I
PHYSICAL REVIEW B, VOLUME 65, 075101
Electronic structure, phase stability, and chemical bonding in Th2 Al and Th2 AlH4
P. Vajeeston,1,* R. Vidya,1 P. Ravindran,1 H. Fjellvåg,1,2 A. Kjekshus,1 and A. Skjeltorp2
1
Department of Chemistry, University of Oslo, Box 1033, Blindern, N-0315, Oslo, Norway
2
Institute for Energy Technology, P.O. Box 40, Kjeller, N-2007, Norway
共Received 16 August 2001; published 16 January 2002兲
We present the results of a theoretical investigation on the electronic structure, bonding nature, and groundstate properties of Th2 Al and Th2 AlH4 using generalized-gradient-corrected first-principles full-potential
density-functional calculations. Th2 AlH4 has been reported to violate the ‘‘2-Å rule’’ for H-H separation in
hydrides. From our total-energy as well as force-minimization calculations, we found a shortest H-H separation
of 1.95 Å in accordance with recent high-resolution powder neutron-diffraction experiments. When the Th2 Al
matrix is hydrogenated, the volume expansion is highly anisotropic, which is quite opposite to other hydrides
having the same crystal structure. The bonding nature of these materials is analyzed in terms of density of
states, crystal-orbital Hamiltonian population, and valence-charge-density analyses. Our calculation predicts a
different nature of the bonding between the H atoms along a and c. The strongest bonding in Th2 AlH4 is
between Th and H along c which form dumbbell-shaped H-Th-H subunits. Due to this strong covalent interaction there is a very small amount of electrons present between the H atoms along c. This reduces the
repulsive interaction between the H atoms along c and explains why Th2 AlH4 has a shorter H-H separation
than most other metal hydrides. The large difference in the interatomic distances between the interstitial
regions where one can accommodate H in the ac and ab planes along with the strong covalent interaction
between Th and H are the main reasons for highly anisotropic volume expansion on hydrogenation of Th2 Al.
DOI: 10.1103/PhysRevB.65.075101
PACS number共s兲: 71.15.Nc, 71.20.⫺b, 81.05.Je
I. INTRODUCTION
Hydrides of intermetallics have been extensively studied
because of their applications in rechargeable batteries. Unfortunately, most metals that absorb large amounts of hydrogen are heavy and/or expensive.1 Consequently, there is a
constant search for hydrides that may be suitable for practical applications. First of all, it is very important to understand how crystal structural evolution takes place in the
course of hydrogenation. Numerous studies have been done
to explain observed stabilities, stoichiometries, and preferred
H sites in hydrides of metallic and intermetallic compounds.
Structural studies of hydrides have provided empirical rules2
that can be used to predict the stability of the H sublattice in
a given metal configuration. A survey of stable hydrides
shows that the H-H distance does not go below 2.1 Å 共the
‘‘2-Å rule’’兲 with a minimum ‘‘radius’’ of 0.4 Å for the
intersite to be used for the accommodation of H. These rules
have been used to predict new hydrides whose existence later
has been verified experimentally.1–3
The review of Yvon and Fischer4 states that Th2 AlH4
共Ref. 5兲 and K2 ReH9 共Refs. 4 and 6兲 violate the 2-Å rule, the
shortest H-H separation being 1.79 and 1.87, respectively.
K2 ReH9 is classified among complex transition-metal hydrides, which comprise highly covalent solids with nonmetallic properties. Th2 AlH4 , on the other hand, has metallic
character.
Th2 Al7 together with Zr2 Fe, Zr2 Co, and Zr2 Ni crystallize
in the CuAl2 -type structure, whereas their hydrides form
rather different crystal structures. Zr2 Fe and Zr2 Co form the
isostructural deuterides Zr2 M D5 (M ⫽Fe,Co) 8 with a change
in symmetry from I4/mcm to P4/ncc on deuteration.
Th2 AlH4 共Refs. 5 and 9兲 and Zr2 NiH4.74 共Ref. 10兲 are formed
without any change in the symmetry from their parent struc0163-1829/2002/65共7兲/075101共10兲/$20.00
tures. Th2 AlH4 belongs to the exclusive class which does not
obey the 2-Å rule. The lattice expansion along a and c has
proved to be highly anisotropic on hydrogenation of Th2 Al.
In order to shed light on this effect we need theoretical understanding about the bonding nature in this compound. Further, the understanding of the lattice expansion and distortion
during hydrogenation will be important for the evaluation of
stability of the hydride. So, we have made a detailed study of
Th2 Al and Th2 AlH4 by first-principles calculations.
Two different powder neutron-diffraction 共PND兲 studies
of Th2 AlH4 give different H-H separations, viz., the older5
value is 1.79 Å and the more recent9 value is 1.97 Å. So one
aim of this study has been to solve this discrepancy. In principle, the stability of hydrides can be evaluated directly from
a theoretical study of the total energy. However, owing to the
complexity of the structure of transition-metal hydrides, to
our best knowledge, no reliable theoretical heat of formation
has hitherto been reported.11 Nakamura et al.11 were the first
to calculate heat of formation for this type of hydrides. However, these authors obtained a positive and unrealistically
large heat of formation even for stable La-Ni-based hydrides,
except for La2 Ni10H14 . 12–14 This unfavorable result clearly
indicates that local relaxation of the metal atoms surrounding
the hydrogens must be included in the calculations in order
to predict the structural stability parameters. Hence, our calculations take into account local relaxation by optimizing the
atom positions globally.
We present the electronic structure of Th2 Al and
Th2 AlH4 , obtained by the full-potential linearizedaugmented plane-wave 共FPLAPW兲 method. A central feature
of the paper is the evaluation of the electronic structure and
bonding characteristics on introduction of H into the Th2 Al
matrix. In addition to regular band-structure data, we also
65 075101-1
©2002 The American Physical Society
P. VAJEESTON et al.
PHYSICAL REVIEW B 65 075101
FIG. 1. The crystal structure of Th2 AlH4 . Five Th in facesharing tetrahedral 共bipyramidal兲 configuration surround two hydrogen. Legends to the different kinds of atoms are given on the illustrations.
provide crystal-orbital Hamiltonian population 共COHP兲
共Refs. 15 and 16兲 results to illustrate the chemical bonding in
more detail.
This paper is organized as follows. Details about the involved structure and computational method are described in
Sec. II. Section III gives the results of the calculations and
comparisons with the experimental findings. Conclusions are
briefly summarized in Sec. IV.
II. STRUCTURAL DETAILS
Th2 Al and Th2 AlH4 crystallize in space group I4/mcm
with the lattice parameters a⫽7.618, c⫽5.862 Å for Th2 Al
共Ref. 17兲 and a⫽7.626, c⫽6.515 Å for Th2 AlH4 .9 The crystal structure of Th2 AlH4 is illustrated in Fig. 1. The crystal
structure of Th2 Al contains four crystallographically different interstitial sites, which are the suitable sites for hydrogen
accommodation, 16l and 4b each coordinates to four Th,
32m coordinates to three Th and one Al, and 16k coordinates
to two Th, and two Al. Each 16l-based intersite tetrahedron
shares a common face with another 16l-based tetrahedron,
whereas the 4b-based tetrahedron shares each of its four
faces with 16l-based tetrahedra. Some of the tetrahedral intersites are closely separated owing to the face sharing of the
coordination polyhedra. According to the experimental
findings,5,9 the 16l sites are fully occupied in Th2 AlD4 , and
also the structure is completely ordered.
A. Computational details
In our calculations we use the FPLAPW method in a scalar relativistic version without spin-orbit coupling as embodied in the WIEN97 code.18 In brief, this is an implementation
of density-functional theory 共DFT兲 with different possible
approximations for the exchange and correlation potentials,
including the generalized-gradient approximation 共GGA兲.
The Kohn-Sham equations are solved using a basis of
linearized-augmented plane waves.19 For the exchange and
correlation potentials, we used the Perdew and Wang20
implementation of the GGA. For the potential and chargedensity representations, inside the muffin-tin spheres the
wave function is expanded in spherical harmonics with
l max ⫽10, and nonspherical components of the density and
potential are included up to l max ⫽6. In the interstitial region
they are represented by Fourier series and thus they are completely general so that such a scheme is termed a fullpotential calculation. In the present calculations we used
muffin-tin radii of 2.5, 2.0, and 1.6 a.u. for Th, Al, and H,
respectively.
The basis set includes 7s, 7p, 6d, and 5 f valence and 6s
and 6p semicore states for Th, 3s and 3p valence and 2p
semicore states for Al, and 1s states for H. These basis functions were supplemented with local orbitals21 for additional
flexibility to the representation of the semicore states and for
generalization of the linearization errors. We have included
the local orbitals for Th-6s, Th-6p, and Al-2p semicore
states. In all our calculations we have used the tetrahedron
method on a grid of 102 k points in the irreducible part of the
hexagonal Brillouin zone,22 which corresponds to 1000 k
points in the whole Brillouin zone. The calculations are done
at several cell volumes 共around the equilibrium volume兲 for
both Th2 Al and Th2 AlH4 and corresponding total energies
are evaluated self-consistently by iteration to an accuracy of
10⫺6 Ry/cell. Similar densities of k points were used for the
force minimization and c/a optimization calculations.
In order to measure the bond strengths we have computed
the COHP,16 which is adopted in the tight binding linear
muffin-tin orbital 共TBLMTO-47兲 package.23,24 The COHP is
the density of states weighted by the corresponding Hamiltonian matrix elements, which if negative indicates a bonding character and if positive indicates an antibonding character. The simplest way to investigate the bonding between
two interacting atoms in the solid would be to look at the
complete COHP between them, taking all valence orbitals
into account. However, it may sometimes be useful to focus
on pair contributions of some specific orbitals.
III. RESULT AND DISCUSSION
The H-H separation is one of the most important factors
in identifying the potential candidate for hydrogen storage,
because if the H-H separation is small one can accommodate
more H within a small region. From this point of view,
Th2 AlH4 may be considered as a potential candidate for storing H. To the best of our knowledge no theoretical or experimental attempts have been made to study cohesive properties
like heat of formation (⌬H), cohesive energy (E coh ), bulk
modulus (B 0 ), and its pressure derivative (B 0⬘ ) for this compound. Hence, to our best knowledge, this is the first theoretical attempt to study the ground-state properties and bonding in this compound.
A. Structural optimization from total-energy studies
In order to analyze the effect of hydrogenation on the
crystal structure of Th2 Al and to verify the discrepancy between the experimentally observed H-H separations, we have
optimized the structural parameters for Th2 Al and Th2 AlH4 .
075101-2
ELECTRONIC STRUCTURE, PHASE STABILITY, AND . . .
PHYSICAL REVIEW B 65 075101
FIG. 2. Total energy 共Ry/f.u.兲 共a兲 vs c/a and 共b兲 vs unit-cell
volume for Th2 Al where ⌬E⫽E⫹106 632. The arrow indicates the
theoretical equilibrium.
For this purpose, first we have relaxed the atomic positions
globally using the force-minimization technique, by keeping
c/a and cell volume (V 0 ) fixed to experimental values. Then
the theoretical equilibrium volume is determined by fixing
optimized atomic positions and experimental c/a, and varying the cell volume within ⫾10% of V 0 . Finally the optimized c/a ratio is obtained by a ⫾2% variation in c/a 共in
steps of 0.005兲, while keeping the theoretical equilibrium
volume fixed. It is important to note that the experimentally
observed lattice parameters are almost equal, while the
atomic position for H alone differs between the two experimental results 共according to Bergsma et al.5 H coordinates
are 0.368, 0.868, and 0.137 whereas So” rby et al.9 give
0.3707, 0.8707, and 0.1512兲. The total energy vs cell volume
and c/a ratio curves for Th2 Al and Th2 AlH4 are shown in
Figs. 2 and 3. From these illustrations it is clear that the
equilibrium structural parameters obtained from our theoretical calculations are in very good agreement with those obtained by the recent PND study.9
The optimized atomic positions along with the corresponding experimental values are given in Table I. Table II
gives calculated lattice parameters and interatomic distances,
along with corresponding experimental values for both
FIG. 3. The total energy 共a兲 vs c/a and 共b兲 vs unit-cell volume
for Th2 AlH4 where ⌬E⫽E⫹106 636. The arrow indicates the theoretical equilibrium.
Th2 Al and Th2 AlH4 . The theoretically estimated equilibrium
volume is underestimated by 0.27% for Th2 Al and by 1.8%
for Th2 AlH4 . The underestimation of bond length in the
present study is partly due to the neglect of the zero-point
motion and thermal expansions. The difference between the
experimental values may be due to the poor resolution of the
1961 PND data.5
B. Cohesive properties
The method of calculation for cohesive properties for intermetallic compounds is well described in Refs. 25–27. The
cohesive energy is a measure of the force that binds atoms
together in the solid state. The cohesive energy of a system is
defined as the sum of the total energy of the constituent
atoms at infinite separation minus the total energy of the
particular system. This is a fundamental property which has
long been the subject of theoretical approaches. The chemical bonding in intermetallic compounds is a mixture of covalent, ionic, and metallic bonding and therefore the cohesive energy cannot be determined reliably from simple
models. Thus, first-principles calculations based on DFT
075101-3
P. VAJEESTON et al.
PHYSICAL REVIEW B 65 075101
TABLE I. Atomic positions of Th2 Al and Th2 AlH4 .
Th2 Al
Th
Theory
Experiment a
Experiment b
Experiment c
Al
Theory
Experiment a
Experiment b
Experiment c
Th2 AlH4
x
y
z
x
y
z
0.1583
0.1588
0.6583
0.6588
0.0000
0.0000
0.1632
0.6632
0.0000
0.1656
0.162
0.6656
0.662
0.0000
0.0000
0.0
0.0
0.25
0.0
0.0
0.0
0.0
0.25
0.25
0.3705
0.3707
0.368
0.8705
0.8707
0.868
0.1512
0.1512
0.137
0.0
0.0
0.0
0.0
0.25
0.25
H
Theory
Experiment b
Experiment c
a
Reference 17.
Reference 9.
Reference 5.
b
c
have become a useful tool to determine the cohesive energy
of solids. For the study of phase equilibrium the cohesive
energy is more descriptive than the total energy, since the
latter includes a large contribution from electronic states that
do not play a role in bonding. From our cohesive energy
calculations we get E coh ⫽0.15 and 0.185 eV/atom for Th2 Al
and Th2 AlH4 , respectively, indicating that hydrogenation enhances the bond strength in Th2 Al.
The formation energy (⌬H) is introduced in order to facilitate a comparison of system stability. ⌬H is defined as the
total-energy difference between the energy of the compound
and the weighted sum of the corresponding total energy of
the constituents. The present type of calculation is not suitable for handling molecules. In order to have the correct heat
of formation of the system one has to calculate the total
energy of the compounds and the constituents in the same
way 共i.e., with same exchange-correlation method, radius,
etc.兲. Patton et al.28 calculated the cohesive energy for an H2
molecule with different exchange-correlation potentials. We
have estimated the total energy of the H2 molecule in the
following way. First we have calculated the atomic total energy for H for the spin-polarized case with the same computational parameters we used in the total-energy calculations
for Th2 AlH4 . We have then added the cohesive energy of
TABLE II. Lattice parameters and interatomic distances of Th2 Al and Th2 AlH4 共except for c/a, all values
are in angstrom兲.
Th2 Al
a
c
c/a
Th-H
Th-Al
Th-Th
Al-H
Al-Al
H-H (ac plane兲
H-H (ab plane兲
Th2 AlH4
a
Theory
Experiment
Theory
Experimentb
Experimentc
7.602
5.723
0.753
7.618
5.862
0.769
3.199
3.403
3.219
3.421
2.861
2.931
7.604
6.433
0.846
2.273
3.269
3.509
3.051
3.216
1.945
2.344
7.626
6.515
0.854
2.305
3.278
3.571
3.061
3.257
1.971
2.305
7.629
6.517
0.854
2.387
3.291
3.495
3.072
3.258
1.790
2.495
a
Reference 17.
Reference 9.
Reference 5.
b
c
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ELECTRONIC STRUCTURE, PHASE STABILITY, AND . . .
PHYSICAL REVIEW B 65 075101
TABLE III. Ground-state properties of Th2 Al and Th2 AlH4 .
Parameters
Th2 Al
Th2 AlH4
⫺⌬H(kJ mol⫺1 )
E coh 共kJ mol⫺1 兲
N(E F ) 共states/Ry cell兲
B 0 共GPa兲
B 0⬘
60
43.42
58.42
93.42
3.41
431
124.95
41.13
111.36
3.48
4.540 eV/H2 given in Ref. 28. This will mimic the total
energy for the H2 molecule as that calculated for the bulk
material. ⌬H provides information about the stability of
Th2 Al towards hydrogenation. The calculated ⌬H values for
La-Ni-based hydrides11 were almost twice the
experimental12–14 ⌬H. As the linear muffin-tin orbital atomic
sphere approximation 共LMTO-ASA兲 method was used in
that study, this discrepancy was to be expected because the
internal relaxation of the atoms was not taken into account
and the interstitial potential is not well represented in the
LMTO-ASA method. Therefore, ⌬H calculated by using the
full-potential method should be more reliable. Our calculated
values for E coh and ⌬H are given in Table III. Since ⌬H is
more negative and E coh is higher for Th2 AlH4 than for
Th2 Al, we can conclude that Th2 AlH4 is more stable than
Th2 Al. However, no experimental ⌬H values for Th2 Al and
Th2 AlH4 are available, but we note that our calculated ⌬H
for Th2 AlH4 关⫺107 kJ/共mol H兲兴 is close to the experimentally observed values of other Th-based hydrides, like ThH2
having a ⌬H value of ⫺73 kJ/(mol H).29
From the self-consistent total-energy calculations for
eight different volumes within the range of V/V 0 from 0.75
to 1.10 using a universal model30 of the equation of state, the
bulk modulus and its pressure derivative for Th2 Al and
Th2 AlH4 are evaluated 共see Table III兲. The calculated bulk
modulus for Th2 Al is 93.42 GPa and for Th2 AlH4 , 111.36
GPa. The corresponding pressure derivatives of the bulk
modulus (B 0⬘ ) are 3.41 and 3.48, respectively. The enhancement of B 0 for the hydrogenated phase indicates that hydrogen plays an important role in the bonding behavior of
Th2 AlH4 . In particular, the hydrogenation enhances the bond
strength, and hence the change in volume with hydrostatic
pressure decreases on hydrogenation. This conclusion is consistent with the observation made from our calculated heat of
formation and cohesive energy for Th2 Al and Th2 AlH4 .
C. Anisotropic behavior
For compounds which maintain the basic structural
framework, the occupancy of hydrogen in interstitial sites is
determined by its chemical environment 共different chemical
affinity for the elements in the coordination sphere also results in different occupancy兲. Although the H atom is small
and becomes even smaller by chemical bonding to the host,
it may deform and stress the host metal considerably depending upon the chemical environment. Lattice expansion usually of the order of 5% to 30%, often anisotropic, results
from hydride formation. The record-large volume expansion
FIG. 4. c/a for CuAl2 -type phases and their corresponding hydrides. Lines are guides for the eye.
observed for the change from CeRu2 to CeRu2 D5 共37%兲 is
due to a hydrogen-induced electron transition as shown by
x-ray photoemission spectroscopy measurements.31 A lattice
contraction upon hydrogenation has so far only been observed for ThNi2 to ThNi2 D2 (⫺2.2%). For most hydrides
formed from intermetallic compounds the crystal structure
usually changes with a loss of symmetry.32 In general the
symmetry decreases as a function of hydrogen content and
increases as a function of temperature. However, on hydrogenation of Th2 Al the symmetry remains unchanged.
The volume expansion during hydrogenation of Th2 Al is
12.47% 关 ⌬V/H atom is 10.32 Å 3 兴. This volume expansion
is strongly anisotropic and proceeds predominantly perpendicular to the basal plane of the tetragonal unit cell; ⌬a/a
⫽0.026%, ⌬c/c⫽12.41%. This indicates a relatively flexible atomic arrangement in the 关001兴 direction. In spite of the
isostructurality between Th2 Al, Zr2 Fe 共hydrated: Zr2 FeH5 ),8
and Zr2 Co 共hydrated: Zr2 CoH4.82) 33 the Zr-based compounds
exhibit a quite opposite anisotropic behavior in that their unit
cells expand exclusively along the basal plane. The c/a ratio
plays an important role for the structural properties of intermetallic compounds including metal hydrides. For example,
in the case of Zr2 Fe,8 Zr2 Co,33 Zr2 Ni,10,34 and Th2 Al c/a is
0.878, 0.867, 0.812, and 0.769, respectively, and for the corresponding hydrides Zr2 FeH5 ,10 Zr2 CoH4.82 , 10 Zr2 NiH4.74 , 10
and Th2 AlH4 c/a is 0.810, 0.815, 0.833, and 0.854, respectively 共see Fig. 4兲. The increase in c/a for Zr2 CoH4.82 and
Zr2 FeH5 compared with their corresponding unhydrated parents is smaller than that for other pairs of compounds. On
hydrogenation, the increase in the c/a ratio for Th2 Al is
considerably larger than for Zr2 Ni, which may be the reason
why the former retains its symmetry on hydrogenation. Our
calculations describe well the anisotropic changes in the
crystal structure on hydrogenation of Th2 Al 共see Table II兲.
The c/a ratio increases almost linearly 共Fig. 4兲 on going
from Zr2 Fe to Th2 Al whereas the corresponding hydrides
show the opposite behavior. Hence, it appears that the systematic variation in c/a plays a major role in deciding the
crystal structure for the CuAl2 -type hydrides. When c/a
⬍0.825 the symmetry is changed from I4/mcm to P4/ncc
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P. VAJEESTON et al.
PHYSICAL REVIEW B 65 075101
FIG. 5. Electronic band structure of 共a兲 Th2 Al
and 共b兲 Th2 AlH4 . The Fermi level is set to zero.
on hydrogenation, whereas when c/a⬎ 0.825 the crystal
symmetry is apparently not affected.
D. Electronic structure
In order to understand the changes in the electronic bands
on hydrogenation of Th2 Al we show the energy-band structure for Th2 Al and Th2 AlH4 in Fig. 5. The illustrations
clearly indicate that inclusion of H in the Th2 Al matrix has a
noticeable impact on the band structure, mainly of the valence band 共VB兲. The two lowest-lying broad bands in Fig.
5共a兲 originate from Al-3s electrons. As the unit cell contains
two formula units, eight electrons are additionally introduced
when Th2 AlH4 is formed from Th2 Al. These electrons form
four additional bands 关Fig. 5共b兲兴, and a large deformation of
the band structure is introduced by the hydrogen in the
Th2 Al lattice. When these bands become localized, the lowest lying energy band is moved from ⫺7.34 to ⫺9.20 eV,
and the character of the latter band is changed from Al-3s to
H-1s. The Al-3s bands are located in a wide energy range
from ⫺2.8 to ⫺7.34 eV in Th2 Al and are in a narrow energy range from ⫺2.5 to ⫺4.2 eV in the hydride. The
change in the Al bands on hydrogenation of Th2 Al is apparently associated with the electron transfer from Th to Al. The
H-s bands are found in the energy range from ⫺2 eV to the
bottom of the VB, whereas their contribution at E F is negligibly small indicating bands with more localized character.
The bands at E F are dominated by the Al-3p and Th-6d
electrons in both Th2 Al and Th2 AlH4 . Owing to the creation
of the pseudogap feature near E F , the contributions of the
Al-3p electrons to the bands at the E F level are significantly
reduced by the hydrogenation of Th2 Al.
E. Nature of chemical bonding
1. Density of states
In order to obtain a deeper insight into the changes in
chemical bonding behavior on hydrogenation of Th2 Al we
give the angular-momentum and site-decomposed density of
states 共DOS兲 for Th2 Al and Th2 AlH4 in Fig. 6. DOS features
for Th2 Al and Th2 AlH4 show close similarity. Both exhibit
metallic character since there is a finite DOS at E F . From
the DOS features we see that E F is systematically shifted
toward higher energy in Th2 AlH4 . This is due to the increase
in the number of valence electrons when Th2 Al is hydrogenated. The DOS for both Th2 Al and Th2 AlH4 lies mainly in
four energy regions: 共a兲 the lowest region around ⫺20 eV
stems mainly from localized or tightly bound Th-6p states,
共b兲 the region from ⫺9.25 to ⫺2.5 eV originates from
bonding of H-1s, Al-3p, and Th-6d (Al-3p and Th-6d
states in Th2 Al), 共c兲 the region from ⫺2.5 to 0 eV comes
from bonding states of Al-3p and Th-6d, and 共d兲 the energy
region just above E F 共0 to 3.5 eV兲 is dominated by unoccupied Th-4 f states.
The semicore Th-6p states are well localized and naturally their effect on bonding is very small. On comparing the
Th-6p DOS of Th2 Al and Th2 AlH4 , it is seen that the width
is significantly reduced in Th2 AlH4 owing to the lattice expansion and the inclusion of additional energy levels below
E F . In the VB region, the bandwidth and DOS are larger for
Th2 AlH4 than for Th2 Al. Hydrogenation enhances interaction between neighboring atoms, thereby increases the overlap of orbitals, and in turn results in the enlarged VB width in
Th2 AlH4 . In particular, the strong hybridization between
Th-6d and H-1s states increases the VB width from 7.1 eV
in Th2 Al to 8.4 eV in Th2 AlH4 . H-1s, Al-3p, and Th-6d
states are energetically degenerate in the VB region indicating a possibility of covalent Th-H, Th-Al, and Al-H bonds.
However, the spatial separation of Th-Al 共3.22 Å兲 and Al-H
共3.02 Å兲 is larger than the Th-H separation 共2.26 Å兲. Therefore, covalent bonds between the former pairs are small
whereas there is a significant covalent contribution between
Th and H. In conformity with this, the COHP and chargedensity analyses show directional bonding between Th and H
共see Secs. III E 2 and III E 3兲. The accommodation of H in
the interstitial position between Th and Al creates new bonding states between Th and H. This also increases the Th-Al
distance around 2.2% compared with that in Th2 Al. The consequence of this enhancement is that the Al DOS in the VB
region becomes narrow and the splitting between the Al-3s
and Al-3p states is almost doubled 共see Fig. 6兲. The finite
DOS at E F which gives the metallic character of Th2 Al and
Th2 AlH4 comes from Th-d states in addition to some states
of Al-p character.
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ELECTRONIC STRUCTURE, PHASE STABILITY, AND . . .
PHYSICAL REVIEW B 65 075101
FIG. 6. Total, site, and orbital projected densities of states for 共a兲 Th2 Al and 共b兲 Th2 AlH4 .
Another interesting feature of the total DOS of Th2 AlH4
is the presence of a deep valley around E F which is termed
as a pseudogap. Pseudogap features are formed not only in
crystalline solids35 but occur also in amorphous phases36 and
quasicrystals.37 Two mechanisms have been proposed for the
occurrence of pseudogaps in binary alloys, one attributed to
ionic features and the other to the effect of hybridization.
Although the electronegativity differences between Th, Al,
and H are noticeable, they are not large enough to explain the
pseudogap in Th2 AlH4 . Hence hybridization must be the
cause for the creation of the pseudogap in Th2 AlH4 . There is
also proposed a correlation between the occurrence of
pseudogaps and structural stability,38 in that materials which
possess pseudogaps in the vicinity of E F usually have higher
stability. This correlates with the higher value of ⌬H in
Th2 AlH4 than in Th2 Al 共Table III兲.
