orientational, 17 torsion angle parameters) could be solved.
IV
Zusammenfassung
Im Rahmen der vorliegenden Dissertation wurde untersucht ob periodische Knotenflächen
(PNS) die Chancen für die Lösung von Kristallstrukturen aus Röntgenpulverdaten erhöhen.
Periodische Knotenflächen sind Flächen, die die Nullstellen einer DichteVerteilungsfunktion
im dreidimensionalen Raum verbinden. Mit diesen Flächen lassen sich in beeindruckender
Weise verschiedene physikalische und chemische Eigenschaften innerhalb einer vorher
bekannten Kristallstruktur beschreiben. Wenn es gelänge, solche Flächen für unbekannte
Strukturen einzig und allein aus den gemessenen Daten eines Pulverdiffraktogramms, zu
berechnen, könnte das für die Stmkturaufklärung dieser Strukturen sehr nützlich sein.
Die Intensitäten der einzelnen Reflexe m einem Pulverdiffraktogramm werden hauptsächlich
durch die ElektronendichteVerteilung innerhalb der Einheitszelle bestimmt. Wenn man den
Strukturfaktoren einiger der stärksten Reflexe aus dem Niedrigwinkelbereich (mit hohen d-
Werten) die richtigen Phasen zuordnet, kann eine DichteVerteilung berechnet werden, die der
einer sehr niedrig aufgelösten Elektroncndichtekarte ähnlich ist. Die Fläche aus den
Nullstellen der Verteilungsfunktion trennt hohe und niedrige Elektronendichten der
entsprechenden Kristallstruktur. Somit kann man sagen, dass es sich bei diesen speziellen
Periodischen Knotenflächen um Struktureinhüllende (Structure Envelope) handelt.
Die Reflexe, deren Strukturfaktoren in die Berechnung der Struktureinhüllenden eingehen,
können sehr leicht ausgewählt werden. In den meisten Fällen sind nicht mehr als fünf
Strukturfaktoren notwendig, um eine Fläche zu berechnen, die das Wesentliche einer
Kristallstruktur in niedriger Auflösung beschreibt. Um eine Struktureinhüllende zu erzeugen
müssen die Phasen der Strukturfaktoren bekannt sein. Manchmal genügt es ausschliesslich die
Strukturfaktoren der Reflexe zu benutzen, die den Ursprung definieren. Die entsprechenden
Phasen können dann willkürlich festgelegt werden. In den meisten Fällen aber sind zur
Berechnung einer informativen Strukturcinhüllenden zusätzliche Strukturfaktoren notwendig,
deren Phasen zuerst bestimmt werden müssen.
Zu diesem Zweck ist ein Computerprogramm (SayPerm) entwickelt worden, das die
Pscudoatommethode, die Anwendung der Sayrcgleichung und Phasenpermutationen
kombiniert. Pseudoatome ersetzen Baueinheiten der Kristallstruktur (z.B. Si04 Tetraeder in
einem Silikatgerüst) um atomare Auflösung von Daten mit niedriger Auflösung zu simulieren.
Die Phasenpermutationen werden über Fehlerkorrekturcodcs (ecc's) kontrolliert, die eine sehr
effiziente Abrasterung des durch die permutierten Phasen aufgespannten Raumes
(Phasenraum) gewährleisten. So müssen z.B. fur 14 zentrosymmtrische Strukturfaktoren statt
V
der 2 = 16384 möglichen Phasenkombinationen nur 256 geprüft werden. Trotz der
unvollständigen Abrasterung des Phasenraums weicht jede der resultierenden
Phasenkombination in höchstens zwei Stellen von einer entsprechenden Kombination ab, die
aus einer systematischen Permutation resultieren würde.
Von jeder Phasenkombination wird eine Phasenerweiterung auf weitere Strukturfaktoren
ausgeführt. Dazu wird die Sayregleichung verwendet. Die Qualität der Phasierung wird dann
über die Gültigkeit der Sayregleichung beurteilt. Es hat sich gezeigt, dass auf diese Weise die
Phasen für die Strukturfaktoren der stärksten Reflexe für zentrosymmetrische, nicht zu
komplexe Strukturen (< 10 durch Pseudoatome zu ersetzende Baueinheiten) relativ zuverlässig
berechnet werden können. Ausserdem sollten die Strukturen gut durch Pseudoatome
approximierbar sein. Unter anderen wurde die Methode an zwei Zeolithstrukturen (RUB-3 und
ITQ-1) und an einer aus Oktaedern aufgebauten Struktur (8-A3F3) getestet. Zusätzlich konnte
mit Hilfe des SayPerm-Programmes von einer unbekannten Tri-peptide-Struktur (Sa322,
C32N3O6H53) eine sehr informative Struktureinhüllende berechnet werden.
Eine Alternative zur SayPerm-Methode ist die Anwendung der s.g. Permutationssynthese.
Dabei werden nur sieben Phasen von Strukturfaktoren mit grosser Amplitude, gesteuert vom
Hamming[7,4,3]-Fehlerkorrekturcode, permutiert. Von jedem der 16 permutierten Phasensätze
wird eine Fourierkarte aus den entsprechenden Strukturfaktoren generiert. Wenn bestimmte
chemische Informationen (z.B. Grösse und ungefähre Form eines organischen Moleküls)
einbezogen werden, hat man eine gute Chance, aus den 16 Fourierkarten, die richtige
auszuwählen zu können. Diese Methode ist an der Kristallstruktur des organischen Moleküls
Cimetidin ausprobiert worden. Über die Permutationssynthese konnte auch das mit der
SayPerm-Methode erzielte Ergebnis für die Berechnung einer Struktureinhüllenden für Sa322
bestätigt werden.
Weil eine Struktureinhüllende das Volumen in der Einheitszelle, in dem sich die Atome einer
Kristallstruktur aller Wahrscheinlichkeit nach aufhalten, begrenzt, kann es für den Erfolg einer
im direkten Raum agierenden Kristallstrukturbestimmungmethode von entscheidender
Bedeutung sein. In das auf die Lösung von Zeolithstrukturen spezialisierte Programm FOCUS
wurde die Möglichkeit der Volumenbegrenzung über eine Struktureinhüllende implementiert
und bei einer automatischen Abrasterung der Lösungsraumes (exhaustive gridsearch) an vier
Zeolith Topologien (APD, SGT, RTE und MWW) getestet. Zum Vergleich wurden die
gleichen Tests ohne die Struktureinhüllenden wiederholt. Es zeigte sich, dass bei Benutzung
der Struktureinhüllenden die notwendige Rechenzeit bei zwei der Teststrukturen (APD, SGT)
VI
um zwei Grössenordnungen gesenkt werden konnte. Für das Auffinden der RTE- und MWW-
Topologien war die Anwendung der Struktureinhüllenden sogar Grundvoraussetzung. Ohne
sie war die Suche nach der Struktur nach zwei Tagen noch nicht beendet, wohingegen die
richtigen Topologien beim Einsatz der Struktureinhüllenden schon nach 25 b.z.w. 27 min
beendet war.
Die für die Tri-peptidstruktur sa322 mit der SayPerm-Methode berechnete Struktureinhüllende
wurde für die Lösung der Struktur im direkten Raum benutzt. Dazu wurde das
Computerprogramm SAFE entwickelt, dass einen "Simulated Annealing" Algorithmus mit
einer Beschränkung des Volumens durch eine Struktureinhüllende kombiniert. Das Molekül
wird, gesteuert vom "Simulated Annealing" Algorithmus, durch die Zelle bewegt, um die
Orientierung und Konformation zu finden, aus der der niedrigste gewichtete Profile-/?-Wert
resultiert. Gleichzeitig wird ein Gütefaktor, der die Lage des Moleküls innerhalb der
Straktureinhüllenden beschreibt, ebenfalls über einen "Simulated Annealing" Algorithmus
minimal gehalten, so dass das Molekül dazu tendiert, seine Einhüllende nicht zu verlassen. Mit
dieser Methode konnte die Tri-peptidstruktur mit seinen 41 Nichtwasserstoffatomen und 23
Freiheitsgraden (drei für die Position, drei für die Orientierung und 17 für die freien
Torsionswinkcl) gelöst werden.
— VII —
Table of contents
1.1 Structural investigations of chemical compounds 1
1.2 Stinctural investigations using X-ray powder diffraction techniques 1
1.3 Structure determination from powder data 2
1.3.1 Single-crystal methods applied to powders 2
1.3.2 Structure determination in direct space 4
1.4 Zeolites 5
1.5 Periodic minimal and nodal surfaces 6
1.6 Overview of the project 6
2 Structure envelopes ,7
2.1 Periodic Minimal Surfaces (PMS) and crystal structures 7
2.2 Periodic Nodal Surfaces (PNS) and crystal structures 7
2.3 From a PNS to a crystal structure? 9
2.4 Generation of a structure envelope 11
2.5 Reflection selection for the calculation of a PNS 13
2.6 Application to non-zeolite structures 15
3 Solving the phase problem for structure envelope generation 17
3.1 Introduction 17
3.2 The Sayre equation 20
3.3 The Pseudo-atom method 20
3.4 Phase permutations 22
3.4.1 Sampling the pliase space with error correcting codes (ecc's) 23
4 Phase estimation using the Sayre equation 26
4.1 Introduction 26
4.2 Data collection and reduction 26
4.3 The SayPerm procedure 28
4.3.1 Data preparation 28
4.3.2 Phase extension 29
4.3.3 Phase set evaluation 29
4.4 Test structure RUB-3 (RTE topolpgy) 30
4.4.1 Data measurement and preparation 30
4.4.2 SayPerm inputfile 30
4.4.3 SayPerm run 32
4.4.4 Results 34
4.5 Test structure TTQ-1 (MWW topology) 35
4.5.1 Data measurement and preparation 35
4.5.2 SayPerm run 35
4.5.3 Results 37
4.6 Test structure 9-AlF^ 39
4.6.1 Data preparation 39
4.6.2 SayPerm run 39
4.6.3 Results 40
4.7 Tri-ß-peptide C32N,06H5, (sa322) 42
4.7.1 Measurement, data preparation, andfirst attempts at structure solution 42
4.7.2 SayPerm run and map evaluation 44
VIII
4.7.3 Results 44
4.8 Limitations of the SayPerm Approach 47
5 Phase estimation by the method of permutation synthesis 49
5.1 Introduction 49
5.2 Test structure Cimetidine 50
5.2.1 Data preparation 51
5.2.2 Application of the permutation synthesis 51
5.2.3 Selection of the best Fourier map 54
5.3 Tri-ß-peptide C32N306H53 (sa322) 55
5.3.1 Selection of the best Fourier map 55
5.4 Conclusions 56
6 Determination of zeolite structures using structure envelopes 60
6.1 Introduction 60
6.2 Topology search for zeolite structures with a structure envelope 60
6.3 Test examples 62
6.4 Structure envelopes with the full FOCUS approach 64
6.5 Conclusions 65
7 From structure envelopes to organic crystal structures 66
7.1 Introduction 66
7.2 Direct-space approaches to crystal-structure determination 66
7.2.1 Model generation techniques for organic structures 67
7.2.2 Model modification control 67
7.2.3 Comparison of diffraction data 69
7.3 Simulated annealing, fragment search and structure envelopes 70
7.4 The program SAFE 70
7.4.1 Input 71
7.4.2 Variation of the trial structure 71
7.4.3 Model construction 71
7.4.4 Checking for a chemically reasonable structure 73
7.4.5 Evaluation of the fit to the powder pattern and/or the structure envelope 74
7.4.6 Acceptance ofmoves and temperature control 75
7.5 Structure of the tri-ß-peptide sa322 75
7.5.1 Combination of chemical information with a structure envelope 75
7.5.2 SAFE inputfile ! 77
7.5.3 SAFE run 81
7.5.4 Refinement of the crystal structure 81
8 Conclusions 84
9 Possible developments of the structure envelope approach 86
10 References 88
1 Introduction
1.1 Structural investigations of chemical compounds
The properties of a material are determined primarily by its atomic structure. To elucidate a
structure, various methods are available, but the most powerful methods for structural
investigations are nuclear magnetic resonance spectroscopy (NMR) and X-ray diffraction
techniques. To use the latter in a routine manner, single crystals of sufficient quality and size
are required. Then the whole chemical structure can be derived from the measured data.
Thousands of crystal structures are determined m this way every year. Single-crystal X-ray
diffraction is one of the most commonly used analytical methods in chemical laboratories. If
single crystals are not available, NMR can still provide a considerable amount of structural
information. In particular, connectivity information can be gleaned, even if the compound is in
solution or in the form of a powder. In the last ten years, a rapid development in NMR
techniques has taken place, but even so it is not usually possible to obtain a complete three-
dimensional structure from NMR data.
1.2 Structural investigations using X-ray powder diffraction techniques
A possible solution to this problem is to use powder diffraction techniques. Even if single
crystals cannot be grown, quite often a polycrystalline powder can be obtained. With some
effort, surprisingly detailed structural information can be extracted from the diffraction
patterns of such materials.
In a polycrystalline powder, millions of small crystallites are present in different orientations.
The diffraction pattern from a powder then is simply a superposition of millions of single
crystal diffraction patterns. A powder pattern can be described as a projection of three-
dimensional single crystal diffraction data onto one dimension. A consequence of this
projection is that symmetrically independent reflections with similar öf-spacings (similar
diffraction angle 29) overlap, and this results m a loss of information. Nevertheless, the one-
dimensional powder pattern can be used as a fingerprint of a crystalline compound. It is
possible to compare two compounds or to carry out a quantitative analysis of mixtures, if the
powder patterns of the components are known.
The positions of the reflections in a powder pattern are determined by the lattice constants and
the relative intensities of the reflections by the position of the atoms in the unit cell. The shape
of the peaks in the pattern depends upon the geometry of the instrument and the quality of the
sample. Peak shapes are usually described in terms of a symmetric function (e.g. pseudo-Voigt
1
which is a linear combination of a Gaussian and a Lorentzian function) with a certain full-
width at half maximum (FWHM) and a geometry-dependent asymmetry. From these
parameters (unit cell, atomic positions, and peakshape), a powder pattern can be simulated.
This can be done for any structural model, and the better the match between simulated and the
measured pattern, the more probable the model.
Rietveld (1969) used this concept to develop a whole-pattern structure refinement method.
That development was a crucial step in the advance of structural analysis using powder data.
Given a structural model, the positions of the atoms can be refined using a least squares
algorithm to obtain a better fit between the calculated and the observed patterns. The problems
in this approach are the exact descriptions of the peak shape and the background. Furthermore,
impurities in the powder sample or the presence of disorder can severely hinder or even
prevent a successful Rietveld refinement. The advantage is that overlapping reflections do not
have to be deconvoluted, because each point of the observed and calculated pattern, and not
the individual structure factors, are compared. The better the resolution of the pattern and the
higher the information content, the more reliable the refinement. Structures with up to 100
parameters can be refined routinely, and with more effort up to 200 parameters can be
handled. The Rietveld method is the standard technique for structure refinement using powder
data.
1.3 Structure determination from powder data
For a Rietveld refinement, a starting structural model close to the final structure is required.
This model must first be determined, so the development of methods to solve crystal
structures using the structural information in a powder pattern is of high interest. Many groups
are actively pursuing this theme, and impressive developments in structure solution from
powder data have been achieved. Two basic approaches have been used to date:
(1) Extraction of individual reflection intensities from the powder pattern followed by an
application of conventional single-crystal methods, such as direct methods or Patterson
methods
(2) Direct-space approaches in which powder patterns calculated for (computer generated)
structural models arc compared with the observed data
1.3.1 Single-crystal methods applied to powders
For the application of traditional single-crystal methods, the extraction of the reflection
intensities to obtain single-crystal-like data is the key. The extraction approaches available
use a Rictveld-Jike whole pattern refinement, in which the reflection intensities rather than
3 —
atomic positions are "refined". Pawley (1981) developed a computer algorithm (ALLHKL) for
a least-squares approach, but instabilities arise for strongly overlapping reflections. A more
robust extraction procedure was proposed by Le Bail (1988), in which the intensities are
adjusted in an iterative manner. This approach has been implemented in various Rietveld
packages such as GSAS (von Dreele, 1990), and EXTRAC (Baerlocher, 1990) in XRS
(Baerlochcr & Hepp, 1982).
However, because of the overlap problem, the extracted data are of much lower quality than
single-crystal data, so the chances of a successful application of Patterson or direct methods
decreases dramatically with the degree of overlap. To overcome this problem, various methods
to improve the quality of the extracted data have been developed. The simplest approach is to
equipartition the reflections that overlap, but this necessarily leads to incorrect intensities for
these reflections. In the FIPS approach developed by Estermann & Grämlich (1993), Patterson
maps, generated from extracted structure factor amplitudes are squared and then back
transformed to obtain a better partitioning of the overlapping reflections. This is done in an
iterative manner until an optimum is found.
Sivia & David (1994) used Bayesian statistics to improve the Pawley extraction approach. By
using prior information in the profile-fitting process, the instabilities in the least-squares
algorithm can be eliminated and overlapping reflections can be deconvoluted. An important
aspect of this algorithm, is that a standard deviation can be calculated for each reflection
intensity. A special approach for the deconvolution of systematically (exactly) overlapping
reflections was described by Rius, Miraviriles, Gies & Amigo (1999). Using a modified direct
method sum function, phase refinement and peak deconvolution is performed simultaneously.
Approaches that allow the estimation of the structure-factor amplitude of a reflection using
other reflections in the data set, such as direct-method approaches (e.g. Jansen, Peschar &
Schenk 1992, Cascarno, Favia & Giacovazzo, 1992, Dorset, 1997) or maximum entropy (e.g.
Gilmore, 1996) can also be used for the estimation of the relative intensities of overlapping
reflections.
Despite these advances in the development of analytical deconvolution techniques, a general
solution to the problem of peak overlap has not been found. An alternative to the
computational methods described above, is to address the problem experimentally.
Synchrotron radiation offers X-ray beams with a higher intensity and smaller divergence than
laboratory sources. This allows data to be collected at higher resolution, so overlapping peaks
can be better deconvoluted. Anisotropic thermal expansion (Shankland, David & Sivia, 1997)
_4
can also be exploited. Several data collections are performed at slightly different temperatures.
If the expansion of the three unit-cell axes occurs anisotropically as the temperature increases
and no phase transition occurs, the relative positions of the reflections will change without
affecting the relative intensities significantly. Thus, by extracting reflection intensities from all
patterns simultaneously, a better data set can be obtained.
Another possibility is to exploit texture. One of the most disturbing effects on powder data is
the corruption of the relative intensities as a result of a preferred orientation of the crystallites
in the powder. However, this effect can also be used to advantage. By measuring several
powder patterns of a carefully textured sample at different orientations, more intensity
information can be extracted (Bunge, Dahms & Brokmeier, 1989; Cerny, 1996; Lasocha &
Schenk, 1997; Wessels, Baerlocher & McCusker. 1999). The extraction and treatment of
overlapping reflections is done in such a way that the result is a pseudo-single-crystal data set,
which can be used in a conventional structure solution procedure (e.g. Patterson or direct
methods). The structural model can then be verified with a Rietveld refinement.
