Understanding the Reorientational Dynamics of Solid-State MBH4 (M

Article pubs.acs.org/JPCC

Understanding the Reorientational Dynamics of Solid-State MBH4 (M = Li−Cs) Niels Bindzus,† Fausto Cargnoni,*,‡ Bo B. Iversen,† and Carlo Gatti‡ †

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

ABSTRACT: The reorientational dynamics of crystalline MBH4 (M = Li− Cs) have been characterized with the interacting quantum atom theory. This interpretive approach enables an atomistic deciphering of the energetic features involved in BH4− reorientation using easily graspable chemical terms. It reveals a complex construction of the activation energy that extends beyond interatomic distances and chemical interactions. BH4− reorientations are in LiBH4 and NaBH4 regulated by their interaction with the nearest metal cation; however, higher metal electronic polarizability and more covalent M···H interactions shift the source of destabilization to internal deformations in the heavier systems. Underlying electrostatic contributions cease abruptly at CsBH4, triggering a departure in the otherwise monotonically increasing activation energy. Such knowledge concurs to the fundamental understanding and advancement of energy solutions in the field of hydrogen storage and solid-state batteries. dynamics of the BH4− anion is a key point to fully understand the chemical behavior of the MBH4 compounds and how their physical properties may be tailored. This knowledge will also assist in the broader understanding of BH4− dynamics, occupying defining roles in other promising materials such as the novel bimetallic borohydrides and complex hydride perovskites.17,18 Within this scope, theoretical insights are further motivated by noticeable discrepancies among the experimental activation energies for BH4− reorientation. In accord with nudge-elasticband calculations, quasi-elastic neutron scattering (QENS) and nuclear magnetic resonance (NMR) both indicate that BH4− dynamics are governed by isotropic reorientations around the tetrahedral C2 axes.19−21 No general consensus, however, has been established on the trend of activation energies as a function of the metal cation. For example, Babanova et al. have by means of NMR recently estimated the activation energies to be 0.126 (3), 0.161 (2), 0.138 (4), and 0.105 (7) eV for MBH4, M = Na−Cs.22,23 Contradictions to this trend may be found in a QENS study reporting a higher activation energy for NaBH4 (0.113 (1) eV) than for KBH4 (0.103 (2) eV) and in a Raman study reporting a monotonic decrease from 0.125 (5) to 0.085 (4) eV along the complete series of alkaline metals.20,24 Another important incentive for our studies is the neglect of disorder effects in preceding theoretical investigations.20,25

1. INTRODUCTION The family of alkaline metal borohydrides, MBH4 (M = Li− Cs), represents a technologically important class of materials. Apart from their classical application as reducing agents in organic and inorganic chemistry, they have attracted great interest because of their potential for hydrogen storage and fastionic conduction. The development of operational hydrogenstorage systems remains challenged by slow sorption kinetics and substantial dehydrogenation temperatures.1,2 Current efforts to address these issues involve methods such as anion substitution and nanoconfinement.3−5 Motivated by prospective applications in solid-state batteries, fast-ionic conduction has been extensively studied in LiBH4 and derivatives.6−8 Furthermore, a recent breakthrough opened this research area up to sodium-based systems.9,10 At high temperatures, the four MBH4 (M = Na−Cs) compounds crystallize in a cubic lattice within the Fm3̅m space group.11 The intriguing feature of their crystal structure is thermally activated disorder among the BH4− moieties, which randomly reorient themselves in the directions. On the macroscopic time scale, the hydrogen atoms enclose the boron atom in a cubic arrangement, where each site has a 50% probability of being occupied. The transition to the hightemperature phase occurs at 190 K for NaBH4, 76 K for KBH4, 44 K for RbBH4, and 27 K for CsBH4.12,13 As an outlier in the MBH4 family, LiBH4 adopts a hexagonal high-temperature phase in the P63cm space group at 380 K; however, an isostructural phase may be produced if an external pressure of 4 GPa or higher is imposed.14−16 Unravelling the complex © 2015 American Chemical Society

Received: January 28, 2015 Revised: May 7, 2015 Published: May 7, 2015 12109

DOI: 10.1021/acs.jpcc.5b00899 J. Phys. Chem. C 2015, 119, 12109−12118

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The Journal of Physical Chemistry C

