THE JOURNAL OF CHEMICAL PHYSICS 132, 114504 共2010兲
Structural investigation and thermodynamical properties
of alkali calcium trihydrides
P. Vajeeston,a兲 P. Ravindran, and H. Fjellvåg
Department of Chemistry, Center for Materials Sciences and Nanotechnology, University of Oslo,
P.O. Box 1033 Blindern, N-0315 Oslo, Norway
共Received 6 August 2008; accepted 6 January 2010; published online 16 March 2010兲
The ground-state structure, equilibrium structural parameters, electronic structure, and
thermodynamical properties of MCaH3 共M = Li, Na, K, Rb, and Cs兲 phases have been investigated.
From the 104 structural models used as inputs for structural optimization calculations, the
ground-state crystal structures of MCaH3 phases have been predicted. At ambient condition,
LiCaH3, NaCaH3, and KCaH3 crystallize in hexagonal, monoclinic, and orthorhombic structures,
respectively. The remaining phases RbCaH3 and CsCaH3 crystallize in a cubic structure. The
calculated phonon spectra indicate that all the predicted phases are dynamically stable. The
formation energy for the MCaH3 phases have been calculated along different reaction pathways. The
electronic structures reveal that all these phases are insulators with an estimated band gap varying
between 2.5 and 3.3 eV. © 2010 American Institute of Physics. 关doi:10.1063/1.3299732兴
I. INTRODUCTION
Crystallization plays an important role in various industries as a large-scale technique for separation, purification,
and structure determination. Most of the compounds crystallize at some point during their production process. Knowledge about the crystal structures is a prerequisite for the rational understanding of the solid-state properties of new
materials. The current interest in the development of novel
metal hydrides stems from their potential use as reversible
hydrogen storage devices at low and medium temperatures.
The crystal structure, shape, size, and surface composition of
materials are major factors that control the hydrogen sorption
properties for energy storage applications. To act as an efficient energy carrier, hydrogen should be absorbed and desorbed in materials easily and in high quantities. Also, in order to use them in practical applications, the materials
involved in such compounds should be easily available in
large quantities with cheaper price. Alkali- and alkalineearth-based complex hydrides are expected to have a potential as viable modes for storing hydrogen at moderate temperatures and pressures.1–7 These hydrides 共e.g., LiAlH4,
NaAlH4, etc.1–7兲 have higher hydrogen storage capacity at
moderate temperatures than conventional hydride systems
based on intermetallic compounds. The disadvantage for the
use of these materials for practical applications is the lack of
reversibility and poor kinetics. Recent experimental findings
have shown that the decomposition temperature for certain
complex hydrides can be modified by introduction of
additives3,4 and/or reduction in particle size.8–11 This has
opened up research activities on identification of appropriate
admixtures for known or hitherto unexplored hydrides. Density functional theory has proven to be very useful for prea兲
Electronic
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dicting the reaction thermodynamics of metal hydrides with
known structures,12–24 but reliable prediction of unknown
crystal structures is much more challenging.
As proposed in our earlier communications,25 it should
be possible to form several series of hydrides with alkali and
alkaline-earth metals in combination with group III elements
of the Periodic Table. In this article we will present results
from structural study of the phases MCaH3 共M = Li, Na, K,
Rb, and Cs兲 based on density-functional total energy calculations. In LiCaH3 and NaCaH3 phases, one can store H up to
6 and 4.6 wt %, respectively. The rest of the considered
phases have relatively low weight percent H content. However, these phases may have potential application in electronic industry, as we proposed in our earlier studies on these
type of hydrides.26,27 In addition, these phases may coexist as
by-products/mixed-phases of the multiphase lightweight hydrides during synthesis. Hence, the understanding about the
structural phase stability of these phases is very important. It
should be noted that the structural properties of the MCaH3
phases are not yet known experimentally except that for
RbCaH3 and CsCaH3.28,29 The formation of KCaH3 has also
been reported in Ref. 30. One of the motivations for this
study is to investigate in detail the ground-state atomic arrangements, electronic structures, and the chemical bonding
behavior within the MCaH3 series, in order to check the
stability of these materials for hydrogen storage and other
electronic applications. The structural phase stability at high
pressures for the MCaH3 phases are under examination and
the results will be published in a forthcoming article.
II. COMPUTATIONAL DETAILS
It is well known that the generalized-gradient approximation 共GGA兲31 usually gives better equilibrium structural
parameters as well as energetics for different phases and
hence we have used GGA for all the present calculations.
132, 114504-1
© 2010 American Institute of Physics
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114504-2
J. Chem. Phys. 132, 114504 共2010兲
Vajeeston, Ravindran, and Fjellvåg
The structures are fully relaxed for all volumes considered
using force as well as stress minimization. The projectedaugmented-wave 共PAW兲32 implementation of the Vienna ab
initio simulation package 共VASP兲33 was used for the total energy calculations to establish phase stability and transition
pressures. In order to avoid ambiguities regarding the freeenergy results, the same energy cutoff and a similar k-grid
density for convergence were always used. In all calculations, 500 k points in the whole Brillouin zone were used
for KMnF3-type structure and a similar density of k points
was used for all structural arrangements. A criterion of
0.01 meV atom−1 was placed on the self-consistent convergence of the total energy, and all calculations used a planewave cutoff of 500 eV. The formation energies 共⌬E兲 have
been calculated according to the reaction equations:
MH + CaH2 → MCaH3 ,
共1兲
M + CaH2 + 21 H2 → MCaH3 ,
共2兲
M + Ca + 23 H2 → MCaH3 ,
共3兲
MH + Ca + H2 → MCaH3 .
共4兲
The total energies of M, MH, and CaH2 have been computed for the ground-state structures, viz., in space group
Im3̄m for M, Fm3̄m for MH, and Pnma for CaH2, all with
full geometry optimization. The energy of a free hydrogen
molecule was computed via a hydrogen dimer in a larger
cubic unit cell. Convergence at the 10−4 eV level was
achieved with Ecut = 700 eV in a cell of length 9 Å using the
Fermi smearing technique. The calculated corresponding
value is ⫺6.794 eV/f.u. which is well agrees well with another theoretical simulation.34 The calculated H bond length
in our work is 0.752 Å; the corresponding experimental
value is 0.741 Å,35 as the plane-wave technique of VASP is
better suited to solids rather than atoms and molecules. However, it was proven that use of the same PAW potential and
the same methodology in computing reactance and product
makes for greater accuracy in ⌬H due to cancelation of
errors.34
The Phonon program developed by Parlinski36 was used
for lattice dynamic calculations. The force calculations were
made using the VASP code with the supercell approach and
the resulting data were imported into the Phonon program.