For the straightforward application of Patterson or direct methods, atomic resolution is
required. Unfortunately, the crystallites in a polycrystalline material are often full of defects
and the scattering power of such a powder is poor. The intensities of the reflections fall off
dramatically with decreasing d-values (increasing 28), and only relatively low-resolution data
(i.e. d-values larger than a bond distance) can be obtained. Some special low-resolution
approaches in reciprocal space have been developed. These include the application of the
Sayre equation (Dorset, 1997), the use of the tangent formula derived from Patterson
arguments (Rius, 1993), and the use of the maximum entropy method (e.g. Gilmore, 1996).
However, structure solution at low resolution cannot yet be done routinely.
1.3.2 Structure determination in direct space
An alternative to performing structure solution in reciprocal or vector space, is to use structure
solution strategies in direct space. The crucial point for this method is the active use of
chemical information to compensate for the lower information content of the powder pattern
(overlapping reflections and the lack of data at higher resolution). For example, for organic
molecules, the connectivity is almost always known. Chemically reasonable trial structures
can be generated (by hand or by computer) without reference to the powder data (Harris et al.,
1994). From the trial structure, a powder pattern can be generated, and a figure-of-merit
calculated from the match between the calculated and the observed powder patterns.
Thousands or even millions of such models can be generated and evaluated by computer.
5
Different techniques for the modification of a model are available. For example, the Monte
Carlo method modifies the model randomly (Harris et al., 1994), simulated annealing is a
more sophisticated Monte Carlo approach, where a structural fragment can be translated and
rotated through the asymmetric unit (David, Shankland & Shankland, 1998; Andreev & Bruce,
1998) or a genetic algorithm based on survival-of -the-fittest rules derived from life sciences
(Shankland, David & Csoska 1997; Harris. Johnston, & Kariuki, 1998) can be used. These
rather time consuming approaches have become possible with the increase in computer power
available. Fast computers are able to generate millions of trial structures within a reasonable
time frame and a number of structures have been solved using the methods mentioned above.
Nevertheless, for complex structures (e.g. more then 15 atoms per asymmetric unit or more
then 15 free torsion angles), the number of parameters exceeds the limits given by the
computer power currently available.
A way out this dilemma is to combine methods in reciprocal space with those in direct space.
One example of such a combination is the FOCUS approach (Grosse-Kunstleve, McCusker &
Baerlocher, 1997), which was especially developed for zeolite structures. Here diffraction data
are used in a Fourier recycling loop and the chemical information is used to interpret the
Fourier maps automatically. Other techniques where reciprocal and direct space methods are
combined are Patterson search methods where a Patterson map generated from a model
modified in direct space is compared with that calculated from the measured data (e.g. Stout &
Jensen, 1989). This comparison can also be done in reciprocal space (e.g. Rossmann & Blow,
1962). It is also possible to use a combination of both (Rius & Miravitlles, 1987).
1.4 Zeolites
Zeolites are microporous materials with three-dimensional four-connected framework
structures with the composition T02 (where T is a tetrahedrally coordinated atom such as Si,
Al, Ga, P etc.). The tetrahedrally coordinated T-atoms are connected to four neighboring T-
atoms via oxygen bridges to form a framework with channels and/or cages, which arc filled
with cations, water, and/or organic species. These guest species can be exchanged or removed,
and this is a crucial property for the application of zeolites in industrial processes (e.g. as
adsorbents, exchangers, molecular sieves, catalysts). Usually, zeolites can only be synthesized
in form of a polycrystalline material, so their structures have to be determined without the
benefit of single-crystal diffraction data. Structural information is essential to the
understanding of a zeolite"s technologically important properties, so there is considerable
interest in developing powder methods for zeolite structure analysis. Interest in extending the
6
limits of zeolite structure determination from powder data was the initial motivation for this
project.
1.5 Periodic minimal and nodal surfaces
Various classes of crystal structures can be described using periodic minimal surfaces (PMS)
(Anderson, Hyde, Larsson & Lidin, 1988) or periodic nodal surfaces (PNS) (Schnering &
Nesper, 1987). Both can be used to highlight chemical, physical, or structural properties, and a
PNS can be generated using just a few parameters (a PMS generation requires more
sophisticated mathematics). In the case of zeolites, for example, a PNS can be used to delineate
the form of the framework. That is, all of the T-atoms lie on the surface, so the channel system
and/or cages of the zeolite can be clearly discerned. If such a surface could be generated for an
unknown zeolite structure, solving the structure would be reduced from a three-dimensional to
a two-dimensional problem (i.e. the decoration of the curved surface).
1.6 Overview of the project
The starting point of this study was the investigation of the feasibility of using PNS in the
determination of complex zeolite structures. Unfortunately, preliminary studies revealed no
rules for the generation of the PNS that would be appropriate for subsequent decoration.
However, the investigation did show that a different kind of PNS, which enveloped the zeolite
framework structure could be generated using just the information in the powder diffraction
pattern. Once this had been established, this "structure envelope" was used in combination
with direct-space algorithms to accelerate the structure solution process.
7
2 Structure envelopes
2.1 Periodic Minimal Surfaces (PMS) and crystal structures
Periodic minimal surfaces (PMS) were first derived by Gergonnc, Riemann, and Schwarz
(Schwarz, 1890) in the last century. These surfaces are the simplest of the hyperbolic surfaces
and are defined as having a mean curvature of zero at each point on the surface. That is,
K, +k2 = 0 (2-1)
where iq and K2 are the principal curvatures or maximum curvatures of opposite sign at one
point. They are described in differential geometry as objects in non-Euclidean space. Neovius
(1883) discovered that certain PMS are related to one another by the Bonnet transformation,
which bends the surface without stretching it. A PMS transformed in this way is also a PMS,
and several new PMS have been so derived.
In 1976, Scriven ( 1976) suggested that PMS could serve as models for liquid crystal structures.
This idea was further developed by Larsson, Foutell and Kragh (1980). Later, the relationships
between minimal surfaces and solid crystal structures were recognized. For example, a PMS
can be used to separate the interpenetrating networks in Cu20 or ice VII (Mackay, 1979) or to
indicate the diffusion pathways in crystal structures (Andersson, Hyde, Larsson & Lidin,
1988).
The relationship between PMS and zeolite structures was first recognized by Mackay (1979).
Hyde (1993) later established that the atoms of a framework structure lie on or near a minimal
surface. That is, a zeolite framework can be considered to be a decoration of a two-
dimensional non-Euclidean object. Even the Bonnet Transformation can be applied to
transform a PMS decorated by a particular zeolite structure to a PMS describing another
zeolite structure. For example, the crystal structures of analcime and sodalite are connected by
such a transformation of decorated surfaces.
2.2 Periodic Nodal Surfaces (PNS) and crystal structures
Unfortunately, the determination and mathematical description of the PMS for an arbitrary
space group is a nontrivial exercise. However, the concept of periodic nodal surfaces (PNS),
introduced by von Schnering & Nesper (1987). provides an alternative to the complicated
mathematics. These surfaces can closely resemble PMS and have the advantage of being
8
somewhat easier to calculate. While they no longer have the elegant description within non-
Euclidean space, they do provide a straightforward link between hyperbolic surfaces and
common crystallographic formalisms. They can be generated using a few structure factors with
I •» * ^
magnitudes \F(h) and phases a(h) in a Fourier summation over all equivalents of a few h
(von Schnering & Nesper, 1991).
t}(!) = J2 F(/o|cos[2ic(/! t-)-a(/01 (2-2)
Equ. 2-2 is used to produce a density distribution r)(x). Normally, the structure-factor
I * Iamplitudes \F(h)\ are simply set to an arbitrary value of 1.0. The points at which the density is
zero describe the PNS. Such a surface can (but does not necessarily) closely resemble a PMS.
In this way the well-known Gyroid PMS (Fig. 2-1), for example, can be approximated if the
structure factor amplitudes LF{110}I and IF{TlO}l are set to 1.0 and the phases a{H0}and
a{ lTO) to 7t/2. However, in most cases, a PNS is not an approximation to a PMS. A PNS is not
even necessarily a hyperbolic surface.
Figuie 2-1 : Simulation ol the gyioid pciiodic minimal surface (PMS) usinga periodic nodal suitace (PNS) calculated using equation (2-2) with
\F{ 1 10}!= ÏF{ 170}I = 1 and a(l 10) = a( l70) = nil
Von Schnering and Nesper (1987) provided many beautiful examples of the description of
structures using PNS. For example, PNS can be used to characterize phase transitions (Leone,
9
1998), they can depict a Coulomb zero potential surface in ionic structures, and they can
describe pathways of ions in an ion conducting process (Fig. 2-2).
In all these cases, the PNS are used to describe different features of materials with known
structures, so the structures could be used as a guide in the calculation of the appropriate
surface. Such surfaces can be very useful to emphasize features of a structure which cannot be
discerned immediately or to overlay a chemical structure with a surface that highlights a
certain property (e.g. equipotential surfaces).
2.3 From a PNS to a crystal structure?
If a PNS is closely related to a crystal structure, it would appear that the PNS should contain a
substantial amount of information about the structure. Thus, if the form of the PNS were
known, it could be used to facilitate the determination of an unknown crystal structure. The
question is whether or not it is possible to generate the appropriate PNS when the structure is
not known and only a powder diffraction pattern is available.
In an attempt to answer this question, a series of known zeolite structures (Table 2-1), which
are known to lie on or near PNS (Hyde. 1993), was investigated. It was hoped that some
objective criteria for the selection of reflections to be used in the summation (2-2) to create an
10
Name* Space group {hkl} IE(hkl)l Phase(°)
ANA la3d 112 1.02 0
APD Cmca 131#
021#
1.07
1.52
0
DDR R3m 101#
003
1.01
0.53
0
180
DOH P6/mmm 11I#
002
021
0.77
0.53
1.20
0
180
180
EDI P4m2 010#
001#
1.44
0.72
0
0
FAU FcBm lll# 0.98 0
GOO CZZA i U0#
111#
002
1.48
0.53
0.78
0
270
180
LEV P3m 012#
110#
1.07
0.94
0
180
MF1 Pnma 011#
102#
301#
200
020
1.19
1.40
1.29
1.16
1.29
180
0
180
180
0
PAU Im3m 033
134
1.95
1.13
180
0
RTE Cllm 110#
111#
201
1.13
1.06
1.01
180
0
180
SGT Mfamd 121#
116
L.50
L.43
0
180
SOD Im3m 011
002
0.91
0.49
0
180
VFI Formern 221#
010#
002
1.49
1.49
1.70
180
180
0
Table 2-1 Zeolite structures tested and the refleetion(s) used to generate the structure
envelope. '•Three-letter codes taken Irom the Atlas ot Zeolite Structure Types. (Meier,01son& Baerlocher, 1906), #Rcflection used to define the origin
_ ll_
appropriate PNS could be established. In each case, a surface that fitted the structure could be
generated, but a set of generally applicable rules for the reflection selection could not be
discerned.
It was obvious that the reflections and their symmetry equivalents must describe all
dimensions of reciprocal space, and that their indices should be low, but the exact combination
needed to generate the surface that best described the framework could not be determined
without using the structural model as a guide. In some cases it was even found that
systematically absent reflections yielded the best surface, (e.g.{100} for sodalite, which is
body-centered cubic). Furthermore, some structures were found to fit a surface with a higher
density level better than they did the surface at zero density. Consequently, the concept of
using the data from the powder diffraction pattern to create a surface that could be decorated
with a 3-connected net was abandoned.
2.4 Generation of a structure envelope
Fortunately, during the testing phase described above, an alternative approach became
apparent. By assigning the correct phases to the strongest low-order reflections, a well-defined
PNS could be generated. For a better estimation of whether a reflection is really strong, the
structure factors F{h) were transformed to normalized structure factors E(h) and these were
used in the summation:
p(x) = ^JE(h)\cos\2n(ti x)-a(ti)\ (2-3)>
h
Within a small tolerance, the PNS connecting points where the density p(x) is equal to zero,
calculated from the roots of this equation, and using only a few strong reflections, was found to
separate the framework atoms of the zeolite test structures from the void space. To verify the
general validity of this observation, a number of zeolites with different symmetries were
examined (Table 2-1). In all cases, the framework was found to lie on just one side of the PNS.
A few examples are shown in Fig. 2-3.
These surfaces, which enveloped the zeolite frameworks, did not resemble PMS, but they
could be generated in a rational manner from the powder data, and they did have the
potentially exploitable property of partitioning the unit cell into regions where atoms were
likely to be found and those where they were not. For CsCl, for example, the PNS generated by
a summation over {100} (Space group Pnßm) using \ElO0\ = 1 and a100 = 0 in Eqn. 2-3
resembles the well-known PMS P-Surface (Schwarz, 1890). This surface (Fig. 2-4a) separates
12
the anions fiom the cations (1 e the zero potential smlaee) That is, the smface has a strong
relationship to the ciystal stiuctuie Howevei, the 100 icllection m the X-tay diffiaction
pattern is extiemely weak If, on the othei hand, 1 10, which is the stiongest of the low-oidei
leflections, is used, a diffeient PNS is pioduced (Fig 2-4b) In this case, the cations and the
anions aie located on the same side of the suiface
The summation (2-3) can be viewed as a seveiely tiuncated Founei seues, and the lesultmg
density distiibution as a veiy-low-iesolution election-density map Howevei, a true E-map
would be calculated using all leflections up to a ceitam <r/-spacing including the E0a0 term, and
not just a handiul of stiong leflections Ihe PNS simply sepaiates the legions of high election
density from those ot low election density In this sense, it snmlai to the molecular envelope
13
used m piotem ciystallogiaphy (Biicognc, 1976) to define the appioximate boundaiy between
the piotem molecule and the solvent in a low-iesolution election-density map (see, fot
example Coulombe & Cyglei, 1997 oi Subbiah, 1993 and tefeiences fheiem) Of couise, the
numbei of leflections needed to genet ate the PNS descnbed heie is significantly lowei than
that used m the piotem case and the suiface does not necessanly have a closed form However,
the two aie closely îelated, so the teim stiuctuie envelope" (Biennei, McCuskei &
Baeilochci, 1997) has been adopted foi this paititionmg PNS
The stiuctuie envelope îeduces the space m the as\mmetnc unit in which the atoms of a ciystal
structme aie likely to be located by a factoi of appioxunately two and its shape imposes seveie
geometnc constiamts on the atomic auangements possible It was hoped that such a stiuctuie
envelope could be used to lacihtate the deteiruination of unknown ciystal structmes for which
no single ciystals aie available
2.5 Reflection selection for the calculation of a PNS
To put the idea into piactice, the limitation of powdei diffiaction data (oveilappmg reflections)
had to be taken into consideration In 12 of the 14 zeolite sttuctuies tested (Tab 2-1), the
leflections used to geneiate the PNS piesent no difficulties They aie the strongest low-index
leflections and they aie at least 0 5 FWFIM (full width at half-maximum) fiom neighboring
leflections, so then intensities could be extiacted leliably fiom the powdei pattern An
advantage of using low-mdcx leflections is the fact that they tend to he m the low-angle (high
14
rf-spacing) region of the powder diffraction pattern, which is less prone to reflection overlap.
In the cases of MFI and PAU, alternatives would have to be considered. Either the relative
intensities of the overlapping reflections would have to be estimated using other methods (e.g.
David, 1987, 1990; Jansen, Peschar & Schenk, 1992; Cascarano, Favia & Ciacovazzo, 1992;
Estermann & Grämlich, 1993; Hedel, Bunge & Reck, 1994), or a different set of reflections
would have to be used. Since there are usually several reflections with \E\ > 1 in the high d-
spacing region of the diffraction pattern, this is not a problem. Fortunately, the exact set of
reflections used to generate the surface proved to be not too critical.
A few rales-of-thumb for the selection of reflections emerged from the preliminary tests:
(1) The lEl-valucs of the reflections selected should be strong (usually \E\ > 1).
(2) The d-values of the chosen reflections should be in the same range as the expected electron
density fluctuations (e.g. the pore size of a zeolite, or the "thickness" of an organic
molecule).
(3) All directions in reciprocal space must be repiesented.
The selection of reflections is demonstrated for the RUB-3 structure (RTE-topology). In Table
2-2, the first ten symmetry-independent reflections are listed. For the envelope calculation,
three strong reflections are needed. Withm these ten reflections, four reflections have an \E\
value larger then 1.0 and are suitable for the envelope generation. Correct phases were
assigned to these reflections and and two possible combinations of three reflections selected
arbitrarily. They were used to calculate two different surfaces, one from the set {110}; {1 lT},
-15
{021} and the other from the set {110}, {111}, {201}. The two resulting surfaces are shown in
Fig. 2-5. Although the surfaces look quite different from one another, in both cases the T-
atoms of the framework are located on only one side of the surface.
hkl d -value IE(h)l
110 9.70 1.48
001 7.25 0.62
200 6.88 0.58
020 6.83 0.79
llT 6.28 1.42
201 5.63 0.04
HI 5.42 0.01
02Ï 4.97 1.27
220 4.85 0.49
201 4.53 1.34
310 4.35 0.98
Table 2-2 Listing ol the first ten symmetry-independent reflections of the RUB-3 pattern
2.6 Application to non-zeolite structures
To test whether or not the concept of structure envelopes could be applied to other classes of
materials, a few organic and inorganic structures were also examined. As for the zeolites, the
atoms were found to be situated on only one side of the curved surface. The envelope
generated using the strongest low-index reflections separates regions of high electron density
from those of low electron density, whatever the chemical composition of the material.
The higher the fluctuations in electron density within the asymmetric unit, the easier it is to
generate the structure envelope. That is, fewer reflections are needed. From this point of view,
zeolite framework structures with their large voids are ideal, but the difference between
bonding and non-bonding contacts in organic compounds is also sufficient for the generation
of a useful structure envelope. To illustrate this fact, the envelope calculated for the organic
molecule Cimetidine (Hädicke, 1978) using only four reflections is shown in Fig. 2-6. All non-
hydrogen atoms of the molecule lie on the positive side of the surface.
For non-centrosymmetric cases, it can be useful to use centrosynimetric projections to limit the
values of the phases to 0° and 180°. In this way Periodic Nodal Lines (PNL) which are the
-16
two-dimensional equivalents of PNS, can be generated within the centrosymmetric planes. The
PNL for a cyclic tetramer of a beta peptide (Seebach et al., 1997) generated from the 220 and
310 reflections in the non-centrosymmetric space group 14 is shown in Fig. 2-7. The presence
of the ring structure and the approximate location of the methyl groups are easily discerned
from the shape of the PNL.
Figure 2-6 The crystal structure of the organic molecule
Cimetidine with the structure envelope calculated using four
strong low-order reflections
17
3 Solving the phase problem for structure envelope generation
3.1 Introduction
If a few strong non-overlapping reflections are present within the low-angle region of the
powder diffraction pattern, the selection of reflections suited for the structure envelope
generation can be done in a straightforward manner. The major problem is the estimation of
the phases for these reflections. Tn some cases, the phase problem can be overcome by using
only origin-defining reflections. Depending on the space group, up to three reflections can be
selected and their phases assigned arbitrarily. Sometimes these reflections alone suffice to
produce a useful structure envelope (see Tab. 2-1). The number of reflections needed for the
envelope calculation is dependent not only on the space group, but also on structural features.
Given the same symmetry, the higher the electron density fluctuations within the unit cell, the
fewer reflections needed to calculate the partitioning surface. Thus, zeolites, with their large
pores, arc particularly well-suited for this approach. For example, the framework structure of
AIPO4-D (Fig. 2-3d) can be enveloped by a surface generated using only the two origin
defining reflections. In this case, just two strong reflections in the low-angle area of the
powder pattern provide surprisingly detailed structural information.
