The effect of relaxation on the structural and electronic properties of a terbium superstoichiometric dihydride

Abstract

The electronic structure and equilibrium properties for the cubic rare earth superstoichiometric dihydride TbH2.25 are calculated using the full-potential linearized augmented plane wave method (FP-LAPW) within the density functional theory (DFT) in the generalized gradient approximation (GGA) and local density approximation (LDA) as implemented in the WIEN2k simulation code at 0 K. The structure of TbH2.25 is stabilized by local atomic relaxations in both approximations. The atomic positions under relaxation calculated with the GGA are closer to the ideal atomic positions than those calculated with the LDA. The GGA calculated equilibrium lattice constant is in excellent agreement with the available experimental data. The calculation of the density of states, electronic charge density and energy band structures show that this superstoichiometric dihydride has a metallic character with a mixed covalent and ionic bonding.

Full text

Chinese Journal of Physics 55 (2017) 2157–2164
Contents lists available at ScienceDirect
Chinese Journal of Physics
journal homepage: www.elsevier.com/locate/cjph
The effect of relaxation on the structural and electronic
properties of a terbium superstoichiometric dihydride
Zahia Ayat
a , ∗, Aomar Boukraa
a
, Bahmed Daoudi
a
, Abdelouahab Ouahab
b
a
Univ Ouargla, Fac. Des Mathématiques et des Sciences de la Matière, Lab. Développement des énergies nouvelles et renouvelables dans
les zones arides et sahariennes, Ouargla 30 0 0 0, Algeria
b
Université Mohamed Khider Biskra, Département des Sciences de la Matière, Faculté des Sciences Exactes et des Sciences de la Nature et
de la Vie, 070 0 0 Biskra, Algeria
a r t i c l e i n f o
Article history:
Received 16 March 2017
Revised 3 June 2017
Accepted 29 June 2017
Available online 30 August 2017
Keywo rds:
Rare-earth dihydrides
TbH
2.25
Density functional theory
Ab initio calculations
WIEN2k
Relaxation
a b s t r a c t
The electronic structure and equilibrium properties for the cubic rare earth superstoichio-
metric dihydride TbH
2.25
are calculated using the full-potential linearized augmented plane
wave method (FP-LAPW) within the density functional theory (DFT) in the generalized gra-
dient approximation (GGA) and local density approximation (LDA) as implemented in the
WIEN2k simulation code at 0 K. The structure of TbH
2.25
is stabilized by local atomic relax-
ations in both approximations. The atomic positions under relaxation calculated with the
GGA are closer to the ideal atomic positions than those calculated with the LDA. The GGA
calculated equilibrium lattice constant is in excellent agreement with the available experi-
mental data. The calculation of the density of states, electronic charge density and energy
band structures show that this superstoichiometric dihydride has a metallic character with
a mixed covalent and ionic bonding.
©2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V.
All rights reserved.
1. Introduction
The depletion of fossil energies has encouraged the search for new energy sources and energy vectors. Hydrogen as a
handy clean energy carrier can be used in a variety of applications [1] , which may be a viable solution to the problem of
possible future energy shortages [2] . However, one of the major problems encountered in this case is related to the storage
of hydrogen. This inflammable gas can be stored in gaseous, liquid or metal hydride forms. The latter way to store hydrogen
is the safest method and presents several advantages [3] , mainly the volume requirements for storing hydrogen is much
smaller and it has a potential to work near ambient pressures and temperatures. One example of metal hydrides, the rare-
earth hydrides, have received much attention due to their potential application for hydrogen storage technology [4,5] ; they
have an ability to absorb and store hydrogen under moderate conditions of temperature and pressure [6] . Since 1950, the
rare earth hydrides have been the object of many investigations due to their interesting properties [7-11] . Comprehensive
reviews have been given by Vajda [12,13] and Schöllhammer [14] . The rare earth metal reacts directly with hydrogen to form
RH
y hydrides, which are normally nonstoichiometric [7] . In this respect the character of the bulk heavy rare-earth could be
changed from a metallic hexagonal close-packed (HCP) (in most cases) solid solution, or α-phase (y ≈0.1), to an even more
metallic CaF
2
-type dihydride in a β-phase (y ≈2), to an insulating or semiconducting HoD
3
-type trihydride in a γ-phase (y
≈3) [15,16] .