Figure 1. Structural overview of C2 and C3 reorientations in the M+BH4− and BH4−BH4− dimer complexes. Each case is rigidly extracted from the crystal and comprises a rotating BH4− together with one of its first-neighboring moieties; symmetry-equivalent neighbors are shown in transparency. The initial positions of the hydrogen atoms are given in blue, whereas new positions emerging after completed rotation are given in red. The C2 branch distinguishes between in-plane and out-of-plane neighbors. A more detailed classification is implemented in the C3 branch, separating the latter into a top and bottom case. In their visualization, the neighbor is located above the rotating moiety. Metal cation (purple) and boron (green).

actual M···B distance from its ionic expectation.22,23 This simple interpretation falls in line with the belief that BH4− dynamics primarily are governed by the shortest M···H interactions.20,22,23,30−32 More elaborate insights into the key mechanisms are afforded by our model cluster approach previously outlined. It highlights the role of changes in M···H distances, that is, M···H interaction energies, and of lengthened H···H distances, that is, presumably less destabilizing H···H interactions. At this level of understanding, however, it remains unsettled whether the dominant role of M+···BH4− destabilizations implies that the reorientational energy barriers are governed by M···H interactions and among these only from the shortest ones as generally believed. It is impossible to deduct a definite answer on such matters merely from consideration of interatomic distances. For instance, four M···H distances participate in the C2 pathway of the M+BH4− dimer complexes. Since M···H interactions ought to be attractive and energy-stabilized by a distance decrease, the reorientational energy barrier must necessarily be the cumulative result of destabilizations from lengthened distances exceeding gains from reduced distances. Furthermore, another point remains even more puzzling. In addition to interaction energies, the overall energy balance also depends on internal energy changes within the moieties themselves. These are a direct consequence of perturbations caused by the modifications in the intergroup interactions. Such effects entail that even the M···B interaction energy might exhibit significant

These were restricted to NaBH4 and KBH4, assuming a simplified structural representation based on identically aligned BH4− moieties. In a preceding study, we expanded upon this by addressing the energetic implications of structural disorder in the complete series of MBH4 (M = Li−Cs) compounds.26 The in-crystal energy barriers for C2 and C3 reorientation were successfully rationalized in terms of in vacuo calculations on M+BH4− and BH4−BH4− dimer complexes (structural overview in Figure 1). This discloses that the barriers derive from a dominant destabilization in the M+···BH4− interaction, which along the alkaline series is counteracted by increasingly smaller energy gains from relaxations in the H···H network. Experimental evidence of this chemical interplay was recently provided in a second study, mapping the atomic charges and the entire bonding network in the crystal structure of KBH4.27 Both the periodic and in vacuo computations identify a maximum in the activation energies positioned at RbBH4.26 Interestingly, this opposes the most recent NMR measurements, reporting a maximum shifted to KBH4.22,23 This difference might be attributed to the adoption of a static approach in the calculations, although the cumulative information from previous NMR studies cannot be ignored. These are limited to a narrower temperature range but imply comparable activation energies for KBH4 and RbBH4, followed by a significant decrease for CsBH4.28,29 Babanova et al. correlated the nonmonotonic variation in their determined activation energies to the deviation of the 12110


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The Journal of Physical Chemistry C alterations despite its fixed distance. Clearly, similar considerations apply to the role of H···H interaction energies and internal BH4− deformations in causing the reorientational energy gain of the BH4−BH4− systems. These challenging issues are afforded in the present work, revealing the factors leading to the reorientational energy barriers. Following a top-down approach, we employ the interacting quantum atom (IQA) theory to map the energy landscape at the moiety and atomistic level.33,34 This probes the fundamental chemistry by a rigorous decomposition of the overall energy in terms of a balance between atom−atom interaction energies and atomic self-energies. Augmenting the descriptive conclusions of our first study,26 the IQA analysis enables us to infer a firm rationale for the reorientational energy profiles and the different impacts of the alkaline metals. The main analysis focuses on the energetics of the M+BH4− clusters in Section 3, whereas secondary effects from the BH4−BH4− clusters are addressed in Section 4.

