DOI: 10.1021/cg900656z
The Madelung Constant of Organic Salts
2009, Vol. 9
4834–4839
Ekaterina I. Izgorodina,* Uditha L. Bernard, Pamela M. Dean, Jennifer M. Pringle, and
Douglas R. MacFarlane*
School of Chemistry, Monash University, Clayton, Victoria 3800, Australia
Received June 11, 2009; Revised Manuscript Received August 18, 2009
ABSTRACT: The Madelung constant is a key feature determining the lattice energy of a crystal structure and hence its stability.
However, the complexity of the calculation has meant that it has previously not been readily available for complex structures, for
example for organic salts. We propose a new robust method for calculating Madelung constants of such structures based on a
generalized numerical direct summation approach. The method is applied to various organic salts from the ionic liquid and
pharmaceutical fields. The values calculated are seen to be a unique feature of the crystal structure, reflecting the positioning of
the ions in the unit cell and being sensitive to ion pairing. The difference in Madelung constants between different polymorphs of
a compound is also shown.
Introduction
The much-discussed Madelung constant is an important
fundamental aspect of our understanding of the solid state. It
describes the total interaction energy of an ion in a crystalline
lattice, with all of the other ions, and hence provides us with an
understanding of the origins of the lattice energy of an ionic
crystal. The Madelung constant of various simple inorganic
salts has been the focus of much attention for nearly 100 years
and a variety of approaches to its calculation have been
developed.1-14 However, these approaches are not easily
generalized to provide a mechanism for the calculation of
the Madelung constant of more complex ionic crystals, for
example, organic salts, and hence the detailed understanding
that it provides is largely lacking in respect of such crystals. On
the other hand, complex organic salts have become a major
area of interest in recent years for a variety of reasons. For
example, about 50% of known pharmaceutical compounds
are produced as organic salts15 and their solid-state phase
behavior in terms of polymorphism is of very great concern
from the point of view of stability and bioavailability.16 In a
different arena, the field of ionic liquids has generated a huge
range of new organic salts17-20 and an intense interest in
understanding and predicting the melting points of these
salts.21,22 As an offshoot of that field, a number of these salts
exhibit plastic phases in their crystalline state and these are of
growing interest as solid-state ion conductors.23
The phase behavior of these organic compounds is often a
subtle combination of long-range and short-range interactions. The short-range interactions are relatively easy to
identify and understand when a crystal structure is available.
Tools such as the Hirshfeld surface have been of great
assistance in this respect.24 However, the long-range interactions are electrostatic and to understand their relative importance one effectively needs to be able to obtain the Madelung
constant for the crystal structure.21 Hence, as part of our
effort in designing and understanding the properties of organic salts for a variety of applications, we sought to develop a
*Corresponding authors. E-mail: [email protected]
(E.I.I.), [email protected] (D.R.M.). Tel.: þ61-3-9905-8639
(E.I.I.), þ61-3-9905-4540 (D.R.M.). Fax: þ61-3-9905-4597.
pubs.acs.org/crystal
Published on Web 09/18/2009
general method of calculating the Madelung constant directly
from a known crystal structure. We describe here the basis of
the approach we have developed and a number of example
calculations for organic salts of interest in both the ionic liquid
and pharmaceutical fields. The electrostatic energy contribution to the lattice energy derived from these Madelung constants is compared with a variety of inorganic salts and the
relative importance of electrostatic interactions is discussed.
The idea of the Madelung constant as a measure of the net
electrostatic interaction energy, Ees, in ionic solids was introduced by Madelung in 1918.1 He summed all electrostatic
interactions experienced by an ion in the crystal structure and
then expressed the total as
Ees ¼
Mqcation qanion
4πε0 dmin
ð1Þ
where dmin is the distance to the nearest counterion. If one
considers the electrostatic binding energy of a single ion pair
IP
IP
then Ees = MEes
and hence M represents the degree
to be Ees
to which the lattice of ions is more stable than the isolated
collection of ion pairs would be. For stable crystal structures
M > 1 and the larger M is, the more stable the crystal
structure will be. For simple inorganic salts Ees makes up a
large part of the lattice energy, UL, of the salt:
UL ¼ Ees þESR
ð2Þ
where ESR is the energy of short-range interactions including
atom-atom repulsive interactions. In the case of organic salts,
these short-range interactions include a number of important
attractive components including, for example, π-π interactions, hydrogen bonding, and van der Waals interactions.
