computer programs
786 doi:10.1107/S0021889807029238 J. Appl. Cryst. (2007). 40, 786–790
Journal of
Applied
Crystallography
ISSN 0021-8898
Received 8 March 2007
Accepted 14 June 2007
# 2007 International Union of Crystallography
Printed in Singapore – all rights reserved
SUPERFLIP – a computer program for the solution
of crystal structures by charge flipping in arbitrary
dimensions
Luka
´
s
ˇ
Palatinus*‡
2
and Gervais Chapuis
Laboratoire de Cristallographie, Le Cubotron, Ecole Polytechnique Fe
´
de
´
rale de Lausanne, 1015 Lausanne,
Switzerland. Correspondence e-mail: palat@fzu.cz
SUPERFLIP is a computer program that can solve crystal structures from
diffraction data using the recently developed charge-flipping algorithm. It can
solve periodic structures, incommensurately modulated structures and quasi-
crystals from X-ray and neutron diffraction data. Structure solution from
powder diffraction data is supported by combining the charge-flipping algorithm
with a histogram-matching procedure. SUPERFLIP is written in Fortran90 and
is distributed as a source code and as precompiled binaries. It has been
successfully compiled and tested on a variety of operating systems.
1. Introduction
Modern crystallographic computing methods require significant
computational effort and are therefore useful only in combination
with a computer program that implements them. This is especially
true for the algorithms that use iterative procedures and computa-
tionally expensive methods like the discrete Fourier transform on
large grids.
One such algorithm is the charge-flipping method, which is a
structure solution method for the reconstruction of scattering
densities from diffraction data. Since its publication (Oszla
´
nyi &
Su
¨
to , 2004, 2005), it has been successfully applied to a wide range of
crystallographic problems, including the solution of simple structures
(Wu et al., 2004), structures with high pseudosymmetry (Oszla
´
nyi et
al., 2006), modulated structures (Palatinus, 2004; Zun
˜
iga et al., 2006;
Palatinus et al., 2006) and quasicrystals (Katrych et al., 2007), and
structure solution from powder diffraction data (Baerlocher, Gramm
et al., 2007; Baerlocher, McCusker & Palatinus, 2007).
The algorithm itself is surpisingly simple. In its original version it
has been developed for the reconstruction of electron density from
X-ray diffraction data. The electron density is described by discrete
sampling on a regular grid in the unit cell. The structure solution
proceeds in iterative cycles. The iteration is initialized by assigning
random phases to the experimental structure amplitudes. From this
trial solution an electron density is calculated by inverse discrete
Fourier transform. This electron density is modified so that all grid
points with density below a certain positive threshold are multiplied
by 1 (flipped). New temporary structure factors are calculated by
discrete Fourier transform of this modified density. The phases of
these temporary structure factors are combined with the experi-
mental amplitudes and this set of structure factors enters the next
cycle of iteration. is the only parameter of the algorithm.
A modification of the algorithm has been developed (Oszla
´
nyi &
Su
¨
to , 2005) that improves the probability of convergence by a special
handling of the weak reflections. If a reflection is considered weak,
the calculated amplitude of its temporary structure factor is retained
and its phase is shifted by =2. This induces an additional perturba-
tion in the phase space and leads to an improved performance of the
algorithm. For more details on the charge flipping algorithm, see
Oszla
´
nyi & Su
¨
to (2004 or 2005). Recently, the algorithm was gener-
alized to the reconstruction of scattering densities that are not strictly
positive, such as those from neutron scattering (Oszla
´
nyi & Su
¨
to ,
2007).
The important ingredient of the algorithm is that it does not use the
symmetry of the structure during the density reconstruction. The
density is treated as if it had symmetry P1 and the symmetry in the
density is induced only by the symmetry in the diffraction pattern.
This makes the structure solution insensitive to possible ambiguities
in the determination of the space group. On the other hand, the
density reconstructed by charge flipping is randomly shifted and the
origin of the space group must be located afterwards.