The intrabasin terms yield self-energies for the individual moieties, whereas the interbasin terms form the interaction energy for unique pairs of moieties. Note that in the IQA approach the interaction energy includes contributions from pairs of atoms not connected through a bond path or, if the system is outside of equilibrium, an atomic interaction line.35 The en, ee, and nn subscripts refer to electron−nucleus, electron−electron, and nucleus−nucleus interactions between superscripted pairs of basins. Although the IQA approach permits the implementation of any type of fuzzy and exhaustive space decompositions, the present study adopts the exhaustive QTAIM partitioning to uniquely define each TA and not just their total sum. By introducing reference energies based on the isolated monomers, the corresponding deformation energies are derived as A ref A A Edef = Eself − Eself

In the present study, the isolated M+ and BH4− moieties serve as suitable points of reference. Equation 5 yields to the binding energy of the system in terms of a balance between internal deformations (or promotion, usually positive) and intergroup interactions (overall negative)

2. THEORETICAL AND COMPUTATIONAL METHODS 2.1. Interacting Quantum Atom Theory. Within Bader’s quantum theory of atoms in molecules (QTAIM),35 the total energy, E, of a system is partitioned into additive atomic contributions E = −T = −∑ TA = A

E bind =

A ∑ Eself

∑ EintAB

+

A

E int = Vcl + Vxc =

⎛ dr1 ⎜⎜T̂ − ΩA ⎝

∑∫ A

+

1 2

∑∫ A ,B

ΩA

dr1

∑ B

ZB ⎞ ⎟ρ (r1; r1) r1B ⎟⎠ 1

∫Ω dr2 B

ρ2 (r1, r2) r12

+

∑ A>B

ZAZB RAB

(3)

where ΩA denotes that the integration extends over the basin of moiety A. By grouping these energy terms into meaningful objects, the energy decomposition of eq 2 emerges E=

∑ (TA + VenAA + VeeAA) + ∑ (VnnAB + VenAB + VenBA A

+ VeeAB) =

A>B A ∑ Eself A

+

∑ EintAB A>B

(6)

A>B

∑ VclAB + VxcAB

(7)

This complex energy dissection constitutes the IQA theory and is a powerful tool to probe the underlying chemistry of functional materials. For instance, its applications have concurred to the understanding of hydrogen, halogen, and transition-metal bonding,37−41 revealed steric repulsion in molecular systems,42,43 and elucidated the origin of high metastability in an endohedral complex.44 2.2. Computational Details. The wave functions were computed at the RHF level using the GAMESS-US computational code.45 The RHF, rather than the DFT method, is used as the former enables an exact partitioning of the pair function into a classical Coulombic and a pure quantum mechanical exchange term due to antisymmetry. The IQA electron− electron interaction energy is therefore decomposed in similar manner by replacing the Vxc and VAB xc terms in eq 7 with Vx and VAB x . All-electron Def2-TZVP basis sets are assigned to all atoms.46 The consistency with higher quality B3LYP and CCSD(T) computations using identical basis sets verifies the energy trends determined at the Hartree−Fock level of theory.26 For each compound, local symmetry-breaking necessitates the consideration of four M+BH4− and seven BH4−BH4− clusters. These cover the starting, middle, and ending point of the C2 and C3 pathway of a central BH4− moiety with respect to its nearest neighbors (Figure 1). Their geometries are rigidly extracted from the periodic computations published in our first paper.26 For matter of consistency, the Fm3̅m structure is imposed on the outlier system of LiBH4. The disregard of second-order neighbors is justified by their

(2)

A>B

∑ EintAB

A>B

Within the present application of the IQA methodology, the labeling scheme of A and B covers any moiety composed of a single or a collective set of atoms. Taking both electronic and nuclear contributions into account, eq 2 follows from the total self-consistent-field energy expressed in terms of the first- and second-order density matrices, ρ1, ρ236 E=

+

Interactions are read in a chemically relevant scale by decomposing the second-order density matrix into its Coulombic and exchange-correlation contribution, ρ2 = ρC2 + ρxc 2 . This enables the intermoiety energy to be divided into a AB AB BA AB classical, VAB cl = Vnn + Ven + Ven + VC , and a nonclassical exchange-correlation term, VAB , yielding xc

(1)

A

where A represents the atomic basins of the system. The wellknown zero-flux recipe delimits the portion of space belonging to each atomic basin and provides a unique definition for the kinetic energy, TA, of these subdomains.35 This fundamental partitioning, however, conceals the interatomic interactions within a single additive contribution and applies only to equilibrium configurations, that is, those where no net forces act on the nuclei and where 2T = −V. These shortcomings are eliminated in the IQA approach by introducing an energy partitioning composed of one- and two-body terms33,34 E=

∑ EdefA A

∑ EA

(5)

