Journal of Alloys and Compounds 450 (2008) 327–337
First-principles investigations of the MMgH3
(M = Li, Na, K, Rb, Cs) series
P. Vajeeston∗ , P. Ravindran, A. Kjekshus, H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
Received 20 April 2006; received in revised form 16 October 2006; accepted 28 October 2006
Available online 11 December 2006
Abstract
The structural stability of the MMgH3 (M = Li, Na, K, Rb, Cs) series has been investigated using the density-functional projector-augmentedwave method within the generalized-gradient approximation. Among the 24 structural arrangements used as inputs for the structural optimization
calculations, the experimentally known frameworks are successfully reproduced, and positional and unit-cell parameters are found to be in good
agreement with the experimental findings. The crystal structure of LiMgH3 has been predicted, the most stable arrangement being the trigonal
LiTaO3 -type (R3c) structure, which contains highly distorted octahedra. The formation energy for all members of the MMgH3 series is investigated
along different reaction pathways. The electronic density of states reveals that the MMgH3 compounds are wide-band-gap insulators. Analyses of
chemical bonding in terms of charge density, charge transfer, electron-localization function, Born effective charge, and Mulliken population show
that these hydrides are basically saline hydrides similar to the parent alkali-/alkaline-earth mono-/di-hydrides.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Electronic structure; Crystal structure of LiMgH3 ; Hydrogen storage materials; Alkali and alkaline-earth-based hydrides
1. Introduction
The crystal structures of the pure elements and most of the
binary compounds have been frequently studied and are well
characterized. On turning to ternary compounds, however, the
amount of knowledge is considerably less extensive (with an
estimated 10% coverage of structural information), and for quaternary and multi-component phases the structural knowledge
is extremely poor. For hydrides, owing to the complexity in
structural arrangements and difficulties involved in establishing hydrogen positions by X-ray diffraction methods, structural
information is very limited [1].
The light elements of groups I–III of the periodic table, e.g.,
Li, Be, Na, Mg, B, and Al, form a large variety of complex
hydrides which are interesting from a hydrogen storage point
of view because of their light weight (the number of hydrogen
atoms per metal atom being in many cases around 2). Alkaliand alkaline-earth-based complex hydrides are expected to have
a potential as viable modes for storing hydrogen at moderate
∗
Corresponding author: Tel.: +47 22855613; fax: +47 22855565.
E-mail address: ponniahv@kjemi.uio.no (P. Vajeeston).
0925-8388/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jallcom.2006.10.163
temperatures and pressures [2–8]. These hydrides (e.g., LiAlH4 ,
NaAlH4 , etc. [2–8]) have higher hydrogen storage capacity at
moderate temperatures and lower cost 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. World-wide a lot of research
activity is in progress with the aim to identify new materials with
high hydrogen content and suitably low operating temperature.
At present we focus on new materials like MM H3 (M = alkali
and M = alkaline-earth element, e.g., LiBeH3 , NaMgH3 , etc.)
and M AH5 (A = group III elements, e.g., MgAlH5 , BaAlH5 ,
etc. [9]). In this article we will present results from a structural
study of the phases MMgH3 (M = Li, Na, K, Rb, Cs). Nowadays Mg-based phases receive special attention because of light
weight and low cost. Other compounds of the MM H3 family are
under examination and the results will be published in forthcoming articles. One motivation for this study was to investigate the
atomic arrangements, electronic structures, and bonding nature
within the MMgH3 series in detail in order to check the stability
of these materials for hydrogen storage applications.
One special aspect associated with incorporation of hydrogen
into metals and intermetallics is enhancement of superconductivity (e.g., PdH has Tc ≈ 10 K, whereas superconductivity is
328
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
absent in Pd [10]). The search for metals exhibiting superconductivity at high temperatures has been going on over a
much longer period than the current activities appear to indicate.
Metallic hydrogen, owing to low mass and high electron density,
has been predicted to be a high-temperature superconductor with
transition temperature ranging between 140 and 260 K [11]. If
MMgH3 proves to be metallic one may well expect superconductivity with Tc in the range predicted for metallic hydrogen.
Hence, also for this reason it seems worthwhile to investigate
the electronic structure of the MMgH3 series.
Indeed Overhauser [10] suggested already in 1987 that metallic LiBeH3 could have an electron density as high as that in
metallic hydrogen and the material could accordingly be a hightemperature superconductor. A number of electronic-structure
calculations has, in fact, already been performed on LiBeH3
[12–16] using a variety of computational approximations. All
these calculations conclude with insulating behavior and a large
gap at the Fermi level (EF ). Li et al. [17] studied the electronic
structure of LiMgH3 , NaMgH3 , and LiCaH3 and reported that
LiMgH3 is an insulator whereas the latter compounds are metals.
In fact, according to a simple chemical picture one may expect
that these compounds are ionic insulators (M and/or Mg/Ca
donating electrons to H). In order to check such expectations
one needs accurate electronic-structure calculations and in these
endeavors one must keep in mind that electronic structure is very
sensitive to crystal structure.
The rest of this paper is organized as follows. Section 2 gives
appropriate details about the computational methods, the results
of the calculations are presented and discussed in Section 3, and
conclusions are summarized in Section 4.
2. Calculation details
Total energies have been calculated by the projected augmented wave (PAW) [18] implementation of the Vienna
ab initio simulation package (VASP) [19]. The generalizedgradient approximation (GGA)[20] were used to obtain accurate
exchange and correlation energies for particular configurations
of atoms. The PAW potentials explicitly treat one valence electron situations for H (1s1 ) and Li (2s1 ), seven for Na (2p6 , 3s1 ), K
(3p6 , 4s1 ) and Rb (4p6 , 5s1 ), eight for Mg (2p6 , 3s2 ), and nine for
Cs (5s2 , 5p6 , 6s1 ). Ground-state geometries were determined by
minimizing stresses and Hellman–Feynman forces with the conjugate gradient algorithm, until forces on all atomic sites were
less than 10−3 eV Å−1 . Brillouin zone integration are performed
with a Gaussian broadening of 0.1 eV during all relaxations. In
order to span a wide range of energetically accessible crystal
structures; cell volume, cell shape, and atomic positions were
relaxed simultaneously in a series of calculations made with
progressively increasing precision. A final high accuracy calculation of the total energy was performed after completion of the
relaxations with respect to k-point convergence and plane-wave
cutoff. From the various sets of calculations it was found that
for the KMgH3 structure 512 k points in the whole Brillouin
zone with a 500 eV plane-wave cutoff are sufficient to ensure
optimum accuracy in the computed results. A similar density of
k points were used for the other structures considered. In order
to avoid ambiguities regarding the free-energy results we have
always used the same energy cutoff and a similar k-grid density for convergence in all structural variants tested. The present
type of theoretical approach has recently been applied successfully [21–23] to reproduce ambient- and high-pressure phases
in our earlier studies. The Mulliken-population analyses have
been made with the help of CRYSTAL03 [24] code in which
we used 5-11G, 6-11G, 8-511G, HAYWSC-31, HAYWSC-31,
HAYWSC-31G, and 8-61G basis sets for H, Li, Na, K, Rb,
Cs, and Mg, respectively. For the Born-effective-charge analyses we utilized the Berry-phase approach [25] implemented
in the VASP code together with local codes for pre- and postprocessing. The formation energies (E) have been calculated
according to the reaction equations:
MH + MgH2 → MMgH3
(1)
M + MgH2 + 21 H2 → MMgH3
(2)
MH + Mg + H2 → MMgH3
(3)
M + Mg + 23 H2 → MMgH3
(4)
The ground-state energies of M, MH, and ␣-MgH2 have been
computed for the ground-state structures, viz. in space group
Im3̄m for M, Fm3̄m for MH, and P42 /mnm for ␣-MgH2 , all
with full geometry optimization.
