Structural and Electronic Properties of the Hydrogenated ZrCr2 Laves

Feb 17, 2010 - Institute of Nuclear Technology and Radiation Protection. ... Crystal-orbital overlap population studies, performed for the C15 structu...
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J. Phys. Chem. C 2010, 114, 4221–4227

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Structural and Electronic Properties of the Hydrogenated ZrCr2 Laves Phases H. J. P. van Midden,*,† A. Prodan,† E. Zupanicˇ,† R. Zˇitko,† S. S. Makridis,‡ and A. K. Stubos§ Jozˇef Stefan Institute, JamoVa 39, SI-1000 Ljubljana, SloVenia, Department of Mechanical Engineering, UniVersity of Western Macedonia, Kozani, GR-50100, Greece, and Institute of Nuclear Technology and Radiation Protection, NCSR Demokritos, Athens, GR-15310 Greece ReceiVed: October 23, 2009; ReVised Manuscript ReceiVed: January 26, 2010

A series of hydrogenated cubic C15 and hexagonal C14 ZrCr2 Laves phases were studied by means of semiempirical extended Hu¨ckel tight-binding, ab initio density functional theory methods and maximally localized Wannier functions, with a goal to find the most energetically favorable positions of interstitial H atoms in the host unit cells. We consider situations with one or two H atoms per primitive cell. Crystalorbital overlap population studies, performed for the C15 structure, show repulsion between two hydrogen atoms in close proximity. This is in accord with the ab initio calculations, performed for the hydrogenated C14 and C15 structures, which clearly favor two separated hydrogen atoms instead of the formation of moleculelike pairs in five-coordinated trigonal-bipyramidal environments. 1. Introduction There is a growing interest lately in the hydrogen fuel alternative1 and particularly in the hydrogen storage2,3 and its optimization.4 Structural considerations and studies of the structure-to-properties relationship in the case of metal hydrides,5–9 especially in the case of the Laves AB2,10,11 Haucke AB5,12–15 and some ternary intermetallic phases,16 may lead to a better understanding of the hydrogenation processes and their dynamics. These works deal mainly with electronic structures, because they are believed to play a dominant role in these processes. A better understanding of the electronic structure in the vicinity of the Fermi level and its modification by the presence of hydrogen may eventually lead to more efficient devices with higher storage capacities, better energy-to-weight ratios, and lower working temperatures. It was recently suggested2 that the interstitial H may partially lose its electron, leaving behind a small mobile proton. Another possibility may be that the hydrogen enters the host structure as mobile weakly bonded molecules by opening up paths along double face-sharing tetrahedral interstices that are able to accommodate H2 molecules as units. In this case, the molecule, if energetically favorable, may diffuse stepwise, displacing during each step only one of its two atoms. To distinguish between the possible scenarios, it is necessary to study the occupancy of the hydrogen levels (i.e., the degree of the electron loss) and the hybridization (bonding) between the hydrogen atoms and the host-lattice atoms. Two theoretical studies were recently published on the C15 hydrogenated structures.17,18 The first deals with a series of ZrX2 (X ) V, Cr, Mn, Fe, Co, Ni) compounds, while the second is devoted to TiCr2. According to these ab initio calculations, hydrogen occupies in ZrCr2 preferably the largest 2A2B interstices, while other possible intersticies seem to be preferred in other members of the series and must be involved in the hydrogen diffusion process in ZrCr2 as well. * To whom correspondence should be addressed. [email protected]. † Jozˇef Stefan Institute. ‡ University of Western Macedonia. § Institute of Nuclear Technology and Radiation Protection.

