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Automation of Structure Analysis. Part II. Automation of the Evaluation

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ANGEWANDTE CHEMIE
VOLUME 5
* NUMBER 3
MARCH 1966
PAGES 267-330
Automation of Structure Analysis
Part 11. Automation of the Eva1uation’“I
BY PROF. DR. W. HOPPE
ABTEILUNG FUR RONTGENSTRUKTURFORSCHUNG AM MAX-PLANCK-INSTITUT FUR EIWEISSUND LEDERFORSCHUNG, M W C H E N
AND
ABTEILUNG FUR STRUKTURFORSCHUNG AM
PHYSIKALISCH-CHEMISCHEN INSTITUT DER TECHNISCHEN HOCHSCHULE,
M W C H E N (GERMANY)
The extent to which it is possible to automate the determination of a structure from the data
collected with automatic measuring equipment is discussed. The development of the automatic evaluation is synchrt;nous with that of digital computer programs. - Even the simple
calculation o f structure factors from the intensities (i.e. the analysis of the geometrical and
physical intensity factors) takes several weeks when the many thousand reflections are processed “manuaily”, i.e. with a desk calculator. A large electronic computer, on the other
hand, executes these calculations in a few seconds. Nevertheless, even the largest computers
available at present are too small for many of the complex steps in the calculation and for
the quantity of data involved. Evaluation methods which, owing to long computing times,
can at present only be tested on relatively simple structures will become more attractive
with the advent of machines with computing times in the nanosecond range.
Introduction
2. Program Systems
The development of methods for. the evaluation of Xray btructural data can be divided into three steps:
1. Individual Programs for Routine Calculations
The primary aim of the crystallographer was to simplify
the routine work, in order to be able to concentrate on
the important steps of the evaluation. As soon as the
first electronic computers made their appearance,
programs were developed for the calculation of structure factors from an approximation model or for the
calculation of Patterson and Fourier syntheses. It soon
became clear, however, that calculation with the aid of
these independently compiled programs
is laborious,
- since the input formats are different and so involve timeconsuming manual operations (re-punching, short intermediate calculations).
~
[*I For Part I (Automation of the Measuring Process) see Angew.
Chem. 77, 484 (1965); Angew. Chem. Internat. Edit. 4, 508
(1965).
Angew. Chem. internnt. Edit.
/ VoI. 5 (1966) 1 No. 3
The next step, therefore, was the establishment of program systems in which a wide variety of routine programs could be carried out and combined, the only
manipulation required being the insertion of a few
control cards. Table 1 shows the programs that can be
dealt with at present in a program system developed at
our Institute. A system of this type must obviously be
flexible, in order that new programs may be introduced at
any time. The sub-programs themselves can naturally
be obtained from a wide range of sources.
Table 1. Computer programs used in our sqstem.
a) Scaling, Lorentz polarization-correction in various recording methods
b)
statistics
C) Correction for absorption
d) Modification of the structure factors for the Patterson synthesis
e) Patterson and Fourier syntheses
f ) Drawing- Dropram
f o r Patterson and Fourier syntheses
. g ) Structure factor and least squares calculations
h) Bond lengths and angles and other characteristics of the structure
i) Convolution calculations
j) Calculation of convolution molecules
k) Scanning the Patterson structure with convolution molecules
267
3. Automatic Evaluation
It will ultimately be asked whether the programs in a
system can be linked by other sub-programs in such a
way that the structure is worked out automatically from
the results of the measurements. At present, however,
there is no universal process for the determination of
structures by X-ray analysis; there are simply a number
of promising processes, as well as beginnings that could
lead to one or more universal and direct methods
for the determination of structures.
The mathematical problem of the evaluation can be defined by Eq. (1):
m
m
m
In this equation [I], pxyz is the quantity sought, i.e. the
electron density in the unit cell. The maxima of the
electron-density function mark the positions of the
atoms. For the numerical evaluation of this equation
(three-dimensional Fourier synthesis), it is possible to
measure the values of l F h k l l (absolute values of the
structure factors), but not its components Ahkl and B h k l
(obtained by multiplication by cos@ or sin@) (the
“phase problem”;
is the phase difference between
primary and scattered waves). In principle, therefore,
there are an infinite number of ways in which an electron density pxyz can be calculated from a given F distribution. However, only one of these distributions is
chemically meaningful, and that is one involving almost
spherically symmetrical electron-density peaks with
chemically reasonable bond lengths and bond angles [21.
The crystallographer can try to narrow down the
diversity of structures by considering only chemically
meaningful structures, or alternatively he can measure
indirectly some of the ratios A/B by comparing
at least two sets of structure factors. Some of the methods of evaluation are quite involved. However, in the
cases that show the greatest promise for practical applications, the fundamentals can be understood without
calculation, as will be seen below.
It is remarkable that automatic structure determination
became feasible only after three-dimensional methods
(i.e. the calculation of a three-dimensional electrondensity distribution, instead of projections of the
electron density) had gained general acceptance. This
may at first seem paradoxical, since three-dimensional
work involves far more calculation than two-dimensional
work. It was found, however, that the much larger vol_
_
~
~
[ l ] In practice the summation limits of h, k, and I are only of the
order of 10 (instead of m). Owing to the triple summation, however, the range of Fh,k,l is of the order of 1000.
[2] We shall not be concerned here with the problems of homometry (different chemically possible structures with the same set
of absolute values of the structure factors).
268
ume of information available greatly simplifies many of
the problems. Structures which, with two-dimensional
methods, can be solved (if at all) only with skill and
intuition, take an almost predetermined course when
tackled with three-dimensional methods.
It is expedient to carry out the structure analysis in
three steps:
A. Establishment ot an approximation model
B. Development of the final model from the approximation model
C. Refinement of the structure.
The approximation model generally contains only some
of the atoms present (e.g. only the heavy atoms in the
case of the heavy-atom method), and the positions of a
few of these may still be wrong. The final model gives a
qualitatively correct picture of the complete structure;
errors of a few tenths of an Angstrom in the interatomic distances are permissible.
This division is also useful in the so-called direct methods. The object of these methods is the calculation of
the correct A:B ratios (phases) without recourse to
intuitive evaluation steps; however, at least in the present form of the methods, the results are so vague that
they give at best an “approximation model”, the interpretation of which may be rather difficult.
A. Establishment of a n Approximation Model
1. Heavy-Atom Method
The determination of structures is greatly facilitated if
the unit cell contains one or a few heavy atoms, and the
heavy-atom method has for many years been one of the
most successful techniques available to X-ray structural
analysts. It is only natural, therefore, to base our attempted automation on this method.
The positions of the heavy atoms are easy to find, even
in a complex organic structure. This is a consequence of
a special property of the so-called Patterson function
PXYZ [Eq.
The formula for Pxyzis very similar to the Fouriersynthesis formula (1) for the electron density p,,. Unlike pxyzrhowever, P,, can be calculated directly from
the observed results, since Eq. (2) contains only
the absolute values of the structure factors F h k l .
The maxima of the Patterson function Pxyzcharacterize
the lengths and directions of the interatomic vectors
in the crystal [31. Since the intensities of the maxima are
proportional to the product of the numbers of electrons in the atoms joined by the interatomic vector, the
heavy atom-heavy atom maxima must be particularly
high. This will be illustrated for G(2,Y-biphenylylene)[3] Cf. W. Hoppe, Angew. Chem. 69, 659 (1957).
Angew. Chem. internal. Edit.
1 Vol. 5 (1966) J No. 3
Np-cyano-Na-(p-iodophenyl)azomethinimine,
the structure of which was determined at our Institute[4,4al.