3. Results and discussion
Twenty-four potentially applicable structure types have
been used as inputs in the structural optimization calculations
for the MMgH3 compounds (Pearson structure classification
notation in parenthesis): KMnF3 (tP20), GdFeO3 [NaCoF3
(oP20)], KCuF3 (tI20), BaTiO3 [RbNiF3 (hP30)], CsCoF3
(hR45), CaTiO3 [CsHgF3 (cP5)], PCF3 (tP40), KCuF3 (tP5),
KCaF3 (mP40), NaCuF3 (aP20), SnTlF3 (mC80), KCaF3
(mB40), LiTaO3 (hR30), KCuF3 (oP40), CaCO3 (mP20),
KNbO3 (tP5), KNbO3 (oA10), KNbO3 (hR5), LaNiO3 (hR30),
CaTiO3 (oC10), FeTiO3 (hR30), SrZrO3 (oC40), BaRuO3
(hR45), and α-CsMgH3 (Pmmm) [26,27]. From the above structural starting points, full geometry optimization has been carried
out without any constrains on the atomic positions and unit-cell
parameters.
3.1. Structural findings
The studied compounds take quite different crystal structures.
Calculated equilibrium structural parameters are presented in
Table 1 (note that the specified structure types refer to the trial
structures used as inputs) and the thereof derived interatomic
distances are given in Table 2. Table 2 also includes bondvalence [32] sums (BVS) for M and Mg calculated from the
expression:
RM/Mg,j − dM/Mg,j
BVSM/Mg =
exp
(5)
b
j
where RM/Mg is the bond-valence parameter for the M–
H/Mg–H bond, dM/Mg the corresponding observed interatomic
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
329
Table 1
Optimized equilibrium structural parameters, bulk modulus (B0 ), and pressure derivative of bulk modulus (B0 ) for the MMgH3 (M = Li, Na, K, Rb, Cs) series
Compound (structure type; space group)
Unit cell (Å)
Positional parameters
B0 (GPa)
B0
LiMgH3 (LiTaO3 type; R3c)
a = 4.958
c = 13.337
Li (6a): 0.0, 0.0, 0.2887;
Mg (6a): 0, 0, 0
H (18b): 0.0376, 0.3626, 0.5637
39.8
3.1
NaMgH3 (GdFeO3 type; Pnma)
a = 5.4525 (5.4634)a
b = 7.6952 (7.7030)a
c = 5.3683 (5.4108)a
Na (4c): 0.0209, 1/4, 0.006 (0.030, 1/4, 0.006)a
Mg (4b): 0, 0, 1/2
H1 (4c): 0.503, 1/4, 0.093 (0.524, 1/4, 0.081)a
H2 (8d): 0.304, 0.065, 0.761 (0.292, 0.042, 0.793)a
38.4
3.6
KMgH3
(CaTiO3 type; Pm3̄m)
a = 4.0295 (4.023)b
K: 0, 0, 0
Mg: 1/2, 1/2, 1/2
H: 0, 1/2, 1/2
35.6
3.7
RbMgH3 (BaTiO3 type; P63 /mmc)
a = 5.9068 (5.9030)c
c = 14.3261 (14.3158)c
Rb1 (4f): 1/3, 2/3, 0.0949 (0.0949)c
Rb2 (2b): 0, 0, 1/4
Mg1 (2a): 0, 0, 0
Mg2 (4f): 1/3, 2/3, 0.6532 (0.8459)c
H1 (12k): 0.1651, 2x, 0.5797 (0.1661, 2x, 0.5797)c
H2 (6h): 0.5236, 2x, 1/4 (0.5247, 2x, 1/4)c
29.8
4.1
␣-CsMgH3
(prototype; Pmmn)
a = 9.9922 (9.9734)d
b = 6.1405 (6.1222)d
c = 8.5768 (8.5562)d
Cs1 (2a): 1/4, 1/4, 0.1797 (0.182)d
Cs2 (4f): 0.4713, 1/4, 0.6797 (0.476, 1/4, 0.642)d
Mg1 (2b): 1/4, 3/4, 0.6354 (0.642)d
Mg2 (4f): 0.8897, 1/4, 0.0728 (0.6080, 1/4, 0.067)d
H1 (2b): 1/4, 3/4, 0.0942 (1/4, 3/4, 0.098)d
H2 (4c): 0, 0, 0
H3 (4e): 1/4, 0.9851, 0.4970 (1/4, 0.990, 0.494)d
H4 (4e): 1/4, 0.9508, 5/6 (1/4, 0.953, 0.835)d
H5 (4f): 0.5513, 1/4, 0.2922 (0.550, 1/4, 0.300)d
17.3
5.1
a
b
c
d
Experimental value from Ref. [28].
Experimental value from Ref. [29].
Experimental value from Ref. [30].
Experimental value from Ref. [31].
distance(s), and b = 0.37 (all in Å) the “universal” bond valence
constant. For the convenience of the readers the Goldschmidt tolerance factors (t) for the perovskite-like phases of the MMgH3
series are listed in Table 2. In order to avoid confusion in the presentation we will consider the structural findings for the different
compounds separately and discuss common aspects toward the
end of this section.
3.1.1. Structural features of LiMgH3
The crystal structure of LiMgH3 is hitherto not experimentally determined. Among the considered structures, an
LiTaO3 -type atomic arrangements [Table 1, Fig. 1(a)] occurs at
the lowest total energy (Fig. 2). However an FeTiO3 -type variant appears energetically very close to the ground-state phase.
The involved energy difference between these two phases is
only 0.01 eV. The LiMgH3 structure consist of corner sharing
MgH6 octahedra (see Fig. 1(a)). From the interatomic Mg–
H distances (Table 2) and H–Mg–H angles (ranging between
76.8◦ ;and 104.9◦ ) it is evident that the MgH6 octahedra are
highly distorted. The BVS for Mg came out close to the formal valence 2 as expected. Li is surrounded by six H atoms,
but here the Li–H distances are little long to confirm the
formal valence 1 by the BVS. The shortest H–H separation
in the LiMgH3 structure exceeds 2.58 Å, and is comparable
with the H–H separation found in other compounds of this
series.