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There are two criteria given in literature that are to be taken into account. The first, “Westlake criterion”, requires that a sphere occupying a tetrahedral interstitial site should have a radius of at least 0.037 nm.19 According to the second, “Switendick criterion”, two hydrogen atoms occupying tetrahedral sites should be at least 0.21 nm apart.20–22 However, these empirical rules can under certain circumstances be violated.23 It was further shown that the total energy of metal-hydrogen clusters as a function of the H-H separation reveals two minima, the first for H-H separations, which correspond to the bond lengths of the H2 molecule and the second for the stable metal-hydrogen spacing.24 Because the distances between the centers of the face-sharing metal tetrahedra in ZrCr2 are comparable to the intramolecular H-H distance, formation of H2-like pairs cannot be a priori excluded. Our goal is to find the most favorable hydrogen environments in the AB2 metal hydrides. In the present work we evaluate the hydrogenated structures of the cubic C15 and the hexagonal C14 structural types on the basis of structural considerations and ab initio energy calculations. In particular, an ab initio calculation is applied to compare the stability of a series of cubic and hexagonal AB2 structures with one or two hydrogen atoms per unit cell, which occupy tetrahedral interstices either in close proximity or sufficiently separated to avoid their interaction. We believe that such calculations can help to understand and possibly predict reaction pathways during hydrogen adsorption and desorption. In the present work we first describe the electronic properties of the hydrogenated Laves AB2 structures by means of a tightbinding method. Because individual H atoms occupy similar environments in both cubic C15 and hexagonal C14 host structures, the electronic properties were calculated for two hydrogenated C14 structures only. Next, the interaction between two H atoms as a function of their positions in the host structures is studied by means of ab initio calculations. For comparison reasons with some earlier results three hydrogenated C15 structures are considered first. These calculations are followed by a detailed analysis of a variety of combinations of single H atoms and their pairs in the hexagonal C14 unit cell. These ab initio density functional theory methods are expected to reveal

10.1021/jp9101288  2010 American Chemical Society Published on Web 02/17/2010

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TABLE 1: Positions of the A (Zr) and B (Cr) Atoms with all Possible Tetrahedral Interstitial Positions, which can Accomodate H Atoms in the Hexagonal C14 AB2 Structure Typea atom

Wyckoff

x

y

z

H coord.

Zr Cr1 Cr2 H1 H2 H3 H4 H5 H6 H7

(4f) (2a) (6h) (4f) (4e) (12k) (12k) (6h) (6h) (24l)

1/3 0 0.83050 1/3 0 1/2 0.25140 0.54090 0.58190 0.29000

2/3 0 0.66100 2/3 0 1/2 0.12572 0.45902 0.79095 0.32300

0.06290 0 1/4 2/3 5/16 1/8 0.64073 3/4 1/4 0.92500

1A3B 4B 2A2B 1A3B 2A2B 2A2B 2A2B

Zr4Cr8(H68) (C14 type): a ) 0.509 nm, c/a ) 1.598, s.g. P63/ mmc (no. 194). a

energetically favorable H environments and might clarify some open questions regarding H dynamics during charging and discharging processes in these compounds. In the present work only bulk hydrogenation effects were considered. Calculations of the surface effects25 would have to take into account much larger unit cells and also the vacuum layer, which was under present conditions not tractable. 2. Electronic Properties of the Hydrogenated AB2 Structures 2.1. C14 Hydrogenated Structure of ZrCr2. The C14 (MgZn2) structure type of ZrCr2 can be derived from a hexagonal close-packed array of Cr atoms, with half of the Cr4 tetrahedra replaced by Zr atoms. Cr atoms form Cr5 trigonal bipyramids (i.e., two face-sharing tetrahedra). Half of these form cornerconnected continuous chains along the hexagonal c-axis. The structure can also be considered as being formed of large ZrCr12 truncated tetrahedra, whose all four faces are capped by Zr atoms with Zr-Zr distances of about 0.32 nm. Cr atoms are also 12coordinated, forming two different kinds of polyhedra, both with 6 Cr and 6 Zr atoms at their corners. It is known that in the hydrogenated MgZn2 type of structure hydrogen atoms occupy tetrahedral interstices.3 Most of these coordination tetrahedra in the discussed AB2 structures form face sharing pairs with their neighbors, forming five-coordinated trigonal bipyramids. In case of ZrCr2 there are seven possible independent hydrogen positions per unit cell, which are multiplied by symmetry into a maximum of 68 tetrahedrally coordinated interstices altogether. With all hydrogen positions occupied, the corresponding unit cell formula would be Zr4Cr8H68 (Table 1). Among all the H-centered tetrahedra, only the H2 ones are coordinated solely by Cr atoms. These form corner-connected chains of trigonal bipyramids with every alternate one rotated by 60° with regards to its neighbors (Figure 1). The H1 tetrahedra, formed of one Zr and three Cr atoms, form similar trigonal bipyramids along the hexagonal c-axis. Contrary to the H2 bipyramids, every second H1 bipyramidal pair is alternatively replaced by hexagonal disk-like features, formed of three H5 and three H6 tetrahedra. Both are formed of two Zr and two Cr atoms and alternate around the hexagonal c-axis. The H1 and H2 trigonal bipyramids are capped at their remaining three faces with six H3 and six H4 tetrahedra, respectively, the first formed of two Zr and two Cr and the second of one Zr and three Cr atoms. They form a kind of double “stellae quadrangulae”. The H3 and H4 tetrahedra do not form trigonal bipyramids of their own. Finally, there is a