(4) 12111


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The Journal of Physical Chemistry C negligible influence on the activation energies.26 Note that the geometry of the BH4−BH4− dimer complexes also depends on the reference metal cation owing to variations in B−H, B···B, and H···H distances. One could object that our cluster models do not take the atomic environment of the crystal into account; however, for all intents and purposes they do. First, as previously explained, the adopted geometries correspond to the crystal arrangements, ensuring that the geometrical effect of the bulk is considered. Second, because we focus on energy differences rather than on their absolute values, our models do not neglect the crystal environment, but only the effect of its change along the rotational path. This has proved to be an acceptable approximation as our in vacuo models nicely interpret the energy profiles obtained for C2 and C3 reorientations in the crystal.26 We also note that charge neutrality is violated in the BH4−BH4− complexes, which might cause charge distributions that deviate from the crystal. We checked that such differences are indeed not relevant, as the crystalline charge of the BH4− unit almost equals the formal value of −1. Even more importantly, we emphasize that our simple model approach allows the lack of charge neutrality to affect the energy of all rotation points as long as the induced changes are equally distributed. A similar reasoning also applies to the corresponding charge distributions. The IQA calculations were carried out with the PROMOLDEN code.47 The IQA integration employs 1200 radial points and an angular Lebedev quadrature consisting of 974 points inside the β-sphere. Outside the β-sphere (up to 10 Å from the nucleus), the number of radial points is maintained, whereas the angular Lebedev quadrature increases to 5810 points. To reproduce the interaction energies, it is necessary to linearly scale the IQA energy terms in such manner that the total kinetic and potential energies exactly match the corresponding values obtained from the RHF computations. This postcorrection assumes that numerical integration errors, in percentage, have a similar effect on all atomic energy quantities and their counterparts for the total system.

3. IQA DECOMPOSITION IN M+BH4− 3.1. C2 Reorientation. The fundamental mechanisms of C2 reorientations are probed by analyzing the energetics of the inplane M+BH4− dimer complexes. If perturbed by a crystalline environment, the transition state depends on local disorder as well as the type of metal cation.26 Nonetheless, approximate transition states constructed from a 45° C2 rotation of the BH4− moiety are adequate for a qualitative understanding of the energetic features (Figure 1). Figure 2a illustrates the development of the C2 energy barrier and its underlying contributions at the moiety level. Within the IQA formalism, the energy barrier corresponds to the alteration in the binding energy, ΔEbind. Consistent with periodic calculations,26 the BH4− moiety reorients freely in Li+BH4−, after which rotational friction emerges. This energy barrier increases continuously until peaking for Rb+BH4− and subsequently dropping to an intermediate level for Cs+BH4−. Along this series of alkaline metals, the IQA dissection reveals a complex construction of the reorientational energy barrier. The only reliable feature is the counterbalancing of unfavorable energy alterations by a stronger M···H interaction for the hydrogen atom approaching the metal cation, inducing a substantial increase in the overall exchange stabilization, ΔVx (Table 1). The electrostatic interactions are contained in the Vcl

Figure 2. Reorientational modifications in the Interacting-QuantumAtom quantities for the three different types of M+BH4− dimer complexes. (a) Transformation of the 0° in-plane cluster into its C2 transition state, whereas it is the C3 counterpart for (b) the 0° bottom and (c) the 0° top cluster.

term, whose changes contribute significantly to the onset of the energy barrier with the exception of the final dimer complex, Cs+BH4−. At this end point, their abrupt decrease accounts for the nonmonotonic departure in the activation energies. Summing up the two quantities, ΔVx and ΔVcl, results in the 12112


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The Journal of Physical Chemistry C Table 1. Interacting-Quantum-Atom Decomposition of the M+BH4− Dimer Complexesa M+BH4− Li

Na

K

Rb

Cs

0° 45° 60° 60° 0° 45° 60° 60° 0° 45° 60° 60° 0° 45° 60° 60° 0° 45° 60° 60°

C2 IP C3 B C3 T C2 IP C3 B C3 T C2 IP C3 B C3 T C2 IP C3 B C3 T C2 IP C3 B C3 T

EM def

4 EBH def

Edef

Vcl

Vx

Eint

Ebind

dMH

VMH cl

VMH x

EMH int

DIMH

−0.18 −0.16 −0.10 −0.16 −0.17 −0.13 −0.05 −0.16 −0.01 0.12 0.25 −0.01 0.01 0.16 0.29 0.03 0.07 0.13 0.36 0.06

0.45 0.42 0.51 0.38 0.42 0.38 0.47 0.37 0.42 0.37 0.60 0.34 0.48 0.42 0.69 0.34 0.50 0.55 0.69 0.36

