A Theoretical Study on the Rotational Motion and ... - ACS Publications

Article pubs.acs.org/JPCC

A Theoretical Study on the Rotational Motion and Interactions in the Disordered Phase of MBH4 (M = Li, Na, K, Rb, Cs) Niels Bindzus,† Fausto Cargnoni,‡ Bo B. Iversen,† and Carlo Gatti*,‡ †

Center for Materials Crystallography, Department of Chemistry and iNANO, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark ‡ CNR-ISTM Istituto di Scienze e Tecnologie Molecolari, via Golgi 19, 20133 Milano, Italy S Supporting Information *

ABSTRACT: The rotational motion in the high-temperature disordered phase of MBH4 (M = Li, Na, K, Rb, Cs) is investigated utilizing two complementary theoretical approaches. The first one consists of high-level periodic DFT calculations which systematically consider several instantaneous representations of the structural disorder. The second approach is based on a series of in vacuo calculations on molecular complexes suitably extracted from the crystal and chosen as to possibly disentangle the energetic factors leading to the observed rotational barriers. The results of the first part demonstrate that the motion of the BH4− anion is dominated by 90° reorientations around the 4-fold symmetry axes of the cubic crystal, and depending on the instantaneous structural disorder activation energies are found to be between 0.00 and 0.31 eV for LiBH4, 0.05 and 0.26 eV for NaBH4, 0.16 and 0.27 eV for KBH4, 0.22 and 0.31 eV for RbBH4, and 0.21 and 0.32 eV for CsBH4. The increasing rotational barriers as well as the movement of the transition state from 7° to 44° observed along the series of alkaline metals, M = Li−Rb, appear to be simply accounted for by an analysis of the energy profiles for the C2 rotation of a BH4− group in M+−BH4− and BH4−−BH4− in vacuo complexes. The energy gained from the introduction of disorder shows a trend opposite to that of the rotational barriers as it decreases along the Li−Rb series. Similar considerations apply to the C3 rotational motion of the BH4− anion, which likewise has been studied in the crystal and in the in vacuo molecular complexes. CsBH4 deviates from the systematic trends observed for LiBH4−RbBH4. Depending on the structural starting point of the rotation, its C2 rotational barriers are found to be slightly higher or slightly lower than for RbBH4, whereas its energy gain due to the introduction of disorder is found to be positioned between that of KBH4 and RbBH4. The C3 rotational barriers of CsBH4 are instead significantly smaller compared to those of RbBH4 and even marginally below those of KBH4. compounds, which crystallize in a cubic lattice within the Fm3̅m (225) space group; see Figure 1.6 Their crystal structure is orientationally disordered as the BH4− anion is able to position itself in two different ⟨111⟩ orientations. Heat capacity studies suggest transition to a low-temperature phase at 190 K for NaBH4, 76 K for KBH4, 44 K for RbBH4, and 27 K for CsBH4.7,8 By means of neutron and X-ray diffraction experiments, the low-temperature phase of NaBH4 and KBH4 has been structurally characterized as having ordered BH4−

1. INTRODUCTION Alkaline metal tetraborohydrides represent a promising class of complex hydrides for hydrogen storage in mobile applications due to their high gravimetric and volumetric hydrogen content. However, their practical usage is impeded by slow sorption kinetics and substantial dehydrogenation temperatures, which are 380 °C for LiBH4, 400 °C for NaBH4, 500 °C for KBH4, 600 °C for RbBH4, and 660 °C for CsBH4.1,2 Hence, an active field of research concerns the development of methods, e.g., anion substitution and nanoconfinement, aiming to destabilize MBH4 and, thereby, ameliorating its hydrogen storage properties.3−5 Our study considers the high-temperature, isomorphous phases of the four MBH4 (M = Na, K, Rb, Cs) © 2013 American Chemical Society

Received: September 4, 2012 Revised: January 8, 2013 Published: January 10, 2013 2308

dx.doi.org/10.1021/jp308777n | J. Phys. Chem. C 2013, 117, 2308−2316

The Journal of Physical Chemistry C

Article

assumed.12,13 The simplified structure consists of BH4− anions positioned in identical orientations and originates if the space group Fm3̅m is reduced to F4̅3m. Application of the nudge elastic band method confirmed that the C2 axis is the most favorable rotation axis for the BH4− anions.12 In this paper, we report a theoretical study on the rotational motion in the high-temperature isomorphous phases of MBH4 (M = Na, K, Rb, Cs) that contributes with a detailed insight into the BH4− dynamics of these compounds. Relative to previous, theoretical investigations,12,13,18 we consider the complete series of the isomorphous MBH4 compounds as well as several instantaneous representations of the orientational disorder. The series of MBH4 compounds is expanded with LiBH4, which, for the sake of a meaningful comparison along the series of alkaline metals, is constructed in a hypothetical Fm3̅m structure. Each representation of the disorder is computed by high-level periodic ab initio DFT calculations employing the CRYSTAL06 code.19 Additionally, a complementary DFT and CCSD(T) study using the Gaussian03 code20 is performed on a series of in vacuo molecular complexes extracted from the crystal and chosen as to dissect the energetic factors leading to the observed rotational barriers. The qualitative results of these computations provide a rationalization of the relative height and specific angular location of the rotational barriers in the crystal along the alkaline series of M = Li, Na, K, Rb, Cs.