The Madelung constant can in principle always be calculated by choosing an ion as the origin and then summing all
possible electrostatic interactions with surrounding ions at
increasing distances from the origin. An (effectively) infinite
series of contributions, the Madelung series, is generated in
which each term represents electrostatic interactions with ions
at a particular distance from the origin. Since Madelung’s
original work, most of the calculations have followed the
same direct summation approach aiming to generate a series
expansion for the Madelung constant, which converges to a
r 2009 American Chemical Society
Article
Crystal Growth & Design, Vol. 9, No. 11, 2009
4835
Table 1. Melting Points (Tm), Number of Ions, Calculated and Literature Madelung Constants (M) and Electrostatic Energy Densities (EED) of Traditional
Crystals Together with Number of Ion Pairs in the Unit Cell (N), Crystal Structure Type and Space Group
salta
Tm,
°C
N
crystal type
ZnS
CaF2
CaCl2
NaCl
CsCl
[NMe4][BF4]
[C3mpyr][Cl]d
[Mep][Sacc]
[Prop][Br]
[C1mpyr][NTf2]
[C2mpyr][tos]
[C4mpyr][tos]
[C4mim][Cl]
[C4mim][Cl]
[C1mpyr][NMes2]d
[C4mpyr][NMes2]d
[C4mpyr][NTf2]
170037
141839
77239
80139
64539
44341
24543
18845
16745
13246
12048
11548
6641
4141,e
4051
3652
-1846
4
4
2
4
1
2
8
4
8
4
2
2
4
4
4
8
4
cubic
cubic
orthorhombic
cubic
cubic
tetragonal
orthorhombic
orthorhombic
tetragonal
monoclinic
monoclinic
triclinic
orthorhombic
monoclinic
monoclinic
orthorhombic
orthorhombic
space group
F43m
Fm3m
Pnnm
Fm3m
Pm3m
P4/nmm
Pbcn
P212121
P4/n
P21/c
Pn
P1
Pna21
P21/c
P21/n
Pbca
P 21 21 21
number
of ionsb
M
1.4 107
2.1 107
1.1 107
1.8 104
3.5 106
8.0 105
1.7 106
2.0 106
5.4 104
8.9 105
2.4 106
1.0 106
4.7 106
2.4 106
3.4 105
3.5 106
5.6 105
1.640 ( 0.006c
2.524 ( 0.010c
2.364 ( 0.005c
1.7476
1.768 ( 0.007c
1.6859
1.5358
1.1841
1.4857
1.3583
1.408 ( 0.004c
1.2963
1.379 ( 0.004c
1.3447
1.1932
1.1639
1.2854
M (literature)
Ees,
kJ mol-1
EED 108, J/m3
source of
crystal structure
1.63812
2.51942
2.36540
1.74762
1.76272
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
-3899.2
-2974.1
-2432.6
-862.6
-690.0
-489.6
-549.9
-406.0
-451.7
-416.6
-519.5
-504.1
-474.2
-450.3
-413.5
-424.9
-423.1
1633
1213
477
318
163
40.8
38.0
10.9
13.2
18.6
22.4
19.1
32.8
31.0
21.7
17.9
15.5
38
38
40
38
38
42
44
45
45
47
49
49
50
50
52
52
53
a
Abbreviations have the following meanings: Cnmpyr = (Cn-alkyl)methylpyrrolidinium, C4mim = butylmethylimidazolium, Mep = mepenzolate,
Prop = propanthalate, Sacc = saccharinate, tos = tosylate, Mes = CH3SO3-, Tf = CF3SO3-. Structures of the propanthalate and mepenzolate ions
are shown in Figures 2 and 3. b Number of ions needed in the calculation to reach convergence of the lattice energy using the EUGEN method. c Cases
where the unit cell has a nonzero dipole moment and therefore convergence reached using the Harrison method. d DSC measurements carried out from
-150 to 150 °C at a scan rate of 10 °C min-1. Onset temperatures are reported in all cases. e A broad peak characteristic of plastic crystal transitions.