0.27 0.26 0.40 0.22 0.24 0.24 0.42 0.20 0.41 0.49 0.85 0.33 0.50 0.58 0.98 0.37 0.57 0.69 1.05 0.42

−5.29 −5.24 −5.32 −5.18 −4.74 −4.62 −4.52 −4.71 −4.28 −4.20 −4.18 −4.27 −4.10 −4.03 −4.06 −4.06 −3.94 −3.94 −3.97 −3.89

−0.27 −0.33 −0.48 −0.21 −0.40 −0.48 −0.72 −0.31 −0.55 −0.63 −0.88 −0.44 −0.52 −0.59 −0.81 −0.42 −0.53 −0.59 −0.81 −0.44

−5.56 −5.57 −5.80 −5.38 −5.14 −5.11 −5.24 −5.02 −4.83 −4.83 −5.05 −4.71 −4.61 −4.62 −4.87 −4.48 −4.47 −4.53 −4.78 −4.34

−5.29 −5.30 −5.40 −5.16 −4.90 −4.86 −4.82 −4.81 −4.42 −4.35 −4.20 −4.38 −4.12 −4.04 −3.89 −4.11 −3.90 −3.84 −3.73 −3.91

2.41 2.04 1.77 2.41 2.56 2.19 1.92 2.56 2.80 2.43 2.17 2.80 2.94 2.57 2.32 2.94 3.12 2.76 2.51 3.12

−4.75 −5.56 −6.03 −4.82 −4.30 −4.85 −5.09 −4.36 −3.90 −4.36 −4.58 −3.94 −3.71 −4.10 −4.36 −3.72 −3.54 −3.93 −4.21 −3.54

−0.11 −0.27 −0.43 −0.12 −0.17 −0.40 −0.63 −0.18 −0.24 −0.51 −0.77 −0.25 −0.23 −0.47 −0.72 −0.23 −0.23 −0.47 −0.71 −0.24

−4.86 −5.83 −6.46 −4.94 −4.47 −5.25 −5.72 −4.54 −4.14 −4.86 −5.36 −4.19 −3.93 −4.57 −5.08 −3.95 −3.77 −4.40 −4.93 −3.78

0.028 0.059 0.084 0.030 0.040 0.084 0.124 0.042 0.056 0.108 0.153 0.058 0.053 0.102 0.145 0.055 0.057 0.104 0.150 0.058

a

In-plane, bottom, and top conformations are abbreviated as IP, B, and T. Total quantities are listed without a superscript, which otherwise depicts the moiety or interaction in question. Delocalization Index (DI) and the distance, d (Å), are included for the shortest H···M interaction. All energies in electronvolts.

change of the overall interaction energy, ΔEint, which apart from Na+BH4− and Cs+BH4− approaches zero due to cancellation effects. Comparison of ΔEint and ΔEbind unveils that the former parallels the energy barrier only for the two lightest complexes, while modifications in the intrabasin energy terms undertake the main role for the heavier systems. This defining transition in the fundamental energetics is represented by an abrupt growth in the change of the total deformation energy, ΔEdef. After increasing to a level comparable to ΔVcl in K+BH4− and Rb+BH4−, it becomes the only source of destabilization in Cs+BH4−. The chemical nature of the reorientational energy barrier is closely connected to the diverse behavior of the deformation energy. Contrary to the formation of the dimer complexes from the isolated monomers (see details in Supporting Information), charge transfer (CT) induced during BH4− reorientation is negligibly small because the QTAIM charges merely vary within a few millielectrons. Alterations in Edef must therefore originate from charge reorganization (CR) taking place within the moieties themselves.48,49 The deformation energy of the BH4− moiety undergoes favorable changes during reorientation around Li+−Rb+ (Figure 2a). For the lightest systems, this gain is neutralized by destabilizing deformations in the metal cation. These destabilizations, however, rise substantially in the heavier systems of K+ and Rb+, assuming a commanding role in the construction of the energy barrier. In line with the deviating behavior displayed by its interaction energy, the end dimer complex of Cs+BH4− once again stands out with destabilizing deformations of equal magnitude in both its constituents. Extrapolating the molecular insights into the crystal level, it is feasible to infer the following important conclusions. The global interaction energy is the key component in the lightest systems, single-handedly accounting for the free reorientation in LiBH4 and for the onset of the energy barrier in NaBH4. In contrast, modifications in the underlying mechanisms trigger the entry of dominant CR effects in the heavier crystals,