3.1.2. Structural features of NaMgH3
NaMgH3 takes a distorted perovskite structure (Fig. 1(b))
analogous to the GdFeO3 type [33,31]. Consistent with the
experimental findings this atomic arrangement has lower energy
than the considered alternatives (see above) and the calculated
unit-cell dimensions and positional parameters at 0 K and ambient pressure are in good agreement with the room temperature
experimental findings (see Table 1). The deviations from the
experimental unit-cell parameters a and b are almost zero and
the underestimation of 1.2% in the c direction is typical for
the state-of-art approximation of density-functional theory. The
findings shows that one can reliably reproduce/predict structural
parameters for even quite complex atomic arrangements with
this type of approach. As seen from Table 2 the MgH6 octahedra in NaMgH3 are somewhat distorted (H–Mg–H angles in the
range 88.2–91.8◦ ). A similar structure with the same type distortions is incidently also found for NaMgF3 [31]. In general, the
hydride and fluoride families mostly show pronounced analogies in structural respect. The BVS for Mg is somewhat lower
than the formal valence of 2 and the BVS for Na is distinctly
higher than the formal valence of 1 (see Section 3.3).
3.1.3. Structural features for KMgH3
KMgH3 crystallizes in an ideal perovskite-type (Pm3̄m;
see Fig. 1(c)) structure [34,35]. In our theoretical simulations
the ideal perovskite-type atomic arrangement came out with
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P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
Table 2
Goldschmidt tolerance factors (t) and interatomic distances (multiplicity in parenthesis) for the MMgH3 (M = Li, Na, K, Rb, Cs) series; bond-valence sum (BVSM/Mg )
appears just below the corresponding distances
Compound
t
Distance (Å)
LiMgH3
0.77
Li–H: 1.934 (3×)
Li–H: 2.073 (3×)
BVSLi = 0.97
Mg–H: 1.922 (3×)
Mg–H: 2.008 (3×)
BVSMg = 1.86
H–H: 2.576 (2×)
H–H: 2.690 (2×)
H–H: 2.691(×)
H–H: 2.803 (2×)
H–H: 2.846 (2×)
H–H: 2.966 (2×)
Li–Mg: 2.794 (1×)
Li–Mg: 2.919 (3×)
NaMgH3
0.87
Na–H1: 2.302 (1×)
Na–H2: 2.326 (2×)
Na–H1: 2.472 (1×)
Na–H2: 2.640 (2×)
Na–H2: 2.701(2×)
BVSNa = 1.31
Mg–H2: 1.967 (2×)
Mg–H2: 1.968 (2×)
Mg–H1: 1.976 (2×)
BVSMg = 1.83
H1–H2: 2.755 (2×)
H1–H2: 2.764 (2×)
H1–H2: 2.813 (2×)
H1–H2: 2.821(2×)
Na–Mg: 3.203 (2×)
Na–Mg: 3.281 (2×)
Na–Mg: 3.332 (2×)
Na–Mg: 3.473 (2×)
KMgH3
1.01
K–H: 2.849 (12×)
BVSK = 1.58
K–H: 2.015 (6×)
BVSMg = 1.62
H–H: 2.849 (8×)
K–Mg: 3.49 (8×)
RbMgH3
1.04
Rb1–H1: 2.963 (6×)
Rb1–H2: 2.975 (6×)
Rb2–H1: 2.951 (3×)
Rb2–H2: 2.962 (6×)
Rb2–H2: 3.037 (3×)
BVSRb = 1.47
Mg1–H2: 2.047 (6×)
Mg2–H1: 2.008 (3×)
Mg2–H2: 2.016 (3×)
BVSMg = 1.48
H1–H1: 2.535 (2×)
H1–H2: 2.920 (4×)
H2–H2: 2.847 (2×)
H2–H1: 2.920 (2×)
H2–H2: 2.943 (2×)
Rb1–Mg1: 3.584 (2×)
Rb1–Mg2: 3.677 (6×)
Rb2–Mg2: 3.513 (4×)
Rb2–Mg1: 3.674 (3×)
␣-CsMgH3
1.09
Cs–H1: 3.157 (2×)
Cs–H1: 3.163 (2×)
Cs–H1: 3.170 (2×)
Cs–H1: 3.312 (4×)
Cs–H1: 3.494 (2×)
BVSCs = 1.34
Mg1–H3: 1.869 (2×)
Mg1–H5: 2.079 (2×)
Mg1–H4: 2.098 (2×)
Mg2–H5 1.971 (1×)
Mg2–H2: 1.990 (2×)
Mg2–H1: 2.000 (1×)
Mg–H: 2.029 (2×)
BVSMg = 1.69
H1–H4: 2.554 (2×)
H2–H4: 2.894 (2×)
H2–H5: 2.983 (2×)
H3–H3: 2.887 (1×)
H3–H4: 2.893 (1×)
Cs1–Mg2: 3.715 (2×)
Cs1–Mg2: 4.008 (4×)
Cs2–Mg2: 3.78 (1×)
Cs2–Mg1: 3.78 (1×)
Cs2–Mg1: 3.791 (2×)
Cs2–Mg2: 3.902 (2×)
distinctly lower energy than the other trial structures tested.
The calculated positional and unit-cell parameters agree well
with the experimental findings. It is interesting to record that
during the theoretical simulations many of the initially assumed
different trial structures (more than half of the 24 variants
tested) relaxed toward the ideal perovskite-type structure (viz.
strongly emphasizing that this particular atomic arrangement
is energetically favorable for KMgH3 ). Generally, the A-site
cations in an ideal perovskite structure (ABX3 ) are larger than
the B-site cations and similar in size to the X-site anions. In
relation to KMgH3 we note that the tabulated ionic radius for
K+ is larger than the radii for Mg2+ and H− , but the Mg2+
radius is only around half of the H− radius [27,36]. This findings
appear to emphasize the significance of size factors associated
with both cations and anions (in particular H− ) for stabilization
of an ideal perovskite arrangement. In KMgH3 , K is surrounded
by 12 H in cuboctahedral coordination at a distance of 2.85 Å
and Mg is octahedrally coordinated to six H at a distance of 2.02
Å, H is surrounded by two Mg and four K, and the shortest H–H
separation is 2.85 Å. The BVS for KMgH3 continues the trend
from M = Li and Na in that the BVS for M increases and that
for Mg correspondingly decreases (see Sections 3.1.6 and 3.3).