Figure 1. Seven possible tetrahedral interstices in the ZrCr2 structure. See text for details. The unit cell of the host C14 structure is shown.

set of rather deformed H7 tetrahedra, formed of two Zr and two Cr atoms, again arranged into trigonal bipyramids, which fill the remaining space in such a way that all tetrahedra share faces with their neighbors. Four H7 and one H4 tetrahedra form an irregular pentagonal disk-like agglomerate. Three sets of such agglomerates are symmetry-related by the 63-axis and aligned parallel to it. The overall distribution of tetrahedrally coordinated H atoms in all mentioned polyhedra reveals an important property, that is, a close proximity of their centers in case they form a pair. These are separated by distances of the order of 0.1 nm only, which is comparable to the separation between the two nuclei of the free H2 molecule, 0.074 nm.26 2.2. Density of States and Crystal Orbital Overlap Population Spectra of the Hydrogenated C14 Structure. Hydrogen absorption and desorption change the electronic properties of the complex structure. We use the extended Hu¨ckel tight binding (EHTB) approximation to determine the influence of hydrogenation on the electronic structure. The method uses semiempirical tight-binding Hamiltonians that employ linear combinations of atomic orbitals. Although total energy calculations are not possible, the method is able to give a rather accurate description of the electronic structure in the vicinity of the Fermi level (EF).27,28 The CAESAR package29 was used to calculate the density of states (DOS) and the crystal orbital overlap population (COOP) spectra of pure and partly hydrogenated hexagonal Zr4Cr8 structure (C14). While the DOS provides information about the band structure and the occupancy of various orbitals, the COOP spectra describe the nature of the bonds between constituent atoms, with positive and negative values indicating bonding and antibonding states, respectively. The DOS and COOP spectra are shown in Figures 2-5 for two cases, that is, for two H atoms in close proximity and positioned further apart. Three H sites, H7 (0.323, 0.290, 0.425), H3 (0.500, 0.500, 0.375), and H7 (0.290, 0.323, 0.575) in the host structure (a ) 0.509 nm, c/a ) 1.598, s.g. P63/mmc (no. 194) were taken into account. Figures 2 and 4 correspond to the structure where the two occupied hydrogen sites are far appart (0.1996 nm), while Figures 3 and 5 correspond to the structure where the two sites are in close proximity, that is, only 0.1109 nm apart. Energies are given relative to EF.

Properties of the Hydrogenated ZrCr2 Laves Phases

J. Phys. Chem. C, Vol. 114, No. 9, 2010 4223 TABLE 2: Integrated Partial Density of States of the H Peaks (in Number of States/Unit Cell) for each Contributing Atom H-H 0.1996 nm

Figure 2. Zoomed-in H peaks with the integral and partial DOS spectra of Zr4Cr8H2 in the inset: both H atoms separated by 0.1996 nm.

Figure 3. Zoomed in H peaks with the integral and partial DOS of Zr4Cr8H2 in the inset: both H atoms only 0.1109 nm appart.

Figure 4. Zoomed-in H peaks with the integral COOP spectra of Zr4Cr8H2 in the inset: both H atoms separated by 0.1996 nm

Figure 5. Zoomed in H peaks with the integral COOP spectra of Zr4Cr8H2 in the inset: both H atoms only 0.1109 nm appart.