In order to allow the wide crystallographic community to take
advantage of the new algorithm, a computer program named
SUPERFLIP has been created. The name is derived from ‘charge
FLIPping in SUPERspace’, because the program allows for density
reconstruction in arbitrary dimensions, thus making it possible to
solve standard periodic structures and aperiodic structures within the
same framework. The program provides a self-contained structure-
solution tool that can be used for automatic structure solution of
simple structures as well as custom-tailored solution of complex
problems.
2. Program description
The principal input to SUPERFLIP is one input file, which contains
the instructions for the program. The program reads in the input file,
processes the instructions found therein and produces the output.
User interaction with the program during the runtime is possible, but
is limited only to changing several parameters of the algorithm. The
principal output of the program is the reconstructed scattering
density. The progress of the program is recorded in a log file.
The input file is in a free format and it contains keywords followed
by one or more values. The only information that must be always
contained in any input file is the lattice parameters, the expected
‡ Permanent address: Institute of Physics, Academy of Science of the Czech
Republic, Na Slovance 2, 182 21 Prague, Czech Republic.
symmetry of the structure, the dimensions of the grid and the list of
reflections. The program has many other options which allow a
detailed control of the execution; however, default values work in
most cases.
The program run can be divided into three major steps:
(a) Reading the data and checking for consistency. The program
reads the instructions from the input file. It checks the consistency of
the symmetry operations, namely if each matrix represents a valid
rotation matrix and if the operations form a complete space group. In
the next step, the list of reflections is expanded according to the given
symmetry to produce a full list of reflections in the whole sphere. If
Friedel pairs are present in the input file, their intensities are aver-
aged, because the charge-flipping algorithm does not take the
anomalous scattering into account and therefore the Friedel pairs are
not independent even in a non-centrosymmetric structure. At the end
of the data processing, the coverage of reciprocal space as a function
of sin = can be calculated to provide the user additional means of
checking the completeness and consistency of the data and symmetry.
(b) Charge flipping. This is the central part of the program. The
expanded list of reflections is assigned random starting phases and
the charge-flipping iteration is started. The cycle is repeated until
convergence is reached or until the maximum allowed number of
iteration cycles is exceeded. The crucial step in this stage is the proper
selection of , which will be discussed in x2.1.
(c) Searching for the proper position of the symmetry operations in
the density. It has already been mentioned that charge flipping
reconstructs the electron density without taking the space-group
symmetry into account. However, the symmetry is contained in the
reflection intensities and possibly in the systematic absences. Thus,
the resulting electron density will exhibit, albeit approximately, the
correct space-group symmetry, but with an origin arbitrarily placed in
the unit cell. SUPERFLIP therefore searches the resulting density for
the location of the symmetry operations and, optionally, averages the
electron density according to the symmetry. The details of this
process are described in x2.2.
In the following sections some issues of particular importance for
the performance of the program are discussed in detail.
2.1. Searching the proper value of d
The flipping threshold is the only free parameter of the original
algorithm and its selection is critical for the success of the algorithm.
No method is currently known for predicting the optimal value of
a priori, as it depends in a complicated manner on the resolution of
the data, the thermal parameters and the type of atoms present in the
cell. Fortunately, extensive testing of charge flipping on a variety of
structures has allowed us to design an empirical method for deter-
mining .
The method is based on the observation of the behaviour of the
total charge in the current density and the amount of the flipped
charge. The total charge in the map c
tot
is obtained as a sum
P
i
over
all grid points, or simply as the Fourier coefficient Fð0Þ of the current
density. The total flipped charge c
flip
is defined as c
flip
¼
P
i
<
j
i
j,
where the summation runs over all density pixels i with
i
<. It has
been observed that for proper values of the ratio c
tot
=c
flip
tends to lie
close to 0.9 and this value is independent of the input data, in
particular independent of the grid size, data resolution, symmetry and
unit-cell volume.
This method was implemented in SUPERFLIP in the following
way. A trial value of is selected so that with this value 80% of the
pixels of the initial random density will be flipped. Ten iteration cycles
are performed with this trial value in order to pass the few transi-
tional iteration cycles at the beginning of the iteration and reach a
stable plateau. After ten cycles the ratio c
tot
=c
flip
is checked. If it falls
in the interval h 0:8; 1:0i, the current value of is accepted and the
iteration continues. Otherwise is decreased, if c
tot
=c
flip
< 0:8, or
increased, if c
tot
=c
flip
> 1:0, and the procedure is repeated with the
modified . Needless to say, the user also has the possibility of
defining manually by giving its value in the input file.
