electronic reprint
Journal of
Applied
Crystallography
ISSN 1600-5767
Modelling pair distribution functions (PDFs) of organic
compounds: describing both intra- and intermolecular
correlation functions in calculated PDFs
Dragica Prill, Pavol Juhás, Martin U. Schmidt and Simon J. L. Billinge
J. Appl. Cryst. (2015). 48, 171–178
c International Union of Crystallography
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J. Appl. Cryst. (2015). 48, 171–178
Dragica Prill et al. · Modelling PDFs of organic compounds
research papers
Journal of
Applied
Crystallography
ISSN 1600-5767
Modelling pair distribution functions (PDFs) of
organic compounds: describing both intra- and
intermolecular correlation functions in calculated
PDFs
Received 28 July 2014
Accepted 1 December 2014
Dragica Prill,a Pavol Juhás,b Martin U. Schmidta* and Simon J. L. Billingeb,c*
This article is dedicated, in memoriam, to our
beloved colleague Professor Dr Erich F. Paulus.
# 2015 International Union of Crystallography
a
Institute of Inorganic and Analytical Chemistry, Goethe University, Max-von-Laue-Strasse 7, 60438
Frankfurt am Main, Germany, bCondensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA, and cDepartment of Applied
Physics and Applied Mathematics, Columbia University, New York 10027, USA. Correspondence
e-mail: [email protected], [email protected]
The methods currently used to calculate atomic pair distribution functions
(PDFs) from organic structural models do not distinguish between the
intramolecular and intermolecular distances. Owing to the stiff bonding
between atoms within a molecule, the PDF peaks arising from intramolecular
atom–atom distances are much sharper than those of the intermolecular atom–
atom distances. This work introduces a simple approach to calculate PDFs of
molecular systems without building a supercell model by using two different
isotropic displacement parameters to describe atomic motion: one parameter is
used for the intramolecular, the other one for intermolecular atom–atom
distances. Naphthalene, quinacridone and paracetamol were used as examples.
Calculations were done with the DiffPy-CMI complex modelling infrastructure.
The new modelling approach produced remarkably better fits to the
experimental PDFs, confirming the higher accuracy of this method for organic
materials.
1. Introduction
The knowledge and understanding of material structures at
the atomic scale has been fundamental for numerous discoveries in almost all natural sciences throughout the past
century. An understanding of the atomic arrangement within
the material gives one an opportunity to predict its properties.
For crystalline materials, this has been possible since the
pioneering work of Laue (Friedrich et al., 1913) and the
Braggs (Bragg & Bragg, 1913). However, when the structures
of interest are not periodic, or the order exists only on a short
length scale, the powerful tools of crystallography break down
(Billinge & Levin, 2007). Many modern technological applications require the use of materials with varying degrees of
disorder. However, significant information can be obtained
from total scattering experiments through the analysis of the
atomic pair distribution function (PDF). This method has been
used to study glasses and liquids for about 80 years (Debye &
Menke, 1930; Warren, 1990; Egami, 1990). Recently, the PDF
method has proven to be very successful in the structural
analysis of complex inorganic materials such as poorly crystalline, nanocrystalline, disordered and even well ordered
compounds (Billinge & Kanatzidis, 2004; Proffen & Kim, 2009;
Brühne et al., 2008). The PDF, GðrÞ, gives the probability of
finding pairs of atoms separated by a distance r. It is experimentally obtained by a Fourier transform of the corrected and
J. Appl. Cryst. (2015). 48, 171–178
normalized total scattering structure function SðQÞ. The SðQÞ
function is obtained by removing self-scattering from the
coherent scattered intensity per atom, IðQÞ, and dividing by
the average squared atomic scattering factor, h f ðQÞ i2 ,
according to (Egami & Billinge, 2012)
SðQÞ ¼
IðQÞ h f ðQÞ2 i
þ 1:
h f ðQÞ i2
ð1Þ
Finally, the reduced pair distribution is obtained from (Farrow
& Billinge, 2009)
GðrÞ ¼ ð2=Þ
QRmax
Q½SðQÞ 1 sinðQrÞ dQ
ð2Þ
Qmin
¼ 4r½ðrÞ 0 0 ðrÞ:
ð3Þ
Here, ðrÞ is the microscopic pair density, 0 is the average
number density and 0 is the characteristic function of the
diffracting particles. For bulk samples, 0 has a value of 1, but
it has an r dependence for nanoparticles or individual molecules. 0 is the autocorrelation function of the particle shape
and may be measured directly by small-angle scattering
(Farrow & Billinge, 2009). The magnitude of the scattering
vector, Q, is given by Q ¼ 4 sin =, where is half of the
scattering angle and the radiation wavelength (Egami &
Billinge, 2012).
