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Electrochemical properties of crystallized dilithium squarate: insight from
dispersion-corrected density functional theoryw
Christine Frayret,*a Ekaterina I. Izgorodina,b Douglas R. MacFarlane,b
Antoine Villesuzanne,c Anne-Lise Barrès,a Olivier Politano,d Didier Rebeixd and
Philippe Poizota
Received 26th November 2011, Accepted 20th June 2012
DOI: 10.1039/c2cp41195d
The stacking parameters, lattice constants, and bond lengths of solvent-free dilithium squarate
(Li2C4O4) crystals were investigated using density functional theory with and without dispersion
corrections. The shortcoming of the GGA (PBE) calculation with respect to the dispersive forces
appears in the form of an overestimation of the unit cell volume up to 5.8%. The original
Grimme method for dispersion corrections has been tested together with modified versions of this
scheme by changing the damping function. One of the modified dispersion-corrected DFT
schemes, related to a rescaling of van der Waals radii, provides significant improvements for the
description of intermolecular interactions in Li2C4O4 crystals: the predicted unit cell volume lies
then within 0.9% from experimental data. We applied this optimised approach to the screening
of hypothetical framework structures for the delithiated (LiC4O4) and lithiated (Li3C4O4) phases,
i.e. oxidized and reduced squarate forms. Their relative energies have been analysed in terms of
dispersion and electrostatic contributions. The most stable phases among the hypothetical models
for a given lithiation rate were selected in order to calculate the corresponding average voltages
(either upon lithiation or delithiation of Li2C4O4). A first step towards energy partitioning in view
of interpretating crystal phases relative stability in link with (de)-intercalation processes has been
performed through the explicit evaluation of electrostatic components of lattice energy from
atomic charges gained with the Atoms in Molecules (AIM) method.
I.
Introduction
Li-ion batteries (LiBs) presently operate using inorganic insertion
compounds. The abundance and materials life-cycle costs of such
batteries may present issues in the long term with foreseeable
large-scale applications. Consequently, in parallel to research on
regular inorganic-based LIBs, a possible alternative could be
foreseen in the use of organic-based electrode materials. To date,
various classes of redox-active organic compounds have been
already investigated vs. Li+/Li0 including, for example, conducting polymers,1,2 sulfides3 and disulfides4–6 materials, stable
neutral radicals,7–9 materials derived from electron-donating/accepting molecules (either in the form of polymers or
a
LRCS-CNRS, Universite´ de Picardie, 33, Rue Saint-Leu, 80039
Amiens Cedex, France. E-mail: [email protected];
Fax: +33 322827590; Tel: +33 322827586
b
School of Chemistry, Monash University, Clayton,
Victoria 3800, Australia
c
ICMCB-CNRS, Universite´ de Bordeaux 1, 87,
Av. Dr. A. Schweitzer, 33608 Pessac Cedex, France
d
LRRS-CNRS, Universite´ de Bourgogne, 9, Av. Savary,
21078 Dijon Cedex, France
w Electronic supplementary information (ESI) available. See DOI:
10.1039/c2cp41195d
11398
Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
low-molecular-weight crystalline compounds)10–24 or metal
clusters.25 However, in comparison with inorganic electrodes,
which have benefited from 30 years of intensive research, the
use of organic materials for energy storage is still in its early
stages. Moreover, despite rising interest for this research field,
little is known about their inner electrochemical reactivity in
the solid state in contrast to the well documented field of
molecular electrochemistry.
Switching from inorganic to organic matter-based electrodes
could provide a considerable breakthrough in the field of
functional materials, with the possible advent of cleaner
and sustainable energy systems provided that precursors
originate from the biomass.12,13,26 Great experimental efforts
are made by our group12–16 to reach such a challenging
objective, with an active search for reliable and efficient
redox-active organic solids reacting at both high and low
potentials vs. Li+/Li0 (for positive and negative electrodes,
respectively). In most of these experiments, (mono- or
polycyclic) quinone-based materials or oxocarbon salts were
envisaged. The benzoquinone skeleton has the potential of a
two-electron redox reaction whereas oxocarbon dianions
[(CnOn)2, where n = 3: deltate; n = 4: squarate; n = 5:
croconate and n = 6: rhodizonate], recognized by West in the
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1960s as a new group of organic compounds,27 contain several
redox-active carbonyl groups.
However, batteries fabricated from organic compounds as
active positive/negative electrodes or with an organic positive
electrode and lithium metal as anode were most often characterized by several drawbacks including either low capacity
or insufficient output voltage, as well as low stability of
charge–discharge cycles. There is therefore still a place for
research of the adequate materials. The accelerated discovery
of appropriate matrices relies on the establishment of the key
parameters, which govern especially the electrochemical
features. Modeling of these materials using density functional
theory (DFT) calculations has proven to be a pertinent tool
for the study of crystalline inorganic materials used as an
electrode in Li-ion batteries (see for example ref. 28–37).
However, to our knowledge, it has not been applied yet to
the newly developed crystallized organic phases. Such lack of
simulation of solid organic matrices may originate from
experimental difficulties in supplying crystal structures as
discussed hereafter. Computational studies in the field of
organic electrodes for batteries have dealt mainly with molecular
features such as HOMO–LUMO properties or the calculation of
energy levels for clusters of molecules.20,21
In complement to the molecular approach, modeling of
solid organic matrices with the potential promise of identifying
strategies of concomitant ideal functionalization and crystallinetype structure to optimize the (de)intercalation properties of
the organic matrices would thus enable to assist and guide the
exploratory experimental work initiated in this new research
area. However, for most cases, the difficulty of the envisaged
modeling lies in the lack of reliable X-ray structure from in situ
data or single crystal phases, most often for the delithiated
phase but sometimes also for the lithiated one. Apart from
final elucidation of the selection rules for the ideal electrode
material, the information obtained from DFT calculations
relies on the geometry optimizations. Full relaxation of the
delithiated phase starting from the unique knowledge of
the lithiated structure should indeed allow some proposition
of the most favourable structural arrangement for the
unknown delithiated crystalline phase, including Li positions
(or reversely, if the delithiated phase is the only one, which is
observed from the experiment). In particular cases, in which
none of the lithiated or delithiated phases have been characterized by X-ray crystallography, the DFT simulation is a
quite challenging task, since starting structural models should
be used as a trial guess. Despite the difficulty of this kind of
task, applying DFT calculations in order to provide proposition of structural arrangement might be of interest for some of
the materials studied in our group for which one has to face
this problem. Before tackling such challenging DFT computations, it appears to be of crucial importance to assess first the
ability of first-principle calculations (and methodologies to
improve them, as described below) to properly describe this
type of materials. The present study was thus applied to the
crystalline dilithium squarate. The relative ability of several
common and newer exchange–correlation (XC) functionals as
well as the dispersion-corrected DFT method to account for
both inter-/intra-molecular structural features and electrochemical features is addressed in this system. The structure
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of solvent-free dilithium salt of squarate is well known from
the experiment and therefore represents an excellent benchmark for the future studies of related organic matrices.