∗Corresponding author.
E-mail address: zahia07 ayat@yahoo .fr (Z. Ayat).
http://dx.doi.org/10.1016/j.cjph.2017.06.017
0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

2158 Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–216 4
In rare earth superstoichiometric dihydride systems RH
2 + x
, it may be observed that the repulsion between the x octa-
hedral hydrogen (H
oct
) atoms may lead to their eventual ordering. For low enough temperatures and high enough concen-
trations of excess hydrogen ( x ≥0.10), the order appeared to be long range, whereas the arrangement appeared to be short
range ordered at small x values ( x < 0.10) [17] . Khachaturyan [18] has predicted theoretically two possible superstructures
for the RH
2.25
composition: the first one is tetragonal and doubled along the c axis and the second is cubic. The latter
structure was used to study CeH
2.25
[19] , PuH
2.25
[20] and GdH
2.25
[21] . However, none of these first-principles theoretical
works had included the effect of local atomic relaxation inside the cell, an effect which was included in the ab initio works
of Jorge Garcés et al. to study YH
x in 2005 [22] , more α-RH systems (R = Sc , Y, Ho, Er, Tm, and Lu) in 2009 [23] and HCP
M-H systems (M = Sc , Y, Ti, Zr) in 2010 [24] .
In the case of the superstoichiometric TbH
2.25
dihydride, which is the subject of the present work, no first principles
study has been carried out to investigate its electronic structure for the hydrogen sublattice symmetry (in the second super-
structure of Khachaturyan), even though the experimental lattice parameters were determined by X-ray diffractometry (XRD)
on TbH
2 + x
(x < 1) [25] together with GdH
2 + x
, DyH
2 + x
and YH
2 + x
materials, but without suggesting any type of structure.
It is known that the density-functional theory (DFT) [26,27] is not only used to understand the observed behavior of
condensed matter, but increasingly more often to predict properties of compounds that have not yet been determined ex-
perimentally. Consequently, it is expected that these predictions will be used to plan future experiments in a rational way
or to analyze recent measurements.
Therefore, in this paper, we shall present the complete results of a detailed electronic structure calculation of the super-
stoichiometric compound TbH
2.25
, based on ab initio calculations at 0 K within density functional theory (DFT). In addition,
we shall look for the role of local atomic relaxation inside the unit cell using both the GGA and LDA approximations.
2. Computational method
Density functional theory calculations were performed using the WIEN2k package [28] with the full potential linearised
augmented plane wave (FP-LAPW) method. The generalized-gradient approximation of Perdew, Burke, and Ernzerhof (GGA
96) [29] and the local density approximation (LDA) [30] were adopted to describe the exchange and correlation effects. The
cut-off energy, which defines the separation energy between the core and valence states, was taken to be −8 Ry, while
the Tb ( 5s
2
5p
6
5d
1
6s
2
) and H ( 1s
1
) orbitals were treated as valence states. We did not treat the f orbitals of Tb as valence
electrons but as core electrons because, in rare earths, the 4f electrons, being very close to the core, are expected to be
chemically inert, i.e. they cannot hybridize with the other s , p , and d valence electrons anymore and are perfectly local-
ized [31] . Self consistency is obtained using 10 0 0 k-points in the irreducible Brillouin zone (IBZ), without considering the
spin polarization. All atoms were fully relaxed until the forces were less than 0.01 eV/
˚
A. The self-consistent calculations are
considered to be converged only when the total energy of the system changes by less than 10
−3 Ry.
3. Results and discussion
3.1. Equilibrium properties
The conventional unit cell for the superstoichiometric dihydride TbH
2.25
with the P m
¯
3 m space group ( No. 221 ) is shown
in Fig. 1 (by using Xcrysden [32] ).