Thereafter the full Hessian was determined and the phonon
density of states 共DOS兲 was calculated. The 2 ⫻ 2 ⫻ 1 共for
LiCaH3 and NaCaH3兲 and 2 ⫻ 2 ⫻ 2 共for KCaH3, RbCaH3,
and CsCaH3兲 supercells were constructed from the optimized
structures for the force calculations. The Hessian 共harmonic
approximation兲 was determined through numerical derivation using steps of 0.03 Å in both positive and negative directions of each coordinate to estimate the harmonic potentials. The sampling of the phonon band structure for the
calculation of phonon DOS was set to “large” in the Phonon
program36 with a point spacing of 0.005 THz.
To gauge the bond strength, we have used the bond overlap population 共BOP兲 values on the basis of the Mulliken
population implemented in the CASTEP code.37 For the
CASTEP computation we used the optimized VASP structures
as input with norm-conserving pseudopotentials and the
GGA exchange correlation functional proposed by Perdew,
Burke, and Ernzerhof were used.31
III. RESULTS AND DISCUSSION
Thirty potentially applicable structure types 共ABX3; A
and B represent the first and second elements in the following structures and X is either O, F, or H兲 have been used as
starting inputs in the structural optimization calculations for
the MCaH3 compounds 共Pearson structure classification notation in parenthesis兲: KMnF3 共tP20兲, GdFeO3
关NaCoF3共oP20兲兴, KCuF3 共tI20兲, BaTiO3 关RbNiF3共hP30兲兴,
CsCoF3 共hR45兲, CaTiO3 关CsHgF3共cP5兲兴, PCF3 共tP40兲,
KCuF3 共tP5兲, KCaF3 共mP40兲, NaCuF3 共aP20兲, SnTlF3
共mC80兲, KCaF3 共mB40兲, LiTaO3 共hR30兲, KCuF3 共oP40兲,
PbGeS3 共mP20兲, CaKF3 共mP20兲, KNbO3 共tP5兲, KNbO3
共oA10兲, KNbO3 共hR5兲, LaNiO3 共hR30兲, CaTiO3 共oC10兲,
FeTiO3 共hR30兲, SrZrO3 共oC40兲, BaRuO3 共hR45兲,
␣-CsMgH3 共Pmmm兲, CaSiO3 共mP60兲; CaSiO3 共aP30兲,
MgSiO3 共mS40兲, YBO3 共oS60兲 USPEX-1 共C2; evolutionary
search38兲, and USPEX-2 共P1; evolutionary search38兲.38–40
The present type of theoretical investigations are highly successful to predict the ground-state structure of hydrides24,41
and other materials.42,43 The reliability of the calculations
depends upon the number of input structures considered in
the calculations. Though it is a tedious process to select input
structures from the 2545 entries for the ABX3 composition in
the Inorganic Crystal Structure Database 共ICSD兲,39 which
also involves tremendous computations, several compounds/
phases have the same structure type, and in some cases, have
only small variations in the positional parameters 共of certain
atoms兲. These possibilities are omitted because during the
full geometry optimization, in spite of using different positional parameters, the structures mostly are converted to
similar type of structural arrangements. In this particular
composition, only 28 structure types have unique structural
arrangements. In order to cover a wide range of structural
modification from the above structural starting points we
have made the computation in the following way: the first
step is to obtain equilibrium volume for this particular
chemical composition in different modifications we have
made symmetry constrained stress minimization for all these
modifications. The lattice parameters optimized starting
structure models have been divided into four class of structure models 共in total of 4 ⫻ 30= 120 structure models兲: 共1兲
the original starting structures 共i.e., ABX3兲; 共2兲 the original
starting structures have been transformed into P1 symmetry
considered as P1-ABX3 model; 共3兲 the A and B element have
been interchanged 共BAX3 model兲; and 共4兲 BAX3 model structures have been further transferred into P1 symmetry
共P1-BAX3 model兲. All the above mentioned structure models
have been fully relaxed 共minimization of force and stress兲
without any constrains on the atomic positions and unit-cell
parameters. It should be noted that, during the structural optimization, some of the initial structures are converted into
other high/low symmetry structure types which are not included in the above models. The calculated total energy as a
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114504-3
J. Chem. Phys. 132, 114504 共2010兲
Structure of alkali calcium tri-hydrides
H
Na
Ca
(a)
(b)
(c)
(d)
FIG. 1. Theoretically predicted crystal structures for 共a兲 LiCaH3, 共b兲
NaCaH3, 共c兲, KCaH3 共d兲 RbCaH3, and CsCaH3.
function of volume has been fitted to the so-called universal
equation of state 共EOS兲44 to calculate the bulk modulus and
its pressure derivative.
1. Structural features of the MCaH3 phases
Among the considered structure models for LiCaH3, a
PbGeS3 derived model 1–4 atomic arrangements occur at the
lowest total energy and all these phases are having exactly
the same total energy. Our symmetry analysis shows that,
during the structural relaxation processes, the low symmetry
triclinic and monoclinic structural modifications of PbGeS3
transform into the somewhat high symmetry hexagonal 关Fig.
1共a兲; R3̄c兴 phase. Such a situation arises often when we perform full geometry optimization. It is often observed that,
instead of relaxing to the local minimum, the system relaxes
to the global minimum, as it is the case here. The LiCaH3
structure consists of corner-sharing LiH6 octahedra. From the
interatomic Li–H distances 共1.843 Å兲 and H–Li–H angles
共ranging between 88.4 and 91.6°兲, it is evident that the LiH6
octahedra are slightly distorted. Ca is surrounded by 9 H
atoms, the Ca–H distances vary from 2.32 to 2.57 Å. The
shortest H–H separation in the LiCaH3 structure exceeds
2.57 Å, and is consistent with the H–H separation found in
other complex/metal hydrides.
According to the structural optimization calculations for
the NaCaH3 phase, USPEX-1-derived models 1–4 atomic arrangements have the lowest total energy. After the structure
relaxation the high symmetry monoclinic structure 共C2 transforms into low symmetry triclinic phase 关P1̄; Table I, Fig.
1共b兲; hereafter ␣-NaCaH3兴. In addition with the USPEX-1derived models PbGeS3-derived models 1–4 共during the
structure relaxation, PbGeS3-derived models 1–4 transform
into R3c rhombohedral phase; hereafter meta-NaCaH3兲 are
energetically closer to the USPEX-1-derived models 共the involved energy difference is 0.06eV兲 and both structure models are also having almost similar equilibrium volume. In the
␣-NaCaH3 phase Na and Ca is surrounded by 5 and 7 hydrogen atoms, respectively. The calculated average Ca–H
and Na–H distances are 2.33 and 2.30 Å, respectively. The
bond angle between H–Ca–H vary from 70.7 to 105.8° and
H–Na–H vary from 74.5 to 105.5°. In the meta-NaCaH3
phase, both Na and Ca are surrounded by six H in cornersharing octahedral coordination, with NaH6 octahedra highly
distorted and CaH6 octahedra slightly distorted. The calculated average Ca–H and Na–H distances are 2.35 and 2.53 Å,
respectively. Each H is surrounded by two Ca and two Na,
and the shortest H–H separation is above 2.51 Å.