Standard implementations of density functional theory
describe well strongly bound molecules and solids but fail to
describe long-range van der Waals attractions (vdW). Van der
Waals energies are related to mutually induced and correlated
dipole moments.38 Contrary to the Hartree–Fock model,
which does not consider electron correlation effects, DFT
calculations should, in principle, give the exact description of
ground state energy, including the vdW energy, if the true
functional is known. However, practical implementations
relying on the local density approximation (LDA) and the
generalized gradient approximation (GGA) fail to reproduce
the physics of vdW interactions at large separations with little
or no overlap of atomic electron densities.39 As a result, DFT
calculations usually overestimate the lattice parameters along
the stacking direction for organic crystals or layered materials.40,41
Recently, a number of semi-empirical approaches have been
taken to incorporate correction schemes for London-type
dispersive interactions into DFT methods and thus improve
common approximate functionals. Various models have been
proposed, including the DFT-D2 and DFT-D3 methods
by Grimme and co-workers,42–44 van der Waals density functionals (vdW-DF) by Lundqvist, Langreth and co-workers,45–47
the dispersion-corrected atom-centered pseudopotential method
by Rothlisberger and co-workers48,49 and the vdW-TS method
by Tkatchenko and Scheffler.50 In this study, we described the
vdW interactions using the DFT-D2 method,41 which was
recently applied with success to various systems.51–53
As already mentioned, establishing selection rules is the key
to making technological advances with this particular energy
storage platform made of organic electrodes. This will require
an understanding of the underlying interactions and how these
are modified during the electrochemical lithium extraction
(oxidation) or uptake (reduction). In addition to the structural
and electrochemical properties examination, an evaluation of
Coulombic (electrostatic) long-range interactions was thus
performed through the explicit calculations of Madelung
constants and electrostatic lattice energies using the newly
developed expanding unit-cells generalized numerical (EUGEN)
method.54
II. Crystal structures
The crystal structure of solvent-free dilithium squarate
(Li2C4O4) has been determined by R.E. Dinnebier et al.55
Crystallographic data at room temperature of Li2C4O4 are:
a = 7.1073(3) Å, b = 9.5627(4) Å, c = 3.2973(1) Å, b =
101.11(1)1, V = 219.91(1) Å3, SG C2/m, Z = 2. Representations of the squarate anion, C4O42, and the molecular
orientations within the unit cell are depicted in Fig. 1a and
b, respectively. The squarate layers in the Li salt of squarate
are superimposed (see Fig. 1a) such that the squarate dianions
stack along the c-axis with an interplanar distance of
3.0995 Å.55 The lithium cations lie in the 4h Wyckoff position
(0; 0.3515(8); 0.5) and are surrounded by distorted tetrahedra
formed by oxygen atoms (dmin = 1.949 Å; dmax = 1.959 Å) of
the squarate dianions, each Li+ ion being linked to four
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Fig. 1 (a) Atomic labeling scheme for the squarate molecule. The lattice vectors a, b, and c define the monoclinic cell of the Li2C4O4 crystal.
(b) Unit cell molecular packing arrangement for dilithium squarate viewed down the (i) a-axis, (ii) b-axis, and (iii) c-axis.
Fig. 2 Collected X-ray diffraction data of Li2C4O4 at lower temperature than room temperature (RT) (down to 150 K) (* corresponds to the
peaks emanating from the sample holder).
distinct C4O42.55 We have collected X-ray diffraction data of
Li2C4O4 powder from ambient to lower temperatures in order
to verify that there was no phase transformation at low
temperature (Fig. 2). Di-lithium squarate (Li2C4O4) was
synthesized according to the procedure described by
Shanmukaraj et al.56 X-ray powder diffraction was performed
using a Bruker D8 Advanced diffractometer equipped with a
Linxeye detector and with Cu radiation (l1 = 1.54056 Å, l2 =
1.54439 Å). The thermal behavior was followed in situ by
performing temperature controlled X-ray diffraction in the
above-mentioned X-ray diffractometer using a low temperature Anton Parr TTK450 chamber cooled by liquid nitrogen.
Each pattern was recorded at constant temperature under a
vacuum, between 2y = 71 and 2y = 751 with steps of 0.0191 s1.
Between each programmed temperature, the heating rate was
5 1C min1. This experiment demonstrates us that no phase
transition is observed down to 150 K, therefore affording more
ensuring of the validity of the calculation from DFT at
0 K concerning the nature of phase into consideration. The
powder patterns were refined using the Rietveld method57 as
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Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
implemented in the FullProf program.58 Rietveld refinement
was performed on a limited 2y range (till 421) to avoid the
presence of extra peaks from the sample holder (see Fig. S1
and Table S1 of ESIw). Refined parameters for Li2C4O4 at
150 K are the following: a = 7.1231(7) Å, b = 9.5945(10) Å,
c = 3.3023(2) Å, b = 100.86(1)1, V = 221.65(3) Å3, space
group: C2/m. Rp = 10.1%; Rwp = 11.7%; RBragg = 1.64%.
The crystal structure of lithium salts of radical compounds
obtained by either the one-electron oxidation (i.e. Li+, C4O4)
or reduction (i.e. 3Li+, C4O43) of Li2C4O4 (denoted as
‘‘LiC4O4’’ and ‘‘Li3C4O4’’, respectively) are unknown from
the experiment. Therefore, the calculation should be able to
provide hypothesis of structural arrangement for these two
phases issued from the sole knowledge of crystallographic data
for dilithium squarate.
Four ways of lithium extraction from Li2C4O4 (crystals A,
B, C and D, see Fig. 3) were considered in order to propose a
model for LiC4O4. In crystal A, the positions (0.5; 0.8515; 0.5)
and (0.5; 0.1485; 0.5) from the 4h Wyckoff position of Li2C4O4
were removed, whereas in crystal B, the positions (0; 0.3515; 0.5)
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Fig. 3 Four models for the lithium extraction from Li2C4O4 (crystals A, B, C and D for LiC4O4 before full-geometry optimization in which Li
removed from Li2C4O4 are indicated within dotted circles).
Fig. 4 Two models for the lithium insertion into Li2C4O4 (crystals E and F before full-geometry optimization in which additional Li is indicated
in dark).
and (0; 0.6485; 0.5) were selected as non-occupied sites. Crystal
C was generated by removing positions (0; 0.3515; 0.5) and
(0.5; 0.1485; 0.5) while in crystal D, Li ions were deleted from
the positions (0; 0.3515; 0.5) and (0.5; 0.8515; 0.5).
Two models of lithium intercalation within Li2C4O4 were
considered, thus forming Li3C4O4 phases (i.e. crystals E
and F, see Fig. 4). In crystal E, the positions (0; 0; 0.5) and
(0.5; 0.5; 0.5) were filled with lithium ions in addition to the
4h Wyckoff position of Li2C4O4 whereas in crystal F, additional alkali ions were placed at the positions (0.25; 0.25; 0)
and (0.75; 0.75; 0).
III.
1.
Theoretical investigation
Methodology and computational details
The structure and relative stability of the various ‘lithiated’
and ‘delithiated’ phases (Li3C4O4, Li2C4O4, and LiC4O4) were
studied by a series of periodic DFT energy minimizations with
several choices of XC functionals: LDA,59 Perdew–Burke–
Ernzerhof (PBE) variant of the GGA,60 PBEsol,61 and
revPBE.62 The revPBE functional is a revised version of
Perdew–Burke–Ernzerhof functional. It was constructed by
optimizing one parameter of the PBE functional against the
exchange energy of noble gas atoms from He to Ar62 and was
designed with the view of giving more accurate energies
for atoms and covalent molecules. It thus improves the
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atomization energies and chemisorption energies over the
PBE functional. PBEsol61 is another revised version of
Perdew–Burke–Ernzerhof functional. In contrast to revPBE,
it was designed for solids and was constructed to restore the
correct second-order expansion for the exchange energy.
PBEsol improves the equilibrium properties of solids and their
surfaces over PBE.
In order to estimate the incidence of dispersion interactions
between the molecules within the crystal, the results of dispersioncorrected PBE functional were compared to those obtained by
using the above-mentioned non-corrected Kohn–Sham density
functionals.