2. Charge density
The analysis of the bonding between the constituents will
give a better understanding about the anisotropic changes in
the structural parameters on hydrogenation of Th2 Al. Figure
7 shows the calculated valence-charge density 共obtained directly from the self-consistent calculations兲 within ab and ac
planes for Th2 AlH4 . The Th, Al, and H atoms are confined to
layers along c, with Th and Al being situated in alternating
075101-7
P. VAJEESTON et al.
PHYSICAL REVIEW B 65 075101
FIG. 7. Valence electron charge-density plot for Th2 AlH4 in the
ab plane 共through H兲 with 40 contours drawn between 0 and 0.25
electrons/a.u.3.
metal layers with hydrogen in between, hence establishing a
sequence of •••-Th-H-Al-H-Th-H-Al-H-Th-••• layers 共see
Fig. 1兲. The H atoms are arranged in a chainlike manner
within the ab plane as also evident from Fig. 7 共lower figure兲. It is interesting to note that the nature of the H-H bonding is quite different along a and c. Although the H-H distance is 2.34 Å within the basal plane and 1.95 Å
perpendicular to the basal plane, the bonding between the H
atoms is not totally dominated by the covalent interactions
along c. In fact, the COHP analyses 共Sec. III E 3兲 shows that
the covalent H-H interaction within the ab plane is larger
than that within the ac plane. Examination of the partial
DOS 共PDOS兲 共see Fig. 6兲 shows that the valence electrons of
Th and Al in Th2 Al are not energetically degenerate in the
VB region and this indicates that there is a degree of ionic
character in their bonding in line with their electronegative
difference. When H is introduced to the Th2 Al lattice the
FIG. 8. Valence electron charge-density plot between the Th and
H atoms for Th2 AlH4 in the ac plane 共through H兲with 40 contours
drawn between 0 and 0.25 electrons/a.u.3.
PDOS value in the VB of Th is drastically reduced whereas
the corresponding PDOS value in the VB of Al is increased.
In conformity with this the integrated charge inside the Al
sphere is around 0.59 electrons 共0.8 electrons according to
the TBLMTO method兲 larger in Th2 AlH4 than in Th2 Al.
This indicates that the ionic character of the bonding between Th and Al is increased on the hydrogenation of Th2 Al.
The bonding between Th and H is predominantly covalent
as evidenced by the finite charge between these atoms 共see
Fig. 8兲. The H-s electrons are tightly bound to the Th-d
075101-8
ELECTRONIC STRUCTURE, PHASE STABILITY, AND . . .
PHYSICAL REVIEW B 65 075101
FIG. 9. COHP for Th2 AlH4 , depicting the contributions from
Th-Al, Th-H, Al-H, and H-H interactions. The COHP for H atoms
in the ab plane and ac plane are given as solid and dotted lines,
respectively.
states, and the Th-H arrangement forms an H-Th-H dumbbell
pattern. Now we will try to obtain a possible explanation for
the short H-H distance within the ac plane of Th2 AlH4 from
the charge-density analysis. The strong covalent interaction
between Th and H in the ac plane 共see Fig. 8, lower part兲
and the dumbbell pattern tend to draw the electrons of H
towards Th leaving only a small amount of electrons between the H along c to repel each other. The main reason for
this short H-H distance is then a reduced repulsion rather
than a bonding interaction between them.
3. COHP
The COHP is an extremely useful tool to analyze covalent
bonding interaction between atoms, the simplest approach
being to investigate the complete COHP between the atoms
concerned, taking all valence orbitals into account. The
COHP between Th-H, Th-Al, Al-H, and H-H in Th2 AlH4 is
given in Fig. 9.
Owing to the very different interatomic distances between
the H atoms in the ab and ac planes, special attention is paid
to the COHP in these planes. Both bonding and antibonding
states are present almost equally in the VB region indicating
that covalent interaction between the H atoms is not participating significantly in the stability of Th2 AlH4 . On the other
hand, the bonding states are present in the whole VB region
in the COHP of Th-Al and Th-H indicating that covalent
interaction between these pairs is contributing to structural
stability. The presence of the large bonding states in the VB
region of the COHP for Th-H along with the enhancement of
the ionic bonding between Th and Al on hydrogenation comply with the larger value of heat of formation for Th2 AlH4
compared with Th2 Al. In order to quantify the covalent interaction between the constituents of Th2 AlH4 we have integrated the COHP curves up to E F for Th-Al, Th-H, and Al-H
giving ⫺0.778, ⫺1.244, and ⫺0.072 eV, respectively. Owing to the presence of both bonding and antibonding states
below E F in the COHP the integrated value for H-H becomes
negligibly small (⫺0.086 and ⫺0.011 eV within the ac and
ab planes, respectively, but as the integrated value of bonding
states
alone
is
⫺0.571
and
⫺0.136 eV, respectively, the bonding H-H interaction is
quite different in the two planes兲. Hence, one can conclude
that the bond strength between the constituents of Th2 AlH4
decreases in the order Th-H⬎Th-Al⬎Al-H⬎H-H.
The experimental5,9 and theoretical studies show highly
anisotropic changes in the lattice expansion on hydrogenation of Th2 Al. According to the crystal structure of Th2 Al the
interatomic distance between the interstitial regions where
one can accommodate H in the ab plane is ⬃2.4 Å. Hence,
there is a large flexible space for accommodation of the H
atoms in this plane without the need to expand the lattice. In
contrast, the interatomic distance between the interstitial regions in the ac plane is only ⬃1.65 Å. So, large expansion
of the lattice along c is necessary to accommodate H within
the ac plane. As a result, even with a short H-H separation of
1.95 Å, a volume expansion of 12.41% is needed when
Th2 AlH4 is formed from Th2 Al. The experimental observation of 0.105% lattice expansion along a and 12.15% along c
is found to be in excellent agreement with the theoretically
obtained values of 0.03% and 12.41%, respectively.
IV. CONCLUSION
This study reports a detailed investigation on the electronic structure, bonding nature, and ground-state properties
of Th2 Al and Th2 AlH4 using first-principles methods. The
following important conclusions are obtained.
共1兲 The calculations show that Th2 Al and Th2 AlH4 are
‘‘formed’’ in the CuAl2 -type crystal structure; the optimized
atomic positions and lattice parameters are in very good
agreement with recent experimental results.
共2兲 Structural optimization gives the shortest H-H separation of 1.95 Å, which is close to the recent experimental
value of 1.97 Å.
共3兲 We observed a highly anisotropic volume expansion of
12.47% of the Th2 Al matrix on hydrogenation to Th2 AlH4 ,
of which 99.76% occurs perpendicular to the basal plane.
共4兲 The large difference in interatomic distance between
the interstitial regions within the ab and ac planes and the
strong covalent interaction between Th and H along c keep
075101-9
P. VAJEESTON et al.
PHYSICAL REVIEW B 65 075101
the H atoms close together in the c direction. These are the
main reasons for the highly anisotropic volume expansion on
hydrogenation of Th2 Al.
共5兲 Charge-density and COHP analyses reveal that the
Th-H bonds are stronger than the H-H bonds and other localized bonds in this structure. The formation of strongly
bonded ThH2 subunits in Th2 AlH4 makes the repulsive interaction between the H atoms smaller along c and this is the
precise reason for the violation of the 2-Å rule.
共6兲 There appears to be a correlation between c/a and the
structural stability of hydrated CuAl2 -type phases. For
phases with c/a⬍0.825, the symmetry changes from
I4/mcm to P4/ncc on hydrogenation, whereas for c/a
⬎0.825, the crystal symmetry is not affected on hydrogenation.
共7兲 Density-of-states and band-structure studies show that
Th2 Al and Th2 AlH4 have nonvanishing N(E F ), resulting in
metallic character. The cohesive energy analysis shows that
Th2 AlH4 is more stable than Th2 Al.
*Electronic address: ponniah.vajeeston@kjemi.uio.no
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1
ACKNOWLEDGMENTS
P.V. gratefully acknowledges Professor Karlheinz
Schwarz, Professor Peter Blaha, Professor O.K. Andersen,
and Professor O. Jepsen for supplying computer codes used
in this study. The authors also acknowledge Dr. Florent
Boucher for useful discussions on the COHP. This work has
received support from The Research Council of Norway
共Program for Supercomputing兲 through a grant of computing
time.
075101-10
II
VOLUME 89, NUMBER 10
PHYSICAL REVIEW LETTERS
2 SEPTEMBER 2002
Violation of the Minimum H-H Separation "Rule" for Metal Hydrides
P. Ravindran,* P. Vajeeston, R. Vidya, A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033, Blindern, N-0315, Oslo, Norway
(Received 5 November 2001; published 20 August 2002)
Using gradient-corrected, full-potential, density-functional calculations, including structural relaxations, it is found that the metal hydrides RTInH1:333 (R La, Ce, Pr, or Nd; T Ni, Pd, or Pt) possess
unusually short H-H separations. The most extreme value (1.454 Å) ever obtained for metal hydrides
occurs for LaPtInH1:333 . This finding violates the empirical rule for metal hydrides, which states that the
minimum H-H separation is 2 Å. The paired, localized, and bosonic nature of the electron distribution at
the H site are polarized towards La and In which reduces the repulsive interaction between negatively
charged H atoms. Also, R-R interactions contribute to shielding of the repulsive interactions between the
H atoms.
DOI: 10.1103/PhysRevLett.89.106403
PACS numbers: 71.15.Nc, 71.20.Eh, 81.05.Je
The most attractive aspect of metal hydrides from a
technological point of view is their potential use as energy-storing materials. Hydrogen as energy carrier and
other visions of the ‘‘hydrogen society’’ are especially
attractive from an environmental point of view, but since
hydrogen is a low-density gas at STP, the storage of the
large quantities required for most applications is a challenge. High-pressure-compressed-gas storage is energy
intensive if high volume efficiency is desired, liquid or
solid hydrogen storage even more so, and all involve
certain hazards. Storage of hydrogen in the form of solid
metal hydrides, from which it can readily be recovered by
heating, is safe and volume efficient.
The amount of hydrogen per volume unit in metal
hydrides is very high; in some cases higher than in liquid
or solid hydrogen, e.g., VH2 stores more than twice the
amount of hydrogen than solid H2 at 4.2 K. It is unfortunate, however, that most metal hydrides are heavy in
relation to the amount of hydrogen they contain. FeTiH2
and LaNi5 H7 , e.g., contain only 1.9 and 1.6 wt. % hydrogen, respectively. Therefore, efforts in hydride research
over the past 25–30 years have been concentrated on
designing new, or modifying known, intermetallic hydrides
to increase the storing capacity and simultaneously adjusting their properties to make them capable of delivering
hydrogen at useful pressures ( > 0:1 MPa) and acceptable
temperatures ( < 425 K) [1]. These aspects are particularly
important for most mobile applications where hydrogen
would be used directly in combustion engines or indirectly
via fuel cells. It has proven difficult to exceed 2 wt. % of
stored hydrogen, and it remains a challenge to increase this
figure if metal hydrides are to become a viable source for
the transportation sector.
The search for efficient hydrogen-storage metal hydrides [2] has to some extent been hampered by the mental
barriers which empirical rules have put on the thinking. For
example, the interstitial hole size that hydrogen is expected
Switendick [3] observed
to occupy should be > 0:40 A.
from a compilation of experimental structure data that the
minimum H-H separation in ordered metal hydrides is >
(‘‘the 2-Å rule’’). This empirical pattern is later [4]
2A
supported by band-structure calculations which ascribe the
effect to repulsive interaction generated by the partially
charged hydrogen atoms. A practical consequence of this
repulsive H-H interaction in metal hydrides is that it puts a
limit to the amount of hydrogen which can be accommodated within a given structural framework. So, if H-H
separations less than 2 Å would be possible, this could
open for new efforts to identify potential intermetallics for
higher hydrogen storing capacity. However, there are indeed metal hydrides which do violate the 2-Å rule, and we
have here identified the origin for such behavior.
RNiIn (R La, Ce, Pr, and Nd) crystallizes in the
ZrNiAl-type structure (space group P62m) and can formally be considered as a layered arrangement with a
repeated stacking of two different planar nets of composition R3 Ni2 and NiIn3 along [001] of the hexagonal unit
cell. If the hydrides of these materials obey the hole-size
demand and the 2-Å rule, one would expect H to occupy
the interstitial 2d site within R3 Ni2 trigonal bipyramid.
However, proton magnetic resonance (PMR) studies suggest [5,6] that H occupies both the 4h and 6i sites or either
of them with H-H distances in the range 1.5–1.8 Å. Recent
powder x-ray and neutron diffraction studies [7] on
RNiInD1:333x (ideally R3 Ni3 In3 D4 ) show that deuterium
occupies the 4h site located on the threefold axis of R3 Ni
tetrahedra that share a common face to form trigonal
bipyramid (Fig. 1). This configuration gives rise to extraordinary short H-H separations of around 1.6 Å [7]. As the
diffraction techniques generally determine the average
structure, neglect of partial H-site occupancies and local
lattice distortions may lead one to conclude with shorter
H-H separations than actually present in the real structure
[8]. Hence, it is of interest to perform structural optimization theoretically.
The present full-potential linear muffin-tin orbital [9]
calculations are all electron, and no shape approximation
to the charge density or potential has been used. The basis
106403-1
© 2002 The American Physical Society
0031-9007=02=89(10)=106403(4)$20.00
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VOLUME 89, NUMBER 10
PHYSICAL REVIEW LETTERS
2 SEPTEMBER 2002
All calculations relate to ideal and fully saturated
hydrides with composition R3 T3 In3 H4 (RTInH1:333 ; R La, Ce, Pr, or Nd, T Ni, Pd, or Pt). For (trivalent) R
the 4f electrons were treated as core electrons (except
for La; 4f). As Ce-4f electrons are known to take different
valence states in intermetallic compounds, different possibilities [12] for the valence states of Ce were considered
in both the hydrides and intermetallic phases during
the structural optimization. For this optimization all
atom positions were relaxed by force minimization and
equilibrium c=a, and volumes were obtained by total
energy minimization. Optimized structural parameters
for the Ce compounds are in good agreement with experimental values (only) when Ce atoms are assumed to be
in the trivalent state. The calculated equilibrium lattice
parameters and the changes between the intermetallic
and corresponding hydride phases are given for selected compounds along with experimental parameters in
Table I.
In general, the calculated lattice parameters are in good
agreement with the experimental values, and the small
differences found may partly be attributed to hydrogen
nonstoichiometry (around 10%) in the experimental studies. The hydrogen-induced lattice expansion is strongly
anisotropic (Table I): a huge expansion along [001]
(c=c 14%–20%) and a smaller intralayer contraction
( a=a 0%–5:8%). The calculated cohesive energy
and heat of formation for the hydrides are larger than for
the corresponding intermetallic phases indicating that it
might be possible to synthesize all these hydrides. The
electronic structure studies show that all considered phases
are in the metallic state consistent with experimental
findings. The calculated R-H, T-H, and H-H distances
are given in Table II along with experimentally available
values. An interesting observation is that all RTInH1:333
materials have unusually short H-H distances. Two explanations have been proposed. Pairing of the hydrogen atoms
(either by molecular H2 -like bonding or by bonding mediated by the intermediate T atom) has been advanced to
explain the anomalous PMR spectrum of CeNiInH1:0 [13].
The second explanation focuses on the significantly shorter
La-La distance in LaNiInH1:333 than in closely related
phases [7], whereby the La atoms (generally R) may
FIG. 1. The RNiInH1:333 -type crystal structure. The R3 Ni2
bipyramid is emphasized with thicker lines.
functions, charge density, and potential were expanded in
spherical harmonic series inside the muffin-tin spheres and
in Fourier series in the interstitial regions. The ratio of
interstitial to unit cell volume is around 0.42. The calculations are based on the generalized-gradient-correcteddensity-functional theory as proposed by Perdew et al.
[10]. Spin-orbit terms are included directly in the
Hamiltonian matrix elements for the part inside the muffin-tin spheres. The basis set contained semicore 5p and
valence 5d, 6s, 6p, and 4f states for La (for Ce the 4f
electrons are treated alternatively as valence and localized
core electrons, whereas the Nd-4f and Pr-4f electrons are
treated as localized electrons using open core approximation), 3s, 3p, 4s, 4p, and 3d for Ni, 4s, 4p, 5s, 5p, and 4d
for Pd, 5s, 5p, 6s, 6p, and 5d for Pt, 5s, 5p, and 5d for In,
and 1s, 2p, and 3d states for H. All orbitals were contained
in the same energy panel. To ensure well-converged wave
functions a so-called multibasis was included, implying the
use of different Hankel or Neuman functions, each attaching to its own radial functions. The-self consistency was
obtained with 105 k points in the irreducible part of the
Brillouin zone. To gauge the bond strength and nature of
bonding we have used crystal orbital Hamiltonian population and electron localization function (ELF) analyses, as is
implemented in TBLMTO-47 [11].
TABLE I. Calculated lattice parameters (a and c in Å) and c=a for LaTInH1:333 and relative variation in unit cell dimensions (in %)
consequent on hydrogenation from RTIn to RTInH1:333 .
Compound
Theor.
a
Expt.
Theor.
Expt.
Theor.
Expt.
a=a
Theor.
Expt.
c=c
Theor. Expt.
V=V
Theor. Expt.
LaNiInH1:333
LaPdInH1:333
LaPtInH1:333
CeNiInH1:333
PrNiInH1:333
NdNiInH1:333
7.2603
7.3501
7.7274
7.4536
7.3783
7.2408
7.3810
7.2921
7.260
7.2255
4.5522
4.8112
4.6903
4.4871
4.4726
4.5560
4.6489
4.6238
4.560
4.5752
0.6270
0.6546
0.6070
0.6020
0.6062
0.6292
0.6399
0.6341
0.6281
0.6332
3:969
5:42
0:04
1:68
2:85
3:72
14.02
16.64
13.98
12.72
13.93
16.75
5.14
4.33
14.00
8.97
7.52
7.60
106403-2
c
c=a
2:76
3:21
3:73
3:92
14.8
16.3
15.4
16.5
8.54
8.98
7.01
7.53
106403-2
VOLUME 89, NUMBER 10
PHYSICAL REVIEW LETTERS
TABLE II.
Calculated interatomic distances (in Å) for RTInH1:333 .
R-H
H-H
T-H
Compound
Theor.
Expt.
Theor.
Expt.
Theor.
Expt.
LaNiInH1:333
LaPdInH1:333
LaPtInH1:333
CeNiInH1:333
PrNiInH1:333
NdNiInH1:333
2.379
2.373
2.475
2.427
2.387
2.350
2.406
2.371
2.350
1.489
1.644
1.618
1.457
1.492
1.493
1.506
1.508
1.506
1.573
1.523
1.454
1.572
1.487
1.492
1.635
1.606
1.562
are not rooted in hydrogen pairing or formation of H2 -like
molecular units.
In order to substantiate this observation further we have
calculated the valence-charge-density distribution in (100)
of LaNiInH1:333 [Fig. 3(a)]. From this figure, it is apparent
that Ni and H form an NiH2 moleculelike structural subunit. Moreover, Fig. 3(a) demonstrates that there is no
substantial charge density distributed between the H atoms.
In order to depict the role of charge transfer, we have
displayed the charge transfer (the difference in the electron
density of the compound and that of constituent atoms
superimposed on the lattice grid) for LaNiInH1:333 within
(100) in Fig. 3(b). From Fig. 3(b) it is clear that electrons
are transferred from La, In, and Ni to the H site. So, there is
a considerable ionic bonding component between H and
the metallic host lattice. The transferred electrons from the
metallic host lattice to the H2 -like subunit of the structure
enter the antibonding levels and give rise to repulsive
interaction. This repulsive interaction between the negatively charged H atoms could explain why the H-H
−0.1725
(c)
1.462
−0.1775
(b)
1.438
−1
∆E (Ry f.u. )
act as a shielding that compensates the repulsive H-H
interaction.
In order to evaluate these possibilities we have calculated the total energy for (hypothetical) LaPtInH1:333 as a
function of H-H separation according to three different
scenarios: (1) Keeping La, Pt, and In fixed in their equilibrium positions, (2) moving La 0.08 Å out of the equilibrium position toward H, and (3) moving La 0.08 Å out of
the equilibrium position away from H. The obtained results
are illustrated in Fig. 2. When La, Pt, and In are in their
optimized equilibrium positions, the equilibrium H-H
separation is 1.454 Å. This scenario corresponds to a lower
total energy than the two alternatives. For scenario 3 we
obtain a shorter H-H separation (1.438 Å) than for the
ground state configuration, and for scenario 2 a correspondingly larger separation (1.462 Å).
As the total energy curves increase steadily on reduction
of the H-H separations, the possibility of stabilization of
hydrogen in the form of molecular H2 -like units seems
completely ruled out. The total energy increases drastically
also for increased H-H separation beyond the equilibrium
value. This is due to increasing repulsive T-H interaction
and decreasing attractive H-H interaction. The considerable changes in the equilibrium H-H distance on R displacement indicate that R in the R3 T2 trigonal bipyramidal
configuration (Fig. 1) acts as a shielding that to some
extent compensates repulsive H-H interactions.
Owing to charge transfer from metal to hydrogen, the
repulsive H-H interaction in metal hydrides are generally
larger than that within the H2 molecule, and this may be the
physical basis for the 2-Å rule. Although minimum H-H
separation (1.945 Å) in Th2 AlH4 [14] is less than 2 Å, it is
much larger than that found in RTInH1:333 . In order to
quantify the bonding interaction between the constituents
in the RTInH1:333 series the integrated crystal orbital
Hamilton population (ICOHP) were calculated. For
example, the ICOHP values up to EF for LaNiInH1:333
are 3:44, 0:14, 0:72, 0:85, 0:86, 1:21, and
0:61 eV for Ni(2c)-H, H-H, La-H, Ni(2c)-In, LaNi(1b), Ni(1b)-In, and Ni(2c)-La, respectively. This indicates that the strongest bonds are those between Ni(2c) and
H. Another important observation is that the bonding interaction between the hydrogens is small, which further
confirms that the short H-H separation in these materials
106403-3
2 SEPTEMBER 2002
−0.1825
−0.1875
(a)
1.454
−0.1925
1.3
1.4
1.5
H−H distance ( Å)
1.6
FIG. 2. Total energy versus H-H distance in LaPtInH1:333 .
(a) All atoms except H are fixed at their equilibrium positions.
(b) La atoms are moved 0.08 Å out of their equilibrium position
toward H. (c) La atoms are moved 0.08 Å out of their equilibrium position away from H.
106403-3
VOLUME 89, NUMBER 10
PHYSICAL REVIEW LETTERS
FIG. 3 (color online). (a) Total charge density, (b) charge
transfer, and (c) electron localization function plot for
LaNiInH1:333 in the (100) plane. The origin is shifted to
1=3; 0; 0, and the charge densities are in e=a:u:3 .
separations in these materials are larger than that in the H2
molecule. If there was strong covalent bonding between Ni
and H, one should expect a significant (positive) value of
charge transfer distribution between these atoms (contributed by both atoms). The absence of such a feature rules
out this possibility. The ELF is an informative tool to
distinguish different bonding interactions in solids [15],
and ELF for LaNiInH1:333 in (100) is given in Fig. 3(c). The
large value of ELF at the H site indicates strongly paired
electrons with local bosonic character. Another manifestation of covalent bonding between Ni and H should have
been paired electron distribution between these atoms. The
negligibly small ELF between Ni and H indicates that the
probability of finding parallel spin electrons close together
is rather high (correspondingly small for antiparallel spin
pairs) in this region confirm metallic bonding consistent
with the result obtained from charge transfer analysis and
the detailed analysis shows that delocalized metallic
Ni(2c)-d electrons are distributed in this region. Even
though the charge distribution between Ni and H looks
like a typical covalent bonding, the charge transfer and
ELF analyses clearly show that the electron distributions
between Ni and H are having parallel spin alignment and
are purely from the Ni site. Hence, chemical bonding
between Ni and H is dominated by metallic components
with considerable ionic weft. The partial density of state
analysis also shows that the H-s states are well separated
from the Ni-d states in the whole valence band and indicates the presence of ionic bonding between Ni and H.
Because of the repulsive interaction between the negatively
charged H electrons, the ELF contours are not spherically
shaped but polarized towards La and In. The localized
nature of the electrons at the H site and their polarization
towards La and In reduce significantly the H-H repulsive
interaction, and this can explain the unusually short H-H
separation in this compound. The ELF between the H
106403-4
2 SEPTEMBER 2002
atoms takes a significant value of 0.35. Considering the
small charge density, this indicates a weak metallic type of
interaction between the hydrogen atoms.
RTInH1:333 constitutes a series with much shorter H-H
separations than other known metal hydrides. We have
shown that the short distances between the H atoms in
such metal hydrides are governed primarily by the polarization of negative charges on H towards the electropositive La and In. We believe that this conclusion is of more
general validity, and may be utilized to search for other
metal hydrides of potential interest as hydrogen-storage
materials.
The authors are grateful to the Research Council of
Norway for financial support and for computer time at
the Norwegian supercomputer facilities.
*Electronic addresses: ponniah.ravindran@kjemi.uio.no
http://folk.uio.no/ravi
[1] A.J. Maeland, in ‘‘Recent Advances in Hydride
Chemistry,’’ edited by R. Poli (North-Holland,
Amsterdam, to be published).
[2] L. Schlapbach, F. Meli, and A. Züttel, in Intermetallic
Compounds: Practice, edited by J. H. Westbrook and R. L.
Fleischer (Wiley, New York, 1994), Vol. 2, pp. 475.
[3] A. C. Switendick, Z. Phys. Chem. 117, 89 (1979).
[4] B. K. Rao and P. Jena, Phys. Rev. B 31, 6726 (1985).
[5] K. Ghoshray, B. Bandyopadhyay, M. Sen, A. Ghoshray,
and N. Chatterjee, Phys. Rev. B 47, 8277 (1993).
[6] M. Sen, A. Ghoshray, K. Ghoshray, S. Sil, and
N. Chatterjee, Phys. Rev. B 53, 14 345 (1996).
[7] V. A. Yartys, R. V. Denys, B. C. Hauback, H. Fjellvåg, I. I.
Bulyk, A. B. Riabov, and Ya. M. Kalychak, J. Alloys
Compd. 330-332, 132 (2002).
[8] K. Yvon and P. Fischer, in Hydrogen in Intermetallics,
edited by L. Schlapbach (Springer, Berlin, 1988).
[9] J. M. Wills, O. Eriksson, M. Alouani, and D. L. Price, in
Electronic Structure and Physical Properties of Solids,
edited by H. Dreysse (Springer, Berlin, 2000), p. 148.
[10] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.
77, 3865 (1996).
[11] G. Krier, O. Jepsen, A. Burkhardt, and O. K. Andersen,
Tight Binding LMTO-ASA Program Version 4.7 (Stuttgart,
Germany, 2000).
[12] A. Delin and B. Johansson, J. Magn. Magn. Mater. 177181, 373 (1998).
[13] K. Ghoshray, B. Bandyopadhyay, M. Sen, A. Ghoshray,
and N. Chatterjee, Phys. Rev. B 47, 8277 (1993).
[14] P. Vajeeston, R. Vidya, P. Ravindran, H. Fjellvåg,
A. Kjekshus, and A. Skjeltorp, Phys. Rev. B 65, 075101
(2002).
[15] A. Savin, R. Nesper, S. Wengert, and T. Fässler, Angew.
Chem., Int. Ed. Engl. 36, 1809 (1997).