Table 1 summarizes the results for TbH
2.25
obtained before (unrelaxed state) and after (relaxed state) the geometrical
optimization of the internal variables with both the GGA and LDA approximations. We can see that the atomic positions
after relaxation calculated with the GGA are closer to the ideal atomic positions than those calculated with the LDA.
The total energy versus volume is fitted by the non-linear Murnaghan equation of state [33] . Under relaxation, the energy
vs. volume curves of TbH
2.25
for both the LDA and GGA approximations are shown in Fig. 2 ( a ) and 2( b ), respectively, whereas
those for the unrelaxed states are plotted in Fig. 2 ( c ) and 2( d ). From this fit, we can obtain the equilibrium lattice parameter
( a
0
), the bulk modulus ( B
0
), its first order pressure derivative ( B
0
’ ), and the total energy ( E
0
). The calculated structural
parameters of TbH
2.25
before and after the relaxation are reported in Table 2 together with other available data. It can be
seen that this relaxation lowers the total energy as expected. To the best of our knowledge, the experimental or theoretical
bulk moduli of this material have not been reported, and there is no direct ab initio theoretical information available for
TbH
2.25
related to the effects of interstitial H atoms on their local atomic environment. Our calculated values can thereby
be considered as a prediction for future investigations.
We find that the structure of TbH
2.25
is stabilized by local atomic relaxations in both approximations, in agreement with
the previous first-principles computational work in other hydrides [22-24] . Thus, the energy gain (relaxed minus unrelaxed
minimum energy) after internal coordinate relaxation is 0.964 meV/cell within the GGA and 2.673 meV/cell within the LDA
approximation.
It is easily seen from Table 2 that the equilibrium lattice parameter after relaxation is in better agreement with the ex-
perimental value in both approximations, but the GGA equilibrium lattice parameter a
0
value is closest to the experimental
value and is smaller than that for the calculated lattice parameters of TbH
2
and GdH
2.25
. This accords with the experimental
observations reported by Ref. [12] , as we know that a
0
decreases with increasing octahedral hydrogen content x [38-41] and
the rare earth atomic number, respectively. As a general rule, H
oct atom addition leads to lattice contraction whereas H
tet

Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–2164 2159
Fig. 1. The compound crystallizes in the CaF
2
fluorite type structure: the large spheres represent rare earth atoms and the small spheres hydroge n atoms
occupying tetrahedral sites and the central octahedral site.
Tabl e 1
Unrelaxe d and relaxed positions of equivalent atoms for the TbH
2.25 structure (space group) in units of lattice parameters ( a, b, c ) for the ( x, y, z )
coordinates, respectively.
Unrelaxe d Relaxed
x y z x y Z
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
Tb
1 0.00 0.00 0.00 0.0 0 0.00 0.00 0.00 0.00 0.0 0 0.00 0.00 0.00
Tb
2 0.50 0.50 0.50 0.50 0.00 0.00 0.50 0.50 0.50 0.50 0.00 0.0 0
0.50 0.50 0.00 0.00 0.50 0.50 0.50 0.50 0.00 0.0 0 0.50 0.50
0.00 0.00 0.50 0.50 0.50 0.50 0.00 0.0 0 0.50 0.50 0.50 0.50
H
3 0.25 0.25 0.25 0.25 0.25 0.25 0.24 4 45753 0.24925265 0.24 4 45769 0.24925265 0.24 4 45769 0.24925265
0.75 0.75 0.25 0.25 0.25 0.25 0.75554247 0.7507435 0.24 4 45769 0.24925265 0.24 4 45769 0.24925265
0.75 0.75 0.75 0.75 0.25 0.25 0.75554247 0.7507435 0.75554231 0.7507435 0.24 4 45769 0.24925265
0.25 0.25 0.75 0.75 0.25 0.25 0.24 4 45769 0.24925265 0.75554247 0.7507435 0.24 4 45769 0.24925265
0.25 0.25 0.75 0.75 0.75 0.75 0.24 4 45769 0.24925265 0.75554247 0.7507435 0.75554231 0.7507435
0.75 0.75 0.75 0.75 0.75 0.75 0.75554247 0.7507435 0.75523367 0.7507435 0.75554231 0.7507435
0.25 0.25 0.25 0.25 0.75 0.75 0.24 4 45769 0.24925265 0.24 4 45769 0.24925265 0.75554247 0.7507435
0.75 0.75 0.25 0.25 0.75 0.75 0.75554247 0.7507435 0.24 4 45769 0.24925265 0.75554231 0.7507435
H
4 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
atoms lead to lattice expansion. The LDA value is clearly smaller than the experimental one, indicating the occurrence of
over-binding in this latter method. This shows that the GGA is the most reliable for the optimized lattice constants.