Both DFT and DFT-D energy minimizations were carried
out with the Vienna Ab initio Simulation Package (VASP).63
In this work, projector augmented wave (PAW) potentials
were used to describe the electron–ion interaction.64 For Li
atoms, the Li_sv pseudo-potential treated the semi-core 1s
states as valence states. The wave functions were expanded in
plane waves with energy below 520 eV. Brillouin zone sampling
was performed by using the Monkhorst–Pack scheme,65 with a
k-point grid of 3 2 7. Simulations of the three phases,
Li3C4O4, Li2C4O4, and LiC4O4, were performed by simultaneously relaxing both the lattice geometry and atomic positions starting from the experimental ones when available
(i.e. in the case of Li2C4O4) without the symmetry constraints
of the space group. The resulting energy was considered to
estimate the redox equilibrium potential corresponding to these
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two biphasic-type electrochemical systems (see Section IV.2). In
addition to these full-geometry optimizations, a relaxation of
atomic positions within the fixed experimental unit cell geometry
was also performed for the case of Li2C4O4. Minimizations
were considered complete when energies were converged to
better than 1 105 eV per atom and maximum residual
forces were lower than 1 103 eV Å1.
As a means to obtain a quantitative measure of net atomic
charges, we have applied a topological analysis of the total
electron density. For this purpose we employed codes included
in the WIEN2k package.66 Following the full-geometry series
of optimizations with the pseudopotential VASP package, we
have therefore performed all electron calculations using the
full potential linearized augmented plane wave (FPLAPW)
method as implemented in WIEN2k.66 The augmented plane
wave plus local orbitals method (APW + lo) was used for
valence states and LAPW was used for the other states. The
previously optimized structures were used as input and the
PBE functional was used for the exchange–correlation terms.
The RMTKmax parameter, which is the product of the smallest
muffin tin radius by the plane wave cutoff, was fixed to 7.
Self-consistent cycles were achieved for 30 k-points in the
irreducible Brillouin zone. The partitioning of the calculated
electron density was then performed using Bader’s ‘‘Atoms in
Molecules’’ (AIM) approach.67–71 Nuclei behave as attractors
and the space containing all the points of the electronic density
whose gradients lead to a nucleus is called an atomic basin.
The electronic population attributed specifically to a given
atom is thus calculated through the integration of the electron
density within its basin. In this study, results of net atomic
charges were used to evaluate Madelung constants from the
predicted crystal structures through the EUGEN code.54 The
carbon atoms were excluded from the calculations and their
charges were added to the adjacent oxygen atoms. The
EUGEN code was modified to account for the fractional
charges on the Li and O atoms and the cases, in which the
oxygen atoms belonged to the same C4O4x molecular fragment to prevent them from repelling each other. The average
Madelung constants on the Li and O atoms and the total
Madelung constant, M, of the LiXC4O4 crystal were calculated
using the following expressions:
Pn
i¼1 Mi ðLiÞqi ðLiÞ
Mave ðLiÞ ¼
X
n
Pn
i¼1 Mi ðOÞqi ðOÞ
ð1Þ
4
Mave ðOÞ ¼
m
M¼
1
ðMave ðLiÞ þ Mave ðOÞÞ
2
where n and m are the number of the Li and O atoms in the
asymmetric unit cell, respectively, and Mi(Li) and Mi(O)
are Madelung constants of the individual oxygen and
lithium atoms in the unit cell, respectively. The electrostatic
lattice energies, EES, were predicted based on the total
Madelung constant and the distance to the nearest neighbour
as follows:
EES ¼
11402
M
4pe0 dmin
Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
2.
Semi-empirical vdW correction
In this work, we used the dispersion correction proposed by
Grimme (Grimme’06 scheme, i.e. DFT-D2 method)43 and
introduced in the VASP package. In this semi-empirical
correction, the total energy of the system is defined as a sum
of the self-consistent Kohn–Sham energy terms as obtained
from the chosen XC-functional (EKS-DFT) and a semi-empirical
correction (Edisp):
EDFT-D =EKS-DFT + Edisp
(2)
The dispersion energy between a pair of atoms at long range
can be expressed as a power series of the interatomic distance,
R72 according to:
Edisp ðRÞ ¼ 1
X
Cn Rn
ð3Þ
n¼6
where n are even numbers, and Cn are dispersion coefficients.
The first term, C6R6, is the dominant contribution, representing
the interaction between instantaneous dipoles,73 and is often
used in practice as the only term of dispersion energy. The
subsequent terms (C8R8, C10R10, etc.) are attributed to interactions between higher-order fluctuating multipole moments.
The Grimme scheme using such pair-wise interaction terms
is defined according to eqn (4):
Edisp ¼ S6
NX
Nat
at 1 X
i¼1
X Cij
6
f ðRij;g Þ
6 dmp
R
ij;g
j¼jþ1 g
ð4Þ
where the energy is the summation over all atom pairs and g
lattice vectors, Nat is the number of atoms in the system, S6 is a
functional-dependent global scaling factor, Cij6 denotes the
dispersion coefficient for atom pair ij, and Rij,g is the internuclear separation of the atom pair. In order to avoid nearsingularities for small R values and double-counting effects of
correlation at intermediate distances, a damping function fdmp
must be used, which is given by:
fdmp ðRij;g Þ ¼
1
1 þ edðRij =Rr 1Þ
ð5Þ
where Rr is the sum of atomic vdW radii.
We have applied this semi-empirical correction to the GGA
(PBE) exchange–correlation functional, as implemented in the
VASP package. For all calculations, the value of d = 20
(dampening parameter) was selected in order to specify the
steepness of the dampening function (eqn (5)). A cut-off radius
of 30 Å for pair interactions was used to truncate the summation over lattice vectors.
Through the proposed parameters (with Cij6 and Rvdw values
taken from Grimme43), hereafter called PBE-D calculations,
the vdW forces might however be overestimated resulting in an
underestimation of the lattice constants. Results may thus be
improved upon modification of the semiempirical vdW correction. The proposed correction by Civalleri et al.74,75 corresponding to a modification of the van der Waals radii, which
allows a softening of the dispersion interaction, was thus
investigated too (hereafter denoted as PBE-D*). It corresponds to a scaling of the Rvdw of heavy atoms by 1.05. This
factor was determined from a manual fitting procedure,
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searching for the best agreement between computed and
experimental cohesive energies on a set of 14 molecular
crystals.74 Additionally, they were validated recently through
a comparative analysis of the intermolecular energy for a data
set including 60 molecular crystals with a large variety of
functional groups.76 In the method labelled as corr-PBED*_0.52 hereafter, the optimized s6 value of 0.52 (lower than
0.75 as proposed by Grimme and used for PBE-D or PBE-D*
calculations) was also employed while leaving the Rvdw fixed at
the PBE-D* values.52 On the other hand, calculations through
the PBE-D, PBE-D* and corr-PBE-D*_0.52 methods used the
C6 coefficient for Li taken from Grimme’s parameters list in
which Li was parameterized as Li atoms, while in this work
examined systems contain Li+ ions (see Section IV.2). Therefore, the corresponding C6 coefficient is expected to be very
small compared to the value determined for Li (1.61). Such an
effect might cause an overestimation of the dispersion interaction. In order to determine the incidence of this parameter,
PBE-D* calculations were repeated by taking a C6 value equal
to zero (calculations labeled PBE-D*-C6(Li) = 0).
IV.
1.
Results and discussion
Geometry optimizations and energetics
1.1. Li2C4O4. The quality of structural reproduction was
evaluated for Li2C4O4. Without taking into account zeropoint energy and thermal correction, comparison of the
calculated structures with the available experimental structure
determined by X-ray crystallography in this work at 150 K
allows for a semi-quantitative assessment of the performance
for each exchange–correlation functional or the DFT-D
method for the same set of computational parameters
(k-points grid/cutoff energy). A ranking of the various methods
can then be proposed according to the degree of reproduction
of inter-/intra-molecular geometry features. The changes in
unit cell dimensions with respect to the experimental X-ray
data for the full-geometry optimizations are given in Table 1.
Relative errors in unit cell volume or monoclinic angle and
lattice dimensions with respect to the experiment for the
various levels of theory are presented in Fig. 5 and 6, respectively. In agreement with previous studies, the intermolecular
interaction is not equally well accounted for within the various
approximations for XC and dispersion correction.