The dimensions of the unit cell, the space group, and
the density of the crystal showed that the asymmetric
unit of the unit cell [51 contains only one molecule, and
hence only one iodine atom. Since this iodine atom contains about eight times as many electrons as the light
atoms (hydrogen is completely ignored at first, because
of its low scattering power), the iodine-iodine maxima
should be easily recognizable above the background of
light atom-light atom maxima, being 64 times as intense as the latter. However, owing to the presence of
heavy atom-light atom maxima (which are eight times as
intense as the light atom-light atom maxima) and the
overlapping of maxima in the background, the difference in intensity is actually rather smaller: the iodineiodine maxima, in fact, are 2.5 times as intense as the
highest background maximum of the Patterson structure
(Table 2).
Table 2. Weights (relative intensities) and parameters (x, y, 2 ) of Patterson maxima in the C-(2,2’-biphenylylene)-Np-cyano-Na-(p-iodophcny1)azomethinirnine structure. which contains one iodine and 23 light
atoms in the asymmetric unit. T h e maxima marked with an asterisk
correspond to interatomic vectors betwcen iodine atoms.
Haight or
maximum
Y
I
0.500
0.286
0.219
0.010
0.000
0.037
0.017
0.030
0.000
0.487
I
0.415
0.500
0.088
0.000
0.000
0.500
0.075
0.500
0.082
0.500
I
0.000
-0.935
0.917
0.725
0.267
0.900
0.800
0.667
40*
35*
0.183
8
0.192
8
20*
9
9
8
8
8
As an illustration, sections through the three-dimensional
Patterson structure in the region of the iodine-iodine
maxima are shown in Fig. 1;the intensity of the maxima
can again be clearly seen [61. The parameters of the positions of the iodine atoms in the unit cell are given directly by the positions of these Patterson maxima. However,
this “iodine crystal structure” is an approximation to the
complete crystal structure, in which the light atoms have
been disregarded. The fact that this approximation
model giving the positions of the heavy atoms is sufficient in most cases to permit the successful execution of
the second step, i.e. the development of the final model,
is of fundamental importance (cf. the discussion of step
B of the evaluation).
[4] In collaboration with R . Huisgen, Organisch-Chemisches Institut der Universitat Miinchen.
[4a] F. Brand, Dissertation, Universitat Miinchen, 1965.
[ 5 ] The asymmetric unit of a unit cell is its smallest part that
cannot be further reduced by the symmetry elements of the space
group. It can be shown that the determination of the positions of
the heavy atoms from the heavy atom-heavy atom maxima becomes particularly easy if the asymmetric unit contains only one
heavy atom. This condition is satisfied in most organic heavyatom structures ( e . g . in vitamin BIZ).
161 The contour maps in this figure were drawn by an automatic
function plotter controlled by a digital computer using a
program developed at our Institute.
Angew. Chern. internat. Edit. 1 Vol. 5 (1966)
1 No. 3
Fig. 1 . Three-dimensional Patterson synthesis of C-(2,2’-biphenylylene)Np-cyano-N,-(p-iodo~henyl)azomethiniminc (CF. Fig. 9). Sections in the
neighborhood of t h e heavy atom-heavy atom maxima. The highest
maxima correspond to iodine-iodine vectors. (a) Patter;on m i x i m u m
of iodine in general position (z -- 55/60); (b) and (c) Harker maxima of
iodine (z = 4/60and 0).
This example indicates how the automation of step A
by the heavy-atom method may by tackled. It is necessary
to develop a computer program which will determine the
positions of the maxima of the function Pxyz and classify
these maxima according to their intensities. In another subprogram, which takes into account the symmetry of the
space group, the parameters of the positions of the intense
Patterson maxima are reduced to the heavy-atom parameters.
There are various ways in which this can be done, but these
cannot be discussed in the present paper. Computer programs
of this type are being developed in our laboratory.
We have discussed this example at some length in
order to show that an approach to automatic evaluation
can be achieved by simply translating the operations
carried out by the crystallographer into computer
language. However, since the machine does not develap
“intuition”, a process of this type can be successful only
if no ambiguities that are difficult to resolve are encountered. The logical principle of the process described
for the determination of the heavy-atom parameters
clearly depends on the condition that the heavy atornheavy atom maxima are sufficiently intense in comparison with tl-e light atom-light atom maxima. In our
next example we shall consider a case in which (for the
two-dimensional calculation) this condition is not
satisfied. This example will also show how it becomes
possible to solve the problem when we turn from twodimensional to three-dimensional methods.
Fig. 2a shows a Patterson projection of potassium
p-nitrophenyl-anti-diazotate. Potassium contains about
three times as many electrons as any of the light atoms.
The potassium-potassium maxima should therefore be
nine times as high as the light atom-light atom maxima,
and three times as high as the light atom-potassium
maxima. Although the dominance of the heavy atomheavy atom maxima is not so pronounced as in the last
example, they should be easily recognizable in the
Patterson synthesis, and it should therefore be possible
to determine the positions of the potassium atoms (determination of the approximation model).
However, the situation is complicated by the superposition of non-potassium-potassium maxima in this
Patterson projection, so that the weight of the heavy
269
double weights. These maxima are clearly also the highest
maxima in the Patterson structure. However, the nonHarker heavy-atom m.aximum is again of the same order
of magnitude as twelve other non-heavy-atom maxima,
owing to certain internal symmetries of the structure in
question. In other words, difficulties are encountered
even in a three-dimensional evaluation if there are no
Harker maxima. If (e.g. in the absence of Harker maxima
in otherwise similar cases) there are a definite number of
possibilities, an attempt can be made to resolve the
ambiguity by testing the various possibilities in turn
with the aid of a computer. Only the correct assumption
will give a chemically meaningful structure.
2. The Convolution-Molecule Method
Fig. 2. Two-dimensional Patterson syntheh for potassium p-nitrophenyl-anti-diazotate (cf. Fig. 9).
(a): b-projection; (b): c-projection. The positions of the heavy atomheavy atom maxima are indicated by dots.
atom-heavy atom maxima is only a fraction of that
of the superimposed maxima. The black dots in the
diagram indicate the potassium-potassium maxima.
The contour lines show clearly the presence of “false”
superposition maxima, which are three to four
times as high. The situation is even worse in another
Patterson projection of the same crystal (cf. Fig. 2b).
It is clear that the heavy-atom method in projections
cannot be used in this case, since neither a crystallographer nor a computer would be able to find the potassium-potassium maxima. A skilled crystallographer
might be able to determine the structure by an indirect
method, e.g. by first establishing the orientation of a
benzene ring, but this would involve a great deal of
effort. An entirely different situation is found when we
consider the three-dimensional Patterson structure.
The left-hand column of Table 3 shows the weights of
the potassium-potassium maxima, the remainder of the
Table those of the highest non-heavy-atom maxima. In
this case the symmetry of the crystal gives rise to socalled Harker heavy-atom maxima [71, which have
Table 3. Weights of the highest maxima in a Patterson synthesis of
potassium p-nitrophenyl-anti-diazotate(Harker maxima are indicated
by an asterisk).
* 5131
* 4448
1959
1
2135
, 2117
i!
11
2099
2067
1908
1889
1825
1814
1695
1665
1626
1608
The convolution-molecule method 181 can to some extent be regarded as an extension of the heavy-atom
method. In the latter, the approximation for an organic
molecule is a heavy atom present in this molecule. In the
convolution-molecule method, this function is taken
over by a molecular fragment having a known steric
configuration (e.g. an aromatic ring system). The position and orientation of this portion of the molecule in
the unit cell are found from the Patterson structure.