Conversely, for the bulk modulus in both the relaxed and unrelaxed states, it is the GGA value which is lower than that
of the LDA as a result of the over-binding characteristic of the LDA. Furthermore, in the GGA, the bulk modulus value of
TbH
2.25
is larger than that of TbH
2
; this behaviour is similar to that found experimentally in the ErH
x system (as shown
in Table 2 ) [37] and theoretically by Refs. [14,20,21] , which indicates a cell volume contraction with increasing hydrogen
content.
Hence in both the relaxed and unrelaxed states, the GGA overestimates the lattice parameter whereas it underestimates
the bulk modulus ( B
0
) in comparison with the LDA, a feature also observed in several similar systems in other simulation
works [42] . It can be concluded that the LDA, with or without relaxation, may incorrectly capture the electronic structure
of cubic TbH
2.25
.
3.2. Electronic properties
From the last section we find that with both approximations the relaxed structure of TbH
2.25
is more stable and the
relaxed equilibrium lattice parameter agrees better with the experimental value. For this reason, all figures presented in this
section are for the relaxed state.

2160 Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–216 4
Fig. 2. Calculated total energy curves for TbH
2.25
as a function of cell volume in the LDA approximation: (a) relaxed state, (c) unrelaxed state. In the GGA
approximation: (b) relaxed state, (d) unrelaxed state.
The calculated electronic band structures at the equilibrium lattice constant for different high-symmetry points in the
Brillouin zone and the total density of states DOS (measured in states per electron-Volt) of TbH
2.25
in GGA and LDA at 0 K
are shown in Figs. 3 and 4 , respectively, where the line at zero eV indicates the Fermi energy.
In Figs. 3 and 4 , it is clear that TbH
2.25
possesses a metallic ground-state, because several bands cross the Fermi level
( E
F
), in agreement with the electrical resistivity measurement interpretations [12] . In both approximations the energy band
structures are qualitatively similar. Indeed, the crossings of bands with the Fermi level are nearly the same, where in the
relaxed state the values of the Fermi energy are 0.56402 Ry in the GGA and a higher 0.59989 Ry in the LDA (as seen
in Table 3 ). Another significant feature of the band structures in the two approximations is the different positions of the
valence bands (at ), where, in the LDA, these shift towards higher energies at the top of the valence band, and towards
lower energies at the bottom of the valence band. All of these indicate a slight increase in the bandwidth, as a consequence
of the reduced lattice parameter. We now turn our attention to the calculated total density of states of TbH
2.25
which has
similar features in both the GGA and LDA (see Figs. 3 and 4 respectively), especially at the Fermi level. However, these
figures show small but non-negligible differences as the peaks in the GGA are fairly sharper and narrower than those of the
LDA, and the total DOS in the LDA moves a little towards lower energies compared to the GGA. Therefore, it is concluded
that the width of the valence states in the LDA is increased due to the lattice contraction. The present study shows that
the density of states on the Fermi level N ( E
F
) in both the relaxed and unrelaxed states is not negligible, as seen in Table 3 ,
and is smaller in the LDA than in the GGA, because the latter causes an under-binding effect of the crystal, yet the crystal
structure is more compact in the LDA. Also from Table 3 , it can be remarked that when the equilibrium lattice parameter
decreases, the Fermi energy increases in both approximations.