Both DOS curves (Figures 2 and 3) appear very similar, except for the isolated H peaks. The splitting of these bands is in the case of two separated H atoms much smaller than when the H atoms are in close proximity, forming an intercalated “molecule”. This is in agreement with expectations, since the closer the atoms, the larger is the splitting of the hydrogen states

H-H 0.1109 nm

atom

1st peak

2nd peak

1st peak

2nd peak

Zr Cr H Total

0.37 0.39 1.24 1.99

0.45 0.56 0.99 2.00

0.24 0.24 1.47 1.95

0.61 0.96 0.43 2.00

TABLE 3: Values of the Integrated Partial Crystal Orbital Overlap Population (in Number of States/Bond) for the Two H Peaks H-H 0.1996 nm

H-H 0.1109 nm

atom

1st peak

2nd peak

1st peak

2nd peak

H-H H-Cr H-Zr Cr-Cr Zr-Zr

2.86 1.41 3.51 0.03 0.06

-5.19 3.37 5.83 0.16 0.22

11.35 1.08 2.56 0.01 0.02

-9.63 2.12 2.81 0.43 0.42

due to the hybridization. In addition to the integral DOS spectrum, partial DOS curves for Zr, Cr, and H are also shown in both figures. The integrated values of these peaks, giving a qualitative indication of the amount of band mixing are given in Table 2. In both cases, the low-energy H peak shows a strong hydrogen character, while the high-energy peak contains more pronounced Zr and Cr contributions. This implies that the H-H bonding is partially indirect via the host lattice atoms. While in the case of H atoms in close proximity the H character of the low-energy peak is increased (from 1.24 to 1.47), the H character of the high-energy peak is decreased, indicating a stronger bonding of H to the host structure. Although the EHTB method cannot give information about the absolute energy of the structures, the similarity of both DOS curves gives an indication that the difference in energy between the two structures is determined by the H peaks. The average energy of the H peaks is 0.2 eV higher in the case of two H atoms in close proximity, indicating that this structure is less favored. The COOP curves in Figures 4 and 5 show the bonding/ antibonding nature of the interaction. Interactions between H and metal atoms at distances up to 0.3 nm (for H-A interaction) and 0.35 nm (for A-B interaction) are taken into account. The integrated partial COOP values for the two peaks are given in Table 3. It appears that most interactions are of a bonding character (i.e., curves have positive values), except for the H-H interaction, which is antibonding in the case of the high-energy peak. It should be emphasized that negative H-H COOP occurs at the high-energy H peak, that is, the one which contains nonnegligible Zr and Cr contributions. In other words, the H-H interaction apears to be repulsive at close H-H separations due to the indirect bonding of the hydrogen atoms via the host lattice atoms. Thus, the calculations show that H atoms prefer to be separated in the host ZrCr2 structure. Because the EHTB method is not able to determine the relative stability of different structures by determining the actual total energy, ab initio calculations were performed next. 3. Ab Initio Energy Calculation of the Hydrogenated C15 and C14 Structures Ab initio calculations were performed by applying the linear muffin-tin orbital (LMTO) method, using the LmtART code

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TABLE 4: Crystallographic Data for the Cubic Zr8Cr16 (C15) Structurea atom Zr Cr a

Wyckoff pos.

x

y

z

(8a) (16d)

1/8 1/2

1/8 1/2

1/8 1/2

TABLE 6: Hexagonal H2Zr4Cr8 Modelsa

a ) 0.721 nm, space group Fd3m (no. 227).

TABLE 5: Cubic H4Zr8Cr16 Modelsa

a The H positions are defined as tetrahedral interstices between the A (Zr) and B (Cr) atoms.