Figure 1. Crystal structure of the high-temperature disordered phase in space group Fm3̅m as well as three instantaneous pictures of the structural disorder, AAAA−AABB. For the three pictures, the A/B orientation of the four BH4− anions is depicted by the four label letters. For each tetrahedron one-fourth of the 12 closest BH4− neighbors are displayed. M (brown), B (green), and H (black).

units located in a tetragonal lattice with P42/nmc symmetry.6,9 In the case of RbBH4 and CsBH4, the low-temperature phases remain to be fully characterized; however, a structural symmetry lowering has been implied for both systems by neutron diffraction data.6 The crystal structures of LiBH4 deviate to those of MBH4, M = Na−Cs. At low temperatures, LiBH4 exhibits an orthorhombic structure in space group Pnma, which at around 380 K transforms into a hexagonal hightemperature phase in space group P63mc.10 The phase transition is accompanied by hydrogen desorption.1 In order to understand the chemical behavior of the MBH4 compounds and how various modification methods perturb it, a thorough elucidation of the dynamical behavior of the BH4− anion and its relation to the crystal structure is essential. Previous experimental studies using quasielastic neutron scattering (QENS) and nuclear magnetic resonance (NMR) confirm that the rotational motion of the BH4− anion is governed by reorientation around the tetrahedral C2 axes.9,11−13 Babanova and co-workers have by means of NMR determined the activation energies to be 0.126(3), 0.161(2), 0.138(4), and 0.105(7) eV for MBH4, M = Na−Cs.9,14 The nonmonotonical variation observed for the activation energy was chemically reasoned by the relative deviation of the actual M···B distance from the sum of the ionic radii of M+ and BH4−. This is in accord with the general belief that the energy barriers for BH4− reorientations in borohydrides are primarily determined by M···H interactions.9,13−15 It should be noted that substantial discrepancies exist among the experimentally determined activation energies. The systematic variation observed by Babanova and co-workers is corroborated by a QENS study and a NMR study, which resulted in activation energies ranging from 0.116(5) to 0.123(5) eV and from 0.151(5) to 0.153(4) eV for NaBH4 and KBH4, respectively.11,16 However, this is contradicted by a Raman study, for which the activation energies were found to decrease from 0.125(5) to 0.085(4) eV along the complete series of MBH4, M = Na−Cs, and by a QENS study, which determined the energy barriers to be 0.113(1) and 0.103(2) eV for NaBH4 and KBH4, respectively.12,17 Theoretical investigations concerning the reorientational motion of BH4− in the disordered phases of the isomorphous MBH4 compounds have up till now been limited to NaBH4 and KBH4, for which a simplified structural representation was

2. COMPUTATIONAL DETAILS 2.1. CRYSTAL06. All-electron Def2-TZVP basis sets21 have been used for H, B, Li, Na, and K, while the Def2-TZVP basis sets of Rb and Cs include a pseudopotential22 for the description of the core electrons. Due to the diffuseness of the outer functions of such basis sets, it was with the exception of H not possible to use the original atomic basis. The most diffuse functions were cut, and in the case of the alkali metals a sp-contraction was added with a Gaussian exponent calculated in accord to the empirical “1/3” rule.19 The B3LYP functional23,24 was applied in the periodic DFT computations, as it was found to reproduce the experimental bond distances of B−H determined by neutron PXRD 6 (powder X-ray diffraction) with a larger accuracy than B3PW,23,25−28 PBE,29 PBE0,30 and PBEsol.31 Further details about the functional selection as well as the final adopted basis sets are given in the Supporting Information. Wave functions were obtained using a shrinking factor of 8 for reciprocal space sampling according to the Pack−Monkhorst method and of eight for defining the Gilat net used in the interpolation of the density matrix as well as in the evaluation of the Fermi energy. Truncation thresholds were set to 10−8, 10−8, 10−8, 10−8, and 10−16 au (hartree) for the Coulomb and exchange series and to 10−6 au for the selfconsistent field convergence of the total energy and eigenvalues. The geometry of the hypothetical LiBH4 structure was fully optimized, while the lattice parameters were constrained to experimental values in the geometry optimization of the others compounds; a(NaBH4) = 6.130 800 Å,32 a(KBH4) = 6.638 442 Å, a(RbBH4) = 6.934 487 Å, and a(CsBH4) = 7.327 100 Å.6,a The lattice parameters of KBH4 and RbBH4 originate from 100 K synchrotron PXRD data collected at SPring8, which remain to be published. The periodic CRYSTAL06 computations provided single-point energies, from which the rotational barriers and energy gains owing to disorder were determined. 2309