limit. Emersleben developed two approaches to these direct
summation calculations: (1) by considering ever expanding cubes and (2) by considering ever expanding spheres with
respect to the ion at the origin.3 However, the direct summation approach becomes intractable for all but the simplest of
inorganic salt structures. There are two ways to overcome this
problem: the Ewald method4 and the Evjen method.5 However, in both methods the sum representing the net electrostatic interactions is conditionally convergent, which means
that the result depends on the order of summation if the unit
cell possesses a nonzero dipole moment.25 A discussion of
such calculations and the various corrections required can be
found in the work of Wood.6 Neither of these methods has
made the evaluation of the Madelung constant straightforward and able to be carried out in a generic way, with a
number of factors affecting the outcome of the calculations.
To improve convergence, modifications to both methods have
been thoroughly studied and introduced.7-14,26,27 Among the
simplest and easily achievable technique is the method introduced by Harrison.28 Importantly, this method does not suffer
from conditional convergence like the Ewald and Evjen
methods; that is, it always converges to the correct Madelung
constant.
In this work, we describe a generic approach to calculating
the Madelung constant that involves a direct numerical summation, and which can therefore be used even with the
complex crystal structures that occur with organic salts. In
principle, it can be used to obtain the Madelung constant for
any known crystal structure including nonorthogonal unit cell
structures and those containing more than one ion pair in
the asymmetric unit, all very common circumstances. The
approach could be integrated into standard crystallography
or crystal structure display software. The unit cells can be of
any shape: cubic, orthorhombic, monoclinic, etc., and contain
an unlimited number of ion pairs. In this sense, the proposed
method can be considered as an expanding unit-cells generalized numerical method, which we refer to as the EUGEN
method. We demonstrate the convergence and accuracy of the
EUGEN calculation for known crystal structures such as
NaCl, body-centered cubic CsCl, CaF2, ZnS, and CaCl2 and
also calculate, for the first time, Madelung constants of
selected examples of organic salts of interest in the ionic
liquid/plastic crystal field and the pharmaceutical salt field.
Results
Accuracy and Convergence. The Madelung constants calculated for a number of simple inorganic and organic salts
are compared in Table 1. The crystal structures of the
two tosylate salts are described here for the first time
(CCDC Refcodes: 713020 for [C2mpyr][tos] and 713021 for
[C4mpyr][tos]). The comparison in Table 1 shows that the
values calculated for the Madelung constants of ZnS, CaF2,
CaCl2, NaCl, and CsCl using the EUGEN method are in
excellent agreement with values reported in the literature. In
the case of the organic salts convergence typically requires
24 iterations on average. The selected convergence threshold
corresponds to only 10-3 kJ mol-1 in the lattice energy.
Loosening the threshold to 1 kcal mol-1 (traditionally
considered to be “chemical accuracy”) further shortens the
calculation time. Nonetheless, the calculation for even the
most complex crystal structures in Table 1 still only requires
a few minutes on a laptop computer.
Organic Salt Madelung Constants. The organic salts studied here are comprised of singly charged ions and, therefore,
a direct comparison with the Madelung constants of NaCl
and CsCl can be made. The Madelung constants of all of the
organic salts studied here are lower, in some cases considerably so, than those of NaCl and CsCl, which is in accord with
their much lower melting points. For example, salts such as
[C1mpyr][NMes2], [C4mpyr][NMes2], and [C4mpyr][NTf2],
with melting points below 100 °C, have Madelung constants
at least 25% lower than that of NaCl. The arrangement of
ions in the unit cell with respect to one another is a major
factor in determining the Madelung constant of the organic
salts, much more so than is the case for the inorganic salts.