2.2. Symmetry of the reconstructed density
A reliable and fast location of the position of the symmetry
elements in the density is crucial for the performance of any charge-
flipping program that has the ambition of being widely used. The
result of the charge-flipping iteration is a density that is randomly
shifted in the unit cell. Moreover, the density obeys the correct space-
group symmetry only approximately. While such density already
contains the structural information, it would be very inconvenient to
interpret it. This is especially true for densities with dimensions
higher than three. Thus, a method must be found to determine
automatically the offset of the origin of the density from the
conventional origin of the space group, so that the density can be
appropriately shifted and then averaged over the symmetry-equiva-
lent grid points.
Perhaps somewhat surprisingly, it is not trivial to determine the
origin shift, especially if the method is required to work in arbitrary
dimensions. In the following paragraphs we describe the method that
is implemented in SUPERFLIP.
Let fRjg be a symmetry operation described with respect to the
conventional origin of the space group. A density
sg
, which contains
this symmetry operation and whose origin is placed at the conven-
tional origin of the space group, fulfills
sg
ðRr þ Þ¼
sg
ðrÞ. A shift of
the density from the conventional origin by a vector s will produce a
density
sh
with
sh
ðrÞ¼
sg
ðr sÞ: ð1Þ
The symmetry operation fRjg is transformed by the shift s to
fRj þ dg with d given by
d ¼ðI RÞs; ð2Þ
where I is the unit matrix. Thus, if the origin shift s is unknown, we
can use d to determine s. Because ðI RÞ is in general not invertible,
a single symmetry operation might not be sufficient to determine all
components of s.
The above expressions are valid if the density
sh
is perfectly
symmetrical with respect to the symmetry operation fRj þ dg.
However, the output of the charge-flipping algorithm is a density
cf
that fulfills the symmetry only approximately. Because it is shifted
from the conventional origin of the space group by an unknown shift
vector, it can be considered an approximation to the density
sh
defined above. The vector d can be found in
cf
by finding the
maximum of the correlation integral between
cf
ðrÞ and
cf
ðRr þ þ dÞ as a function of d. The correlation integral is given by
CðdÞ¼
Z
cf
ðrÞ
cf
ðRr þ þ dÞ dr: ð3Þ
As can be immediately seen from this equation, the correlation
function CðdÞ is given by the convolution
cf
ðrÞ ?
cf
½ðRr þ Þ.
Using the convolution theorem, the correlation function can be
expressed as
CðdÞ¼F
1
½FðHÞF
0
ðHÞ
; ð4Þ
where FðHÞ and F
0
ðHÞ represent the structure factors of
cf
ðrÞ and
cf
ðRr þ Þ, respectively, denotes complex conjugation and F
1
the
computer programs
J. Appl. Cryst. (2007). 40, 786–790 Palatinus and Chapuis
Program for charge flipping 787
inverse Fourier transform. From the properties of the Fourier
transform it follows that F
0
ðHÞ is related to FðHÞ by
F
0
ðHÞ¼FðHRÞ expð2iHÞ: ð5Þ
Using equations (4) and (5), the correlation function CðdÞ of the
discretized electron density can be evaluated efficiently and quickly
by the fast Fourier transform of the product of the structure factors.
The position of the absolute maximum of CðdÞ defines the best
estimate for the vector d, modulo any lattice translation vector.