This “molecular-fragment crystal structure” (as opposed to the “heavy-atom crystal structure”) then forms
the basis for the next step of the structural analysis.
The convolution-molecule method makes use of certain
structural properties of the Patterson function Pxyz.It
can be shown that a molecular fragment having a known
steric configuration gives Patterson maxima with definite
structures. From the mathematical point of view, these
structures are convolution products of the molecular
fragments (convolution molecules). Systematic unravelling of these structures from the Patterson structure, taking into consideration the symmetry of the
space group, yields the orientation and position of the
molecular fragments in the unit cell. The process is
similar in some respects to a jigsaw puzzle. The “picture”
is the Patterson structure, and the “pieces” are the convolution molecules, of which there are two types.
The structure of the first type (convolution molecule with
like indices) depends only on the molecular-fragment
structure, and can therefore be calculated when this
is at least approximately known. By fitting this convolution molecule into the Patterson structure, we find
the orientation parameters of the molecular fragments
which are required for the determination of the second
type of convolqtion molecule (convolution molecule
with mixed indices). If this convolution molecule can
also be fitted correctly into the Patterson structure, we
obtain the translation parameters of the molecular
fragments, and since we know the orientation parameters, we can now find the complete crystal structure
of the molecular fragments. This therefore conclddes
step A of the evaluation (cf. Fig. 3).
[ 8 ] W. Hoppe, Z . Elektrochem. Ber. Bunsenges. physik. Chem.
61, 1076 (1957).
270
Angew. Chem. internnt. Edit. 1 Val. 5 (1966) 1 No. 3
These operations can be carried out graphically in projections [9,101. The use of three-dimensional convolution molecules, however, is very inconvenient. For this reason, the
operations involved in the three-dimensional convolution-
%
Qs
1
1.
3
0
ii
01
01
Fig. 3. The convolution method.
Convolution molecules result from the “convolution” of two molecules.
The principle is shown in Fig. 3a.
molecule method have recently been automated [ I l l . The
rotation and displacement of the convolution molecules
(which are calculated by means of Fourier series, in order to
allow for the finite size of the convolution-molecule maxima)
is carried out systematically by the computer. An interesting
point is the criterion by which the computer judges the fit.
For each orientation and for each translation the convolution
molecules are subtracted at all points in the Patterson space “21
from the overall Patterson function. The sums of the negative
and of the positive differences (or their squares) are then calculated separately. A large negative sum is unacceptable, since
it would mean that the structure in question lacked the required Patterson maxima. On the other hand, a large positive
sum is permissible, since this would simply mean that other
Patterson maxima were present. This could be due to unknown
molecular components, which were not taken into account
in the convolution-molecule evaluation.
As we have shown, the convolution-molecule method
(particularly in the automated three-dimensional form)
can deal with structures of similar complexity to those
studied by the heavy-atom method. In the discussion of
the method of successive Fourier synthesis we shall
describe one of the structures [phyllochlorin ester ( I ) ]
solved by the convolution-molecule method.
This method can also be used for the constitutional
analysis of organic molecules containing no heavy
atoms, as has been shown[131 by the analysis of the
crystal structure of ecdysone, the hormone that controls the pupation of insects. This analysis is remarkable
in that the ‘‘molecular fragment” used was a sterol
skeleton, the folding of which was partly incorrect. In
spite of this the analysis was successful, and the complete chemical constitution, with all the oxygen functions, was found by X-ray analysis. Since the analysis
was carried out on a compound containing no heavy
atoms, the positions of all the atoms were found with
high accuracy, and it was even possible to locate many
of the hydrogen atoms.
Evidently if the “molecular fragment” used in the convolution-molecule method is a heavy atom, this method
becomes the heavy-atom method. In this case the convolution molecules are very simple: they give rise to
single spherically symmetrical maxima. There is no
need to determine an orientation. The programs for the
convolution-molecule method can therefore also be used
to find the positions of heavy atoms. If, for the purposes
of the analysis, the heavy atom has been introduced into
the molecule by means of some large organic group ( e x .
The molecule B is subjected to a parallel displacement to bring it over
each of the vertices of molecule A in return, and the positions of the
atoms OF B are marked (black dots in the drawing). The weight of a point
in the convolution molecule is the product of the weight of the vertex
of A in question a n d the weight of the corresponding point of B. The
molecules in the unit cell of a crystal are generally identical, but their
positions have frequently been altered by symmetry operations.
Figs. 3 b to 3 d show convolution molecules of two molecules which are
identical (3b), linked by a center of symmetry (3c), or linked by a
mirror plane (3d). T h e numbers indicate the multiplicity of coincident
points. I n Figs. 3 b and 3c, rotation of one molecule causes a similar
rotation of the other, and hence leads to rotation of the convolution
molecule with no change i n the structure (convolution molecule with
like indices). In Fig. 3d, o n the other hand, rotation of the molecule
leads t o a change in the structure of the convolution molecule (convolution molecule with mixed indices).
[9] W.Hoppe and G. Wi//,
Z . Kristallogr., Kristallgeometr., Kristallphysik, Kristallchem. 113, 104 (1960).
[lo] M’. Hoppe and R. Rouch, Z. Kristallogr., Kristallgeometr.,
Kristallphysik, I<ristallchem. 115, 141 (1961).
A I I ~ C MChem.
‘.
iiiterircrt. Edit. ) Vol. 5 (1966) / No. 3
[ I 11 R. Huber, Acta crystallogr. 19, 353 (1965).
[12] In order to reduce the computing time, however, the existing
programs are confined to the maxima of the convolution-molecule functions.
[13] R. Huber and W.Hoppe, Chsm. Ber. 98, 2403 (1965).
27 1
introduction of bromine by esterification of a hydroxy
group with bromobenzoic acid), it is expedient to use
the convolution-molecule technique instead of the
heavy-atom method. The approximation model will
then contain the positions of the light atoms of the
organic residue, as well as the position of the heavy
atom, and the prospects for the second stage of the
evaluation will be improved by the improved approximation model.
Another combination of the convolution-molecule
method with the heavy-atom method may be useful
when the asymmetric unit contains several heavy atoms,
and when the geometrical coordination of some heavy
atoms is known. In the structural analysis of 2,3,4,4tetrachloro-1-0x0- 1,4-dihydronaphthaIene, the convolution-molecule evaluation is based on the known configuration of the four chlorine atoms [141.
A more complicated example is shown in Fig, 4a. In
the structure analysis of the cyclopentadienyl-cycloocta-
tetraene-Co complex p-CsHs(CoC5H5)2[151, it was found
that this compound (triclinic, space group Pi) crystallizes with three molecules in the asymmetric unit. The
unit cell therefore contains 12 heavy atoms, so that the
determination of the heavy-atom structure from the 102
heavy-atom maxima in the Patterson function presents a
considerable problem. The problem was solved by
regarding the two cobalt atoms as forming a diatomic
molecule, in which, on chemical grounds and on the
basis of a preliminary examination of the Patterson
synthesis, the distance between these two atoms must be
between 3.5 and 4 A . Evaluation on this basis by the
convolution-molecule method led to the correct heavyatom structure (Fig. 5a), and hence, via several successive Fourier syntheses (cf. Section B.2), to the complete structure containing the 60 atoms of the asymmetric unit. The analysis showed that the cyclooctatetraene ring is in the boat form with alternating double
bonds (cf. Fig. 4b).
3. The Fourier-Transform Method and the
Diffuse-Scattering Method
Fig. 4 b shows the molecular model of U-CsHx(CoCsH&. State of refinement R1 % 20 %; 4500 independent reflections.
