First-Principles Study of Experimental and Hypothetical Mg(BH4)2

Feb 28, 2008 - Abstract: Boranates (tetrahydroborates) are studied intensively because of their potential use as hydrogen storage materials. In this A...
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J. Phys. Chem. C 2008, 112, 4391-4395

4391

First-Principles Study of Experimental and Hypothetical Mg(BH4)2 Crystal Structures Bing Dai Department of Chemical Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261

David S. Sholl Department of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, and National Energy Technology Laboratory, Pittsburgh, PennsylVania 15236

J. Karl Johnson* Department of Chemical Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261, and National Energy Technology Laboratory, Pittsburgh, PennsylVania 15236 ReceiVed: October 19, 2007; In Final Form: January 9, 2008

We have used first-principles density functional theory to relax the experimentally reported crystal structures for the low- and high-temperature phases of Mg(BH4)2, which contain 330 and 704 atoms per unit cell, respectively. The relaxed low-temperature structure was found to belong to the P6122 space group, whereas the original experimental structure has P61 symmetry. The higher symmetry identified in our calculations may be the T ) 0 ground-state structure or may be the actual room-temperature structure because it is difficult to distinguish between P61 and P6122 with the available powder diffraction data. We have identified several hypothetical structures for Mg(BH4)2 that have calculated total energies that are close to the low-temperature ground-state structure, including two structures that lie within 0.2 eV per formula unit of the ground-state structure. These alternate structures are all much simpler than the experimentally observed structure. We have used Bader charge analysis to compute the charge distribution in the P6122 Mg(BH4)2 structure and have compared this with charges in the much simpler Mg(AlH4)2 structure. We find that the B-H bonds are significantly more covalent than the Al-H bonds; this difference in bond character may contribute to the very different crystal structures for these two materials. Our calculated vibrational frequencies for the P6122 structure are in good agreement with experimental Raman spectra for the low-temperature Mg(BH4)2 structure. The calculated total energy of the high-temperature structure is only about 0.1 eV per formula unit higher in energy than the low-temperature structure.

I. Introduction Complex hydrides of light metals have been considered for the development of low-temperature reversible hydrogen storage systems because they have high volumetric and gravimetric hydrogen storage capacities. Magnesium borohydride [Mg(BH4)2] is an interesting complex hydride having a theoretical hydrogen storage capacity of 14.9 wt %. The first synthesis of Mg(BH4)2 was reported in 1950 by Wiberg and Bauer.1 Since that time, research efforts have been directed toward the synthesis and characterization of Mg(BH4)2.2-9 It has been known for some years that both low- and high-temperature phases exist.6 However, the detailed structures of these phases have remained a mystery until last year. Ceˇrny´ et al.10 and Her et al.11 independently identified the low-temperature structure of Mg(BH4)2 as having hexagonal symmetry with space group P61. The low-temperature phase has 330 atoms in the unit cell (30 formula units) with 55 irreducible atoms. The hightemperature structure was identified by Her et al.11 to have orthorhombic symmetry with space group Fddd and has 704 atoms per units cell (64 formula units) with 27 irreducible atoms. These structures have a surprising level of complexity that is arguably unprecedented for this type of material.10 The reasons * Corresponding author. E-mail: [email protected].

for the complexity of Mg(BH4)2 are not immediately apparent. By way of contrast, the analogous Mg(AlH4)2 structure has P3hm1 symmetry and only 11 atoms per unit cell. Prior to the experimental discovery of the Mg(BH4)2 structure, there were a number of different attempts to predict the structure of Mg(BH4)2 from first-principles density functional methods.12,13 For example, Vajeeston et al.12 predicted Mg(BH4)2 to have Pmc21 symmetry (22 atoms per unit cell) and Nakamori et al.13 proposed Mg(BH4)2 to have P3hm1 (11 atoms per unit cell) or P2/c symmetry (22 atoms per unit cell). Density functional theory (DFT) has proven very useful for predicting the reaction thermodynamics of metal hydrides with known structures,14-33 but reliable prediction of unknown crystal structures is much more challenging.12,13,34-39 Given the complexity of the experimentally determined structure of Mg(BH4)2, it is no surprise that the structures predicted from previous DFT calculations turned out to be incorrect. In this paper, we investigate the energetics and relaxed structures of both the low and high-temperature Mg(BH4)2 crystals in their experimentally observed structures as computed from periodic plane wave ab initio DFT. We compare the relaxed structures to the structures obtained experimentally and also to simpler candidate structures obtained from relaxing analogous materials taken from the International Crystal Struc-