Article

observed between the various computational models are of a negligible order of ca. 10−4 eV. Additionally, they all provide the same rotational position of the transition state, namely 41.1°. Although our energy convergence test was limited to only one compound, one disorder model, and one rotation pathway, the obtained results suggest that second neighbors play a minor role in the determination of the rotational energies. Thus, the 1 × 1 × 1 model seems reliable enough for our purposes, being not appreciably affected by symmetric image artifacts. 2.2. Gaussian03. To rationalize the energy landscape of the C2 and C3 rotations in periodic structures, we rigidly extracted the closest M+−BH4− (M = Li, Na, K, Rb, Cs) and BH4−− BH 4 − complexes from the geometry-optimized crystal structures. The rotation of the BH4− group was sampled with steps of 7.5° and 15° for the C2 and C3 rotations, respectively. Molecular computations have been conducted using the Def2-TZVP basis set for all atoms. However, at variance with periodic computations, we adopted the original formulation of such basis, without cutting the diffuse functions. This choice is intended to balance the finiteness of molecular complexes as compared to periodic systems. Consistent with the periodic case, we performed total energy computations at the B3LYP level of theory. Since the energy differences due to changes in the intermolecular conformations are very small and likely comparable to the accuracy obtainable at the DFT level, we also determined the energy landscapes of C2 and C3 rotations at the CCSD(T) level of theory. The results discussed in section 3.2 refer to the B3LYP Hamiltonian, while a summary of the CCSD(T) data is reported in the Supporting Information. Herein, we demonstrate that all the energy trends observed at B3LYP level and discussed in the following are nicely confirmed by higher quality CCSD(T) computations. Hence, we are confident that the B3LYP Hamiltonian is adequate for our purposes.

In order to facilitate the description of the disorder, all periodic crystal simulations were performed on the conventional unit cell consisting of four formula units. The BH4− tetrahedron can be oriented in two possible orientations, denoted A and B, which are connected by a 90° rotation around one of the tetrahedral C2 axes. If these two configurations are superimposed, they will on average result in the disordered BH4− depicted in the Fm3̅m structure as it contains eight H of occupancy 0.5 at each of the cubic corners. We have considered five instantaneous pictures of the disorder labeled by AAAA, AAAB, AABB, ABBB, and BBBB; the four letters simply depict the A/B orientation of the four anions contained in the unit cell, and the AAAA−AABB crystal structures are illustrated in Figure 1. In the investigation of the rotational motion in section 3.1, the first letter refers to the BH4− tetrahedron being rotated along either the C2 or the C3 axis. Its chemical environment is represented by the three remaining letters as each of them depicts the orientation of one-third of its 12 closest BH4− neighbors. An advantage of this approach is reduced computational cost as the structures of AAAB and ABBB as well as AAAA and BBBB are equal. The introduction of disorder relaxes the repulsive H···H network, but maintains identical M···H interdistances. This leads to energy stabilization for the most favorable state AABB within 0.032−0.416 eV depending on the compound and disorder; see Table 1 and the Results and Discussion section. Table 1. Energy Differences (in eV) between Disordered States, EAABB − EX, as Well as Energy Barriers, Ea, for the Rotational Motion of a Single BH4− Unita EAABB − EAAAA EAABB − EAAAB C2 (A → B) EAAAA a EAAAB a EAABB a EABBB a C3 (A → A) EAAAA a EAAAB a EAABB a EABBB a

LiBH4

NaBH4

KBH4

RbBH4

CsBH4

−0.416 −0.109

−0.276 −0.070

−0.153 −0.039

−0.130 −0.032

−0.140 −0.036

0.003 0.007 0.116 0.309

0.051 0.098 0.168 0.257

0.160 0.193 0.232 0.274

0.215 0.244 0.277 0.312

0.212 0.243 0.279 0.317

0.029 0.060 0.157 0.257

0.290 0.344 0.406 0.472

0.491 0.523 0.558 0.593

0.564 0.592 0.621 0.651

0.485 0.515 0.547 0.580

3. RESULTS AND DISCUSSION 3.1. Rotational Motion. On the basis of periodic computations, we explore the rotational motion of the BH4− anion around the tetrahedral C2 and C3 axes. For both types of rotation, we commence with a BH4− anion in A-orientation and include possible disordered states by systematically varying its chemical environment from AAA to BBB. The resulting rotational energy barriers, Ea, are provided in Table 1, in which the C2 barriers represent the energy required for reorienting a BH4− tetrahedron from A- to B-orientation.12 Likewise, the C3 barriers depict the energy needed for rotating the tetrahedron 120° onto itself, thus preserving its initial Aorientation. Energy barriers heights and their angular locations were obtained by a parabolic fit around the energy maxima of the rotational pathways; further details are provided in the Supporting Information. The activation energies of C2 are, in general, significantly lower than those of C3. Hence, in accord with previous experimental and theoretical publications, our computations imply that the reorientational motion of the BH4− tetrahedra is dominated by C2 rotational motion.11−13 For both C2 and C3 rotation, the energy barrier increases as the disordered states are varied from AAAA to ABBB. Within the accuracy of our adopted approach, the outer boundaries, EAAAA a and EBBBB , define the energy interval in which the activation a energy of the real crystal is located. Furthermore, the Ea of the C2 and C3 rotation increase as we go through the series of alkaline metals, M = Na, K, Rb. In the case of M = Li, the calculated Ea values for the C2 and C3 rotations fit, in general,

a

The superscript of Ea depicts the structural starting point of the given rotation.