The results obtained from structural optimizations for
KCaH3 phase shows that the GdFeO3-derived 1, 3, and 4
model structures proved to have the lowest total energy. It is
interesting to note that during the theoretical simulations,
many of the initially assumed different trial structures relaxed toward the GdFeO3-type structure 共viz. strongly emphasizing that this particular atomic arrangement is energetically more favorable for KCaH3兲. This phase consists of
slightly distorted CaH6 corner-sharing octahedra 关Fig. 1共c兲兴.
The calculated Ca–H distances within the octahedra are 2.26
Å and the H–Ca–H bond angles vary from 89° to 91°. The K
atom is surrounded by 8 H atoms and the calculated K–H
distance varies between 2.83 and 3.15 Å.
The structural optimizations for RbCaH3 phase show
that the KCuF3-derived input structure model has the lowest
total energy. The symmetry analysis for the RbCaH3 phase
shows that the tetragonal structure is transformed into cubic
modification 关Fig. 1共d兲兴. For the CsCaH3 phase, CaTiO3-type
atomic arrangement has the lowest energy than the considered structure types consistent with the experimental findings. The calculated unit-cell dimensions and positional parameters for both RbCaH3 and CsCaH3 at 0 K and ambient
pressure are in good agreement with the room temperature
experimental findings 共see Table I兲.28,29 In RbCaH3 and
CsCaH3, Rb/Cs is surrounded by 12 H in cuboctahedral coordination at a distance of 3.21 and 3.27 Å, respectively. Ca
is octahedrally coordinated to six H at a distance of 2.27 Å in
RbCaH3, and 2.31 Å in CsCaH3. Similarly, in both cases, H
is surrounded by two Ca and four Rb/Cs, and the shortest
H–H separation is 3.21 Å in RbCaH3, and 3.27 Å in CsCaH3.
Although these compounds have isoelectronic configurations for all involved constituents, they stabilize in rather
different crystal structures. The broad features of the structural arrangement vary from hexagonal-orthorhombic-cubic
on moving from Li to Cs along the series. It is worthy to note
that the rotation of CaH6 along 共110兲 and 共111兲 directions
bring orthorhombic and rhombohedral distortion into the lattice. Generally, the variation in the crystal structures of ABX3
compounds can be rationalized in terms of the Goldschmidt
tolerance factor 共t兲. The value of t may be used as an indicator of the tendency for structural transitions and for the
deformation in the octahedral coordination at the B site in a
given perovskite family member.45 The range of 0.8–0.89
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114504-4
J. Chem. Phys. 132, 114504 共2010兲
Vajeeston, Ravindran, and Fjellvåg
TABLE I. Optimized equilibrium structural parameters, bulk modulus 共B0兲, and pressure derivative of bulk modulus 共B⬘0兲 for the MCaH3 共M = Li, Na, K, Rb,
or Cs兲 series. The experimental values are given in the parenthesis.
Compound
共structure type; space group兲
Lattice parameters
共Å兲
B0
共GPa兲
B⬘0
LiCaH3 共R3̄c , 167兲
a = 5.2240
c = 12.4814
Li共6b兲:0.0, 0.0, 0.5
Ca共6a兲: 0.0, 0.0, 0.25
H共18e兲: 0.2197, 0.3333, 0.5833
36
3.9
NaCaH3 共P1̄兲
a = 6.8555
b = 7.1940
c = 6.8040
␣ = 118.59°
␤ = 117.01°
␥ = 29.78°
Na共2i兲: 0.7257, 0.0131, 0.2737
Ca共2i兲: 0.1609, 0.6291, 0.7517
H1共2i兲: 0.9004, 0.5172, 0.6503
H2共2i兲: 0.3627, 0.8224, 0.0155
H3共2i兲: 0.2330, 0.8404, 0.3466
27
2.6
meta-NaCaH3 共R3c兲
a = 5.8632
c = 15.2174
Na共6a兲: 0.0, 0.0, 0.7826
Ca共6a兲: 0.0, 0.0, 0.0014
H共18b兲: 0.9878, 0.2819, 0.3980
25
2.3
KCaH3 共Pnma兲
a = 6.3219
b = 8.9293
c = 6.3073
K共4c兲: 0.0165, 0.25, 0.9966
Ca共4b兲: 0.0, 0.0, 0.5
H1共4c兲: 0.9860, 0.25, 0.4439
H2共8d兲: 0.2219, 0.9705, 0.2218
24
3.9
RbCaH3 共Pm3̄m ; CaTiO3兲
a = 4.5427 共4.547兲a
Rb共1a兲: 0.0, 0.0, 0.0 共0.0, 0.0, 0.0兲a
Ca共1b兲: 0.5, 0.5, 0.5 共0.5, 0.5, 0.5兲a
H共3c兲: 0.5, 0.5, 0.0 共0.5, 0.5, 0.0兲a
24
3.8
CsCaH3 共Pm3̄m ; CaTiO3兲
a = 4.6297 共4.609兲b
Cs共1a兲: 0.0, 0.0, 0.0 共0.0, 0.0, 0.0兲b
Ca共1b兲: 0.5, 0.5, 0.5 共0.5, 0.5, 0.5兲b
H1共3c兲: 0.5, 0.5, 0.0 共0.5, 0.5, 0.0兲b
23
3.8
Positional parameters
a
Experimental values from Ref. 28.
Experimental values from Ref. 29.
b
should be an indicator of tetragonal or orthorhombic distortion of the cubic symmetry, for cubic arrangement t should
be in the range of 0.89–1.00, and t above 1 should single out
hexagonal 共trigonal兲 stacking variants of the perovskite family. The t values for the compounds under investigation varies from 0.59 to 0.97 关0.59 共LiCaH3兲, 0.78 共NaCaH3兲, 0.86
共KCaH3兲, 0.92 共RbCaH3兲, and 0.97 共CsCaH3兲兴. Consistent
with the Goldschmidt empirical rule, t takes a value close to
one for cubic RbCaH3 and CsCaH3 where undistorted perovskite arrangements contain perfect octahedra. For the KCaH3
with t = 0.86, distorted octahedra occur within the orthorhombic structure. Even though LiCaH3 and NaCaH3 have smaller
t value than the defined t factor, their ground-state structures
are still perovskitelike frameworks. It indicates that either the
Goldschmidt tolerance factor may not be suitable to describe
the structure of hydrides or the assumed value of the H− radii
共1.4 Å for 6 coordination兲 may be wrong. It should be noted
that this t factor was successfully used to explain the structural modifications along the MBeH3 and MMgH3 共M = Li,
Na. K, Rb, and Cs兲 series.46,47 This indicates that the predicted LiCaH3 and NaCaH3 phases may have different bonding nature, and in particular, they may not be pure ionic
compounds 共see the bonding analysis below兲.