doi:10.1107/S1600576714026454
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171
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Most PDF studies found in the literature to date deal with
inorganic materials (Billinge & Kanatzidis, 2004; Billinge,
2008; Young & Goodwin, 2011). More recently, the PDF
method has been gaining traction in the study of molecular
materials in crystalline and amorphous organic compounds.
The studies can be split into two classes, where the PDFs are
obtained over a very limited Q range (Sheth et al., 2005;
Newman et al., 2008) and a wide Q range (Nollenberger et al.,
2009; Schmidt et al., 2009; Billinge et al., 2010; Dykhne et al.,
2011). It is now understood that the very limited Q range of
the early studies results in peaks in the PDF that do not appear
at the positions of atomic pair separations in the material
(Dykhne et al., 2011), making them of limited value and poorly
controlled as it is difficult to apply the corrections uniquely.
However, high-quality PDFs can be readily obtained from
organic compounds and it is expected that PDF analysis will
become an important tool in organic materials studies. It is
therefore important to be able to model accurately the PDFs
from molecular systems, which is the topic of this paper.
The PDF analysis of organic samples currently consists of
qualitative and quantitative comparisons of different data sets,
using the PDF curve as a fingerprint in the comparison of local
structures in crystalline, nanocrystalline and amorphous
samples (Billinge et al., 2010; Dykhne et al., 2011). By visual
inspection of the curves, it is possible to obtain qualitative
information about the arrangement of neighbouring molecules, packing patterns and the sizes of crystalline domains
(Schmidt et al., 2009). The fingerprinting consists of comparisons between two measured PDFs from a reference and a test
sample (Billinge et al., 2010), or a calculated reference PDF
with a measured test sample. Differentiation between cocrystals and mixtures is also possible, with some phase quantification of mixtures feasible (Davis et al., 2013). However, to
this point, quantitative modelling of molecular crystals has not
been widely explored (Rademacher et al., 2012).
The PDF can be understood as a summation of atom–atom
contributions between all pairs of atoms i and j within the
structure model (up to a maximum distance). Each contribution is weighted corresponding to the scattering power of the
two involved atoms i and j. In the experimental PDF data, the
peaks have a certain width as a result of displacements of
atoms from their average position as well as from the finite Q
range that is used to Fourier transform the data. Prior PDF
studies on inorganic materials have shown that the peaks at
low r are systematically sharper than the peaks in the high-r
regions. This is due to the correlated motion of atoms (Jeong et
al., 1999, 2003): strongly bonded atoms tend to move together,
dependent on each other. These motional correlations tend to
die out smoothly with increasing distance in inorganic solids
and are therefore relatively easy to handle in modelling
programs by applying an r-dependent peak broadening
correction (Proffen & Billinge, 1999). Far-neighbour atoms
move independently of each other and their motion is
uncorrelated, resulting in broader PDF peaks.