In principle, the energies and barriers along the rotational motion of a single BH4− unit could be affected by the symmetric image of such motion in the neighboring cells. In order to check whether our adopted computational model (conventional cell with four formula units) is able to yield realistic energetic results, we calculated the energy profiles and barriers for the C2 rotation of BH4− in KBH4 using cells of increasing size: 1 × 1 × 1, 2 × 1 × 1, 1 × 1 × 2, and 2 × 2 × 2. Along this series of cells, the number of formula units increases as 4, 8, 8, and 32. The initial AAAA disorder model was adopted in all these energy convergence tests. For the directions with doubled cell size, wave functions were obtained using a halved shrinking factor of 4 for reciprocal space sampling and also of four for defining the Gilat net used in the interpolation of the density matrix. As shown in Figure 1 and Table 2 of the Supporting Information, energy differences 2310



Article

the trend described above with increasing atomic number of the alkali metal. However, those for the C2 rotations are either vanishingly small, when the structural starting point is of the AAAA or AAAB type, or, instead, out of such trend for the ABBB type. The negligible energy barriers found for the AAAA and AAAB structure types corroborate why LiBH4 does not exist in the disordered Fm3̅m crystal structure. CsBH4 neatly deviates from the trend of the other alkali metals, as its calculated Ea values for C2 rotations are comparable to those of RbBH4, while those for C3 rotations are significantly smaller than those of RbBH4 and even slightly smaller than those of KBH4. The theoretical variation of Ea partially agrees with the different experimental studies. We concur with the experimental observations that Ea increases from NaBH4 to KBH4.9,11,16 This relationship is also supported by the DFT computations performed by A. Remhof and co-workers.12 However, discrepancies are found for the cases of RbH4 and CsBH4. Based on Raman scattering and NMR, it has been reported that the Ea gradually decreases from KBH4 to RBH4, which opposes our theoretical trend.14,17 In-depth discussion on this issue is delayed to paragraph 3.3, after presenting the results of the in vacuo calculations. Interestingly, an almost perfect inverse relationship exists between the rotational energy barriers and the energy gained by introducing disorder. The relationship is illustrated in Table 1, in which the energy gain due to disorder is demonstrated with respect to the most energetically favorable disordered state, AABB.b Because of the chosen reference, all listed values for the energy gain are negative, denoting that, though stabilized by disorder, all other investigated structures are lesser stabilized than the AABB structure. Due to larger unit cells, the H···H interactions become weaker and the differences of their energy contributions for different disorder configurations become increasingly smaller down the series of alkaline metals.c Hence, the magnitude of the energy gain due to disorder turns out to exhibit a behavior opposite to that of the C2 and C3 energy barriers. The behavior of the energy gain is consistent with the measured phase transition temperatures, which decrease with increasing size of the alkaline metal from Na to Cs. Indeed, the transition is thought to be driven by reducing the number of H···H interactions in the low-temperature phase.9 The larger the cationic size is, the larger the H···H distances are in the high- and low-temperature phases and the lower the enthalpy difference is between these two phases. Thus, the relative transition temperatures reflect the gradual lowering of the enthalpy difference. We shift our focus to the energy landscape of the C2 rotation and the transition between the possible disordered states. The relevant energy curve is illustrated in Figure 2, in which the left and right half depict two separate 90 °C 2 rotations corresponding to the transition between AAAA and AAAB and between AAAB and AABB, respectively. Note that this figure contains all the C2 rotational barriers listed in Table 1, since the activation energy for the backward transition, AAAB → AAAA, is equivalent to that of the ABBB → BBBB transition. Similarly, the rotational barrier for the AABB → ABBB transition equals that of the AABB → AAAB backward transition. Figure 2 shows that the location of the transition states (TS) depends on the disorder and compounds. The TS for rotation in an AAA environment is found at 6.5°, 34.3°, 41.1°, 42.6°, and 42.7° for LiBH4, NaBH4, KBH4, RbBH4, and CsBH4, respectively. Likewise, the TS for rotation in an AAB

Figure 2. Energy landscape for the C2 rotation of a single BH4− unit in the different disordered structures. With the BH4− anion in an initial A-orientation, the left and right halves depict two separate 90 °C2 rotations in an AAA and AAB environment, respectively. The energies are plotted with respect to the energy of the ordered AAAA structure.

environment is positioned at 10.0°, 41.0°, 43.5°, 44.3°, and 44.2°. Interestingly, except for M = Li, the different angular position of the TS can be related to minima in the interdistances between neighboring atoms. The M···H distances are independent of the disorder and have a minimum positioned at 45°. The complete picture of the H···H distances is comprehensive; however, for our discussion it suffices to consider the case with the strongest H···H network, i.e., the AAAA → AAAB transition. During this transition a minimum among the H···H distances is reached at 30°. These two minima enclose the angular area in which the TS is found. However, one cannot clearly ascribe the existence of the energy barrier to the simple effects of the single shortest M···H and H···H distances. Indeed, if taken alone the former effect would cause an energy dip rather than a barrier due to the different sign in the net charges of the H and M atoms. This reasoning assumes that no significant change in the net charges occurs during the rotation that is able to overcome the effect of the internuclear distance decrease.d On the other hand, the study of the C2 rotation of BH4− in the BH4−−BH4− complex in vacuo (see infra) suggests that the destabilizing effect of the H···H contacts due to their large negative charge is slightly released rather than enhanced along such rotation. Clearly, many energetic contributions, besides those of the closest interactions, concur in determining the energy profiles along the BH4− rotational motion, and their disentanglement is a cumbersome and very delicate task. 3.2. In Vacuo Computations. The energy landscape for C2 and C3 rotations of the BH4− anion in the MBH4 crystals is of great interest, but its interpretation is challenging because of the many competing energy contributions. Nonetheless, we show here that the energy profiles obtained from the corresponding C2 and C3 rotations of a BH4− group in M+−BH4− and BH4−− BH4− in vacuo complexes enable one to rationalize the results obtained for the disordered crystals. All the considered molecular in vacuo dimer complexes and their geometrical labeling are exemplified in Figure 3a−h. The energy profiles for C2 rotation of a single BH4− anion from an initial A-orientation to a final B-orientation in three 2311