In order to explain the chemical bonding in TbH
2.25
, we have calculated the total and partial density of states (PDOS), as
shown in Fig. 5 .
From Fig. 5 , the conduction band is mainly dominated by unoccupied s , p and d states from terbium and s from the
hydrogen in octahedral sites (H
oct
). The Fermi level is completely dominated by the D -e
g states of terbium. The upper
valence states situated between −1.95949 eV and E
F
in the GGA and −1.93051 eV and E
F
in the LDA are dominated by the

Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–2164 2161
Tabl e 2
Calculated equilibrium lattice constant a
0
(in ˚
A), bulk modulus B
0
(in GPa), its first order pres-
sure derivative B
0
’, and total energy (Ry), for the GGA and LDA compared to other available
data.
Method a
0 B
0 B
0
’ E
0
TbH
2.25 GGA Unrelaxed 5.2558 61.631 3.526 −93,762.408612
Relaxed 5.2548 61.7312 4.8942 −93,762.409576
LDA Unrelaxed 5.1032 72.5047 2.6947 −93,700.517516
Relaxed 5.1056 72.3416 3.5847 −93,700.520189
TbH
2.24 Exp. 5.2308
a
TbH
2 GGA 5.2993
b 59.9989
b 1.9 613
b
GdH
2.25 GGA Unrelaxed 5.299
c 62.4385
c 3.0152
c
LDA Unrelaxed 5.143
c 80.3544
c 3.1906
c
GdH
2.26 (6) Exp. 5.284
d
GdH
2 GGA 5.326
e 53.1873
e 4.0861
e
ErH
1.9 5 Exp. 67 ±3
f 9 fixed
f
ErH
2.091 Exp. 73 ±4
f 8 fixed
f
a Ref. [12] -Expt.;
b Ref. [34] ;
c Ref. [21] ;
d Ref. [35] -Expt.;
e Ref. [36] ;
f Ref. [37] -Expt.
Fig. 3. Density of states (right panel) and electronic band structure along the high-symmetry directions (left panel) of TbH
2.25
in the GGA, the Fermi energy
being at 0 eV.
Tabl e 3
Fermi energy (Ry) and density of states at the Fermi level (in states/Ry) for TbH
2.25
in the two approaches GGA and LDA.
Fermi energy N(E
F
) N
s
(E
F
) N
p
(E
F
) N
d
(E
F
) N
d-eg
(E
F
) N
d-t2g
(E
F
) N
Htet-s
(E
F
) N
Hoct-s
(E
F
)
GGA Unrelaxed 0.56297 39.69 0.00 0.17 2.14 1.9 3 0.21 0.08 0.16
Relaxed 0.56402 39.80 0.00 0.17 2.13 1.9 2 0.21 0.08 0.16
LDA Unrelaxed 0.60011 37.21 0.00 0.20 2.18 1.9 5 0.23 0.07 0.14
Relaxed 0.59989 37.87 0.00 0.19 2.13 1.91 0.22 0.07 0.13

2162 Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–216 4
Fig. 4. Density of states (right panel) and electronic band structure along the high-symmetry directions (left panel) of TbH
2.25
in the LDA, the Fermi energy
being at 0 eV.
Tb D -e
g states. Whilst it can be seen that in low energy levels ranging between −8,70,792 eV and −1.95949 eV in the GGA
and between −8,89,663 and −1.93051 eV in the LDA, TbH
2.25
dihydride exhibits hybridization mainly between the Tb d -t
2g
states and both H 1s orbitals, implying directional (covalent) bonding in Tb D -t
2g
-H
tet (bonding distance is 2.2672 ˚
A in the
GGA and 2.1653 ˚
A in the LDA) and a degree of ionic character in Tb D -t
2g
- H
oct (bonding distance is 2.6258 ˚
A in the GGA
and 2.5537 ˚
A in the LDA).