To determine the position s of the conventional origin of the space
group, it is sufficient to locate a common origin for a set of generators
of the space group. For each generator g
i
¼fR
i
j
i
g; i ¼ 1; ...; N
gen
,a
corresponding vector d
i
can be found using the method described
above. We then obtain a set of equations for the origin shift s in the
form
d
i
þ t
i
¼ðI R
i
Þs; i ¼ 1; ...; N
gen
; ð6Þ
where t represents any lattice translation vector. If t
i
could be
omitted, equation (6) would represent a set of D:N
gen
linear equa-
tions (D being the dimension of the electron density, normally three)
for the components of the origin shift s. These equations could be
solved by standard methods of linear algebra. However, the infinitely
many translation vectors t
i
cannot be omitted, because each of them
corresponds to an equivalent, but in general different, position of the
generator g
i
. A combination of t
i
that leads to a common solution s
cannot be found a priori. Luckily, the number of possible t
i
can be
restricted, because at least one solution for s lies in the first unit cell,
i.e. we have for the components of s, s
j
2h0; 1i; j ¼ 1; ...; D. With
this restriction, the infinite number of possible translation vectors t
i
for each g
i
can be reduced to a finite, and usually small, number N
trans
i
vectors, such that equation (6) with these translation vectors substi-
tuted for t
i
has a solution in the first unit cell. A common solution s
can result from any combination of choices of the translation vectors,
leading to
Q
N
gen
i¼1
N
trans
i
sets of linear equations. A solution that satis-
fies any of these sets of equations is a possible space-group origin s.
The set of equations (6) can be overdetermined, and in such a case
s must be determined by a least-squares procedure. If the set of
equations (6) is underdetermined, any of the infinitely many solutions
represents an acceptable origin shift.
In practice, the number of generators and the number of accep-
table translation vectors are usually small, and therefore also the total
number of sets of equations
Q
N
gen
i¼1
N
trans
i
is small. Moreover, it is not
necessary to solve all sets, because the calculation can be stopped
as soon as the first solution is found. As a result, the procedure
described in this section is rather fast and takes only a few seconds or
even a fraction of a second on an ordinary computer.
2.3. Handling of higher-dimensional cases
It has been shown by Palatinus (2004) that the charge-flipping
algorithm can be generalized to incommensurately modulated cases
in a straightforward manner. Similarly, it has been shown by Katrych
et al. (2007) that charge flipping can be used for the solution of
structures of quasicrystals. Both these types of structures are usually
described in a higher-dimensional space (superspace), and the real
structure in physical space is obtained as a three-dimensional section
of this higher-dimensional structure. The number of dimensions is
four or more for incommensurately modulated structures, five for
decagonal quasicrystals and six for icosahedral quasicrystals. The only
modification to the original charge-flipping algorithm is that for
higher-dimensional structures the reflections are indexed by D inte-
gers (D > 3) instead of three, and the scattering density is described
on a D-dimensional discrete grid. The space-group symmetry must
correspondingly be defined by the D D rotation matrices and D-
dimensional translation vectors, as is usual in the field of modulated
structures and quasicrystals. Note that the method for the symmetry
search described in x2.2 is completely independent of the dimen-
sionality of the structure, and can be used directly for the higher-
dimensional cases.
In order to be as general as possible and adopt any type of
structure, even those yet undiscovered, SUPERFLIP is written so
that the number of dimensions can be completely arbitrary. This is
true both for the total dimensionality of the structure and for the
dimensionality of the physical space. Thus, SUPERFLIP can be also
used to solve one- and two-dimensional problems, possibly incom-
mensurately modulated, as well as any other problem of any
dimensionality.
3. Additional features of the program
The program offers a broad range of features that are extensively
described in the user manual. Here, we give only a summary of the
most frequently used options that have not been described above,
together with a short description.
(a) Apart from the charge-flipping algorithm, SUPERFLIP can
also use the low-density elimination method (Shiono & Woolfson,
1992). This method is closely related to charge flipping, but differs in
the density-modification step.
(b) The reflection input can have variable format. Any of the
following items can be used: intensity, amplitude, real and imaginary
components of the structure factor, intensity/amplitude + phase. The
order of the entries can be also defined by the user. This makes it
possible to import virtually any reflection list without any modifica-
tions.
(c) The user has the option of providing a reference density.
SUPERFLIP will then shift the reconstructed density so that it
matches the reference. This is indispensable if several calculations are
to be compared directly, because the space group usually has more
possible origins in the unit cell, but these origins are not equivalent
points of the structure. Thus, the structures derived from different
charge-flipping runs need not necessarily have the same origin, and if
they need to be compared directly they must be shifted to a common
origin using a reference density.