By fitting the total energy as a function of cell volume
using the so-called universal EOS 共Ref. 44兲 the bulk modulus 共B0兲 and its pressure derivative 共B0⬘兲 are obtained 共Table
I兲, but no experimental data are yet available for comparison.
The bulk modulus decreases monotonically when we move
from Li to Cs and on the other hand its pressure derivative is
almost constant 共except for NaCaH3兲. The variations in B0 is
accordingly correlated with variations in the size of M and
consequently also with the cell volume. Compared to
intermetallic-based hydrides, these compounds have low B0
values, implying that they are soft and easily compressible.
The soft character of the MCaH3 materials arises from the
weaker bonding interaction between M and Ca. Also due to
the soft nature, one may expect that destabilization of some
of the hydrogen atoms from the matrix may be feasible.
However, the Ca–H bonds are likely to be strong. From this
point of view, one can expect that the MCaH3 phases also
release hydrogen at elevated temperature only. Hence, these
materials may not be suitable for on-board transportation applications. However, additive substitution/共particle size control兲 may reduce the decomposition temperature considerably as it was observed in alanates. Further research is
needed in this direction.
B. Formation-energy considerations
Formation enthalpy is the best aid to establish whether
theoretically predicted phases are likely to be stable and also
such data may serve as guides for possible synthesis routes.
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114504-5
J. Chem. Phys. 132, 114504 共2010兲
Structure of alkali calcium tri-hydrides
TABLE II. Calculated hydride formation energy 共⌬H; in kJ mol−1兲 according to Eqs. 共1兲–共4兲 for the MCaH3 series.
CsCaH3
0.10
0.05
Compound
⌬H1
⌬H2
⌬H3
⌬H4
RbCaH3
LiCaH3
NaCaH3
KCaH3
RbCaH3
CsCaH3
17.74
34.45
17.03
10.32
12.84
⫺70.01
⫺10.55
⫺25.62
⫺24.67
⫺24.26
⫺237.25
⫺177.79
⫺192.86
⫺191.91
⫺191.50
⫺149.51
⫺132.79
⫺150.21
⫺156.93
⫺154.4
Density of States (1/THz)
0.10
0.05
KCaH3
0.10
0.05
meta-NaCaH3
α− NaCaH3
0.10
0.05
In this study, we have considered four possible reaction pathways 关Eqs. 共1兲–共4兲兴, and the associated formation enthalpy
listed in Table II are estimated from the calculated total energies without temperature effect. In general, synthesis of
MCaH3 compounds from an equiatomic MCa matrix is not
possible as the alkali metals and calcium are immiscible in
the solid and liquid state.48 Schumacher and Weiss49 have
suggested that the ternary MMgH3 hydrides can be synthesized directly by a reaction between M and Mg in hydrogen
atmosphere at elevated temperatures. A similar approach
may also be valid for the MCaH3 series. However, most of
the MMgH3 compounds have also been synthesized from the
appropriate combination of binary hydrides.29,50–52 This
shows that, among the considered reaction pathways, tracks
1 and 3 are experimentally verified for MMgH3 series,29,50–52
but for the MCaH3 phases, reaction path 1 is not possible and
the path 3 may be a possible route. On the other hand, tracks
2 and 4 are not yet verified for any of the MM ⬘H3 phases
共M ⬘ = any alkali-earth atom兲 and they are open for verification or rejections.
The results show that reaction pathways 2–4 give rise to
an endothermic reaction for the MCaH3 compounds. Hence,
preparation of MCaH3 from MH and CaH2 共pathway 1兲 is
not likely to be successful. It should be noted that CsCaH3
has been synthesized from its binary hydrides under 200 bar
H2 pressure.29 All MCaH3 compounds are seen to exhibit
high formation energies according to pathway 3. Hence, for
all the studied phases, pathway 3 is energetically more favorable than other paths and we suggest that it should be possible to synthesize/stabilize these compounds using CaH2
and M by passing H2. One can easily verify the validity of
the proposed reaction pathways 关Eqs. 共1兲–共4兲兴. The difference in ⌬H4 − ⌬H1 = ⌬H3 − ⌬H2 corresponds to the formation
energy of CaH2 which amounts to −167.25 kJ mol−1, in
good agreement with the measured value of
−181.521⫾ 9.2 kJ mol−1.53 Temperature effect has not been
included in the present calculations 共but note that one can
reliably reproduce the formation energy of H2 without including thermal vibrations兲.
In order to understand the stability of the predicted
phases, we have calculated phonon DOS for the equilibrium
structures for these phases, which are shown in Fig. 2. For all
these compounds, no imaginary frequency was observed indicating that all the predicted structures are ground-state
structures for these systems, or at least they are dynamically
stable.54 It should be noted that, for the NaCaH3 phase, both
␣ and metamodifications are dynamically stable, hence we
have displayed total phonon DOS for both phases in Fig. 2.
LiCaH3
0.10
0.05
0
10
20
Frequency (THz)
30
40
FIG. 2. Calculated total phonon DOS for MCaH3 compounds.
The structure of CsCaH3 is already known, but that for the
other phases are not yet identified experimentally. As the
predicted phases are expected to be stable, we need experimental verification. Both LiCaH3 and NaCaH3 have almost
similar phonon DOS. Similarly the remaining phases
KCaH3, RbCaH3, and CsCaH3 also exhibit almost similar
phonon DOS. Hence we have displayed in Fig. 3 only the
partial phonon DOS for LiCaH3 and CsCaH3. In both cases
the partial phonon DOS is plotted along three directions, x, y,
and z. For Li/Cs and Ca atoms, the vibrational modes along
the x, y, and z directions are identical. On the other hand, the
FIG. 3. Calculated partial phonon DOS for 共a兲 LiCaH3 and 共b兲 CsCaH3
phases.