106403-4
III
PHYSICAL REVIEW B 67, 014101 共2003兲
Short hydrogen-hydrogen separation in RNiInH1.333 „RÄLa, Ce, Nd…
P. Vajeeston,1,* P. Ravindran,1 R. Vidya,1 A. Kjekshus,1 H. Fjellvåg,1,2 and V. A. Yartys2
1
Department of Chemistry, University of Oslo, Box 1033, Blindern, N-0315, Oslo, Norway
2
Institute for Energy Technology, N-2027 Kjeller, Norway
共Received 2 July 2002; revised manuscript received 19 September 2002; published 15 January 2003兲
First-principle studies on the total energy, electronic structure, and bonding nature of RNiIn (R⫽La, Ce, and
Nd兲, and their saturated hydrides (R 3 Ni3 In3 H4 ⫽RNiInH1.333) are performed using a full-potential linear
muffin-tin orbital approach. This series of phases crystallizes in a ZrNiAl-type structural frame-work. When
hydrogen is introduced in the RNiIn matrix, anisotropic lattice expansion is observed along 关001兴 and lattice
contraction along 关100兴. In order to establish the equilibrium structural parameters for these compounds we
have performed force minimization as well as volume and c/a optimization. The optimized atomic positions,
cell volume, and c/a ratio are in very good agreement with recent experimental findings. From the electronic
structure and charge density, charge difference, and electron localization function analyses the microscopic
origin of the anisotropic change in lattice parameters on hydrogenation of RNiIn has been identified. The
hydrides concerned, with their theoretically calculated interatomic H-H distances of ⬃1.57 Å, violate the ‘‘2-Å
rule’’ for H-H separation in metal hydrides. The shortest internuclear Ni-H separation is almost equal to the
sum of the covalent radii. H is bonded to Ni in an H-Ni-H dumbbell-shaped linear array, with a character of
NiH2 subunits. Density of states, valence charge density, charge transfer plot, and electron localization function
analyses clearly indicate significant ionic bonding between Ni and H and weak metallic bonding between H-H.
The paired and localized electron distribution at the H site is polarized toward La and In which reduces the
repulsive interaction between negatively charged H atoms. This could explain the unusually short H-H separation in these materials. The calculations show that all these materials have a metallic character.
DOI: 10.1103/PhysRevB.67.014101
PACS number共s兲: 81.05.Je, 71.15.Nc, 71.20.⫺b
I. INTRODUCTION
Hydrogen is considered as an ideal fuel for many types of
energy converters. However, neither storage of hydrogen as a
compressed gas nor as a cryogenic liquid appears suitable
and economical for most types of potential applications. In
this respect hydrogen storage in the form of a metal hydride
is a promising alternative with many attractive features.1,2
Over the past few decades, a major challenge which still
remains, is to identify optimal candidates in intermetallics
for such hydrogen storage. Rare-earth 共R兲 alloys seem promising, owing to a high hydrogen capacity per volume unit and
an ability to absorb hydrogen under moderate conditions.1,3
The hydrogen absorption properties of these alloys are very
much dependent on the constituents, and metal-hydrogen
bonding interactions play a major role in the stability of the
hydrides. In order to optimize an intermetallic phase for a
certain application, an improved understanding of the role of
individual alloy constituents in relation to electronic and
structural properties is desirable. Several empirical models4
have been proposed for the heat of formation and heat of
solid solution of metal hydrides, and attempts have been
made to rationalize the maximum hydrogen absorption capacity of certain alloy matrices.5–7 These models infer that
the metal-hydrogen interaction depend both on geometric
and electronic factors.
Numerous phases between transition metals and nonmetals can accommodate hydrogen in the form of solidsolution or stoichiometric hydride phases. The amount of
hydrogen per unit volume in metal hydrides is very high; in
fact in some cases higher than in liquid or solid dihydrogen,
e.g., VH2 stores more than twice the amount of solid dihydrogen at 4.2 K. It is unfortunate, however, that most metal
0163-1829/2003/67共1兲/014101共11兲/$20.00
hydrides are heavy in relation to the amount of hydrogen
they contain. The crystal structures of these phases are often
complex and there are several potential interstices that might
accommodate the hydrogen depending on factors like the
size and shape of the interstitial site, chemical nature of the
surrounding atoms, and the distances to coordinating atoms
and hydrogen neighbors.8,9
Structural studies of intermetallic hydrides have revealed
empirical rules that can be used to predict features of the
hydrogen sublattice in a given matrix.10,11 The ‘‘2-Å rule’’ is
one such guideline, which states that the H-H distance11 in a
metal hydride must exceed 2 Å, and there is also theoretical
evidence12 in support of this rule. The nonmetallic, complex
hydride K2 ReH9 共Refs. 13 and 14兲 appears to provide an
example of violation of the rule, with an H-H separation of
1.87 Å, whereas recent experimental15 and theoretical16 results for Th2 AlH4 agree on a closest H-H separation of
around 1.95 Å.
The recent experimental findings for deuterides with the
ZrNiAl-type structural matrix (LaNiInD1.225 , CeNiInD1.236 ,
and NdNiInD1.192) prove that very short D-D distances of
about 1.5–1.6 Å are indeed possible.17 The reason for this
behavior is not yet understood, but it is expected that more
insight may provide new ideas for how hydrogen can be
packed in an efficient way in an alloy matrix. Nuclear magnetic resonance 共NMR兲 study on CeNiInHx (x⫽1 and 1.6兲
共Refs. 18 and 19兲 and PrNiInH1.29 共Ref. 20兲 have given independent indications of H..H pairing in these phases 关1.48
Å for the H-H distance in CeNiInHx 共Ref. 18兲 and 1.5 to 1.8
Å in PrNiInH1.29]. Recent powder neutron diffraction 共PND兲
data suggested that the H-H interaction is mediated via triangular R 3 structural units, where strong R-R bonds prob-
67 014101-1
©2003 The American Physical Society
PHYSICAL REVIEW B 67, 014101 共2003兲
VAJEESTON et al.
ably ‘‘shield’’ the direct H-H interaction.17 However, for the
isostructural phase LaNiInHx there appears to be no evidence
of hydrogen pairing.21
We have recently shown22 the violation of 2-Å rule in
RTInH1.333 共where T⫽Ni, Pd, or Pt兲 and the present paper is
the full account of electronic structure and bonding behavior
in this class of metal hydrides. In this paper we present the
results of accurate full-potential linear muffin-tin orbital calculations on RNiIn (R⫽La, Ce, and Nd兲 and RNiInH1.333 .
The main scope of the study is to reproduce the experimentally observed short H-H separations and, if so, to understand
the reasons behind this behavior. Also it is of interest to
identify reasons for the anisotropic lattice expansion during
hydrogenation of RNiIn.
Details about structural aspects and computational methods are described in Sec. II. Section III gives the results of
the calculations and comparisons with experimental findings.
The most important conclusions are briefly summarized in
Sec. IV.
II. STRUCTURAL ASPECTS AND COMPUTATIONAL
DETAILS
A. Structural features
ABC aluminides usually crystallize with the ZrNiAl-type
structure.23,24 On substituting Al with larger In atoms in
R-based analogs, the ab plane of the hexagonal structure
expands considerably relative to the c axis. This makes hydrogen absorption more favorable owing to enlarged interstitial sites. In RNiIn phases, both the 4h and 6i sites are
candidates for hydrogen absorption.18,19 Recent experimental
results show that 4h site is fully occupied in R 3 Ni3 In3 H4
hydrides. In these saturated RNiInH1.333 hydrides the H atoms are located inside R 3 Ni tetrahedra that share a common
face, thereby forming a R 3 Ni2 trigonal bipyramid.
Both the RNiIn intermetallic phases and the RNiInH1.333
hydrides crystallize in a hexagonal ZrNiAl-type structure
with space group P6̄2m, structural details being summarized
in Table I and illustrated in Fig. 1. The unit cell contains 13
atoms of which Ni occupies two different crystallographic
sites, Ni(1b) in the 1b position and Ni(2c) in the 2c
position.
TABLE I. Optimized atomic coordinates for RNiIn and
RNiInH1.3333 . Ni atoms occupy two different sets of atomic positions: Ni(1b) in 1b at 0,0,1/2 and Ni(2c) in 2c at 1/3,2/3,0 and
2/3,1/3,0.
LaNiIn
La
In
LaNiInH1.3333
La
In
H
CeNiIn
Ce
In
CeNiInH1.3333
Ce
In
H
NdNiIn
Nd
In
NdNiInH1.3333
Nd
In
H
Experiment 共Ref. 17兲
x
y
z
x
Theory
y
z
0.5866
0.2475
0
0
1/2
0
0.5940
0.2560
0
0
1/2
0
0.6036
0.2444
1/3
0
0
2/3
1/2
0
0.6728
0.6035
0.2437
1/3
0
0
2/3
1/2
0
0.6759
0.5880
0.2480
0
0
1/2
0
0.5940
0.2560
0
0
1/2
0
0.6077
0.2507
1/3
0
0
2/3
1/2
0
0.6752
0.6013
0.2462
1/3
0
0
2/3
1/2
0
0.6737
0.5886
0.2496
0
0
1/2
0
0.5940
0.2560
0
0
1/2
0
0.6013
0.2483
1/3
0
0
2/3
1/2
0
0.6723
0.6013
0.2462
1/3
0
0
2/3
1/2
0
0.6737
B. Computational details
The theoretical approach is based on the generalizedgradient approximation with the Perdew et al.25 proposed exchange correlation of density-functional theory. The KohnSham equation was solved by means of a full-potential linear
muffin-tin orbital method.26 The calculations were relativistic
including spin-orbit coupling and employed no shape approximation to the charge density and potential. Spin-orbit
terms are included directly in the Hamiltonian matrix elements inside the muffin-tin spheres. The basis functions,
charge density, and potential were expanded in spherical harmonic series inside the muffin-tin spheres and in Fourier
series in the interstitial regions. The spherical-harmonic expansion of the charge density, potential, and basis functions
were carried out up to l ⫽ 6. The tails of the basis functions
FIG. 1. The crystal structure of RNiInH1.333(R⫽La, Ce, and
Nd兲. Legends for the different kinds of atoms are given in the illustration. The linear H-Ni(2c)-H array is marked with thicker connecting lines.
014101-2
PHYSICAL REVIEW B 67, 014101 共2003兲
SHORT HYDROGEN-HYDROGEN SEPARATION IN . . .
outside their parent spheres are linear combinations of Hankel or Neumann functions depending on the sign of the kinetic energy of the basis function in the interstitial regions.
For the core-charge density, the Dirac equation is solved selfconsistently, i.e., no frozen core approximation is used. Our
calculations concern ideal and fully saturated hydrides with
the composition R 3 Ni3 In3 H4 (RNiInH1.333). The ratio of the
interstitial to unit-cell volume is around 0.42. The basis set
contained semicore 5p and valence 5d, 6s, 6p, and 4 f
states for La 关for Ce the 4 f electrons are treated as valence
and localized core electrons, whereas Nd-4 f electrons are
treated as localized electrons using the open core approximation兲. In the open core approximation we treated the f electrons as localized and removed their contribution in the valence band. This is equivalent to the local density
approximation (LDA)⫹U approach with U⫽⬁ for the f
electrons. A similar type of approach was successfully used
in Ref. 27兴, 4s, 4 p, and 3d states for Ni, 5s, 5p, and 5d
states for In, and 1s, 2p, and 3d states for H. All orbitals
were contained in the same energy panel. A so-called multibasis was included, to ensure a well-converged wave function, implying the use of different Hankel or Neuman functions each attaching to their radial functions. This is
important in order to obtain a reliable description of the
higher-lying unoccupied states and lower-lying semicore
states. Integration over the Brillouin zone was done using
‘‘special-point’’ sampling,28 and self consistency was obtained with 105 k points in the irreducible part of the Brillouin zone of the hexagonal Bravais lattice, which corresponds to 768 k points in the whole Brillouin zone. Test
calculations were made for the double number of k points to
check for convergence, but the optimized c/a ratio for 105
and 210 k points for LaNiInH1.333 are essentially same.
Hence 105 k points were used for the optimization of c/a,
unit-cell volume, and atomic positions as well as the calculation of electron density. For the density of states 共DOS兲
calculations, the Brillouin-zone integration was performed
by means of the tetrahedron method.29
To gauge the bond strength we have used crystal orbital
Hamiltonian population 关COHP 共Ref. 30兲兴 analyses, as is
implemented in the TBLMTO-47 package.31,32 The COHP,
which is the Hamiltonian population weighted density of
states, is identical to the crystal orbital overlap population. If
the COHP is negative, it indicates bonding character,
whereas a positive COHP indicates antibonding character.
The bulk moduli have been obtained using the so-called
universal-equation-of-state fit for the total energy as a function of volume.
III. RESULTS AND DISCUSSION
Structural optimizations were carried out in order to understand the anisotropic expansion effect during incorporation of H in the RNiIn matrix and to verify the experimental
H-H separation in RNiInH1.333 . Experimental structural information for the RNiIn phases was used as input. First, a
relaxation of atomic positions globally using the forceminimization technique 共forces are minimized up to 0.002
mRy/a.u.兲 was done, keeping the experimental c/a and unit-
FIG. 2. Total-energy curves for RNiIn and RNiInH1.333 as
functions of V/V 0 . LaNiIn (E⫽⫺31 803⫹⌬E), LaNiInH1.333
(E⫽⫺31 805⫹⌬E), CeNiIn (E⫽⫺32 539⫹⌬E), CeNiInH1.333
(E⫽⫺32 540⫹⌬E),
NdNiIn
(E⫽⫺34 068⫹⌬E),
and
NdNiInH1.333 (E⫽⫺34 069⫹⌬E).
cell volume (V 0 ) fixed. Thereafter the theoretical equilibrium volume was determined by fixing the optimized atomic
positions and the experimental c/a, while varying the unitcell volume between ⫺15% and 10% of V 0 共see Fig. 2兲.
Next c/a was optimized by a ⫾2% variation of c/a 共in steps
of 0.02兲 while keeping the theoretical equilibrium unit-cell
volume fixed 共Fig. 3兲. The theoretically obtained structural
parameters are presented along with experimental data in
Tables I and II. The corresponding interatomic distances are
tabulated in Table III. Finally, using the theoretically obtained structural parameters for the RNiIn phases as a starting point, hydrogen was inserted into the 4h site and the
entire structural optimization procedure was repeated. Our
optimized atomic positions 共Table I兲 and lattice parameters
共Table II兲 are in very good agreement with the recent PND
results,17 and the small differences found may partly be attributed to non-stoichiometry with respect to hydrogen
which experimentally is of the order of 10%. Hence the calculations confirm the unusual shortest H-H separation in
RNiInH1.333 . We also find that the volume expansion 共LaNiIn: 2.54 Å 3 /H, CeNiIn: 4.45 Å 3 /H and NdNiIn:
3.93 Å 3 /H) on hydrogenation is highly anisotropic, with a
large lattice expansion along 关001兴 (⌬c/c⫽12.7– 16.7 %)
and a small lattice contraction along 关100兴 (⫺⌬a/a⫽1.7–
4.0%兲. The results presented in the rest of the paper are
014101-3
PHYSICAL REVIEW B 67, 014101 共2003兲
VAJEESTON et al.
cell volumes for Ce in different electronic configurations
for CeNiInH1.333 : 238.48 Å 3 for 4 f 2 localized, 215.89 Å 3
for 4 f 1 localized, and 194.02 Å 3 for 4 f 1 valence. The
4 f 1 -localized value 215.89 Å fits very well with an experimental unit-cell volume of 212.93 Å 3 , indicating that Ce
exists as Ce3⫹ with one 4 f electron well localized in
CeNiInH1.333 共similarly for CeN:In兲.
Using the universal-equation-of-state fit37 for the total energy as a function of the unit-cell volume, the bulk modulus
(B 0 ) and its pressure derivative (B ⬘0 ) are obtained 共Table II兲.
B 0 for LaNiIn and CeNiIn decreases on hydrogenation,
which can be explained as a consequence of the volumeexpansion during hydrogenation. In the case of NdNiIn B 0
increases on hydrogenation, indicating that the introduction
of hydrogen in the NdNiIn lattice enhances the bond strength
such that it overcomes the volume expansion effect. There
are no experimental bulk moduli available for these
materials.
A. Ni-H and H-H separation
FIG. 3. Total-energy curves for RNiIn and RNiInH1.333 as a
function of c/a. 共Also see the caption to Fig. 2兲.
based on the theoretical equilibrium lattice parameters.
CeNiIn is a valence-fluctuating system with Kondo-like
behavior.33 Hydrogenation of the isoelectronic CeNiAl
phase34,35 关like CeNiAlH 2.04 共Ref. 36兲兴 induces a localization
of the Ce-4 f electrons. In order to establish the valence of
Ce in CeNiIn and CeNiInH1.333 , we have made total energy
calculations as a function of cell volume 共see Fig. 4兲 for
different electronic configurations 关e.g., a trivalent state with
4 f electrons as valence electrons 共designated 4 f 1 valence兲,
one 4 f electron as localized in the core state 共designated 4 f 1
localized兲 and two 4 f electrons as localized in the core state
共designated 4 f 2 localized兲兴 using constrained densityfunctional calculations. From the minima in the total-energy
curves 共see Fig. 4兲 we have obtained the equilibrium unit-
Compared to binary metal-hydride structures which are
characterized by a few relatively simple and usually highly
symmetrical configurations, ternary metal-hydride structures
show a great variety of complex and often low-symmetric
configurations. In the latter class hydrides, one may find
metal-hydrogen distances close to the sum of the covalent
radii concerned 共e.g., for ␤ ⬘ -MgNiH4 the experimental Ni-H
distance is 1.49 Å, as compared with the sum of the covalent
radii 1.47 Å兲.
In the RNiInH1.333 phases the hydrogen 4h site is fully
occupied and all H atoms have the same environment in the
crystal lattice. The Ni(2c)-H distance is 1.457–1.493 Å depending on R 共see Table III兲 which matches the sum of covalent radii for Ni and H. This may be associated with the
H-Ni-H (NiH2 -molecule-like兲 subunits which occur in these
structures. Another interesting feature of these structures is
the short H-H separations. Such situations may occur when
the two H atoms concerned form an occupied H-H bonding
state with the empty anti-bonding states38 above the Fermi
level (E F ). The thus resulting structural H 2 ‘‘dimers’’ located inside the alloy matrix may give rise to highly unusual
TABLE II. Calculated lattice parameters 共in Å兲, c/a, variation in a (⌬a/a), c (⌬c/c), and volume (⌬V/V) on hydrogenation 共in %兲,
density of states at the Fermi level 关 N(E F ) in states Ry⫺1 f.u.⫺1 ], bulk modulus (B 0 in GPa兲 and its pressure derivative(B 0⬘ ) for RNiIn and
RNiInH1.333 .
a
c
c/a
⌬a/a
⌬c/c
⌬V/V
N(E F )
B0
B ⬘0
LaNiIn
Theor.
Expt.
共Ref. 17兲
LaNiInH1.333
Theor.
Expt.
共Ref. 17兲
CeNiIn
Theor.
Expt.
共Ref. 17兲
CeNiInH1.333
Theor.
Expt.
共Ref. 17兲
NdNiIn
Theor.
Expt.
共Ref. 17兲
NdNiInH1.333
Theor.
Expt.
共Ref. 17兲
7.5604
3.9924
0.5281
—
—
—
38.22
70.38
4.12
7.2603
4.5522
0.6270
⫺3.97
14.02
5.15
35.30
69.45
4.08
7.5807
3.9806
0.5251
—
—
—
38.90
86.24
2.88
7.4536
4.4871
0.6020
⫺1.68
12.72
8.97
38.10
81.67
3.48
7.5207
3.9023
0.5189
—
—
—
43.74
76.20
4.35
7.2408
4.5560
0.6292
⫺3.72
16.75
7.60
28.13
86.03
4.12
7.5906
4.0500
0.5336
—
—
—
—
—
—
7.3810
4.6489
0.6399
⫺2.76
14.8
8.54
—
—
—
7.5340
3.9750
0.5276
—
—
—
—
—
—
014101-4
7.2921
4.6238
0.6341
⫺3.21
16.3
8.98
—
—
—
7.5202
3.9278
0.5223
—
—
—
—
—
—
7.2255
4.5752
0.6332
⫺3.92
16.50
7.53
—
—
—
PHYSICAL REVIEW B 67, 014101 共2003兲
SHORT HYDROGEN-HYDROGEN SEPARATION IN . . .
TABLE III. Interatomic distances 共in Å兲 and ICOHP 共in eV兲 for RNiIn and RNiInH1.333 .
Theor.
RNiIn
Expt. 共Ref. 17兲
ICOHP
Theor.
RNiInH1.333
Expt. 共Ref. 17兲
ICOHP
R⫽La
Ni(2c)-H
H-H
La-H
Ni(2c)-In
La-Ni(1b)
Ni(1b)-In
Ni(2c)-La
—
—
—
2.8993
3.1251
2.7358
3.0290
—
—
—
2.8688
3.0359
2.8062
3.0783
—
—
—
⫺1.26
⫺0.67
⫺1.24
⫺0.66
1.4891
1.5734
2.3609
2.7990
2.8791
2.8857
3.1835
1.5065
1.6350
2.4064
2.8490
2.9262
2.9390
3.2441
⫺3.44
⫺0.14
⫺0.72
⫺0.85
⫺0.86
⫺1.21
⫺0.61
R⫽Ce
Ni(2c)-H
H-H
Ce-H
Ni(2c)-In
Ce-Ni(1b)
Ni(1b)-In
Ni(2c)-Ce
—
—
—
2.9045
3.1229
2.7375
3.0320
—
—
—
2.8474
3.0133
2.7692
3.0407
—
—
—
⫺1.22
⫺0.54
⫺1.19
⫺0.57
1.4573
1.5721
2.4271
2.8427
2.9237
2.9192
3.2102
1.5086
1.6061
2.3708
2.8026
2.9070
2.9268
3.2124
⫺3.32
⫺0.22
⫺0.79
⫺0.87
⫺0.88
⫺1.33
⫺0.63
R⫽Nd
Ni(2c)-H
H-H
Nd-H
Ni(2c)-In
Nd-Ni(1b)
Ni(1b)-In
Ni(2c)-Nd
—
—
—
2.7667
2.9785
2.9075
3.1572
—
—
—
2.7308
2.9331
2.9415
3.1690
—
—
—
⫺1.14
⫺0.25
⫺1.21
⫺0.39
1.4928
1.5699
2.3499
2.7731
2.8866
2.9015
3.1771
1.5064
1.5618
2.3421
2.7704
2.8870
2.9038
3.1793
⫺3.34
⫺0.23
⫺0.69
⫺0.84
⫺0.65
⫺1.35
⫺0.64
behaviors.39 This could have been the case for the
RNiInH1.333 phases where the experimental findings as well
as our structural optimization study show H-H separations of
around 1.57 Å. However, a detailed theoretical analysis 共see
below兲 reveals quite a different type of bonding situation
between the H atoms.
B. Electronic structure
In general a hydrogen atom modifies the electronic structure of a host alloy by the creation of metal-hydrogen bonding states, a shift of the Fermi level, a change in the width of
bands, and/or a modification of the lattice symmetry. The
calculated band structures for LaNiIn and LaNiInH1.333 are
shown in Fig. 5. These illustrations clearly indicate that the
insertion of H in the LaNiIn matrix has a noticeable impact
on the band structure, mainly in the valence-band 共VB兲 region. Three low-lying bands 关see Fig. 5共a兲 where a single s
band and a doubly degenerated p-band are seen at the ⌫
point兴 originate mainly from In-5s and Ni-4s electrons. Hybridized In-5p, Ni-3d, and La-5d bands are present in an
energy range from ⫺3 to ⫺1 eV, and the electrons corresponding to these bands are mainly participating in the
chemical bonding. Similar band structures are obtained for
CeNiIn, NdNiIn, and their hydrides 共not shown兲. The unoccupied La-4 f states are found in the conduction-band region
around 2.5 eV above E F . As the unit cell contains three
formula units, four electrons are additionally introduced
when LaNiInH1.333 is formed from LaNiIn. Therefore two
additional s bands are present in the lower part of the VB
共from ⫺8.5 to ⫺3 eV兲 in LaNiInH1.333 . The lowest-energy
band in Fig. 5共b兲 near the ⌫ point 共behaves almost like a
free-electron band兲 corresponds to one of these H-1s bands.
The other H-1s band is well dispersed and hybridized with
the rest of the VB in the region from around ⫺6 to
⫺3 eV. A cluster of well-localized bands in the VB around
⫺2 eV originates from the Ni-3d electrons. Owing to the
introduction of extra electrons in the lowest portion of the
VB, the hybridized bands are moved toward E F by the addition of hydrogen. As seen from Fig. 5 several bands cross
E F , and hence these phases will exhibit metallic behavior.
This is further confirmed by the total DOS profiles which
show a finite number of electrons at the Fermi level.
C. Nature of chemical bonding
Insight into the nature of the chemical bonding may provide a clearer picture of the reasons for the short Ni-H and
H-H separations in these phases. In order to identify the type
of bonding and to gain more knowledge we have analyzed
the DOS, charge density, electron localization function
共ELF兲, and COHP characteristics.
014101-5
PHYSICAL REVIEW B 67, 014101 共2003兲
VAJEESTON et al.
FIG. 6. Total DOS for RNiIn and RNiInH1.333 (R⫽La, Ce, Nd兲.
FIG. 4. Total energy vs V/V 0 for CeNiInH1.333 , with different
possible valence states for Ce. The experimental volume from diffraction studies is V 0 ⫽212.93 Å 3 共Ref. 17兲.
1. DOS
All the DOS curves for RNiIn and RNiInH1.333 show
close similarities 共Fig. 6兲. A striking feature of Fig. 6 is that
the hydrogenated phases have a pseudogap, i.e., a deep val-
ley closer to E F , most pronounced for CeNiInH1.333 . The
strong Ni(2c)-H interaction is mainly responsible for this
pseudogap. In general, a gain in total energy can be obtained
when E F lies in the vicinity of a pseudogap.40 On moving
from La to Nd, the additional f electrons are treated as localized electrons that do not participate in the chemical bonding. However, due to the variation in the interatomic distances, there are small differences in the broadening of the
DOS. The metallic character of all these phases mainly origi-
FIG. 5. Energy bands 关 E(k) 兴 for 共a兲 LaNiIn
and 共b兲 LaNiInH1.333 . High-symmetry directions
in the Brillouin zone are marked. The Fermi
energy is set to zero.
014101-6
PHYSICAL REVIEW B 67, 014101 共2003兲
SHORT HYDROGEN-HYDROGEN SEPARATION IN . . .
FIG. 8. Valence-electron-density plot for LaNiIn in the 共100兲
plane 关the origin being shifted to 共0.303,0.198, 0.315兲兴. 25 contours
between 0 and 0.075 electrons/a.u. 3 . Ni refers to Ni(2c).
FIG. 7. Site- and orbital-projected DOS’s for LaNiIn and
LaNiInH1.333 .
nates from the finite contributions to the DOS at E F from
R-5d, Ni-3d, and In-5p states.