The relaxation with LDA leads to an underestimation of the
volume of the cell (of 7.2%), resulting from the contractions
of a-, b- and c-axes. In the PBE full optimization without
dispersion corrections, an expansion of 3.4% was observed
along the a- and c-axis, leading to an overestimation of the
volume of +5.8%. Therefore, such observation is a clear
indication that the PBE functional does not account correctly
for the stacking interactions in this crystal, which occurs along
both a- and c- axes due to the orientation of the molecular
packing. The results of the optimization with the PBEsol
functional are in much better agreement with the experiment,
with an overestimation of the unit cell volume lower than 1%.
The full-geometry optimization with revPBE gives rise
to the poorest results, with an overestimation of the volume
by +12.2%. These observations confirm that pure DFT
functionals (be it LDA, PBE or rev-PBE) show serious
deficiencies in properly describing molecular crystals, in which
the dispersion contribution is not negligible. The structural
description afforded by the PBEsol functional is the only one
among the pure DFT functionals that we tested, which is able
to produce the least errors for lattice constants. These results
are consistent with previous observations concerning molecular crystals. For instance, Todorova and Delley77 observed
that the PBEsol functional has a signed mean deviation of
o1% for normal crystals and molecular crystals up to the
dipole class. Similarly to our study, they evidenced that such
functional performs more favorably than PBE for the weak
interactions in molecular solids. This improvement can be
ascribed to the better treatment of medium-range correlation
provided by the PBEsol over PBE. However, according to the
Table 1 Optimized lattice parameters, a, b, c, a, b and g, unit cell volume, V, and inter-plane distance, d, for Li2C4O4 (discrepancies between
experimental (issued from this work at 150 K) and optimized values are given in parentheses)
Compound
Method
a (Å)
b (Å)
c (Å)
a (deg.)
b (deg.)
g (deg.)
V/Z (Å3)
d (Å)
Li2C4O4
Expta
Exptb
LDA
7.1073 (3)
7.1231 (7)
6.8936
(3.22%)
7.3654
(+3.40%)
7.1778
(+0.77%)
7.5453
(+5.93%)
7.0842
(0.55%)
7.1305
(+0.10%)
7.2092
(+1.21%)
7.1304
(+0.10%)
9.5627 (4)
9.5945 (10)
9.4476
(1.53%)
9.6560
(+0.64%)
9.6087
(+0.15%)
9.7835
(+1.97%)
9.6485
(+0.56%)
9.5986
(+0.04%)
9.6240
(+0.31%)
9.5990
(+0.05%)
3.2973 (1)
3.3023 (2)
3.1852
(3.68%)
3.3940
(+2.78%)
3.3094
(+0.22%)
3.4953
(5.84%)
3.3096
(+0.22%)
3.3205
(+0.55%)
3.3393
(+1.12%)
3.3206
(+0.55%)
90.0
90.0
90.0
101.11 (1)
100.86 (1)
97.63
(3.20%)
103.71
(+2.83%)
101.40
(+0.54%)
105.37
(+4.47%)
99.41
(1.44%)
100.36
(0.50%)
101.49
(+0.62%)
100.35
(0.51%)
90.0
90.0
90.0
109.95
110.83
102.81
(7.24%)
117.25
(+5.79%)
111.88
(+0.95%)
124.40
(+12.24%)
111.59
(+0.69)
111.78
(+0.86)
113.52
(+2.43)
111.79
(+0.87)
3.0995
3.16 (2)
2.9392
(6.99%)
3.2378
(+2.46%)
3.1125
(1.50%)
3.3658
(+6.51%)
3.0611
(3.13%)
3.0940
(2.09%)
3.1368
(0.73%)
3.0941
(2.09%)
PBE
PBEsol
revPBE
PBE-D
PBE-D*
corr-PBE-D*_0.52
PBE-D*-C6(Li) = 0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
Ref. 55, Rp = 11.46%; Rwp = 14.57%; RF = 7.86%; R2F = 10.3%, reduced w2 = 0.55 as defined in GSAS (Larson, von Dreele, 1994).
work (Section II), Rp = 10.1%; Rwp = 11.7%; RBragg = 1.64% as defined in the FullProf program.58
a
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b
This
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Fig. 5 Relative errors (%) of the DFT or DFT-D calculated values to
the experimental ones (at 150 K) for the unit cell volume V and the
monoclinic angle in the Li2C4O4 phase.
Fig. 6 Relative errors (%) of the DFT or DFT-D calculated values to
the experimental ones (at 150 K) for the optimized lattice parameters,
a, b, and c in the Li2C4O4 phase.
recent work of Al Saidi et al.,78 the introduction of dispersion
in the formalism of the PBEsol functional leads to poorer
results than the PBE-D2 or PBE-TS treatment.
Using the set of dispersion parameters suggested by Grimme
(PBE-D), the unit cell volume was only slightly overestimated
(by +0.7%), with a slight expansion along the b- and c-axes
(+0.6/0.2%, respectively), while the a-axis undergoes a slight
contraction (0.6%). As expected, due to the molecular
orientation related to the directions of packing, inclusion of
long-range dispersion corrections in the DFT calculations has
a little effect on the treatment of the b-lattice parameter. For
PBE-D, the agreement with the experimental structure is
already much better for the dilithium structure studied in this
work than for other materials, especially those containing
hydrogen bonding chains. For instance, large over-compensation
of the dispersion forces has been noticed for the naproxen drug
with this method (DV/V = 10.5%).52 Such underestimation
of the lattice parameters in this compound can be ascribed to
the well-known hydrogen bond over-binding by PBE-D.
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Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
The full-geometry optimization with PBE-D* also gives rise
to a very satisfying result with a slight overall expansion of the
unit cell volume (+0.9%). With PBE-D*, the a-lattice
parameter is slightly overestimated (+0.1%). The results
of corr-PBE-D*_0.52 are inferior to those associated with
PBE-D or PBE-D*, the overestimation of the unit cell volume
being higher (+2.4%), with associated larger extents of lattice
parameter discrepancies with the experiment. In addition to a
better reproduction of lattice parameters for PBE-D*, the lowest deviation for the monoclinic angle is also found for this latter
method. Finally, calculations by using the PBE-D*-C6(Li) = 0
method, in which a C6 value equal to zero was selected for Li
resulted in practically equivalent results for both unit cell
volume and lattice parameters values compared to those
gained from PBE-D* in such a way that the incidence of this
parameter can be neglected for this compound (Table 1).
In addition to the direct comparison between experiment
and full-geometry optimizations, the change in unit cell energy
between the fixed-unit cell optimization and full-geometry
provides an indication of the degree of representation of the
various interactions within the unit cell. Negligible changes in
full-geometry optimization energies with respect to the fixed
one (DE) can be correlated to the same effect on all atoms
whether or not the atomic coordinates were optimized within
the fixed or fully optimized lattice dimensions. Minimal DE is
thus desirable in order to correctly represent molecular interactions in the system. As expected from the large departure
from the experimental structure for the full-geometry optimization, a prohibitive value is naturally found for the revPBE
functional (DE = 18.78 kJ mol1 per formula unit (p.f.u.)).
Similarly, noticeable values of this difference in energy are also
found for LDA and PBE functionals, with DE = 6.48 and
3.94 kJ mol1 p.f.u., respectively. Much lower DE values are
observed with the PBEsol functional (DE = 0.16 kJ mol1
p.f.u.) and with the three DFT-D methods (i.e. PBE-D,
PBE-D*, and corr-PBE-D*_0.52), the lowest one being
observed for PBE-D* (DE = 0.57, 0.03 and 1.63 kJ mol1
p.f.u. for the three DFT-D methods, respectively).