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J. Chem. Phys. 132, 114504 共2010兲
Vajeeston, Ravindran, and Fjellvåg
TABLE III. Calculated standard hydride formation energy 共⌬H; in kJ mol−1兲
according to Eq. 共1兲 for the MCaH3 series. The vibrational energies are
given at 300 K. ⌬HZPE refers ZPE correction to the net reaction, ⌬HZPE+vib
refers to the corrections due to ZPE plus vibrational contributions, and ⌬H1
refers electronic contributions to the reaction Eq. 共1兲. For calculation of
standard enthalpy of formation 共⌬H兲 the following corresponding binary
hydride values 共in kJ mol−1兲 were utilized: LiH= −90.621; NaH= −56.421;
KH= −57.821; RbH= −47.421; CsH= −49.9; MgH2 = −76.2; and CaH2 =
−181.521. Experimentally known formation energy is given in the parenthesis.
Compound
LiCaH3
␣-NaCaH3
KCaH3
RbCaH3
CsCaH3
NaMgH3
⌬HZPE
⫺0.48
⫺6.9
⫺3.6
⫺3.5
⫺1.97
⫺0.41
⌬HZPE+vib
0.37
⫺4.8
⫺1.5
⫺1.6
⫺1.4
⫺0.18
⌬H1
17.74
34.45
17.03
10.32
12.84
⫺10.71
⌬H
⫺271.77
⫺242.74
⫺240.84
⫺230.54
⫺232.82
⫺143.4 共⫺144兲
modes along x direction in the H site in LiCaH3 is slightly
different from those along the y and z directions 关see Fig.
3共a兲兴. In contrast, the H modes along the x and y are identical
for CsCaH3 and that along the z direction is much different
from the x and y directions 关see Fig. 3共b兲兴. Since the mass of
H atom is much smaller than that of M or Ca atom, Fig. 2
shows that the high frequency modes above 10 THz are
dominated by H atom, and the low frequency modes below
10 THz are mainly dominated by M 共except Li兲 and Ca atoms. The center of the Ca mode frequency in all these cases
is always present around 5 THz. If one moves from Li to Cs
due to the mass difference between the M atoms, the vibrational mode frequencies from M are systematically shifted
toward low frequency region.
The standard hydride formation energy 共⌬H兲 is calculated according to Eq. 共1兲 with the use of the ⌬H for the
following corresponding binary hydrides 共all values are in
kJ mol−1兲: LiH: ⫺90.621; NaH: ⫺56.421; KH: ⫺57.821;
RbH: ⫺47.421; CsH: ⫺49.9; and CaH2: ⫺181.521.53 The
vibrational energies are given at 300 K and the calculated net
zero point energy 共ZPE兲 for the reaction Eq. 共1兲 are listed in
Table III. The vibrational contribution to the formation energy is clearly shown to be smaller than that from the ZPE.
Also, for all these phases, the vibrational contribution to the
formation energy is negligible compared to that from electronic contribution 共⌬H1兲. In order to compare the numerical
values, we have also included the values for NaMgH3 phase,
for which the formation energy is already known
experimentally.53 The calculated formation energy for
NaMgH3 according to Eq. 共3兲 is −119.80 kJ mol−1. This
value is considerably smaller than the experimentally known
value of −144 kJ mol−1.53 In order to have reasonably good
standard formation energy, we have adopted the following
empirical method where the formation energy is estimated
from the sum of the experimental formation energy of the
constituent
binary
hydrides+ electronic contribution
+ ⌬HZPE+vib. Using this empirical method we obtained ⌬H
value of −143.4 kJ mol−1 for NaMgH3 which is in very good
agreement with the experimentally observed value. In order
to establish the validity of this empirical method, we have
made systematic estimation of ⌬H using this approach not
2.0
EF
CsCaH3
1.0
2.0
DOS (states/e.V f.u.)
114504-6
RbCaH3
1.0
2.0
KCaH3
1.0
2.0
NaCaH3
1.0
2.0
LiCaH3
1.0
-7.5
-5
-2.5
0
2.5
Energy (eV)
5
7.5
FIG. 4. Calculated total electronic DOS for MCaH3 phases. The Fermi level
is set at zero energy and marked by the vertical dotted line.
only for the NaMgH3 phase, but also for several known ternary saline, as well as complex hydrides.55 Hence, one can
expect that the MCaH3 series may also have the tabulated
⌬H values, and their magnitude indicates that these phases
are quite stable. Further experimental investigations are
needed to verify this prediction.
C. Chemical bonding
In previous studies47,56,57 on hydrides we have demonstrated that several theoretical tools are needed in order to
draw more assured conclusions regarding the nature of the
chemical bonding. From the bonding analysis of MBeH3 and
MMgH3 series we found46,47 that all these phases are basically saline hydrides similar to the parent alkali-/alkalineearth mono-/dihydrides. Generally one may expect that the
MCaH3 phases may also have similar bonding behavior.
The total electronic DOS at the equilibrium volumes for
the ground-state structures of the MCaH3 compounds are displayed in Fig. 4 and site projected DOSs for LiCaH3 and
CsCaH3 are shown in Fig. 5. All MCaH3 compounds have
finite energy gap 共Eg; varying between 2.5 and 3.3 eV兲 between the valence band 共VB兲 and the conduction band and
hence they can be considered as insulators. It is commonly
recognized that the theoretically calculated Eg for semiconductors and insulators are strongly dependent on the approximations used and in particular on the exchange and correlation terms of the potential. In general, the Eg values obtained
from the density-functional calculations are always underestimated compared to the experimentally measured values.
From the band structure 共not given in the manuscript兲 we
found that these materials having indirect band gap 共except
KCaH3兲. According to pure ionic picture, the insulating behavior of these materials can be explained as follows: in each
formula unit, one electron from M fills one of the three originally half-filled H-s orbitals and the other two are filled by
electrons from Ca resulting in a completely filled VB and
accordingly an insulating behavior. A similar feature was
found in most of the complex hydrides 关e.g., MAlH5 共M
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114504-7
J. Chem. Phys. 132, 114504 共2010兲
Structure of alkali calcium tri-hydrides
EF
0.10
H
H
0.05
Ca
Ca
DOS (states/eV f.u.)
H
H
0.10
H
H
H
H
H
Ca
1.00
Li
H
-5
-2.5
(a)
0
2.5
5
7.5
Energy (eV)
Cs
H Cs
H
H
H
(b)
Ca
Ca
(d)
FIG. 6. Calculated 关共a兲 and 共b兲兴 valence-electron-charge density and 关共c兲 and
共d兲兴 ELF plots for LiCaH3 and CsCaH3, respectively.
H
0.10
0.05
DOS (states/eV f.u.)