When working with molecular compounds, handling the
correlated motions becomes more complicated. In molecular
systems, there are a variety of bonding interactions that differ
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Dragica Prill et al.
in strength by orders of magnitude: covalent bonds (very
strong and rigid), hydrogen bonds (less strong), electrostatic
interactions (less strong) and van der Waals interactions
(weak). The atoms on the same molecule do not vibrate
independently, since they are bonded to their neighbouring
atoms with strong and rigid covalent bonds. On the other
hand, the molecules in the crystal are mostly held together by
less strong electrostatic interactions, hydrogen bonds and soft
van der Waals interactions or even with only van der Waals
interactions. Therefore, the width of the PDF peaks can be
vastly different for pairs of atoms located in the same molecule
versus atoms located in two different molecules. This presents
a special challenge to model self-consistently the PDF data
from molecular solids over a wide range of r. Here we present
a rather straightforward workaround that will work with
existing modelling codes whilst producing high-quality fits.
We will use three examples to demonstrate the new
procedure: (1) naphthalene (C10 H8 ; Cruickshank, 1957), (2)
-quinacridone (C20 H12 N2 O2, Pigment Violet 19; Paulus et al.,
2007) and (3) paracetamol (C8 H9 NO2 ; Bouhmaida et al.,
2009). Naphthalene and quinacridone are completely planar
molecules with known structures determined from singlecrystal X-ray diffraction. Naphthalene is mainly used as a
precursor to other chemicals. The molecules are held together
by van der Waals interactions only (see Fig. S1 in the
supporting information1). Quinacridone is the most important
pigment for red–violet shades. Four phases are known to date
(I , II , and ). The red–violet phase is used for the
coloration of plastics and coatings (Herbst & Hunger, 2004).
In quinacridone the molecules are connected by van der Waals
interactions and hydrogen bonds. In the phase, each molecule is bound to two neighbouring molecules by two N—
H O C hydrogen bonds each. One half of the resulting
chains are parallel to the ½110 direction and the other half to
the ½11 0 direction. In the ½010 direction the molecules stack
on top of each other ( stacking). This results in a high
number of close van der Waals contacts supported by
Coulomb interactions between the partial charges of the
molecules. In the ½001 direction there are weak van der
Waals interactions between C—H groups only (see Fig. S2)
(Paulus et al., 2007).
Paracetamol is a widely used mild analgesic and antipyretic.
Its structure was also determined from single-crystal X-ray
diffraction. The paracetamol molecule is not completely
planar. The amide group is inclined to the benzene ring by
22.60 . Along ½100 the molecules are connected by N—
H O C hydrogen bonds between the amide groups.
Additionally, O—H O C hydrogen bonds connect the
molecules in the ½001 direction, leading to a zigzag sheet
parallel to (010). These sheets are stacked along the b axis (see
Fig. S3).
Molecules with more internal degrees of freedom than in
our examples will substantially increase the complexity of the
systems and thus the interpretation of the PDFs.
1
Supporting information for this article is available from the IUCr electronic
archives (Reference: PO5017).
Modelling PDFs of organic compounds
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J. Appl. Cryst. (2015). 48, 171–178
research papers
It is important to be able to model quantitatively both intraand intermolecular effects since important physical information is present in both signals. For example, the temperature
dependence of PDF peak widths in the intramolecular PDF
will give information about the lattice dynamics. It is observed
in IR or Raman spectra that intramolecular vibrations have
generally considerably higher wavenumbers than intermolecular vibrations (phonons), corresponding to stronger
force constants and smaller vibrational amplitudes, and
information about this is contained in sharp low-r PDF peaks
but not in crystallographic Debye–Waller factors. On the other
hand, thermal expansion of a molecular crystal with increasing
temperature is caused by increases of intermolecular distances
Figure 1
Observed powder pattern of naphthalene (molecular structure as inset).
due to the anharmonicity in the intermolecular vibrations. The
molecular geometry changes only slightly with temperature
(Oddershede & Larsen, 2004). This can also be studied using
the PDF.