KBH4−CsBH4. The CR activation is a direct consequence of the greater electronic polarizabilities of K+−Cs+ and of the more shared M···H interactions, details in Sections 3.3 and 3.4. Lastly, the monotonic increase in the reorientational energy barrier ceases at CsBH 4 in response to the sudden disappearance of electrostatic contributions. 3.2. C3 Reorientation. To probe the energetically more expensive reorientations around the C3 axes, it is necessary to consider both a top and bottom M+BH4− dimer complex owing to symmetry breaking among the first-neighboring metal cations (Figure 1). The transition state corresponds in all cases to a 60° C3 rotation of the BH4− moiety,26 and the induced alterations are graphically summarized in Figure 2b,c. The energetics of the bottom dimer complexes resemble to some extent those of C2 reorientations, as one of its nearest M···H distances is shortened, too (Table 1). This structural change, however, is even more pronounced for the C3 case, leading to stronger perturbations of the two interacting moieties. In consequence, unfavorable deformations contribute considerably to the reorientational energy barrier in even the lightest dimer complexes. At variance with the mechanism behind C2 reorientation, both constituents consistently experience destabilizing deformations. In the case of the top dimer complexes, no shortening of the M···H distances takes place (Table 1) and a different energy landscape therefore arises. First of all, the activation energy decreases monotonically and even turns into an energy gain for the final dimer complex, Cs+BH4−. In an almost mirrored picture, the energy barrier results from unfavorable changes in both the exchange and classical component of the interaction energy, which are counteracted by stabilizing deformations. The BH4− component governs the latter. Merging the energetic insights from the top and bottom dimer complexes, the total C3 energy barrier materializes as a complex quantity, whose exact reconstruction at no point can be ascribed to a single dominant term. Interesting conclusions 12113


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The Journal of Physical Chemistry C may, nevertheless, be drawn for the heavy metal regime, K+− Cs+, as the overall energy contribution from the top complex diminishes to a negligible level. The molecular insights from the dominant bottom clusters corroborate that unfavorable deformations within both moieties govern the onset of the energy barrier in KBH4−CsBH4 crystals. They furthermore account for the energy barrier peak at RbBH4 predicted by periodic calculations.26 These features concur closely with those of the C2 reorientation; however, discrepancies transpire when comparing the underlying landscape. First of all, the higher energy cost of C3 reorientations is the result of augmented deformations as both constituents are destabilized. Second, none of the subcomponents experience drastic changes. The nonmonotonic departure in the C3 activation energy can therefore not be attributed to an isolated event. 3.3. Deformation Energy. The IQA dissections previously presented taught us that the importance of internal deformations rises abruptly in the transition to the heavier complexes of K+−Cs+. The fundamental nature of this interesting phenomenon is explored by contemplating the complete series of 0° conformations (Table 1). The deformation energy suffered by the metal cation upon complex formation increases progressively down the series of alkaline metals; however, a distinct jump separates the initial region, Li+−Na+, from the final region, K+−Cs+. A stabilizing deformation characterizes the former as the gain in electron− nucleus energy more than compensates for the overall increase in kinetic energy and electron−electron repulsion. Such compensation ceases in the final region, where the energetic balance starts to shift in favor of the destabilizing terms. M The characteristic domains of Edef are rationalized by considering its underlying CT and CR component (Table 2).48,49 These two phenomena are decoupled by a two-step