The calculated partial density of states 共PDOS兲 共Ref. 41兲
is a useful tool to analyze the nature of the chemical bonding
in solids. In order to follow the changes in electronic structure on hydrogenation, Fig. 7 displays the calculated PDOS
for LaNiIn and LaNiInH1.333 . In the lower portion of the
PDOS curve for LaNiIn there occurs a gap 共from ⬃⫺5 to
⫺4.2 eV兲, below which In-s and Ni-s states are present. The
energy range ⫺4.2 eV to E F carries a large number of electronic states with mainly La-5d, Ni-3d, and In-5p characters. These states are energetically degenerate, which implies
that it is possible to form covalent Ni(1b)-R, Ni(2c)-R,
R-In, Ni(1b)-In, and Ni(2c)-In bonds. The interatomic
Ni(1b)-In and Ni(2c)-In distances are much shorter than
Ni(2c)-R 共Table III兲. Hence the formation of covalent bonding between Ni(1b) or Ni(2c) and In is favorable both from
energetical and spatial points of view. The unoccupied states
are dominated by La-4 f contributions, in particular above
⬃1 eV.
When H is introduced into the RNiIn matrix the atoms are
somewhat rearranged 共see Tables I and III兲 in order to accommodate the hydrogens, and the energy levels are modified accordingly. NMR studies21 on LaNiInHx showed that
N(E F ) does not vary appreciably with the H content. The
calculated N(E F ) values for LaNiIn and LaNiInH1.333 are not
significantly different, and this observation is consistent with
the NMR findings. In CeNiIn also N(E F ) does not change
considerably upon hydrogenation. In contrast, N(E F ) for
NdNiIn is drastically decreased upon hydrogenation 共see
Table II兲. One common feature of the electronic structure of
these hydrides is the occurrence of H states at the bottom of
the VB. The inclusion of the additional H-s states in the
energy range from ⫺8 to ⫺3 eV changed not only this portion of the DOS, but also systematically shifted the E F toward the unoccupied states in the unhydrogenated phases.
Moreover the energy gap between ⫺5 and ⫺4.2 eV disappears on going from the alloy matrix to the corresponding
hydride. Another interesting consequence for the DOS upon
hydrogenation is that In-s states become broadened as a result of the reduced interatomic distance between Ni(2c) and
In 共i.e., following the lattice contraction along a兲. H-s, In-s,
Ni-d, and La-p states are energetically degenerate in an energy range from ⫺8.2 to ⫺3.4 eV in LaNiInH1.333 , implying
a possible covalent bonding contribution for the combinations In-H, Ni-H, La-H, and H-H. Since the Ni(2c)-H separation is very short this combination becomes more favorable
for covalent bonding than the others. However, more
Ni(2c)-d electrons are accumulated near E F ; on the other
hand, H-s states are well localized in the bottom of the VB,
as a result possibility to form ionic bonding between Ni(2c)
and H is more probable than covalent.
By the addition of hydrogen into the LaNiIn matrix the
DOS’s of Ni(1b) and Ni(2c) are both considerably broadened in an energy range from ⫺8 to ⫺4 eV. This is mainly
due to the Ni(2c)-H interaction in the latter case and the
reduction in the Ni(1b)-La distance in the former. However,
the changes in DOS for Ni(2c) are more extensive than
those for Ni(1b) by hydrogen addition, which is mainly due
to the reduction in the Ni(2c)-In distance apart from the
Ni(2c)-H interaction.
2. Charge density analysis
In order to understand the microscopic origin of the short
Ni(2c)-H and H-H separations we have made valence
charge density analyses in different crystal planes for the
alloy matrix as well as the hydrogenated phases. Figures 8
014101-7
PHYSICAL REVIEW B 67, 014101 共2003兲
VAJEESTON et al.
FIG. 9. Valence-electron-density plot for LaNiInH1.333 in the
共100兲 plane 关the origin being shifted to 共1/3,0,0兲兴. 25 contours between 0 and 0.075 electrons/a.u. 3 . Ni refers to Ni(2c).
and 9 show the 共100兲 plane for LaNiIn and LaNiInH1.333 ,
respectively. Similar results were obtained for the analogous
Ce and Nd phases.
The electronegativity difference between Ni and In is only
0.1, which indicates that covalent interaction between these
atoms is more probable than ionic interaction. This is indeed
confirmed by the charge density analysis 共Fig. 8兲, which
shows that there is finite electron density present between
Ni(2c) and In. In fact, as will be argued later, the Ni(2c)-In
bond is stronger than the other interatomic bonds in RNiIn.
Ni(1b)-R and In-R have a mixed 共partial covalent, partial
ionic兲 character which may be attributed to the electronegativity difference of 0.5 between R and Ni as well as between
R and In. Alternating Ni(2c) and In layers 共see Fig. 8兲 have
no charge accumulated between them indicating a interstitial
void for potential accommodation of hydrogen.
When hydrogenation takes place the interstitial site is
shifted 共a small lattice contraction along a) toward Ni(2c)
共see Tables I and II兲 and a Ni(2c)-H bond is formed. Ni(2c)
and H form a dumbbell-like linear arrangement along 关001兴
which results in an appreciable lattice expansion along c.
This rationalizes the anisotropic changes in the lattice. Another noteworthy finding is that the interatomic distance between Ni(2c) and H is almost equal to the sum of the covalent radii of Ni and H, which gives the H-Ni(2c)-H
arrangement a distinct character of a linear NiH2 moleculelike structural subunits 共see Fig. 9兲. The same type of bonding is present in Na 2 PdH 2 , 42 and the Pd-H separation here
共1.68 Å兲 is close to the Ni(2c)-H separation in RNiInH1.333 ,
共1.46 –1.49 Å兲, whereas the H-H separation in Na 2 PdH 2
共3.35 Å兲 is much larger than in RNiInH1.333 共1.570–1.573
Å兲.
The covalent bond strength of Ni(2c)-In is reduced upon
hydrogenation in spite of the reduction in the interatomic
distance from 2.89 to 2.79 Å 共see Sec. III C 4兲. This may be
rationalized as another consequence of the formation of the
NiH2 structural unit which takes up some Ni(2c)-3d electrons and prevent them from participating in the covalent
bonding between Ni(2c) and In. Although the H-H separation is very short, their mutual interaction is weak since very
little electronic charge is present between them 关see Fig.
9共b兲兴. The main reason for this behavior is again the NiH2
structural subunit, which strongly involves the H-1s electron
FIG. 10. 共a兲 Charge transfer and 共b兲 electron localization function plot for LaNiInH1.333 in the 共100兲 plane 关origin is shifted to
共1/3,0,0兲兴.
in the bonding to Ni(2c). Hence an insufficient amount of
electronic charge is left for repulsion between the hydrogen
atoms which consequently can approach each other rather
closely.
Our spin-polarized calculations for LaNiIn and
LaNiInH1.333 show that these systems always converge into a
nonmagnetic state, implying a nonmagnetic ground state. We
know that valence electrons participate either in bonding or
in magnetism. As in this case Ni(2c) and Ni(1b) electrons
participate in the bonding, a quenching of the magnetic moment results. However, in the Nd and Ce systems one may
expect finite magnetic moments from localized 4f electrons.
But we have not considered this magnetic aspect in our calculations.
3. Charge transfer and ELF analysis
To depict the role of the charge-transfer effect we have
displayed the charge-transfer plot for LaNiInH1.333 in 共100兲
in Fig. 10共a兲. The charge-density transfer contour is the selfconsistent electron density of the solid in a particular plane,
minus the electron density of the free atoms in overlapping
regions. This enables one to observe how the electrons are
redistributed in a particular plane in the real crystal 共compared to the free atoms兲 due to the bonding between them.
From Fig. 10共a兲 it is clearly seen that electrons are transferred from La, In, and Ni to H, resulting in ionic bonding
between H and the host lattice.
The ELF is another useful tool to distinguish different
bonding interaction in solids.43,44 The value of the ELF is
limited to the range 0 to 1. High value of the ELF corresponds to a low Pauli kinetic energy, as can be found in
covalent bonds or lone electron pairs. The ELF for
LaNiInH1.333 in 共100兲 is displayed in Fig. 10共b兲. The large
value of the ELF at the H site indicates strongly paired electrons. In between Ni(2c) and H, delocalized metallic
Ni(2c)-d electrons are distributed; hence the ELF is low.
Due to the repulsive interaction between H’s, the ELF contours are not spherically shaped, but polarized toward La and
In sites, which can explain why these materials have unusual
short H-H separation 共i.e., the polarization of electrons at the
014101-8
PHYSICAL REVIEW B 67, 014101 共2003兲
SHORT HYDROGEN-HYDROGEN SEPARATION IN . . .
FIG. 11. COHPs for LaNiInH1.333 ; referring to the short distances corresponding to the combinations Ni(2c)-H, H-H, La-H,
Ni(2c)-In, Ni(1b)-La, Ni(1b)-In, In-H, and Ni(2c)-La.
H site toward La and In give less electrons to participate in
the repulsive interaction between the H atoms兲. The ELF
value between Ni and In is 0.7; a typical value for covalent
bonding, consistent with the conclusion arrived at from our
charge-density analysis. A finite ELF 共0.345兲 is present between H-H bonds, which indicates that the interaction between the H-H bonds is weak metallic.
4. COHP
A simple way to investigate the bond strength between
two interacting atoms in a solid is to look at the complete
COHP between them, taking all valence orbitals into account. In order to understand the bonding pattern further,
results from such COHP analyses for LaNiInH1.333 are shown
in Fig. 11 for all possible interactions within a 3.5-Å range.
This illustration shows that VB comprises mainly bonding
orbitals 共negative COHP兲 and that antibonding orbitals are
found some ⬃3 eV above E F . Integrated COHP 共ICOHP兲
values up to E F are included in Table III for all phases studied. The most notable feature being the remarkable strength
of the Ni(2c)-H interaction 共⫺3.32 to ⫺3.44 eV in ICOHP兲
compared with the other bonds.
As measured by ICOHP, the bonding interaction
Ni(2c)-In is reduced upon hydrogenation 共from around
⫺1.20 to ⫺0.85 eV兲. The bonding ICOHP values for the
short H-H separations are very small, around ⫺0.04 eV 共not
listed in Table III, thus supporting the already advanced inference that there is no significant covalent bonding interaction between the H atoms 共see Sec. III C 2兲. The low ICOHP
value reflects the fact that both bonding and antibonding
states are present below E F , but even if one takes into ac-
count only the bonding states, ICOHP remains low 关⫺0.14 to
⫺0.23 eV, which is much smaller than ICOHP for
Ni(2c)-H]. Hence both COHP and charge-density analyses
agree that the H-H interaction is considerably weaker than
the Ni(2c)-H interaction. Our conclusion therefore disagrees
with NMR findings for CeNiInHx and PrNiInHx , 18 –20 which
concluded that H . .H pairing is the main reason for the unusually short H-H separation in these hydrides. On the basis
of the PND results17 it has been speculated that the H-H
interaction is shielded by R-R interaction. However, our
COHP study shows that the R-R interaction is only from
⫺0.61 to ⫺0.62 eV in ICOHP 共not included in Fig. 11兲
which is 5– 6 times smaller than for the Ni(2c)-H interaction
and closer to the bond strength for R-H and R-Ni(2c).
The experimental and theoretical studies show highly anisotropic lattice expansion on hydrogenation of RNiIn. Now
let us try to understand the reason for this anisotropic lattice
expansion. According to the crystal structure of RNiIn, the
possible positions for hydrogen accommodation are 6i and
4h sites. However, from a hole size point of view the 4h site
is more favorable than the 6i site 共the hole size for the 4h
site is ca. 0.4 Å, whereas that for the 6i site is less than 0.34
Å兲. The optimized atom position of hydrogenated compounds show that the structural deformation does not lead to
any substantial rearrangement of metal atoms in the basal
plane. None of the atoms are significantly shifted in x and y
coordinates from those of the intermetallic RNiIn. This may
be because all atoms are bonded strongly 共from the COHP
study兲 in the ab plane; hence there is no room for H in the
Ni(2c),In plane. When H occupies a 4h site the atoms try to
rearrange themselves to have a minimum energy configuration. Hence the only possibility to expand the lattice is along
the 关001兴 direction. Our charge-density study clearly indicates the formation of Ni-H-Ni linear chains along 关001兴,
implying an expansion along the c axis. The charge-transfer
plot shows that during the formation of a hydride phase,
some charges transfer from the electron-rich metal atoms to
the H site. This may lead to a contraction in the ab plane,
hence resulting in anisotropic lattice changes upon hydrogenation.
5. H-H interaction
In order to elucidate the present findings further we made
the following model calculations. First we fixed all structural
variables, except those for the H position at their theoretically derived equilibrium values. Then we changed the H-H
separation 共moving the hydrogen atoms either toward or
away from each other, i.e., we allowed H to move in the z
direction alone, keeping x and y parameters fixed兲, and calculated the total energy as a function of the H-H separation.
Such a H displacement is equivalent to a shortening or enlargement of the Ni(2c)-H distance depending upon the actual shift of the hydrogen. A shortening of the H-H distance
corresponds to a decrease in the Ni(2c)-H interaction and an
increase in the repulsive interaction between H atoms. The
thus obtained total energy as a function of H displacement is
014101-9
PHYSICAL REVIEW B 67, 014101 共2003兲
VAJEESTON et al.
tal structure or whether strong Ni(2c)-H interactions can
also be generalized to other intermetallics. In order to test
this hypothesis and furthermore to identify potential hydrides
with short H-H separations we are currently considering the
effect on the H-H separation by replacement of Ni by other
metals. The results will be published in a forthcoming paper.
IV. CONCLUSION
FIG. 12. Total energy vs H-H distance in LaNiInH1.333 (E
⫽⫺31805⫹⌬E). 共a兲 All atoms, except H, are fixed at their equilibrium positions. 共b兲 La atoms are moved 0.05 Å out of their equilibrium position toward H. 共c兲 La atoms are moved 0.05 Å out of
their equilibrium position away from H.
shown in Fig. 12共a兲. The minimum in the total energy corresponds to the equilibrium H-H separation obtained by the
structural optimization procedure.
In order to consider the shielding mechanism suggested in
Ref. 17, we have made two sets of additional model calculations. The total energy has been calculated as a function of
the H-H separation, keeping Ni and In fixed in their equilibrium positions, by either 关Fig. 12共b兲兴 moving La 0.05 Å out
of its equilibrium position toward H or 关Fig. 12共c兲兴 moving
La 0.05 Å out of its equilibrium position away from H. The
lowest total energy is obtained when all atoms, except H, are
kept at their equilibrium positions 关Fig. 12共a兲兴. The equilibrium H-H separations are 1.573, 1.555, and 1.587 Å, for the
three models in Fig. 12共a兲, 共b兲, and 共c兲, respectively.
Let us now try to understand the variation in total energy
with H-H separation in these compounds. A shortening of the
H-H separation corresponds to a reduction in the Ni(2c)-H
interaction and an enhanced repulsive interaction between
the H atoms. As the total energy curves increase steadily
upon a shortening of the H-H separations, the possibility of
stabilizing hydrogen in the form of molecular H 2 -like units
seems to be completely ruled out. The total energy also increases drastically for increasing H-H separations beyond the
equilibrium value. This is due to a decrease in the covalent
H-H interaction and the increasing repulsive Ni(2c)-H interaction. The considerable changes in the equilibrium H-H distance on R displacement indicate that R in the bipyramidal
configuration 共see Fig. 1兲 acts as a shielding, and to some
extent compensates for the repulsive H-H interaction. Calculations show that when the R-H separation is reduced the H
atoms are allowed to come closer to each other.
It would be interesting to study whether the short H-H
separation is due to special aspects of the ZrNiAl-type crys-
We have carried out investigations of the electronic structure and bonding in RNiIn and RNiInH1.333 (R⫽La, Ce or
Nd, with a ZrNiAl-type basic framework兲 using generalizedgradient-corrected full-potential density-functional calculations, and have arrived at the following conclusions.
共1兲 The optimized lattice constants exhibit a highly anisotropic lattice expansion 共13–17 %兲 along 关001兴 and a small
contraction 共⫺1.7 to ⫺4.0%兲 along 关100兴 upon hydrogenation of RNiIn, in very good agreement with experimental
findings. The optimized atomic coordinates, unit-cell volumes, and c/a ratios are in very good agreement with experimental findings.
共2兲 All these compounds violate the so-called ‘‘2-Å rule’’
for metal hydrides. RNiInH1.333 is found to have the shortest
H-H separation hitherto reported for hydrogenated alloys.
共3兲 Charge-density and ELF studies show a weak metallic
type of interaction between the hydrogen atoms. A chargetransfer plot clearly indicates that electrons are transferred
from La, In, and Ni to H. Hence a strong ionic bonding is
present between H and the host lattice. The short distances
between H atoms in such metal hydrides are governed primarily by the polarization of negative charges on H toward
the electropositive La and In.
共4兲 Model calculations show that H-H interaction is
strongly repulsive, which makes an explanation based on formation of H 2 molecular subunits in the structure highly improbable. Considerable changes in the equilibrium H-H distance upon R displacement indicate that R in the bipyramidal
configuration acts as a shielding and to some extent compensates for the repulsive H-H interaction.
共5兲 RNiIn and RNiInH1.333 have a finite number of electrons at E F , and are accordingly classified as metals.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Professor John Wills
and Professor O.K. Andersen for allowing to use their computer codes and Professor Andreas Savin and Dr. Florent
Boucher for useful communications on ELF and COHP, respectively. P.V. and P.R. also acknowledge the Research
Council of Norway for financial support and the grant of
computing time on Norwegian supercomputers. P.R. wishes
to thank Professor Olle Eriksson, Dr. Per Andersen, and Dr.
Hakan Hugosson for useful communications.
014101-10
PHYSICAL REVIEW B 67, 014101 共2003兲
SHORT HYDROGEN-HYDROGEN SEPARATION IN . . .
*Electronic
address: ponniahv@kjemi.uio.no; URL: http://
www.folk.uio.no/ponniahv
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014101-11
IV
PHYSICAL REVIEW B 70, 014107 (2004)
Search for metal hydrides with short hydrogen–hydrogen separation: Ab initio calculations
P. Vajeeston,* P. Ravindran, H. Fjellvåg, and A. Kjekshus
Department of Chemistry, University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
(Received 7 August 2003; revised manuscript received 24 March 2004; published 21 July 2004)
The present investigation is a part of a series on metal hydrides with extraordinary short H u H separations.
The electronic structure, chemical bonding, and ground state properties of RTIn 共R = La, Ce, Pr, Nd; T
= Ni, Pd, Pt兲 and their saturated hydrides R3T3In3H4 共=3RTInH1.333兲 are systematically studied using the
full-potential linear muffin-tin-orbital method. The effect of the metal matrix on the H u H separation in
RTInH1.333 is analyzed in terms of chemical bonding, and bond strength is quantitatively analyzed using the
crystal-orbital-Hamilton population. Force and volume optimizations reveal that all these hydrides violate the
“2-Å rule.” The insertion of hydrogen in the metal matrix causes highly anisotropic lattice changes; a large
expansion along c and a small contraction in the a direction. Among the 12 studied hydrides the hypothetical
LaPtInH1.333 phase exhibits the shortest H u H separation 共1.454 Å兲. The optimized unit-cell parameters and
atomic coordinates fit very well with the experimental findings for RNiInH1.333, R = La, Ce, and Nd. Examination of the effect of the metal matrix on the H u H separation in RTInH1.333 suggests that on a proper choice
of alloying element one may be able to reduce the H u H separation below 1.45 Å. The H u H separation is
reduced significantly by application of pressure.
DOI: 10.1103/PhysRevB.70.014107
PACS number(s): 81.05.Je, 71.15.Nc, 71.20.⫺b
I. INTRODUCTION
The present series of papers1,2 concerns metal hydrides
with extraordinary short H u H separations. The reversible
storage of hydrogen in the form of intermetallic hydrides has
several advantages over storing as gaseous and liquid
hydrogen.3,4 However, major a disadvantage of hydrides as
storage media over other options is the considerable weight
of the metal matrices and their relatively high cost. Hence, it
seems desirable to investigate other materials for such applications at affordable cost, sufficient storage capacity, and
functional reliability. Although accounting for structures and
properties of prospective hydrogen storage materials still offers a tremendous challenge to improve the amount of stored
hydrogen, the last two decades have seen considerable
progress in the understanding of the basic problems. With the
development of advanced theoretical tools, the gathering of
knowledge from fields that tended to ignore each other in the
past, and the progress in computer technology, one can make
reliable predictions of properties for even quite complex
materials.5
Numerous studies have been carried out to explain observed stabilities, stoichiometries, and preferred hydrogen
sites in various kinds of hydrides.6,7 It has been suggested
that limiting factors in terms of minimum hole size 共0.40 Å兲
and H u H distance 共2.1 Å兲 in stable hydrides should be
prime selection criteria for guidance about hydrogen absorption capacity.7–10 Hence, reduction in H u H separation may
be one means to enhance the hydrogen content at minimum
matrix volume and mass. From experimental and theoretical
studies it has been established that the separation between
hydrogen atoms in interstitial matrix sites is ruled by repulsive H-to-H interactions, which in practice is assumed to
prevent the hydrogen atoms to approach one another closer
than 2.1 Å (Switendick’s criterion).7,9,10 Nuclear magnetic
resonance (NMR) study on CeNiInHx 共x = 1 to 1.62兲11,12
0163-1829/2004/70(1)/014107(12)/$22.50
and PrNiInH1.2913 has revealed that the H u H separation in
these alloys is between ⬃1.5 and 1.8 Å. It has been suggested that both the 4h and 6i interstitial sites or either of
them are occupied by hydrogen atoms in a ZrNiAl-type matrix. Although PrNiInH1.29 has been obtained as a stable
phase, no detailed structural examination has so far been
reported. A recent low-temperature neutron diffraction
study14 gives evidence that H atoms occupy only the 4h site
in an ordered arrangement in the ZrNiAl-type structure of
R3Ni3In3H4 (=3RNiInH1.333; R = La, Ce, and Nd hereafter referred to with the latter simple matrix-based formula) with a
H u H separation of ⬃1.6 Å. En passant it may be mentioned that in some disordered hydrides (like K2ReH99,15),
due to the partial occupancy of the hydrogen sites, the H u H
separation becomes appreciably lower than 2.1 Å.
In previous works1,2 on RNiIn and RNiInH1.333 共R
= La, Ce, Nd兲 the origin of the anisotropic volume expansion on hydrogenation has been explained, and the presence
of strong interaction between the transition metal 共T兲 and
hydrogen in RNiInH1.333 is shown to prohibit the electrons at
the hydrogen atoms from participation in repulsive interaction between two such arranged H atoms (see Fig. 1 and for
more details Refs. 1 and 2). Moreover, also the R u R bond
grid contributes to the shielding of the repulsive interaction
between the H atoms. These observations have motivated us
to search for new hydrides with short H u H separation in
RTIn matrices and in particular to identify the effect of isoelectronic substitution of Ni by Pd and Pt. In the present
contribution we consider the crystal and electronic structures
of nine other members of the series (R = La, Ce, Pr, Nd; T
= Pd, Pt; plus PrNiInH1.333) in addition to the earlier reported
combinations with T = Ni, viz. all together 12 phases. Furthermore the influence on the H u H separation by application of high external pressures is considered.
The rest of this paper is organized as follows. Section II
gives appropriate details about the computational methods,
70 014107-1
©2004 The American Physical Society
PHYSICAL REVIEW B 70, 014107 (2004)
VAJEESTON et al.
FIG. 1. (Color online) Structural fragment of RTInH1.333 共R
= La, Ce, Pr, Nd; T = Ni, Pd, Pt兲, showing the trigonal bipyramidal coordination of 3 R and 2 T共2c兲 around 2 H atoms. Legends to
the different kinds of atoms are given on the illustration. The
completion of the linear H u T共2c兲 u H units is indicated at the top
and bottom of the illustration.
structural results of the calculations are presented and discussed in Sec. III, the nature of chemical bonding is analyzed
in Sec. IV, and the conclusions are summarized in Sec. V.
II. DETAILS OF THE CALCULATIONS
We have used the full-potential linear muffin-tin-orbital
(FPLMTO) method16 within the generalized-gradient approximation (GGA),17 in which the Kohn-Sham equations
are solved for a general potential without any shape approximation. For the local density approximation (LDA) calculations, we have used the exchange-correlation function proposed by Hedin and Lundqvist,18 which allows accurate
calculation of the total energy as a function of volume. The
unit cell is divided into nonoverlapping muffin-tin spheres,
inside which the basis functions were expanded in spherical
harmonics up to a cutoff in angular momentum at ᐉ = 6. The
basis functions in the interstitial region (i.e., outside the
muffin-tin spheres) are Neumann or Hankel functions depending upon the kinetic energy. The ratio of interstitial to
unit-cell volume is around 0.42. The basis set contained
semicore 5p and valence 5d, 6s, 6p, and 4f states for La. For
Ce, Pr, and Nd, the 4f electrons are treated as localized core
electrons. In rare earth compounds, the equilibrium volume
strongly depends on the valence of the f-electron-carrying
element. For the present series of compounds this is a point
of concern for the Ce-containing compounds. In the earlier
communication2 it is shown that only when one assumes Ce
in the 3+ state in CeNiIn and CeNiInH1.333 agreement is
obtained between the calculated equilibrium cell volume and
the experimental cell volume. Hence, for the rest of the Ce-
containing compounds in the series it was simply assumed
that the valence state is Ce3+. The open core approximation
was used and hence contributions of the 4f electrons to the
valence band are removed. This is equivalent to the LDA
+ U approach with U = ⬁ for the f electrons (successfully
used in Ref. 19). 4s, 4p, and 3d orbitals were used for Ni; 5s,
5p, and 5d for In; semicore 4p and valence 5s, 5p, and 4d
for Pd; and semicore 5p, valence 6s, 6p, and 5d for Pt; and
1s, 2p, and 3d for H. All orbitals were contained in the same
energy panel. A so-called multibasis was included to ensure a
well-converged wave function, implying the use of different
Hankel or Neuman functions each attaching to their radial
functions. This is important in order to obtain a reliable description of the higher-lying unoccupied states and low-lying
semicore states. Integration over the Brillouin zone (BZ) was
done using “special-point” sampling,20,21 and selfconsistency was obtained with 105 k points in the irreducible part of the BZ, which corresponds to 768 k points in the
whole BZ. Test calculations were made for a doubled number of k points to check for convergence, but the optimized
c / a ratio for 105 and 210 k points for LaNiInH1.333 turned
out essentially the same. Hence, 105 k points were used consistently for the optimization of c / a, unit-cell volume, and
atomic positions as well as for calculations of electron density. For the density of states (DOS) calculations, the BZ
integration was performed by means of the tetrahedron
method.22
Equilibrium volumes and bulk moduli were extracted
from the calculated energy versus volume data by fitting to
the “universal equation of state” proposed by Vinet et al.23
Virtually the same results were obtained by fitting to the
Birch24 or Murnaghan25 equations. For the calculation of the
bulk modulus, the theoretically estimated c / a value was kept
constant.
To calculate electron-localization-function plots we used
the first-principles self-consistent, tight-binding, linear
muffin-tin orbital (TB-LMTO) method of Andersen and
Jepsen.26 The von Barth–Hedin27 parameterization is used
for the exchange correlation potential within the localdensity approximation. In the present calculation, we used
the atomic sphere approximation. The calculations are semirelativistic (i.e., without spin-orbit coupling, but all other
relativistic effects included) taking also into account combined correction terms. The BZ k-point integrations are
made using the tetrahedron method on a grid of 403 k points
in the irreducible part of BZ. In order to get more insight into
the chemical bonding, we have also calculated the crystalorbital-Hamiltonian population (COHP)28,29 in addition to
the regular band structure. The COHP is the DOS weighted
by the corresponding Hamiltonian matrix elements, a positive sign of which indicating bonding character and negative
antibonding character. The detailed mathematical expressions for the COHP are, e.g., found in Ref. 28. The electronlocalization function (ELF) was calculated according to the
TB-LMTO code.