For example, [NMe4][BF4], Figure 1, crystallizes in a cubic
system with very uniformly arranged positions of the ions
with respect to all of their neighbors. To simplify inspection of
the structure, we show in Figure 1b a version in which only the
4836
Crystal Growth & Design, Vol. 9, No. 11, 2009
Izgorodina et al.
Figure 1. Packing diagrams for [NMe4][BF4] viewed down the c-axis showing (a) full unit cell contents (the BF4 ions are shown with their
measured disorder) and (b) charged atoms only.
Figure 2. Packing diagrams for [Prop][Br] as viewed down the c-axis showing (a) full unit cell contents and (b) charged atoms only.
charged atoms are visible. As a result of the highly symmetrical packing the Madelung constant, at 1.69, is only very
slightly lower than that of NaCl (M = 1.7476) or CsCl (M =
1.7627). On the other hand, Figure 2 shows the structure of
propanthaline bromide ([Prop][Br]); this compound is a
muscarinic acetylcholine receptor antagonist sold under the
trade name Pro-Banthine. Figure 2b shows that the charged
atoms are positioned in clusters in the unit cell, with distinct
ion-pairing also present (i.e., one counterion in much closer
proximity than any other). This lowers the Madelung constant considerably, to 1.49. To put this effect in context, a
tight ion pair that is well isolated in a large neutral species
from neighboring ion pairs could have a Madelung constant
approaching 1.0. This situation is illustrated to an extent by
mepenzolate saccharinate ([Mep][Sacc]) in Figure 3; the
mepenzolate cation is a skeletal muscle relaxant sold commercially as the Br- salt under the trade name Cantil. In this
structure, we see more distinct ion pairs and a correspondingly lower Madelung constant of 1.18. The fact that this
compound has a melting point as high as 188 °C, despite this
very low Madelung constant, is a clear sign of the strong
anion-cation π-π stacking that is present in this salt.
For ionic liquids containing the same anion and the cation
with the alkyl chains of different length we can easily discern
that the melting point decreases in accord with the Madelung
constant decrease. For example, this trend is observed for the
following pairs of ILs: (1) [C1mpyr][[NMes2] and [C4mpyr][[NMes2] and (2) [C1mpyr][[NTf2] and [C4mpyr][[NTf2].
However, comparison of the two pairs with each other shows
that the Madelung constant of [C4mpyr][[NTf2] is larger than
Article
Crystal Growth & Design, Vol. 9, No. 11, 2009
4837
Figure 3. Packing diagrams for [Mep][Sacc] as viewed down the a-axis showing (a) full unit cell contents and (b) charged atoms only (the
negatively charged center is shown as a larger symbol to indicate the fact that the charge is likely to be delocalized to some extent in this anion).
that of [C4mpyr][[NMes2] and yet the melting point of the
former is lower by about 50 degrees. This finding further
reinforces the importance of additional specific interactions
(such as hydrogen bonding, van der Waals interactions,
π-π stacking interactions etc.) in the lattice energy, UL, of
the salt. Therefore, accurate estimation of these interactions
is needed before a correlation between UL and the melting
point can be established.
Also listed in Table 1 is the electrostatic contribution, Ees,
to the lattice energy calculated from the Madelung constant
using eq 1. The values for the inorganic structures range from
more than 1000 kJ/mol for the divalent metal salts, to values
around 800 kJ/mol for NaCl. The values for the organic salts
are typically much lower than these, reflecting the lower
electrostatic interaction involved.
Charge Delocalization. Charge delocalization in molecular
ions potentially places fractional charges on several atoms.