10.1021/jp710154t CCC: $40.75 © 2008 American Chemical Society Published on Web 02/28/2008

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tures Database (ICSD).40 These calculations provide information on several questions that could not be answered from the published experimental analysis of the structures, such as how the charge distribution in Mg(BH4)2 differs from similar materials, to what extent the BH4 tetrahedra relax (they were held fixed in the experimental structure determination), and how simpler hypothetical Mg(BH4)2 structures compare structurally and energetically with the observed experimental structures. II. Computational Methods Plane wave DFT calculations were performed with the Vienna Ab initio Simulation Package (VASP).41,42 Ionic cores were described by the projector augmented wave method.43,44 Electron exchange and correlation effects were described using the generalized gradient approximation with the Perdew-Wang 91 functional.45 The cutoff energy for the plane wave expansion was set to 425 eV. This energy is sufficiently high to give wellconverged structures and total energies. The Brillouin zone for each structure was sampled with a Monkhorst-Pack mesh using a k-point spacing of less than 0.05 Å-1. Convergence of the energies with the number of k points was checked for the proposed structures. Geometry optimizations for bulk structures were performed using the full unit cell in all cases and by allowing all atomic positions and all cell parameters, including the cell volume and shape, to vary. The positions of all atoms were relaxed until the force on each of the atoms was smaller than 10-2 eV/Å. We performed an additional energy calculation on the optimized structures using the tetrahedron method with Blo¨chl corrections to get accurate total energies.46 The vibrational frequencies of the candidate structures taken from the ICSD were calculated to examine their stabilities and to compare with experimental Raman spectra for Mg(BH4)2. The frequencies were calculated within the harmonic approximation by a finite difference estimation of the Hessian matrix. A step size of 0.02 Å was used in these calculations. The vibrational frequencies of the optimized low-temperature structure were also calculated by displacing only the irreducible atoms, rather than all the atoms in the unit cell.

Figure 1. Unit cell of the optimized low-temperature structure of Mg(BH4)2. The top figure is the view along the c axis, and the bottom figure is the view along the a axis. The green, pink, and white spheres represent Mg, B, and H atoms, respectively.

III. Results and Discussion The two experimentally determined low-temperature structures of Mg(BH4)210,11 are nearly identical. We have relaxed both of these structures using DFT and found that they converge to essentially the same structure, shown in Figure 1. Our relaxed structure belongs to the P6122 space group, which has higher symmetry than the experimentally observed P61 space group. There are six symmetry operations in space group P61 and an additional six symmetry operations in the P6122 space group. The numbers of irreducible atoms in the unit cells of the P6122 and P61 structures are 29 and 55, respectively. The atomic coordinates of the relaxed P6122 structure are given in the Supporting Information. The relaxed lattice parameters are a ) b ) 10.27 Å and c ) 36.80 Å, which are in good agreement with the experimentally determined lattice parameters from C ˇ erny´ et al.,10 a ) b ) 10.318 Å and c ) 36.998 Å, and Her et al.,11 a ) b ) 10.341 Å and c ) 37.086 Å. It is difficult to distinguish between P61 and P6122 based on powder diffraction data alone because differences in these two structures appear as small variations in the reflection intensities, which can be hidden behind texture and bad grain statistics in powder diffraction data.47 However, refinement of the thermal and selected profile parameters using our P6122 structure indicates that the P61 structure is a better fit to the experimental data.48 Single-crystal data may be needed to definitively answer the

Figure 2. Comparison of experimental11 and simulated XRD patterns for the low-temperature structure of Mg(BH4)2.

question of whether the actual experimental structure belongs to the P61 or P6122 space group. However, it is likely that the P6122 structure identified by DFT is stable at low temperatures and that there is a transition to the P61 structure somewhere below room temperature. We have relaxed the experimentally determined high-temperature structure11 with DFT. In contrast to the low-temperature hexagonal structure, the high-temperature phase is orthorhombic, belonging to the Fddd space group, but with a larger unit cell containing 704 atoms. The optimized structure retains the Fddd space group of the experimentally determined structure. The calculated lattice parameters are a ) 36.90 Å, b ) 18.83 Å, and c ) 10.65 Å, while the experimental values11 are a ) 37.07 Å, b ) 18.65 Å, and c ) 10.91 Å. The relaxed atomic coordinates are given in the Supporting Information. We have simulated the X-ray powder diffraction (XRD) patterns of the optimized structures. The simulated and experimentally measured11 XRD patterns for the low-temperature structures are plotted in Figure 2. The agreement between these two patterns is excellent; this indicates that there are only subtle differences between the P61 and P6122 XRD patterns.