3.1.4. Structural features of RbMgH3
RbMgH3 is the first known hydride reported to crystallize
with the BaTiO3 -type structure [a so-called 6H variant of the
hexagonal perovskite; P63 /mmc; see Fig. 1(d)] [30]. Also in
this case the theoretical simulations successfully established
the experimental structure as the ground state, and calculated
atomic coordinates and unit-cell parameters (see Table 1) fit very
well with the experimental findings. In cubic perovskites (like
KMgH3 ), the smaller cation (in this case Mg2+ ) is in octahedral
coordination (Fig. 1(c)) and these octahedra are linked only at
corners. In RbMgH3 the MgH6 octahedra are linked not only
at corners, but also some faces are shared. The Rb ions have
12-fold coordination, with different coordination polyhedra
for the two crystallographically different Rb sites (Fig. 1(d)).
Rb1 experiences twinned cuboctahedral coordination, whereas
Rb2 takes the usual cuboctahedral coordination. The Rb–H and
Mg–H distances vary mutually only little whereas the H–Mg–H
angles cover a fairly broad range from 78.3◦ ; to 94.6◦ . Again
the BVS follows the pattern from the first part of the MMgH3
series: BVS for M increasing and BVS for Mg decreasing from
KMgH3 to RbMgH3 .
3.1.5. Structural features of CsMgH3
Experimental studies have established that CsMgH3 occurs
in two different modifications. The high-pressure form (prepared at 30 kbar) of CsMgH3 is trigonal (BaRuO3 type; R3̄m)
and is designated ␤-CsMgH3 [37]. The intermediate-pressure
(prepared at 150 bar) ␣ modification is orthorhombic (Pmmn)
[27]. The ␤ modification is closely related to the perovskitetype structure whereas the ␣ modification takes a different
structural arrangement. According to the theoretical simula-
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
331
Fig. 1. (a) Predicted crystal structure of LiMgH3 (LiTaO3 type; R3c). Ground-state crystal structure for (b) NaMgH3 (GdFeO3 type, deformed perovskite; Pnma),
(c) KMgH3 (CaTiO3 type, ideal perovskite; Pm3̄m), (d) RbMgH3 (BaTiO3 type, hexagonal perovskite; P63 /mmc), and (e) ␣-CsMgH3 (prototype; Pmmn). (f)
Magnified illustration of the three condensed MgH6 octahedra of the ␣-CsMgH3 structure.
tions ␣-CsMgH3 occurs at much lower energy than ␤-CsMgH3 .
Moreover, several of the other tested trial structures proved to
occur at lower energies than ␤-CsMgH3 , some of these are, in
fact, energetically close to the ␣ modification. According to the
present calculations the energy difference between the ␣ and ␤
modifications is ca. 1.1 eV f.u.−1
The ␣-CsMgH3 structure comprises MgH6 octahedra which
are connected via four corners to two-dimensional slabs (Fig.
1(e)). This atomic arrangement is entirely different from the
frameworks found in rest of the MMgH3 series, in particular with
respect to the condensation of three MgH6 octahedra into one
structural “building block” (Fig. 1(f)). Cs are surrounded by 12
H in anti-cuboctahedral configuration with Cs–H bond distances
in the range 3.16–3.49 Å (Table 2). Repulsive Mg–Mg interaction is proposed as the reason for the off-center displacement of
Mg in the MgH6 octahedra. The variation in Mg–H distances
(Table 2) and the scatter in H–Mg–H bond angles (72.0–101.1◦ )
show that the MgH6 octahedra are highly distorted. ␣-CsMgH3
break the trend in the development of BVS along the MMgH3
series in that BVS for M is found to have decreased from Rb
to Cs and BVS for Mg likewise to have increased. This could
of course have been a rather trivial consequence of the fact that
␣-CsMgH3 does not belong to the perovskite family or indicate
errors in the interatomic M–H and Mg–H distances (dM/Mg ; see
Table 1) and/or inaccuracies in the corresponding bond-valence
parameters (RM/Mg ). However, the situation is similar for the
␣ and ␤ modifications which more points toward an effect of
the element combination Cs, Mg, and H. In support of this view
we draw attention to a recent study [38] of volume increments
in the crystal structures of hydrides. Precisely parallel to the
present finding the volume increments for the MH series increase
monotonically from M = Li to Rb for thereafter to decrease significantly to Cs. These changes in trends are clearly associated
with an alteration in the polarizability of the electron cloud on
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P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
Fig. 2. Cell volume vs. total energy curves for LiMgH3 in different possible
structural arrangements (structure-type inputs are specified on the illustration).
For clarity only the nine variants with the lowest total energy are documented
on the illustration. Arrow marks the crossover point where a pressure-induced
transition is predicted (30.69 Å3 f.u.−1 corresponding to 3.8 GPa).
going from Rb to Cs, but further work is needed to uncover the
mechanism(s) involved.
3.1.6. Structural features of the MMgH3 series
The calculated and experimental equilibrium volumes per
formula unit for the entire MMgH3 series indicate that, different from the usual picture for other saline hydrides, the molar
volumes of these compounds are smaller than the weighted sum
of the volumes for the comprising binary hydrides (Table 3).
The volume reduction upon the formation of the ternary hydride
is smallest for LiMgH3 and largest for KMgH3 . As expected
the equilibrium cell volume per formula unit increases monotonically along the MMgH3 series (Fig. 3). Although these
compounds have isoelectronic configurations for all involved
components they stabilize in different crystal structures. The
broad features of the structural arrangement vary from Li to
Cs along the series: trigonal–orthorhombic–cubic–hexagonal–
orthorhombic. The variation can be rationalized in terms of the
Goldschmidt tolerance factor (t). The value of t may be used
Fig. 3. Variation in cell volume and bulk modulus with the ionic radius of M
along the MMgH3 series.
as an indicator of the tendency for structural transitions and
for specification of the deformation in the octahedral coordination at the B site in a given perovskite-family member [39]. For
cubic arrangement t should be in the range 0.89–1.00, the range
0.8–0.89 should be indicative of tetragonal or orthorhombic distortion of the cubic symmetry, and t above 1 should single out
hexagonal (trigonal) stacking variants of the perovskite family.
The t values for the compounds under investigation varies (see
Table 2) from 0.77 (LiMgH3 ) to 1.09 (CsMgH3 ). Consistent
with the Goldschmidt empirical rule, t takes a value close to one
for cubic KMgH3 and this undistorted perovskite arrangements
contains perfect octahedra. For the cases with t below one (viz.
M = Li and Na) there occur distorted octahedral and for those
with t above one (viz. M = Rb and Cs) one finds the expected
hexagonal symmetry. In the case of CsMgH3 the ground-state
structure deviates appreciably from a perovskite-like framework. The Goldschmidt tolerance concept rests on packing of
touching spherical objects which obeys certain size relations.