The Fourier-transform method is related in some
respects to the convolution-molecule method. If the
molecular structure is known, it is possible to calculate
the scattering diagram of the molecule, and then to try
to match this with the intensity distribution of the X-ray
reflections 1161. This leads to orientation parameters and
(in a somewhat more complicated procedure) to translation parameters. Difficulties are encountered in the
analysis of molecular fragments, since, in contrast to
the convolution-molecule method, the influence of
atoms that are neglected cannot be compensated. This
method is therefore less suitable for constitutional
analysis.
In a recent variant, the Fourier-transform method is
used in the evaluation of thermal diffuse scattering from
molecular crystals 1171. Whereas the crystal reflections
form a discontinuous distribution, the diffuse scattering
gives rise to a continuous blackening on the X-ray
photographs (background scattering component). It has
been shown that the diffuse scattering in molecular
crystals is modulated approximately in accordance with
a superposition of the scattering diagrams of the molecules in the unit cell. We can therefore try to match the
scattering diagrams of the molecules with the distribution of the diffuse scattering instead of with the reflection distribution. Since the distribution is continuous,
the information obtained is more comprehensive. It is
possible, for example, to estimate the sizes and shapes of
molecules before the structural analysis is commenced.
No details can be given in the present paper“71. It is
naturally also possible to combine the Fourier-transform method with the convolution-molecule method.
[I41 W. Hoppe and R . Rauch, 2. Kristallogr., KristallgeDmetr.,
Kristallphysik, Kristallchern. 115, 141 (1961). - The threedimensional refinement of the structure (R % 15 Oh) has not yet
been published.
[ I 51 E. F. Paulus, Dissertation, Universitiit Munchen, 1965.
[16] A. Hettich, Z. Kristsllogr., MineralDg. Petrogr., Abt. A ,
90, 483 (1935); P . P. Ewald, ibid. 90, 493 (1935).
[I71 Cf. W. Ho;ipe ii7 R. Brill: Fortschritte der Strukturforachung
mit Beugunpsmethoden. Friedr. Vieweg, Braunschweiz 1951,
pp. 90-166.
FllL
Fig. 4. The convolution-molecule method in structures with a known
heavy-atom configuration. Fig. 4 a shows the structure of the ‘cCoz
molecules” in w-C8H8(CoCsHS)2 in the three-dimensional Fourier
synthesis of the heavy atoms in a projection along the b-axis. The lines
joining the ends of the “ C O molecules”
~
are dotted.
272
Angew. Chem. internnt. Edit. , Vol. 5 (1966)
No. 3
Since the diffuse scattering provides additional experimental data, a combination of this kind can often
make an analysis much easier (cf. the evaluation of the
diffuse scattering in ecdysone [181).
4. Image-Seeking Methods
In the convolution-molecule method, the Patterson
structure is broken down into sub-structures, which
depend on, but are not identical with, the molecular
structure. Taking an entirely different approach, we can
also regard the Patterson structure as being composed of
n images of the crystal structure which have been
mutually displaced [191. It would obviously be possible
to recognize the crystal structure directly from the Patterson structure, if the Patterson maxima of such an
image could be marked (cf. Fig. 5 ) . This can be easily
done for the Patterson structure of a crystal with a
[1(8151
Fig. 5 . Image seeking in the Patterson function.
T h e Patterson function P(p) of the structure p can be regarJed as resulting i r o n the superposition of identical images of the structure p. I n the
image-seeking method, all the Patterson maxima in the Patterson structure that belong t o any image of the structure (marked by crosses) are
sought.
center of symmetry, if two Patterson structures, which
are mutually displaced by an interatomic vector that
occurs only once (unique), are superimposed 1201. Patterson maxima that coincide belong to the same “image”
of the crystal structure. In the case of acentric structures, however, we first obtain a double image consisting
of the structure and its centrosymmetric image. Superposition of this double image on another Patterson
function, followed by marking of coincident maxima
again gives the structure (or its centrosymmetric image).
The process described is independent of model considerations; its use obviously does not even require a
knowledge of the empirical formula. This is therefore a
direct method, which combines steps 1 and 2 of the
evaluation, and which leads directly to the complete
structure.
Unfortunately, however, this is applicable only to
structures containing small numbers of atoms. If we
consider an organic structure consisting of light atoms,
and with about 100 atoms in the unit cell, it is necessary
to unravel a Patterson structure containing about
loo00 Patterson maxima. Since the Patterson maxima
have a finite volume, it is hopeless to try to separate
these 10000 maxima in the small volume of the unit cell.
[I81 M’. Hoppe and R Huber, Chem. B x . 98,2353 (1955).
[I91 ti is the number of atoms in the untt cell.
[201 In centrosymmetric stiucturej thc,e are vxtors b:t,v;?n
atoms which are linked by a center of symnetry.
Angew. Chem. intertiat. Edit.
i Vol. 5
(1966)
/ No. 3
It is not possible to find unique Patterson maxima[21]
(required at the beginning of the method), nor can
unique Patterson maxima be marked by coincidence as
the positions of atoms in the multiple superpositions.
The situation changes, however, when the structure
contains one or more heavy atoms. The maxima in the
Patterson structure then fall into the height classes
(heavy atom-heavy atom maxima, heavy atom-light
atom maxima, light atom-light atom maxima) discussed
earlier. If single Patterson maxima corresponding to
distance vectors between heavy atoms can be used for
the interpretation of the Patterson structure, it can be
shown that the crystal structure can be found from the
heavy atom-light atom maxima alone [221.
Evaluation by this method has two advantages. First,
it is possible to find the heavy atom-heavy atom vectors
even in complex structures, owing to their great weight,
and second, the number of heavy atom-light atom Patterson maxima is proportional to the number, n, of
light atoms. Since the heavy atom-light atom maxima
are higher than the light atom-light atom maxima, and
since their number is of the same order as the number of
light atom maxima, they should be visible separately,
even in a complex structure, against a diffuse background of the very numerous light atom-light atom
maxima. As we have already mentioned, separability of
the Patterson maxima is essential to the success of
superposition methods. Many relatively complicated
heavy-atom structures have been resolved by this
method.
It is admittedly desirable, even in these cases, t o determine
the positions of the heavy atoms before analysis in the same
manner as in the heavy-atom method. Only then is it possible,
for example, t o decide in limiting cases which Patterson
maxima correspond t o heavy atom-heavy a t o m vectors; this
is d o n e by means of the Harker maxima (cf. Table 3). In
cases of this type, however, the superposition method may be
regarded simply as a special case of the heavy-atom method,
in which the determination of the approximation model
again consists in the determination of the heavy-atom
structure.
Since the reliability of evaluation by superposition increases
with the number of superpositions (as has been pointed out,
at least two superpositions are required in the c i s e of a n
acentric structure), t h e use of this method for structures
consisting entirely of light atoms is more reliab!c when a
large number of unique light atom-light atom vectors are
known. If the structure of a part of the molecule is known,
the positions of the light atoms can be found by the convolution-molecule method. Unique light atom-light a t o m
vectors can then be calculated from these positions, and a n
attempt can be made t o use these as a basis for a structural
analysis by the superposition technique.
To summarize, the use of the superposition method for
complex structures is likely to be successful if the re[21] Multiple Patterson maxima occur when a crystal structure
includes several interatomic vectors of identical length and
direction. Only superpositions on single Patterson maxima lead
to coincidence-marking of double or single images of th? structure.