The group trends of decreasing IE and increasing electronic polarizability, α, are able to account for the entry of significant CR contributions. Such IE lowering is a well-known fact from elementary physical chemistry textbooks and reproduced by our computations, whereas the enhancement of α is exemplified by experimental determinations on alkaline chloride crystals as the influence of the crystal environment is negligible for cationic cases (Table 2).50,51 These two quantities hold apparent relationships to the CT and CR component because IE modulates the energy change caused by CT, while α measures how easily the electron density is reorganized. The dominating CT effects for Li+ and Na+ are therefore a direct consequence of their high IE and low α. Appreciable alterations in IE and α activate the CR effects in the heavy-atom domain of K+−Cs+. Directing the focus to the other constituent of the dimer complexes, it is observed that the BH4− moiety consistently experiences deformations that are destabilizing and much larger than its counterpart, EM def (Table 1). This feature most likely originates from concurrent CT and CR destabilizations, as suggested by the loss of electrons compared with the fully ionized reference and by the sizable electronic polarizability of BH4−, which for MBH4 crystals has been experimentally established to exceed that of Cs+.52 Owing to the commanding 4 role of EBH def , the total deformation energy raises the binding energy of all dimer complexes. Such behavior concurs with the classical IQA picture, outlining that upon interaction binding results from the competition between destabilizing deformations suffered by the interacting moieties and stabilizing intermoiety contributions.37 3.4. Interaction Energy. The two-body components of the C2 and C3 activation energies are explored in greater detail by combining the relevant energetics of all the different systems and conformations. This produces an overall picture composed of a consistently attractive binding energy between M+ and BH4−, which ranges from about −5 to −4 eV (Table 1). The governing component is the balance between repulsive M···B and attractive M···H interactions, as the deformation energy merely constitutes ∼5% of the binding energy. The evolution of the M+···BH4− interaction is quantified by the series of 0° conformations. As the metal cation gradually changes from Li+ to Cs+, the binding energy deteriorates because the electrostatic interactions diminish in response to growing interdistances. Contrasting this progressive decrease, the importance of Vx ascends from Li+ to K+ before leveling out at −0.5 eV for the two final clusters. The growth of Vx is inherently linked to the electronic polarizability of the alkaline metals and their CR as stronger perturbations of the moieties accompany more shared interactions. Among the atomic quantities, it is essential to fully characterize the M···H interaction as many speculations regarding its role in BH4− dynamics have previously been published.20,22,23,30−32 Figure 3 establishes that the five sets of EMH int share a similar trend, remaining attractive over the entire range of computed distances. The metal cation and the exact B−H···H arrangement therefore occupy minor roles in the determination of the M···H interaction energy, which to a good approximation is regulated by the internuclear distance. At the shortest separations, this simple relationship is upheld owing to cancellation between the two components of the interaction energy. Proceeding down the group of alkaline metals, the MH increase in VMH cl is counterbalanced by a more stabilizing Vx . The latter entails that at a given distance the partially shared

Table 2. Charge Transfer (CT) and Charge Reorganization (CR) Components of the Cation Deformation in the 0° Conformationsa M+BH4−

EM def

IE

q(M)

CT

CR

αb

LiBH4 NaBH4 KBH4 RbBH4 CsBH4

−0.179 −0.173 −0.006 0.012 0.068

5.341 4.949 4.008 3.739 3.488

+0.967 +0.959 +0.957 +0.969 +0.975

−0.176 −0.202 −0.173 −0.117 −0.087

−0.003 0.030 0.167 0.129 0.155

0.19 1.11 5.77 9.95 17.66

a

Ionization energy (IE) is calculated as the difference between the Hartree-Fock energy of M and M+. All energies are in electronvolts, while the electronic polarizability, α, is in atomic units. bValues from ref 51.

process that describes the transformation of the metal cation from its isolated to its interacting state by an initial CT followed by an isoelectronic redistribution of such charge. In accordance with previous approaches, the CT component is estimated with (q(M) − 1)·IE(M), where IE is the first ionization energy of the neutral atom and q is the QTAIM charge of the metal cation.48,49 Interestingly, the deformation energy of Li+ and Na+ is regulated by the CT component in correspondence with the initial indication, given by their dominant electron−nucleus contribution. The situation, however, changes notably for the final region, K+−Cs+, as the CR component suddenly assumes a comparable magnitude. Mutual cancellation leads to a near-zero deformation energy for K+ and Rb+. 12114


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Figure 4. Bonding evolution for the M+···BH4− and the shortest M···H interaction in terms of a simplistic overlap parameter (δ, pink) and the relative contribution of the exchange energy (Vx/Eint, purple and green).

and atomic level for Li+ to Rb+. This conclusion, however, is restricted to the molecular regime as the similar levels of K+− Cs+ suggest that crystal effects may impact their mutual relationship.