Nevertheless, further phases must usually be determined. Either fewer reflections are needed
to define the origin than are required for the calculation of the envelope, and/or the strongest
reflections cannot be used as origin defining reflections (e.g. structure semi-invariants). This
phase determination is not trivial. For a routine phase determination by direct methods, atomic
resolution is needed. However, the degree of reflection overlap in a powder pattern increases
with diffraction angle, so only in the low-angle (low-resolution) region can the intensities be
extracted reliably. Of course, if this were not the case, structure determination from powder
data would not be a problem. Consequently, alternative techniques that are less dependent on
resolution are required. For example, two approaches have been described:
(1) Maximum Entropy (e.g. Gilmore, 1996). The structure factors of a basis set of reflections
{H} are used as constraints to generate an entrop-maximized map. Fourier transformation
of the map then allows, the amplitudes and phases of other reflections outside the set {H}
to be estimated (extrapolation). A certain number of phases in the basis set are permuted to
generate several possible sets, which are the nodes in the phasing tree. As a figure-of- merit
for the correctness of a phase set, the "likelihood gain", which is indicative of the
agreement between the observed and calculated structure amplitudes of reflections not
included in {H}, and the student-t-test are used.
18-
(2) Modified tangent formula (Rius, 1993; Rius, Sane, Miravitlles, Amigo & Reventes, 1995a;
Rius, Sane, Miravitlles, Gies, Marier, & Oberhagemann, 1995b; Rius, Miravitlles &
Allmann, 1996; Rius, Miravitlles, Gies & Amigo 1999). Phases of a randomly phased set
of structure factors can be refined using a modified tangent formula derived from Patterson
function arguments. This formalism is similar to that used in a Patterson search for finding
the orientation of a given strucural fragment. Instead of rotation angles, the phases of
normalized structure factors are refined. This shows a stable behavior even under
conditions of low resolution (about 2.2 A). The best refined phase sets are selected using
conventional combined figures-of-merit (CFOM).
Both are multisolution approaches. With the amount of computer power now readily available,
the generation and examination of thousands of phase sets is possible within a reasonable time
frame. Besides the space group ambiguity, the major problem is that the lower the resolution
and the number of reflections, the more difficult it is to find criteria to identify the best phase
set. Common figures-of-merit used in structure-determination algorithms developed for higher
resolution data are often inappropriate for very low-resolution data. Consequently, other
criteria must be found.
Currently, the likelihood gain mentioned above seems to be one of the most powerful figures-
of-merit for low-resolution data. It has yielded impressive results in the evaluation of electron
density maps in protein crystallography, where the resolution is much lower than the distance
between two carbon atoms. Another criterion can be the "peakiness function" (Stanley, 1986),
where the integral of the cubed electron density should reach a maximum for the map
generated with the best phase set. An alternative approach to evaluating electron density maps
is to judge them by eye. This can be quite a powerful method if the appearance of structure
fragments (e.g. small organic molecules) is known.
Despite advances in finding new criteria, the techniques currently available do not provide a
general solution to the problem of low-resolution data.
For the structure envelope generation, phasing approaches that function with a relatively small
number of structure factor amplitudes at very low resolution are needed. The combination of
direct methods (Sayre equation) with the pseudo-atom technique has been succesfully applied
to very-low-resolution electron-diffraction data from proteins (Dorset, 1997). An adaptation of
this method to powder diffraction data in combination with a multisolution approach (phase
permutation) appeared to be a promising avenue to achieve a reliable phase determination of a
19
few structure factors with strong amplitudes. This would allow the generation of a correct
structure envelope that could facilitate the determination of of an unknown crystal structure.
20
3.2 The Sayre equation
In 1952, a paper was published by Sayre with the title: "The Squaring Method: a New Method
for Phase Determination" (Sayre, 1952). For a structure of identical and non-overlapping
atoms, the electron density p(x) and its square p"(x) arc quite similar. The only difference is
that the peaks of the latter function are sharper then those of the former. The structure factors
SCI*
of the "squared" structure F (h) can be calculated using the equation
Fiq(h) = Q(h)F(h)(3""1}
where Q(h) is a function describing the change in the atomic shape from the true to the
squared atom. Using this concept, Sayre derived the fundamental relationship
F(|) = -^..pr^F^-fc) (3-2)
Q(h)V I
where Fis the volume of the unit cell. By applying this equation, it is possible to calculate each
structure factor F(h) from all other structure factors having a triplet relationship with F(h).
Furthermore, the phase of a structure factor can be obtained from other structure factors, and
this is the basis for phase extension from a starting set of phases.
An advanced form of the Sayre equation was developed for crystal structures consisting of
two kinds atoms by adding a cubic term (Woolfson, 1954). Woolfson demonstrated its validity
for a one-dimensional test structure, but so far no useful application to a three-dimensional
structure has been reported.
In most cases, the Sayre equation is transformed into the Sayre~Hugb.es equation (Sayre 1980),
where the structure factors are replaced by normalized structure factors. Strictly speaking, the
Sayre equation is valid only for structures of identical and resolved atoms, but it holds
reasonably well over a large range of conditions (Glover et al. 1983). Sayre (1972) showed that
the atomic shape function 6 can be modified to compensate for data incompleteness. This
equation, in combination with other approaches, has proved to be a powerful tool in the
structure determination of polymer structures from powder data (Dorset, 1996), or even
protein structures from electron-diffraction data (Sayre, 1972; Dorset, 1997).
33 The Pseudo-atom method
To use the Sayre equation as a phase extension tool for data with much lower then atomic
resolution, the data have to be transformed m such a way that atomic resolution is simulated.
21
The assembly of globular subunits in a protein, for example, can be treated as pseudo-atoms
for the normalization of the observed electron diffraction intensities (Harker, 1953). It was
demonstrated by Dorset (1997), with data from bacteriorhodosin, that a multisolution approach
via the Sayre-Hughes equation could be used to estimate phases to 6 A resolution. A two-
dimensional pseudo-atom with the carbon scattering curve was used to describe the projection
of a-helices along their axes in a two-dimensional projection of the unit cell. To compensate
for the difference between the diameter of a carbon atom and that of an a-helix. the unit cell
axes were scaled down by a factor of 10.
Figure 3-1 Simulated powder patterns (CuKa) ol the zeolite ITQ-t structure (thin black line)and the corresponding pseudo atom structure (thick gray line). Each tetrahedron of ITQ-1 is
replaced by a pseudo-atom with 30 electrons. In the low angle area (up to 4 Â, 22 °26), the
patterns are roughly the same. At higher 28-angles. the intensities of the pseudo-atom structure
pattern decrease faster, because the scattcnng curve of the pseudo-atoms falls off faster than
does that of an SiO 4 tetrahedron.
The pseudo-atom strategy can also be used to facilitate structure determination from powder
data. The Si04 tetrahedra in a zeolite, for example, can also be treated as pseudo-atoms. The
diameter of the pseudo-atom corresponds to the distance between two Si atoms. If the powder
pattern of the original structure is compared with that of the pseudo-atom structure (Fig. 3-1),
it can be seen that the intensity relationships are roughly the same in the low angle region. At
higher angles (<7-values smaller than the diameter of the pseudo-atoms), the intensities in the
powder pattern of the pseudo-atom structure decrease faster than do those of the real structure.
By assuming that the structure consists of pseudo-atoms and omitting data with d-values lower
TO
than the pseudo-atom diameter, the requirement of atomic resolution is met and the pseudo
atom structure can be solved by direct methods. The pseudo-atom approach is well suited for
the estimation of the phases of low-resolution data for structures consisting of one kind of
building unit (e.g. octahedra, tetrahedra). Fragments in organic molecules whose non-
hydrogen atoms have a similar scattering power (e.g. C. N. O), can also be replaced by pseudo-
atoms with an appropriate diameter.
3.4 Phase permutations
For phase extension, a basis set of reflections whose phases are known (from the origin
defining reflections and enantiomorph) is defined. Additional phases can also be obtained by
other techniques. For example, phase information of reflections from a particular projection of
a crystal structure might be obtained from a Fourier transformation of a HRTEM (high-
resolution transmission electron microscopy) image. Any additional phases in the starting
phase set have to be permuted. The disadvantage of this method is the dramatic increase in the
number of possible phase sets with the number of permuted reflections. In the centrosymmetric
case, sign permutations are sufficient. So, if N phases are permuted systematically, 2 phase
sets are generated (e.g. 16384 permutations for 14 reflections). From each starting set, a phase
extension and a calculation of a figure-of-merit have to be performed. This is very time
consuming, but for fast computers still feasible within a reasonable time. However, for a non-
centrosymmetric structure, permutation of fourteen acentric phases using quadrant
permutation (tc/4, 3n/4, 5n/4, 7rc/4), the computer must calculate and evaluate billions of phase
extensions, and that exceeds the computer capacity of most laboratories. An alternative to
systematic permutation, is the use of a random number generator to produce the starting phase
set. Given a sufficiently large number of random sets, phase space can be sampled rather
exhaustively. Currently this is the favored technique for large sets of reflections.
If the phases are permuted systematically, all points in phase space, which has the dimensions
of the number of phases permuted, are visited. If only a portion of the points are to be visited,
it is of interest to sample the space in steps large enough to be efficient, but small enough to
see all relevant features. If, for example, the phase space is filled with closest-packed spheres
of a certain size, each sphere covers a piece of phase space, and a visit to each sphere is the
most efficient way of sampling the phase space.
At least two techniques have been developed to perform such efficient sampling.
(1) Magic integers. This method was first used by White and Woolfson (1975) and later
23
refined by Main (1977). By using the Fibonacci sequence as "magic integers" 16385
systematically permuted phase sets, each consisting of 14 signs or of quadrant phases from
7 non-centrosymmetric reflections, can be reduced to 128. The gain in efficiency is
obviously considerable. The magic integer representation of the phases is implemented in
several widely used computer program e.g. in the MULTAN (Debaerdemaker, Tate, &
Woolfson, 1985), XTAL (Hall, King & Stuart, (1995), and SHELX (Sheldrick, 1993).
(2) Error-correcting codes (ecc's). Where data in digital form are transmitted, errors occur and
to keep these to a minimum, ecc's were developed (Hamming, 1947; Shannon, 1948;
Golay, 1949). These codes are widely used and are implemented in modern digital
telephones, CD-devices, and receivers for satellite signals.
3.4.1 Sampling the phase space with error correcting codes (ecc's)
An error correcting binary code consists of a subset of 2 combinations of n binary digits (0 or
1) among the 2n possible n-bit words (codewords). If an information source only uses the
codewords of a given ecc, a receiver can cheek whether or not an n-bit word received is a
legitimate one. If the received codeword is corrupted, the codeword in the code, which differs
in the fewest places (closest codeword) from the received word is used to correct the error.
Since transmission errors, in which the fewest bits have been corrupted, are the most likely to
have occurred, this correction is usually quite successful.
A very important group of the ecc's are linear. They consist of the linear span of the 2 linear
combinations of k //-dimensional binary vectors (generators) formed with coefficients 0 or 1
under modulo 2 arithmetic (Fig. 3-2). The code is denoted by [n, k, d], where d is the
"minimum distance" (Hamming distance) of the code (i.e. the lowest number of differences
between any two codewords of the code).
The Hamming [7, 4, 3| code can be produced using a generator matrix. Each binary value in
the ecc is obtained from the product under modulo 2 of the generator matrix and the matrix
from all combinations of four binary digits (Fig. 3-2).
An error correcting code can also be used as the basis for an efficient phase-permutation
procedure (Bricogne, 1997). Woolfson (1954) demonstrated such a method (called
permutation synthesis), in which sixteen combinations of signs of structure factors were
generated for 7 centrosymmetric reflections. This was done in such a way that none of the 128
possible combinations of the signs differed from the sixteen combinations in more than one
place. In the sixteen resulting two-dimensional Fourier maps, expected features of a structure
24
00 0 0 0 0 0 0 0 0 0
00 0 1 1110 0 0 1
0 0 10 0 1 10 0 10
0 10 0 1 0 1 0 1 1 0
10 0 0 110 10 0 0
0 0 11 10 0 0 0 1 1
0 10 1 1 10 10 0 0
mod(2)0 10 0 10 1
10 0 1 I 0 1 0 1 0 0 0 0 1 10 0 0
0 110 0 110 0 10 110 0 110
10 10 J l i o 0 0 1_ 10 1 10 10
110 0 generator matrix 0 11110 0
1110 0 0 0 1110
110 1 1 0 0 I 1 0 1
10 11 0 I 0 1 0 1 1
0 111 0 0 10 111
.1 1 I i 1 1 I 1 I 1 1
Matrix from all combinations
consisiting of four binary digitsHamming[7, 4, 3] code
Figure 3-2 Generation of the Hammmg|7,4,3] code using the corresponding generator matrix.
1 10 0 0 0 0 0 0 0 0 0 0 1 10 1 1 10 0 0 10
1 0 1 0 0 0 0 0 0 0 0 0 0 0 i 1 0 1 1 1 0 0 0 1
10010000000 0 0 101 10 1110 00
I 0 0 0 I 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0
1 0 0 0 0 i 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0
10 0 0 0 0 10 0 0 0 0 0 0 0 0 10 110 111
1 0 0 0 0 0 0 1 0 0 0 0 0 I 0 0 0 1 0 1 1 0 1 1
1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 I 1 0 1
I 0 0 0 0 0 0 0 0 1 0 0 0 I 1 I 0 0 0 I 0 1 10
1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 I I 0 0 0 1 0 I 1
10 0 0 0 0 0 0 0 0 0 10 10 1 1 10 0 0 10 1
0 000000000001 I llllll I lll_
higurc 3-3 Generator matrix ot the Go lay [24, 12, 8] code
OS
(fluorine) were sought, and in one map were found. In the generation of these sixteen sign
combinations, the 7-dimensional phase space was sampled rather completely. Woolfson did
not note the link to the error correcting codes, but his code for the generation of the 16 sign
combinations is the same as the Hamming [7, 4, 3] ecc.
The Golay [24, 12, 8] code (Golay, 1949), one of the most powerful codes, provides a way of
varying the signs of 24 reflections, such that one of the 4096 combinations (of 224 = 16777216
possible sets) will have a maximum of four incorrect signs. The generator matrix for the Golay
[24, 12, 8] code is shown in Fig. 3-3.
The Golay code can also be used to permute the quadrant phases of 12 acentric reflections. In
this case, the phase of each reflection is assigned using two digits rather than one within each
codeword (e.g. 0,0:7t/4; 1.0: 3ir/4: 1,1: 5xc/4; 0,1:77t/4).
A good introduction to this fascinating subject, which is also understandable to non-
mathematicians can be found in Bricogne (1997).
-26
4 Phase estimation using the Sayre equation
4.1 Introduction
The combination of the Sayre equation, the pseudo-atom method and phase permutation using
ecc's has been implemented in the program SayPerm. The program is written in ANSI-C and
all space group information needed is provided by Sglnfo (Grosse-Kunstleve, 1996). The
program uses a multisolution approach, where a large number of phase sets are generated by
phase permutation and, extended and ranked according to various figures-of-merit by
assuming the validity of the Sayre equation. The primary figure-og-merit is the validity of the
Sayre equation itself. It was hoped that the phases of the few reflections required to generate a
structure envelope could be established in this way.
4.2 Data collection and reduction
Before SayPerm can be applied, powder diffraction data must first be collected. Depending on
the absorption coefficient of the material and its tendency towards preferred orientation, either
Debye-Scherrer transmission (capillary) or Bragg-Brentano reflection (flat-plate) geometry
can be used. The positions of the peaks in the diffraction pattern can usually be found with a
second-dcrivative-based automatic peak-search program (e.g. Alexander, 1973). The cell
parameters can then be derived from these positions. To do this, there are a number of different
indexing programs (Visser, 1969; Werner, Erikson & Westdahl, 1985; Taupin, 1989; Boutif &
Louer, 1991), which apply different strategies to the problem. The space group is then
determined from systematically absent reflections. However, because of the overlap of
reflections, indexing and space group determination are often ambiguous and several
possibilities must be taken into consideration. At this stage, the symmetry information and
unit-cell dimensions are used to extract the intensities for all reflections in the pattern. For the
determination of the scale factor, a Wilson plot is used. With intensities extracted from powder
data, the Wilson plot often deviates significantly from the ideal straight line and sometimes
even negative overall displacement (thermal) factors are obtained. In such cases, fixing the
displacement factor at a reasonable value and estimating the scale factor by using this
assumption is recommended (Grosse-Kunstleve. 1996). Because the information content of a
powder pattern is much lower than that of a single-crystal data set, ambiguous results can be
produced. This restricted reliability should be kept in mind in the course of subsequent
evaluations.
For all examples presented in this study, the intensity extraction was performed using the
27
The SayPerm Procedure
'obs
structure factor amplitudes extracted
from the powder pattern
IZ,F -
N FP°mî -ü(Sm20AlVr
obs
MN
B: overall displacement factor
8: Bragg angleA.: wavelength
ff. scattering factor of atom /'
Z. : atomic number of atom j
1|r
' point
Structure factor amplitudes of a
structure consisting of point atoms
'pseudo
\
FT
A
I \ Ppseudo
M/ i
j
^ r^ "
^pseudoStructure factor amplitudes for a
structure consisting of atoms with the
pseudo-atom density curve
phase assignementto a prescribed number of £V>scll(]0 t
I phase permutation
using ecc
phase extension
to next set of reflections using Sayre
equationt
1 output
consistency test
ranking accordingR-value calculation
J
Figme 4-1
28 —
EXTRAC module (Baerlocher, 1990) in the XRS-82 Rietveld refinement package (Baerlocher
& Hepp, 1982). To scale the structure factor amplitudes, the XTAL (Hall, King & Stuart,
1995) module GENEV (Hall & Subramanian, 1995) was used.
4.3 The SayPerm procedure
4.3.1 Data preparation
The whole SayPerm procedure is shown in the flow diagram in Fig. 4-1.The cell parameters,
the space group, the scaled structure-factor amplitudes, and the overall displacement factor are
used as input to the SayPerm program. To satisfy the condition that the data have atomic
resolution and the structure be an equal-atom one, the structure-factor amplitudes must be
modified. First, each is divided by the displacement-factor function e~ and the
variation of the scattering factors with sin 9/A to simulate a point-atom structure. Combining
Eqn. 3-1 and 3-2 gives
F^(h) = ~^F(k)F(h~k)(4-1)
To make this equation valid for lower resolution data, pseudo-atoms having an electron-den¬
sity distribution
,>\ i-bnr) (4-2)
arc used. The parameter b describes the half width of the function (i.e. the size of the pseudo-
atom). From this function, the scattering curve for a spherical pseudo-atom is calculated by a
Fourier transformation of the density curve p PH,ut}0
°° *>
°°
,-,f > ^ j ——TC/ï2 j
, f (2mit >) ,> . r sin271/? • r 2,> 1Kb)
r„,.„j. = p c d r = 4K \ p,„ .,. . -—rdr= ——e
(4-3)Jpseudo }ypseudo J rpseudo
7t > ^ 1^
J
The point-atom data are multiplied by this curve to give F(h)->
FS(t(h) on the same scale, the squared pseudo-atom
on a relative scale. To obtain
P(r-r7.