This can be clearly understood from the electronic charge density contours along the (110) plane, as shown in Fig. 6 .
The charge-density was presented only for the GGA method, because it is similar to that of the LDA method with a small
difference. In Fig. 6 , it is clear that appreciable charge density exists in the outer regions of Tb and H
tet atoms with a slight
deformation in the direction of these nearest-neighboring atoms. This feature confirms that the bonding between Tb and
H
tet atoms is certainly covalent, a fact confirmed by the hybridization analysis. At the same time, it is clear that very little
electronic charge is shared between Tb and H
oct
, where most of the valence electrons of H
oct
are tightly bound around their
atoms, and this implies that the bond has some ionic character (the electron density is much weaker in the middle of the
bond). Therefore, the Tb-H bonds in TbH
2.25
have a mixed (covalent-ionic) character, as is found in several metal hydrides
[19-21,43-45] . Another point of interest is the existence of a little charge in the interstitial regions away from the bonds,
which gives a metallic character to this compound.
4. Conclusion
In the present study, ab-initio calculations were performed to investigate the structural and electronic properties for the
cubic superstoichiometric TbH
2.25
dihydride using the FP-LAPW method in the local density approximation (LDA); and the
generalized gradient approximation (GGA) for the exchange correlation as implemented in the WIEN2k code at 0 K. The
atomic positions under relaxation calculated with the GGA are closer to the center of the ideal atomic positions than those
calculated with the LDA. The structure of TbH
2.25
is stabilized by local atomic relaxations. The equilibrium lattice parameter
under relaxation is in better agreement with the experimental value in both approximations, the GGA value being the best.
The bulk modulus is determined by fitting the energy vs. cell volume curve to the Murnaghan equation of state. Under
relaxation, the GGA value is smaller by 14 .67% than that in the LDA. The electronic band structure, the density of states
and electronic charge density confirm the metallic character of TbH
2.25
. The total density of states in both approximations
shows important differences in the vicinity of the valence band, where, in both the relaxed and unrelaxed states, the GGA
presents a tendency for under-binding in this material. Atomic relaxation shows that the Fermi energy changes in all cases

Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–2164 2163
Fig. 5. The calculated total and partial density of states for TbH
2.25
in the GGA (right panel) and in the LDA (left panel), the Fermi energy being at 0 eV.
Fig. 6. Calculated valence-electron-charge density contour (in electrons per ˚
A
3
) of TbH
2.25
in the (110) plane.
are inversely proportional to those of the equilibrium lattice parameter. The electronic charge density has been plotted and
shows a covalent character for the Tb-H
tet bonds and an ionic character for the Tb-H
oct bonds.
References
[1] A. Izanlou , M.K. Aydinol , An ab initio study of dissociation adsoption of H
2
on FeTi surfaces, Int. J. Hydrogen Energy 35 (2010) 16 81 .
[2] L. Schlapbach , A. Züttel , Hydrogen-storage materials for mobile applications, Nature 414 (2001) 353 .
[3] M. Bououdina , D. Grant , G. Wa lker ,Review on hydrogen absorbing materials-structure, microstructure, and thermodynamic properties, Int. J. Hydrogen
Energy
31 (2006) 177 .

2164 Z. Ayat et al. / Chinese Journal of Physics 55 (2017) 2157–216 4
[4] J.N. Huiberts , R. Griessen , J.H. Rector , R.J. Wijngaarden , J.P. Dekker , D.G. de Groot , N.J. Koeman , Yttrium and lanthanum hydride films with switchable
optical properties, Nature 380 (1996) 231–234 .
[5] P.J . Kelly , J.P. Dekker , R. Stumpf , Theoretical prediction of the structure of insulating YH
3
, Phys. Rev. Lett. 78 (1997) 1315 .
[6] M. Gupta , J.P. Burger , Electronic structure of rare-earth hydrides: LaH
2
and LaH
3
, Phys. Rev. B 22 (12) (1980) 6074 .
[7] W.G. Bos , K.H. Gayer , The rare earth hydrides, J. Nucl. Mat. 18 (1966) 1 .