First-Principles Study of Mg(BH4)2 Structures

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TABLE 1: Properties of the Seven Hypothetical Mg(BH4)2 Structures Having the Lowest Energies (Etot/fu is the Total Energy Per Formula Unit in eV; and ∆Etot/fu is the Energy Relative to the Optimized Low-Temperature Structure on a Per Formula Unit Basis, in Units of eV) prototype structure

Etot/fu

Al2CdCl8 B2CaF8 Al2Cl8Sm Mo2O8Zr Cu2O Al2Cl8Ti B2F8Mn

-44.57 -44.55 -44.37 -44.30 -44.10 -44.09 -44.07

∆Etot/fu

space group

space group number

number of atoms/u.c.

0.17 0.19 0.37 0.44 0.64 0.65 0.67

P1a1 Pbca P12/a1 Pmn21 Pn3hm Pnn2 Pnma

7 61 13 31 224 34 62

22 88 22 22 22 22 44

It is interesting to compare the complex experimental crystal structures with simpler alternate structures. We identified 30 possible structures by searching the ICSD40 for all compounds having AB2X8 stoichiometry and substituting Mg, B, and H at the positions of A, B, and X, respectively. In addition, we also investigated the Cu2O structure, where BH4 groups take the Cu sites and Mg atoms take the O sites, as suggested by C ˇ erny´ et al. in their Supporting Information.10 Each of these alternate structures were fully relaxed with DFT. We also computed the vibrational frequencies of each of these structures to test the stability of the relaxed structure. The seven systems having the lowest energies per formula unit are listed in Table 1. The first six structures listed in Table 1 were found to be stable, meaning that all of their vibrational frequencies were positive. The last structure, based on the prototype material B2F8Mn, had multiple imaginary modes. The Pmc21 structure proposed by Vajeeston et al.12 has the same total energy as our Al2CdCl8(P1a1) structure. The P3hm1 structure proposed by Nakamori et al.13 has slightly higher energy than the Mo2O8Zr (Pmn21) structure in Table 1. The energy per formula unit for our optimized P6122 structure was -44.74 eV, so our calculations are consistent with the experimental observation that this crystal structure is more stable than any of the alternatives listed in Table 1. The calculated total energy of the high-temperature structure is -44.60 eV per formula unit. This is only about 0.1 eV higher than the optimized low-temperature structure and is also lower in energy than any of the hypothetical structures listed in Table 1. We have analyzed the bond lengths and bond angles for the experimental structures, the relaxed structures, and the six lowest hypothetical energy hypothetical structures. Key bond lengths and bond angles for these structures are reported in Table 2. The experimental structures were obtained from diffraction data by constraining the BH4 groups to be ideal tetrahedra.10 The DFT relaxed structures did not have this constraint. We see from Table 2 that the relaxed experimental structures have a wider range of H-B-H angles than the experimental structures, indicating a lower level of ideality in the BH4 tetrahedra than accounted for in the fitting procedure for both the low- and high-temperature structures. Note, however, that the B-H bond lengths in the optimized structures remain very nearly ideal (i.e., all equal). Both the B-H and Mg-H relaxed bond lengths increase from their experimentally determined values. The distribution of the Mg-B bond distances in the optimized structures is considerably more narrow than those in experiments. The distribution of B-Mg-B angles is slightly narrowed in the relaxed low-temperature structure relative to the experiments, but the optimized high-temperature structure has a slightly broadened distribution of B-Mg-B angles compared with experiments.