=e(-2hnr) (4-4)1 v
'pseudo
is Fourier transformed to obtain its scattering curve
r w C~mh i) > f sq su\2nh-r 2 ,> 1p e d r = 471 p
,v—— r d r= -——— e
J 'pwndo J fpseudo , J > F>u fü2nh r *j2b>Jh
1 ,i
and is multphed by the point-atom data.
29
The diameter of the pseudo-atom can be varied in the program and should correspond to the
diameter of the building units of the crystal structure being described by the pseudo-atom.
Furthermore, the pseudo-atom diameter should not be smaller than the minimum d-value of the
data.
4.3.2 Phase extension
The prepared set of reflections is divided into a basis reflection set {Hi} and up to three
additional sets. The basis set consists of structure factors for which the phases are either known
(from origin defining reflection rules, enantiomorph's) or assumed to be known. The latter are
generated by phase permutation. The number of phases permuted depends upon the computer
power and/or on the permutation method. For example, if the phases are permuted using an
ecc, only a prescribed number of phases can be permuted. From a basis phase set {H]}, the
phases are now extended to a second phase set {Kt} using the equation
sei > 1_,
» » > (4-6)
An lvalue describing the discrepancy between the calculated Fs4Sayre
(structure factor
amplitudes of the squared pseudo-atom structure calculated using the Sayre equation) and the
"observed" robs
(structure factor amplitudes of the squared pseudo-atom structure derived
from measured data) is then calculated.
-i^
sq
T,
sq
(4-7)R = —
F
sq
- FS(]
obs Sayre
1fU'
"1(1,
ob\
The |F' (000)1 's are excluded from this calculation. If this i?-value is lower than the value
defined in the input file, the sets {K]} and {Hj} are merged to form a new phase set {H2}. The
phases from {H2} are then again extended to the next phase set {K2}. This procedure is
repeated until each subset of structure factors is phased.
4.3.3 Phase set evaluation
Once all structure factors have been assigned phases, a consistency test follows. Here again,
each phase (excluding the o.d.r.'s and enantiomorph) and structure factor amplitude is
calculated from all other structure factors and the phases are modified if nessessary. This is
repeated for several cycles until the Z?-valuc has converged. This /lvalue is the main criterion
used to rank the resulting phase sets. The best reflection sets are listed in a file, and Fourier
maps are then calculated.
30
The selection of the best phase set follows. If there are several "best" phase sets with similarly
low R-values, each must be evaluated. A good indicator in the centrosymmetric case is the
ratio of negative to positive signs. If no atomic positions are expected at the origin, this ratio
should be about 1:1, otherwise large electron densities will appear at the origin of the Fourier
map. Phase sets can also be discarded if the corresponding Fourier maps contradict known
chemical information (e.g. the experimentally determined size and dimensionality of a
zeolite's channel system, or the approximate shape of an organic molecule). A few test
applications of the SayPerm algorithm are described in the following sections.
4.4 Test structure RUB-3 (RTE topolpgy)
4.4.1 Data measurement and preparation
Powder data from a calcined sample of RUB-3 were kindly provided by Marier, Grünewald-
Lüke, & Gies (1995). They solved the stracture originally by model building and combined it
with a Rietveld refinement.
chemical composition S124^4 8
lattice parameters a = 14.039Â: b == 13.602A;c == 7.428Ä; ß = 102.22°
space group C 21m
asymmetric unit 3 Si, 6 0
Tabic 4-1 Data for the ciystal stiucture of RUB-3
The lattice parameters and the space group (Cllm) were determined from the powder pattern,
and then the structure factor amplitudes were extracted and the scale factor was determined to
be k = 0.47. For this calculation, the overall displacement factor was fixed at B = 2.5Â2.
4.4.2 SayPerm input fde
The input file used for the SayPerm run is shown in Fig. 4-2. The lines beginning with "#" are
comments and are not read. In the first two lines, the space group (SpaceGroup) and the unit
cell parameters (uniteell) are defined.
In the next line, the number of grid points along the a, b, and c directions for the generation of
a Fourier map are defined (GndDimensions). The distance between the grid points should be
about 0.5À. The next three values define the number of the unit cells in each dimension in
which the map should be generated.
The MaxRvaiue line gives the maximum 7?-value (Eqn. 4-7) allowed after a phase extension
from one subset or a group of subsets to another subset. If the R-value exceeds this value, the
31
# RUB-3 measured data
UnitCell 14.098 13.670 7 .431 90.000 102.421 90.000
SpaceGroup C2/m
#
GridDimensi on 24 24 16 1 1 1
#
MaxRvalue 30.0 30.0 50.0
Fscale 0.47
TempFac 2.S
PseudoAtom 3.1
PermCode hamming
#
# h k 1 F(hkl)| phaso
0 0 0 0 720.00 0.0
0 1 1 0 421.103 0.0
1 0 0 1 217.007 0.0
1 2 0 0 272.476 0.0
1 0 2 0 307.76b 0.0
0 i 1 -1 390.415 0.0
2 2 0 -1 80.467 0.0
2 1 1 1 15.969 0.0
1 0 2 1 323.793 0.0
1 2 2 0 138.311 0.0
1 2 0 1 503.206 0.0
1 3 1 0 177.443 0.0
1 2 2 -1 371.249 0.0
1 1 3 0 322.760 0.0
1 3 1 -1 372.827 0.0
2 t 3 -f L69.782 0.0
2 2 0 J 22.89 1 0.0
1 0 0 2 372.082 0.0
2 1 3 1 7.07L 0.0
2 1 1 -2 133.503 0.0
1 2 0 -2 244. L68 0.0
2 3 1 L 254.073 0.0
1 4 0 0 387.085 0.0
1 0 4 0 408.797 0.0
2 4 0 -1 227.077 0.0
2 1 1 2 425.507 0.0
2 3 3 0 208 .931 0.0
2 0 2 2 101.400 0.0
2 2 -1 -2 282.814 0.0
2 3 3 L 164,560 0.0
2 3 L - 2 16.248 0.0
2 0 4 Ï 2 62 .324 0.0
2 4 2 0 2 6 6.816 0.0
2 2 4 0 251.641 0.0
2 4 2 -1 258.818 0. 0
2 2 0 2 130 .077 0.0
2 2 4 -1 342 .254 0.0
2 4 0 1 US. 684 0.0
2 1 3 -2 16.4 52 0.0
2 4 0 z. 275.2 93 0.0
2 3 3 1 212 .73 5 0. )
2 2 4 J 112 .067 0.0
Figure 4-2 SayPerm input file of RUB-3 test calculation, data up to a resolution of 2.73Â were used
— 32
calculation using the current permuted phase set is interrupted and discarded and the next
phase set is generated. The first value is the highest Ä-value allowed for phase extension from
the first two sets (labelled 0 and 1 in the reflection list) to the subset with the flag 2. The
second it-value is the highest value allowed after the extension from subsets 0, 1 and 2 to
subset 3, and the third for phase extension from the first four to any remaining subsets. An
appropriate choice of these values can drastically reduce the run time. The scale factor
(Fscale) is defined in the next line. This factor is usually taken from the Wilson plot and used
to scale the unsealed structure amplitudes. The value for displacement factor B (TempFac)
should be the same as that assumed for the scale-factor estimation. PseudoAtom defines the
diameter (in Ä) of the pseudo-atom and is defined by Jl/nb where b is the ^-parameter from
Eqn. 4-2, and Permcode the code to be used for the permutation. The options for the latter are
hamming, golay 01" permsynth.
A doubled Hamming |7, 4, 3J code is used to generate 256 phase permutations of fourteen
(instead of seven) centrosymmetric structure factors. For the doubled Hamming [7, 4, 3] code,
all combinations of the codewords of two single Hamming [7, 4, 3] are generated (16 = 256).
A maximum of two signs are incorrect if all 214 = 16348 sign combinations are compared with
the 256 codewords of the doubled code. This doubled Hamming [7, 4, 3] code is initialized
with the word " hammi ng ".
In the next lines, the reflection data are given. The first column of the reflection line contains
the flag defining the subset. The phases of reflections having the flag "0" are fixed. This is
used for the phases of F(000), the origin defining reflections (o.d.r.) and any other structure
factors with known phases. The phases of the reflections with the flag "1" are to be permuted.
The first phase extension is done to the reflections labeled "2". the second extension to those
labelled "3" and the third to those with a flag higher than "3". The other values in the reflection
line are the indices hkf the unsealed structure factor amplitudes, and the phase. Unknown
phases can be set at an arbitrary value. They will be replaced in the output file by the phases
obtained from the phase extension.
4.4.3 SayPerm run
The first 41 reflections from the reflection list for calcined RUB-3 (Figure 4-2) were used. This
corresponds to a resolution of 2.73 Ä. The 110 and llT reflections were selected as origin
defining reflections, in view of the low number of reflections, the permutations were
determined by the doubled Hammingl7. 4, 31 code and only one phase extension from a
reflection subset (0 and 1) was carried out to another subset (2). Because the allowed .R-values
-33
(MaxRvalue) were set relatively high, a phase extension was calculated for each permutation.
Fig. 4-3 shows the i?-value obtained for each phase permutation and extension. The lowest R-
value was found for the 112th permutation.
0 80
0.60 -
g 0 40i
0 20
•>.
'»Vf %*
V'-/* *.
. . .a. •.
• *
.%•»
«V /*
a #J
0 00
*•. V*. :•'
/50 100 150 200
permutation number
250
Figuic 4-3 A'-\alues ol all 256 peitnutations and extensions toi RUB-3 The bestÄ-
valuc A' = 0 192 was obtained fi om the 112th peimutation set (marked by the arrow)
Figuic 4-4 RUB-3 stiuctuie and isosuiface at 807r ot the highest Founei peak of the
map calculated itom the estimated phases
34
h k / psq i/viZ F(k)F(h~-k) (j)Sayre Tcorrect
0 0 0 2400.615 2400.615 0 0
1 1 0 575.490 582.169 0 0
0 0 1 285.055 287.220 TC Tt
2 0 0 354.354 2 3 8.677 Tt Tt
0 2 0 399.674 271.777 71 Tt
1 1 _1 497.718 432 .403 0 0
2 0 -1 99.629 31.200 0 0
1 1 1 19.544 29.622 0 Tt #
0 2 1 384.331 475.402 0 0
2 2 0 162.53 5 194.740 Tt Tt
2 0 1 574.250 5 6 6.662 0 0
3 1 0 198.574 2 18.898 Tt TC
2 2 -1 415.354 410.33 0 0 0
1 3 0 360.158 333.409 Tt Tt
3 1 -1 405.402 3 2 9.878 Tt Tt
1 3 -1 17 6.731 92.459 0 Tt #
2 2 1 23.605 3 8.53 6 n Tt
0 0 2 373.211 461.221 0 0
1 3 1 7.066 12.63 6 0 0
1 1 -2 132.636 64.020 0 Tt #
2 0 -2 240.451 351.053 0 0
3 i 1 244.464 208.09 1 0 Tt #
4 0 0 373.023 246. 833 Tt 0 #
0 4 0 391. 687 232.624 Tt Tt
4 0 -1 217.006 265.841 Tt TC
1 1 389.456 322.833 0 0
3 3 0 190.911 2 6.2 60 Tt 0 #
0 2 2 91.9 05 44.376 7X Tt
2 2 -2 251.664 195.532 TC Tt
3 3 -1 146.106 13 4.42 6 0 0
3 1 -2 14.3 42 41.804 0 0
0 4 1 229.733 178.880 Tt Tt
4 2 0 232.340 193.470 0 0
2 4 0 218.133 111.747 0 0
4 2 -1 223.499 2 69.067 Tt Tt
2 0 2 108. 820 22.770 Tt 0 #
2 4 -1 282.519 215.324 Tt Tt
4 0 1 9 6.3 52 32.845 Tt 0 #
1 3 -2 13.330 72.315 Tt Tt
4 0 -2 217.956 i.i J. J . Z 2) J TC Tt
3 3 1 167. 131 143.109 Tt Tt
2 4 1 85.264 128.357 Tt Tt
Table 4-2 Result of the phase extension from fourteen permuted phases for the RUB-3 structure
factors, The R-value is calculated from the values m column four and five. Rows marked with a "#"
have incorrectly estimated phases
4.4.4 Results
The set having the lowest R-value was selected, and a Fourier map was calculated using the
pseudo-atom structure factor amplitudes and the extended phases. An isosurface at 80% of the
highest Fourier peak value structure is shown with the RUB-3 in Fig. 4-4. The T-atom
35
positions are readily apparent. The different diameters of the isosurface enclosures are caused
by the imperfect phase estimation from such low resolution data. By reducing the isovalue for
the isosurface, the bond directions between the T-atoms become visible, and the approximate
positions of the oxygen atoms can be detected. In the selected reflection set, comparison
between the structure factor amplitudes of the squared pseudo-atom structure and the
amplitudes calculated from the Sayre equation gives an ft-value of 0.192 (Fig. 4-3).
All permutations and evaluations of the reflection set took 15 min CPU time. Out of 39 phases
(41 minus two o.d.r.), 32 were determined correctly (Table 4-2). This is sufficient to determine
the topology of the framework, so, of course, a structure envelope could also be generated very
easily from a few strong reflections. This was an encouraging result, but RUB-3 is a simple
structure (three T-atoms in the asymmetric unit). For a more complicated one, such a similarly
reliable phase estimation cannot be expected. However, because the phases of the strong
structure factors are likely to be estimated correctly, it should still be possible to generate a
useful structure envelope.
4.5 Test structure ITQ-1 (MWW topology)
4.5.1 Data measurement and preparation
The first structure with the topology MWW (aluminosilicate MCM-22) was derived by model
building from high-resolution electron micrographs, and refined with synchrotron powder
diffraction data by Leonowicz et al. (1994). Later Camblor et.al (1998) described the synthesis
and refinement of the pure silica MWW-type zeolite ITQ-1 (Table 4-3). Those data, collected
on the Swiss Norwegian beamline (SNBL) at the European Synchrotron Radiation Facility
(ESRF) in Grenoble were also used for the SayPerm test calculations. The intensities were
extracted and the scale factor determined.The displacement factor was fixed at B = 2.0 À".
chemical formula S172OJ44
lattice parameters a= 14.209Ä, c = 24.969Ä
space group P6/mmm
asymmetric unit 8 Si, 13 0
Table 4-3 Data lor the structure of ITQ-1
4.5.2 SayPerm run
Data up to a resolution of 2.7 À were input into the SayPerm procedure. The 72 reflections
were divided into 4 subsets for a stepwise phase extension (Fig. 4-5). Using the doubled
Hamming [7, 4, 3] code, 256 phase permutations were examined and a Fourier map was
36
# Ti tie ITQ--1 measured data
Uni tCe Ll 14 .209 14 .209 24.969 90.000 90.000 120.000
Spa ceGroup P6/mmm
#
Gri dDimension 2t 2 8 48 1 1 1
#
MaxRvalue 0 3 0.3 0. 3
Fscale 0 68
TempFa 2 0
PseudoAtom 3 1
PermCode hamming
#
# h k 1 |F(hkl) [ phase
0 0 0 0 2159.00 0.0
2 0 0 1 79.40 0.0
1 0 0 2 286.83 0.0
1 1 0 0 191.91 0.0
1 1 0 1 15 8.63 0.0
0 1 0 2 17 0.2 6 0.0
2 0 0 3 96.48 0.0
2 1 1 0 36.14 0.0
2 1 0 3 45.73 0. 0
2 1 1 1 63.74 0.0
1 0 0 4 3 3 4.73 0.0
2 1 1 2 88.35 0.0
1 2 0 0 158.75 0.0
2 2 0 1 91.30 0.0
2 1 0 4 62.53 0.0
1 2 0 2 118.59 0.0
2 1 1 3 52 .66 0.0
2 0 0 5 72.97 0.0
2 2 0 3 54.22 0.0
2 1 1 4 0.00 0.0
2 2 1 0 41.7 6 0.0
2 1 0 5 63.80 0.0
2 2 1 1 32.86 0.0
1 2 0 4 102.92 0. 0
2 2 1 2 66.53 0.0
1 0 0 6 221.67 0.0
1 3 0 0 138.91 0.0
2 1 1 5 88.87 0.0
2 2 1 3 27.44 0.0
1 3 0 1 142.54 0.0
1 1 0 6 130.48 0.0
1 3 0 2 215.27 0.0
2 2 0 5 6 1.02 9.0
1 2 1 4 13 6.3 9 0.0
2 3 0 3 3 3.35 0 . 0
2 1 1 6 52.3 7 0. 0
1 0 0 7
•
264.90 0.0
Figure 4-5 SayPcrm input tile for the TTQ-1 test calculation, data up to resolution of 2.7Â were used
calculated from the reflection set having the lowest R-value.The run was repeated using the
Golay [24, 12, 8] code with reflection flags changed accordingly (24 phases were permuted).
Here the word golay is used in the permcode line of the input file.
o>/
h k / Fï? l/V^F(hF(h-k) ^Sayre rcorrect
0 0 2 307.953 352.458 Tt Tt
1 0 0 205.859 143.586 Tt TC
1 0 1 168.873 224.816 0 0
1 0 2 177. 171 132.834 0 0 odr
0 0 4 328.0/5 1 ^ 3.2 n 5 Tt Tt
2 0 0 155.040 1 o 5 .213 Tt Tt
2 0 2 112.353 9 2.5 8 6 Tt Tt
2 0 4 89.014 108.875 0 0
0 0 6 186.646 2 79.695 Tt 0 #
3 0 0 116.02 8 184.017 0 0
3 0 1 118.159 1L4.132 Tt Tt
1 0 6 106.482 30.600 0 Tt #
3 0 2 174.429 L44.45Ï Tt Tt
2 1 4 107.398 (•3 .875 0 Tt #
0 0 7 2 02.07 6 120.578 It TC
Table 4-4 List of reflections for ITQ-1 with permuted phases. The fixed phased odr is indicated. Rows
with a "#" mark the incorrect phases. The calculated and "observed" values (fourth and fifth column)of reflections used to generate a structure envelope generation should agree reasonably well
4.5.3 Results
The highest electron densities were found very close to framework atoms (Fig. 4-6). Four T~
atoms can be seen directly, but if the structure were unknown, it would not be possible to
construct the framework structure by surveying the Fourier map alone. The same test was
repeated using the Golay [24, 12, 81 code. Here more phases (24) were permuted. That is, the
phase space was sampled with a smaller stepsize. In the case of ITQ-1, this more accurate scan
did not provide phases that allowed a more meaningful Fourier map to be generated. Of the 69
stracture factor phases, 23 were incorrectly determined. However, all strong low-order
reflections, which would be used to generate a stracture envelope, were phased correctly
(Table 4-4). A very informative structure envelope (Fig. 4-7) could be calculated from just
four reflections, and this was subsequently used to limit the search volume in a direct space
structure determination procedure (Sect. 6.3). Evaluation of the second best phase set showed
that the corresponding Fourier map, generated using all structure factors, could also be
considered to enclose the ITQ-1 structure, but it had a shift of origin relative to the Fourier map
of the best reflection set. Thus, fixing the origin with origin defining reflections does not
guarantee that all maps calculated from the different phase sets will have the same origin.
Fourteen permuted structure amplitudes have more power to fix the origin than do the few
odr's. Because the phase sets do not necessarily have the same origin, phase statistics (e.g the
most likely phase for a given structure factor estimated from its frequency of occurrence in the
best phase sets) cannot be applied in a straightforward manner.