We illustrate this phenomenon with naphthalene, whose
molecular structure and powder diffraction pattern are shown
in Fig. 1 and whose measured PDF is shown in Fig. 2.
To gain insight into how intramolecular and intermolecular
atom–atom distances appear in the PDF, we define the largest
intramolecular atom–atom distance as l and the shortest
intermolecular atom–atom distance as i. For the case of
naphthalene, i ’ 3:5 Å. In the r region below i ’ 3:5 Å, the
PDF curve contains only intramolecular atom–atom distances
with sharp narrow peaks. In the high-r region beyond l ’ 5 Å
the PDF curve contains only intermolecular atom–atom
distances which yield broader peaks in the PDF curve. In the
region between i and l the sharp intramolecular and broad
intermolecular peaks coexist, preventing the clean separation
of the signals based simply on r.
This region is especially challenging to model using ‘smallbox’ approaches such as PDFgui (Farrow et al., 2007), where
PDF peak widths are determined by convolving the structure
with Gaussian-like peaks to account for the thermal motion.
The usual r-dependent peak-width methods fail. This is clearly
evident in the fit shown in Fig. 3, where a traditional modelling
approach has been attempted using the PDFgui program.
It is clear that a new modelling approach has to be developed to be able to describe self-consistently the low-, intermediate- and high-r regions correctly for organic samples. The
new procedure needs to distinguish between the intra- and
intermolecular atomic pairs so that the PDF peaks from each
can be evaluated using different atomic displacement parameters. Here we present a straightforward way to calculate the
Figure 2
Experimental PDF of naphthalene. In the r region below 3.5 Å only
intramolecular C C distances are contributing, in the region
3.5 Å < r l (l is the largest intramolecular atom–atom distance)
intramolecular and intermolecular distances overlap, and in the r region
beyond l only intermolecular distances are contributing to the PDF curve.
In the case of bigger molecules, the range of overlapping distances is
much larger.
J. Appl. Cryst. (2015). 48, 171–178
Figure 3
Experimental PDF (blue circles) of naphthalene. The calculated PDF
(red line) of the structural model was calculated using the standard
method, refining one thermal parameter (Biso ), one r-dependent peakwidth parameter 2 and lattice parameters. This is suitable for fitting the
broad peaks in the high-r region. The corresponding difference curve is
shown in green below.
Dragica Prill et al.
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Modelling PDFs of organic compounds
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PDF more accurately by using separate isotropic displacement
models for intra- and intermolecular distances. In contrast to
Rademacher et al. (2012), we do not use a supercell model.
2. Experimental
2.1. X-ray powder diffraction and data preparation
Naphthalene and paracetamol were purchased from Sigma
Aldrich (99% purity) and used without further purification.
-Quinacridone was obtained from Clariant GmbH. X-ray
powder diagrams of all samples were measured at 300 K at the
X17A beamline of the National Synchrotron Light Source at
Brookhaven National Laboratory, using a two-dimensional
PerkinElmer amorphous silicon detector. Samples were
packed in cylindrical polyimide capillaries 1 mm in diameter.
The capillaries were sealed with clay at both ends. A monochromatic incident X-ray beam 0.5 by 0.5 mm in size was used,
conditioned using an Si(311) monochromator to have an
energy of 67.42 keV ( = 0.1839 Å). The detector was
mounted orthogonal to the beam path with a sample-todetector distance of 204.2 mm, as calibrated with an LaB6
standard sample. Multiple scans were performed on each
sample to achieve a total exposure time of 30 min. The twodimensional diffraction data were integrated and converted to
intensity versus Q using the software FIT2D (Hammersley et
al., 1996) (Figs. 1, 4 and 5). For the PDF analysis the data were
corrected and normalized (Egami & Billinge, 2012) using the
program PDFgetX3 (Juhás et al., 2013) to obtain the total
scattering structure function, FðQÞ, and pair distribution
function, GðrÞ. The data were truncated at a finite maximum
value of the momentum transfer Qmax , which was optimized to
avoid large termination effects whilst maximizing the signalto-noise ratio. The values Qmax = 19.5 Å1 for naphthalene,
Qmax = 19.0 Å1 for quinacridone and Qmax = 18.7 Å1 for
paracetamol were found to be optimal.