Cs H
EF
0.15
Ca
0.15
0.10
0.05
Cs
0.15
0.10
0.05
Total
1.50
1.00
0.50
-10
Ca
Ca
Total
2.00
(b)
(c)
H
0.025
H Cs
Ca
Ca
(a)
0.075
0.050
Cs
H
H
Li
-7.5
Cs H
Li
H
0.05
Ca
Ca
-5
0
Energy (eV)
5
10
FIG. 5. Calculated partial DOS for 共a兲 LiCaH3 and 共b兲 CsCaH3 phases. The
Fermi level is set at zero energy and marked by the vertical dotted line; s
states are shaded and p states are marked by continuous line.
= Mg, Ba兲 M 3AlH6 共M = Li, Na, K兲,58,59 and MAH4 共A
= B , Al, Ga兲兴.60,61 On moving from LiCaH3 to CsCaH3, the
calculated DOSs are significantly changing. In LiCaH3 the
VB region is relatively broad and it systematically narrows
when one moves to NaCaH3, KCaH3, RbCaH3, and CsCaH3
共Fig. 4兲. This may be due to the variation in the coordination
number of Ca and also the increase in Ca–H bond length. A
similar feature was also observed in MBeH3 and MMgH3
phases.46,47 It should be noted that the electronic structure of
LiCaH3 has been investigated along with several light weight
hydrides by Khowash et al.62 Because of the different crystal
structure the estimated Eg value for the LiCaH3 phase is
共around 3.5 eV兲 much higher than the present investigation.
In order to elucidate the bonding situation more properly,
we have calculated the partial electronic DOS 共PDOS兲 for
MCaH3. As seen from Fig. 5共a兲, the PDOSs for Li and Ca
show very small contributions from s and p states in the VB.
This demonstrates that valence electrons are transferred from
the Li and Ca sites to the H sites. The small Li- and Ca-s
states present at the VB are energetically degenerate with the
H-s states. Compared with LiCaH3, a significant change in
the VB feature can be found in KCaH3, RbCaH3, and
CsCaH3. As the VB features are almost having close similarity, we have displayed the PDOS for CsCaH3 only. In all
phases Ca-p and H-s states are present in the vicinity of the
Fermi level at the VB. In KCaH3, RbCaH3, and CsCaH3, the
Ca-s and −p states are well-separated. In contrast, these
states for LiCaH3 and NaCaH3 are energetically degenerate
almost in the whole VB region, indicating hybridization interaction.
In order to gain further understanding about the bonding
situation in MCaH3 compounds, we turn our attention to
charge density and electron-localization-function 共ELF兲
plots. Again the different members of the series exhibit similar features and in view of that we have only documented
such plots for LiCaH3 and CsCaH3. Figures 6共a兲 and 6共b兲
show the charge-density distribution at the Li, Cs, Ca, and H
sites, from which it is evident that the highest charge density
resides in the immediate vicinity of the nuclei. Further, the
spherical charge distribution shows that the bonding between
Cs–H and Ca–H have predominantly ionic character. On the
other hand, in the case of LiCaH3 关Fig. 6共a兲兴, the interaction
between Li–H is almost ionic 共spherical charge distribution兲
with noticeable directional character. The nature of charge
distribution seen in Fig. 6共b兲 appears to be typical for ionic
compounds.25 A common distinction between the bonding in
MCaH3 series and the situation in the MAlH4 and M 3AlH6
series is that the interaction between Ca and H has more
ionic character than that between Al and H.57 In KCaH3,
RbCaH3, and CsCaH3 the electron population between M
and the CaH6 units is almost zero 共viz., charges are depleted
from this region兲, which reconfirms that the interaction between M and CaH6 units is virtually purely ionic in these
compounds.
The calculated ELF plot 关Fig. 6共b兲; for more details
about ELF see Refs. 63–65兴 shows a maximum of ca. 1 at
the H site and these electrons have a paired character. The
ELF values at the M and Ca sites are very low. The inference
from this observation is that charges are transferred from the
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114504-8
J. Chem. Phys. 132, 114504 共2010兲
Vajeeston, Ravindran, and Fjellvåg
TABLE IV. Calculated 共BC 共given in terms of e兲 and energy band gap 共Eg in
electron volts兲 for the MCaH3 series.
LiCaH3
␣-NaCaH3
KCaH3
RbCaH3
CsCaH3
Atom
BC
Eg
Li
Ca
H
Na
Ca
H
K
Ca
H
Rb
Ca
H
Cs
Ca
H
+0.79
+1.45
⫺0.75
+0.83
+1.41
⫺0.84
+0.84
+1.56
⫺0.80
+0.76
+1.51
⫺0.75
+0.67
+1.53
⫺0.73
2.5
BOP
0.4
Compounds
Ca-H
M-H
0.3
0.2
0.1
3.2
LiCaH3
NaCaH3
KCaH3
RbCaH3
CsCaH3
Compound
3.0
FIG. 7. Variation in the Ca–H and M – H 共M = Li, Na, K, Rb, and Cs兲 BOP
along the MCaH3 series.
3.2
3.1
M and Ca sites to the H sites and there are certainly very few
paired valence electrons left at the M and Ca sites. A certain
polarized character is found in the ELF distribution at the H
sites in all the complex hydrides we have investigated
earlier.57 Similarly, in the MCaH3 phases, the ELF distribution is not spherically symmetric at the H site. But the polarization is considerably lower in MCaH3 phases than the
aluminum based hydrides.
In an effort to quantify the bonding and estimate the
amount of electrons on and between the participating atoms,
we have made Bader topological analysis. Although there is
no unique definition to identify how many electrons are associated with an atom in a molecule or an atomic grouping in
a solid, it has nevertheless proved useful in many cases to
perform Bader analyses.66–68 In the Bader charge 共BC兲
analysis, each atom of a compound is surrounded by a surface 共called Bader regions兲 that run through minima of the
charge density and total charge of an atom is determined by
integration within the Bader region. The calculated BC for
the MCaH3 series is given in Table IV. The BC for M and H
in the MCaH3 compounds indicate that the interaction between M and H is almost ionic 共in all cases around one
electron is transferred from M to H兲. This finding is consistent with the DOS and charge density analyses. Within the
CaH6 units, Ca donates nearly 1.5 electrons in KCaH3,
RbCaH3, and CsCaH3 and around 1.4 electrons in LiCaH3
and NaCaH3 phases to the H site, which is much smaller than
in a pure ionic picture. This is partly associated with the
small covalency present between Ca and H and also may be
due to the artifact of making boundaries to integrate charges
in each atomic basin using Bader’s “atoms in molecule” approach. However, consistent with the charge and DOS analysis, the BC analysis always qualitatively shows that M and
Ca atoms donate electrons to the H site.