This effect is relatively easy to deal within the EUGEN
method if the details of the charge delocalization are known,
or can be estimated. The case of the tosylate anion illustrates
the simplest approach; the value listed in Table 1 is the
average of the Madelung constants calculated by assuming
that the unit negative charge is present on each of the oxygens
in turn, producing values: 1.387, 1.500, and 1.338. A more
sophisticated approach would place partial charges on the
three oxygens simultaneously and we will address this approach in a future paper. The fact that the effective Madelung constants of the three oxygens in the lattice are so
different indicates that they experience quite different electrostatic environments and one of the oxygens in particular
(O3 in Figure S3, Supporting Information) experiences more
attraction than the others. This may itself have an impact on
the actual electron distribution in the ion within the crystal and
the electric field vector, which is implicitly obtainable from the
EUGEN method, could be used in an advanced quantum
chemical calculation to further investigate this effect.
Polymorphism. The Madelung constant itself is effectively
a geometric parameter. While each space group will have a
unique Madelung constant associated with it when the
structure involves close packed ions, this will not necessarily
be true where the ions are molecular species. Two salts
crystallizing in the same space group may have different
Madelung constants if the charged atoms have different
locations in the molecular ions. For organic salts, then, the
Madelung constant is very much a unique feature of each
individual compound. Where polymorphism exists for an
organic salt, each polymorph will also very likely have a
different Madelung constant. The change in lattice energy
that the change in Madelung constant describes therefore
becomes an important aspect of understanding the phase
4838
Izgorodina et al.
Crystal Growth & Design, Vol. 9, No. 11, 2009
transformation. A good example of this can be seen in the
case of the polymorphs of 1-butyl,3-methyl imidazolium
chloride (C4mimCl) listed in Table 1. The Madelung constant of the monoclinic form is distinctly lower, and in energy
terms this amounts to about 24 kJ/mol difference in lattice
energy (other factors being equal); as a result it would be
expected that this difference in lattice energy could drive a
solid-solid transformation of this form to the more stable
orthorhombic form. As observed by Holbrey et al.41 the
orthorhombic form is indeed more thermodynamically
stable than the monoclinic form.
take place through regions of, for example, hydrocarbon
chains; the assumption that the material is isotropic also does
not necessary hold. Measurements and investigations of the
components of dielectric constants in organic liquids and
solids are ongoing in several groups32-36 and will assist in
exploring this issue further. However, the EUGEN approach
allows this issue to be explicitly accounted for in the calculation by inclusion of different values for the average dielectric
constant, or potentially a spatially varying dielectric constant.
Discussion
In this paper, we have introduced an expanding unit-cells
generalized numerical (EUGEN) method for the calculation of
the Madelung constants of salts with a known crystal structure.
The approach has been validated against Madelung constants
available in the literature for inorganic salts. Madelung constants
of a number of organic salts of interest in the ionic liquid/plastic
crystal and pharmaceutical fields are reported here for the first
time and it was shown that the values were lower than those of the
inorganic salts, falling in the range of 1.16 to 1.69. Organic salts
with highly symmetrical packing, such as [NMe4][BF4], can display Madelung constants almost as high as that of NaCl. On the
other hand, distinct ion pairing in a crystal, as observed in the
crystal structure of [Mep][Sacc], results in a considerable decrease
of the Madelung constant, approaching 1.0. The Madelung
constants calculated are currently only a first approximation,
however the EUGEN approach paves the way for more advanced
calculations, which might properly account for any charge delocalization in the ions or the effective screening of electrostatic
interaction by other regions of the molecule.
The Madelung constant allows us to calculate an electrostatic energy density (EED) for the structure:
EED ¼
Ees N
ðJ=m3 Þ
V
ð3Þ
where V is the unit cell volume and N is the number of ion pairs
in the unit cell. The EED values are also reported in Table 1.