2.2. Structural models
The structural models used for the calculations were taken
from the literature: naphthalene from the work of Cruickshank (1957), -quinacridone from Paulus et al. (2007) and
paracetamol from Bouhmaida et al. (2009). The corresponding
reference codes in the Cambridge Structural Database (Allen,
2002) are NAPHTA11 for naphthalene, QNACRD05 for
quinacridone and HXACAN27 for paracetamol. During the
calculations and refinements, for simplicity the H atoms were
not taken into consideration, which is not expected to affect
the fits too much owing to the weak scattering of H atoms.
3. Method development
3.1. Standard method for calculating the PDF using an
r-dependent peak broadening
For a known structural model, the PDF is calculated using
the relation (Egami & Billinge, 2012)
"
#
1 X X fi f j
Gc ðrÞ ¼
ð4Þ
ðr rij Þ 4r0 :
Nr i j6¼i f ðQÞ 2
The sum iterates over all pairs of atoms i and j separated by
distance rij within the structural model. The scattering powers
of atoms i, j are fi, fj ; h f ðQÞi is the average atomic scattering
factor and N the number of atoms in the structural model.
Equation (4) generates peaks at every position r where the
structural model has pairs of atoms separated by this distance.
In order to model the peak broadening due to the atomic
displacement parameter, caused by displacement of atoms
from their average position due to thermal motion and static
disorder, the ðr rij Þ function in equation (4) is convolved
with a Gaussian-like profile function. To account for a limited
Qmax range in the measurement, the curve is also convolved
with the Fourier transform of a step function terminated at
Qmax (Proffen & Billinge, 1999). In the standard approach,
Figure 4
Observed powder pattern of -quinacridone (molecular structure as
inset).
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Dragica Prill et al.
Figure 5
Observed powder pattern of paracetamol (molecular structure as inset).
Modelling PDFs of organic compounds
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J. Appl. Cryst. (2015). 48, 171–178
research papers
there is no difference in the atomic displacement parameter
value for PDF peak widths of atom pairs within one molecule
and thermal vibrations of atoms belonging to two different
molecules. This is not an issue for conventional powder
diffraction, which does not distinguish between intramolecular
and intermolecular motion, but it is a very important effect in
the PDF, as evident in Figs. 2 and 6. The latter figure shows the
calculated PDF in the case where the correlated motions are
not taken into account.
3.2. New approach to calculate PDFs of molecular systems
Here we describe a simple approach that will allow us to
include both the sharp intramolecular and broad intermolecular PDF peaks self-consistently in the model. The
model uses two different isotropic displacement parameters B,
one small B value if the PDF peak is between atoms that
belong to the same molecule (Bintra ) and a larger one if the two
atoms belong to different molecules (Binter ). This description
covers the low-r range (r < 3.5 Å, only sharp peaks), the
medium-r range (3.5 Å < r l, mixture of sharp and broad
peaks) and the large-r range (r > l, only broad peaks) with
only one additional parameter. We use the program DiffPyCMI (http://www.diffpy.org; Farrow et al., 2010) for the
calculations. Since the program does not a priori distinguish
between intra- and intermolecular distances, we use a superposition of multiple PDFs that are calculated with different
displacement parameters as follows:
(i) calculation of the PDF of a single molecule, Gm (i.e. of
the intramolecular distances only), using the small isotropic
displacement parameter Bintra ,
(ii) calculation of the PDF of the total crystal, Gc (all
distances), using the larger isotropic displacement parameter
Binter suitable to fit the high-r peaks,
(iii) calculation of the PDF of a single molecule, G0m (i.e. of
the intramolecular distances only), using the same larger
isotropic parameter as in (b),
(iv) summation of the calculated PDFs using the relation
Gtot ¼ Gm þ Gc G0m .