Figure 3. Calculated vibrational frequencies of the P6122, Al2CdCl8, and B2CaF8 structures. The solid curve is the experimental Raman spectrum of Mg(BH4)2.10

Comparing the total energies of the hypothetical structures and the optimized experimental structures suggests that the Mg-B bond lengths are important to the total energies. The two lowest-energy hypothetical structures have roughly the same average Mg-B bond length as the optimized low-temperature structure (2.41-2.43 Å vs 2.40 Å). The Mg-B distances of all of the other higher energy structures are either too long or too short. Among the four lowest energy structures, the optimized low-temperature structure has the most narrow Mg-B distance distribution. The relaxed high-temperature structure, however, has much shorter Mg-B bond distances (2.36 Å). An interesting outcome from Table 2 is that there are multiple structures available to Mg(BH4)2 that have very similar total energies. An analogy can be drawn between this observation and similar analyses of zeolitic materials. Silica is known to have a number of very different structures, all with about the same energy. These structures include quartz and a large collection of porous zeolitic materials.49-54 For these siliceous materials, variations in the synthesis procedure of the material can change the crystal structure of the final material. The mechanisms underlying this process are not completely understood. In contrast to the siliceous zeolites, however, there is as yet no experimental evidence that any of the simpler crystal structures of Mg(BH4)2 proposed here or by others12,13 have been prepared or observed. It was suggested by Her et al.11 that the more covalent nature of the BH4 unit versus AlH4 may contribute to the complexity of the Mg(BH4)2 structure compared with that of the compositionally similar Mg(AlH4)2. Mg(AlH4)2 has P3hm1 symmetry with space group number 164 and contains only 11 atoms in the unit cell. We used Bader charge analysis55,56 to obtain the average charge distribution of atoms in Mg(AlH4)2 and the lowtemperature Mg(BH4)2 phase; these are reported in Table 3. According to our calculations, each Al atom gives up about 2.15 electrons compared with 1.53 electrons for the B atoms. Likewise, the charge on H atoms in the Mg(AlH4)2 structure is -0.74, compared with -0.59 for hydrogens in Mg(BH4)2. Hence, B-H bonds are more covalent than Al-H bonds. These charge calculations are consistent with the hypothesis put forward by Her et al.11 but does not explain how more covalent bonding leads to more complex structures.

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TABLE 2: Key Bond Lengths and Bond Angles from the Experimental High-Temperature Structure (Exp-H11), Low-Temperature Structures (Exp-L11 and Exp-L10), DFT Optimized High- (Opt-H) and Low- (Opt-L) Temperature Structures, and Six Hypothetical Structures (Labeled Using Their Prototype Materials)a Etot/fu ∆Etot/fu MgB MgH BH HBH BMgB

max min av max min av max min av max min av max min av

exp-H11

opt-H

2.49 2.34 2.39 1.88 1.70 1.76 1.02(1) 1.02(1) 1.02(1) 109.47 109.47 109.47 118.24 101.91 109.36

-44.60 0.14 2.40 2.32 2.36 2.03 1.98 2.00 1.23 1.21 1.22 116.45 104.84 109.52 119.59 99.81 109.44

exp-L11

2.57 2.28 2.42 1.97 1.76 1.85 1.12 1.12 1.12 109.47 109.47 109.47 137.30 91.80 109.60

exp-L10

ppt-L

Al2CdCl8

B2CaF8

Al2Cl8Sm

Mo2O8Zr

Cu2O

Al2Cl8Ti

2.53 2.31 2.42 1.89 1.80 1.85 1.18(1) 1.18(1) 1.18(1) 109.47 109.47 109.47 130.77 82.80 109.58

-44.74 0.00 2.41 2.38 2.40 2.03 2.00 2.02 1.23 1.22 1.22 115.89 105.68 109.52 128.77 93.86 109.81

-44.57 0.17 2.48 2.39 2.43 1.98 1.98 1.98 1.24 1.20 1.22 115.95 101.18 109.50 120.34 98.54 109.48

-44.55 0.19 2.49 2.35 2.41 1.96 1.95 1.95 1.24 1.20 1.22 114.29 104.21 109.49 119.70 90.97 108.92

-44.37 0.37 2.69 2.67 2.68 2.04 2.04 2.04 1.22 1.22 1.22 112.22 106.57 109.41 179.72 75.57 102.62

-44.30 0.44 2.49 2.44 2.47 1.96 1.96 1.96 1.23 1.20 1.22 114.87 101.25 109.41 112.67 96.73 102.04

-44.10 0.64 2.51 2.38 2.47 1.96 2.06 2.01 1.23 1.21 1.22 115.17 105.54 109.44 143.71 103.75 120.00

-44.09 0.65 2.34 2.34 2.34 1.97 1.93 1.95 1.24 1.21 1.22 113.80 107.55 109.44 180.00 179.95 179.97

a All bond lengths (MgB, MgH, and BH) are in Å, and bond angles (HBH and BMgB) are in degrees. E tot/fu is the total energy per formula unit in eV. ∆Etot/fu is the energy relative to the optimized low-temperature structure on a per formula unit basis, in units of eV.