A. Structural details
The hydrogen storage capacity of an alloy is limited in the
first place by the number of interstial sites in the host lattice.
014107-2
PHYSICAL REVIEW B 70, 014107 (2004)
SEARCH FOR METAL HYDRIDES WITH SHORT…
Often there exist different interstitial sites which can be occupied by hydrogen (note: as atomic H not as molecular H2),
e.g., tetrahedral and octahedral. However, owing to electrostatic limitations, most of the interstitial sites remain unoccupied on hydrogenation, and consequently the formulae for
such hydrides usually take noninteger indices for the hydrogen content.30 The ZrNiAl-type structure (space group
P6̄2m) is common among the ABC intermetallics, more than
300 examples being recorded.31 In the RTIn lattice the most
favorable sites for the hydrogen have tetrahedral surroundings: (a) 6i at x, 0, z etc. 共x ⬇ 0.25, z ⬇ 1 / 3兲, coordinated by
three Rs and one In; (b) 4h at 1 / 3 , 2 / 3 , x etc. 共x ⬇ 2 / 3兲,
coordinated by three Rs and one T共2c兲 (illustrated in Fig. 1);
and (c) 12l at x , y , z etc. 共x ⬇ 0.44, y ⬇ 0.16, z ⬇ 2 / 9兲, coordinated by two Rs, one In, and one T共1b兲. The shortest
4h–4h and 6i–6i separations occur along the c axis of the
hexagonal structure. Recent experimental findings show that
H prefer only the 4h among the available sites.14 However,
the complete filling of such arranged tetrahedral sites with
hydrogen has been considered unlikely on the basis of the
requirements of the hole size and the “2-Å rule”.32 All calculations reported here assume ideal and fully saturated hydrides with the formula 3RTInH1.333.
III. STRUCTURAL ASPECTS
In order to understand the role of the R and T components
in the RTIn metal matrix for the unusual H u H separation
and the anisotropic volume expansion on hydrogenation a
structural optimization study has been carried out. The
ZrNiAl-type structure with the experimental lattice parameters and atomic positions for the appropriate compounds14
were chosen as the starting point. First all variable atom
coordinates were relaxed globally (by keeping c / a and cell
volume constant) using the force minimization technique.
The atomic coordinates were then fixed at their optimized
values while the unit-cell volume were varied in steps of 5%
within ±15% from the optimized value. The thus obtained
total energy versus unit-cell volume curve was fitted to a
polynomial and the minimum, which represents the theoretical equilibrium volume, was established. The theoretical c / a
was, in turn, determined from the minimum in total energy
versus c / a at fixed atomic coordinates and equilibrium unitcell volume. Finally, once again we relaxed the atom coordinates globally by fixing c / a and unit-cell volume. The thus
obtained atomic coordinates differ less than 1% from the first
established values. The calculated data are in good agreement with the experimental findings (see Tables I and II).
The next step was to introduce the hydrogen atoms in the 4h
position of the optimized metal matrix. This results in local
strain, which implies that the entire, just-outlined, structural
optimization process had to be repeated. The then obtained
atomic coordinates and unit-cell dimensions agree very well
(see Tables I and II) with recent powder neutron diffraction
findings for the phases with T = Ni.14 The lattice changes during the hydrogenation are highly anisotropic, on the average
16% expansion along c and a small contraction (average
⬃2.5%) along a, the largest anisotropy being found within
the T = Pd series.
On going from La to Nd the calculated unit-cell volume
decreases gradually, a trend that is recognized also in the
experimental data (see Fig. 2). A similar trend was observed
by Muller, Blackledge, and Libowitz34 in RH2 phases, which
in that case, was attributed to the lanthanide contraction. Experimentally all intermetallic mother phases except CePtIn
have been identified,14,31 whereas most of the hydrogenated
phases are yet structurally little explored (see Tables I and
II).
In order to check possible effects of different computational approaches we have performed structural optimizations for the RPtInH1.333 phases [among which the shortest
H u H separations are found] using both LDA and GGA
exchange correlation functionals (see Table III). Due to the
overestimation of bond characteristics by LDA the calculated
H u H seprations come out smaller by this approach than
those obtained by GGA. For rest of the compounds we only
used the GGA formalism and the following discussion is
solely based on the GGA calculations.
The calculated unit-cell volume versus total energy is fitted to the “universal equation of state” in order to estimate
the bulk modulus 共B0兲 and its pressure derivative 共B⬘0兲 for
these RTIn and RTInH1.333 phases (see Table IV). On going
from the RNiIn mother phases to the corresponding hydrides,
B0 is reduced (except for NdNiIn), which can be understood
as a consequence of the volume expansion during hydrogenation. On the other hand, on going from the RPdIn and
RPtIn series to their hydrides, B0 increases (except for
LaPtIn). This may be due to enhanced bond strength in the
RTIn matrix (in an extent that overcomes the volume expansion effect) on introduction of the hydrogen. No experimental data for B0 of these compounds are available.
IV. CHEMICAL BONDING
A. DOS
From the characteristic features of the DOS, one may be
able to rationalize (see, e.g., Ref. 35) the bonding in the RTIn
series and changes introduced in the bonding upon the hydrogenation. No electronic structure calculations have apparently hitherto been undertaken for RTIn phases. In general all
members of the RTIn and RTInH1.333 series (see Table IV)
have a finite number of electrons at the Fermi level 共EF兲,
which classifies them as metals. As examples the DOS for
LaPtIn and LaPtInH1.333 is shown in Fig. 3. For LaPtIn it is
convenient to divide the DOS into regions: region I
共−8 to − 5 eV兲 with well localized In-s and Pt-s states and
some hybridized La-s states; region II 共−5 eV to EF兲 with
hybridized In-p, Pt-d, and La-d states; and region III (above
EF) with unoccupied states. By studying the changes in these
regions on hydrogenation one can extract valuable information about the changes in bonding character upon hydrogenation. Region II where the In-p, Pt-d, and La-d states are
energetically degenerate, signals an appreciable degree of covalent bonding. The electronegativity (Pauling scale) differences between In and Pt (0.5) and In and La (0.6) are much
smaller than between La and Pt (1.1), which indicate that the
tendency to form covalent bonding between the former pairs
014107-3
PHYSICAL REVIEW B 70, 014107 (2004)
VAJEESTON et al.
TABLE I. Calculated a , c (in Å), and c / a for RTIn and RTInH1.333; and changes ⌬a / a, ⌬c / c, and ⌬V / V (in %) on hydrogenation.
a
c
⌬a / a
c/a
⌬c / c
⌬V / V
Compound
Theor.
Expt.
Theor.
Expt.
Theor.
Expt.
Theor.
Expt.
Theor.
Expt.
Theor.
Expt.
LaNiIn
LaNiInH1.333
LaPdIn
LaPdInH1.333
LaPtIn
LaPtInH1.333 (LDA)
LaPtInH1.333 (GGA)
CeNiIn
CeNiInH1.333
CePdIn
CePdInH1.333
CePtIn
CePtInH1.333 (LDA)
CePtInH1.333 (GGA)
PrNiIn
PrNiInH1.333
PrPdIn
PrPdInH1.333
PrPtIn
PrPtInH1.333 (LDA)
PrPtInH1.333 (GGA)
NdNiIn
NdNiInH1.333
NdPdIn
NdPdInH1.333
NdPtIn
NdPtInH1.333 (LDA)
NdPtInH1.333 (GGA)
7.5604
7.3771
7.7716
7.3501
7.7277
7.5262
7.7274
7.5807
7.4536
7.8113
7.4494
8.0112
7.6419
7.8250
7.5950
7.3783
7.8674
7.4107
7.8042
7.5766
7.7595
7.5207
7.2408
7.7898
7.3519
7.7610
7.5362
7.6980
7.5905a
7.3810a
7.7290b
7.6950b
7.5340a
7.2921a
7.7290b
7.5410b
7.2600c
7.3790b
7.6522b
7.5202a
7.2255a
7.6830b
7.6360b
-
3.9924
4.6254
4.1250
4.8112
4.1144
4.6261
4.6903
3.9806
4.4871
4.1257
4.8676
3.9800
4.6467
4.7101
3.9259
4.4726
4.0257
4.8041
4.0706
4.6283
4.6934
3.9023
4.5560
3.9735
4.7895
3.9815
4.6172
4.6750
4.05000a
4.6489a
4.1330b
4.1250b
3.9750a
4.6238a
4.1330b
3.9500b
4.5600c
4.0430b
4.0455b
3.9278a
4.5752a
3.9970b
4.0100b
-
0.5281
0.6270
0.5307
0.6546
0.5324
0.6147
0.6070
0.5251
0.6020
0.5282
0.6534
0.4892
0.6081
0.6019
0.5149
0.6062
0.5117
0.6483
0.5216
0.6109
0.6049
0.5189
0.6292
0.5101
0.64515
0.5130
0.6127
0.6073
0.5336a
0.6399a
0.5347b
0.5361b
0.5276a
0.6341a
0.5347b
0.5238b
0.6281c
0.5265b
0.5287b
0.5223a
0.6332a
0.5202b
0.5251b
-
−2.43
−5.42
−2.44
−0.04
−2.76
−3.21
−3.73
−3.92
-
15.86
16.64
12.44
13.98
12.72
17.98
16.75
18.34
13.93
19.34
13.70
15.30
16.75
20.54
15.97
17.40
14.80
16.3
15.40
16.50
-
10.31
4.33
6.65
14.00
8.97
7.30
6.24
12.098
7.52
5.88
6.69
13.98
7.60
7.37
6.35
15.52
8.54
8.98
7.01
7.53
-
aRef.
bRef.
cRef.
−1.68
−4.63
−4.61
−2.32
−2.85
−5.81
−2.92
−0.06
−3.72
−5.62
−2.90
−0.08
14
31
13
of elements is more favorable than that between La and Pt.
Indium is surrounded by four transition-metal atoms (two in
2c and another two in 1b site) and six rare-earth atoms. Owing to the crystal field splitting the In-p state separates into
two parts (see the partial DOS of In in Fig. 4) between
−5 to − 2.5 eV the pz electrons mainly dominate whereas
the non-bonding px and py electrons reside at the top of the
valence band (VB). It is the well-separated pz states that
participate in the covalent bonding with the energetically
nearest Pt-d states.
Figure 3 shows that when hydrogen is inserted in the
LaPtIn matrix, new states are formed around −9.5 to
− 2.5 eV. Hence the VB is shifted to lower energy and the EF
is pushed slightly upward. It is commonly accepted that on
insertion of hydrogen in a metal matrix, the hydrogen will
absorb some valence electrons from the nearest electron-rich
metal atom(s), viz. a charge-transfer effect. In the RTInH1.333
series H absorbs electrons from all the metal atoms of the
mother matrix (see Sec. IV B). In general, the so-called corelevel shifts observed in photo-electron spectra during formation of metal hydrides36 may be regarded as direct evidence
of charge transfer from metal atoms to the hydrogens. In the
LaPtIn-to-LaPtInH1.333 case, the La-5p level is lowered by
⬃0.8 eV; (present at around −17 eV; not shown in Fig. 4);
similar shifts being reported37 in rare-earth hydrides. The detailed features of the partial Pt DOS for T共1b兲 and T共2c兲
show some distinction due to their different atomic environments. The total width of the VB is lower in LaPtIn than
LaPtInH1.333 (similar for the other pairs of compounds studied). This is due to the fact that hydrogen enhances the interaction between neighboring atoms, and thereby increases
the overlap of the orbitals concerned. Further, the location of
the H-s state at the bottom of the VB also contributes to an
increase in the width of the VB. In particular the contribution
of Pt共2c兲 to the VB enhances the width from 8 to 9.5 eV,
whereas contributions of Pt共1b兲 and In on the other hand,
014107-4
PHYSICAL REVIEW B 70, 014107 (2004)
SEARCH FOR METAL HYDRIDES WITH SHORT…
TABLE II. Optimized atomic coordinates for RTIn and RTInH1.333 phases.a
RTIn
T = Ni
R = La
In
H
R = Ce
In
H
R = Pr
In
H
R = Nd
In
H
T = Pd
R = La
In
H
R = Ce
In
H
R = Pr
H
R = Pr
In
H
R = Nd
In
H
T = Pt
R = La
In
H
R = Ce
In
H
R = Pr
In
H
R = Nd
In
H
RTInH1.333
Theor.
x
Expt.
x
Theor.
x
Expt.
x
Theor.
z
Expt.
z
0.5866
0.2475
0.594
0.256
0.588
0.248
0.594
0.256
0.6035
0.2437
1/3
0.6013
0.2462
1/3
0.585
0.248
0.594
0.256
0.594
0.256
1/2
0
0.6728
1/2
0
0.6752
1/2
0
0.6663
1/2
0
0.6723
1/2
0
0.6759
1/2
0
0.6737
1/2
0
0.5886
0.2496
0.6036
0.2444
1/3
0.6077
0.2507
1/3
0.6059
0.2568
1/3
0.6013
0.2483
1/3
0.5930
0.2537
0.594
0.256
0.5885
0.2501
0.5895
0.594
0.5895
0.2529
0.594
0.256
0.5913
0.2535
0.594
0.256
0.5874
0.2533
0.594
0.256
0.5908
0.2525
0.594
0.256
0.5894
0.2516
0.594
0.256
0.5904
0.2548
0.594
0.256
1/3
0.6013
0.2462
1/3
0.6008
0.2441
1/3
0.6055
0.2517
1/3
0.6015
1/3
0.6015
0.2440
1/3
0.60414
0.2695
1/3
1/2
0
0.6583
1/2
0
0.6636
1/2
0.6663
1/2
0
0.6624
1/2
0
0.65045
0.6020
0.2460
1/3
0.6061
0.2456
1/3
0.6034
0.2486
1/3
0.6035
0.2513
1/3
1/2
0
0.6551
1/2
0
0.6554
1/2
0
0.6595
1/2
0
0.6639
1/2
0
0.6737
a
R in 3g共x , 0 , 1 / 2兲, T共1b兲 in 1b共0 , 0 , 1 / 2兲, T共2c兲 in 2c共1 / 3 , 2 / 3 , 0兲, In in 3f共x , 0 , 0兲, and H in
4h共1 / 3 , 2 / 3 , z兲.
would have reduced the width of the VB from
⬃8 to 7.5 eV. The latter modification reflects the increase
in the Pt共1b兲 u In separation following from the insertion of
the additional H atoms between the Pt共2c兲 – Pt共2c兲 linear
configuration.2
The interatomic Pt共2c兲 u H distance almost equals the
sum of the covalent radii of Pt and H (not only in
LaPtInH1.333, but also in the rest of the compounds). The
H-s, In-s, and Pt共2c兲-d electrons are energetically degenerate
in the energy range between −9.75 and −2.50 eV. According
to the electronegativity differences of 0.0, 0.5, and 1.2 for
Ptu H, Inu H, and Lau H, respectively, covalent bonding
between Pt and H is more likely than between La and H.
Careful inspection of the partial DOS of Pt共2c兲 and H in Fig.
014107-5
PHYSICAL REVIEW B 70, 014107 (2004)
VAJEESTON et al.
FIG. 2. Experimental (쎲) and calculated (䊊) unit-cell volume
for R3Ni3In3H4 versus ionic radius33. Experimental data for
PrNiInH1.29 from Ref. 13 and for the other phases from Ref. 14.
4 shows accumulation of d electrons in the range
−4.25 to − 1 eV, whereas most of the H electrons are localized in the range −9.75 to − 2.35 eV, indicating that the interaction between H and Pt is far from perfect covalent.
B. Charge density and charge transfer
The unusually short H u H separations in the RTInH1.333
phases (Table III) are a motivation for further penetration
into the nature of their bonding. The variation in the electron
distribution on replacement of R or T components may be
expected to give hints about factors which influence the
bonding between the H atoms as well as between H and T.
1. Charge density
The charge-density contours in the (100) plane of LaPtIn
and LaPtInH1.333 are displayed in Figs. 5 and 6(a), respectively. Owing to the pronounced mutual similarities between
the charge density within the RTIn and RTInH1.333 series we
present only illustrations for the charge density of LaPtIn and
LaPtInH1.333 (Figs. 5 and 6), the latter hydride having moreover the shortest H u H separation among the investigated
hydrides. The finite electron distribution present between
Pt共2c兲 and In (Fig. 5) clearly indicates that the interaction
between these atoms has an appreciable covalent component.
The R u Pt共1b兲 and R u In bonds have mixed (partial cova-
lent and partial ionic) character also suggested by their different electronegativities.
In order to gain insight in the impact on the bonding interaction between metal atoms of the mother matrix upon
hydrogenation Fig. 6(a) shows the charge density for
LaPtInH1.333. The interesting aspect is that, the charge density of Pt共2c兲 is directed toward H, leading to the formation
of dumb-bell shaped H u Pt共2c兲 u H units, linked like a
chain along [001]. The interatomic separation between
Pt共2c兲 and H in LaPtInH1.333 corresponds nearly to the sum
of the covalent radii of Pt and H. The same trend (viz. very
short T u H separations) is observed in all the studied
phases. The H - 1s electrons participate in pairing mediated via Pt共2c兲 to a molecule-like structural 共H u Ptu H兲
unit as well as in H…H interactions with neighboring
H u Ptu H
units
(viz.
a
linear
arrangement
. . .H u Ptu H . . . H u Ptu H. . .). Examination of the partial
DOS (in Fig. 4) shows that the valence electrons of Pt and H
are not fully energetic degenerate in the VB region indicating
a degree of ionic character in the Pt-to-H bonding. This interpretation is supported by the charge-transfer map [Fig.
6(b)]. The bond strength of Pt共2c兲 u In is reduced upon hydrogenation as reflected in the increased interatomic distance
from 2.93 to 2.97 Å, and further confirmed by the COHP
analyses. An important difference between the complex hydrides and the present type of hydrides lies in the nature of
the interaction between the H atoms. Most of the complex
hydrides have nonmetallic character, for example, in LiAlH4
there is strong covalent interaction between Al and H (within
the AlH4 subunits) and very weak H u H bonds between two
AlH4 subunits.38 In RTInH1.333, on the other hand, there is a
finite electron density distributed between the TH2 subunits
implying a considerable H u H interaction.
2. Charge transfer plot
A convenient and illustrative way to represent and analyze
the bonding effects in the RTInH1.333 series is to use chargetransfer plots. The charge-transfer contour is the selfconsistent electron density in a particular plane, ␳comp., minus
the electron density of the overlapping free atoms, ␳o.f.a., i.e.,
⌬␳共r兲 = ␳共r兲comp. − ␳共r兲o.f.a. ,
共1兲
which allows one to visualize how electrons are redistributed
in a particular plane compared to free atoms due to the bonding in the compound.
Since the charge-transfer plots for all RTInH1.333 phases
show close qualitative similarity, the following considerations are limited to LaPtInH1.333. Figure 6(b) shows that
charge has been depleted from the La, Pt, and In sites and
transferred to H, but the magnitude of charge transfer varies
in different directions of the crystal. Most charge has been
transferred from In, but not isotropically and the resulting
charge distribution at the In site is certainly not spherically
symmetric. There is a significant amount of extra electron
density around the H atoms, and also an increased amount of
electrons in the interstitial region between the Pt共2c兲 and In
atoms. The charge transfer from the metal components to the
H atoms leads to an appreciable ionic component in the
014107-6
PHYSICAL REVIEW B 70, 014107 (2004)
SEARCH FOR METAL HYDRIDES WITH SHORT…
TABLE III. Interatomic distances (in Å), and ICOHP (in eV) for RTInH1.333 phases.
RuH
TuH
HuH
Compound
Theor.
Expt.a
ICOHP
Theor.
Expt.a
ICOHP
Theor.
Expt.a
ICOHPb
LaNiInH1.333
LaPdInH1.333
LaPtInH1.333 (LDA)
LaPtInH1.333 (GGA)
2.3993
2.3727
2.4018
2.4748
2.4064
-
−0.710
−0.747
−0.631
−0.642
1.5132
1.6440
1.6167
1.6177
1.5065
-
−3.211
−2.829
−3.595
−3.539
1.5983
1.5230
1.3923
1.4543
1.6350
-
−0.121
−0.319
−0.442
−0.423
CeNiInH1.333
CePdInH1.333
CePtInH1.333 (LDA)
CePtInH1.333 (GGA)
2.4271
2.4238
2.4532
2.5098
2.3708
-
−0.791
−0.691
−0.605
−0.534
1.4573
1.6373
1.6011
1.5883
1.5086
-
−3.322
−2.838
−3.659
−3.621
1.5721
1.5925
1.4440
1.4637
1.6061
-
−0.224
−0.256
−0.326
−0.313
PrNiInH1.333
PrPdInH1.333
PrPtInH1.333 (LDA)
PrPtInH1.333 (GGA)
2.3873
2.3979
2.4298
2.4955
-
−0.629
−0.668
−0.489
−0.582
1.4924
1.6217
1.6012
1.5979
-
−3.397
−2.895
−3.615
−3.645
1.4874
1.5602
1.4252
1.4970
-
−0.307
−0.281
−0.351
−0.365
NdNiInH1.333
NdPdInH1.333
NdPtInH1.333 (LDA)
NdPtInH1.333 (GGA)
2.3499
2.3682
2.4160
2.4817
2.3499
-
−0.690
−0.528
−0.555
−0.384
1.4928
1.6740
1.6025
1.5711
1.5064
-
−3.341
−2.652
−3.635
−3.614
1.4928
1.4411
1.4118
1.5323
1.5618
-
−0.231
−0.579
−0.396
−0.384
aFrom
Ref. 14
value up to bonding states.
bICOHP
n
bonds to H. The charge transfer to the interstitial regions
between Pt共2c兲 and In reflects covalent bonding.
␶ = 1/2 兺 兩 ⵜ ␺i兩2
共5兲
i
C. ELF
ELF is a ground-state property that discriminates between
different kinds of bonding interaction for the constituents of
solids.39–41 In the implementation for density-functional
theory, this quantity depends on the excess of local kinetic
energy t p originating from the Pauli principle.
ELF = 兵1 + 关t p共r兲/t p,h共␳共r兲兲兴2其−1 ,
共2兲
t p = ␶ − 1/8关共ⵜ ␳兲2/␳兴
共3兲
where:
is the Pauli kinetic energy density of a closed-shell system.
共ⵜ␳兲2 / 共8␳兲 is the kinetic energy density of the bosonic-like
system, where orbitals proportional to 冑␳ are occupied. t p is
always positive and, for an assembly of fermions, it describes the additional kinetic energy density required to satisfy the Pauli principle. The total electron density, ␳:
n
␳ = 兺 兩 ␺ i兩 2
i
as well as the kinetic energy density:
共4兲
are computed from the orbitals, ␺i. In both equations, the
index i runs over all occupied orbitals.
According to Eq. (2) the ELF takes the value one either
for a single-electron wave function or for a two-electron singlet wave function. In a many-electron system, ELF is close
to one in regions where electrons are paired in a covalent
bond, or for the unpaired lone electrons of a dangling bond,
while the ELF is small in low-density regions. In a homogeneous electron gas ELF equals 0.5 at any electron density,
and ELF values of this order in homogeneous systems indicates regions where the bonding has a metallic character. The
estimated ELF value of ca. 0.38 between the nearest neighbor hydrogens indicates metallic-like interaction. The ELF
value between Pt共2c兲 and H is relatively small. It is interesting to note that our test calculations for Ni, Co, and Cu
metals shows that such low ELF values are generally characteristic of transition metals (d electrons). This clearly indicates that metallic Pt共2c兲-5d electrons are mainly present
between Pt共2c兲 and H, and hence the apparently appreciable
Pt共2c兲 u H bond strength (seen by the COHP analysis) originates from a metallic-like situation with additional small
ionic interactions between the atoms concerned. The large
014107-7
PHYSICAL REVIEW B 70, 014107 (2004)
VAJEESTON et al.
TABLE IV. Calculated ground-state properties: N共EF兲 (in
states/Ry cell), bulk modulus B0 (in GPa), and its pressure derivative B0⬘.
Compound
N共EF兲
B0
B⬘0
LaNiIn
LaPdIn
LaPtIn
38.22
39.33
28.42
70.4
74.9
85.4
4.1
4.2
4.3
LaNiInH1.333
LaPdInH1.333
LaPtInH1.333
35.30
24.10
37.40
69.5
86.1
77.6
4.1
4.2
4.3
CeNiIn
CePdIn
CePtIn
38.90
47.67
26.15
86.2
71.3
80.8
2.9
4.4
4.4
CeNiInH1.333
CePdInH1.333
CePtInH1.333
38.10
30.03
37.56
81.7
84.6
86.4
3.5
4.4
4.3
PrNiIn
PrPdIn
PrPtIn
44.53
42.21
27.06
73.4
74.1
83.7
4.4
4.4
4.5
PrNiInH1.333
PrPdInH1.333
PrPInH1.333
55.58
33.34
33.36
71.1
101.0
88.4
4.0
4.4
4.3
NdNiIn
NdPdIn
NdPtIn
43.74
50.69
27.46
76.0
74.7
86.0
4.4
4.8
4.4
NdNiInH1.333
NdPdInH1.333
NdPtInH1.333
28.13
32.88
30.93
86.0
89.7
101.8
4.1
4.5
4.5
FIG. 3. Total DOS for LaPtIn (top panel) and LaPtInH1.333 (bottom panel).
ELF value at the H site indicates the presence of largely
paired electrons. Owing to the repulsive interaction between
the negatively charged H electrons, the ELF contours are not
spherically shaped but rather polarized toward La and In.
The localized nature of the electrons at the H site and their
polarization toward La and In reduce significantly the
H-to-H repulsive interaction and this can explain the unusually short H u H separation in this series of compounds.
D. COHP
Implications of COHP is that a negative value indicates
bonding and positive antibonding states. In order to get insight in the bond strength between two interacting atoms in a
solid, one should examine the entire COHP between them,
taking all valence orbitals into account.29 The COHP contours depend on the number of states in a particular energy
interval. The integrated COHP (ICOHP) curve up to the EF is
the total overlap population of the bond in question.
Figure 7 shows COHP curves of the T u H and H u H
bonds for PrTInH1.333. When one includes hydrogen in the
RTIn matrix the estimated ICOHP value for the T u In bond
is changed from ca. −0.8 to − 1.1 eV. This appears to reflect
transfer of electrons from both T and In to the H site, which
reduces the covalent interaction between T and In upon hydrogenation. All RTInH1.333 phases studied have short T u H
separations (1.457 to 1.674 Å; see Table III) approaching
the sum of the covalent radii of T and H and indicates the
presence of covalent bonding. However, our charge-density,
charge-transfer, and ELF analyses show that nonbonding T
-d electrons are found between T and H, and the bonding
interaction between T and H is dominated by the transferred
charge. Another important observation is that all Ni-based
phases have shorter T u H distances than those containing Pt
and Pd (in fact the T u H distances follow the sequence:
Ni⬍ Pt⬍ Pd), which is consistent with the sequence of the
covalent radii of the involved elements. On comparing the
estimated ICOHP values for the different combinations (like
R u H, R u In, T共2c兲 u In, T共1b兲 u In, and H u H), it is
seen that these combinations take a peak value at T共2c兲 u H,
which ultimately indicates that the interaction between T and
H is stronger than the other interactions. The charge-density,
charge-transfer, and ELF analyses show that this interaction
does not have a specific bonding character and indeed exhibit
metallic interaction with a small ionic component. A similar
type of bonding situation is reported42 for LaNi5H7.