The variation of the calculated interlayer spacing (i.e. interplane distance), d, as a function of the level of theory, is
summarized in Table 1. As seen from this table, the calculated
d values are not satisfying for pure DFT results except for
PBEsol, which properly represents the interlayer distance with
a slight underestimation by 1.5%. PBE-D, PBE-D* and
corr-PBE-D*_0.52 also slightly underestimate the experimental value by 3.1%, 2.1% and 0.7%, respectively.
Once more, we remark that using the PBE-D*-C6(Li) = 0
method leads exactly to the same result as for PBE-D* even
for the stacking parameter.
In order to be complete, the discussion of the geometry
optimization shall include the consideration of intramolecular
bond lengths for crystalline Li2C4O4, which are given in
Table 2. Indeed, even if the impact of the intramolecular
geometry on the electronic structure is weak, this directly
affects the structure and energy of the systems in the computations, which are meaningful especially for the intercalation
potential estimation. A general trend of underestimation of the
bond length of the C1QO1 double bond can be noticed.
Similarly, the bond length of the C2QO2 double bond is also
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Table 2 Intramolecular bond lengths (Å) for the Li2C4O4 crystal. Root mean square deviation (RMSD) values with respect to experimental
reference data of this work at 150 K
Compound
Bond length
Expta
Li2C4O4
C–C
C1–O1
C2–O2
O1–Li
O2–Li
RMSD
1.472
1.333
1.263
1.983
1.923
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a
(9)
(14)
(12)
(15)
(11)
LDA
PBE
PBEsol
revPBE
PBE-D
PBE-D*
corr-PBE-D*_0.52
1.462
1.255
1.250
1.922
1.903
0.046
1.475
1.267
1.260
2.020
1.985
0.044
1.472
1.263
1.257
1.984
1.960
0.036
1.482
1.273
1.265
2.073
2.031
0.069
1.473
1.265
1.259
1.988
1.979
0.040
1.473
1.265
1.258
1.984
1.969
0.037
1.474
1.266
1.259
1.990
1.976
0.038
This work (Section II), Rp = 10.1%; Rwp = 11.7%; RBragg = 1.64%.
slightly underestimated across all calculations except in rev-PBE,
which exhibits a slight overestimation. The simple bond C–C is
very well reproduced by using the DFT-D methods, while the
interatomic distance between O and Li is underestimated in
LDA and overestimated for all other computations. In agreement with the experimental observation, the dianion rings in
Li2C4O4 computed through the PBE-D* method are planar
with all the C–C and CQO bonds in each ring being either
equal or nearly equal. The tiny departure from the CQO bond
length equalization can be analysed as a slight deviation from
the complete aromaticity in this structure. However, the
aromaticity of oxocarbons dianions is still a matter of debate.
For instance, the work of Aihara79 predicted all oxocarbon
dianions but deltate to be substantially nonaromatic with
negligibly small resonance energies. Moreover, from an analysis
of vibrational spectroscopy, assignment of IR-Raman spectra in
the squarate salts M2C4O4 (M = Li, Na, K and Rb) series
indicates an enhancement of electronic delocalization and consequently in the degree of aromaticity for salts with larger ions.80
The overall agreement between the different levels of theory
and experiment is characterized by the root mean square
deviations (RMSDs) with respect to the experimental data,
which quantify the deviations from the experimental bond
lengths. The relative ability of the various XC functionals or
DFT-D methods to reproduce bonding distances for Li2C4O4
follows the same ranking than already found for the lattice
parameters reproduction. The PBEsol and PBE-D* provide
the most correctly described intramolecular geometry (with
RMSD = 0.036 Å and 0.037 Å, respectively). Again, among
the various DFT-D methods, the PBE-D* comes out to be the
best one.
According to the aforementioned results, we consider that in
the case of Li2C4O4, the overall quality of structural reproduction is the best one for the PBE-D* level of theory. The
crystals of LiC4O4 and Li3C4O4 are characterized by the same
kind of interactions, and therefore calculations for the LiC4O4
and Li3C4O4 phases were first performed using the PBE-D*
level of theory in order to identify the most stable phase
among various possibilities. Once the most stable phase was
identified, the incidence of choosing other DFT or DFT-D
methods (among the ones giving the lowest discrepancies with
the experiment for the geometry of Li2C4O4) on the intercalation potential value was also studied (see Section IV.2).
1.2. LiC4O4 and Li3C4O4. Due to excellent agreement with
experimental data for Li2C4O4, the identified methodology
(PBE-D*) applied to the LiC4O4 and Li3C4O4 phases can then
be utilized to calculate the materials electrochemical properties
from the total energy values issued from all of these phases (see
Section IV.2). In the special case of this study, this necessitates
to propose some structural models for crystalline LiC4O4 and
Li3C4O4. For these two phases, results gathered for PBEsol are
also compared to the PBE-D* calculations.
Following the procedure presented in Section III, we have
first compared the optimized lattice constants (Table 3) and
the energetic profile (Fig. 7) of the various models for the
delithiated phase, LiC4O4 in PBE-D* calculations. The delithiation process results in two degenerate phases, with two distinct
space groups: crystals A and B, on one hand and C and D, on
the other hand, that are characterized by the P2/m and P2/c
space group, respectively. The relaxed crystal structure of
these phases is shown in Fig. 8. Crystals A and B exhibit an
Table 3 Optimized lattice parameters, a, b, c, a, b and g, unit cell volume, V, and inter-plane distance, d, for LiC4O4 by using either the PBE-D* or
the PBEsol level of theory. (Departure from optimized values for Li2C4O4 is given in parentheses). Difference in total energy, DE, between the four
LiC4O4 models A, B, C or D in the PBE-D* calculation (the most stable phase has been set to zero)
LiC4O4
Crystal
DE (eV)
a (Å)
b (Å)
c (Å)
a (deg.)
b (deg.)
g (deg.)
V/Z (Å3)
d (Å)
PBE-D*
A
0.63
C
0
0
102.00
(+1.63%)
102.06
(+1.69%)
99.62
(0.74%)
99.6300
(0.73%)
105.654
(+4.20%)
99.62
(1.75%)
115.99
(+3.77%)
116.05
(+3.82%)
102.80
(8.03%)
102.80
(8.03%)
119.00
(+6.36%)
102.82
(8.10%)
3.0817/3.2744
(0.40%)/(+5.83%)
3.0827/3.2769
(0.37%)/(+5.91%)
3.0868 (0.23%)
D
3.4700
(+4.50%)
3.4702
(+4.51%)
3.4172
(+2.91%)
3.4182
(+2.94%)
3.5052
(+5.92%)
3.4182
(+3.29%)
90.0
0.63
9.5266
(0.75%)
9.5275
(0.74%)
7.4822
(22.05%)
7.4803
(22.07%)
9.5432
(0.68%)
7.4825
(22.13%)
90.0
B
7.1738
(+0.61%)
7.1783
(+0.67%)
8.1556
(+14.38%)
8.1557
(+14.38%)
7.3888
(+2.94%)
8.1548
(+13.61%)
PBEsol
B
C
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90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
90.0
3.0873 (0.22%)
3.0866/3.3912
(0.83%)/(+8.95%)
3.0865 (0.84%)
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Fig. 7 Relative total energy (p.f.u.) with and without dispersion, dispersion energy (p.f.u.) for the set of four relaxed crystal structures of LiC4O4
(phases issued either from A/B or C/D) and for the set of two relaxed crystal structures of Li3C4O4 (phases issued from E or F) in PBE-D*
calculations. The most stable phase has been set to zero for both LiC4O4 (left) and Li3C4O4 materials (right).