[8] P. Tessier , Hydrogen storage in metastable Fe-Ti Thesis, McGill University, Montreal, Canada, 1995 .
[9] E. Shalaan , H. Schmitt ,
K-H. Ehses , On the optical properties of gadolinium hydride systems, Thin Solid Films 489 (2005) 330 .
[10] I. Tereshina , S. Nikitin , W. Suski , J. Stepien-Damm , W. Iwasieczko , H. Drulis , et al. , Structural and magnetic properties of Dy
2
Fe
17
H
x
(x = 0 and 3) single
crystals, J. Alloys Compd. 404–406 (2005) 172 .
[11] I. Aruna , B.R. Mehta , L.K. Malhotra , S.M. Shivaprasad , Stability and hydrogenation of “bare” gadolinium nanoparticles, Adv. Funct. Mater 15 (1 ) (2005)
131 .
[12] P. Vajda , Hydrogen in rare earth metals, in: K.A.
Gschneidner, L. Eyring (Eds.), Handbook On the Physics and Chemistry of Rare Earths, vol. 20, North
Holland, Amsterdam, 1995, pp. 207–291 .
[13] P. Vajda , Rare earth (and actinide) dihydrides: structural, electronic and magnetic properties, in: P. Vajda, J.-M. Costantini (Eds.), Properties of Fluorite
Structu re Materials, 2013, p. 31. chapter 2 .
[14] G. Schöllhammer , F. Karsai , P. Herzig , Fluorite-type rare earth hydrides studied from first principles, in: P. Vajda, J.-M. Costantini (Eds.), Properties of
Fluorite Structure Materials, 2013, p. 1. chapter 1 .
[15] P. Vajda , Hydrogen ordering and magnetic phenomena in (mostly rare-earth based)
metal-hydrogen systems: highlights and open problems, Phys. B
289-290 (20 0 0) 435 .
[16] B. Stalinski , Electric conductivity of lanthanum hydride LaH
2
-LaH
3
, Bull. Acad. Pol. Sci. Class III 5 (1957) 10 01 .
[17] T.J. Udovic , J.J. Rush , I.S. Anderson , Neutron spectroscopic comparison of
β;-phase rare earth hydrides, J. Alloys Compd. 231 (1995) 138 .
[18] A.G. Khachaturyan , The problem of symmetry in statistical thermodynamics of substitution and interstitial ordered solid solutions, Phys. Status Solidi
(b) 60 (1973) 9 .
[19] B.Y. Ao , X.L. Wang , P. Shi , P.H. Chen , X.Q. Ye , X.C. Lai , J.J. Ai , T. Gao , Lattice contraction of cerium hydrides from first-principles LDA + U calculations,
Int. J. Hydrogen Energy 33 (2012) 510 8 .
[20] B.Y. Ao , P. Shi , Yon g Guo , T. Gao , The abnormal lattice contraction of plutonium hydrides
studied by first-principles calculations, Chin. Phys. B 3 (22)
(2013) 037103 .
[21] Z. Ayat , A. Boukraa , B. Daoudi , First-principles study of the electronic structure of a superstoichiometric rare earth dihydride GdH
2.25
, J. Rare Earths 34
(1) (2016) 66 .
[22] J. Garcés , J.L. Gervasoni , P. Vajda , Relaxation of internal stress field and hydrogen ordering on YH
x
, J. Alloys Compd. 404-406 (2005) 126 .
[23] J. Garcés , R. González , P. Vajda , First-principles study of H ordering in
α-RH systems (R = Sc , Y, Ho, Er, Tm, and Lu), Phys. Rev. B 79 (2009) 054113 .
[24] J. Garcés , P. Vajda , H ordering in hcp M-H systems (M = Sc , Y; Ti, Zr), Int. J. Hydrogen Energy 35 (2010) 6025 .
[25] M. Chiheb , J.N. Daou , P. Vajda , Lattice parameters of the β-phase rare-earth hydrides RH
2 + x
(R = Y, Gd, Tb, Dy), Z. Phys. Chem. 179 (1993) 271 .