(version 6.50)30,31 in the full potential (FP) approximation. The exchange-correlation functional of the density functional theory was taken after Vosko, Wilk, and Nussair,32 together with gradient corrections according to Perdew, Burke, and Ernzershof.33 The calculations were performed in the semirelativistic approximation with the spin-orbit coupling. Muffin-tin radii of 0.255 and 0.243 nm were taken for the two independent Cr, 0.3028 nm for Zr, and 0.12 nm for H atoms, respectively. The program used automatically determines the required number of k points in the Brillouin zone. Total energy self-consistency of 0.01 mRy and charge-density self-consistency of 1 × 10-4 e/(a.u.)3 were attained at the end of the iterations. 3.1. Calculation Procedure. The total energies were calculated for several hydrogenated structures based on both, the cubic C15 and the hexagonal C14 structures. One or two intercalated H atoms were added, either separated or positioned into two adjacent tetrahedral intersticies, which form a pentagonal bipyramid. For comparison with the available data17 a primitive unit cell, that is, a quarter of the cubic C15 Zr2Cr4 cell with one H atom at three different crystallographic positions, was calculated first. The crystallographic data of the C15 structure are listed in Table 4. Before determining the influence of the additional H atom the total energy of pure ZrCr2 was calculated. By minimizing the energy of the system a value of 0.721 nm was determined as the lattice parameter of the cubic C15 structure, which compares well to the experimental value of 0.7208.17 Next, the effect of intercalated hydrogen atoms was studied by considering simulated cells with one atomic H per two ZrCr2 formula units (Zr2Cr4H). The slight increase in the size of the unit cell due to the presence of the H atom, experimentally determined to be less than 1%, was ignored. As shown by test calculations with 1 and 2% larger unit cells in case of the model 4 (see Table 6), the difference in energy is very small, which

a The H positions are defined as tetrahedral interstices between the A (Zr) and B (Cr) atoms.

justifies the usage of unrelaxed unit cells in all case considered, that is, with only one or two hydrogen atoms per calculated C14 unit cell (see Table 8). The relative stability of the models was studied and compared with the published results. The binding energy of one H atom per two formula units of ZrCr2 is given as17

EB ) Ediff - 0.5E(H2) ) 2[E(ZrCr2H0.5) - E(ZrCr2)] - 0.5E(H2)

(1)

with E(H2), E(ZrCr2H0.5), and E(ZrCr2) being the energies of H2, ZrCr2H0.5, and ZrCr2, respectively. E(H2) is the sum of the energy of two widely separated H atoms, calculated using the LmtART code and taking into account a dissociation energy for H2 of 432 kJ/mol H2 (i.e., 4.48 eV per molecule). Three models of the cubic (C15) Zr8Cr16H4 and seven models of the hexagonal (C14) Zr4Cr8H2 with H atoms in different

Properties of the Hydrogenated ZrCr2 Laves Phases

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TABLE 7: Calculated Total and Derived Binding Energies (EB) for Pure and Hydrogenated C15 ZrCr2 Models with H in Different Crystallographic Positionsa structure/model

E(Zr2Cr4H) [kJ/mol]

Ediff [kJ/mol] (to pure C15)

EB [kJ/mol] (+1/2E(H2))

EB17b

/4(C15) ) Zr2Cr4 /4(C15 + 1H(2A2B) 1 /4(C15 + 1H(4B) 1 /4(C15 + 1H(1A3B)

-29936083.92 -29937608.29 -29937545.92 -29937587.67

0.00 -1524.37 -1462.00 -1503.75

0.00 -103.79 -41.42 -83.17

-33.5 -1.4 -25.2

1

1

a A lattice constant of a ) 0.714 nm was calculated by relaxing the pure ZrCr2 structure. This value was kept constant for the hydrogen intercalated structures. b For comparison reasons, the last column contains results from the literature.17

TABLE 8: Calculated Total and Derived Binding Energies (EB) for Pure and Hydrogenated C14 ZrCr2 Models with H in Different Crystallographic Positions structure/model Zr4Cr8 (C14 type)