Fig. 8
Relaxed structures of crystals A, B, C and D.
augmentation of the c-lattice parameter (+4.5%) with respect
to the one of Li2C4O4, while the phase undergoes only a slight
reduction of the b-axis (0.8%) and a slight increase of the
a-lattice parameter (+0.60.7%). The volume of the cell
increases by +3.8% when the compound is delithiated in this
manner. After the full geometry optimization, Li ions are
fourfold coordinated to oxygen atoms within a distorted
tetrahedron (dmin = 1.983 Å; dmax = 2.015 Å) in which
distances are augmented with respect to the ones characterizing Li2C4O4 after optimization with PBE-D* (dmin = 1.969 Å
and dmax = 1.984 Å). On the other hand, for the other
delithiation schemes (crystals C and D), stronger modifications are applied to the Li2C4O4 phase: a decrease of 8.0% of
the unit cell volume is observed after lithium extraction. In
these structures, the a-axis is reduced by 22.1%, whereas the
b- and c-lattice parameters are increased by 14.4% and 2.9%,
respectively. In such a structure, the Li ions lie in a distorted
octahedron of oxygen atoms, with bond lengths of 2.242 Å
and 1.966 Å for the Li–Oequatorial and Li–Oapical bonds,
respectively.
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Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
From the energetic perspective highlighted in Fig. 7, a first
observation can be drawn concerning the classification of
crystals in terms of relative stability: total energy excluding
dispersion part presents the same trend compared to that
characterizing the total energy integrating dispersive terms.
Therefore, a first indication of semi-quantitative ranking can
already be reached without taking into account the dispersion
correction since the nature of the most stable phase is preserved by the adjunction of dispersive term. We can remark
that the P2/c phase is much more stable by 0.63 eV p.f.u., with
respect to the P2/m one (Fig. 7). This higher stabilization
originates both from the total energy value without dispersion
and from the van der Waals one. Extent of dispersion energy
amounts to 0.7 and 0.8 eV p.f.u. for P2/m and P2/c phases,
respectively, which is slightly smaller compared to 1 eV p.f.u.
in the case of the Li2C4O4 crystal.
Upon lithium intercalation in Li2C4O4 (i.e. Li3C4O4), a
volume decrease of 9% is observed for crystal E, the extent
of the lattice parameter diminution (for b- and c-axes) or
augmentation (for a-axis) with respect to the Li2C4O4 phase
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Table 4 Optimized lattice parameters, a, b, c, a, b and g, unit cell volume, V, and inter-plane distance, d, for Li3C4O4 by using either the PBE-D*
or the PBEsol level of theory. (Departure from optimized values for Li2C4O4 is given in parentheses). Difference in total energy, DE, between the
two Li3C4O4 models E or F in the PBE-D* calculation (the most stable phase has been set to zero)
Li3C4O4
Crystal
DE
(eV)
PBE-D*
E
1.89
F
0
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PBEsol
E
F
a (Å)
b (Å)
c (Å)
a (deg.)
b (deg.)
g (deg.)
V/Z (Å3)
d (Å)
7.4663
(+4.71%)
7.7819
(+9.14%)
7.4613
(+3.95%)
7.7998
(+8.67%)
9.1129
(5.06%)
9.6020
(+0.04%)
9.2989
(+3.22%)
9.6266
(+0.19%)
3.1096
(6.35%)
3.6228
(+9.10%)
3.0964
(6.44%)
3.5713
(+7.91%)
102.09
(+13.43%)
90.0
104.26
(+3.89%)
110.40
(+10.00%)
103.05
(+1.63%)
110.94
(+10.37%)
83.15
(7.61%)
90.0
100.00
(10.54%)
126.87
(+13.50%)
103.08
(7.87%)
125.22
(+11.92%)
2.8464
(8.00%)
3.6159
(+16.87%)
2.8780
(8.15%)
3.5673
(+14.61%)
99.335
(+10.37%)
90.0
84.603
(6.00%)
90.0
Fig. 9 Relaxed structures of crystals E and F.
being quite isotropic (Table 4). The symmetry of the phase is
lowered (space group P1% , Fig. 9). Li ions lie either in a
distorted tetrahedron for initially occupied positions (dmin =
1.889 Å and dmax = 2.029 Å) or in a distorted octahedral
environment for additional positions (with d(Li–Oap) = 2.266 Å
and d(Li–Oeq) = 2.351 or 2.159 Å).
The other phase resulting from the full-geometry optimization
(crystal F) is characterized by a noticeable volume augmentation of +13.5% while the structure is maintained in the
monoclinic system (space group C2/m) due to the symmetry
related to the initial positioning of additional Li ions. In this
system, the b-axis is practically not affected by the relaxation
of atoms within the cell whereas the a- and c-lattice parameters
both undergo a large increase (by +9.1%). For the relaxed F
phase (Fig. 9), an environment similar to the one characterizing
Li in Li2C4O4 is kept for already existing Li ion positions (CN 4,
dmin = 1.959 Å and dmax = 2.078 Å), while the additional Li
ions are maintained in between two parallel squarate ions
through cation–p interactions (in that sense, the d parameter
does not correspond in this crystal to a true inter-plane
distance related to the stacking of molecules). This second
relaxed phase is less stable than the first one (crystal originating
from model E) by 1.89 eV p.f.u. (Fig. 7). The extent of
dispersion energy in these phases is higher than the one
characterizing the two models of delithiated Li2C4O4 and
Li2C4O4 itself, with values of 1.2 and 1.1 eV p.f.u. for the
most stable and less stable Li3C4O4 models, respectively.
In the delithiated phase B after PBEsol full-geometry optimization (Table 3), Li ions are still fourfold coordinated to
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oxygen atoms within a distorted tetrahedra (dmin = 1.977 Å;
dmax = 2.035 Å) in which distances are not very far from those
originating from PBE-D* optimization (dmin = 1.983 Å and
dmax = 2.015 Å). Similarly to the PBE-D* calculation, crystal
C undergoes a reduction of 8.1% in the PBEsol calculation.
The resulting stacking parameter, d, of 3.0865 Å is nearly
identical to the one found for the PBE-D* modeling (3.0868 Å).
The sixfold-coordination of Li ions within a distorted octahedron of oxygen atoms is also preserved, with the bond
lengths of 2.241/2.242 Å and 1.967 Å for Li–Oequatorial and
Li–Oapical bonds, respectively.
For crystal E treated with the PBEsol functional (Table 4),
the unit-cell volume is slightly larger (103.08 Å3) than that of
the PBE-D* calculation (100.00 Å3). The inter-plane distance,
d = 2.8780 Å, is also enhanced in comparison with the result
originating from PBE-D* calculation (2.8464 Å). As already
observed for these latter calculations, Li ions lie either in a
distorted tetrahedron for initially occupied positions or in
distorted octahedral environment for additional positions.
The interatomic distances are respectively (dmin = 1.905 Å
and dmax = 1.991 Å) for the first case of environment and
(d(Li–Oap) = 2.266 Å and d(Li–Oeq) = 2.572 or 2.207 Å) for
the second one. Therefore, interatomic distances in the sixfold
environment are longer in the PBEsol calculations compared
to those in the PBE-D* ones, while similar distances are found
in the tetrahedral environment.
For crystal F in PBEsol calculation, the distorted tetrahedral
environment characterizing non-additional Li ions is characterized by the following interatomic distances: dmin = 1.948 Å and
Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
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dmax = 2.067 Å (not far from those observed after PBE-D*
relaxation). This phase is still the less stabilized one, but in
PBEsol calculation, the extent of stabilization of crystal E as
compared with crystal F amounts to 1.44 eV p.f.u. and is thus
less pronounced than for PBE-D* calculation (1.89 eV p.f.u.).
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2. Intercalation potentials, DOS, AIM study and electrostatic
lattice energies
Calculated intercalation potentials are compared with the
currently available experimental data. Experimental values
of intercalation potentials for oxidation (0.9 V56) and
reduction (3.9 V81) of Li2C4O4 are extracted from previous
electrochemical analyses (galvanostatic measurements).