Hence, preference for different packing modes found in these
saline hydrides can readily be rationalized in terms of the ionic
size of M+ , Mg2+ , and H− . The common feature for the MMgH3
compounds is the octahedral MgH6 configuration which, irre-
Table 3
Cell volume (V in Å3 f.u.−1 ; experimentally observed lattice parameters are used except for LiMgH3 ), relative difference compared to the sum of the volumes for
the corresponding binary hydrides (V = 30.81 Å3 f.u.−1 for MgH2 ), calculated hydride formation energy (H; in kJ mol−1 ) according to Eqs. (1)–(4), and energy
band gap (Eg in eV) for the MMgH3 series
Compound
V
LiH + MgH2
LiMgH3
NaH + MgH2
NaMgH3
KH + MgH2
KMgH3
RbH + MgH2
RbMgH3
CsH + MgH2
CsMgH3
46.96
45.87
59.40
56.93
76.57
65.11
85.17
73.15
95.97
87.60
V
−1.38
H1
4.048
H2
H3
H4
Eg
−83.67
−59.75
−147.50
3.98
−4.15
−11.00
−56.01
−74.80
−119.80
3.45
−14.97
−32.23
−74.87
−96.02
−138.67
2.67
−14.12
−28.75
−63.74
−92.55
−127.54
2.81
−8.7
−31.99
−69.09
−95.79
−132.88
2.98
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
spective of the M component, exhibits an interatomic Mg–H
distances of around 2.00 Å (average Mg–H ranging between 1.92
Å in LiMgH3 and 2.02 Å in CsMgH3 ). On the other hand, the
average M–H distance increases monotonically from 1.93 Å in
LiMgH3 to 3.16 Å in CsMgH3 and the shortest H–H separation
varies between 2.57 and 2.98 Å.
At first sight the BVS suggests that the effective valence of
Mg is approaching the expected value 2 in LiMgH3 and NaMgH3
and that the discrepancies may be attributed to systematic errors
in the empirical foundation of Eq. (5). However, in the rest of
the serie the BVS for Mg comes out with a value below 2. The
situation for the M component is rather opposite; whereas the
BVS for Li is close to the normal valence 1, the BVS for M for
the rest of the MMgH3 compounds always takes a much higher
value than 1. These findings are opposite to the outcome of the
our Mulliken-population analyses (see Section 3.3). Although
the individual values of the BVS for M and Mg deviate from the
expected nominal value 1 and 2, respectively, their sum remains
close to 3 [between 2.83 (LiMgH3 ) and 3.20 (KMgH3 ); see Table
2]. These observations appear to constitute a suitable basis for
fruitful bond-distance considerations (see Section 3.3).
Bulk modulus values for the MMgH3 series are listed in Table
1 but experimental data are not available for comparison. The
bulk modulus decreases monotonically when we move from Li
to Cs and its pressure derivative increase correspondingly (Fig. 3
upper panel). The variations in B0 and B0 are accordingly correlated with variations in the size of M and consequently also with
the cell volume. Comparison of B0 for the MMgH3 , MAH4 , and
M3 AlH6 compounds shows that the former and latter series have
similar B0 values whereas the MAH4 compounds carries lower
B0 values. Hence, the MMgH3 compounds are relatively harder
than the MAH4 compounds. However, the magnitude of B0 for
MMgH3 still classifies these materials as easily compressible.
The soft character of the MMgH3 materials arises from a high
degree of ionic character in the chemical bonding. However,
although these materials are soft, we expect that high energy
is required to strip hydrogen atoms from the MMgH3 matrix
(similar to the findings for the MAH4 and M3 AlH6 series).
333
MgH2 (pathway 1) is not likely to be successful and this may be
the reason why this phase has been not yet identified experimentally. All MMgH3 compounds are seen to exhibit high formation
energies according to pathway 4 (certainly much higher than
according to the experimentally established route along pathway 1 for M = Na–Cs). For LiMgH3 , pathway 4 is clearly the
energetically most favorable and we suggest that it should be
possible to synthesis/stabilize this compound along this route.
One can easily check the validity of the proposed reaction
pathways (Eqs. (1)–(4)). The energy difference H3 − H1 =
H4 − H2 corresponds to the formation energy of MgH2
which amounts to −63.8 kJ mol−1 in good agreement with
the measured value of −76.2 ± 9.2 kJ mol−1 [40]. Temperature
effect have not been included in the present calculations (but
note that one can reliably reproduce the formation energy of H2
without including thermal vibrations). Hence, the energy quan-
3.2. Formation-energy considerations
In general, synthesis of the MMgH3 compounds from an
equiatomic MMg matrix is not possible as the alkali metals and
magnesium are immiscible in the solid and liquid state. Schumacher and Weiss [35] have suggested that the ternary hydrides
can be synthesized directly by a reaction between M and Mg in
hydrogen atmosphere at elevated temperatures. However, most
of the MMgH3 compounds have been synthesized from the
appropriate combination of binary hydrides [30,31,34,37]. This
shows that, among the considered reaction pathways, tracks 1
and 4 are experimentally verified whereas tracks 2 and 3 are
open for verification or rejections.
In order to identify the thermodynamically most feasible of
the reaction pathways 1–4, formation energies have calculated
and listed Table 3. The results show that pathways 1–4 gives rise
to an exothermic reaction for the MMgH3 compounds (except
for LiMgH3 ). Hence, preparation of LiMgH3 from LiH and
Fig. 4. (a) Calculated total DOSs for the MMgH3 compounds and (b) partial
DOS for NaMgH3 in the ground-state structures; s states are shaded. The Fermi
level is set to zero energy and marked by the vertical dotted line.
334
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
tities for the different pathways (Table 3) may be regarded as
reliable and reasonably supported by experimental findings.
3.3. Chemical bonding
In previous studies [41,42] on hydrides we have demonstrated
that several theoretical tools are needed in order to be able
to draw more assured conclusions regarding the nature of the
chemical bonding. In this section we will try to characterize the
chemical bonding in the MMgH3 series by the same procedures.
The total density of states (DOS) at the equilibrium volumes
for the ground-state structures of the MMgH3 compounds are
displayed in Fig. 4(a) and site projected DOSs for LiMgH3 are
shown in Fig. 4(b). All MMgH3 compounds have finite energy
gaps (Eg between 2.67 and 3.98 eV) between the valence band
(VB) and the conduction band (CB) and are hence, proper insulators. According to textbook chemistry the insulating behavior
can be explained as follows: per formula unit, one electron from
M fill one of the three originally half-filled H-s orbitals and the
other two of these orbitals are filled by electrons from Mg, resulting in complete filling of the VB and accordingly in insulating
behavior. The MAlH5 (M = Mg, Ba) [9], M3 AlH6 (M = Li,
Na, K) [43,44], and MAH4 (A = B, Al, Ga) [45,46] compounds
also exhibit similar insulating behavior. This suggests that, one
can generally expect wide band gaps in hydrides with octahedral coordinated structural units. On moving from LiMgH3 to
CsMgH3 , owing to the enlargement of the interatomic distances,
the calculated width of VB degreases from 6.2 to 3.6 eV.
The presently calculated band gap for KMgH3 is around
0.8 eV lower than Eg reported in another theoretical paper [47]
which, however, used a different computational approach and
different exchange correlations. It should also be noted that electronic structure cluster calculations [17] for LiMgH3 report a
band gap of around 1 eV. However, according to the DOS in Fig.