014107-8
PHYSICAL REVIEW B 70, 014107 (2004)
SEARCH FOR METAL HYDRIDES WITH SHORT…
and charge-density analyses agree that the H u H interaction
is considerably weaker than the T共2c兲 u H interactions.
When we compare the H u H interaction in these series, the
Pt-based phases have stronger interactions (ICOHP between
−0.313 to − 0.423 eV)
than
the
others
(except
NdPdInH1.333).
E. Effect of pressure on c / a and H A H separation
FIG. 4. Partial DOS for LaPtIn (left panel) and LaPtInH1.333
(right panel).
The ICOHP values for the H u H separations in the
RTInH1.333 series are very small, around −0.04 eV, thus supporting the already advanced inference that there is no significant covalent bonding interaction between the H atoms
(Sec. IV B 1). The low ICOHP value reflects the fact that
both bonding and antibonding states are present below EF,
but even if one takes into account only the bonding states,
ICOHP remains low [−0.14 to − 0.23 eV, which is much
smaller than the ICOHP for T共2c兲 u H]. Hence, both COHP
In order to explore the effect of pressure on the H u H
separation we have made a high-pressure study on
LaNiInH1.333 up to 400 GPa. The pressure was evaluated
from the derivative of the total energy with respect to volume, i.e., calculated energy versus volume data are fitted to a
universal equation of state from which the bulk modulus and
its pressure derivative are calculated.23 The calculated H u H
separation and the shrinkage in a and c (viz. ⌬a and ⌬c with
respect to the equilibrium values) as a function of pressure
shows a considerably and nonlinear variation as a function of
pressure. The H u H separation (Fig. 8) has become ca.
1.1 Å at 400 GPa. Similarly, a and c also decrease in the
same manner but the magnitude of the shrinkage varies with
the direction, which is in agreement with recent highpressure experiments.43 The difference between ⌬a and ⌬c
may be classified as somewhat insignificant at lower pressure
(say up to 40 GPa; ⌬a − ⌬c = 0.16 and 0.63 Å at 20 and
200 GPa, respectively). Because of the structural …H-NiH…H-Ni-H… configuration in the linear chains along c one
should of course have anticipated such anisotropic lattice
expansion/contraction upon application of pressure. It should
be noted that a recent experimental study43 shows there is no
pressure-induced structural transition in this material up to
40 GPa. Hence, we have assumed that the ambient pressure
phase will be stable at higher pressures also. The calculated
total energy is increased monotonically without discontinuities and indications of local minima upon increasing pressure. This ultimately implies that formation of molecular-like
H2 units in this particular structure does not occur at least up
to 400 GPa, even though the H u H separation has approached the bond distance in the H2 molecule 共0.746 Å兲.
F. Is the RTInH1.333 compounds really metal hydrides?
FIG. 5. (Color online) Calculated valence-electron-density map
for LaPtIn in the (100) plane. (1.0E-2 is an abbreviation for 1.0
⫻ 10-2 etc.)
The TH2 structural subunit in the RTInH1.333 compounds
resembles a linear complex molecule and the adjacent subunits are also arranged in a linear fashion as often found in
structures of linear complex molecules. For completely noninteracting molecules one would have anticipated that the
“2-Å rule” should have been applicable, but for the present
compounds this is certainly not the case. However, these
compounds are not complex hydrides, for the following reasons:
1) The two H atoms within a given trigonal bipyramid
(see Fig. 1) are not completely isolated from each other as
would have been the case in complex hydrides.
2) In RTNiInH1.333 there are considerable interactions between the H atoms as evidence by the fact that the H u H
distance also varies considerably on displacement of the R
atoms.1,2 If the TH2 units had been mutually completely non-
014107-9
PHYSICAL REVIEW B 70, 014107 (2004)
VAJEESTON et al.
FIG. 6. (Color online) Calculated (a) valence-electron-density map, (b) charge transfer map, and (c) ELF map for LaPtInH1.333, in (100).
(1.0E-2 is an abbreviation for 1.0⫻ 10-2 etc.)
interacting, then the H u H distance should not have been
sensitive to displacements of R.
3) If the RTInH1.333 compounds should be viewed as
complex hydrides44 with isolated TH2 molecular units, then
the desorption of hydrogen from the host lattice may be expected to require high temperature. However, experimental
studies show14 that desorption of hydrogen from RTInH1.333
takes place in a rather low-temperature range around 100° C,
viz. typical for a metal hydride.
4) In general, complex hydrides possess nonmetallic
behavior,45 but the electronic structure calculations show that
RTInH1.333 behave like metal.
5) Unlike many complex hydrides, the RTInH1.333 series
can be prepared by the traditional methods used for the synthesis of intermetallic hydrides.
6) Complex metal hydrides have molecular character
with stoichiometric hydrogen contents. On the contrary, in
the RTInH1.333 phases the hydrogen content can be varied
like in metal hydrides.14,32
7) The COHP analyses show that a considerable COHP is
FIG. 7. COHP for PrTInH1.333 (T = Ni, Pd, Pt) describing the T u H and H u H interactions.
014107-10
PHYSICAL REVIEW B 70, 014107 (2004)
SEARCH FOR METAL HYDRIDES WITH SHORT…
FIG. 8. Shrinkage in lattice parameters (upper panel) and H u H
separation (lower panel) for LaNiInH1.333 as a function of applied
pressure.
accommodated between the hydrogen atoms indicating that
they are weakly interacting.
V. CONCLUSION
This study concerns the electronic structure and physical
properties of RTIn and their saturated hydride phases
共RTInH1.333兲 investigated by first principle total-energy calculations. The following conclusions are made:
1) The optimized unit-cell volume, c / a, and atomic coordinates are in good agreement with experimental findings for
the parent alloy phases. For the RTInH1.333 compounds experimental data are only available for T = Ni that indeed show
good agreement with the calculated data. The introduction of
*Electronic
address: ponniahv@kjemi.uio.no; URL: http://
folk.uio.no/ponniahv
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(12.4%–20.5%), and a small contraction along
a 共−0.04 to − 5.8% 兲.
2) When one replaces the Ni by Pd or Pt in RTInH1.333;
the observed H u H separation becomes shortened, which
appears to suggest that one with a proper combination of R
and T (and perhaps also substitution for In) can obtain phases
with even shorter H u H separations.
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direct H u H interaction is weak compared with the T u H
interactions. Hence, the shortest distance between the H atoms in RTInH1.333 is governed primarily by the polarization
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and In atoms.
5) High-pressure simulations show that formation of more
H2-like units in RTInH1.333 matrices is not feasible even at
ultra-high pressures (up to 400 GPa).
6) The occurrence of very short T u H and unusual
H u H separations would at first sight suggest that the
RTInH1.333 phases belong to the complex hydride family.
However, a careful analysis of all facts show that these compounds should be classified as normal metal hydrides.
7) All RTInH1.333 phases have nonvanishing density of
states at the Fermi level and should accordingly exhibit metallic conductivity.
ACKNOWLEDGMENTS
P.V. and P.R. gratefully acknowledge Professor O.K.
Andersen and Professor J. Wills for being allowed to use
their computer codes and the Research Council of Norway
for financial support. This work has also received support
from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time.
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014107-12
V
Indicator for site preference of hydrogen in metal, alloy, and intermetallic frameworks
P. Vajeeston,∗ P. Ravindran, R. Vidya, A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
(Dated: November 10, 2004)
Systematic examinations of ZrNiAl-type phases suggest that the electron-localization function
(ELF) can predict hydrogen locations in metal, alloy, and intermetallic frameworks. Hydrogen incorporated into an ZrNiAl-type framework will occupy different interstitial sites depending on the
chemical near surrounding of the sites. From trends in the ZrNiAl-type family it appear that whenever the ELF indicates maximum value in a certain interstitial region of the structural framework
that site stand forth as a likely location for hydrogen incorporation. The findings may be generalized
to a site-preference rule which appears to originate from a desire to attain charge levelling. This
may be a useful approach in the search for potential hydrogen-storage materials, and the ELF is a
suitable tool for pursuing such problems.
PACS numbers: 62.50.+P,61.50.Ks,61.66.Fn
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 multicomponent phases the structural knowledge is extremely
poor. For hydrides, owing to the complexity in the structural arrangements and the difficulties involved in establishing hydrogen positions in metal, alloy, and intermetallic matrices from X-ray diffraction methods, structural
information is very limited.
metallic hydrides. This empirical experience is summarized in the so-called “2-Å rule”.
Nowadays a lot of attention is paid to identify potential candidates for safe and easy hydrogen storage for
the energy sector. The major challenges in the development of new hydrogen-storage materials are (with particular reference to batteries and fuel cells) to improve
the ratio of the stored-energy quantity to the weight of
the storage medium, desorption kinetics, cycle life, and
ready availability of the storage system, all at a cost acceptable for the users. Different kinds of intermetallic
phases have been considered, such as LaNi5 (with maximum storage capacity 1.4 wt %), TiFe (1.9 wt %), Zrand Ti-based Laves phases (around 1.8 wt %), various
Mg-based materials as well as numerous composites. Hydrides of metal, alloy, and intermetallic phases have been
the prime targets for these endeavors because experience
has shown that these hydrides have relatively low absorption/desorption temperatures (viz. offer favorable
operating conditions). However, the weight percentage
of stored hydrogen is yet too limited [1, 2].
The main theme of the present investigation has
been to predict hydrogen position(s) in phases with the
ZrNiAl-type structure using density-functional calculations, and triggered by the findings, to understand why
hydrogen takes different sites within this series of isostructural phases. It is recognized that the electronlocalization function (ELF) is a useful tool to characterize chemical bonding in solids. For example, Savin et
al.[8] have maintained that it is possible to identify the
site preference for hydrogen in the f cc Ca structure using ELF analysis. The present study has made use of the
same approach in a systematic examination of the hydrogen location in more than 95 hydride phases orginating
from the ZrNiAl-type framework.
A key challenge in the research on hydrogen-storage
materials is to pack hydrogen atoms or molecules close
together. The closest approach between two hydrogen
atoms in a crystalline framework is limited by the H-toH repulsion and it is commonly believed that two hydrogen atoms cannot be located closer together than some
2.0 Å [3]. This regularity appears to be true for the absolute majority of precisely described structures of inter-
A recent study [4] of hydrogen incorporation in the
RNiIn (R = La, Ce, Nd; generally a rare-earth element) series shows that these hydrides violate the ”2Å rule” vigorously, and a theoretical follow-up investigations [5, 6] suggest that the H-H separation may be
reduced to even below 1.5 Å by other element combinations. The RNiIn phases crystallize in the ZrNiAl-type
structure (Fig. 1), which comprises a family of more
than 300 individuals [7]. Turning to the corresponding
hydrides, only a few is identified experimentally.
The theoretical simulations are based upon densityfunctional theory (DFT) [9] with plane-wave basis
sets using the Vienna ab initio simulations package (VASP) [10], which calculates the Kohn-Sham
ground state via an iterative band-by-band matrixdiagonalization scheme and charge-density mixing [10].
All calculations employed the generalized-gradient approximation (GGA) of Perdew and Wang [11]. The valence electrons were explicitly represented with projectoraugmented plane-wave (PAW) [12] pseudo potentials.
The ions are relaxed toward equilibrium until the
Hellmann-Feynman forces are less than 10−3 eV/Å. Brillouin zone integration is performed with a Gaussian
2
broadening of 0.1 eV during all relaxations. All calculations are performed with 768 k points in the whole
Brillouin zone and a 600 eV plane-wave cutoff. The preference for hydrogen localization in the 2d and 4h sites
of space group P 62m with full occupancy have been systematically tested for the considered ZrNiAl-type phases
(more surveying studies have been made for the 3g site).
Thus all together theoretical simulation for 285 combinations have been performed with full geometry optimization and without any constrains on atomic positions and
unit-cell parameters. Equilibrium volumes and unit-cell
parameters were extracted from calculated energy versus volume data by fitting to the “universal equation of
state” proposed by Vinet et al. [13]. The hydrogen formation energy (∆E) was calculated from the relation:
∆E =
1
1
[E(RT M Hx ) − E(RT M )] − E(H2 )
x
2
(1)
where x refers to the hydrogen content per formula unit
[viz. x = 0.6667, 1.333 or 2.333 for the filled-up site(s)
under consideration], T is a transition metal, M an element in The periodic table, E(RT M Hx ) represents the
energy of the hydride phase, E(RT M ) the energy of the
intermetallic phase, and E(H2 ) the energy of the dihydrogen molecule (−6.729 eV).
In the ZrNiAl-type structure (maintaining the general
formula RT M , while abolishing the above strict specifications of R and T ) R is placed in 3g (x,0,1/2), T in 1b
(0,0,1/2) and 2c (1/3,2/3,0), and M in 3f (x,0,0) of space
group P 62m. The unit cell has eight different interstitial
sites centered around 2d, 2e, 3f , 3g, 4h, 6i, 6k, and 12l.
Of these, the tetrahedral holes around 4h (surrounded
by 3R and 1T ), the trigonal bipyramidal holes around 2d
(surrounded by 2R, 1T , and 1M ), and the distorted octahedral holes around 3g (surrounded by 3R, 1T , and 2M )
are the most interesting in relation to hole size and symmetry. The present study has focused on 4h and 2d as the
most probable sites for incorporation of hydrogen. Even
though we have examined all together 95 ZrNiAl-type
phases it is convenient to concentrate the presentation of
the findings on two typical examples, where theoretical
calculations and experiments show that hydrogen prefers
different sites.
The hole size for all interstitial sites of the ZrNiAl
structure (as derived from structural parameters [14] and
atomic radii) show that the 2d and 12l sites give rise to
the largest interstitial holes; 0.49 and 0.43 Å, respectively.
The experimental study shows that hydrogenation of ZrNiAl leads to ZrNiAlH0.53 , with the hydrogen atoms on
the 2d site. The theoretically optimized structural parameters for the ZrNiAl matrix came out in very good
agreement with the experimental values (Table 1). In order to localize possible hydrogen positions in the hydride
phase, the ELF for ZrNiAl was systematically visualized
in different crystal planes. The results showed that elec-
trons only have a tendency to accumulate at the 2d site
in the 100 plane (Fig. 2a; ELF above 0.6 at 2d and much
smaller values at the other interstitial sites). Hydrogen
at full occupancy was subsequently placed in the 2d site
and atomic positions and unit-cell parameters were optimized. The thus obtained optimized structural parameters fit very well with the experimental data (within the
3 % limit typical for DFT calculations). Calculations performed on the assumption of accommodation of hydrogen
in the other interstitial sites gave poorer fits to the experimental values and less favorable hydride-formation energies. The shortest H-H separation in the ZrNiAlH0.667
phase obeys the 2 Å rule. The insertion of hydrogen in
the metal matrix causes highly anisotropic changes of the
crystal lattice; a large expansion along c and a small contraction along a.
Exactly the same course was used in a systematic
attempt to locate hydrogen positions in the isotypic
LaNiIn phase. Hole-size considerations for the experimental structural parameters of LaNiIn gave in this case
prominence to 3g (hole radius 0.63 Å), 4h(0.47 Å), and 3f
(0.45 Å) as possible interstitial sites for accommodation
of hydrogen (much small hole radii were derived for the
other interstitial sites). In order to approach the localization task theoretically, unit-cell dimensions and atomic
positions for the LaNiIn structure were first optimized.
These calculations gave parameter values in close agreement with the experimental findings (Table 1). The next
step was to derive the ELF and to visualize this function
in different planes of the LaNiIn structure. Inspection of
these ELF maps revealed that electrons are accumulated
at the 4h site (ELF above 0.6 at the 4h site and negligible ELF at all other interstitial sites; Fig. 2c). The
subsequent optimization calculations for the corresponding filled-up LaNiInH1.333 structure show that this is indeed a very likely atomic arrangement for the hydride,
with again quite good agrement between experimental
and theoretical structural variables (Table 1). Calculations based on accommodation of hydrogen in the other
interstitial sites gave also in this case poorer fit to the
experimental data and less favorable hydride-formation
energies. Since the hydrogen atoms take different positions in the ZrNiAl and LaNiIn matrices it is not surprising that the shortest H-H separation in the hydride
phases differs. However, superficially viewed the distinction seems surprisingly large. ZrNiAlH0.667 obeys the “2Å rule” whereas LaNiInH1.333 breaks it in sovereignty.
Like the ZrNiAl-to-ZrNiAlH0.667 conversion the hydrogenation of LaNiIN to LaNiInH1.333 also causes highly
anisotropic lattice changes; contraction of a and expansion of c.
The chemical environment around a given interstitial
site clearly plays a decisive role for its ability to accommodate hydrogen. On this background it was considered of interest to go one step further and visualize the
ELF for LaNiInH1.333 (see Fig. 2d). With the 4h site
3
TABLE I: Optimized unit-cell dimensions (in Å) and positional parameters, hydrogen formation energy (in kJ/mole), and
shortest H-H separation (in Å) for selected ZrNiAl-type phases [space group P 62m; sites: R in 3g (x,0,1/2), T in 1b (0,0,1/2)
and 2c (1/3,2/3,0), and M in 3f (x,0,0); for hydrogen: 2d (2/3, 1/3, 1/2), 4h (1/3, 2/3, z), and 3g (x, 0, 1/2)].
Compound
ZrNiAl
ZrNiAlH0.666
LaNiIn
LaNiInH1.333
LaNiInH2.333
ThCoAl
ThCoAlH1.333
ThNiIn
ThNiInH1.333
YNiIn
YNiInH0.666
a
Unit cell (a and c)
6.8738 (6.9152)a
3.5652 (3.4703)a
6.8651 (6.7225)a
3.6579 (3.7713)a
7.6046 (7.5906)b
4.0850 (4.0500)b
7.3853 (7.3810)b
4.6761 (4.6489)b
7.3828 (7.3874)c
4.8436 (4.6816)c
7.1711 (7.0460)d
4.0313 (4.0364)d
7.0854
4.3207
7.3815 (7.3673)d
4.1690 (4.1170)d
7.1519
4.3664
7.4536 (7.4860)d
3.8272 (3.7840)d
7.4611
3.8182
Positional parameters
R: x = 0.5904 (0.591)a ; M : x = 0.2482 (0.251)a
−∆E
H-H separation
R: x = 0.5947 (0.592)a ; M : x = 0.2477 (0.246)a
H(2d)
R: x = 0.5866 (0.5940)b ; M : x = 0.2475 (0.256)b
46.59
3.77
R: x = 0.6036 (0.6035)b ; M : x = 0.2444 (0.2437)b
H(4h): z = 0.6728 (0.6759)b
R: x = 0.5982 (0.603)c ; M : x = 0.2547(0.247)c
H(4h): z = 0.3182 (0.317)c ; H(3g): x = 0.2193 (0.226)c
R: x = 0.5861; M : x = 0.2266
46.26
1.63 (1.64)
35.04
1.75 (1.72)
R: x = 0.6124; M : x = 0.2399
H(4h): z = 0.6620
R: x = 0.5835; M : x = 0.2443
10.322
1.40
R: x = 0.6008; M : x = 0.2514
H(4h): z = 0.6658
R: x = 0.5890; M : x = 0.2536
28.09
1.45
R: x = 0.5912; M : x = 0.2539
H(2d)
64.36
3.72
Experimental value from Ref. 14
b Experimental value from Ref. 15
c Experimental value from Ref. 15 with
D in 4h (96 % occupancy)
and 3g (36 % occupancy)
d Experimental value from Ref. 7
in
4h
to
H
yd
2d
ro
ge
na
tio
n
2d
Expansion
Contraction
(d)
to
in
n
Expansion
io
at
4h gen
ro
yd
H
(b)
3g
(f)
(c)
Contraction
(a)
(e)
FIG. 1: (Color online) (a) The ZrNiAl-type crystal structure. Legends for the different kinds of atoms are given in the
illustration. The 2d (trigonal bipyramidal interstice), 4h (tetrahedral interstice), and 3g (distorted octahedral interstice) sites
are indicated by open circles and indicated by arrows. (b) The empty trigonal bipyramidal interstice. (c) The empty tetrahedral
interstice. (d) The trigonal bipyramidal interstice filled with hydrogen. (e) Two face-sharing tetrahedral interstices filled with
hydrogen. (f) The empty (distorted) octahedral interstice.
filled up by hydrogen another interstitial site (3g) with
an ELF value of around 0.6 appears (while other interstitial sites still show negligible ELF levels). A recent study of deuteration of LaNiIn under pressure (pD2
= 4.6 bar) has unveiled a new phase, LaNiInD1.64 , in
which deuterium occupies both the 4h (96% filled) and
3g (36% filled) sites. The optimization calculations for
the corresponding filled-up LaNiInH2.333 structure gave
unit-cell dimensions and positional parameters in satisfactory agreement (viz. within the expected accuracy
for DFT calculations) with the experimental findings for
LaNiInD1.64 (Table 1). The effect of partial versus full
4
4h
3g
1
2d
0.75
0.5
0.25
0.001
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
FIG. 2: (Color online) Calculated ELF plot for (a) ZrNiAl, (b) ZrNiAlH0.667 , (c) LaNiIn, (d) LaNiInH1.333 , (e) ThCoAl, (f)
ThNiIn, (g) ThPdIn, and (h) YNiIn in (001) plane. Different crystallographic sites (2d, 4h, and 3g) are marked by arrows.
The iso-surface values corresponding to 0.6 ELF value. Legends for the different kinds of atoms are given on the illustration.
occupancy of hydrogen positions appears to be of subordinate importance for the structural details. The formation energy for hydrogenation from LaNiInH1.333 to
LaNiInH2.333 (Table 1) confirms that this conversion is
thermodynamically acceptable as already demonstrated
by experiment. As a consistency check optimization calculations and formation energy estimates were also made
for other combinations of fully occupied interstitial sites.
The outcome (Table 1) confirms that the interstitial site
combination 4h and 3g is the only realistic possibility
in this case. Inspired by the ability of ELF to predict
the existence of LaNiInH2.333 , ELF maps were were prepared for ZrNiAlH0.667 . However, additional sites with
electron accumulation did not show up in the ELF maps
(see Fig. 2b) and calculations of formation energies (positive ∆E; not documented) confirm that ZrNiAlH0.667
should be that the most hydrogen-rich phase.
It was indeed the success of ELF as a guide to the
secrets of the just discussed hydride phases that led us
to undertake a more systematic study of 95 members
of the ZrNiAl-type family. The working hypothesis was
that when the ELF value at a certain interstitial site exceeds 0.5 then this site will favor accommodation of hydrogen. The full account of our findings will be published
elsewhere, since space limitations only permit a brief
overview here. Statistically seen the ELF criterion predicts that hydrogen prefers the 2d site in 43 % of the studied RT M phases (e.g., YPdIn, ThRhSn, and ZrCoGa)
and the 4h site in 12 % (e.g., CeNiIn, PrNiIn, and NdNiIn). For 5 % of the RT M phases (e.g., YNiIn, ScMnSi,
and ThNiGa) the ELF is unable to provide a clear-cut
distinction between the 2d and 4h sites. The ELF for
these phases is generally blurred in interstitial regions
and the ELF level at 2d and 4h reflects a more metalliclike situation. For 40 % of the studied RT M phases (e.g.,
ZrAsOs, ScPdGe, and ThIrAl) the ELF predicts that attempted hydrogenation will be unsuccessful. These predictions compare reasonably well with the corresponding
calculated formation energies, which show that 44 % of
the 95 examined RT M phases have a clear preference
for the 2d site, 10 % for the 4h site, 8 % undecided, and
38 % unamendable for hydrogen. Table 1 includes predicted crystal-structure and formation-energy data for
three more or less arbitrary chosen hydride phases based
on the ZrNiAl-type matrix. As examples of ELF maps,
Fig. 2e-f shows illustrations for ThCoAl, ThNiAl, and
YNiIn. The above selection of ZrNiAl-type phases should
permit a first experimental check of the predicting power
of what we like to proposed as a site-preference rule for
hydrogen in metal, alloy, and intermetallic matrices: Accumulation of electronic charge in a certain interstitial
region of a given structural framework gives prominence
to that site as a likely location for hydrogen occupation.
The ELF serve as a simple, but very efficient indicator
for visualization of interstitial regions with electron accumulation.
5
Finally a few words about the origin of the sitepreference rule. A hydrogen atom can take up an electron
and in this way complete its valence shell. The thus obtained H− ion can collect the additional electron from
any electron source and the heap-up of electrons in the
more “common-land” interstitial regions are very proper
for such a charge-levelling process. Another possibility
for charge-levelling is that a hydrogen atom gives up its
own electron to the collective interstitial-site electron well
[thus forming an H+ ion (viz. a bare proton)] and arrange
itself on the interstitial site. If more than one interstitial
site show electron accumulation in a particular matrix,
it seems likely that first hydrogen atoms will go to the
site with the largest amount of localized electrons. As
a result of the filling of this site electron localization in
other sites may either increase or decrease. In the former
case a new site may be available for hydrogen filling, say,
when exposed high hydrogen pressures.
In conclusion it is shown that although the structural
arrangement of the intermetallic matrix is the same, hydrogen may take different sites depending upon the chemical environment. The site preference of hydrogen to a
certain interstitial site is decided by the electron accumulation at that site. Hydrogen prefer to occupy interstitial sites where the ELF take a maximum. The origin
of this preference may be visualized as either a desire
to complete the valence shell of the hydrogen atom by
electrons from the interstitial excess store (viz. the H−
picture) or complementary as proton immersed in a bath
of the electrons originally possessing the site plus electrons given up by formation of the bare proton (viz. the
H+ picture). The site-preference rule can be used to design metal, alloy, and intermetallic phases for testing as
potential hydrogen-storage materials.
The authors gratefully acknowledge the Research
Council of Norway for financial support and for the computer time at the Norwegian supercomputer facilities.
∗
Electronic
address:
ponniahv@kjemi.uio.no;
URL: http://www.folk.uio.no/ponniahv
[1] G. Sabdrock, J. Alloys Compd. 293-295, 877 (1999).
[2] G. Sabdrock and G. Thomas, IEA/DOE/SNL Hydride
Databases, http://hydpaek.ca.sandia.gov
[3] A.C. Switendick, Z. Phys. Chem. 117, 89 (1979).
[4] V.A. Yartys, R.V. Denys, B.C. Hauback, H. Fjellvåg,
I.I. Bulyk, A.B. Riabov, and Ya. M. Kalychak, J. Alloys
Compd. 330-332, 132 (2002).
[5] P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, and
H. Fjellvåg, Phys. Rev. Lett. 89, 106403 (2002).
[6] P. Vajeeston, P. Ravindran, H. Fjellvåg, and A. Kjekshus, Phys. Rev. B 70, 014107 (2004).
[7] P. Villars and L. D. Calvert, Pearson’s Handbook of
Crystallographic Data for Intermetallic Phases, 2nd edition, American Society for Metals, Materials Park, OH.,
1986.
[8] A. Savin, R. Nesper, S. Wengert, and T.F. Fässler,
Angew. Chem. Int. Ed. Engl. 36, 1808 (1999).
[9] P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864
(1964).
[10] G. Kresse and J. Hafner, Phys. Rev. B 47, R6726 (1993);
G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15
(1996).
[11] J.P. Perdew, in : Electronic Structure of Solids, P. Ziesche, H. Eschrig (Eds.), Akademie Verlag, Berlin, 1991;
J.P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54,
16533 (1996), J.P. Perdew, K. Burke, and M. Ernzerhof,
Phys. Rev. Lett. 77, 3865 (1996).