[26] P. Hohenberg , W. Kohn , In homogeneous electron gas, Phys. Rev. 136 (1964) B864 .
[27] W. Kohn , L.J. Sham , Self-consistent equations including exchange and correlation effects, Phys. Rev. A 140 (4A) (1965)
A1133 .
[28] P. Blaha , K. Schwarz , G. Madsen , D. Kvasnicka , J. Luitz , Wien2k, an Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Proper-
ties, Techn. Universität, Vienna, Austria, 2001 ISBN 3-9501031-1-2 .
[29] J.P. Perdew , K. Burke , M. Ernzerhof , Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865 .
[30] J.P. Perdew , Y. Wang , Accurate and simple analytic representation of the electro-gas correlation energy, Phys. Rev. 45 (1992) 13 24 4 .
[31] P. Vajeeston , R. Vidya , P. Ravindran , H. Fjellvåg , A. Kjekshus
, A. Skjecltorp , Electronic structure, phase stability and chemical bonding in Th
2
Al and
Th
2
AlH
4
, Phys. Rev. B 65 (2002) 075101 .
[32] A. Kokalj, XCrysDen - a new program for displaying crystalline structures and electron densities, Comp. Mater. Sci. 28 (2003) 155. Code available
from http://www.xcrysden.org .
[33] F.D. Murnaghan , The compressibility of media under extreme pressures, Proc. Na tl. Acad. Sci. 30
(9) (1944) 244 .
[34] Z. Ayat, A. Boukraa. unpublished results.
[35] M. Ellner , H. Reule , E.J. Mittemeijer , Unit cell parameters and densities of the gadolinium dihydride GdH
2 + x
, J. Alloys and Compd. 279 (1998) 179 .
[36] Z. Ayat , B. Daoudi , A. Ouahab , A. Boukraa , Electronic structure of the rare earth dihydride GdH
2
, Metallofiz. Noveishie Tekhnol. 37 (5) (2015) 593 .
[37] T. Palasyuk , M. Tkacz , P. Vajda , High pressure studies of the erbium-hydrogen system, Solid State Commun. 135 (2005) 226 .
[38] Y. Wang , M.Y. Chou , Energetics and lattice contraction of
β;-phase YH
2 + x
, Phys. Rev. B 49 (1994) 10731 .
[39] C.E. Holley , R.N.R. Mulford , F.H . Ellinger , W.C. Koehler , W.H . Zachariasen , The crystal structure of some rare earth hydrides, J. Phys. Chem. 59 (1955)
1226 .
[40] H. Müller , P. Knappe , O. Greis , Lattice
parameters of hydrides and deuterides of La, Pr, and Nd, Z. Phys. Chemie Neue Folge 114 (1979) 45 .
[41] P. Knappe , H. Müller , Phase relationships in the cerium-hydrogen and cerium-deuterium systems, Z. Anorg. Allg. Chem. 487 (1982) 63 .
[42] J. Ai , T. Liu , T. Gao , B. Ao , Firs principles study of electronic structure and metal-insulator transition of plutonium dihydride and trihydride, Comput.
Mater. Sci. 51 (2012) 127 .
[43] J.G. Roquefer , S.F. Matar , J.L. Bobet , Stability of the hydrides REMgNi
4
H
4
(RE = Y, Gd) from first principles, Int. J. Hydrogen Energy 35 (2010) 7858 .
[44] P. Vajeeston , P. Ravindran , H. Fjellvåg , A. Kjekshus , Search for metal hydrides with short hydrogen- hydroge n separation: Ab initio calculations, Phys.
Rev. B 70 (2004) 014107 .
[45] P. Vajeeston , P.
Ravindran , R. Vidya , A. Kjekshus , H. Fjellvåg , V.A. Yar tys ,Short hydrogen-hydrogen separation in RNiInH
1.333
(R = La, Ce, Nd), Phys. Rev.
B 67 (2003) 014101 .