A schematic of these calculation steps is shown in Fig. 7.
PDF refinements were conducted as usual, i.e. the lattice
parameters, isotropic displacement parameters and scale
factor were all refined. Note that in the new approach it is
important to fix to zero (no effect) all the correlated motion
parameters such as and 2 that are available in PDFgui.
Atom positions were kept constant for either protocol to
ensure that the occurring differences arise because of different
displacement parameters.
4. Results and discussion
Figure 6
Experimental PDFs (blue circles) of (a) naphthalene, (b) quinacridone
and (c) paracetamol. The calculated PDFs (red lines) of the corresponding structural models were calculated using the standard method,
applying one thermal parameter (Biso ), one r-dependent peak-width
parameter 2 and lattice parameters. The corresponding difference curves
are shown in green below.
J. Appl. Cryst. (2015). 48, 171–178
The experimental PDFs of naphthalene, quinacridone and
paracetamol were fitted first using PDFgui and standard
approaches used to fit inorganic compounds and second using
the new modelling approach described above. The Rw values
clearly show a significant improvement in fit using the new
approach, as evident in Fig. 8 and reported in Tables 1–3.
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Table 1
Summarized results of PDF analysis for naphthalene.
Bintra represents the isotropic displacement parameter for atom–atom
distances within a molecule and Binter the isotropic displacement parameter
for atom–atom distances between two molecules. Rw is the PDF fit residuum.
Space group
a (Å)
b (Å)
c (Å)
( )
2
Biso (Å2)
Bintra (Å2)
Binter (Å2)
Rw
GoF
Crystal data
Standard approach
New approach
P21 =a
8.235 (5)
6.003 (10)
8.658 (10)
122.92 (8)
–
–
–
–
–
0.952
P21 =a
8.25 (3)
5.95 (2)
8.67 (4)
122.91 (37)
1.731 (9)
2.56 (13)
–
–
0.428
–
P21 =a
8.25 (3)
5.98 (2)
8.71 (2)
122.83 (30)
–
–
0.232 (16)
3.72 (32)
0.179
–
This is particularly good given the fact that no atomic
positions were allowed to vary in the refinement. As expected,
the refined atomic displacement parameter is rather small for
PDF peaks from pairs of atoms within the same molecule
(Bintra ), compared to the thermal motion for contributions of
pairs of atoms belonging to two different molecules in the
structure (Binter ) (Tables 1–3). During refinements, the 2
parameter, which corrects the effect of the correlated motion,
was always refined to values close to zero. Obviously, the peak
widths are described sufficiently well by Bintra and Binter that
the correction by 2 could be omitted. As seen in Fig. 8, there
are still minor differences between the experimental and
calculated data due to the simplicity of the present model.
Figure 7
Calculated PDF of crystalline naphthalene, illustrating the calculation
steps outlined in the text. (a) Gm , the PDF of an isolated molecule
calculated using the small isotropic displacement parameter Bintra . (b) Gc ,
the PDF of the crystal calculated using the larger isotropic displacement
parameter Binter . (c) G0m , the PDF of the isolated molecule calculated
using the same large displacement Binter . (d) The inter-molecular GðrÞ
obtained from Gc G0m . (e) Gtot , the corrected total PDF.
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Dragica Prill et al.
Figure 8
The blue curves are the experimental PDFs of (a) naphthalene, (b)
quinacridone and (c) paracetamol. The red curves represent the
corresponding calculated PDFs using the new approach with two
isotropic displacement parameters. The green curves depict the fit
difference, while the red lines at the bottom show the fit difference from
the standard approach.
Modelling PDFs of organic compounds
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J. Appl. Cryst. (2015). 48, 171–178
research papers
Table 2
Summarized results of PDF analysis for quinacridone.