[12] P.E. Blöchl, Phys. Rev. B 50, 17953 (1994); G. Kresse
and J. Joubert, ibid. 59, 1758 (1999).
[13] P. Vinet, J. Ferrante, J. R. Smith, and J. H. Rose, J.
Phys. C 19, L467 (1986); P. Vinet, J. H. Rose, J. Ferrante, and J. R. Smith, J. Phys. Condens. Matter 1, 1941
(1989).
[14] M. Yoshida, E. Akiba, Y. Shimojo, Y. Morii, and F.
Izumi, J. Alloys Compd. 231, 755 (1995).
[15] R. V. Denys, A. B. Riabov, V. A. Yartys, B. C. Hauback,
and H. W. Brinks, J. Alloys Compd. 356-357, 65 (2003).
[16] D. G. Westlake, J. Less-Common Met. 103, 203 (1984).
VI
VOLUME 89, N UMBER 17
PHYSICA L R EVIEW LET T ERS
21 OCTOBER 2002
Pressure-Induced Structural Transitions in MgH2
P. Vajeeston,* P. Ravindran, A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033, Blindern, N-0315, Oslo, Norway
(Received 15 May 2002; published 8 October 2002)
The stability of MgH2 has been studied up to 20 GPa using density-functional total-energy
calculations. At ambient pressure -MgH2 takes a TiO2 -rutile-type structure. -MgH2 is predicted to
transform into -MgH2 at 0.39 GPa. The calculated structural data for - and -MgH2 are in very good
agreement with experimental values. At equilibrium the energy difference between these modifications
is very small, and as a result both phases coexist in a certain volume and pressure field. Above 3.84 GPa
-MgH2 transforms into -MgH2 , consistent with experimental findings. Two further transformations
have been identified at still higher pressure: (i) - to -MgH2 at 6.73 GPa and (ii) - to -MgH2 at
10.26 GPa.
DOI: 10.1103/PhysRevLett.89.175506
The utilization of high-pressure technology, which can
reach the range of giga Pascal (GPa), has made considerable progress in experimental studies of hydrogenstorage materials. It has been demonstrated for a number
of metal-hydrogen systems that application of high pressure is an effective tool to produce vacancies in the host
metallic matrix, which in turn leads to various novel
properties [1]. The Mg-based alloys possess many advantageous functional properties, such as heat resistance,
vibration absorption, recycling ability, etc. [2]. In recent
years, therefore, much attention has been paid to investigations on specific material properties of Mg alloys.
Magnesium is an attractive material for hydrogen-storage
applications because of its light weight, low manufacture
cost, and high hydrogen-storage capacity (7.66 wt. %). On
the other hand, owing to its high operation temperature
(pressure plateau of 1 bar at 525 K) and slow absorption
kinetics, practical applications of magnesium-based alloys have been limited. However, it has recently been
established that improved hydrogen absorption kinetics
can be achieved by reduced particle size and/or addition
of transition metals to magnesium and magnesium hydrides [3]. -MgH2 often occurs as a by-product in highpressure syntheses of technologically important metal
hydrides such as Mg2 NiH4 . Hence, a complete characterization of -MgH2 , in particular, knowledge about its
stability at high pressure is highly desirable. Present
theoretical investigation assumes importance as the highpressure x-ray diffraction and neutron diffraction studies
are unable to identify the exact position of hydrogen
atoms owing to its very low scattering cross section.
This contribution represents the first theoretical report
on the high-pressure properties of MgH2 as evident from
ab initio density-functional calculations.
-MgH2 crystallizes with TiO2 -r-type (r rutile)
structure at ambient pressure and low temperature [4,5].
At higher temperatures and pressures, tetragonal -MgH2
transforms into orthorhombic -MgH2 . Recently Bortz
et al. [6] solved the crystal structure of -MgH2 (-PbO2
175506-1
0031-9007=02=89(17)=175506(4)$20.00
PACS numbers: 62.50.+p, 61.50.Ks, 61.66.Fn
type) on the basis of powder neutron diffraction data
collected at 2 GPa. The - to -MgH2 transition pressure
is not yet known. Owing to the difficulties challenged in
establishing hydrogen position in a metal matrix by x-ray
diffraction, high-pressure information on MgH2 is scarce.
The aim of the present investigation is to remedy this
situation by examining MgH2 theoretically at high pressure in 11 closely related structural configurations.
We used the ab initio generalized gradient approximation (GGA) [7] to obtain accurate exchange and correlation energies for a particular structural arrangement. The
structures are fully relaxed for all the volumes considered
in the present calculations using force as well as stress
minimization. Experimentally established structural data
were used as input for the calculations when available.
The full-potential linear muffin-tin orbital (FP-LMTO)
[8] and the projected augmented wave (PAW) implementation of Vienna ab initio simulation package ( VASP) [9]
were used for the total-energy calculations to establish
phase stability and transition pressures. Both methods
yielded nearly the same result; e.g., the transition pressure
from - to -MgH2 came out as 0.385 and 0.387 GPa from
FP-LMTO and VASP, respectively. Similarly, almost identical values were obtained for ground-state properties
such as bulk modulus and equilibrium volume. Hence,
for the rest of the calculations we used only the VASP code
because of its computational efficiency. 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 calculations 500 k points
in the whole Brillouin zone were used for -MgH2 and
a similar density of k points for the other structural
arrangements. The PAW pseudopotentials [10] were
used for all our VASP calculations. A criterion of at least
0:01 meV=atom was placed on the self-consistent
convergence of the total energy, and the calculations
reported here used a plane wave cutoff of 400 eV.
In addition to the experimentally identified and -modifications of MgH2 , we have carried out
© 2002 The American Physical Society
175506-1
VOLUME 89, N UMBER 17
PHYSICA L R EVIEW LET T ERS
calculations for several other possible [11] types of structural arrangements (see Table I and Figs. 1 and 2). The
calculated total energy vs volume relation for all these
alternatives are shown in Fig. 1. In order to get a clear
picture about the structural transition points, in Fig. 2 we
have displayed the Gibbs free-energy difference between
the pertinent crystallographic structures of MgH2 with
reference to -MgH2 as a function of pressure. The equi 3 =f:u: for - and
librium volumes (30.64 and 30:14 A
-MgH2 , respectively) are within 1% of the experimental
values, indicating that the theoretical calculations are
reliable. The calculated positional parameters (Table I)
are also in excellent agreement with the experimental
data. The calculated values for the bulk modulus (B0 ,
see Table I) vary between 44 to 65 GPa for the various
structural arrangements. Among the 11 possibilities considered, the Ag2 Te type leads to the smallest B0 value and
the AlAu2 type to the highest.
The calculated transition pressure for the - to
-MgH2 conversion is 0.39 GPa (Fig. 3) and as the free
energy of the two modifications is nearly the same at the
equilibrium volume (Fig. 1), it is only natural that these
phases coexist in a certain volume range [6]. Structurally
- and -MgH2 are also closely related, both comprising
Mg in an octahedral coordination of 6 H which in turn are
21 OCTOBER 2002
linked by edge sharing in one direction and by corner
sharing in the two other directions. The chains are linear
in tetragonal -MgH2 and run along the fourfold axis of
its TiO2 -r-type structure, while they are zigzag shaped in
-MgH2 and run along a twofold screw axis of its
orthorhombic -PbO2 -type structure. The octahedra in
-MgH2 are strongly distorted. The pressure induced
-to- transition involves reconstructive (viz. bonds are
broken and reestablished) rearrangements of the cation
and anion sublattices. The occurrence of a similar phase
transition in PbO2 suggests that thermal activation or
enhanced shear is of importance [14].
The subsequent phase transition from - to -MgH2
occurs at 3.84 GPa. Bortz et al. [6] found no evidence for
the formation of such a modification up to 2 GPa,
whereas Bastide et al. [4] found that at higher pressure
(4 GPa) and temperature (923 K) there occurs a new
phase (viz. in accordance with our findings). However, the H positions are not yet determined experimentally and, hence, our optimized structural parameters for
-MgH2 should be of particular interest. The volume
3 =f:u: On furshrinkage at the transition point is 1:45 A
ther increase of the pressure to around 6.7 GPa, -MgH2
is predicted to transform into -MgH2 (orthorhombic
3 =f:u: In the
Pbc21 ) with a volume shrinkage of 1:1 A
TABLE I. Optimized structural parameters, bulk modulus (B0 ), and its pressure derivative (B00 ) for MgH2 in the different
structural arrangements considered in the present study.
Structure
Space group
type
a
TiO2 -r ; -MgH2
b
Mod. CaF2 ; -MgH2
b
At 3.84 GPa
b
-PbO2 ; -MgH2
P42 =mnm
b
Pa3
Unit cell (Å)
b
c
4.4853
(4.501
4.7902
4.4853
4.501
4.7902
2.9993
c
3.010 )
4.7902
Pbcn
4.4860
(4.501
Pbc21
4.8604
Pnma
5.2804
P63
3.2008
P21 =c
4.8825
Fm3 m
I41 =amd
Pbca
4.7296
3.7780
9.3738
Pnnm
4.4810
b
At 0.39 GPa
b
Ortho; -MgH2
a
b
At 6.73 GPa
AlAu2 ; -MgH2
b
b
At 10.2 GPa
InNi2
b
Ag2 Te
b
b
CaF2
b,f
TiO2 -a
b
AuSn2
CaCl2
b
Positional parameters
B0 (GPa)
Mg (2a): 0, 0, 0
51
c
d
e
H (4f): 0.3043, 0.3043, 0 (0.304, 0.304, 0 )
55 ; 50
Mg (4a): 0, 0, 0; H (8c): 0.3417, 0.3417, 0.3417
56
Mg (4a): 0, 0, 0; H (8c): 0.3429, 0.3429, 0.3429
c
5.4024
4.8985
Mg (4c): 0, 0.3314, 1=4 (0, 0.3313, 1=4 )
48
c
c
5.4197
4.9168 )
H (8d): 0.2717, 0.1085, 0.0801 (0.2727, 0.1089, 0.0794 )
Mg (4c): 0, 0.3307, 1=4; H (8d): 0.2710, 0.1073, 0.0797
4.6354
4.7511
Mg (4a): 0.0294, 0.2650, 3=4;
60
H1 (4a): 0.3818, 0.0976, 0.8586; H2 (4a): 0.2614, 0.5584, 0.0298
Mg (4a): 0.0294, 0.2665, 3=4;
H1 (4a): 0.3933, 0.1090, 0.8475; H2 (4a): 0.2586, 0.5576, 0.0305
3.0928
5.9903
Mg (4c): 1=4, 3=4, 0.6033; H1 (4c): 0.3610, 1=4, 0.4250;
65
H2 (4c): 0.4794, 1=4, 0.8345
Mg (4c): 1=4, 3=4, 0.6075; H1 (4c): 0.3602, 1=4, 0.4260;
H2 (4c): 0.4765, 1=4, 0.8300
3.2008
5.9870
Mg (2b): 1=3, 2=3, 1=4; H1 (2a): 0, 0, 0;
59
H2 (2b): 1=3, 2=3, 3=4
4.6875
5.0221
Mg (4e): 0.2630, 0.4788, 0.2117;
44
( 99:25)
H1 (4e): 0.0718, 0.2039, 0.3818; H2 (4e): 0.4309, 0.7430, 0.5021
4.7296
4.7296
Mg (4a): 0, 0, 0; H (8c): 1=4, 1=4, 1=4
59
3.7780
4.7108
Mg (4a): 0, 0, 0; H (8e): 0, 0, 0.2101
45
4.8259
4.7798 Mg (8c): 0.8823, 0.0271, 0.2790; H1 (8c): 0.7970, 0.3765, 0.1651;
58
H2 (8c): 0.9738, 0.7433, 0.5207
4.4657
3.0064
Mg (2a): 0, 0, 0; H (4g): 0.3076, 0.2991, 0
58
B00
3.45
3.52
3.07
3.68
3.72
3.42
3.31
3.52
3.19
3.45
3.45
r rutile.
b
To elucidate variations of atomic positions with pressure, parameter values are given at equilibrium volume as well as at transition
pressures.
c
Experimental value [6].
d
Theoretical value [12].
e
Theoretical value [13].
f
a anatase.
a
175506-2
175506-2
VOLUME 89, N UMBER 17
TiO2−r ; α−MgH2
Mod. CaF2; β−MgH2
α−PbO2; γ−MgH2
Ortho.; δ−MgH2
AlAu2; ε−MgH2
InNi2
Ag2Te
CaF2
TiO2−a
AuSn2
CaCl2
0.125
0
−0.125
Energy (eV/f.u)
PHYSICA L R EVIEW LET T ERS
−0.25
−0.55
−0.6
−0.65
−0.7
−0.75
(d)
22
23
24
−0.8
−0.375
−0.85
−0.5
(c)
(b)
25 26 27 28 29
−0.625
−0.75
−0.85
(d)
(a)
(c)
−0.875
−0.9
(b)
(a)
22.5
27.5
32.5
37.5
3
volume (Å /f.u)
29 30 31 32
3
volume (Å /f.u)
FIG. 1. Calculated unit-cell volume vs total-energy relations
for MgH2 in actual and possible structural arrangements as
obtained from VASP. Magnified versions of the corresponding
transition points are shown on right-hand side of the figure.
pressure range 6.7–10.2 GPa, the structural arrangements
-MgH2 , AuSn2 , and -MgH2 are seen to lie within a
narrow energy range of some 10 meV (Fig. 1). This closeness in energy suggests that the relative appearance of
these modifications will be quite sensitive to, and easily
ε
δ
2
21 OCTOBER 2002
affected by, other external factors such as temperature
and remnant lattice stresses. A transformation to -MgH2
(AlAu2 -type structure also called cotunnite-type) is
clearly evident from Fig. 1 with a volume change of
3 =f:u: at the transition point (see also Figs. 2 and
1:7 A
3). It is interesting to note that similar structural transition sequences are reported for transition-metal oxides
such as HfO2 [11]. Recently, the cotunnite-type structure
of TiO2 (synthesized at pressure above 60 GPa and high
temperatures) has been shown [15] to exhibit an extremely high bulk modulus (431 GPa) and hardness
(38 GPa). The present study predicts that it should be
possible to stabilize MgH2 in the AlAu2 -type structure
as a soft material (B0 65 GPa) above 10.26 GPa, a
prediction which should be of considerable interest.
The calculated unit-cell volume difference between the
involved phases at their equilibrium volume relative to
the phase is 0.50 (for ), 2.94 (for ), 3.60 (for ), and
3 =f:u:, which is approximately 1.6%, 9.5%,
5.96 (for ) A
11.8%, and 19.5%, respectively, smaller than the equilibrium volume of the phase. The energy difference
between phase and the , , , and phases at their
equilibrium volume is 0.81, 43, 66, and 197 meV=f:u:,
respectively, which is considerably smaller than for similar types of phase transitions in transition-metal dioxides
[11]. The known stabilization of the high-pressure phase
of TiO2 by using high-pressure –high-temperature synthesis [15] indicates that it may be possible to stabilize the
high-pressure phases of MgH2 due to their small energy
differences and (partly) reconstructive phase transitions.
Such stabilization of MgH2 high-pressure phases at room
temperature would reduce the volume considerably, and
in the extreme case the expected volume reduction would
be ca. 19.5% for -MgH2 compared with -MgH2 . This
β
10.26
37.5
6.73
1
γ
0.5
0.39
3.84
cell volume (Å /f.u.)
α
0
3
Difference in Gibbs free energy (kJ/mol)
1.5
−0.5
α
32.5
0.39
γ
3.84
27.5
β
−1
6.73
−1.5
δ
10.26
22.5
0
2.5
5
7.5
10
12.5
ε
15
Pressure (GPa)
5
0
5
10
15
20
Pressure (GPa)
FIG. 2. The stabilities of various known and hypothetical
MgH2 phases relative to -MgH2 as a function of pressure.
The transition pressures are marked by arrows at the corresponding transition points.
175506-3
FIG. 3. Pressure vs volume relation for MgH2 . Pressure
stability regions for the different modifications (see Table I
and text) are indicated.
175506-3
VOLUME 89, N UMBER 17
PHYSICA L R EVIEW LET T ERS
1.5
1.25
EF
α − MgH2
−1
DOS (states eV )
1
MgH2 (total)
Mg − total
Mg − s
Mg − p
H− total
H −s
0.75
0.5
0.25
−7.5
−2.5
1.5
1
−1
DOS (states eV )
1.25
2.5
EF
MgH2 (total)
Mg − total
Mg − s
Mg − p
H − total
H −s
7.5
γ − MgH2
0.75
21 OCTOBER 2002
-to--MgH2 are found to be in very good agreement
with the experimental findings. We predict that further
compression of -MgH2 will lead to a phase transition to
-MgH2 (orthorhombic; Pbc21 ) at 6.73 GPa and that the
AlAu2 -type structure of a hypothetical modification
stabilizes above 10.26 GPa. Our total-energy study suggested that the formation of the high-pressure phase
could also be possible by appropriate preparation methods. This may reduce the volume requirement by ca.
19.5% compared with -MgH2 . In agreement with experiments, -MgH2 turns out to be an insulator, and the
theoretical treatment shows that the high-pressure modifications also should exhibit insulating behavior.
The authors gratefully acknowledge the Research
Council of Norway for financial support and for the
computer time at the Norwegian supercomputer facilities
and R. Vidya for critical reading of this manuscript.
0.5
0.25
−7.5
−2.5
2.5
7.5
Energy (eV)
FIG. 4. The electronic total and partial density of states for
- and -MgH2 .
would imply an increased volumetric storage capacity of
hydrogen for MgH2 . Of interest would be to explore the
possibility of stabilizing the high-pressure phases by
chemical means. At the same time, this may possibly
open up for improved kinetics with respect to reversible
hydrogen absorption/desorption. Hence, the observation
of high-pressure phases in MgH2 may have technological
significance.
The calculated energy gap [from the top of the valence
band (VB) to the bottom of the conduction band] is 4.2
and 4.3 eV for - and -MgH2 , respectively. According to
Fig. 4, the width of VB is 7.5 and 8.3 eV in - and
-MgH2 , respectively. The increased width of VB in
-MgH2 is due to the reduction in the Mg-H separation
(1.93 Å in -MgH2 vs 1.91 Å in -MgH2 ). An experimental UV absorption study [12] of -MgH2 gave an
absorption edge at 5.16 eV, whereas an earlier theoretical
electronic structure investigation [12,13] reported a band
gap around 3.4 eV. Hence, our calculated energy gap value
is in better agreement with the experimentally measured
value than the earlier values. In fact, the present underestimation of the band gap by about 17% is typical for the
accuracy obtained by first principle methods for semiconductors and insulators. Such distinctions probably
originate from the use of GGA-based exchange correlation functionals.
In conclusion, the calculated structural parameters
for - and -MgH2 are in excellent agreement with
the experimental findings. We found that the ground
state of -MgH2 becomes unstable at higher pressure
and the calculated transition pressures for -to-- and
175506-4
*Electronic address: ponniahv@kjemi.uio.no
http://www.folk.uio.no/ponniahv
[1] Y. Fukai and N. Okuma, Phys. Rev. Lett. 73, 1640 (1994).
[2] S. Orimo and H. Fujii, Appl. Phys. A 72, 167 (2001).
[3] Y. Chen and J. S. Williams, J. Alloys Compd. 217, 181
(1995).
[4] J. P. Bastide, B. Bonnetot, J. M. Letoffe, and P. Claudy,
Mater. Res. Bull. 15, 1215 (1980).
[5] W. H. Zachariasen, C. E. Holley, Jr., and J. F. Stamper, Jr.,
Acta Crystallogr. A16, 352 (1963).
[6] M. Bortz, B. Bertheville, G. Bøttger, and K. Yvon, J.
Alloys Compd. 287, L4 (1999).
[7] J. P. Perdew, in Electronic Structure of Solids, edited by
P. Ziesche and H. Eschrig (Akademie, Berlin, 1991),
p. 11; J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B
54, 16 533 (1996); J. P. Perdew, S. Burke, and
M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
[8] J. M. Wills and B. R. Cooper, Phys. Rev. B 36, 3809
(1987); D. L. Price and B. R. Cooper, ibid. 39, 4945
(1989).
[9] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993);
G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15
(1996).
[10] P. E. Blöchl, Phys. Rev. B 50, 17 953 (1994); G. Kresse
and J. Joubert, ibid. 59, 1758 (1999).
[11] J. E. Lowther, J. K. Dewhurst, J. M. Leger, and J. Haines,
Phys. Rev. B 60, 14 485 (1999); J. Haines, J. M. Léger,
and O. Schulte, J. Phys. Condens. Matter 8, 1631 (1996).
[12] B. Pfrommer, C. Elsässer, and M. Fähnle, Phys. Rev. B
50, 5089 (1994).
[13] R. Yu and P. K. Lam, Phys. Rev. B 37, 8730 (1988).
[14] J. Haines, J. M. Leger, and O. Schulte, J. Phys. Condens.
Matter 8, 1631 (1996).
[15] L. S. Dubrovinsky, N. A. Dubrovinskaia, V. Swamy,
J. Muscat, N. M. Harrison, R. Ahuja, B. Holm, and
B. Johanson, Nature (London) 410, 653 (2001); N. A.
Dubrovinskaia, L. S. Dubrovinsky, R. Ahuja, V. B.
Prokopenko, V. Dmitriev, J. P. Weber, J. M. OsirioGuillen, and B. Johanson, Phys. Rev. Lett. 87, 275501
(2001).
175506-4
VII
APPLIED PHYSICS LETTERS
VOLUME 84, NUMBER 1
5 JANUARY 2004
Structural stability of BeH2 at high pressures
P. Vajeeston, P. Ravindran,a) A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
共Received 4 September 2003; accepted 11 November 2003兲
The electronic structure and structural stability of BeH2 are studied using first-principles
density-functional calculation. The calculated structural parameters for ␣ -BeH2 at the equilibrium
volume are in very good agreement with experiments. At higher pressures ␣ -BeH2 successively
undergoes four structural transitions: 共i兲 ␣- to ␤ -BeH2 at 7.07 GPa; 共ii兲 ␤- to ␥ -BeH2 at 51.41 GPa;
共iii兲 ␥- to ␦ -BeH2 at 86.56 GPa; and 共iv兲 ␦- to ⑀ -BeH2 at 97.55 GPa 关an effective two-phase 共␥ and
␦兲 region is found at 73.71– 86.56 GPa兴. Density of states studies reveal that BeH2 remains
insulating up to 100 GPa whereupon anomalous changes are seen in the band-gap region with
increasing pressure. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1637967兴
Metal hydrides are of great scientific and technological
interest in view of their potential applications, e.g., for hydrogen storage, in fuel cells and internal combustion engines,
as electrodes for re-chargeable batteries, and in energyconversion devices. Hydrides for hydrogen storage need to
be able to form hydrides with a high hydrogen-to-metal mass
ratio, but should not be too stable, so that the hydrogen can
easily be released without excessive heating. Beryllium and
magnesium and beryllium-magnesium-based hydrides contain a relatively high fraction of hydrogen by weight, but
need to be heated to at least 250 to 300 °C in order to release
the hydrogen. An improved understanding of the stability of
metal hydrides is a key to rationally investigate and design
potential hydrogen-storage materials.
The alkali–metal and alkaline–earth–metal hydrides
represent series with largely ionic bonding. The highpressure behavior of the alkali–metal monohydrides is expected to parallel that of the alkali–metal halides.1 However,
there is no systematic high-pressure study on the alkaline–
earth–metal hydrides. The incorporation of hydrogen into
metals as a means to enhance superconductivity is an interesting aspect. For example, PdH exhibits superconductivity
with a transition temperature of ⬃10 K whereas superconductivity is absent in Pd.2 Owing to low mass and high electron density, ‘‘metallic hydrogen’’ has been predicted to
show superconductivity with a transition temperature between 140 and 260 K.3 For example, if BeH2 becomes metallic when subjected to high pressures one can entertain the
possibility that its properties could resemble those of metallic hydrogen. This was another motive for the present highpressure study of BeH2 .
BeH2 is commonly considered a covalent hydride with a
postulated polymeric crystal structure made up of H-bridged
chains. However, mainly owing to experimental difficulties
in the synthesis of the material, the structure has long remained unknown.4 Recently, crystalline BeH2 has been synthesized and the structure has been established as bodycentered orthorhombic by synchrotron-radiation-based
powder x-ray diffraction.5 No electronic structure calculation
is available for this particular phase.
To examine the structural stability of BeH2 , we have
a兲
Electronic mail: ponniah.ravindran@kjemi.uio.no
used density-functional theory 共DFT兲6 within the generalized
gradient approximation 共GGA兲,7 as implemented with a
plane-wave basis in the Vienna ab initio simulations package
共VASP兲.8 Results are obtained using projector-augmented
plane-wave 共PAW兲9 potentials provided with the VASP. The
ions are relaxed toward equilibrium until the Hellmann–
Feynman forces are less than 10⫺3 eV/Å. Brillouin zone
integration are performed with a Gaussian broadening of 0.1
eV during all relaxations. For the BeH2 -type structure, we
have performed the calculations with 512 k points in the
whole Brillouin zone and a 600 eV plane-wave cutoff. 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 for all structural variants
studied. The present type of theoretical approach has recently
been successfully applied10,11 to reproduce ambient- and
high-pressure phases computationally.
For our theoretical simulation we have considered 24
different types of structural variants. The involved structure
FIG. 1. Cell volume per formula unit vs free energy for BeH2 in different
possible structural modifications. In order to keep the illustration reasonable
simple only results for the 15 most relevant structure types are shown. The
remaining variants are found at higher energies and are not included in the
illustration. Transition points are marked with arrows, the letters 共a兲–共d兲
correspond to those in Fig. 2.
0003-6951/2004/84(1)/34/3/$22.00
34
© 2004 American Institute of Physics
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Appl. Phys. Lett., Vol. 84, No. 1, 5 January 2004
Vajeeston et al.
35
FIG. 3. Calculated total density of state for ␣-, ␤-, ␥-, ␦-, and ⑀ -BeH2 .
Partial DOSs of H1 and Be1 in ␣ -BeH2 are shown in the inset.
FIG. 2. Calculated 共a兲 pressure vs volume relationship and 共b兲 stability
diagrams for hypothetical BeH2 phases 共difference in Gibbs free energy
⌬G) relative to ␣ -BeH2 . Transition points are marked by arrows, the letters
共a兲–共d兲 correspond to those in Fig. 1.
types are BeH2 , MnF2 , XeF2 , KF2 , NaF2 , SnF2 , LiF2 ,
BeF2 , PdF2 , MgF2 , HfH2 , ␥ -SnF2 , TiO2 (anatase),
TiO2 (brookite), TiO2 (rutile), modified CaF2 , ␣ -PbO2 ,
␦ -MgH2 , AlAu2 , InNi2 , TeAg2 , CaF2 , AuSn2 , and
CaCl2 . 12 Among the 24 structure types considered here, the
experimentally observed BeH2 modification has the lowest
total energy 共see Fig. 1兲 and the calculated structural parameters are also in very good agreement with the observations.