The equilibrium voltage, V, relative to lithium metal for
lithium insertion into a host material M can be estimated
according to:
VðxÞ ¼
GðMÞ þ xGðLiÞ GðLix MÞ
xz
ð6Þ
where G is the Gibbs free energy per formula unit (p.f.u), and z
is the elementary charge per lithium ion (z = 1).
By neglecting small changes in entropy (TDS) and volume
(PDV), which are expected to be relatively small,82 the Gibbs
free energy (DG) term can be approximated by the internal
energy (DE) and the average potential value can be expressed
as:
VðxÞ ¼
EðMÞ þ xEðLiÞ EðLix MÞ
xz
ð7Þ
where E refers to the total energy p.f.u.
After the geometry optimizations of both lithiated
(Li3C4O4) and delithiated (LiC4O4) phases, the average
lithium insertion potential can thus be extracted from
eqn (7) in which x = 1, since one lithium ion p.f.u. is
considered to be (de)-inserted from Li2C4O4.
By taking into account the results of the previous section
(IV.1.2), the phase with the highest stability among the various
studied models has been selected for both delithiation (phase
originating from the full-geometry optimizations of crystals C
or D) and intercalation of lithium (relaxed phase issued from
crystal E). In order to appreciate the incidence of choice of the
functional or dispersion-corrected method on the oxidation
or reduction of Li2C4O4, the two average potentials were
evaluated within the most suited set of methods considered
in this work (PBEsol, PBE-D, PBE-D*, corr-PBE-D*_0.52)
(Fig. 10a and b). For the evaluation of the average potential
value within dispersion-corrected methods, semi-empirical
treatment of the dispersion was only restricted to the lithiated
or delithiated phases while bcc Li was treated with the PBE
functional. Otherwise, as proven from test calculations that we
performed for Li bcc metal in the PBE-D* treatment, geometry features are less close to the experimental one and the error
generated in the estimation of the cohesive energy for this
latter would affect the intercalation voltage estimation.
Indeed, within the PBE-D* formalism, the relaxed a lattice
parameter exhibits a discrepancy with respect to the experiment
as large as 6.7% (against 2.05% for the PBE-D treatment).
Applying dispersion terms to the bcc Li metal thus leads as
expected to clear overestimation of these long range forces.
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Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
Fig. 10 Comparison of calculated and experimental potentials56,81
for (a) the delithiation and (b) the intercalation of Li2C4O4.
As a result of dispersion treatment, the energy of the Li phase
is B0.12 eV more stable than in the PBE calculation. By
including such corrections, the electrochemical potential
should thus be lowered by B0.1 eV compared to the values
presented hereafter concerning calculations with the PBE
treatment of bcc Li (which are more physically reasonable).
Concerning the reduction potential, PBE-D and PBE-D*
exhibit low discrepancies with experiment, of +0.04 V and
0.11 V, respectively. A much higher difference is observed for
the corr-PBE-D*_0.52 method and the PBEsol functional,
which both underestimate the potential (by 0.33 and 0.62 V,
respectively). For the oxidation potential, the lowest discrepancies are still observed for PBE-D and PBE-D* (DV =
0.17 V for both levels of theory). The corr-PBE-D*_0.52 result
is not far away from these lowest discrepancies. Once more,
PBEsol calculation provides the poorest results among the
restricted set of the probed methodologies. Despite the lack of
knowledge of the crystal structure for Li3C4O4 and LiC4O4
phases, Fig. 10a and b thus highlight the good agreement
between experimental and calculated intercalation potentials
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with the DFT-D computation (especially in the case of PBE-D*)
for both oxidation56 and reduction81 of the dilithium squarate.
Therefore, predictions with this method are indeed at a
0.1–0.2 V scale of precision, which seems to be a typical for
current computational studies of intercalation voltages.83,84
For the various Li2C4O4, LiC4O4 and Li3C4O4 optimized
PBE-D* structures, we calculated the density of states (DOS)
by using the all electron code, Wien2k, within the PBE
formalism, as shown in Fig. 11(a–e). Due to the well-known
limitation of DFT to predict the band gap energy, it is not
possible to stand that determined Eg corresponds to accurate
experimental values but this energy is however given as first
indication. A semiconductor behavior is found for the
Li2C4O4 phase, with a band gap lower than 3 eV (2.4 eV).
On the other hand, the different LiC4O4 and Li3C4O4 structures appear to be metallic, with no band gap (i.e., some DOS
are present at the Fermi level). The densities of states thus
show that removal of lithium as well as lithium intercalation in
Li2C4O4 introduces new states in the gap of the Li2C4O4
phase. In the case of the delithiated crystals (LiC4O4 structures),
lithium de-intercalation from the Li2C4O4 matrix makes the
band located within the range between 1 and 0 eV with
respect to the Fermi level to shift up. Therefore, such a band
now crosses the Fermi level. For crystals A or B the bandwidth
of such a band is not modified while for crystals C or D it is
slightly enhanced. For the intercalated phase (Li3C4O4 structures), the DOS of crystal F suggest a definite localization of
the excess charge in link with the appearance of a very
localized state below the Fermi level while such a kind of
localized state appears above the Fermi level in the case of
crystal E, in which a band of moderate N(E) crosses EF.
Net atomic charge (NAC) for the set of atoms within the
crystal phases was obtained through electron density partitioning within the AIM approach by subtracting electron
density population within the atomic basin and atomic number
value for each species. For all phases (except the crystal F), the
net Li charge corresponds to B+0.9, indicating a full-oneelectron transfer from Li to the adjacent carbonyl group,
leaving a total charge of the squarate anion of 1.8 (total
sum of NAC for C and O atoms), near from the formal value.
In the crystal F, the Li ion positioned in between two p systems
loses a slightly lower amount of electronic charge: i.e. qLi B +0.8.
As expected from previous studies of both inorganic and
organic materials,30,85–90 atomic electron populations for C
and O atoms belonging to the squarate ions lie in between
those expected for purely covalent and ionic bonds, due to the
iono-covalent character of bonding. Concerning NACs in
the delithiated and lithiated phases, we should mention that
Fig. 11 Total density of states in arbitrary units, N(E), as a function of E EF, of Li, C and O atoms for (a) A or B, (b) C or D, (c) Li2C4O4,
(d) E and (e) F crystal phases.
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Table 5 Average Madelung constants of the Li and O atoms (Mave), total Madelung constant (M), distance to the nearest neighbour (dmin in Å)
and electrostatic lattice energy (EES in kJ mol1) calculated from the predicted crystals structures of LiC4O4, Li2C4O4 and Li3C4O4
Material
Method
Crystal
Mave(Li)
Mave(O)
Ma
LiC4O4
PBEsol
PBE-D*
PBE-D*
PBEsol
PBE-D*
PBEsol
PBE-D*
PBE-D*
C
B
C
0.913
0.892
0.918
2.000
2.010
3.238
3.304
3.461
0.742
0.953
0.720
2.515
2.523
5.141
5.200
4.886
0.828
0.923
0.819
2.258
2.266
4.190
4.252
4.174
Li2C4O4
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Li3C4O4
a
E
E
F
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.002)
(0.002)
(0.001)
dmin
EES
1.9666
1.9838
1.9663
1.9600
1.9686
1.9051
1.8890
1.9589
586.3
648.0
580.2
1604.9
1604.1
3046.3
3136.5
2968.8
The errors in the calculations of total Madelung constants are included in parentheses.
the PBE functional is expected to over-delocalize the unpaired
electron characterizing each of the Li3C4O4 and LiC4O4
structures, due to the incorrect description of the selfinteraction (self-interaction error) causing extra repulsion.