TABLE 3: Average Charge Distribution in Mg(AlH4)2 and Mg(BH4)2 Calculated from Bader Charge Analysis55,56 Mg Al or B H

Mg(AlH4)2

Mg(BH4)2

1.58 2.15 -0.74

1.64 1.53 -0.59

We plot the calculated vibrational frequencies of the two lowest-energy hypothetical structures (Al2CdCl8 and B2CaF8) and the P6122 structure in Figure 3. Note that all of the calculated frequencies are positive, indicating that the structures are true minima. The experimental Raman spectrum10 is also plotted in the figure for comparison. The calculated vibrational modes can be separated into two groups: one group having frequencies above 2000 cm-1 and another group with frequencies between 1000-1500 cm-1. These groups correspond to B-H stretching and B-H bending modes, respectively. No modes are observed over 2500 cm-1 in either the experimental spectrum or the calculated P6122 structure; in contrast, the two hypothetical structures exhibit a few modes at about 2550 cm-1. These high-frequency modes for Al2CdCl8 and B2CaF8 structures involve single B-H bond stretches, with the other H atoms in the BH4 group remaining almost stationary. In contrast, the modes between 2000 and 2500 cm-1 involve mainly asymmetric stretching of multiple B-H bonds. Moreover, the highfrequency modes in Al2CdCl8 and B2CaF8 are a direct result of the asymmetry in the BH4 groups of those structures (i.e., nonideal tetrahedral structures). We see from Table 2 that the Al2CdCl8 and B2CaF8 structures have B-H bond lengths that are both longer and shorter than those in the P6122 structure. The high-frequency modes in the two hypothetical structures involve stretching of the shortest B-H bonds (1.20 Å). Thus, the calculated frequencies from the P6122 structure gives the best agreement with the experimental spectrum because it has BH4 groups that are more nearly ideal tetrahedra. IV. Conclusions In this paper, we have relaxed the experimentally reported crystal structures for the low- and high-temperature phases of Mg(BH4)2 using first-principles density functional theory. These complicated structures contain 330 and 704 atoms per unit cell for the low and high-temperature phases, respectively. Our DFT relaxed low-temperature structure has P6122 symmetry, which

is a supergroup of the experimentally observed P61 symmetry. The higher symmetry obtained in DFT may be the actual crystal structure, given that it is very difficult to distinguish between P61 and P6122 based on powder diffraction data. It is also possible that the P6122 is the correct low-temperature structure and that there is a transition to the P61 phase below room temperature. Single-crystal data are needed to resolve this question. Several much simpler hypothetical structures for Mg(BH4)2 have been proposed that have DFT total energies that are close to the low-temperature ground-state structure. In this respect, Mg(BH4)2 may be similar to silica, which has a larger number of different crystal structures having about the same energy. It is still not exactly clear why the very complicated 330 atom structure is preferred over the alternative structures, although the having correct Mg-B and B-H bond lengths and nearly ideal BH4 tetrahedra seem to be important factors. Calculated vibrational frequencies for Mg(BH4)2 in the P6122 structure are in good agreement with the experimental Raman spectrum. The calculated total energy of the high-temperature structure is only about 0.1 eV per formula unit higher than that of the low-temperature structure and has an energy that is lower than any of the proposed hypothetical structures. Acknowledgment. We gratefully acknowledge Radovan C ˇ erny´, Job Rijssenbeek, Ji-Cheng Zhao, Yan Gao, and Klaus Yvon for many helpful discussions. This work was supported by the DOE Grant No. DE-FC3605GO15066 and was performed in conjunction with the DOE Metal Hydride Center of Excellence. Calculations were performed at the University of Pittsburgh Center for Molecular and Materials Simulations. Supporting Information Available: The atomic coordinates and unit cell parameters for the optimized low- and hightemperature experimental Mg(BH4)2 structures. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Wiberg, E.; Bauer, R. Z. Naturforsch., B 1950, 5, 397. (2) Pinto, A. M. F. R.; Falca˜o, D. S.; Silva, R. A.; Rangel, C. M. Int. J. Hydrogen Energy 2006, 31, 1341. (3) Bremer, M.; No¨th, H.; Warchhold, M. Eur. J. Inorg. Chem. 2003, 111. (4) Konoplev, V. N. Russ. J. Inorg. Chem. 1980, 25, 964.

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