The calculated cell parameters differ by less than 1% from
the experimental values, which again confirm that the present
type of calculation is reliable and accurate. This bodycentered orthorhombic unit cell contains twelve BeH2 formula units and the primitive cell six formula units. Both Be
and H occupy two different types of sites 共see Table I兲. The
structure consists of BeH4 tetrahedra linked by hydrogens at
the corners, and there exists no known analog among other
compounds with tetrahedral building blocks. The packing in
the structure is not especially dense; consequently it is anticipated that it should be possible to produce a form with
higher density. When ␣ -BeH2 is exposed to external pres-
sures it transforms to the denser TiO2 (anatase)-type,
␤ -BeH2 modification. The calculated total energy versus
volume relation in Fig. 1 shows that several pressure-induced
structural transitions occur in BeH2 upon increasing pressure. Since it may be hard to identify the transition points
from Fig. 1, Fig. 2共a兲 gives the calculated pressure–volume
relation, and Fig. 2共b兲 gives the Gibbs free energy difference
relative to ␣ -BeH2 for the pressure-induced structural arrangements under consideration.
The application of pressure 共Fig. 2兲 transforms ␣- to
␤ -BeH2 at 7.07 GPa with a calculated volume discontinuity
at the transition point of ca. 19.2%; ca. 26.4% equilibrium
volume difference at ambient pressure 共the corresponding
energy difference being ⬃0.4 eV). Such a huge pressureinduced volume collapse is rather uncommon among hydrides as well as inorganic compounds in general. However,
large volume collapses 共9 to 20%兲 under pressure are observed for lanthanides and actinides associated with valence
transition or localized-to-delocalized transition of f electrons. In LiAlH4 there has also been established an unexpectedly large 共17%兲 volume collapse at a structural transition
point.13 ␤ -BeH2 remains stable over a wide pressure range
共7.07–51.41 GPa兲. At 51.41 GPa ␤ -BeH2 transforms to the
␥ modification with a CaCl2 -type structure which in turn is
converted into an effective two-phase region with the coexistence of ␥ -BeH2 and TiO2 (rutile)-type ␦ -BeH2 , at 73.71
GPa which is then converted into a single-phase region of
␦ -BeH2 共86.56 to 97.55 GPa兲. Above 97.55 GPa ␦ -BeH2 is
transformed to the PdF2 -type ⑀ modification. In the pressure
range 10 to 75 GPa ␤-, ␥-, and ␦ -BeH2 fall within a narrow
energy range of some 10 meV 共Fig. 1兲 and above 97.55 GPa
␤-, ␥-, ␦-, and ⑀ -BeH2 all fall in a similar narrow energy
range. This closeness in energy suggests that the relative appearance of these modifications will be quite sensitive to,
and easily affected by, other external factors like temperature
and remnant lattice stresses. Several pressure-induced structural modifications are predicted11 for the iso-electronic compound MgH2 共recently also verified by high-pressure
experiments14兲. However, the structural transition sequences
in BeH2 and MgH2 are entirely different,11 and quite different from the findings for the alkali-monohydride series15
where systematics in the pressure-induced phase transitions
has been observed.
The calculated values for the bulk modulus (B 0 ; see
Table I兲 vary between 23.79 and 94.74 GPa for the various
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36
Vajeeston et al.
Appl. Phys. Lett., Vol. 84, No. 1, 5 January 2004
TABLE I. Optimized structural parameters, bulk modulus (B 0 ), pressure derivative of bulk modulus (B ⬘0 ), and calculated energy band gap (E g ) for BeH2 in
different possible structural arrangements 共space group and structure types for the different phases are specified in parentheses兲. Unless otherwise specified
structure data refer to equilibrium.
Phase
␣ -BeH2
(Ibam)
␤ -BeH2 (I4 1 /amd;
TiO2 -anatase兲
␥ -BeH2
( Pnnm; CaCl2 )
␦ -BeH2 ( P4 2 /mnm;
TiO2 -rutile兲
⑀ -BeH2
( P2 1 3; PdF2 )
a
Coordinates
B 0 共GPa兲
B 0⬘
E g 共eV兲
Be1 (4a) : 0, 0, 1/4; Be2 (8 j) : 0.1678, 0.1207, 0 共0.1699, 0.1253, 0兲a
H1 (16k) : 0.0720, 0.1842, 0.1361 共0.0895, 0.1949, 0.1515兲a
H2 (8 j) : 0.3097, 0.2790, 0 共0.3055, 0.2823, 0兲a
Be (4a) : 0, 0, 0
H (8e) : 0, 0, 0.2281
Be (2a) : 0, 0, 0
H (4g) : 0.3054, 0.1964, 0 共0.2989, 0.2030, 0兲b
23.79
4.24
5.51
73.78
3.22
2.39
85.00
3.91
4.07
85.17
3.44
4.57
94.74
3.35
5.55
Cell constants 共Å兲
a⫽8.9823 共9.082兲a
b⫽4.1563 共4.160兲a
c⫽7.6455 共7.707兲a
a⫽3.2178
c⫽6.7895
a⫽3.8427
b⫽3.8334
c⫽2.3003
a⫽3.8400
c⫽2.2986
a⫽3.9715
Be (2a) : 0, 0, 0
H (4 f ) : x, x, 0; x⫽0.3045 共0.2975兲b
Be (4a) : 0, 0, 0; H1 (4a) : x, x, x; x⫽0.3522 共0.3588兲b
H2 (4a) : x, x, x; x⫽0.6477 共0.6412兲b
Experimentally observed values from Ref. 5 at ambient.
Atomic coordinates at the transition point.
b
structural arrangements. Among the five involved structure
variants ␣ -BeH2 shows the smallest B 0 value and the
⑀ -BeH2 the highest 共Table I兲, which appears to be a consequence of the variation in the equilibrium volumes 共monotonically decreasing from 23.79 Å 3 /f.u. for ␣ -BeH2 to
15.66 Å 3 /f.u. for ⑀ -BeH2 ).
The calculated total density of states 共DOS兲 for the different modifications of BeH2 共at the transition pressures for
␤-, ␥-, ␦-, and ⑀ -BeH2 and at the equilibrium volume for the
␣ phase兲 in Fig. 3 shows that all involved phases have a
nonmetallic character with finite band gaps (E g ). From ␣- to
␤ -BeH2 the calculated band gap is reduced from 5.51 to 2.39
eV, whereas from ␤- to ⑀ -BeH2 the estimated band gap is
drastically increased from 2.39 to 5.55 eV 共see Table I兲. In
isoelectronic MgH2 11 the estimated band gap is monotonically reduced on increasing pressure. The variation in the
band gap established for the BeH2 phases certainly is not
common, but has for example been established for WO3 , 16
where the findings have been attributed to the tilting of octahedra in the pressure modifications. The calculated
valence-band width is increased from 6.5 eV in ␣ -BeH2 to
21.2 eV in the ⑀ phase, which may be attributed to the reduction in the bonding interatomic distances upon increasing
pressure. Usually the hydrides of alkali and alkaline–earth
metals have ionic bonding which reflects the low ionization
energy of these metals. For example, LiH is a wide-gap insulator with the H-1s band ca. 5 eV below the Li-2s conduction band.17 In a hypothetical simplified ionic picture for
BeH2 the Be valence electrons would have been transferred
to the H atoms, but the calculated DOS shows that Be-2s
and H-1s states are energetically degenerate in the energy
range ⫺6.25 to 0 eV 共see the inset of Fig. 3兲 and the valence
electron charge density 共not shown兲 confirms a finite electron
distribution between the Be and H atoms, viz. reflecting the
covalent interaction which is also responsible for the stabilization of the low symmetric structure of ␣ -BeH2 . Hence it is
not surprising that BeH2 significantly deviates from the other
alkaline–earth dihydrides. Contrary to the MgH2 phases
which gradually undergo conversion from ionic to more metallic character upon application of pressure, BeH2 changes
from covalent to ionic bonding character upon application of
pressure. This should explain why BeH2 behaves differently
from the rest of the alkaline–earth dihydride series.
In summary, BeH2 becomes unstable upon application of
pressure, and at higher pressures ␣ -BeH2 transforms to ␤-,
␥-, ␦-, and ⑀ -BeH2 . Up to 100 GPa BeH2 remains an insulator, hence the possibility of obtaining high-pressure phases
with superconducting properties is ruled out. There occurs a
huge volume collapse at the ␣-to-␤ phase transition and only
smaller volume changes at the ␤-to-␥, ␥-to-␦, and ␦-to-⑀
transitions. The band gap attains a minimum value at the
␤ -BeH2 phase.
The authors gratefully acknowledge the Research Council of Norway for financial support and for the computer time
at the Norwegian supercomputer facilities and R. Vidya for a
critical reading of the manuscript.
1
St-J. Duclos, Y. K. Vohra, A. L. Ruoff, S. Filipek, and B. Baranowski,
Phys. Rev. B 36, 7664 共1987兲.
2
A. W. Overhauser, Phys. Rev. B 35, 411 共1987兲.
3
N. W. Ashcroft, Phys. Rev. Lett. 21, 1748 共1968兲.
4
D. A. Armstrong, J. Jamieson, and P. G. Perkins, Theor. Chim. Acta 51,
163 共1979兲.
5
G. S. Smith, Q. C. Johnson, D. K. Smith, D. E. Cox, R. L. Snyder, and R.
S. Zhou, Solid State Commun. 67, 491 共1988兲.
6
P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 共1964兲; W. Kohn and
L. J. Sham, Phys. Rev. A 140, 1133 共1965兲.
7
J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 16533 共1996兲.
8
G. Kresse and J. Joubert, Phys. Rev. B 59, 1758 共1999兲.
9
P. E. Blöchl, Phys. Rev. B 50, 17953 共1994兲.
10
P. Vajeeston, P. Ravindran, R. Vidya, A. Kjekshus, and H. Fjellvåg, Appl.
Phys. Lett. 82, 2257 共2003兲.
11
P. Vajeeston, P. Ravindran, A. Kjekshus, and H. Fjellvåg, Phys. Rev. Lett.
89, 175506 共2002兲.
12
Inorganic Crystal Structure Database 共Gmelin Institut, Germany, 2001兲.
13
P. Vajeeston, P. Ravindran, R. Vidya, A. Kjekshus, and H. Fjellvåg, Phys.
Rev. B 共to be published兲.
14
P. Vajeeston, P. Ravindran, H. Fjellvåg, B. C. Hauback, S. Furuseth, and
A. Kjekshus 共unpublished兲.
15
H. D. Hochheimer, K. Strössner, and W. Hönle, J. Less-Common Met.
107, L13 共1985兲.
16
G. A. de Wijs, P. K. de Boer, and R. A. de Groot, Phys. Rev. B 59, 2684
共1999兲.
17
F. E. Pretzel, G. N. Rupert, C. L. Mader, E. K. Storms, G. V. Gritton, and
C. C. Rushing, J. Phys. Chem. Solids 16, 10 共1960兲.
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VIII
PHYSICAL REVIEW B 68, 212101 共2003兲
Huge-pressure-induced volume collapse in LiAlH4 and its implications to hydrogen storage
P. Vajeeston,* P. Ravindran, R. Vidya, H. Fjellvåg, and A. Kjekshus
Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway
共Received 27 August 2003; published 5 December 2003兲
A detailed high-pressure study on LiAlH4 has been carried out using the ab initio projected augmented
plane-wave method. Application of pressure transforms ␣ - to ␤ -LiAlH4 ( ␣ -NaAlH4 -type structure兲 at 2.6 GPa
with a huge volume collapse of 17%. This abnormal behavior is associated with electronic transition from Al-s
to -p states. At 33.8 GPa, a ␤ to ␥ transition is predicted from ␣ -NaAlH4 -type to KGaH4 -type structure. Up
to 40 GPa LiAlH4 remains nonmetallic. The high weight percent of hydrogen, around 22% smaller equilibrium
volume, and drastically different bonding behavior than ␣ -phase indicate that ␤ -LiAlH4 is expected to be a
potential hydrogen storage material.
DOI: 10.1103/PhysRevB.68.212101
PACS number共s兲: 81.05.Je, 71.15.Nc, 62.50.⫹p, 71.20.⫺b
Metal hydrides which can accommodate more than 3
wt % hydrogen have been targeted in the Japanese WE-NET
project MITI.1 The parallel international cooperative project
under IEA Task-12 is set up to develop storage materials
which can store more than 5 wt % hydrogen. Several interstitial metal hydrides operate at around room temperature,
but their reversible hydrogen storage capacity is limited to at
most 2.5 wt %.2 Recent interest is directed toward ternary
aluminum hydrides as potential materials with enhanced
storage capacity 共e.g., LiAlH4 and NaAlH4 with 10.6 and 7.5
wt % theoretical hydrogen content, respectively兲 as solidstate sources for hydrogen cells 共e.g., fuel reservoirs兲 etc.
Hence, LiAlH4 and NaAlH4 could be viable candidates for
practical usage as on-board hydrogen storage materials.
However, a serious problem with these materials is poor kinetics and lacking reversibility with respect to hydrogen
absorption/desorption. Improved understanding of the processes which occur in these hydrogen-containing materials
during uptake and release of hydrogen are of considerable
interest. Recent experimental evidences show that LiAlH4
and NaAlH4 after being subjected to mechano-chemical processing under ambient conditions in the presence of certain
transition-metal catalysts3– 6 rapidly release 7.9 and 5.6 wt %
of H, respectively. This represents nearly four to five times
more stored hydrogen than LaNi5 -based alloys which are
presently used in nickel-based hydride batteries. The detailed
crystal structure of LiAlH4 is known, but a systematic highpressure study has not yet been reported. A theoretical investigation of LiAlH4 assumes importance as high-pressure
x-ray and neutron diffraction studies will experience difficulties in identifying more accurate positions for the hydrogen
atoms. The present study concerns the phase stability and
electronic structure of LiAlH4 using first-principles ab initio
calculations.
LiAlH4 crystallizes in the monoclinic ␣ -LiAlH4 -type
structure with space group P2 1 /c and four formula units per
unit cell.7 Four hydrogen atoms are arranged around aluminum in an almost regular tetrahedral configuration. The
structure consists of 关 AlH4 兴 ⫺ units well separated by Li⫹
ions. The Al-H distances vary between 1.59 and 1.64 Å, the
Li-H separations between 1.83 and 1.97 Å, and the arrangement of the lithium ions gives rise to one short Li-Li distance
of ca. 3.1 Å.
0163-1829/2003/68共21兲/212101共4兲/$20.00
Seven closely related potential structure types have been
considered for the present theoretical modeling: ␣ -LiAlH4
共monoclinic; P2 1 /c), 7 ␣ -NaAlH4 共tetragonal; I4 1 /a), 8,9
␤ -LiBH4 共hexagonal; P6 3 mc), 10 NaGaH4 共orthorhombic;
Cmcm), 11 NaBH4 共cubic, Fm3m), 12 SrMgH4 共orthorhombic; Cmc2 1 ), 13 and KGaH4 共orthorhombic; Pnma). 14
For the total-energy calculation we have used the projected augmented plane-wave 共PAW兲 共Ref. 15兲 implementation of the Vienna ab initio simulation package 共VASP兲.16
The generalized-gradient approximation 共GGA兲 共Ref. 17兲
was used to obtain the accurate exchange and correlation
energy for a particular atomic configuration. The structures
were fully relaxed for all volumes considered in the present
calculations using force as well as stress minimization. Experimentally established structural data were used as input
for the calculations when available. For ␣ -LiAlH4 we have
used 500 k points in the whole Brillouin zone. In order to
avoid ambiguities regarding the free-energy results we have
always used the same energy cutoff and corresponding
k-grid densities for convergence in all calculations. A criterion of at least 0.01 meV/atom was placed on the selfconsistent convergence of the total energy, and the calculations reported here used a plane-wave cutoff of 600 eV.
In agreement with the experimental observations we
found that the lowest energy configuration among the seven
considered possibilities for LiAlH4 is the already established
ambient pressure/temperature ␣ -LiAlH4 -type structure 共Fig.
1兲. The calculated unit-cell volume and atom coordinates fit
very well 共within 1.5%; Table I兲 with the experimental
findings.7
A similar theoretical approach has recently been applied18
successfully to reproduce the ambient-pressure and highpressure phases for MgH2 共and also to predict two further
phases at higher pressures兲. In fact recent high-pressure
experiments19 have reproduced the theoretically predicted
pressure-induced structural transitions in MgH2 . We have
identified two potential high-pressure modifications of
LiAlH4 : At 2.6 GPa ␣ -LiAlH4 共prototype structure兲 transforms to ␤ -LiAlH4 ( ␣ -NaAlH4 type兲 and a subsequent transition from ␤ - to ␥ -LiAlH4 (KGaH4 type兲 is established at
33.8 GPa 共Fig. 2兲. In order to get a clearer picture of the
structural transition points we have displayed 共see the inset
of Fig. 2兲 the Gibbs free-energy difference 共relative to
68 212101-1
©2003 The American Physical Society
PHYSICAL REVIEW B 68, 212101 共2003兲
BRIEF REPORTS
FIG. 1. Calculated cell volume vs free energy for LiAlH4 in
actual and possible structural arrangements 共structure types being
labeled on the illustration兲. Arrows make 共a兲 ␣ → ␤ and 共b兲 ␤ → ␥
transition points.
␣ -LiAlH4 ) for the pertinent crystal structures of LiAlH4 as a
function of pressure. The experimental high-pressure/hightemperature study of Bulychev et al.20 found that the ␣ to ␤
transition occurs at a static pressure of 7 GPa and a temperature of 250–300°C. When the temperature was increased to
500°C, the same study20 reports that a third LiAlH4 modification occurs. Experimental structural data for ␤ - and
␥ -LiAlH4 are not available, but the present calculated findings are included in Table I together with the available experimental parameters for ␣ -LiAlH4 .
FIG. 2. Graphical representation of equation of state data for
LiAlH4 . The stabilities of the ␤- and ␥ -LiAlH4 phases relative to
␣ -LiAlH4 as a function of pressure are shown in the inset. Transition points are marked with arrows and numerical pressure values
共in gigapascal兲 are stated.
At the ␣ to ␤ transition point for LiAlH4 the estimated
difference in cell volume is ca. 17% 共Fig. 2兲. The Raman
scattering measurement on the high-pressure phase also indicates the presence of a large volume collapse at the transition point.21 Comparison on the basis of the equilibrium volumes for ␣ - and ␤ -LiAlH4 shows an even larger volume
difference 共viz. a huge value of 22%兲. For example, in
NaAlH4 the calculated volume difference at the transition
point between the ␣ and ␤ phases is less than 4%.9 Another
TABLE I. Optimized structural parameters, bulk modulus B 0 and its pressure derivative B ⬘0 for LiAlH4 in different structural
arrangements.
Phase
Unit-cell dimensions 共Å兲
Atom coordinates
B 0 共GPa兲
B 0⬘
␣ -LiAlH4
( P2 1 /c)
a⫽4.8535(4.8174) a
b⫽7.8259(7.8020) a
c⫽7.8419(7.8214) a
␤ ⫽111.878° (112.228°) a
Li : .5699, .4652, .8245 共.5603, .4656, .8266兲a
Al : .1381, .2017, .9319 共.1386, .2033, .9302兲a
H1 : .1807, .0986, .7630 共.1826, .0958, .7630兲a
H2 : .3542, .3723, .9777 共.3524, .3713, .9749兲a
H3 : .2361, .0810, .1146 共.2425, .0806, .1148兲a
H4 : .7948, .2633, .8717 共.7994, .2649, .8724兲a
Li : 共0, 1/4, 5/8兲b,c; Al : 共0, 1/4, 1/8兲b,c
H : 共.2527, .4237, .5413兲b 共.2492, .4191, .5429兲c
Li : 共.2428, 1/4, .2467兲b 共.2497, 1/4, .2502兲c
Al : 共.5120, 1/4, .8221兲b 共.5002, 1/4, .7361兲c
H1 : 共.3067, 1/4, .9617兲b 共.2815, 1/4, .9617兲c
H2 : 共.7162, 1/4, .9631兲b 共.7189, 1/4, .9467兲c
H3 : 共.4889, .9833, .2943兲b 共.4998, .9173, .3279兲c
12.95
4.10
25.64
4.35
14.25
4.85
␤ -LiAlH4
( ␣ -NaAlH4 type; I4 1 /a)
␥ -LiAlH4
(KGaH4 type; Pnma)
a
a⫽4.6611 b 共4.7312兲c
c⫽10.5219 b 共10.7161兲c
a⫽6.4667 b 共5.4421兲c
b⫽5.3478 b 共4.4843兲c
c⫽6.5931 b 共5.5225兲c
Experimental value from Ref. 7.
Calculated value at equilibrium.
Calculated value at transition point.
b
c
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PHYSICAL REVIEW B 68, 212101 共2003兲
BRIEF REPORTS
important feature of the ␣ to ␤ LiAlH4 transition is the small
energy difference 共Fig. 1兲 between the involved phases 共only
11.56 meV f.u.⫺1 or 1.154 kJ mol⫺1 , which is indeed much
smaller than that found for other hydrides18 and oxides22兲.
Hence, LiAlH4 significantly deviates from the other isoelectronic compounds in the ABH4 (A⫽Na, K, Rb, or Cs; B
⫽Al or Ga兲 series.23 It should be noted that our findings
support the experimental observation of a high-pressure
phase that has been stabilized at ambient pressure by
quenching.21 The relatively small equilibrium volume of
␤ -LiAlH4 along with its high weight content of hydrogen
imply an increased hydrogen storage capacity and therefore
it would be of interest to explore the possibility of stabilizing
this phase by chemical means, and perhaps also find a way to
improve the kinetics of reversible hydrogen absorption/
desorption because the bonding behavior of the ␤ phase is
drastically different from the ␣ phase. It should be noted that
the ␤ - and ␥ -LiAlH4 have almost the same volume at the ␤
to ␥ transition point.
We have calculated the total energy for 12 different volumes for each of the modifications ␣ , ␤ , and ␥ . By fitting
the total energy as a function of cell volume using the socalled universal equation of state24 the bulk modulus B 0 and
its pressure derivative B 0⬘ are obtained, but no experimental
data for comparison are yet available. Among the three structures identified for LiAlH4 as a function of pressure 共Figs. 1
and 2兲, ␤ -LiAlH4 has almost twice the bulk modulus of the
␣ and ␥ modifications. Comparison of the equilibrium volumes of these modifications shows that ␤ has lower equilibrium volume than the ␣ and ␥ phases and this is the main
reason for its larger bulk modulus.
The density of states 共DOS兲 of ␣ -, ␤ -, and ␥ -LiAlH4 are
shown in Fig. 3. A common feature of these three phases is
their nonmetallic character with finite energy gaps 共4.71,
4.25, and 3.95 eV, respectively兲. Measurements of resistivity
as a function of pressures25 suggested that the boron containing compounds LiBH4 and NaBH4 form new phases at
higher pressures 共indicated by jumps in the resistivity25兲,
whereas LiAlH4 and RbBH4 showed no evidence of
pressure-induced phase transitions 共almost linear pressure vs
resistivity relationships兲. The present identification of two
structural transitions in LiAlH4 therefore could indicate that
lacking resistivity evidence of the transitions reflects the
nonmetallic nature and the similar sized band gaps of the ␣ ,
␤ , and ␥ phases. However, regarding the conductivity of
LiAlH4 two other reports are also found in the literature.
Alder and Christian26 report that the resistivity of LiAlH4
becomes reduced by a factor of some 104 at 5 GPa and
Griggs et al.27 report lacking metallic conduction in this
pressure range. The latter finding complies with the presently
established insulating nature of LiAlH4 up to 40 GPa.
The DOS of ␣ -, ␤ -, and ␥ -LiAlH4 differ noticeably
mainly in the valence band 共VB兲 region. In the total DOS of
the ␣ phase, the VB is split in two separate regions 共region I:
⫺6.2 to ⫺4 eV, region II: ⫺3.5 eV to E F ) with a ca. 0.4 eV
gap between the two regions. Al-s states are mainly found in
region I. The total DOS in region II is contributed by Al-p,
H-s, Li-s, and Li-p states. In general the Al-s and -p states
FIG. 3. Calculated density of states 共DOS兲 for ␣ - 共at equilibrium兲, ␤ - 共at transition pressure; 2.6 GPa兲, and ␥ -LiAlH4 共at transition pressure; 33.8 GPa兲. Fermi levels are set at zero energy and
marked by dotted vertical lines; s states are shaded.
are well separated whereas the Li-s and -p states mainly
appear in region II. The Al-p and H-s states are energetically
degenerate in region II, which clearly facilitates the formation of the hybridization prerequisite for the occurrence of
the covalently bonded 关 AlH4 兴 ⫺ subunits in the crystal structure. When we go from ␣ -LiAlH4 to the ␤ and ␥ phases the
gap in the VB region disappears, which may reflect the increase in the hybridization interaction. The changes in the
DOS for Li are rather insignificant between ␣ - and
␤ -LiAlH4 whereas the DOS for Al are markedly different. In
␤ -LiAlH4 we find more mixing of the s and p states for Al.
The s-to-p electronic transition within the Al atom in the ␤
phase causes the huge volume collapse during the ␣ to ␤
phase transition. Moreover, on going through the ␤ to ␥
transition the tetrahedral environment of Al in the ␤ modification is changed to a strongly deformed octahedral environment in the ␥ modification 共four H atoms at distances of
1.23–1.36 Å and two further H atoms at some 2.25 Å). The
identification of AlH6 -configured units in the ␥ phase is consistent with the measured infrared spectra.20
The electron localization function 共ELF兲 is a powerful
tool to visualize different types of bonding in solids.28 The
value of ELF is limited to the range 0–1. High value of the
ELF corresponds to a low Pauli kinetic energy, as can be
found in covalent bonds or lone electron pairs. The ELFs for
␣ - and ␤ -LiAlH4 displayed in Fig. 4 clearly convey that
212101-3
PHYSICAL REVIEW B 68, 212101 共2003兲
BRIEF REPORTS
FIG. 4. 共Color online兲 Electron localization
function for ␣ - 关left picture; 共100兲 plane兴, ␤ 关middle picture; 共001兲 plane兴, and ␥ -LiAlH4
关right picture; 共110兲 plane兴.
关 AlH4 兴 ⫺ forms distinct covalently bonded units in these
modifications; well separated from other 关 AlH4 兴 ⫺ anions
and Li⫹ cations. The high ELF along the Al-H bonds reflects
its covalent character whereas the almost negligible ELF between 关 AlH4 兴 ⫺ and Li⫹ confirms the ionic bonding. The
ELF analysis accordingly corroborates the traditional chemical intuition of the bonding in ␣ - and ␤ -LiAlH4 . The right
picture of Fig. 4 mirrors the increase from four to six hydrogen atoms in the coordination around Al in ␥ -LiAlH4 and
emphasizes the correspondingly more composite bonding
situation in this modification.
In summary, on application of pressure ␣ -LiAlH4 transforms to ␤ -LiAlH4 at 2.6 GPa and this transition is associated with a 17% volume collapse, apparently originating
*Electronic
address: ponniahv@kjemi.uio.no; URL: http://
folk.uio.no/ponniahv
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from electronic transition of Al-s to -p states. Above 33.8
GPa the ␤ phase transforms to ␥ -LiAlH4 with a negligible
change in volume, but with an increased coordination number of Al from four to six. The electronic density of states
confirms that all these phases have nonmetallic character up
to 40 GPa. The energy difference between ␣ - and ␤ -LiAlH4
is small, the ␣ to ␤ transition pressure is relatively low, the
equilibrium volume for ␤ -LiAlH4 is low 共implying efficient
storage of hydrogen兲, and the relative weight content of hydrogen is high, and hence, the ␤ phase stands out as a promising candidate for hydrogen storage.
The authors gratefully acknowledge the Research Council
of Norway for financial support and for the computer time at
the Norwegian supercomputer facilities.
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