38
Figure 4 6 ITQ I stiuctuie and the isosuilacc at 80% ol the maximum
Founei peak The highest densities aie located at the positions ol loui Si-
atoms
Figuie 4 7 Stiuctuie tmclopc loi ITQ 1 calculated liom the {002},{100} {101} and {102} leflections whose phases weie estimated using
the SayPeim pioccduic
39
4.6 Test structure 6-AlF3
4.6.1 Data preparation
The pseudo-atom approximation should not only work for structures consisting of tetrahedra.
Pseudo-atoms can also be applied to other coordination polyhedra. To test this, a structure
containing octahedrally coordinated Al atoms was investigated.
In 1995 Herron et al. reported the preparation of two new A1F3 structures consisting of corner
sharing octahedra. One of these structures (9-AlF^) (Tab. 4-5) was selected as a test example.
unit cell formula A1|6F48
lattice parameters a= 10.184 Â, r = 7.172 A
space group P4nmm, choice 2
asymmetric unit 4 Al, 7 F
Table 4-5 Strutural data (or 9 AIF\
No experimental data were available, so a powder pattern was simulated. The relevant
parameters for the simulation are listed in Tab. 4-6. The peak shape in the simulated pattern
corresponds approximately to that of data obtained from a laboratory diffractometer. From this
pattern, the intensities were extracted and the unsealed structure factor amplitudes determined.
In this way, peak overlap could be taken into account. The scale factor (k = 1.8) was estimated
using a fixed overall displacement factor (B = 3.5 Â~).
20-range 3 - 50°
stepsize 0.02°
polarization ratio 1.0
FWHM = U+VtmQ + lTtan""8 (7=0.005; U =0; W=0
Asym = al + tfVtanG + a^/tan"9 a; = -0.005; a2 = 0.003; a3 = 0
profile peak shape pseudo-Voigt
peak range in FWHM 10
Lorentz fraction 0.5
Table 4-6 Parameters used to calculate the powder pattern ot the 0-A1P2,
4.6.2 SayPerm run
The input file is shown in Figure 4-8. The reflections 201 and 310 were selected to be the
odr's, and the doubled Hamming [7,4,3] code was used to permute 14 phases (label 1). The
pseudo-atoms used had a diameter of 3.4 A. which is approximately that of an A1F6
40
octahedron. The permutation having the lowest i?-vaiue (R= 0.202) was selected and a Fourier
map was calculated from the corresponding structure factors.
# Title A1F3 Chem. Mater. 1995, 7, 75-83
#
UnitCell 10.184 10.184 7.173 90.000 90.000 90.000
SpaceGroup P4/nmm .2
#
GridDimension 28 28 M 2 2 2
#
MaxRvalue 0 22 0 .22 0.22
Fscale 1 857
TempFac 3 5
PseudoAt om 3 4
PermCode hamming
#
# h k 1 JF(hkl)| phase
0 0 0 0 639.955 0.0
2 1 1 0 34.22 0.0
2 0 0 1 16.22 0.0
1 1 0 1 73.52 0.0
1 2 0 0 91.52 0.0
1 1 1 1 66.06 0.0
0 2 0 1 169.79 0.0
1 2 1 1 117.84 0.0
1 2 2 0 131.32 0.0
2 0 0 2 55.5 8 0.0
1 1 0 2 143.86 0.0
0 3 1 0 149.67 0.0
1 2 2 1 160.06 0.0
1 1 1 2 103.59 0.0
1 3 0 1 166.76 0.0
1 3 1 1 44.11 0.0
1 2 0 2 86.05 0.0
2 2 I 2 22.96 0.0
1 3 2 1 40.48 0.0
2 4 0 0 31 .54 0.0
2 2 2 2 3 6.15 0.0
1 3 0 2 53 .14 0.0
3 3 3 0 16.7 6 0.0
3 4 0 1 11.79 "1 AJ . \J
4 3 1 2 15.75 0.0
1 0 0 3 109.22 0.0
4 4 1 1 17 .64 0.0
4 1 0 3 16.67 0.0
Figure 4-8 SayPerm input file for the 0-AlF^ test calculation Reflections up to a resolution of 2.5 À
were used.
4.6.3 Results.
The maximum densities of the map were be found at or near (+0.5Â) the positions of the Al
atoms (Fig. 4-9). Only seven, relatively weak reflections of the 25 input were phased
incorrectly (Tab. 4-7). This would be quite a good starting point for the calculation of a
41
h k l psq \/V^F(k)F{h-k) ^Sayre ^correct
0 0 0 824.995 824.995 0 0
1 1 0 21.683 27.932 0 0
0 0 1 10.270 13.681 Tt 0 #
1 0 1 44.473 60.372 Tt Tt
2 0 0 52.929 25.42 0 Tt Tt
1 1 1 3 8.177 48.385 Tt Tt
2 0 1 89.562 107.2 66 0 0
2 1 1 59.385 67.699 Tt Tt
2 2 0 63.271 93.094 0 0
0 0 2 26.702 71. 2 8 6 0 0
1 0 2 66.029 59.747 0 0
3 1 0 65.819 60.025 0 0
2 2 1 70.337 53.219 0 0
1 1 2 45.424 40.348 Tt Tt
3 0 1 70.011 64.33 7 0 0
3 i 1 17.692 22.627 0 Tt #
2 0 2 34.440 47.48 8 0 0
2 i 2 8.779 12.23 3 Tt Tt
3 2 1 14.158 9.660 0 Tt #
4 0 0 10.547 3 0.3 87 0 0
2 2 2 12.053 n c; g cj Tt 0 #
3 0 2 16.927 8.32 6 0 Tt #
3 3 0 5.115 11.750 0 0
4 0 1 3.596 3.023 0 0
3 1 2 4.793 4.605 0 0
0 0 3 33.120 28.557 0 Tt #
4 1 1-L. 5.140 11.415 Tt Tt
1 0 3 4.829 7.13 3 0 Tt #
Table 4-7 List of the phaseset from the SayPerm procedure for e-AlF? with the lowest R-value.
Figure 4-9 Fourier map calculated from all reflections involved in the phasingprocedure. The isosurface has a value of 85% of that of the strongest Fourier peak
42
stracture envelope followed by structure solution in direct space. The positions of the
remaining F atoms could also be deduced by considering the distances between the Al atoms
4.7 Tri-ß-peptide C32N306H53 (sa322)
4.7.1 Measurement, data preparation, and first attempts at structure solution
The tri-ß-peptide sa322 (Tab. 4-8) is highly insoluble and therefore difficult to recrystallize
(Abele, 1999). Consequently, only a polycrystalline material could be obtained. Powder
diffraction data (Fig. 4-10) were collected in transmission mode (0.3 mm capillary) on a high-
resolution laboratory powder diffractometer (Stoe STADI P) using CuKal radiation and a
small linear position sensitive detector. The crystallinity of the powder was sufficient to obtain
data up to a resolution of 1.8 A (50°29). A subsequent measurement using synchrotron
radiation (SNBL at ESRF) did not provide data of higher resolution. Furthermore, the sample
appeared to change during that measurement. The positions of the peaks moved during the data
collection. Consequently, these data were not used.
Sum formula C^N^OgHsj}
structure formula
AI h 1 H ä H \
Table 4-8 Chemical formulae of the sa322 molecule
The pattern collected using the laboratory instrument was indexed using the program TREOR
(Werner, Erikson & Westdahl, 1985), and could be confirmed with the program DICVOL
(Louer, 1992). With both approaches, a high figure-of-merit, indicative of the correctness of
the cell (61.03Â; 11.18Â; 5.08Â; 90.0°; 90.0°; 90.0°) was obtained. The most probable space
groups (P21212; P2i2121) were established by examining the data for systematically absent
reflections and using the information that the compound investigated is enantiomerically pure
(i.e. centrosymmetric space groups could be excluded from consideration). Assuming a density
of about lg/cm3, it could be calculated that the asymmetric unit would contain one molecule.
To test whether or not the structure could be solved using a direct methods program in a
straightforward manner, the intensities for both space groups (P2i2121 and P21212) were
extracted and input to the EXPO (Altomare et al, 1997) program. A crystal structure
consistent with the chemical information available could not be recognized from the solutions
43
120.0
100.0
80.0
C
2- 60.0
>
«
0)
40.0
20.0
0.0
^•J J\
2.5 7.5
NI 1.
vA_
12.5 17.5
2-theta
llllll mu un II m m l IIIIIIIIIIIII NIM m
_____
_____^^22.5
10.0
9.0
8.0
«2 7.0c0)
>6-°
tu
5.0
4.0
3.0
2.0
I'
*
n iv>
111 u 'j
'„. « .»
iiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiii iiiiiiHiiiiiii'iiiiiiiiiiiiiwiiniiiiiiiiiiiiiiiiniiniiiiiiuniiiHiiiiiiiiHiiiMiiniiHiiiiiii
27.5 32.5 37.5 42.5
2~theta
47.5 52.5
Figure 4-10Measuted powdei pattern and indexing (61 O^A, 11 18A, 5 08A, 90 0°, 90 0°, 90 0°,
7J21212,)oUa322
44
presented by EXPO. This was certainly caused in part by size of the structure and the low
quality and the insufficient resolution of the data.
4.7.2 SayPerm run and map evaluation
The unit cell obtained from the indexing has quite a long (61.03 A) a-axis and a very short
(5.08 A) r-axis. Thus, it could be expected that the projection along the short c-axis would
show the main features of the structure such as the shape and the packing of the molecules.A
further advantage of such a projection is that both possible space groups have identical
projections (plane group p2gg). That meant that the selection of the space group could be
postponed until further information was obtained.
The full set of extracted intensities for the space group P21212j was used for the estimation of
the scale factor (k = 0.40). Then, the data of the reflections hkO up to a resolution of 2.5Â were
input to the SayPerm program (Fig. 4-11). For the SayPerm run, the centrosymmetric space
group Pnmm, which has the same projection along the c-axis as P2]212 and P21212]_, was
assumed. In the case of organic compounds, it is not usually possible to assign a pseudo-atom
to a particular group of atoms in the organic molecule. Nevertheless, assuming a pseudo atom
with a diameter of 2.5 A (approximately the diameter of a methyl group) can succeed.
Fortunately, the non-hydrogen atoms (C. N, O in sa322) have similar scattering power, so an
equal atom structure could be assumed. This approximation is of course of lower quality than
the tetrahedron or octahedron replacement, but it was hoped that the validity of the Sayre
equation would be sufficient to find a phase combination, from which a two-dimensional
molecular envelope could be determined. 24 phases were permuted using the Golay [24,12,8]
code. Phases of origin defining reflections were not fixed. The input file is shown in Fig. 4-11.
Because the procedure for a permutation phase set is interrupted if the #-value after the first
phase extension exceeds the value specified in the input file (Rvalue), only 81 permutation
sets were evaluated in full. The run required only 10 minutes CPU time to complete the 4096
phase sets.
4.7.3 Results
The 81 sets were ranked by the R-value, and a Fourier map was calculated from the best set. In
this Fourier map (calculated using all reflections), it was impossible to recognize any
relationship between the appearance of the map and a reasonable packing of molecules.
Following the rules described in Sect. 2.5, seven strong reflections (Fig. 4-12) were selected
and a two dimensional structure envelope generated (Fig. 4-13). A chemically reasonable form
45
# Title sa322 STOE measurement, sayperm inputUnitCell 61.010 11 191 5.085 90.000 90.000 90.000
SpaceGroup Pnnm
#
GridDimension 120 20 10 12 1
#
MaxRvalue 0.25 0.25 0.3 0
Fsccile 0.405
TempFac 3 .0
Ps eudoAtom 2.5
PermCode golay
#
# h k 1 JF(hkl)| phase
0 0 0 0 1220.00 0.00
2 2 0 0 82.90 0.00
2 4 0 0 81.42 0.00
2 1 1 0 48.03 0.00
2 2 1 0 68.43 0.00
2 6 0 0 123.57 0.00
1 3 1 0 197.65 0.00
1 4 1 0 498.32 0.00
1 5 1 0 367.89 0.00
1 8 0 0 245.88 0.00
1 6 1 0 2 07.2 6 0.0 0
2 7 1 0 74.40 0.0 0
1 8 1 0 291.95 0.00
1 10 0 0 383.80 0.00
1 9 1 0 295.26 0.00
2 0 2 0 26.27 0.00
1 1 2 0 303.23 0.00
1 2 2 0 213.83 0.00
1 3 2 0 167.33 0.00
2 10 1 0 1.41 0.00
1 4 '"i
0 214.29 0.0 0
1 12 0 0 544.82 0.00
1 5 2 0 452.67 0.00
1 11 i 0 313.41 0.00
1 6 2 0 104.32 0.00
1 7 2 0 165.96 0.00
1 14 0 0 137.66 0.00
1 9 2 0 13 0.54 0.00
1 10 2 0 270.27 0.0 0
1 14 1 0 173.46 0.00
1 11 2 0 143 .01 0.00
1 16 0 0 6 5 6.90 0.00
2 12 2 0 151.26 0.00
2 1 3 0 101.55 0.00
3 2 3 0 152.03 0.0 0
3 3 3 0 194.46 0. 0 0
3 4 3 0 122.74 0.00
Figure 4-11 Input file for the SayPerm run for phase estimation of the M0 reflections. Data up to a
rcso lution of 2.5 A were used
46
# Titl
#
61.
e sa322 reflections for envelope generation
010 11 191 5.085 90.000 90.000 90 000
# h k 1 |F(hkl) | phase
# 0 0 0 0 1220 000 0 00
# 2 2 0 0 2 9 293 0 00
# 2 4 0 0 28 770 0 50
# 2 ] 1 0 16 972 ^ 50
# 2 2 1 0 24 180 3 00
# 2 6 0 0 43 664 0 50
# 1 3 1 0 69 841 1 5 0
1 4 1 0 176 085 0 00
1 5 L 0 129 996 0 50
1 8 0 0 86 383 3 5 0
# I 6 1 0 73 2 3 7 ^ 00
# 2 7 1 0 26 >9 0 ot
L 8 1 0 103 163 0 50
1 10 0 0 135 612 3 50
1 9 1 0 L04 332 3 03
# 2 0 2 0 9 283 1 0 0
1 1 2 0 L07 148 3 00
# 1 2 2 3 75 558 T 00
# 1 3 2 0 59 127 0 00
# 2 10 1 0 0 498 ^00
# 1 4 2 0 75 J21 n 00
# L 12 0 0 192 516 0 00
# 1 6 2 0 159 9 5 \ e 5 0
# 1 11 1 0 1 L0 74 6 00
# 1 6 2 0 36 86„ 0 00
# 1 / 2 0 58 643 0 00
Figuie4 12 Beginning of the îeflection list with the lowest R-valuc (0 21) The list is pait of the
SayPeim output fil c The stiuctui e envelope ( Fig 4 13) was calculated fiom the leflections
without #'
Figuie 4 J 3 Two dimensional stiuctuie envelope toi the sa322 molecule
Two unit cells along the b axes aie displa\ed
47
for the sa322 molecule became visible. In this case the structure envelope can be considered to
be a molecular envelope because it has a closed form. This envelope could then be used to set
the molecule at a sensible starting position and to restrain it to a limited region of the unit cell
in a simulated annealing procedure (Sect. 7.5).
4.8 Limitations of the SayPerm Approach
The examples described demonstrate the potential of the SayPerm program. In these favorable
cases, the positions of T-atoms (RUB-3), useful phase information for the calculation of a
structure envelope (ITQ-1 and sa322), and the Al-position (0-AlF-ri could be obtained.
As mentioned at the beginning of this chapter, the Sayre equation holds over a large range of
conditions. However if the approximation is poor, the Sayre equation is no longer valid, so of
course, its validity cannot be used as a sensible figure-of-merit for the best phase set. The
approximation of coordination polyhedra with spherically shaped pseudo-atoms is sufficient
to make direct methods applicable at low resolution (2.5 - 3.0 A) for small-to-medium-sized
structures. The more pseudo-atoms in the asymmetric unit (e.g. the more complex the
structure), however, the lower the quality of the approximation. Tests with veiy complex
zeolite structures (e.g. ZSM-5 with 12 T-atoms in the asymmetric unit) have shown that the
SayPerm approach becomes less reliable as the complexity increases.
Theoretically, the SayPerm approach should work for non-centrosymmetric structures as well.
However, the number of possible phase combinations is much higher than for the
centrosymmetric case, and phase estimations for test cases have not been successful to date. A
number of possible approaches to improving the method to address the problems of non-
centrosymmetric structures remain to be explored. The main problem is to find the best phase
set from the many sets produced, so the development of a more selective figures-of-merit is
necessary to enable the SayPerm approach to handle non-centrosymmetric structures.
In addition to the size of the structure, the chemical composition is crucial to the applicability
of the Sayre equation. For the original form of the Sayre equation (Sayre, 1952) an equal atom
structure is assumed. That means that, the pseudo-atoms should also be equal. This works well
for structures containing groups of atoms with equal or similar scattering factors, but as the
diversity increases, the validity of the Sayre equation decreases. Woolfson (1958) published an
extended form of the Sayre equation valid for structures with two kinds of atoms. He
demonstrated the validity of the extension using a one-dimensional structure. In the scope of
this work, some initial tests with this extension for real three-dimensional crystal structures
48
were carried out. Zeolite structures with different T-Atoms (e.g. Ga and P) were examined,
but no notable improvement in the results of the phasing procedure with respect to those of the
original SayPerm approach could be achieved. Nevertheless, the implementation of
Woolfson's extension to the Sayre equation was left in the SayPerm program.
49
5 Phase estimation by the method of permutation synthesis
5.1 Introduction
Statistical approaches are not feasible if the number of objects used for the statistical
evaluation is too low. Thus, for a phase estimation, the number of reflections involved must
exceed a certain limit. Otherwise, any statistical calculation, however clever it might be, will
fail.
An alternative to statistical arguments is the visual evaluation of Fourier maps, generated using
different phase combinations. If the size and shape of structure fragments are known, objects
having an appropriate shape can be sought. Any chemical information such as the connectivity
of a molecule or a zeolite's pore size can be used in this process. Unfortunately, multisolution
approaches can produce millions of phase sets, so, evaluating by eye is not reasonable. The
selection of the correct phase set must be automated using direct space arguments (e.g.
Stanley, 1986). It would be a challenge to develop a computer algorithm that could replace a
visual map evaluation, and there are encouraging developments in pattern matching
algorithms. However, many problems must be solved before usable algorithms for
crystallographic purposes are available. An alternative way is the evaluation of each phase set
by a subsequent search for the structure in direct space. This concept is used in the FOCUS
approach (Grosse-Kunstleve, McCusker & Baerlocher, 1997). Here the program searches for a
reasonable zeolite framework within the peaks of the Fourier maps generated from randomly
phased structure factors.
As shown in Chap.2, a few reflections with the correct phases are sufficient to describe the
coarse features of a crystal structure. If the phases arc unknown but some features of the
structure can be assumed based on chemical knowledge, there is a chance that the correct
phase set can be found by permuting the phases of just a few reflections and evaluating the
resulting maps. The more phases permuted, the more distinct the features of the stracture in the
map. Ecc's can be used to allow a higher number of phases to be permuted while keeping the
number of maps to be evaluated low.