(d) The reconstructed density can be tested for the presence of
user-defined symmetry operations. Because the density is recon-
structed in P1, it is possible to analyse it for the presence of different
symmetry operations. Using this option it is, for example, possible to
distinguish a mirror plane from a glide plane or to test for the
presence of an inversion centre.
(e) The value of that is optimal for convergence is not optimal for
obtaining a density as noise-free as possible. Therefore, it is possible
to follow the ‘structure-solving’ iteration by several cycles of ‘density-
polishing’ iteration, which suppresses the noise in the density
considerably and leads to better definition of the weak features and
more symmetrical densities.
(f) Difficult structures sometimes need several attempts before the
iteration converges. In extremely difficult cases, the quality of the
reconstruction can also vary from one calculation to another.
Therefore, it is possible to repeat the whole calculation automatically
many times and save only the best results.
(g) It turns out that the symmetry analysis of a density can be
interesting independently of the charge-flipping process. Therefore,
SUPERFLIP can perform a symmetry analysis of a user-supplied
computer programs
788 Palatinus and Chapuis
Program for charge flipping J. Appl. Cryst. (2007). 40, 786–790
density without performing the charge-flipping iteration. The
symmetry analysis can be performed on any density distribution
supplied in one of the formats supported by SUPERFLIP.
(h) Charge flipping can be used not only as a structure-solution
method but also as a structure-completion method (Oszla
´
nyi & Su
¨
to ,
2004). In practice this means that the starting phases of the structure
factors need not be random, but may correspond to a partial structure
model. Charge flipping can then be used for structure completion as a
very efficient alternative to the usual Fourier recycling. It is possible
to provide the non-random starting model either as a model density
or by specifying the phases of the input structure factors.
(i) SUPERFLIP supports the band-flipping variant of the charge-
flipping algorithm (Oszla
´
nyi & Su
¨
to , 2007), which allows for the
solution of structures with negative scattering densities, mainly from
neutron scattering experiments.
(j) The standard version of the algorithm uses a fixed value of
during the whole iteration. It has been suggested by Wu et al. (2004)
that a dynamic could be used instead. In this version, is rede-
termined in each cycle so that a constant proportion of the density
pixels are flipped. SUPERFLIP supports both the static and the
dynamic handling of .
(k) SUPERFLIP supports the histogram-matching procedure to
enhance the performance of the structure solution from powder
diffraction data. The procedure uses an expected histogram of the
density to modify the density during the iteration and to repartition
the intensities of the overlapping reflections. For a detailed descrip-
tion of the procedure and an overview of the solved structures, see
Baerlocher, McCusker & Palatinus (2007).
4. Output description
The reconstructed density can currently be saved in three formats.
The first is the binary ‘m81’ format of the crystallographic computing
system JANA2000 (Petr
ˇ
ı
´
c
ˇ
ek et al., 2000) which can be directly read by
JANA2000 and viewed by its plotting module CONTOUR. The
second supported format is the X-PLOR format. X-PLOR is a
software package for structural biology (Bru
¨
nger, 1992). This is an
ASCII format that can be easily transformed into any other format if
needed. This format can be read, for example, by the software
package CHIMERA (Pettersen et al., 2004) to produce three-
dimensional isosurface plots of the density. The third supported
format is the m80 format of JANA2000, which contains a list of the
structure factors corresponding exactly to the density. This format is
very concise in terms of file size. It can be used by the FOURIER
module of JANA2000 to produce arbitrary sections through the
electron density. The m81 format is designed for densities up to six
dimensions. The X-PLOR format has been generalized to arbitrary
dimensions, but only its three-dimensional version can be read by
other software. The m80 format can adopt structures with a
completely arbitrary dimensionality. Finally, it should be noted that
additional formats can be added at any time upon request.
5. Analysis of the output density
SUPERFLIP itself does not perform any analysis of the resulting
electron density in terms of locating the atomic positions and
assigning the elements to the atomic maxima. However, a separate
program, EDMA, exists that can be used to perform the analysis.