To get a better understanding of the interaction between
the constituents, the BOP values are calculated on the basis
of the Mulliken population. The BOP can provide useful information about the bonding property between the two atoms. A high BOP value indicates a strong covalent bond,
while a low BOP value indicates an ionic interaction. The
calculated BOP values for the Ca–H and M – H results are
displayed in Fig. 7. From Fig. 7, it is seen that the BOP
values for the Ca–H bonds in all the five hydrides vary between 0.15 and 0.49. Similarly, the calculated BOP values
for the M – H vary between 0.01 and 0.16. It should be noted
that the Ca–H BOP values are close to that for ionic 共0.39兲
Na–H bond in NaH, but lower than that for the covalent C–C
bond in diamond 共1.08兲 and H–Al bonds in NaAlH4 共0.88兲
and Na3AlH6 共0.62–0.64兲. Therefore, the Ca–H bonds in
these hydrides have dominant ionic character, and M – H interactions are much weaker. Figure 7 clearly indicates that
where one goes from LiCaH3 to CsCaH3, the Ca–H interaction becomes stronger, on the other hand, the M – H interaction gets weaker. This is one of the reasons why the Ca–H
distance is reduced on going from LiCaH3 to CsCaH3.69
IV. CONCLUSION
The crystal, electronic structure, and thermodynamical
properties of the MCaH3 共M = Li, Na, K, Rb, and Cs兲 series
have been studied by state-of-the-art density-functional calculations. For the experimentally known CsCaH3 phase, the
ground-state structure has been successfully reproduced
within the accuracy of the density-functional approach. The
ground-state crystal structures for MCaH3 共M = Li, Na, K,
Rb, and Cs兲 phases have been predicted by performing structural optimization of a number of different structure types
using force as well as stress minimizations. The predicted
crystal structures for LiCaH3 and NaCaH3 are found to have
rhombohedral and triclinic structures, respectively, with insulating behavior. KCaH3 stabilizes in orthorhombic and
RbCaH3 and KCaH3 stabilize in cubic structures. Formation
energies for the MCaH3 series are calculated for different
possible reaction pathways. For all these phases, we propose
that synthesis from elemental M and Ca in hydrogen atmosphere should be a more feasible route. The phonon DOS of
the lattices are calculated by using a direct force constant
method and it shows that all the predicted phases are dynamically stable. A semiempirical method for estimation of
standard enthalpy of formation for MCaH3 hydrides is proposed, which gives the best value for the known NaMgH3
phase. All the studied phases are wide-band-gap insulators
and the insulating behavior is associated with well localized,
paired s-electron configuration at the H site. The chemical
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114504-9
Structure of alkali calcium tri-hydrides
bonding character of these compounds is predominantly
ionic according to analyses of DOS, charge density, ELF,
BCs, and BOP.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the Research Council of Norway under Grant No. 460829 and European Union
Seventh Frame Work program under the “NanoHy” 共Grant
No. 210092兲 project. The authors gratefully acknowledge the
Research Council of Norway for financial support and for
computer time at the Norwegian supercomputer facilities.
J. Block and A. P. Gray, Inorg. Chem. 304, 4 共1965兲.
J. A. Dilts and E. C. Ashby, Inorg. Chem. 11, 1230 共1972兲; J. P. Bastide,
B. Monnetot, J. M. Letoffe, and P. Claudy, Mater. Res. Bull. 20, 999
共1985兲.
3
B. Bogdanovic and M. Schwickardi, J. Alloys Compd. 253, 1 共1997兲.
4
B. Bogdanovic, R. A. Brand, A. Marjanovic, M. Schwikardi, and J. Tölle,
J. Alloys Compd. 302, 36 共2000兲.
5
H. W. Brinks, B. C. Hauback, P. Norby, and H. Fjellvåg, J. Alloys
Compd. 351, 222 共2003兲.
6
C. M. Jensen and K. Gross, Appl. Phys. A 72, 213 共2001兲.
7
H. Morioka, K. Kakizaki, S. C. Chung, and A. Yamada, J. Alloys Compd.
353, 310 共2003兲.
8
S. Orimo, H. Fujii, and K. Ikeda, Acta Mater. 45, 331 共1997兲.
9
A. Zaluska, L. Zaluski, and J. O. Strom-Olsen, J. Alloys Compd. 288,
217 共1999兲.
10
J. Huot, G. Liang, and R. Schultz, Appl. Phys. A: Mater. Sci. Process. 72,
187 共2001兲.
11
J. Huot, J. F. Pelletier, L. B. Lurio, M. Sutton, and R. Schulz, J. Alloys
Compd. 348, 319 共2003兲.
12
C. Wolverton, V. Ozolinš, and M. Asta, Phys. Rev. B 69, 144109 共2004兲.
13
J. F. Herbst and L. G. Hector, Appl. Phys. Lett. 88, 231904 共2006兲.
14
S. V. Alapati, J. K. Johnson, and D. S. Sholl, J. Phys. Chem. B 110, 8769
共2006兲.
15
X. Ke and I. Tanaka, Phys. Rev. B 71, 024117 共2005兲.
16
M. Aoki, K. Miwa, T. Noritake, G. Kitahara, Y. Nakamori, S. Orimo, and
S. Towata, Appl. Phys. A: Mater. Sci. Process. 80, 1409 共2005兲.
17
J. K. Kang, J. Y. Lee, R. P. Muller, and W. A. Goddard, J. Chem. Phys.
121, 10623 共2004兲.
18
O. M. Løvvik, S. M. Opalka, H. W. Brinks, and B. C. Hauback, Phys.
Rev. B 69, 134117 共2004兲.
19
S. Orimo, Y. Nakamori, G. Kitahara, K. Miwa, N. Ohba, T. Noritake, and
S. Towata, Appl. Phys. A: Mater. Sci. Process. 79, 1765 共2004兲.
20
K. Miwa, N. Ohba, S. Towata, Y. Nakamori, and S. Orimo, Phys. Rev. B
71, 195109 共2005兲.
21
Y. Song, R. Singh, and Z. X. Guo, J. Phys. Chem. B 110, 6906 共2006兲.
22
N. Ohba, K. Miwa, M. Aoki, T. Noritake, S.-i. Towata, Y. Nakamori, S.-i.
Orimo, and A. Züttel, Phys. Rev. B 74, 075110 共2006兲.
23
D. J. Siegel, C. Wolverton, and V. Ozolinš, Phys. Rev. B 75, 014101
共2007兲.
24
P. Vajeeston, P. Ravindran, H. Fjellvåg, and A. Kjekshus, J. Alloys
Compd. 363, L7-11 共2003兲; P. Vajeeston, P. Ravindran, R. Vidya, H.