Woolfson (1954) permuted seven phases using the Hamming [7, 4, 3] code to confirm a
projection the of the p-nitroaniline crystal structure. This approach was called "permutation
synthesis". The Hamming [7, 4, 3] code has 16 phase combinations instead of 128, which
would be produced by a systematic phase permutation. From one of these 16 maps, the
structure projection could be recognized easily. The permutation synthesis has even been
50
applied to obtain a low-resolution structural characterization of a protein structure (Glycos,
1998). Unfortunately, permutation synthesis is not suited for non-centrosymmetric cases,
because the number of maps produced would be too large if the phases of a sufficiently high
number of structure factors were permuted. However, a centrosymmetric projection could be
useful, if the important features can be recognized from the projection.
The advantage of a permutation synthesis is that phase information can be obtained for just the
few reflections from which the structure envelope is generated. For other phasing methods, a
larger number of reflections (up to a certain resolution) are required and this can be an
handicap if the peak overlap is severe. The same rules used for the reflection selection for the
generation of a structure envelope (Chap.2) can be used. The only one difference is that in the
case of a permutation synthesis, the number of reflections is fixed, depending on the ecc used
for the phase permutation. For example, the Hamming [7, 4, 3] code has seven signs per
combination, and this corresponds to the number of the permuted centrosymmetric phases. A
maximum of three odr's are used for the map generation. With the Hamming [7, 4, 3] code, 16
phase sets arc produced and consequently sixteen maps must be examined. In favorable cases
the correct map can be recognized. The permutation syntheses has been implemented in the
SayPerm program.
5.2 Test structure Cimetidine
For historical reasons, Cimetidine has become a standard example for demonstrating new
approaches to structure determination from powder data (e.g. Cernik et al., 1991). It was first
synthesized by Haedicke et al. (1978) and its structure determined from single crystal data
(Table 5-1).
chemical formula H3C
\ H
/^ /\ ^\ /N\ ^N,+Ch6
HN T ||\===N N-CN
lattice parameters 10.699Â. 18.818Â, 6.825À, 111.284°
space group P21/>/asymmetric unit IOC, 6N. 1 S, 16 H
Table 5-1 Data tor the Cimetidine structure
The Cimetidine molecule is quite flexible, with nine free torsion angles. Therefore, structure
determination in direct space is a challenge (e.g. Csoka, 1997). A structure envelope could be
quite useful in limiting the number of possible conformations and orientations of the molecule
51
in trial structures. Attempts to determine the phases at a resolution of about 2.5 Â using the
SayPerm approach failed, because the organic structure does not consist of building units that
can be approximated by a single spherically shaped pseudo-atoms. Furthermore, the scattering
power of the sulfur atom and the carbon or nitrogen atoms are too different to be treated as a
single kind of pseudo-atom. Thus, the permutation synthesis approach appeared to be an
appropriate one.
5.2.1 Data preparation
The powder pattern used for the extraction of the intensities was provided as a test powder
pattern with the EXPO package (Altomare et al, 1997). Reflection intensities were extracted.
A normalization of the structure factor amplitudes was not necessary because only the ratio of
the strongest structure factor amplitudes influences the result of the calculations.
# Cimetidine st rongest extracted reflections
Unite ?11 14.09 8 13. 670 7.43 1 90.000 102.421 90.000
Space
#
OnctD
jroup P2/n
intension 24 24 L6 2 2 2
PermC
#
#
ode permsynth
h k ] |F(hkL) | phase
1 -1 2 1 362.791 0.0 00
1 0 2 1 390.237 0.000
1 2 0 0 490.322 0.000
1 1 L 1 342.079 0.000
1 2 L 0 342.975 0.000
0 -1 4 1 469.635 0.000
0 0 4 1 912.867 0.000
0 1X 3 1 409.342 0.000
J 2 4 0 564.758 0.00 0
1 3 I 0 286.929 0.00 0
Fig ire 5-1 SayPerm input file a permutation synthesis for Cimetidine
5.2.2 Application of the permutation synthesis
For a permutation synthesis, up to 10 strong reflections (odr's plus permuted phases) from the
low angle region of the powder pattern can be used. These reflections should not overlap. The
reflections were selected following the same rules (Sect. 2.5) as for the generation of a
structure envelope. The space group Plln was assumed, and the phases of three origin defining
reflections (141, 041, 131) were fixed to be 0.0. Then seven other reflections were selected for
phase permutation. The input file for SayPerm is shown in Fig. 5-1. To initialize the
permutation synthesis "permsynth" is used in the PermCode line. The program permutes the
phases using the Hamming [7,4.3 ] code and generates a Fourier map for each of them.
52
era
Cr
to
CT
fO
g
ryi
S1
O
Pi-
CD
Ra
s-
3
&
HCi
CT
ST
ito
a
s-
cr
o
er
os
#
iSJ
-J
— 53 —
es so
e
-3
"5
S
5
O
S
1o
<0
60"xf
OS
54
These maps were viewed using the graphics program ISOSURFACE ,a module implemented
in the CERIUS2 (MSI, 1997) package. The fact that chemical structures could be also
generated within the CERIUS2 package was an important reason for choosing to use this
software. The maps resulting from the SayPerm run are displayed in Fig. 5-2 and 5-3.
5.2.3 Selection of the best Fourier map
If three-dimensional maps are to be evaluated, software and hardware that allow a three
dimensional visualization on the screen is highly recommended. The most useful approach is
to look at the isosurfaces. The isovalue should be set such that the coarse features of the
structure can be recognized. For a small organic molecule structure, this value is usually about
50% of the highest positive value on the map. The following criteria proved to be useful to find
the most probable structure envelope:
(1) If only one molecule is expected in the asymmetric unit, the map should show only a single
shape
(2) The envelopes should be packed in an effective manner within the unit cell.
(3) If no atoms are expected at the origin, no density should be present there.
(4) The shapes should be consistent with the known connectivity of the molecule.
If a map fulfilling these requirements can be found, it is probably the correct envelope. The
map should be evaluated for a range of isovalues (e.g. 50% to 70% of the highest map density),
otherwise it is not possible to decide whether or not the isosurfaces tend to describe discrete
entities.
The codewords of the Hamming [7,4,3] code differ in at most one place from one of the 128
possible codewords generated by systematic permutation. Thus, it is possible, that no
acceptable map can be found if the phase of an important reflection (e.g one that strongly
influences the appearance of the map) is wrong. In this case, it can be useful to change the
phases of the origin defining reflections, so that the sign relationships between the reflections
change. This might be lead to a new sign combination that better fits the correct one.
The maps generated by the permutation synthesis for the Cimetidine crystal structure are
shown in Fig. 5-2 and 5-3. The surfaces in all maps have the same isovalue (60% of the highest
density in the map). Maps 1, 2, 3, 9,13, and 16 can be excluded because they show continuous
regions of high density. For the cimetine structure, individual objects can be expected. Maps 4,
7, 11, 12, and 14 can also be discounted, because they show more than one shape, and only a
single shape would be expected. The maps 6. 8 and 15 show objects which cannot
55
accommodate any conformation of the Cimetidine molecule. The decision between the two
remaining maps 5 and 10 is not easy. If the they are viewed in three dimensions (stereo mode)
it can be seen that the packing of the isosurfaces in map 5 is a little more effective (see also
Fig. 5-4), but actually, a definitive decision cannot be made. The phase sets of both maps
should be taken into account in further steps of structure solution.
5.3 Tri-ß-peptide C32N306H53 (sa322)
In Sect. 4.7, the phasing of a few strong low-order reflections for sa322 using all reflections up
to a resolution of 2.5A was described. In this case, where the molecule can be treated as an
equal atom structure, the accuracy of the pseudo-atom approximation for an organic molecule
sufficed for the Sayre equation to be used as a figure-of-merit. All conditions for a successful
application of a permutation synthesis are also given. Therefore, it was used to confirm the
results obtained earlier using the Golay code in the SayPerm program. Information about the
data collection and preparation were given in Sect. 4.7. To obtain results that were more
independent, a slightly different reflection set was used for the permutation synthesis.
5.3.1 Selection of the best Fourier map
All of the 16 maps generated are displayed in the Fig. 5-6 and 5-7. The isolines of densities
lower than 50% of the highest density in the map are not shown. The options for each map are
set at the same values. Following the rules given in Sect. 5.2.3, map 4 appeares to be is a
56
# sa322, inputfil e for the permu tation synthesesUnitCell 61.010 11.191 5.085 90.00 90.00 90.00
SpaceGroup
#
GridDimension
P21/n
120 20 10 12 1
PermCode
#
perms /nth
#
# h k 1 |F(hkl)] phase
1 4 1 0 498.32 0.0 0
1 5 1 0 3 67.8 9 0.00
0 8 1 0 2 91.95 C.5 0
1 10 0 0 3 83.80 0.00
0 9 1 0 295.26 0.00
1 1 2 0 3 03.2 3 0.00
1 4 2 0 214.29 0.00
1 12 0 0 544.82 0.00
1 5 2 0 452.67 0.00
Figure 5-5 SayPerm input file for the permutation synthesis. The origin defining (0 in the first column)reflections were phased according to the phases obtained from the SayPerm ran (Figure 4-12) to obtain
maps with the same origin
promising candidate for a structure envelope for the sa322 molecule. All of the remaining
maps contradict the assumption that the coarse features of a single molecule should be seen in
the map. Either differently shaped objects (maps 5, 9, 14, 16) or continuous regions of high
density (maps 1,2,3,6,7,8, 10, 11, 12, 13, 15) instead of individual objects are present. The
shape and packing of the objects in the fourth map correspond to those of the map generated
using the SayPerm procedure in section 4.7.2.
5.4 Conclusions
In most cases, the connectivity of a small organic molecule is known either from the synthesis
procedure and/or spectroscopic analyses (e.g. NMR). This information can be used to
recognize the appropriate form in a Fourier map. Permutation synthesis corresponds more
closely to the structure envelope approach than does the phase estimation with the Sayre
equation, because, as for the structure envelope generation, just a few strong low-order
reflections are taken into account. For small organic structures, permutation synthesis is a
suitable tool for obtaining information about the molecular packing and the coarse molecular
shape. Unfortunately, solution cannot be achieved in every case, but at least the number of
possible phase combinations can be limited.
For inorganic crystal structures, criteria for map evaluation are more difficult to formulate
because the connectivity cannot usually be predicted as easily as it can for a small organic
molecule. However, it would be interesting to investigate whether chemical information, such
(1)
(5)
(2)
(6)
(3)
(7)
(8)
en
Figu
re5-6The
firstei
ghtFouriermapgeneratedbythepermsynthpr
oced
ure.
The
selectedmolecularen
velo
peismarkedby
ablackframe
(9)
00
|ß^
ff໫
(16)
b>Ä
<S1»#?|^»i
<
Figure5-7Maps9-16ofthepermutation
synthesesofsa322
59
as, for,l example the pore size or HRTEM images from zeolite structures would suffice to
allow the correct map to be selected.
For low-resolution phasing approaches, the selection of an appropriate starting phase set is a
crucial point. The more phases (in addition to those of the odr's) available, the higher the
probability of a successful phase extension and refinement. Such promising phase sets can be
determined using permutation synthesis.
60
6 Determination of zeolite structures using structure envelopes
6.1 Introduction
To build a zeolite framework model on one side of a structure envelope, an intelligent model-
building algorithm is needed. Chemical information such as the fact that zeolite frameworks
consist of corner-sharing T04 tetrahedra forming a 3-dimensional 4-connected framework, can
be used in an automatic procedure. Grosse-Kunstleve, McCusker & Baerlocher (1997)
presented an algorithm designed especially for this problem. It is implemented as a subroutine
of the program FOCUS (Grosse-Kunstleve, 1996). Given a list of potential atomic positions,
this topology-search algorithm seeks possible framework structures that are consistent with the
space group and the cell parameters. For small structures, the list of potential nodes can be a
simple grid of points within the asymmetric unit, and that is sufficient to find a chemically
reasonable framework. As the size of the structure (i.e. number of T-atoms in the asymmetric
unit) increases, the time required for such a gridsearch becomes formidable. Computers
currently available are not capable of completing such an exhaustive search procedure within a
reasonable timeframe. In the FOCUS program, this difficulty is addressed by implementing a
Fourier recycling loop. It was hoped that by introducing a structure envelope to define the
region within the asymmetric unit in which the framework was to be sought, the topology
search for even more complex structures would become feasible.
6.2 Topology search for zeolite structures with a structure envelope
For an exhaustive gridsearch, each grid point within the asymmetric unit is treated as a
potential node atom (T-atom). The search procedure is divided into two stages. First a list of
potential bonds (bondlist) is prepared for each grid point. Then a backtracking algorithm
operates on these bondlists seeking 3-dimensional 4-connected nets with appropriate
interatomic distances.
To test the effectiveness of using a structure envelope as a mask, an additional subroutine was
implemented in the original algorithm (Grosse-Kunstleve, 1997b).
Only those gridpoints located on the positive side of the structure envelope were considered in
the topology search (in Fig. 6-1, the negative side of the envelope is shaded). This reduces the
the number of possible atom positions by a factor of two. Starting from one Pivot atom (black
point in in Fig. 6-1), a bond list is created with all gridpoints on the positive side of the
envelope and with approximately the correct T-T distance (3.1Ä) from the Pivot atom. In Fig.
6-1 these atoms lie on the dark grey circle. Each point on the positive side of the envelope is
60
6 Determination of zeolite structures using structure envelopes
6.1 Introduction
To build a zeolite framework model on one side of a structure envelope, an intelligent model-
building algorithm is needed. Chemical information such as the fact that zeolite frameworks
consist of corner-sharing T04 tetrahedra forming a 3-dimensional 4-connected framework, can
be used in an automatic procedure. Grosse-Kunstleve, McCusker & Baerlocher (1997)
presented an algorithm designed especially for this problem. It is implemented as a subroutine
of the program FOCUS (Grosse-Kunstleve, 1996). Given a list of potential atomic positions,
this topology-search algorithm seeks possible framework structures that are consistent with the
space group and the cell parameters. For small structures, the list of potential nodes can be a
simple grid of points within the asymmetric unit, and that is sufficient to find a chemically
reasonable framework. As the size of the structure (i.e. number of T-atoms in the asymmetric
unit) increases, the time required for such a gridsearch becomes formidable. Computers
currently available are not capable of completing such an exhaustive search procedure within a
reasonable timeframe. In the FOCUS program, this difficulty is addressed by implementing a
Fourier recycling loop. It was hoped that by introducing a structure envelope to define the
region within the asymmetric unit in which the framework was to be sought, the topology
search for even more complex structures would become feasible.
6.2 Topology search for zeolite structures with a structure envelope
For an exhaustive gridsearch, each grid point within the asymmetric unit is treated as a
potential node atom (T-atom). The search procedure is divided into two stages. First a list of
potential bonds (bondlist) is prepared for each grid point. Then a backtracking algorithm
operates on these bondlists seeking 3-dimensional 4-connected nets with appropriate
interatomic distances.
To test the effectiveness of using a structure envelope as a mask, an additional subroutine was
implemented in the original algorithm (Grosse-Kunstleve, 1997b).
Only those gridpoints located on the positive side of the structure envelope were considered in
the topology search (in Fig. 6-1, the negative side of the envelope is shaded). This reduces the
the number of possible atom positions by a factor of two. Starting from one Pivot atom (black
point in in Fig. 6-1), a bond list is created with all gridpoints on the positive side of the
envelope and with approximately the correct T-T distance (3.1Â) from the Pivot atom. In Fig.
6-1 these atoms lie on the dark grey circle. Each point on the positive side of the envelope is
Figute 6-1 Exhaustive gndseaich limited by tt structure envelope.
treated, in turn, as new Pivot atom, until all possible topologies have been found.
For the tests, a simple circa 0.5A grid with a 0.5 Â tolerance on interatomic distances
(represented by the thickness of the dark grey circle in Fig. 6-1) is used. All calculations were
performed on a Silicon Graphics Solid Impact R10000 computer. The procedure consisted of
the following steps (Fig. 6-2):
(1) A set of origin defining reflections with high lEI-values and large ^-spacing, which do not
overlap in 29 were selected.
(2) If additional reflections were required, their phases were determined using the program
SayPerm.
(3) A structure envelope was generated using these reflections.
(4) An exhaustive gridsearch using the topology search on only one side of the structure
envelope was performed.
(5) The geometries of any topologies found were optimized using a distance least squares
procedure.
(6) If the search was not successful, additional high lEl-value, large <:/~spaeing reflections were
added.
62
Calculation of the
structure envelopeittf
*
1
exhaustive FOCUS grid search
additional
reflections
iAny nets found? no
yes
1Can the framework geometry
be optimized? (DLS)
no
yes
lRefinement of
atomic positions
Figure 6-2 Combination of an exhaustive gudseatch with a structure envelope
6.3 Test examples
Four zeolites structures, A1P04-D (APD topology). Sigma-2 (SGT topology), RUB-3 (RTE
topology), and ITQ-1 (MWW topology), with orthorhombic, tetragonal, monoclinic, and
hexagonal crystal systems, respectively, were used as test examples for the procedure shown in
Figure 6-2. For A1P04-D (APD), two origin defining reflections can be chosen, and the two
with the highest lEI-values and rf-spacings (021 and 131) were selected. These sufficed to
produce a useful structure envelope (Fig. 2-3 d). For Sigma-2 (SGT) the 121 reflection was
selected to define the origin, but the structure envelope generated with just this reflection did
not allow the structure to be found, so the phase of a second strong reflection, 116, was
determined using the SayPerm approach outlined in Chap. 2, and both reflections were then
used to generate a useful structure envelope. The grid searches were completed in 6 min CPU
63
time for APD and in 1.8 min for SGT. In both cases the DLS geometry optimization of the
topologies generated allowed the correct topologies to be singled out very clearly. For
comparison, grid searches were also performed over the whole asymmetric unit without the
structure envelope as a mask, and these took 43 and 123 min CPU time, respectively.
Zeolite Space Number of Number of CPU Time (min)
Topology Group T-Atoms Reflections with Envelope w/o Envelope
APD Cmca 2 2 6 43
SGT 14\/amd 3 2 1.8 123
RTE C2/m 4 3 25 >2880
MWW P6/mmm 8 4 27 >4320
Table 6-1 The effect of using a structure cmelope mask on computing time
In the lower symmetry case of RUB-3 (RTE). more then just the two origin-defining
reflections were required. A list of the ten reflections with the largest û?-spacings is given in
Table 2-2 on page 15. Initially, 110 and llT were selected for the generation of a structure
envelope. Flowever the grid search with this mask did not yield a satisfactory framework
topology. Consequently, a third reflection, 201, was added to the origin defining ones. As an
alternative method to the SayPerm approach, two surfaces were generated: one with the phase
for (201) set at 0° and one with it set at 180°. For the former combination, none of the 56
topologies generated had a satisfactory geometry. For the latter (structure envelope is shown in
Fig. 2-5b), the RTE topology emerged as the only geometrically sensible one out of the 15
generated. The two grid searches with the structure envelope each required 25 min CPU time.
The search without the structure envelope was still not finished after more than 48h.
In Sect. 4.5, the phasing of some strong reflections having large ^/-spacings for the test
structure ITQ-1 (MWW) was demonstrated. The structure envelope generated from just the
origin defining reflection 102 did not allow the structure to be found. New structure envelopes
using more reflections were generated, but the grid searches with these masks did not yield any
reasonable framework topologies. Only when four reflections, 102, 100, 101, and 002
(structure envelope is shown in Fig. 4-7) were used did the grid search yield a sensible result.
Of the 24 topologies generated, only the MWW topology was found to have a satisfactory
geometry. The search was repeated without the structure envelope and was not still finished
after 72h.