4 of Ref. [17], there is no gap between VB and CB. There is,
in fact, a distinct electron distribution present at EF . Moreover,
in contrast with our theoretical prediction, Li et al. [17] found
that NaMgH3 is a metal. One of the deficiencies in Ref. [17] is
the use of an almost ideal cubic type of clusters, with a limited
number of atoms fixed at their crystallographic positions. The
identified discrepancies illustrate how sensitive the electronic
structure is to the geometry adopted for the calculations. It is
also commonly recognized that theoretically calculated energy
gaps 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, compared to
experimental band gaps density-functional calculations always
underestimate band gaps significantly. Normally GGA gives better band-gap values. Our experience from similar calculations
for MgH2 , is that the band-gap discrepancy between theory and
experiment for such hydrides should be within 12% [21]. The
local-density approximation is basically designed to describe
ground-state properties of systems. Although it reasonably predicts excited state properties of materials, calculated band gaps
always come out smaller than values obtained from experimental
optical spectroscopy. In order to treat the many-body problems
correctly one should go beyond the usual density functional
approximations by using GW approximation [48]. However,
more accurate calculated values for the band gaps are outside
the scope of the present work.
As mentioned above one can formally discuss the MMgH3
series in terms of pure ionic or mixed iono-covalent bonding. We
infer that the interaction between the M and MgH6 units is pure
ionic whereas the interaction between Mg and H contains an
iono-covalent component. According to the electronegativities
of the constituents, the bonding in these materials should have
a dominant ionic character.
In order to elucidate the bonding situation more properly we
have calculated the partial density of states (PDOS) for MMgH3 .
Owing to close similarity between the PDOSs for the different
members of the series, Fig. 4(b) only documents the findings
for LiMgH3 . As seen from this illustration the PDOSs for Li
and Mg show very small contributions from s and p states. This
demonstrates that valence electrons are transferred from the Li
and Mg sites to the H sites. The Mg-s and -p states are energetically well separated in the VB region whereas the Li-s and -p
states are energetically degenerate.
In order to gain further understanding of the bonding situation
in the MMgH3 compounds we turned to charge-density, chargetransfer, and electron-localization-function (ELF) plots. Again
Fig. 5. (Color online) Calculated plots of (a) valence-electron-charge density, (b) charge transfer, and (c) ELF for KMgH3 . The illustrations refer to the (1 1 0) plane.
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
the different members of the series exhibit similar features and
accordingly we have only documented such plots for KMgH3 in
Fig. 5. According to the charge-density distribution at the K, Mg,
and H sites, it is evident that the highest charge density resides in
the immediate vicinity of the nuclei. As also evidenced from the
almost spherical charge distribution the bonding between K and
H is virtually pure ionic and between Mg and H predominantly
ionic. The type of charge distribution seen in Fig. 5 appears
to be typical for ionic compounds [49]. The main distinction
between the bonding in the MMgH3 series and the situation in
the MAlH4 and M3 AlH6 series is that the interaction between
Mg and H has more pure ionic character than that between the Al
and H [42]. The electron population between K and the MgH6
units is almost zero (viz. charges are depleted from this region),
which reconfirms that the interaction between K and MgH6 units
is virtually pure ionic.
Fig. 5(b) depicts the charge transfer (i.e., the electron distribution in the compound minus the electron density of the
corresponding overlapping free atoms) in KMgH3 . This illustration further reconfirms that charge has been depleted from the K
and Mg sites and transferred to the H sites. The overall message
is that KMgH3 is to be regarded as an ionic substance. However,
near the Mg sites, the magnitude of the transferred charge varies
in different directions of the crystal, which implies that there
must be a small but significant amount of covalent character in
the Mg–H bonds.
The calculated ELF plot (Fig. 5(c); for more information
about ELF see Refs. [50–52]) shows a predominent maximum of
ca. 0.99 at the H site and these electrons have a paired character.
The ELF value at the K and Mg sites is very low. The inference
is that charges are transferred from the inhabitants of these sites
to the H sites and there are certainly very few paired valence
electrons left at the K and Mg sites. A certain polarized character is found in the ELF distribution at the H sites in all complex
hydrides we investigated earlier [42]. The ELF distribution is on
the contrary quite isotropic in MMgH3 . This provides another
indication for a higher degree of ionic character in the MMgH3
series than in the MAlH4 and M3 AlH6 series.
In an other attempt to quantify the bonding and estimate the
amount of electrons on and between the participating atoms we
have made Mulliken-population analysis. Although there is no
unique definition to identify how many electrons that 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 population analyses. Due to its simplicity, the Mulliken-population
[53] scheme has become the most popular approach. However, the method is more qualitative than quantitative, providing
results that are sensitive to the atomic basis. The calculated
Mulliken charges are reported in Table 4 for MH, MgH2 , and
MMgH3 . The commonly recognized nearly pure ionic compound LiH gave Mulliken effective charges (MEC) of +0.98e
for Li and −0.98e for H. The corresponding MEC values for
MgH2 are +1.87e for Mg and −0.93e for H. The overlap populations between Li+ and H− in LiH and Mg2+ and H− in
MgH2 are close to zero, as expected for ionic compounds. In
accordance with textbook chemistry, the actual values shows
that MgH2 is slightly less ionic than LiH. The MEC for M
335
Table 4
Mulliken-population analysis for MH, MgH2 , and MMgH3 compounds
Compounds
Atom
MEC
Overlap population
LiH
Li
H
+0.98
−0.98
−0.003 (Li–H)
NaH
Na
H
+0.96
−0.96
−0.004 (Na–H)
KH
K
H
+0.97
−0.97
−0.004 (K–H)
RbH
Rb
H
+0.97
−0.97
−0.005 (Rb–H)
CsH
Cs
H
+0.95
−0.95
−0.005 (Cs–H)
MgH2
Mg
H
+1.87
−0.93
−0.040 (Mg–H)
LiMgH3
Li
Mg
H
+1.005
+1.964
−0.99
−0.017 (Li–H)
−0.015 (Mg–H)
NaMgH3
Na
Mg
H
1.079
1.953
−1.009
−0.046 (Na–H)
−0.022 (Mg–H)
KMgH3
K
Mg
H
+1.152
+1.962
−1.038
−0.030 (K–H)
−0.010 (Mg–H)
RbMgH3
Rb
Mg
H
+1.033
+1.927
−0.992
−0.025 (Rb–H)
−0.001 (Mg–H)
␣-CsMgH3
Cs
Mg
H
+1.073
+1.945
−1.072
−0.077 (Cs–H)
−0.008 (Mg–H)
The Mulliken-effective charges (MEC) are given in terms of e.
and H in the MMgH3 compounds indicates that the interaction
between M and MgH6 is ionic (in all cases around one electron
transferred from M to MgH6 ). Within the MgH6 unit, Mg consistently donates nearly two electrons to the H sites. The overlap
population between Mg and H is very small evidencing that the
interaction between Mg and H is highly ionic. The latter feature is one of the main bonding distinction between the MMgH3
series and the MAlH4 and M3 AlH6 series where there is found
a significant overlap population between Al and H within the
AlH4 /AlH6 units [42,9].