EDMA was originally developed as part of the BayMEM suite (van
Smaalen et al., 2003) to analyse electron densities obtained by the
maximum entropy method, but it has also been extended for the
analysis of densities obtained by SUPERFLIP. EDMA analyses the
density for maxima and assigns atomic types to these maxima
according to the qualitative or quantitative chemical composition
supplied by the user. EDMA can output the result of the analysis as a
CIF, an INS file for the SHELX program suite or an m40 file for
JANA2000. EDMA can also analyse the electron density of modu-
lated structures.
6. Software and hardware requirements and availability
The program is written in standard Fortran90 and can thus be
compiled and run on any system with a working Fortran90 compiler.
The program requires a FFTW3 library for the fast Fourier transform
(Frigo & Johnson, 2005; http://www.fftw.org). This library must be
installed on the system before SUPERFLIP is compiled. SUPER-
FLIP also makes use of the LAPACK linear algebra library
(Anderson et al., 1999; http://netlib.org/lapack/). However, the
necessary code is contained in the installation package of SUPER-
FLIP and the LAPACK libraries do not have to be installed sepa-
rately.
Apart from being available as a source code, precompiled binaries
of SUPERFLIP are available for the operating systems Windows and
MacOS X. These binaries are self-standing and do not need any
supplementary libraries to run.
The program does not have any special hardware requirements.
The program requires typically several MB or tens of MB of memory
to handle common three-dimensional structures. However, the
required memory can exceed 1 GB if very large structures are
considered, especially large modulated structures or quasicrystals.
SUPERFLIP is freely available for download at http://superspace.
epfl.ch/superflip. The complete installation package contains the
source code, documentation and sample input files. Precompiled
binaries for Windows and MacOS X are also available for download.
The same webpage also allows the download of the density-analysis
program EDMA as source code or precompiled binaries. Please note,
however, that despite being available on the SUPERFLIP web page,
EDMA is not part of SUPERFLIP.
7. Conclusions
SUPERFLIP is a computer program for the solution of crystal
structures using the charge-flipping method. The program has been
applied to a broad range of crystal structures, including complex
modulated structures and quasicrystals. It has been also used to solve
several structures from powder diffraction data, some of them
extremely complex (Baerlocher, Gramm et al., 2007; Baerlocher,
McCusker & Palatinus, 2007). The aim of the program is to provide
an alternative to the established structure-solution software. The
program is designed so that it can solve simple structures fully
automatically, but at the same time provide sufficient versatility and
transparency to allow for the solution of complicated cases. One of
the strengths of the program is the way of analysing the symmetry of
the reconstructed density.
SUPERFLIP, in combination with the density-analysis program
EDMA, provides a self-contained structure-solution system. In order
to facilitate user interaction with these programs, an interface for
SUPERFLIP and EDMA is available in the crystallographic
computing system JANA2000 (Petr
ˇ
ı
´
c
ˇ
ek et al., 2000) and in the
CRYSTALS package (Betteridge et al., 2003). These interfaces
provide a convenient way of creating the input files for SUPERFLIP
and EDMA.
computer programs
J. Appl. Cryst. (2007). 40, 786–790 Palatinus and Chapuis
Program for charge flipping 789
We would like to acknowledge the cooperation with Christian
Baerlocher, who provided many ideas and valuable feedback, espe-
cially concerning support for powder diffraction. We thank Dieter
Schwarzenbach and Marc Schiltz for fruitful discussions concerning
the symmetry-search algorithm. Feedback from Michal Dus
ˇ
ek and his
help with the production of precompiled binaries is also gratefully
acknowledged. We are also grateful to Arie van der Lee for his
comments and for writing the interface for the CRYSTALS package.
We have truly appreciated all comments and suggestions from all
users of SUPERFLIP. This work was supported by the Swiss National
Science Foundation, grant No. 20-105325, and by the Grant Agency
of the Czech Republic, grant No. 202/06/0757.
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computer programs
790 Palatinus and Chapuis
Program for charge flipping J. Appl. Cryst. (2007). 40, 786–790