-64
6.4 Structure envelopes with the full FOCUS approach
Encouraged by the obvious advantage offered by structure envelopes in an exhaustive grid
search, a more demanding test was performed. The structure envelope mask was used in
conjunction with the full power of FOCUS. The FOCUS program is a combination of Fourier
recycling, topology search, and sorting algorithms.
(1) Fourier recycling. Strong structure factor amplitudes extracted from a powder pattern are
assigned random phases and a Fourier map is calculated. The strongest peaks in this map
are interpreted automatically using the chemical information input. This model (simple
atom assignment, a fragment of a framework, or a complete framework structure) is used
to calculate new phases, which are then assigned to the corresponding observed structure
factor amplitudes. These and the phases, in turn, are used to calculate a new Fourier map.
This is done in several cycles until the phases converge.
(2) Topology search. From the list of peaks from the Fourier map (potential atomic positions) a
list of potential bonds is made and a 3-dimensional, 4-connected framework is sought (as
described in Chap. 6.2). If an exhaustive grid search is to be performed, the peak list is
replaced with a list of grid point coordinates, and no Fourier recycling is done.
(3) Sorting of topologies. The frameworks produced by the topology search are classified and
sorted by evaluation of site multiplicities, loop configurations (Meier & Olson, 1992;
Fischer, 1971), and coordination sequences (Branner, 1979; Grosse-Kunstleve, Brunner &
Sloane, 1996) to decide whether or not two frameworks produced by the search algorithm
are equivalent. For a typical FOCUS run using the Fourier recycling procedure, the
framework that is found most often is usually the correct one.
In contrast to the combination of a structure envelope with a simple grid search where all grid
points outside the envelope were simply excluded from the search, for the combination with
the Fourier recycling loop, those peaks within the envelope were given more weight (i.e.
moved up in the peaklist) than those outside the envelope, but the latter were not excluded
completely. Whether or not a peak is included in the topology search depends upon its position
in the peaklist, because only the highest peaks (to a prescribed level) are used.
This strategy was used to solve the very complex structure ZSM-5 (MFI topology, 12 atoms in
the asymmetric unit, space group Pnma). High resolution synchrotron powder diffraction data
collected on the SNBL at the ESRF in Grenoble were used for the intensity extraction. The
structure envelope was calculated using five reflections, Oil, 102, 301, 200, and 020, three of
which were origin-defining (see also Tab. 2-1). Without the mask, FOCUS found the MFI
topology 50 times in eight hours (no other topologies were found). With the mask, the time for
50 MFI topologies to be found could be reduced to three hours.
6.5 Conclusions
A structure envelope defines a region in the asymmetric unit in which the framework atoms are
likely to be located. This means that a search program has fewer potential atomic positions to
check, and will therefore generate more reasonable models in a shorter period of time. The
search for the correct model becomes much more selective and a large number of calculations
can be saved. In some test cases in which only geometrical knowledge was used (exhaustive
grid search), the structure envelope restriction was crucial to the structure determination (e.g.
RTE, MWW). Other tests on zeolite framework structures demonstrated that the use of
envelopes can reduce the amount of computer time required by as much as two orders of
magnitude (Table 6-1). Weighting the peak positions from a Fourier map using a structure
envelope mask, was also found to reduce the computer time required when Fourier recycling
was applied, and this might be critical for the solution of more complex structures.
66
7 From structure envelopes to organic crystal structures
7.1 Introduction
In view of the very encouraging results obtained for the use of a structure envelope in zeolite
structure solution, their applicability to other classes of materials was considered. From a
structure envelope alone, a structure solution cannot be achieved. The lack of resolution must
be compensated with chemical information such as bond lengths, bond angles, torsion angles
and/or data of higher resolution. So far, in contrast to protein crystallography, the use of
envelopes in combination with direct space search methods has not been used for powder data.
Usually structure determination is attempted either in reciprocal space or in direct space. Both
approaches have advanced rapidly in recent years but neither has become dominant. For this
reason, the development of methods that take advantage of both direct and reciprocal space
concepts is appealing. The program FOCUS discussed in the previous chapter, does just this
for zeolite structure. (Grosse-Kunstleve. McCusker & Baerlocher, 1997).
A structure envelope could be used in structure solution from powder data in the same way a
molecular envelope is used by protein crystallographers. Basically, both fields have the same
major problem: the lack of reliably interprétable data at higher resolution and consequently too
many free parameters for the amount of data available. Although the immediate impression is
that the world of protein crystallography must be completely different from that of powder
diffraction, there are commonalities. In the last few years, there have been some advances in
low-resolution phasing (e.g. Gilmore, Henderson & Bricogne, 1991, Dorset, 1997, Rius,
Miravitlles & AUmann, 1996) and its combination with direct-space structure determination
(Tremayne, Dong & Gilmore, 1998), and these could be relevant to powder diffraction
methodology.
7.2 Direct-space approaches to crystal-structure determination
Instead of determining a structure from the peaks in a Fourier map calculated from reflections
phased in some way (e.g. direct methods), another approach can be applied. An arrangement of
electron densities can be generated without any consideration of the diffraction data. This can
be done without chemical knowledge (electron density modification) (e.g. Subbiah, 1993) or
with chemical knowledge (model building). From the model or the generated electron density
distribution, diffraction data are calculated and compared with the measured data. The better
the two sets match the higher the probability that the model or the electron distribution is
correct. So far, the approach of electron density modification has only been applied to organic
67
macromolecules and single crystal data, but in principle this approach should also be
applicable to other materials and powder data.
The first crystal structure solutions from powder data were done by building models by hand.
With the rapid development in computing power, more automated approaches can now be used
to build models. Not only can models be generated by the computer, they can also be modified
automatically. The diffraction pattern of each model can be calculated and compared with the
observed data simultaneously. In this way, it is possible to build and evaluate thousands or
even millions of models within a reasonable amount of time.
7.2.1 Model generation techniques for organic structures
The technique used for model building depends on the kind and complexity of the structure to
be solved. Generally two approaches are used:
(1) By hand. A chemically reasonable model is built either as a real model made of plastic,
wood or metal or on the computer screen using molecular-modeling software. Some of
these programs allow a fragment to be moved around the cell in real time while the powder
pattern is calculated simultaneously. This allows reasonable models for simpler structures
such as small rigid-body molecular structures to be determined (Seebach et al., 1997).
(2) By computer. The connectivity, bond lengths, and bond angles are input into a computer
program, which is able to move one or more organic molecules around the asymmetric
unit. When the position and orientation of the molecules are similar to those of the real
structure, a good fit between calculated and measured data is achieved. It is also possible to
construct these molecules in a flexible manner, so that orientations of parts of the molecule
can be varied separately by setting flexible torsion angles (David, Shankland, Shankland,
1998, Andreev & Bruce, 1998. Chernychev & Schenk, 1998).
7.2.2 Model modification control
The number of possibilities that have to be evaluated for a structure solution depends on the
complexity of the structure and the previous knowledge about the structure. The aim is to find
a combination of structural parameters providing the best agreement between the measured
diffraction pattern and that calculated for the model. For small problems, it is feasible to
evaluate the possibilities rather exhaustively m a reasonable amount of time. However, if the
number of unknown structural parameters is too high, a more intelligent way of determining
the correct parameter combination must be found. In principle the following, which are also
applied in other fields of science, can be used.
(1) Grid search. A space having the same number of dimensions as unknown parameters is
68-
defined (e.g. three for the position of the structural fragment in the asymmetric unit and
three for its orientation). Then a grid is placed in this space. Each grid point represents a set
of parameters describing a trial stracture. These points can be tested selectively or
exhaustively and each trial structure is evaluated. Thus, exhaustive searches are limited to
relatively small structures. For example, each torsion angle in an organic molecule would
increase the dimensionality of the space, so the number of grid points increases
dramatically with every additional parameter. A grid search has been applied to rigid-body
organic molecules (e.g. Chernychev & Schenk, 1998).
(2) Monte Carlo. Here each structural parameter <j) is modified using a random number m
between 0 and 1
è = m- Aè (7~i)
where A(f> is the range allowed for o. The atoms or atom groups are placed in the
asymmetric unit in a random fashion. This is usually done using chemical knowledge such
as the connectivity of an organic molecule (e.g Harris, Tremayne, Lightfoot & Bruce, ,
1994) or the Coulomb potentials in a structure with charged fragments (Putz & Schoen,
1999). A large number of models are generated, and it is hoped that the structural
parameters of one of these models are close to those of the correct structure.
(3) Metropolis Monte Carlo. Here random numbers also play an important role, but the
structural parameters (f> are not generated from scratch for each new random number m, but
are derived from the previous values. That is
<ta» = told + '» A^ (7-2)
where A<\> is the limit of the change for <]). Tn this way, the parameter <|)oW is changed to §new
and a decision (Metropolis importance sampling technique) is made whether or not this
change will be performed (Metropolis et al., 1953). The criterion used is the value of the
figure-of-merit X (e.g. the agreement between the measured and calculated data or the fit of
the structure to the structure envelope). The parameters are changed from 4>ö/j to §new
if (Xue»<%o,d) or f„<exp^^/^) (7~3)
where n is a random number between 0 and 1 (independent of m) and AT is given. This
second condition enables the system to escape from a local minimum and gives the
Metropolis approach a crucial advantage over minimization using a least-squares
algorithm.
69
(3) Simulated annealing. Simulated annealing involves the same approach as Metropolis
Monte Carlo with the difference that AT, which can be viewed as the temperature in an
annealing process, is slowly decreased during the course of the ran. The probability that
the exponential term in Eqn. 7-3 is larger than the random number n decreases, so fewer
and fewer modifications of the model (Eqn. 7-2) are accepted. At some point, AT is
decreased to a value where the structure becomes "frozen". With an appropriate schedule
for the annealing process, this technique is a powerful method for finding the optimal
combination of structural parameters (David, Shankland, Shankland, 1998).
(4) Genetic algorithm. The genetic algorithm uses strategies borrowed from genetics in life
sciences. First, a "population" of trial structures is generated. From these structures, the
next generation is generated by combining the "genetic material" (structural parameters)
from two parents in the original population to form two "children". From these new
models, the "fittest" (those structures providing the best agreement between calculated and
observed data) are selected as parents for the next generation. Mutations are introduced
randomly to avoid the development of the population to a local minimum. Once a
sufficiently high number of generations have been produced, structural parameters
providing a good agreement between measured and calculated data will be found (Harris,
lohnston & Kariuki, 1998; Shankland, David & Csoka 1997).
(5) Error correcting codes. Theoretically, error correcting codes can also be used to control
model modification (Bricogne, 1997). With the Golay [24,7,4] code, for example, one
could control the variation of torsion angles in a molecule. This approach has not yet been
put into practice, but its potential is obvious.
7.2.3 Comparison of diffraction data
Once a model has been built, diffraction data must be simulated for comparison with the
observed data. There are two principal approaches:
(1) Whole pattern approach. The agreement between the observed and simulated patterns is
calculated point for point. Usually this fit is calculated using the weighted /^-factor {Rwp) or
the goodness-of-fit function %".
(2) Extracted intensities. The observed structure-factor amplitudes are calculated from the
reflection intensities extracted from the powder diffraction pattern. These are then
compared with those calculated for the trial structure. This approach can only be used if
estimated standard deviations (csd's) are available for the extracted intensities (e.g. Sivia
& David 1994), or if overlapping reflections are treated as a single observation. If the esd's
70
are known, the figure-of-merit for the agreement between the simulated and the
experimental data is equivalent to that calculated using the whole pattern approach (Rwp ).
The calculation of the powder pattern for a trial structure is one of the most time-consuming
steps in direct-space approaches to structure determination. For a whole-pattern comparison
the structure-factor amplitudes have to be convoluted with the peak-shape function and the
intensity at each point of the powder pattern calculated. In the case of extracted intensities,
only the structure-factor amplitudes have to be calculated, so the agreement factor can be
evaluated considerably faster (David, Shankland & Shankland, 1998).
All direct-space approaches to structure determination of small organic structures from powder
data are combinations of the methods described above. In each case, models are generated,
modified according to some scheme and evaluated.
7.3 Simulated annealing, fragment search and structure envelopes
The combination of simulated annealing and fragment search has proven to be one of the most
powerful methods for solving relatively complex organic crystal structures from powder data
(e.g. David, Shankland & Shankland, 1998; Andreev & Brace, 1998). The only degrees of
freedom in the search are the torsion angles, the orientation, and the position of the molecule(s)
within the asymmetric unit. The bond lengths and angles are well known from thousands of
other crystal structures and can be included as known chemical information. Though this
approach limits the number of free parameters, it is still a challenge to find the combination of
parameters which describes the global minimum for the profile R-value (Rwp) or the goodness-
of-fit function. Further limitations in the number of tree parameters and/or the range in which
they can vary could be used to advantage. A structure envelope describes the coarse features of
the molecular crystal structure, and could be used to limit the range of variations of most free
parameters. Furthermore, the structure envelope provides information that can be used to find
optimal starting values for the free parameters that arc close to the global minimum of the
parameter space.
7.4 The program SAFE
To implement the structure envelope approach in a simulated-annealing-controlled fragment
search, the program SAFE (Simulated Annealing and Fragment search within an Envelope)
was developed. It is written in ANSI C and linked to the Sglnfo program library (Grosse-
Kunstleve, 1998) for all space-group information. The simulation of the powder pattern from
atomic coordinates and the lvalue calculations are performed by the XRS (Baerlocher &
71
Hepp, 1982) Rietveld package. Fig. 7-1 shows the flowchart of the complete SAFE procedure.
7.4.1 Input
As input, SAFE needs an initial model. This model is either created using a random set of
torsion angles, orientation angles and fractional coordinates, which determine the position of
the molecule, or the model is built using previously obtained information (e.g. stracture
envelope, chemically reasonable hand-built model). SAFE also produces output files that can
be used as input, so a model obtained from a previous run can be used as a starting model for a
new one. The program uses chemical information such as connectivity, known torsion angle
limits, and typical bond lengths and angles. The cell parameters and the space group must be
specified. To control the simulated-annealing process (see Sect. 7.2.2), some parameters such
as the cooling rate are needed. Furthermore, the measured powder pattern and/or the structure
envelope generated, must be input.
7.4.2 Variation of the trial structure
Bond lengths and angles are constrained to be constant, while the torsion angles, the
orientation and the position of the molecule are varied. In this way parameter space is sampled
to find the minimum of a figure-of-merit function. This model modification is described in
section 7.2.2 in point (3) and (4). Each free parameter is changed independently, using its own
random number (m in Eqn.7-2) to generate a new trial structure for evaluation.
7.4.3 Model construction
Because the crystallographic calculations are not performed in this parameter space, the atomic
coordinates must be transformed into those of the crystallographic frame. In the first step, the
coordinates of the parameter space (internal coordinates: /; $; p) are transformed to those of
the orthonormal frame (*£; y£; z„) using a chain-like description. Starting from the origin of
the orthonormal frame, where the first atom of a molecule, consisting of n atoms, is located.
The second atom has a distance / to the first one. The bond between the two atoms lies along
the x-axis. The third atom is connected to the second one (bond length /2). The bond angle (|)2 is
the angle between the first, the second and the third atom. If a fourth atom is connected to the
third one, rp is the torsion angle around the bond between the second and the third atom. In
general :
ll = bond length (atom.(?'-l),atom(/))
<bl = bond angle (atom(/-2), atom(/-1), atom(/))
if = torsion angle (atom(/'-3), atom(/'-2), atom(/-l), atom(/)).
72
Flow diagram for SAFE
XRS
powder data
preparation
CRYLSP
Inputinitial model
SA control parameters
structure envelope and/or
powder data
Move
random torsion angle, position and
orientation variation
T
Construct model
orthonormal and crystallographiccoordinates
ÏTrial structure chemically
reasonable ?
intermolecular
intramolecular d istances
yes
evaluate fitness
powder data envelope
penalty function
Move accepted ?
Metropolis
no
back to previousposition
-no-
lowering of temperatureafter certain number of
accepted moves
yes
Figure 7-1 Flow diagram for the SAFE program
73
The Cartesian coordinates of the n'th atom (x£; y£; z„) in a chain of atoms can then be
calculated using the following recurrent equation first proposed by Annott and Wonacott
(1966)f \
vC
I( = 0
( X\
Il A 0
V, = o )l°l(7-4)
where the transformation matrix A describes the change of orientation of all following bonds,
caused by the torsion angle r\l and bond angle <j)'
-coscV -sino' 0
sin(j)'costV -cos 0'cos p.7 siniy
^ -sin(j)'ship' cos6.'sinr\l cosfl' )
A1 (7-5)
with Z° = 0; (j)°= (j)1=Tc;ri0= ri1 = rp2 = 0. The result of the multiplication of the matrices
A0 to A" gives the rotation matrix which transforms the vector (/", 0, 0) to the bond between
atom(n-l) to atom(w). The orthonormal coordinates ( xc; yc;zc ) are then transformed to
crystallographic ones (xf',yi',:f) via a transformation matrix (e.g Giacovazzo, 1992).
7.4.4 Checking for a chemically reasonable structure
Once the crystallographic coordinates have been calculated, the chemical sense of the model
can be evaluated. Three tests can be performed:
(1) Intramolecular distance. A change in torsion angle can cause non-bonded atoms of the
molecule to come too close to one another.
(2) Intermolecular distance. The molecule changes its position and orientation within the unit
cell, so symmetry equivalent molecules can come too close to one another (< van de Waals
distance). This test is performed only within one unit cell to limit calculation time.
(3) Translational symmetry. The length of a molecule can clearly exceed a lattice constant. In
this case, an inappropriate orientation of the molecule will violate the translational
symmetry.
Trial structures that do not pass the three tests are omitted. These relatively simple checks can
save significant calculation time in subsequent steps. It is possible that a model is so far from
the true structure, that unreasonable intermediate trial structures are required to get to the
global minimum. In this case the tests can be switched off.
4
7.4.5 Evaluation of the fit to the powder pattern and/or the structure envelope
SAFE provides the possibility of minimizing two functions that depend upon free parameters.
The first one is the weighted profile i?-value (Rwp), which describes the fit of the measured
powder pattern to that simulated from the trial structure.
(7-6)R =
'J^wfyfobs^^yfcalc))^rfyfobs-A
Here each point in the measured pattern yfobs) is compared with that of the simulated data
yfcalc). As weighting factor vv-, = y,(obsyl is used. This calculation is done by the XRS
CRYLSP module (See Fig. 7-1). Using other XRS modules (STEPCO, PEAK, SPRING,
DATRDN), the powder data are prepared for their use in the CRYLSP module.
The special feature of SAFFI is the use of a structure envelope for the calculation of a figure-of-
merit. To do this a value from a penalty function P
0 for s, > sUmit
P = s, - y (7-7)V1 limit '
c ^
i limit nun
is calculated, where s, is the density at grid point i in the map used to generate the structure
envelope, and sjimit is the value of the isosurface used to define the envelope, and smin is the
value s,- of the minimum density in the map. The positions of the atoms of the trial structure are
assigned the nearest grid point. Those grid points that lie inside the structure envelope (i.e. S[ >
slimit) are set t0 De 0- The summation of s, over all atoms gives the penalty value P. The lower
this value, the better the model fits the envelope.