Bintra represents the isotropic displacement parameter for atom–atom
distances within a molecule and Binter the isotropic displacement parameter
for atom–atom distances between two molecules. Rw is the PDF fit residuum.
Space group
a (Å)
b (Å)
c (Å)
( )
2
Biso (Å2)
Bintra (Å2)
Binter (Å2)
Rw
GoF
Crystal data
Standard approach
New approach
P21 =a
5.692 (1)
3.975 (1)
30.02 (4)
96.76 (6)
–
–
–
–
–
0.908
P21 =a
5.731 (9)
3.917 (6)
30.05 (5)
96.78 (16)
1.62 (14)
0.76 (10)
–
–
0.405
–
P21 =a
5.709 (13)
3.929 (7)
30.108 (40)
96.33
–
–
0.112 (13)
1.79 (19)
0.281
–
Table 3
Summarized results of PDF analysis for paracetamol.
Bintra represents the isotropic displacement parameter for atom–atom
distances within a molecule and Binter the isotropic displacement parameter
for atom–atom distances between two molecules. Rw is the PDF fit residuum.
Space group
a (Å)
b (Å)
c (Å)
( )
2
Biso (Å2)
Bintra (Å2)
Binter (Å2)
Rw
GoF
Crystal data
Standard approach
New approach
P21 =a
7.0915 (3)
9.2149 (4)
11.6015 (5)
97.8650 (10)
–
–
–
–
–
0.887
P21 =a
7.090 (10)
9.226 (13)
11.630 (16)
97.91 (15)
1.53 (6)
0.95 (8)
–
–
0.307
–
P21 =a
7.091 (11)
9.233 (13)
11.624 (16)
97.74 (14)
–
–
0.205 (25)
1.34 (11)
0.197
–
It is notable that the fit of the calculated PDF to the
experimental PDF curve of naphthalene and paracetamol is
much better than that of -quinacridone. This observation
may be explained by a feature of the crystal structures.
Naphthalene molecules are held together in the crystal by van
der Waals interactions in all spatial directions. Temperaturedependent X-ray analysis shows that the thermal expansion is
almost isotropic too. Hence, the variation of the intermolecular distances during the vibrations should be similar in
all spatial directions. In contrast, quinacridone exhibits two
different types of van der Waals interactions in the ½100 and
½001 directions and hydrogen bonds in a third direction.
Correspondingly, the intermolecular vibrational amplitudes
and the thermal expansion are strongly anisotropic. Hence, we
might expect the variation of intermolecular distances to
depend on the spatial direction. This will require a further
extension of the modelling protocol to allow for anisotropic
displacement parameters in the refinement.
5. Conclusion
This work has shown that for a PDF refinement or a PDF
calculation of molecular systems one needs to distinguish
between intramolecular and intermolecular atom–atom
J. Appl. Cryst. (2015). 48, 171–178
distances, i.e. between the motion of atoms belonging to two
different molecules and the motion of atoms within the same
molecule. Using the program DiffPy-CMI, it is possible to
calculate more accurate PDFs of molecular systems. Our
approach using two different thermal factors to describe the
atomic motion within the molecules and between two molecules results in a significantly better fit of model PDFs for
organic samples. This step is of key importance for the further
development of PDF analysis of organic samples. Local
structural information can be extracted from the PDF of an
organic material and the study of organic materials would
profit from better tools for PDF analysis.
Data collection, development of the DiffPy-CMI modelling
software and PDF simulations were supported by the
Laboratory Directed Research and Development (LDRD)
program 12-007 (Complex Modeling) at Brookhaven National
Laboratory (BNL). X-ray experiments were carried out at the
National Synchrotron Light Source beamline X17A, at BNL.
BNL is supported by the US Department of Energy, Division
of Materials Sciences and Division of Chemical Sciences, DEAC02-98CH10886.
References
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