b

E(Zr2Cr4Hx) [kJ/mol] -29936093.34

(+1/2E(H2) (2H)a

-770.19

-59.90

-119.81

-757.38

-47.09

-94.18

-774.69

-64.40

-128.79

-778.13

-67.84

-135.69

-769.84

-59.55

-119.10

0.00

1H in a Single Tetrahedral Coordination; C14 + 1H atom: Zr4Cr8H1 C14 + 1H(1A3B) (H1) -29936863.53 (2/3, 1/3, 1/3) or (2/3, 1/3, 1/6) C14 + 1H(4B) (H2) -29936850.72 (0, 0, 11/16) or (0, 0, 13/16) C14 + 1H(2A2B) (H5) -29936868.02 (0.5409, 0.45902, 3/4) C14 + 1H(2A2B) (H6) -29936871.47 (0.58190, 0.79095, 1/4) C14 + 1H(2A2B) (H7) -29936863.18 (0.290, 0.323, 0.925) 2H in Two Separated Tetrahedral Coordinations; C14 + 2H Atoms: Zr4Cr8H2 C14 + 2H(2A2B) (H5 and H6) -29937622.83 (0.58190, 0.79095, 1/4; 0.91812, 0.45910, 3/4) 1% larger unit cellc -29937629.71 2% larger unit cellc -29937627.72 C14 + 2H(1A3B) (H1) -29937619.18 (2/3, 1/3, 1/3; 1/3, 2/3, 2/3) C14 + 2H(1A3B and 4B) (H1 and H2) -29937607.51 (2/3, 1/3, 1/3; 0, 0, 11/16) 2H in Trigonal Bipyramidal Coordination; C14 + 2H Atoms: Zr4Cr8H2 C14 + 2H(1A3B) (H1) -29937589.55 (2/3, 1/3, 1/3; 2/3, 1/3, 1/6) C14 + 2H(4B) (H2) -29937541.17 (0, 0, 11/16; 0, 0, 13/16) C14 + 2H(1A3B and 4B) (H4 and H2) -29937587.75 (0.25140, 0.12572, 0.85927; 0, 0, 13/16) C14 + 2H(2A2B) (H5 and H6) -29937571.86 (0.582, 0.791, 1/4; 0.459, 0.541, 1/4)

EB [kJ/mol] (1H)

Ediff [kJ/mol] (to pure C14)

-1529.50

-108.9

-1536.37 -1534.38 -1525.84

-115.79 -113.80 -105.26

-1514.17

-93.59

-1496.22

-75.64

-1447.84

-27.26

-1494.41

-73.83

-1478.52

-57.94

To be compared with the C15 values. b Ediff ) -4.71 [kJ/mol H] in comparison with the cubic ZrCr2 (C15). c For comparison reasons, the model was also calculated with enlarged unit cells. a

tetrahedral positions were calculated. These models are listed in Tables 5 and 6. The hexagonal C14 structure was studied by adding one or two H atoms into four ZrCr2 formula units, resulting in Zr4Cr8H and Zr4Cr8H2 compositions, respectively. Like in the case of the C15 structure, the lattice parameters of the C14 structure without any additional H were determined first by minimizing the total energy. Additional minimization of the total energy by optimizing the internal positions of the atoms in the C14 structure resulted in an energy reduction of approximately 0.5 kJ/mol for ZrCr2. Since this value is small in comparison with the energy differences found for the models considered, it was neglected in the calculations. The calculated lattice parameters (a ) 0.509 nm, c/a ) 1.598) can be compared with the experimental ones (a ) 0.509 nm, c/a ) 1.62).34 Contrary to some previous calculations11 the C14 structure of ZrCr2 is energetically favored in comparison with the C15 structure, as shown by the small negative difference in energy Ediff ) -4.71

kJ/mol. However, the differences in energy found are relatively small. Again, the effects of additional H on the unit cell volume of the C14 structure were neglected. 3.2. Results. 3.2.1. Hydrogenated C15 Structures. The H binding energies for the chosen tetrahedral sites in the cubic (C15) and the hexagonal (C14) AB2 structures are given in Tables 7 and 8. In comparison with the published data17 the energies of the cubic models appear larger, but are in qualitative agreement. The energy of one H atom as calculated by the LMTO method is E(H) ) -0.9176 Ry/atom ) -1204.579 kJ/mol, and the energy of two widely separated H atoms is 2E(H) ) -2409.158 kJ/mol. By adding the H2 dissociation energy (-432 kJ/mol) the total energy for a molecule is E(H2) ) -2841.158 kJ/mol and for a single H atom half of that, that is, 1/2E(H2) ) -1420.579 kJ/mol In the case of the C15 structure, one H atom was added into the primitive unit cell with unit vectors: 0 1/2 1/2, 1/2 0 1/2, 1/2 1/2