[22] This condition must be satisfied for each stage of the evaluation. For example, if an acentric structure contains only two
heavy atoms, there is only a single heavy atom-heavy atom vector. Only the first superposition to obtain the double image over
the heavy atom-light atom maxima c m then be carried out. The
same ambiguity is found in the calculation of a Fourier synthesis
phased with the two heavy atons. “Phased” means that the
phase shift @ is taken into account in the structure factor.
273
quired unique Patterson vectors are deduced from a
previous determination of an approximation model (by
the heavy-atom or folded-molecule methods). The
superposition method is then used to find the final model,
and will therefore be discussed once more in Section B.
the fact that the structure factors of the differentiatingatom approximation structure and of the two isomorphous crystals must form a vector triangle. As can be
seen from Fig. 6, there are two possible solutions. This
embiguity can be eliminated by preparation of a third
aomorphous derivative and repetition of the same
visaluation.
5. “Direct” Methods
The special form of the electron-density function of the
crystal (positivity, separability into atomic electrondensity functions) gives rise to relationships between
structure factors, which lead directly to the phases of
the structure. The methods are therefore automatic in
themselves, since the structure can be directly calculated
and interpreted by a Fourier synthesis of the phased
absolute values of the structure factors. Owing to the
unavoidable error in the calculation of the phases, however, this “electron-density synthesis” is often not very
clear. As was pointed out in the introduction it provides
at best the positions of some of the atoms (“approximation structure”). A structure determination by a direct
method must therefore be followed by step B of the
evaluation, although this should not really be necessary.
Relatively little experience has been gained so far with
direct methods. They have mostly been used in projections, and were successful with relatively simple structures. It is possible that more recent work 123-251 and
programming in three dimensions will extend the scope
of the method to include more complex structures.
6 . Indirect Phase-Measurement Methods
The difficulties in the evaluation due to the fact that the
phases are unknown could, of course, best be eliminated
by measurement of the phases. If a direct measurement
with X-rays, based on interference effects, does not
seem to be possible, the phases can be determined indirectly if additional scattering centers are introduced
into the unit cell. It is often not too difficult to introduce
an additional scattering center. For example, it may be
possible to prepare a chlorine derivative and a bromine
derivative in the case of an organic compound. If the
two derivatives crystallize in the same lattice (isomorphism), the scattering factor of the “additional
scattering center” (in the bromine derivative) corresponds to the difference between the scattering factors
of bromine and chlorine.
If the positions of these differentiating atoms in the unit
cell are known, it is possible to calculate the structure
factors fo of this “differentiating-atom approximation
structure”. The phases of the structure factors of the
isomorphous crystals can then be obtained by construction, as shown in Fig. 6. The construction is based on
[23] P . Main and M . M . Woolfson, Acta crystallogr. 16, 1056
(1963).
[24] W. Hoppe, K . Anzenhofer, and R . Huber: Crystallography
and Crystal Perfection - Proceedings of the Symposium held in
Madras, January 14thP18th, 1963. Academic Press, LondonNew York 1963, p. 51.
[ 2 5 ] W . Hoppe, Acta crystallogr. 16, 1056 (1963).
274
-
m
I
Fig. 6 . Indirect phase-measurement by isomorphous replacement
Construction of the vector characterizing the amplitude IFHI and the
phase @H of a structure factor FH of the isomorphous heavy-atom
derivative, using the vector triangle of the structure factor of the heavyatom approximation structure f and of the two measured absolute values
/ F L : and ~ F Hof
( the structure factors of the two isomorphous derivatives. The intersection of the circles drawn through the two end-points
of f with radii ~ F Land
/
IFH/ represents the solution.
The method of isomorphous replacement was first applied to centrosymmetric structures, in which only
one isomorphous replacement is required. In the case
of acentric structures, isomorphous replacement has so
far been used only in the structure analysis of protein
crystals. The successful analysis of these extremely large
and complex structures was made possible by indirect
phase-measurement by multiple isomorphous replacement.
/
Q
Fig. 7. Determination of the differentiating atoms in isomorphous
protein derivatives. Protein [26, 271: erythrocruorin from Chironomus
tltnmmi; molecular weight 16000; space group P 31.
T h e Figure shows the “Patterson projection of the differences” in the
direction of the trigonal axis, calculated from the squares of the differences between the absolute values of the structure factors /Fslof the
protein-HgClz compound (obtained by diffusion of HgC12 into the protein crystals) and the absolute values of the structure factors IF1 of the
protein with no heavy atom. This Patterson synthesis of the differences
(one asymmetric unit is shown) shows the Patterson structure of the
heavy atoms in the protein (apart f r o m a background that depends o n
the protein). Each protein molecule is bonded t o two heavy atoms in
definite positions. The Patterson point-structure of the heavy atomheavy atom maxima is also shown (crosses) for comparison. The circles
indicate the arrangement of the heavy atoms; note the threefold repetition.
~~
[26] R. Huber, H . Formanek, V . Braun, G . Braunitrer, and W.
Hoppe, Ber. Bunsenges. physik. Chem. 68, 818 (1964).
[27] R . Huber, H. Forriinnek, and 0. Epp, unpublished work.
Angew. Chem. internnt. Edit.
;
Vol. 5 (1966) ] N o . 3
The simple geometrical relationships of Fig. 6 can be
easily programmed for each reflection, so that “automatic phase-measurement’’ in accordance with Fig. 6
raises no fundamental difficulties. The determination of
the approximation structure of the differentiating atoms
is not quite so simple; however, this evaluation can be
largely automated. The real problem with the isomorphous replacement method is chemical in nature:
thus it is necessary to prepare isomorphous heavy-atom
derivatives in which not only the isomorphously substituted atoms are correctly located in the unit cell, but
there must be no disturbance of the isomorphism owing
to small displacements of the unsubstituted atoms.
Fortunately, the large holes in protein structures, which
are filled with solvent molecules, can accommodate
quite large heavy-atom groups without disturbance of
the crystal structure (cf. Fig. 7). Nevertheless, the preparation of suitable derivatives is often very difficult, even
in the case of proteins.
Lack of isomorphism generally prevents the determination of phases with a resolution of better than 2 A [*81.
Although a Fourier synthesis based on these data, in
principle, shows all the atoms, this synthesis cannot be
interpreted directly, owing to the low resolution and the
unavoidable phase error. As in the direct methods, it is
necessary to include an evaluation step which, on the
one hand, leads to the Fourier synthesis with all the
absolute values of the structure factors that can be
found by experiment [29J, and on the other, eliminates
the effect of the phase error (determination of the final
model).
The method of anomalous scattering is related in some
respects to the isomorphic replacement method. It can
be shown that the scattering power of heavy atoms in a
structure can be altered by variation of the wavelength
of the primary beam. This change in scattering power
has the same effect as the replacement of the atom by
another atom. To bring about this change, it is necessary
to use radiation having a wavelength close to that of an
absorption edge of the heavy atom. In acentric structures the change in scattering power is accompanied by a
phase change, which provides a possible method for
indirect phase measurement. The measurement is made
difficult by the fact that the effects are small, so that the
measurements must be very accurate. The anomalousscattering method can be automated by a system fundamentally similar to that used for the indirect phase measurement with multiple isomorphous replacement.
Since both the multiple isomorphous replacement
method and the anomalous-scattering method involve
[28] Definition of resolution : T he smallest interplanar distance
down t o which reflections are to be measured. It can be shown
that this limitation of the range of reflections mcasured
limits the sharpness of the Fourier electron-density synthesis. I n
protein crystals, owing to defects in the crystal structure (small
displacements of the atoms of the protein molecule) reflections
can only be measured down to an interplanar distance of about
1.5 A. An electron-density synthesis calculated with the aid of
these reflection data does not lead t o clear resolution of thc clectron-density maxima of light atoms linked by covalent bonds;
it does, however, permit an unambiguous assignment of the
amino-acid residues.