Born-effective-charge (BEC) analysis is another tool to evaluate bonding characteristics. The BEC were here calculated
using the Berry-phase approach implemented in the VASP code.
The King-Smith and Vanderbilt [25] method was used to calculate the polarizations in the perturbed cells and subsequently
the BEC tensor elements (see Table 5) for the ions involved.
For the MH and MgH2 cases (not listed in Table 5) the diagonal components of the effective charges in a Cartesian frame
satisfy the relation Zxx = Zyy = Zzz , whereas the off-diagonal
components turn out to be negligible. This is as expected for
ionic compounds due to the spherical character of ionic bonds.
For MMgH3 the diagonal components of the effective charges
at the M and Mg sites are also almost equal (Zxx ≈ Zyy ≈ Zzz )
and the off-diagonal components are negligibly small (except
for CsMgH3 ). At the H sites the diagonal components are also
336
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
Table 5
Calculated Born-effective-charge tensor elements (Z) for the constituents of MMgH3 compounds
xx
yy
zz
xy
LiMgH3
ZLi
ZMg
ZH
0.971
1.832
−1.507
0.966
1.822
−1.361
0.951
1.908
−1.450
0.014
−0.116
0.021
0.000
0.000
0.059
0.000
0.000
0.119
0.000
0.001
0.121
0.000
0.000
0.070
0.000
0.000
0.059
NaMgH3
ZNa
ZMg
ZH1
ZH2
1.074
1.808
−0.811
−1.039
1.088
1.799
−1.240
−0.823
1.064
1.797
−0.828
−1.018
0.000
−0.029
0.000
−0.004
0.004
0.123
0.002
0.012
0.012
−0.079
0.020
0.210
0.003
0.076
−0.020
0.204
0.000
−0.108
0.000
0.017
0.004
0.123
0.002
0.012
KMgH3
ZK
ZMg
ZH
1.008
1.899
−0.826
1.008
1.899
−0.826
1.008
1.899
−1.259
−0.008
0.000
−0.097
−0.008
0.000
−0.081
−0.001
0.009
0.000
0.017
−0.073
−0.091
0.013
−0.079
−0.010
−0.024
0.000
−0.087
RbMgH3
ZRb1
ZRb2
ZMg1
ZMg2
ZH1
ZH2
1.069
1.057
2.064
1.684
−0.838
−1.180
1.066
1.057
2.061
1.682
−0.906
−0.823
0.995
1.025
1.944
1.914
−1.044
−0.952
−0.001
0.000
−0.001
0.000
−0.063
0.001
0.000
0.001
−0.003
0.012
0.000
0.023
0.000
0.000
0.000
0.000
−0.005
−0.014
0.000
0.000
−0.001
0.002
0.000
−0.002
0.000
0.001
0.000
0.000
−0.009
0.000
0.000
0.061
−0.003
0.022
0.000
−0.003
␣-CsMgH3
ZCs1
ZCs2
ZMg1
ZMg2
ZH1
ZH2
ZH3
ZH4
ZH5
1.188
1.208
1.550
1.851
−1.107
−0.949
−0.939
−0.935
−1.033
1.226
1.245
1.417
1.649
−0.796
−1.147
−0.963
−0.897
−0.828
1.264
1.136
1.655
1.561
−0.934
−0.836
−0.978
−0.891
−1.006
0.000
0.000
0.000
0.000
0.000
0.244
−0.005
−0.010
0.000
−0.002
−0.003
0.011
0.002
−0.010
−0.152
0.097
0.027
−0.004
0.000
0.065
0.000
−0.149
0.000
0.101
0.000
0.000
0.066
−0.003
0.083
−0.012
−0.161
−0.012
0.104
0.004
0.000
0.016
0.000
0.000
0.000
0.000
0.000
−0.125
0.077
0.007
0.000
−0.002
−0.003
0.011
0.002
−0.010
−0.152
0.097
0.027
−0.004
almost equal, but the off-diagonal components take small but
definite values in most cases. This probably originates from
some exchange owing to sharing of electrons between H and
Mg within the MgH6 units. The BEC analyses accordingly
reconfirms that the M and Mg sites of the MMgH3 series
give up nearly one and two electrons, respectively, whereas
the H atoms correspondingly gain nearly one electron each.
This conclusion therefore largely concurs with the formal ionic
picture. A methodological comment seems appropriate: Our
experience from studies of chemical bonding [9,42] by both
Mulliken-population and Born-effective-charge analyses show
that two approaches yield almost the same conclusion. Our personal view is that Born-effective-charge calculations normally
demand larger computations. Hence, we recommend Mullikenpopulation analysis as a more suitable tool to probe bonding in
complex hydrides.
Finally, one comment on the bond-valence data included in
Table 2. The calculated BVS data from experimental as well as
theoretical interatomic distances suggest that one does not get
the correct picture in this case (except possibly for LiMgH3 ).
A possible reason for this failure may be that the hard sphere
approximation is not valid because of the diffuse and nonspherical nature of the electron distribution at the H site. As
pointed out in Section 3.1.6 the added BVS values for M and Mg
comes close to the expected value 3. The findings for the ideal
perovskite-type structure of KMgH3 (BVS = 3.20) suggest that
yz
zx
xz
zy
yx
the K–H distances have come out too short and the Mg–H distances too long compared with the model situations assumed
for BVS considerations. We believe that it is just the diffuse
and non-spherical character of the electron distribution around
H which give rise to the approximatively equal elongation of the
M–H bonds and shortening of the Mg–H bonds (see also Ref.
[38]). The exponential relation (Eq. (5)) between bond valence
and bond distance then explains why approximately equal shortening in M–H and elongation in Mg–H lead to correspondingly
roughly equal positive and negative increments in BVS for M
and Mg, respectively, and subsequently the observed constancy
in the added BVS values for M and Mg.
4. Conclusion
The crystal and electronic structures of the MMgH3 (M = Li,
Na, K, Rb, Cs) series have been studied by state-of-the-art
density-functional calculations. The ground-state crystal structures have been identified from structural optimization of a
number of structures using force as well as stress minimizations.
For the experimentally known compounds, the ground-state
structures are successfully reproduced within the accuracy of the
density-functional approach. The crystal structure of LiMgH3
has been predicted to be of the LiTaO3 type (note: lacking
inversion symmetry) at 0 K and atmospheric pressure. LiMgH3
should be an ionic insulators, and we predict that it will exhibit
P. Vajeeston et al. / Journal of Alloys and Compounds 450 (2008) 327–337
ferroelectric properties. Formation energies for the MMgH3
series are calculated for different possible reaction pathways.