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0, with a formula unit Zr2Cr4H (corresponding to Zr8Cr16H4 for the four times larger FCC unit cell). The binding energy EB was calculated according to formula (1) in the paper of Hong and Fu.17 The results are listed in Table 7. 3.2.2. Hydrogenated C14 ZrCr2 Structures. In the case of the C14 structure, one or two hydrogen atoms were added to the hexagonal (a × c) unit cell, corresponding to Zr4Cr8H and Zr4Cr8H2. The results are listed in Table 8. 4. Analysis of the Hydrogen Bonding Using the Maximally Localized Wannier Functions To complement our study of the energetics of the hydrogen absorption in ZrCr2, we also considered the local properties of the chemical bonding of hydrogen atoms in real space by calculatingthemaximallylocalizedWannierfunctions(MLWFs)36–38 corresponding to the Bloch energy bands associated with the hydrogen atoms. The MLWFs provide a very compact and accurate local representation of the electronic structure: they give direct insight in the nature of the chemical bonding. In the context of hydrogen absorption, the main question to be answered by this approach concerns the fate of the hydrogen electron upon absorption (i.e., does it remain essentially confined around the hydrogen ion, so that H is more atom-like, or does it become very delocalized, so that H is more ionic) and the possible chemical bonding between the H atoms in the lattice when they occupy neighboring interstitial sites. The calculations have been performed using a density functional theory code using a plane-wave basis and ultrasoft pseudopotentials, as implemented in the PWSCF code in the Quantum Espresso package (http://www.quantum-espresso. org/).39 We have used the Perdew-Burke-Ernzerhof exchangecorrelation functional, 30 Ry kinetic-energy cutoff for wave functions and 300 Ry electron density cutoff, cold smearing by 0.02 Ry and a 6 × 6 × 6 Monkhorst-Pack mesh of k-points in the initial self-consistent-field band-structure calculations. The calculations of MLWF were performed using the wannier90 package (http://www.wannier.org).40 A non-self-consistent calculation with a 4 × 4 × 4 uniform grid of k-points was used for this purpose, and we used an initial projection on orbitals with s symmetry centered on the H ions. The energy window was chosen so as to encompass the partial density of states (PDOS) peaks corresponding to the hydrogen atoms. The MLWF calculation converges in just a few steps. We determined the MLWF both for a single H atom in the unit cell (1 Wannier function), as well as for a pair of H atoms located at different interstitial sites (two Wannier functions). Because the hydrogen levels are rather well separated from other orbital energies (even when the H atoms have their least possible separation and the level splits), the disentanglement of the orbitals poses no difficulties.37 The MLWF that corresponds to a single hydrogen atom located at the tetrahedral-interstitial H2 site (4B coordination) is clearly centered on the hydrogen ion and exhibits the local tetrahedral symmetry of the absorption site (see Figure 6a). The MLWF is essentially confined on the pyramid. As expected, it has s-wave character, but there are also clear signs of the hybridization with the d orbitals of the neighboring Cr atoms. Nevertheless, the results indicate that the electron essentially remains bound on the hydrogen ion rather than spread out in the crystal (the total spread of the MLWF is 0.160 nm). The hydrogen absorbed in ZrCr2 is thus atom-like. For other interstitial hydrogen sites we find similar results. For two hydrogen atoms in the unit cell, we observe that the Wannier functions are essentially unchanged unless the H atoms