[29] Cf. W . Hoppe and J . Go/j’,nanu, Ber. Bunsenges. physik.
Chem. 68, 808 (1964).
Angew. Chern. iiiterircrt. Edit. , Vol. 5 (1966)
No. 3
measurements on heavy-atom derivatives, these m.ay be
included among the heavy-atom methods in the wider
sense.
B. Determination of the Final Model
1. Image-Seeking Functions
In the discussion of step A of the evaluation, we showed
that the image-seeking functions permit the determination (at least in principle) of the positions of all the
atoms. The method used to check for coincidence of
superimposed Patterson maxima in the two mutually
displaced Patterson structures is important in practice (cf.
Fig. 5 ) . The simplest method is to add the superimposed
Patterson functions. Coincident Patterson maxima then
appear to be intensified (‘harked”) in the picture obtained by superposition (additive image-seeking function).
Another method is to multiply the Patterson functions.
When a Patterson maximum is situated over the background (non-coincidence), one of the factors in the
multiplication is zero, and the Patterson maximum disappears in the picture obtained by superposition. In the
case of coincident Patterson maxima, both factors are
finite, and a maximum therefore apperas in the picture
obtained by superposition. It is easy to see that, in the
second of these methods, the background of non-coincident Patterson maxima is eliminated, at least in principle (product function). A third method is simply to
note the lower of the two values of the superimposed
Patterson functions. In this case the background is
again eliminated, since non-coincident Patterson maxima
lie on the background, the value for which is zero, so
that the value recorded in the picture obtained by superposition is equal to zero (minimum function). All three
methods can be easily programmed for electronic computers. There are therefore no obstacles to their use in
routine work.
It should again be noted that in the case of complex
structures, image-seeking methods are likely to be
successful only if heavy atoms are present, owing to the
overlapping of Patterson maxima.
2. Successive Fourier Syntheses
It we calculate a Fourier synthesis with the absolute
values of the measured structure factors, together with
the phases of an incomplete structural model (an approximation model) - e.g. in the case of a heavy-atom
structure, the phases of a structure in which all the light
atoms are neglected - the Fourier synthesis includes not
only the electron-density maxima corresponding to the
atoms inserted, but also maxima corresponding to some
of the atoms that were neglected in the phase calculation. These atoms can then be included in the next
phase calculation (improved approximation model), and
a further Fourier synthesis should yield the maxima due
to more new atoms. This operation is repeated until
275
the positions of all the atoms have been found. Clearly,
the better the phases of the first approximation model,
the more rapidly will this procedure converge.
It is also obvious without calculation that the more
atoms an approximation model contains, or - in the
heavy-atom method - the higher the scattering power
of the heavy atoms, the better is the model. As an illustration, Fig. 8 shows the maxima found in a Fourier
structure. Thus the scattering pouer of potassium, for example, is sufficient to give the positions of all the light atoms
[311 by a single
in potassium p-nitrophenyl-onri-diazotate
Fourier synthesis (cf. Fig. 9).
In structural analyses by the convolution-molecule
method, the basis for the calculation of the first phases
is the approximation structure of the known molecular
Z=C
fa1
n~
K
101
2-
Fig. 9. Potassium p-nitrophenyl-onii-diazotate.
Fig. 9a shows maxima of a three-dimensional Fourier synthesis phased
with potassium (representation as in Fig. 8 ) . Fig. 9 b shows the threedimensional Fourier synthesis obtained when the anzlysis was completed.
T h e diazotate group in onri-configuration is planar (cf. [32]).
Fig. 8. C-(2,2’-biphenylylene)-Np-cyano-Nc(-(p-iodophenyl)azomethinimine (2).
Fig. 8a shows maxima of a three-dimensional Fourier synthesis calculated with the phases of the iodine-atom approximation structure. T h e
numbers in circles give the relative heights of the maxima; those outside
the circles are the y parameters. Note the low “ghost maxima” and the
deformation of the structure. Fig. 8 b shows the structure obtained when
the analysis was completed.
synthesis of C-(2,2’-biphenyly1ene)-Np-cyano-Na-(piodopheny1)azomethinimine phased only with the iodine
atoms (cf. also Fig. 1). Since iodine has a high scattering
power, this synthesis already contains the positions of all
the atoms. The diagram contains a few extra “ghosts”
and break-off maxima, but these can be readily distinguished on chemical grounds. Besides the maxima in
Fig. 8, figures are given to indicate their heights.
For an organic structure of approximately the same size, viz.
5-anilinopenta-2,4-dien-l
-ylideneanilinium bromide (3) [301,
but containing the less strongly scattering bromine, the
Fourier synthesis phased only with bromine did not contain
the maxima for all the atoms. However, insertion of the
light-atom maxima already found into the calculation of the
(3)
next Fourier synthesis gave the complete solution. The
phase-determining power of a heavy atom depends not only
on its scattering power, but also on the size of the light-atom
Fig. 10. Molecular model of phyllochlorine ester in c-projection (a)
m d in b-projection (b), derived from a three-dimensional Fourier
synthesis; refined structure, R, = 1 5 %. Note the marked difference between the chlorine ring and the idealized planar configuration with fourfold symmetry (not shown) used in the convolution-molecule evaluation.
[3 I ] 0. Epp, Diploma Thesis, Technische Hochschule Munchen,
1965.
[30] H.-J. Springer, Dissertation, Technische Hochschule Mun-
[32] R . Huber, R . Langer, and W. Hoppe, Acta crystallogr. 18,
chen, 1965.
467 (1965).
276
Angew. Chem. internat. Edit.
i Vol. 5 (1966) 1 No. 3
fragment. Successive Fourier analysis, apart from giving
the positions of new atoms, also leads to a better picture
of the molecular fragment, the stereochemistry of which
is generally not accurately known. Fig. 10 illustrates this
effect for the structure analysis of phyllochlorineester r33J.
The “molecular fragment” used in the convolutionmolecule evaluation was a planar chlorine ring with fourfold symmetry. The chlorine ring is obviously anything
but planar: apart from a slight curvature of the large
ring system, the two tetrahedral carbon atoms in positions 7 and 8 are strongly displaced from the average
plane.
In the determination of the structure of ecdysone ( 4 ) (cf.
Section A.2), the coprostane skeleton used was improved so far that the position and orientation of the
double bond of an o$-unsaturated ketone could be
discerned. The case of ecdysone is particularly typical,
since a sterol skeleton that was wrongly convoluted at
first was subsequently corrected by the convolutionmodel method, and since the unknown constitution of
the molecule could be established (with differentiation
between C atoms and 0 atoms) by three successive
Fourier syntheses without the aid of chemical information. Fig. 1 1 shows the three-dimensional Fourier synthesis of ecdysone.
0,
3. Improvement of Phases by Modification of the
Electron Density
Fig. 1 I . Three-dimensional Fourier synthesis of the molecular model of
ecdysone.
As in step A of the evaluation, the replacement of twodimensional by three-dimensional methods greatly increases the reliability. This is due not only to the avoidance of undesirable coincident projections of atoms, but
also to the large quantity of information, which leads to
a partial averaging of the phase error. It must be pointed
out, however, that certain positions of the heavy atoms
can sometimes give rise to a false symmetry in the
Fourier synthesis, which can be very difffcult to eliminate.