Article

BH4− and the decrease of the corresponding energy gains obtained by C2 rotation in the BH4−−BH4− dimer complexes. Additionally, rationalization (b) and the declining energy gains depicted in the energy profiles of the in-plane and out-of-plane BH4−−BH4− dimer complexes confirm the inverse relationship observed in section 3.1 between the height of the energy barrier positioned at approximately 45° and the energy gain induced by disorder. The accumulation of the C2 rotational energy differences for the Na to Cs series depicted in Figure 4 suggests that the maximum of the energy barrier in a crystal structure with identically oriented BH4− groups, i.e., the AAAA structure, should be located before 45°. This is indeed the case as illustrated in Figure 2. Moreover, analogous to the energy barrier height, the angular position also moves to higher values with increasing size of the alkaline cation. This behavior is evident from the angular trends shown in Figure 2 and it is a consequence of the differences in the C2 rotational energy profiles, for which the energy barrier present in the M+−BH4− in-plane dimer complex becomes progressively more dominant relative to the energy gains from the BH4−−BH4− dimer complexes. The dominance of the M+−BH4− energy profile is in accordance with the correlation established by Babanova and co-workers between the experimentally NMR determined activation energies for MBH4 (M = Na−Cs) and the relative deviation of the actual M···B distance from the sum of the ionic radii of M+ and BH4−.14 The importance of the M+−BH4− interactions is further underpinned by a NMR study on αMg(BH)4, for which the main features of the coexisting energy barriers could qualitatively be reasoned by the anisotropic coordination of the BH4− group.15 Whether this implies that the barriers for BH4− reorientations are determined solely by M···H interactions, and among these only from the shortest ones, is to some extent discussed in the next paragraph. The dependency of the C2 activation energy on the structural disorder is derived from the rotational energy profiles of the inplane and out-of-plane BH4−−BH4− dimer complexes (see Figure 4). The C2 rotation of a BH4− anion from an initially AA-oriented BH4−−BH4− complex induces an energy gain for both types of complexes. Considering the backward transition, the single BH4− tetrahedron is rotated from B to A conformation starting from an AB-oriented BH4−−BH4− complex. At the approximate position of the TS (≈45°), this results in a smaller energy gain for the out-of-plane dimer complex as compared to the AA case, while even an energy supply is required for the in-plane dimer complex. Since for symmetry reasons the energy profile for BH4− rotation in M+− BH4− complexes is independent of the A/B-orientations, this evinces that the energy barriers for the C2 rotation should increase when the crystal structure contains BH4− groups, whose orientation differs from that of the rotating BH4− group. The trends shown in Table 1 for the C2 energy barriers as a function of disorder confirm that this is indeed the case. For the C3 rotations of a BH4− group in the M+−BH4− complexes, there are two possible arrangements labeled as top and bottom; see Figure 3, c and g. In both conformations the energy profiles depicted in Figure 4 imply that there is an energy barrier located at 60° with the exception of Li+−BH4− bottom, which possesses an energy gain with a minimum positioned at 60°. In the M+−BH4− top complexes the barrier decreases as the atomic number increases from Li to Rb, while in the case of CsBH4 the barrier is slightly higher than found for RbBH4. In the bottom complexes, no energy barrier is found

Figure 3. The molecular in vacuo dimer complexes, which for C2 rotation are (a) M+−BH4− in plane, (b) BH4−−BH4− out of plane and (e) BH4−−BH4− in plane. Regarding C3 rotation they consist of (c) M+−BH4− top, (g) M+−BH4− bottom, (d) BH4−−BH4− top, (h) BH4−−BH4− bottom and (f) BH4−−BH4− in plane. The rotating BH4− unit is emphasized by the relevant rotation axis (dashed line) and its chemical environment is demonstrated by including all neighbors forming equivalent dimer complexes with it. M (brown), B (green) and H (black).

different dimer complexes, Figure 3a,b,e, are illustrated in Figure 4, and appertaining energy values are listed in Table 2. The nonrotating BH4− unit in the BH4−−BH4− complexes is maintained in A-orientation. For all cations, except Li+, the C2 rotation in the M+−BH4− in-plane dimers presents an energy barrier with a maximum positioned at 45°. The height of the barrier rises with increasing atomic number of the alkaline cation until Rb+, after which it slightly decreases for Cs+ to a position between the energy barriers of NaBH4 and KBH4. In the case of Li+−BH4−, the C2 rotation of the BH4− unit induces a comparably smaller energy gain, and the interaction energy profile has a minimum located at 45°. Conversely, the C2 rotation of BH4− in the in-plane and out-of-plane BH4−−BH4− complexes consistently induces a stabilization of the dimers. The energy gain decreases (i.e., becomes less negative) down the series of alkaline metals due to the increase in the distance between the complex moieties. On the basis of these tendencies, we are able to rationalize (a) the negligible barrier found for the AAAA C2 rotation in the LiBH4 crystal; (b) the increase of the same barrier when the alkaline metal is varied from Na to Rb, as the result of the increase in the M+−BH4− rotational energy barrier and the concomitant decrease of the BH4−−BH4− rotational energy gain; and (c) the similar AAAA energy barriers found for the RbBH4 and CsBH4 C2 rotations, which result from the combined effect of a lower energy barrier for the BH4− rotation in the Cs+−BH4− complex as compared to Rb+− 2312



Article

Figure 4. Energy profiles for the C2 and C3 rotation of a single BH4− group in all the in vacuo molecular dimer complexes displayed in Figure 3. For the C2 rotation, the BH4− anion is rotated 90° from an initial A-orientation to a final B-orientation, while for the C3 rotation the BH4− unit is rotated 120° from an initial A-orientation to a final A-orientation. Energies are given in eV with respect to the reference 0° arrangement.

for Li+−BH4−, but a gradually increasing barrier is observed along the series from NaBH4 to RbBH4. For the final compound, CsBH4, it decreases to a position between the energy barriers of NaBH4 and KBH4; see Table 2 and Figure 4.