For LiMgH3 we propose that synthesis from elemental Li and
Mg in hydrogen atmosphere should provide a suitable route.
The MMgH3 compounds are wide-band-gap insulators and the
insulating behavior is associated with well localized, paired
s-electron configuration at the H site. The chemical bonding
character of these compounds is highly ionic according to analyses of DOS, charge density, charge transfer, ELF, Mulliken
population and Born effective charge. In fact the ionic character for the MMgH3 compounds closely matches the bonding
situation in the corresponding binary hydrides.
Acknowledgement
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
The authors gratefully acknowledge the Research Council of
Norway for financial support and for computer time at the Norwegian supercomputer facilities. P.V. also acknowledge Arne
Klaveness for helping to compute the Born effective charges
from the Berry-phase approach.
References
[1] K. Yvon, R.B. King (Eds.), Encyclopedia of Inorganic Chemistry, vol. 3,
Wiley, NY, 1994.
[2] J. Block, A.P. Gray, Inorg. Chem. 304 (1965) 4.
[3] J.A. Dilts, E.C. Ashby, Inorg. Chem. 11 (1972) 1230;
J.P. Bastide, B. Monnetot, J.M. Letoffe, P. Claudy, Mater. Res. Bull. 20
(1985) 997.
[4] B. Bogdanovic, M. Schwickardi, J. Alloys Compd. 253–255 (1997) 1.
[5] B. Bogdanovic, R.A. Brand, A. Marjanovic, M. Schwikardi, J. Tölle, J.
Alloys Compd. 302 (2000) 36.
[6] H.W. Brinks, B.C. Hauback, P. Norby, H. Fjellvåg, J. Alloys Compd. 315
(2003) 222.
[7] C.M. Jensen, K. Gross, J. Appl. Phys. A 72 (2001) 213.
[8] H. Morioka, K. Kakizaki, S.C. Chung, A. Yamada, J. Alloys Compd. 353
(2003) 310.
[9] A. Klaveness, P. Vajeeston, P. Ravindran, H. Fjellvåg, A. Kjekshus, Phys.
Rev. B 73 (2006) 094122.
[10] A.W. Overhauser, Phys. Rev. B 35 (1987) 411.
[11] N.W. Ashcroft, Phys. Rev. Lett. 21 (1968) 1748.
[12] M.R. Press, B.K. Rao, P. Jena, Phys. Rev. B 38 (1988) 2380.
[13] M. Seel, A.B. Kunz, S. Hill, Phys. Rev. B 39 (1989) 7949.
[14] M. Gupta, A.P. Percheron-Guegan, J. Phys. F 17 (1987) L201.
[15] R. Yu, P.K. Lam, Phys. Rev. B 38 (1988) 3576.
[16] J.L. Martins, Phys. Rev. B 38 (1988) 12776.
[17] Y. Li, T.M. Mullen, P. Jena, P.K. Khowash, Phys. Rev. B 44 (1991) 6030.
[18] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953;
G. Kresse, J. Joubert, Phys. Rev. B 59 (1999) 1758.
[19] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) R6726;
G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15.
[20] J.P. Perdew, P. Ziesche, H. Eschrig (Eds.), Electronic Structure of Solids,
Akademie Verlag, Berlin, 1991;
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
337
J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B 54 (1996) 16533;
J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
P. Vajeeston, P. Ravindran, H. Fjellvåg, A. Kjekshus, J. Alloys Compd. 363
(2003) L7.
P. Vajeeston, P. Ravindran, R. Vidya, A. Kjekshus, H. Fjellvåg, Appl. Phys.
Lett. 82 (2003) 2257.
P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvåg, Phys. Rev. Lett. 89
(2002) 175506.
http://www.crystal.unito.it/.
R.D. King-Smith, D. Vanderbilt, Phys. Rev. B 47 (1992) 1651.
Inorganic Crystal Structure Database, Gmelin Institut, Germany, 2001, pp.
1.
G. Renaudin, B. Bertheville, K. Yvon, J. Alloys Compd. 353 (2003) 175.
A. Bouamrane, J.P. Laval, J.-P. Soulie, J.R. Bastide, Mater. Res. Bull. 35
(2000) 545.
H.-H. Park, M. Pezat, B. Darriet, P. Hagenmuller, J. Less-Common Met.
136 (1987) 1.
F. Gingl, T. Vogt, E. Akiba, K. Yvon, J. Alloys Compd. 282 (1999)
125.
E. Rönnebro, D. Noréus, K. Kadir, A. Reiser, B. Bogdanovic, J. Alloys
Compd. 299 (2000) 101.
N.E. Brese, M. O’Keeffe, Acta Crystallogr., Sect. B 47 (1991) 192.
S. Geller, J. Chem. Phys. 24 (1956) 1236.
J.P. Bastide, A. Bouamrane, P. Claudy, J.-M. Letoffe, J. Less-Common
Met. 136 (1987) L1.
R. Schumacher, A. Weiss, J. Less-Common Met. 163 (1990) 179.
R.D. Shannon, Acta Crystallogr., Sect. B 32 (1976) 751.
B. Bertheville, P. Fischer, K. Yvon, J. Alloys Compd. 330–332 (2002) 152.
W. Bronger, R. Kniep, M. Kohout, Z. Anorg. Allg. Chem. 631 (2005)
265.
R.H. Mitchell (Ed.), Perovskites Modern and Ancient, Almaz, Thunder
Bay, Canada, 2002 .
M. Yamaguchi, E. Akiba, R.W. Cahn, P. Hassen, E.J. Kramer (Eds.),
Materials Science and Technology, vol. 3B, VCH, New York, 1994 , p.
333.
P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, H. Fjellvåg, Phys. Rev.
Lett. 89 (2002) 106403.
P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvåg, Phys. Rev. B 71
(2005) 216102.
P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvåg, Phys. Rev. B 69
(2004) 020104 (R).
P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvåg, Phys. Rev. B 71
(2005) 092103.
P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvåg, J. Alloys Compd. 387
(2005) 97.
P. Vajeeston, P. Ravindran, R. Vidya, H. Fjellvåg, A. Kjekshus, Cryst.
Growth Des. 4 (2004) 471.
M. Gupta, A. Percheron-Guegan, Chem. Scr. 28 (1988) 117.
F. Gygi, A. Baldereschi, Phys. Rev. Lett. 62 (1989) 2160.
P. Vajeeston, Theor. Model. Hydrides (2004), ISSN 1501-7710.
A. Savin, A.D. Becke, J. Flad, R. Nesper, H. Preuss, H.G. von Schnering,
Angew. Chem. Int. Ed. Engl. 30 (1991) 409;
A. Savin, O. Jepsen, J. Flad, O.K. Andersen, H. Preuss, H.G. von Schnering,
Angew. Chem. Int. Ed. Engl. 31 (1992) 187.
B. Silvi, A. Savin, Nature 371 (1994) 683.
A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397.
R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833.