The automation of successive Fourier syntheses requires
programs that permit the location of maxima in the electrondensity functions. Programs of this type are relatively easy
to develop, and have recently been introduced into our
program system. In order to be complete, a n iteration program
must also include selection programs, the function of which
1331 W . Huppe, G. Will, J . GaJmann, and H . Weichselsartnri,
unpublished work.
Angew. Chem. internat. Edit.
is to decide which of the maxima obtained are to be interpreted as atoms. This decision can be made by neglecting
maxima with intensities lower than a certain value. It has
been found useful, however, to introduce an apparent occupation density of the atoms, assumed to be propxtional to
the height of the maximum. A variation process applied to
successive calculation cycles involving least squares 1341 not
only leads to an improvement in the position parameters of
the true atoms, but also brings the occupation densities of
these atoms to unity (occupation by a single atom). The occupation densities of maxima that d o not correspond to atoms,
on the other hand, become zero. The new positions found in
this way are used by the computer in the calculation of a new
Fourier synthesis, which is interpreted in a new series of
cycles. Successes achieved by this method have already been
reported [351. The interpretation of Fourier maxima by the
computer may also include the use of stereochemical criteria
in the selection of maxima that correspond to atoms. It is
interesting to note that the trend in these programming tasks
show a steady drift away from classic calculations. Programs
of this type are also known as pattern-recognition programs.
A non-crystallographic example of such a program is the
recognition of hand-written characters. The same characters
written by different persons can differ quite markedly. However, they have certain features in common, which permit
the reader to read a very wide range of scripts. In this case the
computer must imitate decisions that the reader generally
makes subconsciuously. A distorted image of a group of
atoms in a Fourier synthesis containing phase errors can also
be regarded as such a “character”.
1 Vul. 5 (1966) J Nu. 3
Some time ago we reported methods by which the inaccurate phases derived from an approximation model
could be improved without the use of model constructions in the Fourier space. We sought a function which,
when operating on an approximate electron-density
function, would improve the approximation to the true
electron-density function [361. The phases and absolute
values of the Fourier coefficients of the improved electron-density synthesis can then be determined by a
Fourier transformation. Consider a structure in which
all the atoms have the same weight. A better approximation is then obtained by replacement of the absolute
values of the Fourier coefficients by the measured absolute structure factors, since any deviation from the
measured (and hence correct) absolute values of the
structure factors indicates an error. The process is then
applied to this new set of structure factors, and iteration
over several cycles ultimately gives the final structure.
Fig. 12 shows how such functions can be built up. In
principle, the specially constructed function appraises an
electron-density synthesis in the same way as a crystallographer does: new maxima with heights greater than a
certain value are regarded as “probable” atoms, while
maxima below this limit are regarded as “ghosts”. However, the simple yes-no decision is replaced by the more
cautious continuous function. Thus the modified electron-density function contains the new “probable”
[34] See refinement by means of least squares
[35] J. D . Rullett and L. J. Hudgson’ Automatic Methods for
Heavy Atom Phase Determination. Lecture at t h z Laue Symposium, Munich 1962.
[361 W Huppe, R . Huber, and J CaJmann, Acta crystallogr. 16,
A 4 (1963)
277
atom maxima as somewhat higher (but still too low)
maxima, and the “probable” ghost maxima as somewhat lower (but still present) maxima. The relationship
with the method of successive Fourier syntheses is
readily apparent.
The function represented in Fig. 12 has the further
advantage that it can be readily transformed into reciprocal space. It is therefore possible to modify the structure factors directly in order to improve their phases in
I
A
B
A
C
successivecycles without intermediate calculation of electron density syntheses. Fig. 13 shows how well the
method works for structures in which all the atoms are
identical. Fairly small planar test structures have been
chosen in order to minimize the extent of calculations
required.
b12
t
B
a
Fig. 13. Two-dimensional hypothetical test structure (a “methylated
cyclopentadienophenanthrene”) phased with all the atoms, but with
qualitatively incorrect positioning of one of the eighteen atoms. The
false maximum (cross) appears t o be appreciably higher than the true
one (dot). The process of Fig. 10 led t o convergence in eight cycleq, with
complete elimination of the false maximum, and with development of
the true maximum t o the correct height.
The hypothetical test structure in Fig. 13 was phased
with a model in which one out of the eighteen atoms
given was in an entirely wrong position. As can be seen
in the diagram, the maximum corresponding to the
false atom is appreciably higher than the one at the
position of the correct atom, which can scarcely be
distinguished from the background. In considering this
Fourier synthesis, the analyst would be inclined to assume that the structure is correct, and that the only
errors are phase errors due to lack of accuracy of the
coordinates. Only after vain attempts to refine the model
would he realise that his structure must contain a
qualitative error. Even in this case, however, the new
phase-improvement process converges.
The programs, originally designed for two-dimensional
calculations, were subsequently extended to three
dimensions. It was found that even much poorer approximation models lead to convergence in threedimensional calculations. In one case convergence was
obtained with a structure phased with only four out
of ten identical atoms. Moreover, the four given atoms
were displaced by about 0.2 A from their true positions.
Investigations on a similar process for heavy-atom
structures are in progress.
t.03
0
1
05
I
A
5
Ibi
A
1 0 Q-
B
C
3
B
,r
Q
0
10
0.5
Icl
+
Fig. 12. Phase improvement by modification of the electron density.
(a) The result of a Fourier synthesis phased with only some of the atoms
present is shown in the one-dimensional example. The atoms used (A)
appear t o be excessively high, while the atoms (B) omitted are too low.
There is also a background present ( C ) (the correct structure is shown).
(b) A function p * suitable for improvement of the structure
(c) The modified Fourier synthesis obtained on conversion of p into p * .
It can be seen that the excessively high maxima (A) have been lowered,
the low maxima (B) have been raised, and the background ( C ) has been
reduced. It is to be expected that the phases of the structure in Fig. 1 2 c
will be better than those of the structure in Fig. 12a. The corresponding
calculptions can be carried out automatically in iterated cycles by the
convolution process, without intermediate calculation of the Fourier
synthesis.
278
C. Refinement of the Structure
The result of a successful Fourier synthesis is a set of
atomic coordinates which still contain slight errors.
Methods of refining the structure, most of which are
regularly used in a semi-automatic form, were developed
some time ago. We shall deal here only with those that
are particularly suitable for automatic calculation, in
which the experimental values are fitted by the method
of least squares.
Angew. Chem. internut. Edit.
i Vol. 5 (1966) I/ No. 3
1. Least-Squares Refinement of Atomic Parameters
3. Other Methods of Refinement
The equation for the structure factor is:
N
FZ
-
2
++
--f
f j exp 2 xi (h, r, -t A rj)
(3)
j = l
After this equation has been linearized by the method of
least squares, the A< can be found by solution of a
system of linear equations, and the difference between
(Fh)exp and (Fh)& can then be minimized.
The atomic coordinates, the scaling factor, and the
thermal factors of the individual atoms may be used as
parameters. Anisotropic thermal vibrations can be
readily taken into account, and the method of least
squares has the advantage that the residual error of the
varied coordinates can be determined in a satisfactory
manner. The process and the associated computing
programs have for some time been an essential part of
structural analysis.
2. Refinement of Phases
This process has only recently been described 1371, and
like the method of phase correction by modification of
the electron density (Section B.3), has so far been successfully tested only for structures consisting of identical
atoms. In structures of this kind. each structure factor
It has been shown that the refinement methods of difference Fourier synthesis and the method of steepest
descent are related to the refinement of the atomic coordinates using the method of least squares. The difference Fourier synthesis method is often used in structure
analysis. In this method the differences between the
calculated and experimental structure factors are used to
calculate a Fourier synthesis, the course of which in the
vicinity of the atom maxima gives an indication of the
corrections to be applied. In principle, the method of
least squares is completely analogous, but in this case
the gradients of the Fourier synthesis are considered
instead of the atom maxima. It cannot be said whether a
gradient of this kind is subject to any disturbing effects.