Such a failure of adequate description of strongly localized
unpaired charge carriers can be solved by the use of hybrid
exchange–correlation functional. In this work, the AIM
method on these structures (LiC4O4 and Li3C4O4) provides
however a slight differentiation in the electronic charge of the
various C or O atoms between themselves (of the maximum
order of 0.1) while no distinction was found for Li2C4O4. In
these delithiated or lithiated phases, the averaged values are
mentioned hereafter for seek of simplicity. In Li2C4O4, the
NACs of C and O are, respectively, +0.81 and 1.26. By
passing from this phase to the delithiated one (LiC4O4), the
oxidation takes place through the concomitant average
increase of qC (qC = +0.88 for crystal B and +0.90 for
crystal C) and decrease of |qO| (qO = 1.10 for crystal B and
1.12 for crystal C). Similarly, in crystal E (Li3C4O4), the
electronic charge afforded by additional lithium atoms is
redistributed on the O atoms but also on the carbon, with
net charge of carbon and oxygen amounting in average to
+0.64, and 1.32, respectively. For crystal F, due to the
positioning of additional Li, the reduction mostly concern
carbon atoms (qC = +0.58) while O atoms remain as a whole
slightly affected by the introduction of Li (qO = 1.23).
From Table 5, depicting the result of application of the
EUGEN code to the considered phases with the previously
mentioned net atomic charges, one can draw the general
comment that the electrostatic lattice energy component is
linearly increasing by passing from LiC4O4 to Li2C4O4 and
from Li2C4O4 to Li3C4O4. This trend can be correlated to the
enhancement of attractive Li–O interactions contributions
within the crystal as exemplified by the increase in Madelung
constants. Due to the very low difference in charges between
PBEsol and PBE-D* (|Dq| B 0.01), the main distinction in
electrostatic lattice energy should originate from the differences
in interatomic distances after relaxation. With respect to this
point, we should stress that inclusion of dispersion does not
influence the electronic structure or the charge density and has
thus in turn no effect on the NACs. The sole incidence of
dispersion is related to the indirect effect of modifications of
the relaxed geometries.
Within PBE-D* calculation, Table 5 exhibits a somewhat higher contribution of the globally stabilizing electrostatic lattice energy in crystal C as compared with crystal B.
11410
Phys. Chem. Chem. Phys., 2012, 14, 11398–11412
However, crystal C was found more stable than the B phase
owing to total energy values described in Section IV.1.2.
Therefore, the ranking in energy between the two types of
delithiated phases cannot be ascribed to the electrostatic
component, which follows a reverse trend. This may indicate
that other stabilizing contributions such as polarization might
be involved in the differentiation of relative stabilities between
the two delithiated phases. For crystals B and C, the Madelung
constants below one might be interpretated from the fact that
these systems represent a cross-over between highly-ionic and
more covalent systems. With this level of theory (PBE-D*), we
can remark that electrostatic lattice energy is slightly lower for
crystal F as compared with the E phase. Such a trend follows
the relative stability of the two phases. This effect can be
ascribed to distinction in the environment of Li ions. In both
crystals, four of them lie in a pseudo tetrahedral environment.
On the other hand, for crystal E the two additional ones are
located in a sixfold environment of Li ions, while in crystal F, these
two ions are mostly interacting through cation–p interactions.
V. Conclusion
The present work was stimulated by the rising interest in the
development of organic electrodes for Li-ion batteries. The
main challenge in the field is the choice of the ‘right’ molecular
crystal obtained from the biomass that has desirable electrochemical properties without any problem of solubility in the
electrolyte. Theoretical prediction of high-performance novel
organic insertion materials is currently hindered by the lack of
reliable experimental structures. Additionally, prediction of
organic crystal structures based on the knowledge of molecular geometry remains a challenging task. The complexity of
such a calculation is even enhanced in the case of salts. The
difficulty arises from the fact that the sum of various interactions such as weak intermolecular forces (e.g. van der Waals),
H-bonding and p–p interactions as well as Coulomb interactions in organic salts determine the final packing structure.
To date, most of the computational studies regarding these
organic materials were restricted to the molecular state.
However, the control of molecular properties is not sufficient
for successful insertion materials design, as bulk properties
(for example, orbital overlap, packing) depend on the crystal
structure. In this work, we used the dilithium squarate
as benchmark system in order to define the most suited
methodology to account for geometry and electrochemical
properties.
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From this study, DFT-D calculations emerge as a computationally tractable means for assessing organic crystal lithium
salts with non-negligible dispersion interactions in which the
molecules possess both localized and delocalized orbitals and
are devoid of hydrogen bonding. In particular, the original
Grimme model was modified by rescaling van der Waals radii
in the damping function (scaling factor: 1.05 for all atoms),52
whereas the value of the global scaling factor s6 was set to 0.75.
This variation of the original D term, named D*,52 was shown
to be particularly effective for the modeling of the well-known
molecular crystal, Li2C4O4.
It exhibits better agreement in geometry parameters with
experiment compared to the standard PBE-D calculations.
Another methodology using the optimized s6 value for the
organic crystal, the naproxen drug, in conjunction with the
Rvdw fixed at the PBE-D* values,52 i.e. ‘corr-PBE-D*_0.52’,
was also tested. Although results are globally less satisfactory
for the dilithium squarate compared with two other methods
including empirical dispersion corrections, this modulation of
s6 comes out to be more efficient in the case of materials
characterized by H-bonding.91 Among other possible functionals, revPBE leads to large discrepancies with the experiment, while PBEsol is able to reproduce the intermolecular as
well as the intramolecular bonding with accuracy comparable
to that of the vdW-corrected calculations.
Having tested density functional theory with and without
dispersion corrections for the lattice parameters and intramolecular bonds, we have applied the most suited methodologies to the relaxation of selected hypothetical crystal
structures for either delithiated or lithiated phases. In order
to check the optimized crystal structures for intercalated
(Li3C4O4) or deintercalated (LiC4O4) derivatives, i.e. that they
actually correspond to real minima on the PES, we calculated
the phonon frequencies at the G point of the Brillouin zone.
No unstable modes were found for Li3C4O4, whereas a single
unstable mode was found for LiC4O4, at 5.5 meV. This mode
corresponds to rotations of C4O4 units, in the ab plane, similar
to the distorsion found for the F-phase of Li3C4O4. This
signature of a symmetry lowering in LiC4O4 does not affect
our conclusions on the de-intercalation potentials issued
from this phase. However, this might indicate that another
structural model should be involved during the experimental
delithiation process, which is more difficult to identify than
the lithiated one, due to the higher possibilities of lithium
positioning.
A first step towards energy partitioning in view of interpretating
crystal phases relative stability in link with (de)-intercalation
processes has been performed through the explicit evaluation
of electrostatic components of lattice energy. Due to the used
methodology, all possibilities of frameworks were not tested
for these compounds and such DFT calculations cannot
ensure to reach the global minimum for a given material.
The presented studies will thus be extended to more thorough
crystal structure predictions in a forthcoming work.
Acknowledgements
The authors gratefully acknowledge the GCEP sponsors for
financial support of the project within the GCEP program
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(http://gcep.stanford.edu/). We would like to thank the
DSI-CCuB from the University of Bourgogne, the computer
center MCIA of the University of Bordeaux and Pays de
l’Adour and the CINES from Montpellier for allowing us to
access their computer facilities. We gratefully acknowledge
generous allocations of computing time from the CINES. CF
gratefully acknowledges Dr Tomáŝ Bučko from the Comenius
University in Bratislava for his help during installation of
the VASP code. We gratefully acknowledge Dr Jean-Noël
Chotard from the Laboratoire de Réactivité et Chimie des
Solides (UMR-CNRS 7314 Amiens, France) for access to
in situ XRD facilities and for fruitful discussions. EII gratefully acknowledges generous allocations of computing time
from the Monash Sun Grid Cluster at the e-research centre
of Monash University, Australia, on which the Madelung
constant calculations were performed. We thank the Australian
Research Council for funding a Discovery grant to EII and
DRM, an Australian Postdoctoral Fellowship for EII and a
Federation Fellowship for DRM.
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