Fjellvåg, and A. Kjekshus, Appl. Phys. Lett. 82, 2257 共2003兲; P. Vajeeston, P. Ravindran, and H. Fjellvåg, Phys. Rev. Lett. 89, 175506
共2002兲.
25
P. Vajeeston, Theoretical Modeling of Hydrides, PhD thesis, Department
of Physics, Faculty of Mathematics and Natural Sciences, University of
Oslo, 2004.
26
S. Zh. Karazhanov, P. Ravindran, P. Vajeeston, and A. G. Ulyashin, Phys.
Status Solidi A 204, 3538 共2007兲.
27
S. Zh. Karazhanov, A. G. Ulyashin, P. Ravindran, and P. Vajeeston, Europhys. Lett. 82, 17006 共2008兲.
28
H. H. Park, M. Pezat, and B. Darriet, C. R. Acad. Sci. II 共Paris兲 306, 963
共1988兲.
29
F. Gingl, T. Vogt, E. Akiba, and K. Yvon, J. Alloys Compd. 282, 125
共1999兲.
30
A. Bouamrane, J.-Ph. Soulie, and J. P. Bastide, Thermochim. Acta 375,
81 共2001兲.
31
J. P. Perdew, in Electronic Structure of Solids, edited by P. Ziesche and H.
1
2
J. Chem. Phys. 132, 114504 共2010兲
Eschrig 共Akademie, Berlin, 1991兲; J. P. Perdew, K. Burke, and Y. Wang,
Phys. Rev. B 54, 16533 共1996兲; J. P. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 共1996兲.
32
P. E. Blöchl, Phys. Rev. B 50, 17953 共1994兲; G. Kresse and J. Joubert,
ibid. 59, 1758 共1999兲.
33
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 共1993兲; G. Kresse and J.
Furthmuller, Comput. Mater. Sci. 6, 15 共1996兲.
34
L. G. Hector, Jr., J. F. Herbst, and T. W. Capehart, J. Alloys Compd. 353,
74 共2003兲.
35
W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys. 32, 219 共1960兲.
36
K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 共1997兲;
81, 3298 共1998兲.
37
M. D. Segall, P. L. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip,
S. J. Clark, and M. C. Payne, J. Phys.: Condens. Matter 14, 2717 共2002兲.
38
C. H. Hu, A. R. Oganov, Y. M. Wang, H. Y. Zhou, A. Lyakhov, and J.
Hafner, J. Chem. Phys. 129, 234105 共2008兲.
39
Inorganic Crystal Structure Database, Gmelin Institut, Germany, 2006.
40
G. Renaudin, B. Bertheville, and K. Yvon, J. Alloys Compd. 353, 175
共2003兲.
41
P. Vajeeston, P. Ravindran, B. C. Hauback, H. Fjellvåg, S. Furuseth, M.
Hanfland, and A. Kjekshus, Phys. Rev. B 73, 224102 共2006兲.
42
H. L. Skriver, Phys. Rev. B 31, 1909 共1985兲.
43
P. Söderlind, O. Eriksson, B. Johansson, J. M. Wills, and A. M. Boring,
Nature 共London兲 374, 524 共1995兲.
44
P. Vinet, J. H. Rose, J. Ferrante, and J. R. Smith, J. Phys.: Condens.
Matter 1, 1941 共1989兲.
45
Perovskites Modern and Ancient, edited by R. H. Mitchell 共Almaz, Thunder Bay, 2002兲.
46
P. Vajeeston, P. Ravindran, and H. Fjellvåg, Inorg. Chem. 47, 508 共2008兲.
47
P. Vajeeston, P. Ravindran, H. Fjellvåg, and A. Kjekshus, J. Alloys
Compd. 327–337, 450 共2007兲.
48
Binary Alloy Phase Diagrams, 2nd ed. 共ASM International, Materials
Park, OH, 1990兲.
49
R. Schumacher and A. Weiss, J. Less-Common Met. 163, 179 共1990兲.
50
E. Rönnebro, D. Noréus, K. Kadir, A. Reiser, and B. Bogdanovic, J.
Alloys Compd. 299, 101 共2000兲.
51
B. Bertheville, P. Fischer, and K. Yvon, J. Alloys Compd. 330–332, 152
共2002兲.
52
J. P. Bastide, A. Bouamrane, P. Claudy, and J.-M. Letoffe, J. LessCommon Met. 136, L1 共1987兲.
53
NIST-JANAF Thermochemical Tables, J. Phys. Chem. Ref. Data Monogr.
No. 9, 4th ed., edited by M. W. Chase, Jr. 共American Chemical Society,
Washington, DC, 1998兲.
54
P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P. Rudin, Phys. Rev.
Lett. 100, 095901 共2008兲.
55
A. Klaveness, H. Fjellvåg, A. Kjekshus, P. Ravindran, and O. Swang, J.
Alloys Compd. 469, 617 共2009兲.
56
P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, and H. Fjellvåg, Phys.
Rev. Lett. 89, 106403 共2002兲.
57
P. Vajeeston, P. Ravindran, A. Kjekshus, and H. Fjellvåg, Phys. Rev. B
71, 216102 共2005兲.
58
D. J. Singh, Phys. Rev. B 71, 216101 共2005兲.
59
P. Vajeeston, P. Ravindran, A. Kjekshus, and H. Fjellvåg, Phys. Rev. B
71, 092103 共2005兲.
60
P. Vajeeston, P. Ravindran, A. Kjekshus, and H. Fjellvåg, J. Alloys
Compd. 387, 97 共2005兲.
61
A. Aguayo and D. J. Singh, Phys. Rev. B 69, 155103 共2004兲.
62
P. K. Khowash, B. K. Rao, T. McMullen, and P. Jena, Phys. Rev. B 55,
1454 共1997兲.
63
A. Savin, A. D. Becke, J. Flad, R. Nesper, H. Preuss, and H. G. von
Schnering, Angew. Chem., Int. Ed. Engl. 30, 409 共1991兲; A. Savin, O.
Jepsen, J. Flad, O. K. Andersen, H. Preuss, and H. G. von Schnering,
ibid. 31, 187 共1992兲.
64
B. Silvi and A. Savin, Nature 共London兲 371, 683 共1994兲.
65
A. D. Becke and K. E. Edgecombe, J. Chem. Phys. 92, 5397 共1990兲.
66
R. F. W. Bader, Atoms in Molecules: A Quantum Theory 共Oxford University, New York, 1990兲.
67
G. Henkelman, A. Arnaldsson, and H. Jónsson, Comput. Mater. Sci. 36,
354 共2006兲.
68
C. F. Guerra, J.-W. Handgraaf, E. J. Baerends, and F. M. Bickelhaupt, J.
Comput. Chem. 25, 189 共2004兲.
69
R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 共1993兲.
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