Since the closest M+−H− bonds are broken during C3 rotation in the bottom complexes, the magnitude of the energy barriers in these complexes are, in general, substantially larger than the corresponding ones of the top complexes and will therefore 2313



Article

Table 2. Energy Differences (eV) between Different Conformations of M+−BH4− and BH4−−BH4− Complexes with Respect to the Reference 0° Arrangement conformation M+−BH4− in plane top top BH4−−BH4− in plane out of plane top top in plane

rotation

LiBH4

NaBH4

KBH4

RbBH4

CsBH4

C2 45° C3 60° C3 60°

−0.011 +0.105 −0.092

+0.032 +0.073 +0.062

+0.059 +0.032 +0.169

+0.065 +0.013 +0.189

+0.046 +0.021 +0.115

C2 C2 C2 C3 C3 C3

−0.032 −0.060 −0.041 +0.005 −0.036 −0.029

−0.016 −0.032 −0.021 +0.004 −0.019 −0.015

−0.006 −0.015 −0.008 +0.005 −0.007 −0.005

−0.003 −0.009 −0.004 +0.004 −0.004 −0.002

−0.001 −0.005 −0.002 +0.003 −0.002 −0.001

45° 45° 90° 60° 60° 60°

employing the static approach in our calculations. This neglects important effects on the atom−atom contact distances due to vibrations; examples are increased anharmonicity and increased amplitude of libration modes with temperature. Since the transition temperature to the high-temperature phase differs for the various alkaline metals, our static model approximation may admittedly have an impact on the barrier heights and on their relative variation as a function of the cation size. Conversely, it is unlikely that the deficiencies of the DFT model are the cause of the observed discrepancy with experiment, because the higher level CCSD(T) model for the in vacuo case predicts trends of the activation barriers entirely similar to those computed with DFT; see Supporting Information. On the basis of the in vacuo model calculations, as mentioned earlier, we could reasonably suggest that the increase we found for the AAAA C2 rotational barrier on going from Na to Rb is the combined result of an increase of the M+−BH4− rotational energy barrier and of a decrease of the BH4−−BH4− rotational energy gain along this series. Does this result also imply that the barriers for BH4− reorientations are mainly determined by the M···H interactions and, among these, only from the shortest ones, as it is, respectively, believed and assumed?9,14 We consider ourselves unable to give a definitive answer to such a question, nor do we believe that it is possible to deduct a definite answer on such issue simply from the contemplation of trends in the ratio of the M···B distances to the sum of the M+ and BH4− radii. Indeed, four metal M···H distances are involved in the C2 rotational path of the studied M+−BH4− dimer complex; one shortens along the rotation and reaches a minimum at 45°, while another one has a starting value identical to that of the first distance, but it increases sharply to a maximum at 45°. The remaining two distances are significantly longer at 0° and become either slightly shortened or lengthened at 45°. A representative drawing for the four M···H distances is shown for LiBH4 in Figure 3 of the Supporting Information. Since the M···H interaction ought to be attractive and energy stabilized by a distance decrease, the observed energy barrier should result from a combination of an energy destabilization due to increasing M···H distances that dominates over the energy gain originating from the decreasing M···H distances. However, the overall energy balance is due to a delicate balance of changes in the atom−atom interaction energies (two-body terms in the energy decomposition recipes; see, e.g., ref 34) and in the self-atom energies (one-body terms in the same recipes). As these terms are not simple functions only depending on the distance, an influential role may also be played by the alterations in the M···B and H···H interaction