For most salts the EED makes up a large part of the lattice
energy (expressed in unit volume terms) of the structure, and
therefore it is a useful quantity for direct comparison of
different compounds or different polymorphs of the same
compound. This quantity is a more comprehensive measure of
electrostatic interactions in a material, because it takes account of the very different concentrations of charged species
(i.e., number of ions per unit volume) that can occur when
comparing inorganic and various, much larger, organic ions.
The EED values of the organic salts in Table 1 are as much as
2 orders of magnitude lower than those of the inorganic salts,
whereas Ees, which is a molar quantity, varies by only about
1 order of magnitude. In general, the EED trend also shows a
fairly good correlation with melting point until it drops to around
20 108 J/m3, where undoubtedly short-range interactions begin
to make an important contribution to the lattice energy.
In general, due to the nature of most organic salts, such
short-range interactions make a more significant contribution
to the total lattice energy than in inorganic salts. For example,
the attraction due to electron correlation between imidazolium rings is about 28 kJ mol-1 (as estimated at the CCSD(T)
level of theory29), which is a non-negligible contribution to the
total lattice energy of salts based on this cation, for example,
the C4mim salts in Table 1. Since the type of interactions
present in a particular organic salt depends very much on the
nature of the constituting ions, approaches to their quantification, of the type being intensely investigated in the computational chemistry field, are needed if a complete picture of the
lattice energy is to be achieved.30,31
A “second order” issue that exists in all approaches to
calculating long-range electrostatic interactions can also be
dealt with by an appropriate expansion of the EUGEN code
presented here. The Madelung constant is normally calculated
on the basis that the electrostatic interaction between two ions
occurs through a medium of dielectric constant equal to 1. In a
real ionic material, including NaCl, this is plainly not the case,
since ions/atoms of varying polarizability may lie between the
interacting ions. In the classical Madelung analysis and all of
the more modern approaches to the calculation, including the
Ewald summation, this is ignored, partly on the basis that the
attractive and repulsive interactions are equally affected. In a
material consisting of mainly organic cations and anions this
issue becomes more acute since some of the interactions may
Conclusions
Methods
Expanding Unit-Cells Generalized Numerical (EUGEN) Method.
A detailed discussion of the method and the implemented code are
provided in the Supporting Information. The method requires a fully
solved crystal structure as input data, from which a list of the
fractional coordinates of all of the charged atoms in the unit cell
has been produced. The calculation then proceeds in a series of
iterations involving a larger number of ions at each step. At the nth
iteration, the calculation constructs an expanded cell containing
n replicates of the unit cell along each of the unit cell directions. It
then sums all of the pairwise interactions between each ion in the
expanded cell and the central ion. It continues expanding the calculation cell until convergence is reached. As is well-known, such a
calculation is conditionally convergent if the unit cell possesses a
nonzero dipole moment;25 this occurs occasionally in organic salt
crystal structures. The calculation checks for this property and, when
present, adopts the Harrison approach to achieving convergence.
Since the result in this case tends to produce a slow oscillation toward
convergence, the software calculates a moving average (and standard
deviation) of the five most recent iterations and reports this value
when the convergence condition is reached. If the convergence is not
reached after 60 iterations, the calculation stops and the final Madelung constant is calculated as an average of the 5 last iterations.
Acknowledgment. We gratefully acknowledge generous
allocations of computing time from the National Facility of
the National Computational Infrastructure. E.I.I. gratefully
acknowledges the support of the Australian Research Council
for her postdoctoral fellowship, as does D.R.M. for his
Federation Fellowship.
Supporting Information Available: The supporting information
contains a thorough description of the EUGEN method as well as
the program that implements this method. Examples of the input
files needed to run the program are also given. In addition, the
supporting information provides details of the crystal structures,
[C2mpyr][tos] and [C4mpyr][tos], reported here for the first time.
Article
Crystal Growth & Design, Vol. 9, No. 11, 2009
The program code (written in Fortran 95) for the EUGEN method
is available for download from http://www.chem.monash.edu/
ionicliquids. This material is available free of charge via the Internet
at http://pubs.acs.org.
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