The weighted profile /?-value and the envelope penalty value can either be used separately or
in combination to derive the figure-of-merit of a trial structure. If a stracture envelope is used
to find a favorable starting model for a new SAFE run, P alone is used. In the SAFE run itself,
Rwp and P can be combined to calculate the figure-of-merit
F = w}R + w2P (7-8)
where wj and w2 are weighting factors that can be adjusted depending upon the probability that
the envelope is a useful one. In this way, the minimization can be performed while the
molecule is encouraged to stay within the structure envelope.
75
7.4.6 Acceptance of moves and temperature control
Whether a move is accepted or not is decided by the Metropolis importance sampling
technique, described in section 7.2.2. Rwp, P, or F are then used as figures-of-merit in Eqn. 7-3.
If the move is not accepted, a new model is created from the previous one. Otherwise, the
accepted model acts as the starting point for a new modification. After a certain number of
accepted moves, the temperature (AT) in Eqn. 7-3 is decreased to tighten the acceptance
conditions for subsequent moves. As the temperature decreases and fewer models are
accepted, the possibility of escaping from a minimum is more and more difficult. At the end of
the run, a deep minimum is reached, and it is hoped that it is also the global one.
7.5 Structure of the tri-ß-peptide sa322
Application of the SayPerm procedure to the tri-ß-peptide sa322 (Sect. 4.7) allowed a two-
dimensional stracture envelope to be generated. From this projection along the [001] direction,
the coarse shape and the packing of the molecules could be discerned, but that alone did not
suffice to build a model good enough for a Rietveld refinement. The envelope did not provide
information below a resolution of 2.5A, so many conformations of the molecule could fit
inside the envelope. Consequently, an automatic approach to finding the optimal position of
the molecule within the envelope was needed, and the SAFE algorithm was applied.
Furthermore, there were still two space groups (P2|2j2j and P2]2^2) to be evaluated.
7.5.1 Combination of chemical information with a structure envelope
If all 17 torsion angles in the molecule were to be varied freely in the simulated-annealing
process, a large number of chemically unreasonable models would be produced and computing
time wasted. The geometries of many molecular fragments are known from rules of organic
chemistry and from thousands of structure analyses. It is important to include such information
in the input file for the simulated annealing run. In case of the sa322 molecule, the phenyl rings
and the peptide groups must have a planar geometry and so, the corresponding torsion angles
can be fixed.
Furthermore, the structure envelope suggests that the molecule is more or less straight, so
torsion angles or their combinations that would produce a U-turn or a zig-zag form can be
excluded. Torsion angles, not influencing the approximate form and not limited by chemical
rules, were set to vary freely (Fig. 7-2). Altogether, eight torsion angles were allowed to vary
freely (360°) and nine were varied within a limited range (60°). Because of the straight form of
the molecule and the short r-axis, hydrogen bonding between the molecules stacked in this
direction was expected. In order to obtain a chemically reasonable conformation one requires
76
3603-" 60° 6(7 H ?60°
\^
Figure 7-2 Molecule ot sa322 The bonds maiked by an anow aie varied in the SAFE run The others are
fixed. The torsion angles are varied in a range specified by the dcgiee-numbcrs.
that the oxygens are all on the same side (and therefore the three nitrogens on the other).
The starting model for the simulated annealing run was built by hand. A linear sa322 molecule
was moved and rotated withm the cell, until the molecule fitted the two-dimensional structure
envelope (Fig. 7-3). This was done, using molecular modeling software (Cerius2). The starting
parameters input to the SAFE program are shown in Fig. 7-4.
77
7.5.2 SAFE input file
First the space group P212l2l was assumed because it seemed to allow a better packing of the
molecule in the unit cell.
The molecule, consisting of 41 non-hydrogen atoms, was input in terms of internal parameters
(bond lengths, bond angles, and torsion angles). To do that, an input file similar to that for the
MOPAC-program (Steward, 1993) is used. The parameters are listed in the first part of the
input file (Fig. 7-4). AH parameters pertinent to one atom are given in one line:
(1) Atom name. If the description of the molecule begins with an atom located in the middle of
a chain, it is usually necessary to introduce dummy atoms. These are used to define the
initial bond and torsion angles. All atom names beginning with an upper case "D" are
understood to be dummy atoms and are excluded from the powder pattern calculation.
(2) Bond length. The bond length between the current and the previous atom. In the case of the
first atom the bond length is set to 0.0 À,
(3) Bond angle. This is the angle between the two previous atoms and the current one in a
chain. For the first two atoms, the angle is defined to be 0.0°.
(4) Torsion angle. This is defined by the last three and the current atom. The torsion angles for
the first three atoms are set to be 0.0°.
(5) Torsion angle flag. If this is larger than 0, the torsion tingle is varied. Torsion angles having
the same flag are connected and varied in the same manner. For example, the three torsion
angles around the bond connecting a tri-methyl group (-C-(CH3)3), must be changed
simultaneously by the same value, to maintain the geometry of the group.
(6) Maximum torsion angle change. Here the maximum change in the torsion angle in a single
move is fixed. This depends upon the state of the structure solution process. If the current
parameter is thought to be close to the correct one and just a fine tuning is needed, then this
value should be kept low.
(7) & (8) Minimum and maximum torsion angle. These values define the range in which the
torsion angle can be varied. The starting value should be within these limits.
(9) Previous atom in chain. As mentioned in Sect. 7.4.3. the molecule is described in terms of a
main chain (back bone) and connected subchains. In this column the number of the
preceding atom in the chain is given.
(10)Atom where a ring is closed. It is necessary for SAFE two know at which atom a ring is
closed. This information is used, for example, in the internal distance check.
Using this list, it should be possible to describe any connectivity of an organic molecule. It is
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79
recommended that an atom near the centre of gravity of the molecule is selected as the first
atom in the list of atoms, because this atom is used to define the rotation center and the
position of the molecule. The end of the list is marked with the word "End".
In the next section of the input file, information about the starting conditions and the control of
the model modification is given. The starting position of the molecule is defined after the word
"startingPoint ". This point is the starting position in fractional coordinates for the first atom
(or dummy atom) in the atom list. The maximum translation of the molecule in one move is
given in the "DxDyDz" line. "startRotation" is the starting orientation of the molecule. The
whole molecule is rotated around the first atom in the list. In the next line "RotationRange",
the maximum rotation (in degrees) in one move is given.
The next three lines contain the lattice parameters (CellParameters), the space group
(SpaceGroup) and the region (in fractional coordinates) in which the first atom will be moved
(Assunit). This can be done within the asymmetric unit or another region of the unit cell.
In the next lines, the three geometry checks can be switched on or off. These are the intra¬
molecular distance (intraMoiDistcheck), the packing distance check (PackingDistcheck),
and the translational symmetry check (TransiDistcheck) .The minimum allowed intra- and
intermolecular distance can be fixed (in A) in the "intraMinimalDistance" and
"interMinimalDistance" lines.
In the next line (simulAnneal), five control parameters for the simulated annealing procedure
are given. The first value is the minimum number of moves that must be performed before the
"temperature" is reduced. Multiplying this number by the second value gives the number of
accepted moves that are necessary for a reduction of the "temperature". An inexperienced user
will have difficulty choosing an appropriate starting temperature, so SAFE also performs an
automatic temperature fixing. Starting from a very high test starting temperature, a number of
test moves are performed to obtain an optimal start temperature that produces the acceptance
probability (between 0 and 1) specified in the third value. The test starting temperature is the
fourth value in the line. When a sufficient number of moves, given by the first two values, have
been performed, the temperature is reduced by the fifth value. The last line (surface) defines
the stracture envelope mask. The é?;-value (Eqn. 7-7) of gridpoints having a value higher than
the first value given in this line are set to zero for the penalty function calculation. The second
value defines the weight of the penalty function for the calculation of a figure-of-merit (Eqn.
7-4). If this weight is set to 1.0 the calculation of a weighted profile R-value is switched off.
80
The structure envelope is not used if the "surface" line is omitted. For sa322, the weight for
the envelope penalty value was set to 0.3. For the penalty-function calculation the two-
dimensional envelope was extruded to three dimensions to form a columnar envelope.
Q*
U.W
0.35
iii'fi::,"i\:HjJ
i
-
<N
rt 0.30 W—IlMB ? I.Bii i ihiPib
-
+ timWtM*i ;'
'
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1
y -.%
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20000 40000 60000
accepted moves
80000
Figure 7-5 Trend of the figure-of-mcnl calculated from the weighted profile Ä-value
(Rwp) and the envelope penalty value with the number of accepted moves
yr—
Figure 7-6 Molecule before (light) and after (dark.) the simulated annealing ran
81
7.5.3 SAFE run
First some tests were made to find appropriate parameters for the annealing control. The
temperature was slowly reduced and approximately 80000 moves were performed before the
reduced temperature prevented escape from the minimum and then the figure-of-merit value
was minimized within that minimum. Fig. 7-5 shows the trend of the combined figure-of-merit
(0.7 Rwp + 0.3 P) with the number of accepted moves. The difference between the starting
structure and the final one from the SAFE ran is shown in Fig. 7-6.
7.5.4 Refinement of the crystal structure
The atomic coordinates of the final model obtained from the SAFE run were then refined. Both
the crystallographic and the profile parameters are written to a binary data file that could be
used without change for the Rietveld refinement. As the refinement progressed, it became
apparent that the refinement was more stable if the hydrogen atoms were included in the
model. The final parameters of the refinement are given in Tab. 7-1.
Unit cell
Space group P2,2,2la(A) 61.033
b(A) 11.185
c(A) 5.084
Refinement
29 range (°20) used in refinement 2 -42
Number of observations 2057
Number of contributing reflection 268
Number of geometric restrains 100
Number of structural parameters 123
Number of profile parameters 8
^exp 0.056
KP 0.134
RF 0.128
Table 7-1 Experimental anc structural data for sa322
Fig. 7-7 shows the corresponding Rietveld plot. The relatively high ^-values probably result
from an inconsistency in the data. As mentioned earlier, the sample seemed to change during
the synchrotron measurement. A low temperature data collection might allow this problem to
be circumvented. Tn the refinement only data up to 42° 20 were used, because the data at
higher 29 were not consistent to the those of the low angle region. However the stracture
82
100.0
80.0
60.0
40.0
20.0
0.0
i i t
wS-4. 4~rV ^t~^'ÀJ^yV^Y!,'^---,',-.S -
L» -jt MyJL*Jktök
2.5 7.5 12,5 17.5 22.5
2-theta
27,5 32.5 37.5 42.5
Figure 7-7 Sa322 Rietveld plot
83
refined is very likely to be the correct one. Of course, it cannot be absolutely guaranteed that
the stracture is correct in every detail, but the probability that almost all features are correct, is
rather high. The final refined crystal structure and the structure envelope used in the SAFE run
are shown in Fig. 7-8.
The whole procedure, SAFE run and refinement, had to be repeated, using the other possible
space group P2|2|2, but less satisfactory results were obtained.
84
8 Conclusions
A periodic nodal surface (PNS) that partitions a unit cell into regions of high and low electron
density can be calculated from just the data in a powder diffraction pattern. These surfaces
(structure envelopes) envelop the structure and therefore can be useful in direct-space
structure-determination algorithms.
The reflections needed to generate a structure envelope can be selected by following some
simple rules. The intensities of these low-index reflections can usually be extracted reliably,
because the reflections from the low-angle region of the powder diffraction pattern tend not to
overlap. However, to calculate the envelope, the phases of the corresponding structure factors
are needed.
Sometimes, just the origin defining reflections, whose phases can be assigned arbitrarily, are
sufficient to generate an informative structure envelope, but usually the phases of additional
reflections must be determined. To do this, the computer program SayPerm, which can
estimate the phases of structure factors from low resolution (2.5-3.5 Ä) data, has been written.
It combines the pseudo-atom approach to simulate atomic resolution, the Sayre equation for
phase extension and phase set evaluation, and phase permutation using error correcting codes
(ecc's) to obtain an efficient sampling of the phase space. The SayPerm approach works
reliably for small- to medium-sized structures that can be treated reasonably well as equal
pseudo-atom structures. In such cases, the validity of the Sayre equation can be used as a
figure-of-merit to select the best phase set.
An alternative approach is to (i) permute the phases of seven structure factors using the
Hamming [7.4,31 ecc, (ii) generate a Fourier map from each phase set and (iii) evaluate each
map by eye. If chemical information such as the connectivity of a small organic molecule is
known, there is a good chance of recognizing the coarse features of the structure in one of the
16 Fourier maps.
A structure envelope restricts the volume of the unit cell in which atoms are likely to be
located, so the structure determination process can be accelerated dramatically. Stracture
envelopes used as masks in an exhaustive gridsearch combined with a specialized topology
search, allowed zeolite structures to be solved. For comparison, the searches were repeated
without the envelope mask. The use of the envelope reduced the amount of computer time
required by as much as two orders of magnitude, and in two cases proved to be essential for the
structure solution.
85
The shape of a structure envelope restricts the allowed conformations of a organic molecule
considerably. By combining the envelope with a simulated-annealing-controlled structure-
determination procedure, a previously unknown tri-ß-peptide structure (C32N306H53) with 23
degrees of freedom, could be solved. The envelope could be used to obtain suitable starting
models for the simulated annealing run and/or to accelerate the convergence (fit of the
calculated to the measured powder pattern) during the run.
86
9 Possible developments of the structure envelope approach
The potential of using a structure envelope to facilitate structure determination from powder
diffraction data has been demonstrated in this study. Nevertheless, many improvements in the
method can be envisioned
At the moment, the estimation of phases using the the SayPerm program is only feasible for
centrosymmetric structures. With its efficient phase permutation techniques (error correcting
codes), SayPerm should also be applicable to noncentrosymmetric structures. However,
because of the high number of phase combinations, the selection of the best phase set using
just validity of the Sayre equation as the only figure-of-merit is a problem with
noncentrosymmetric structures. Additional criteria must be found.
If a phase set producing a low R-value is produced and the results of the consistency test are
satisfactory, the further potential of the Sayre equation could be implemented in SayPerm.
After phase extension, a limited number of amplitudes could be extended to stracture factors of
overlapping reflections. At least the information whether a reflection is strong or weak should
be derivable from its relationships to the other structure factors (e.g. Jansen, Peschar &
Schenk, 1992, Dorset, 1997). The sum of the intensities of the overlapping reflections in a
cluster could be fixed during the calculations but the partitioning adjusted. If the Ä-value
derived from the validity of the Sayre equation is decreases after such an amplitude extension,
the new reflection intensities are likely to be closer to the correct ones.
A useful structure envelope divides the unit cell into regions where atoms or fragments of a
structure are likely to be found and those where they are not. Thus, it can serve as a guide in
the evaluation of a Fourier map generated from all stracture factors or be used to perform a
density modification. Theoretically, the envelope can be used directly in the phasing process.
After a SayPerm run, the peaks in a Fourier map (from the best phase set) that lie on the
negative side of the envelope could be omitted or weakened (using a penalty function). This
modified map could then be Fourier transformed in order to obtain a new phase set. Then
more reflections (than in the first SayPerm run) could be selected to have fixed phases in the
next SayPerm run. From the structure factors with fixed phases, a new, more detailed envelope
could be generated and used in the next Fourier map modification. This iterative density
modification and phase extension using a structure envelope mask would be similar to the
Shake and Bake algorithm (Miller, Gallo, Khalak & Weeks, 1994) and the solvent flattening
approach (e.g. Hoppe & Gassmann, 1968) used in protein crystallography.
87
Structure envelopes can be used to find a starting position, orientation and conformation of an
organic molecule for a simulated-annealing run. Nevertheless, for a low-quality envelope or a
very complex structure, this starting model can be ambiguous and several starting models must
be tested. Perhaps, this could be done systematically using error correcting codes (ecc's).
Positional and orientational parameters and torsion angles could be selected and varied under
the control of an ecc to generate starting models. Starting from each of these models, a
simulated annealing run could be performed. The final model that is found most often and
whose powder pattern shows the best match to the measured one, is likely to be correct. For
example, using the Golay[24,12,8] code, 64 positions, 64 orientations, and six torsion angles
with four values each could be varied more or less systematically to give 4096 starting models.
Of course, the parameters for the simulated annealing runs must allow the calculations to be
performed within a reasonable time frame. The computing time could be also reduced by using
a structure-envelope penalty function. Simulated-annealing runs from more than one starting
model are also used in protein crystallography.
Just as a structure envelope is closely related to a molecular envelope approach used in protein
crystallography, some of the methods suggested above are also derived from macromolecular
crystallography. It would definitely be worthwhile evaluating other methods used in this field
for their applicability to powder data more systematically. It may even be possible to use
available or slightly modified software from protein crystallography for this purpose.
88
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Curriculum Vitae
Birth
Leipzig, Germany, July 1, 1965
CitizenshipGerman
Marital Status
single
Education
1972 - 1982 Polytechnische Oberschule, Frankfurt (Oder), Freiberg, Leipzig
1982- 1985 Metal-worker apprenticeship
1985 Abitur
1985 - 1987 Military service
1987 - 1989 Employed in a bookshop
1989 - 1994 Stdent at the Universität Leipzig
1994 Diplom in crystallography
1994 - 1995 Research at the Max-Planck-Gesellschaft in Rostock
I started this thesis research in January 1996
93
Dank
Am Ende dieser Arbeit möchte ich allen danken, die direkt oder indirekt, mehr oder weniger,
bewusst oder unbewusst zu deren Gelingen beigetragen haben. Das sind, neben vielen anderen
alle Kollegen des Laboratoriums für Kristallographie an der Eidgenössisch Technischen
Hochschule in Zürich mit denen die Arbeit grossen Spass gemacht hat,
Dr. Lynne McCusker und Dr. Christian Bärlocher die nicht nur begeisternde Chefs, an die ich
mich nicht nur jederzeit mit Fragen und anderen Dingen wenden konnte, sondern die auch
wunderbare Kollegen waren,
Prof. Dr. Walter Steurer, meinem Doktorvater und Gutachter der Dissertation,
Prof. Dr. Jordi Rius für das Begutachten und die wertvollen Hinweise in der Endphase der
Arbeit,
Dr. Thomas Wessels, Dr. Javier de Onate, mit denen es ein Vergnügen war, das Büro zu teilen,
Dr. Tone Meden, der mich mit bewundernswerter Geduld in die Programme, die für die Arbeit
mit Pulverdaten nötig sind, eingeführt hat,
Prof. Dr. Grämlich, Dr. Torsten Haibach, Dr. Michael Estermann und Dr. Jürgen Schreuer, die
auf unzählige Fragen Antworten gegeben haben,
Dr. Ralf Grosse-Kunstleve, der mir bei der Anpassung des FOCUS-Programms für die
StruktLireinhüllenden prompt und ausführlich behilflich war, und dessen Sginfo- und
Atominfomodule unverzichtbare Bestandteile, der im Rahmen dieser Dissertation
entstandenen Programme, sind,
Prof. Dr. Chris Gilmore, Prof. Dr. Kenneth Harris, Dr. Benson Kariuki, die mir Anregungen zu
Ideen, die wichtig für diese Arbeit waren, seseben haben,
Prof. Dr. Hermann Gies für seine ermunternden Worte und das zur Verfügung stellen der
RUB-3 Pulverdaten,
und noch einmal Dr. Lynne McCusker, für das. ganz bestimmt, nicht immer unterhaltsame
Geradebiegen meiner englischen Ausdrucksweise.
Diese Arbeit wurde vom Schweizer Nationalfond (SNF) unterstützt.