play the dominant energetic role. Thus, we anticipate the contribution of the M+−BH4− interactions to the energy barrier in C3 rotations to be very small for Li+−BH4−, subsequently to become larger (i.e., more repulsive) from NaBH4 and RbBH4 and finally to decrease for CsBH4. The BH4−−BH4− complexes can assume three different conformations in C3 rotations labeled as top, bottom, and in plane; see Figure 3, d, f, and h. In these complexes, both BH4− units are in an initial A-orientation, from which one of the BH4− units is rotated 120° onto itself. The BH4−−BH4− top complexes give a small and destabilizing energy contribution at 60°, which is almost insensitive to the atomic number of the cation. The energy profile for the bottom and in-plane BH4−− BH4− complexes indicate that at 60° there is an energy gain, whose amount decreases as the atomic number of the alkaline cation increases. This results from the increasing distance between the complex moieties. Combining the results of the C3 complexes, we expect in the crystal case a minor or negligible energy barrier for C3 rotation in LiBH4. The energy barrier should gradually rise from NaBH4 to RbBH4, and finally decrease for CsBH4. This is indeed the trend observed in the periodic crystal computations, as discussed in section 3.1. 3.3. Reorientational Energy Barriers: Relationships with Recent Experimental Data and with Changes in the Atom−Atom Interactions. As mentioned earlier, the most recent NMR measurements of the 1H and 11B spin−lattice relaxation rates in MBH4 (M = Na, K, Rb, Cs) over a wide range of temperatures (48−400 K) and resonance frequencies (14−90 MHz) suggest that the reorientational energy barrier of the high-temperature phase changes nonmonotonically as a function of the cation size with a maximum found for KBH4.14 The cumulative information from previous NMR studies, limited to a narrower temperature range, suggested instead a comparable energy barrier for KBH4 and RbBH4 and a significant decrease of such barrier for CsBH4.16,36 Our results, both in the solid and in vacuo computations, confirm the existence of a maximum along the alkaline series, though shifted to RbBH4. The RbBH4 maximum is more evident in the in vacuo rather than in the crystal calculations, where, though slightly smaller for CsBH4 in the AAAA and AAAB structures, RbBH4 and CsBH4 exhibit comparable barriers. The results for the model cubic structure of LiBH4 fit with this trend, since the barrier is almost zero in the solid and even reversed for the in vacuo model compounds. The different location of the maximum along the metal series with respect to that found in the most recent NMR experiments may be the result of 2314



Article

Internuclear Li-H distances upon C2 rotation of the BH4− group in the Li+-BH4− in plane complex. This material is available free of charge via the Internet at http://pubs.acs.org.

energies, even if their corresponding atom−atom distances remain the same or almost the same along the rotation, and by the changes in the interaction energies of the directly bonded atoms, i.e., B−H. Likewise, the changes in the self-atomic energies may also concur in determining the barriers. Overall, based on our present model computations, it seems safer and more enlightening to interpret the energy barriers in terms of interactions between the moieties of the complexes, rather than in terms of atom−atom interactions. The latter should not only include the interactions between the atoms of the moieties, but also those within each moiety.

■

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the Danish National Research Foundation for partial funding of this work through the Center for Materials Crystallography (CMC).

4. CONCLUSION In this study, the aim was to accurately assess the dynamical behavior of the BH4− anion in the high-temperature phase of MBH4 (M = Na, K, Rb, Cs) utilizing a more reasonable modeling of the disorder than has ever been accomplished in previous theoretical studies. The Ea for the C2 and C3 rotations have been determined and demonstrated to depend on the local disorder. The outer boundaries of the range of determined Ea define the energy interval in which the Ea of the real crystal is presumably located. The stabilizing energy gain originating from the introduction of disorder is found to be within 0.032− 0.276 eV depending on the compound and disorder. The energy gain and barrier energies show opposite trends down the alkali series. In terms of simple molecular models, we rationalize the Ea trends and the energy profiles as a function of rotation and local disorder. This enables a simple, neat view of the main molecular group mechanisms beneath the observed barriers and profiles. In particular, we provide hopefully convincing evidence that the C2 and C3 rotational barrier energy trends from Na to Rb are the combined result of an increase of the M+−BH4− rotational energy barrier and of a concomitant decrease of the BH4−−BH4− rotational energy gain along this metal series. This result does not necessarily imply that the barriers for BH4− reorientations are determined by the dominance of the overall energy destabilization of the M···H interactions over the energy gain due to the concomitant stabilization of the H···H interactions. Other energetic effects may play an important role. An atomistic and energetic view of the BH4− rotation in the gas phase and in the disordered crystal calls for a further analysis. To this end, we are presently following a dual strategy which combines the evaluation of the changes of bonding and net atomic charges during the rotation through the quantum theory of atoms in molecules with the computation of the corresponding changes in the self-and interaction energies of the atoms through the interacting quantum atom approach.34 We do not anticipate such study to provide a full detailed energetic dissection of the process due to the huge number of interactions and to their minor individual energy changes during rotation. However, it will provide important insights into the atom−atom mechanisms concurring to the observed energy profiles.

■

AUTHOR INFORMATION

■

ADDITIONAL NOTES Owing to the adopted constraint on cell parameters, we did not introduce the customarily used London-type empirical energy correction for dispersion interactions. b This result is not at odds with the low-temperature phase being structurally ordered. The ordered low-temperature phase differs from the AAAA structure in terms of symmetry and the H···H network. c On the basis of electron density parameters, a topological analysis of the electron density highlights the presence of such interactions and their decreasing strength with increasing cell size (manuscript in preparation). As it depends on the adopted approach, it is difficult to assess whether these interactions are energy stabilizing or destabilizing. For instance, within the interacting quantum atom approach34 a positive interaction energy (destabilization) seems to be predicted35 for the observed H···H distances in the full series of MBH4 (M = Li, Na, K, Rb, Cs). However, what matters in the present context is that the energy stabilizing or destabilizing relevance of such interactions clearly decreases with increasing cell size. d Evaluation of changes of the Bader’s net charges along the rotation confirms the validity of such assumption. a