Figure 6. Maximally localized Wannier functions (MLWFs) for (a) single hydrogen atom at the H2 site (4B coordination) and for (b) two hydrogen atoms at the neighboring H2 sites. Zr atoms are gray, Cr atoms are large blue, and H atoms are small blue spheres (only the H atoms are shown in the case of two H atoms). The transparent red and blue sheets are the amplitude isosurfaces (2/V for a single H, 1.25/ V for two H) with positive and negative sign. For producing this figure, we made use of the visualization package xcrysden (http:// www.xcrysden.org/).41

occupy the neighboring interstitial sites. An example on the latter case is presented in Figure 6b, where the hydrogen atoms occupy two neighboring H2 sites. This corresponds to Model 8 presented above, that is, this is the extreme case of the smallest separation between the H atoms. Even here, however, the character of MLWFs is not changed much. We find two MLWFs, each of which corresponds to one of the two H atoms, with the centers of the WFs only slightly shifted toward the second hydrogen ion. This result is significant: it indicates that the two H atoms occupying the nearest-neighbor interstitial sites in a bipyramid cannot be considered to form a H2 molecule (in this case, one would expect the MLWFs to correspond to σ-bonds, that is, two MLWFs centered on the midpoint between the H atoms, with one bonding and one antibonding combination). We thus conclude that hydrogen atoms absorbed in ZrCr2 always have atom-like properties. 5. Conclusion Both the semiempirical EHTB and the ab initio LMTO calculations show that the diffusion of H into the C14 and C15 structures does not take place via the formation of H2 pairs in five-coordinated metal atom environments, characteristic of the Laves AB2 phases. First, the DOS and COOP spectra calculated within the EHTB approximation indicate that there is a rather strong repulsion between the H atoms, if these occupy two adjacent tetrahedral positions in the five-coordinated trigonalbipyramidal environment of metal atoms. Second, the alternative ab initio calculations of the ground-state energy, performed for a series of models with a single H atom or with pairs of H atoms, indicate that two H atoms tend to avoid close proximity. The results confirm some previous works17 and support the suggestion35 that H pairs, resembling H2 molecules in a metal environment, result in an increased ground state energy as compared to two separated H atoms. The energetically unfavorable occupancy of two face-sharing tetrahedra with H atoms in close proximity may also be related to the problem of enhanced H occupancy of all available interstitials and hence to the problem of attaining improved hydrogen storage capacities in such compounds. This important issue remains to be clarified in future studies. In addition, calculation of the MLWFs shows clearly that the hydrogen atom takes on an “atom-like” shape with the electron remaining localized close to the proton. According to the recently published studies,17,18 the largest 2A2B interstices are in some cubic C-15 AB2 compounds (e.g.,

Properties of the Hydrogenated ZrCr2 Laves Phases TiCr2 and ZrCr2), the energetically most favorable ones, while the 4B interstices are the least appropriate ones due to their smaller volumes. However, it was also shown that in other members of the same AB2 structural series the smaller intersticies may be even more appropriate. Also, these must be involved in all hydrogen diffusion processes and they should not be automatically discarded as possible hydrogen sites. In addition, there are four different A2B2 interstices (H3, H5, H6, and H7; see Table 1) that are not energetically equivalent. Further, H-H pairs in close proximity, resembling the H2 molecules, might play a certain role in the diffusion processes. Thus, we found it justified to consider their formation in relation to the generally accepted H-H separation at distances larger than 0.2 nm. That does not mean that H concentration in the AB2 Laves phases can be enlarged through the formation of H2 pairs, but rather that the hydrogen diffusion mechanism can also involve formation of such pairs. The possibility of full hydrogen occupancy is certainly not a real option. Its limits are clearly related to the 0.2 nm rule. However, the minimum in the energy spectrum at H-H distances characteristic for the H2 intramolecular separation might play an important role in the hydrogen diffusion process at low hydrogen concentrations. According to the calculations performed, in the case of ZrCr2, the C14 structure is stabilized in comparison with the C15 structure. However, the difference in energy is small, and in the case of other comparable AB2 compounds, the C15 structure may turn out to be the most favorable state. Acknowledgment. Financial support of the Slovenian Research Agency (ARRS; H.J.P.v.M., A.P., E.Z., and R.Zˇ.), the bilateral cooperation program between the Hellenic Republic and the Republic of Slovenia (GSRT Code 043Γ), and the European Integrated Project NESSHY (SES6-518271; S.S.M., A.K.S.) is gratefully acknowledged. References and Notes (1) (2) (3) (4) Energy (5) (6) (7)

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