In the difference Fourier synthesis, however, the course
of a function is considered over the entire space. It may
therefore be possible to recognize any abnormalities in
the course of the function. For example, if an atom is
omitted in the calculation of the structure factors, the
missing atom appears in the correct place in the difference Fourier synthesis. A qualitatively wrong atom
gives a deep minimum at the wrong position and a
Fourier maximum in the correct position. Like the
phase-refinement method, therefore, difference Fourier
synthesis can also lead to qualitative corrections. No
attempt has yet been made to carry out an automated
difference Fourier synthesis, but this is possible with
the aid of principles similar to those discussed in connection with the automatic evaluation of Fourier syntheses.
D. Closing Remarks
can be represented as the sum of all the other structure
factors, as in Eq. (4). This equation is very similar to
that for the structure factor from the contributions of
the individual atoms present in the structure (3), with
the product F h F h - w of the structure factors in place
of the form factors f j of the atoms.
A particularly interesting point is that - again purely
formally - the coordinate-dependent exponential functions exp 2xi(hrj + Arj) are replaced by ones depending
on the phases of the structure factors. Owing to the
form,al similarity of the two equations, it is possible, in
principle, to use the same mathematical process based on
the method of least squares. This process, however,
no longer leads to correction of the error in the atomic
coordinates Arj, but to the correction of the phase error A@. We have successfully used the new method for
a number of simple test structures. It should be noted
that phase refinement also leads to the appearance of new
atoms and the disappearance of wrong ones in the improved Fourier synthesis. The method gives similar
results in this respect to those obtained by the method
described in Section B involving modification of the
electron density.
[37] W. H o p p e , Z. Kristallogr., Kristallgeometr., Kristallphysik, Kristallchem. 118, 122 (1963); W. Hoppe, R . Huber, and J .
Ga/3mnnn, Acta crystallogr. 16, A 4 (1963).
Angew. Chem. internut. Edit.
/ Vol. 5
(1966)
1 No. 3
Crystal-structure analysis has not yet become a n automatic
process, and it may take some time before structures will be
solved by simply running off a computer program. It is also
likely that only some of the structural problems that arise
will be capable of being solved automatically. However, Xray analysis will undoubtedly become more important in
chemistry, and particularly in constitutional analysis. A
constitutional formula will then frequently form the starting
point of an investigation instead of the end result. It will also
be possible, as is the case in absorption spectroscopy, to
carry out parallel studies on related compounds in a reasonable
time, and so base conclusions not only o n the absolute values
of the bond lenghts and bond angles, but also o n changes in
these parameters.
E. Appendix
Since completion of the manuscript new results have been
obtained, particularly in phase correction by modification
of the electron density[381. Fig. 14 shows the Fourier synthesis of a n acentric, two-dimensional test structure comprising 22 carbon and 2 phosphorus atoms calculated from
the phases of the P atoms. After 30 cycles of phase correction
the Fourier synthesis shown in Fig. 15 was obtained, in which
the arrangement of atoms can be clearly seen (the electron
densities due to the phosphorus atoms have been subtracted).
[38] J . GuJmann, Dissertation, Technische Hochschule Munchen,
1966.
279
Fig. 14. Two-dimensional Fourier synthesis of a hypothetical 24-atom
test structure (a naphthalene-type ring system connected by a carbon
chain to a five-membered ring; short side chains) phased with two
phosphorus atoms. Plane space group pg. The distorted form of the
diagram makes chemical interpretation very difficult.
Three-dimensional phase correction was successfully tested
on two real organic structures: Firstly, on phyllochlorine
ester, C33N402H38, where intitial phasing was m a d e on t h e
basis of a n idealized, plane chlorine ring model, the orientation a n d position of which had been determined by the convolution molecule method (cf. also Fig. 10). Secondly, on
the rubidium salt of t h e “sulfolipid” RbSCg010H17 1391
Fig. 15. Two-dimensional Fourier synthesis of the same structure as
in Fig. 14 after 30 phase-correction cycles. The locations of the two
phosphorus atoms used for initiel phasing are marked by asterisks.
(initial phasing by t h e heavy-atom technique with R b a n d S).
In both cases strong, spurious maxima were introduced into
t h e initial Fourier syntheses on account of symmetry elements; in the course of the phase correction these maxima
disappeared.
Received: July 26th. 1965
[A 481/284 IE]
German version: Angew. Chem. 78, 289 (1966)
Translated by Express Translation Service, London
[391 Y. Olcaya, Acta crystallogr. 17, 1276 (1964).
Some Advances in the Inorganic Chemistry of Bromine [*I
BY DR. V. A. STENGER
SPECIAL SERVICES LABORATORY, THE DOW CHEMICAL COMPANY,
MIDLAND, MICHIGAN (U.S.A.)
Recent advances in the chemistry of bromine and its inorganic compounds are covered
under the headings of technology, properties and reactions, liquid bromine as an inorganic
solvent, analysis, and new or newly studied compounds. Much of what is new in bromine
chemistry is characteristic also of what is new in inorganic chemistry as a whole. That is,
more detailed information about compounds and reactions is becoming available jrom the
application of new instrumental and theoretical techniques.
Technology
Bromine is obtained from ocean water and natural
brines, and to a lesser extent as a by-product in the processing of sodium and potassium chlorides from salt
deposits. Some attention has been given to new ways of
isolating bromine, including ion exchange
and solvent extraction[21, but these have not come into com[ * ] Presented in part at a symposium on the inorganic chemistry
of the halogens, held by the Chemical Society (London) at its
anniversary meeting at the University of Birmingham, England,
April 7th, 1964.
[ l ] R. F. Hein, US.-Pat. 3037845 (June 5th, 1962); 3174828
(March 23rd, 1965); F. J. Gradishar and R. F. Hein, US.-Pat.
3098716 (July 23rd, 1963); L. C. Schoenbeck, US.-Pat. 3075830
(Jan. 29th, 1963); 3 101250 (Aug. 20th. 1963), E. I. duPont de
Nemours.
[2] M . Israel, A. Baniel, and 0. Schachter, Israel Pat. 15408
(Oct. 25th. 1962).
280
mercial use. Practically all of the current production
from ocean water is made by chlorination and volatilization, followed by the collection of bromine either in
sodium carbonate or sulfurous acid solution [3,41. From
the latter, bromine is recovered by rechlorination and
distillation. It may be possible to avoid the need for
chlorine in this step by a new process [51 in which a trace
of butyl nitrite or an alkali metal nitrite in an acidic
medium serves as a catalyst for oxidation by oxygen or
air at 0 to 200 O C and pressures of 1 to 4 atmospheres.
[3] A. G. Sharpe in J . W. Me//or: Comprehensive Treatise on Inorganic and Theoretical Chemistry. Longmans, Green and Co.,
London 1956, Vol. 2, Suppl. 2, p. 689.
[4] V . A . Stenger in Kirk-Othmer: Encyclopedia of Chemical
Technology. 2nd Ed., Wiley, New York 1964, Vol. 3, p. 759.
[5] W. A. Harding and S. G. Hindin, US.-Pat. 3 179498 (April
20th, 1965), Air Productsand Chemicals, Inc., and Northern Natural Gas Co.
Angew. Chem. internat. Edit.
1 Vol. 5 (1966) No. 3
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