■

REFERENCES

(1) Züttel, A.; Rentsch, S.; Fischer, P.; Wenger, P.; Sudan, P.; Mauron, Ph.; Emmenegger, Ch. J. Alloys Compd. 2003, 356−357, 515−520. (2) Grochala, W.; Edwards, P. P. Chem. Rev. 2004, 104, 1283−1315. (3) Vajo, J. J.; Olson, G. L. Scr. Mater. 2007, 56, 829−834. (4) Nielsen, T. K.; Bosenberg, U.; Gosalawit, R.; Dornheim, M.; Cerenius, Y.; Besenbacher, F.; Jensen, T. R. ACS Nano 2010, 4, 3903− 3908. (5) Rude, L. H.; Nielsen, T. K.; Ravnsbaek, D. B.; Bosenberg, U.; Ley, M. B.; Richter, B.; Arnbjerg, L. M.; Dornheim, M.; Filinchuk, Y.; Besenbacher, F.; Jensen, T. R. Int. J. Hydrogen Energy 2011, 36, 15664−15672. (6) Renaudin, G.; Gomes, S.; Hagemann, H.; Keller, L.; Yvon, K. J. Alloys Compd. 2004, 375, 98−106. (7) Johnston, H. L.; Hallett, N. C. J. Am. Chem. Soc. 1953, 75, 1467− 1468. (8) Stephenson, C. C.; Rice, D. W.; Stockmayer, W. H. J. Chem. Phys. 1955, 23, 1960−1960. (9) Babanova, O. A.; Soloninin, A. V.; Stepanov, A. P.; Skripov, A. V.; Filinchuk, Y. J. Phys. Chem. C 2010, 114, 3712−3718. (10) Soulié., J-Ph.; Renaudin, G.; Cerný, R.; Yvon, K. J. Alloys Compd. 2002, 346, 200−205. (11) Verdal, N.; Hartman, M. R.; Jenkins, T.; DeVries, D. J.; Rush, J. J.; Udovic, T. J. J. Phys. Chem. C 2010, 114, 10027−10033.

ASSOCIATED CONTENT

S Supporting Information *

Periodic DFT computations performed on the AAAA to AABB structures for MBH4, M = Li−Cs :a) selection of DFT functional and of the computational model to minimize simulation artifacts; b) details on the estimation of the energy barriers; c) details on basis sets. In vacuo computations: a) CCSD(T) results and their comparison with DFT results; b) 2315



Article

(12) Remhof, A.; Lodziana, Z.; Martelli, P.; Friedrichs, O.; Zuttel, A.; Skripov, A. V.; Embs, J. P.; Strassle, T. Phys. Rev. B 2010, 81. (13) Remhof, A.; Lodziana, Z.; Buchter, F.; Martelli, P.; Pendolino, F.; Friedrichs, O.; Zuttel, A.; Embs, J. P. J. Phys. Chem. C 2009, 113, 16834−16837. (14) Babanova, O. A.; Soloninin, A. V.; Skripov, A. V.; Ravnsbæk, D. B.; Jensen, T. R.; Filinchuk, Y. J. Phys. Chem. C 2011, 115, 10305− 10309. (15) Skripov, A. V.; Soloninin, A. V.; Babanova, O. A.; Hagemann, H.; Filinchuk, Y. J. Phys. Chem. C 2010, 114, 12370−12374. (16) Tsang, T.; Farrar, T. C. J. Chem. Phys. 1969, 50, 3498−3502. (17) Hagemann, H.; Gomes, S.; Renaudin, G.; Yvon, K. J. Alloys Compd. 2004, 363, 126−129. (18) Zavorotynska, O.; Corno, M.; Damin, A.; Spoto, G.; Ugliengo, P.; Baricco, M. J. Phys. Chem. C 2011, 115, 18890−18900. (19) Dovesi, R.; Saunders, V. R.; Roetti, C.; Orlando, R.; ZicovichWilson, C. M.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N. M.; Bush, I. J.; D’Arco, Ph.; Llunell, M. CRYSTAL06 User’s Manual; University of Torino: Torino, Italy, 2006. (20) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hatching, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian03, Revision B.05; Gaussian, Inc.: Wallingford, CT, 2004. (21) Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297−3305. (22) Leininger, T.; Nicklass, A.; Kuechle, W.; Stoll, H.; Dolg, M.; Bergner, A. Chem. Phys. Lett. 1996, 255, 274−280. (23) Becke, A. D. J. Chem. Phys. 1993, 98, 5648−5652. (24) Lee, C. T.; Yang, W. T.; Parr, R. G. Phys. Rev. B 1988, 37, 785− 789. (25) Perdew, J. P. Electronic Structure of Solids; Akademie Verlag: Berlin, 1991. (26) Perdew, J. P.; Yue, W. Phys. Rev. B 1986, 33, 8800−8802. (27) Perdew, J. P. Phys. Rev. B 1989, 40, 3399−3399. (28) Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, 13244−13249. (29) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865−3868. (30) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158−6170. (31) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Phys. Rev. Lett. 2008, 100, 136406. (32) Filinchuk, Y.; Hagemann, H. Eur. J. Inorg. Chem. 2008, 3127− 3133. (33) Grimme, S. J. Comput. Chem. 2006, 15, 1787−1799. (34) Blanco, M. A.; Pendas, A. M.; Francisco, E. J. Chem. Theory Comput. 2005, 1, 1096−1109. (35) Pendás, A. M.; Francisco, E.; Blanco, M. A.; Gatti, C. Chem. Eur. J. 2007, 13, 9362−9371. (36) Tarasov, V. P.; Bakum, S. I.; Privalov, V. I.; Shamov, A. A. Russ. J. Inorg. Chem. 1990, 35, 2096−2099.

2316


A Theoretical Study on the Rotational Motion and ... - ACS Publications

Recommend Documents