4. IQA DECOMPOSITION IN BH4−BH4− 4.1. C2 and C3 Reorientation. As previously reported, misalignment between first-neighboring BH4− moieties lowers the reorientational energy barrier.26 Such stabilization, however, rapidly develops into a secondary effect owing to the continuing increase in interatomic distances from LiBH4 to CsBH4. Local symmetry breaking renders it necessary to consider an in-plane and out-of-plane dimer complex for the C2 pathway to fully describe the interplay between the nearest BH4− moieties, whereas a top, bottom, and in-plane are required for the C3 case (Figure 1). The BH4−BH4− geometry from the NaBH4 crystal is selected as a representative example for demonstrating the general features of the underlying energetics; all energy plots and a full listing of table values are available in the Supporting Information. The IQA energy decomposition illustrated in Figure 5 reveals that the overall stabilization is the outcome of favorable deformations surpassing the loss in interaction energy. The latter is dominated by reductions in Vx, which derive from elongated intermoiety H···H contacts. Deviations to this general picture occur only in the 60° C3 top conformation, as it is the sole case for which the strongest H···H contact coincide with the rotation axis; however, a convenient consequence of this feature is an energetic degeneracy with the ideal 0° reference, allowing the energetics of the 60° C3 top conformation to be neglected. For the final series of conformations extracted from the CsBH4 crystal, divergence issues are, nevertheless, encountered owing to the zero-like nature of the energy modifications. In conclusion, a proper reconstruction of the BH4−BH4− energy barriers consistently necessitates the inclusion of both one- and two-body energy terms. 4.2. Interaction Energy. This paragraph inspects the twobody components of the BH4−BH4− dimer complexes more closely. Starting at the moiety level and employing the ideal 0° geometry from the NaBH4 crystal as an illustrative example, a repulsive binding energy of 3.21 eV is obtained. The collective

Figure 3. Interacting-quantum-atom characterization of the atomic M···H interaction by gathering all information from the ensemble of M+BH4− conformations.

character of the M···H interaction augments with the size of the metal cation. At internuclear separations larger than ca. 3 Å, VMH fades away and EMH x int becomes entirely electrostatic in nature. The detailed IQA characterization of the bonding in the M+BH4− dimer complexes sheds light on the simplistic bonding descriptor introduced by Babanova and coworkers22,23 R M+ + R BH4− δ= dMB (8) where R refers to the ionic radii and dMB to the actual M···B distance. Although this straightforward quantification of the overlap between neighboring M + and BH 4 − moieties successfully accounts for the nonmonotonic variation in one series of experimental activation energies, its chemical meaning remains ambiguous. Figure 4 addresses this issue by applying eq 8 to the 0° conformations and defining the shared character of the M+···BH4− and the shortest M···H interaction as Vx/Eint; other descriptors such as Vx and the delocalization index lead to identical trends (Table 1). Strikingly, the variation in δ parallels Vx/Eint with the exception of the end case, Cs+BH4−. The collapse may be understood in terms of increasing CR effects, which the simple distance relationship of eq 8 ignores. Note that the trend of the moiety interaction transcends to the atomic level as the covalent enhancement is concentrated in the shortest M···H interaction (Table 1, Figure 4). In summary, δ successfully reflects the covalent component at both the moiety 12115


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Figure 5. Interacting-quantum-atom dissection of the BH4−BH4− dimer complex based on the NaBH4 geometry. Setting the 0° conformation as a reference point, energy differences are induced by reorienting a single moiety around either the C2 or C3 axis. Note that the deformation energy distinguishes between the fixed and rotated moiety.

deformation of the equivalent moieties accounts merely for 0.51 eV, granting the leading role to the intermoiety interaction energy with its contribution of 2.71 eV. The latter stems from a delicate balance between B···B repulsion (13.5 eV), intermoiety H···H repulsion (29.0 eV), and intermoiety B···H attraction (−39.8 eV). The H···H interactions are therefore instrumental in defining the repulsive nature of the binding energy. Proceeding down to the atomic level, the homopolar H···H interaction is characterized by combining the relevant IQA quantities from the entire ensemble of computed conformations. In the range of covered distances, its interaction energy stays positive in accord with its expected repulsiveness (Figure 6). This feature is easily understood in terms of dominant electrostatics caused by negatively charged hydrogen atoms; QTAIM indicates net charges of about −0.75 e. Similar to the M···H case, the internuclear separation holds the regulating power as the B−H···H−B conformation marginally impacts the trends shared by the three IQA quantities in Figure 6. The exchange component is activated for internuclear distances below 3.5 Å, accounting for more than −1% of EHH int . It reaches a maximum of −13% at the shortest of the considered distances. If the decrease in the atomic separation is continued, Pendás and coworkers have demonstrated that the magnitude of VHH x eventually overtakes the electrostatic component, thereby 53 triggering a stabilizing EHH int . For the current range of systems, however, the efficient decay of VHH x enables most of the H···H interactions to be adequately described from an electrostatic perspective alone.

Figure 6. Interacting-quantum-atom characterization of the intermoiety H···H interaction by gathering all information from the ensemble of BH4−BH4− conformations.

hydrogen evolution, a problem of uttermost importance in the design of improved solid-state hydrogen storage materials.54−56 The key to understanding the onset and nature of the energy barriers is the destabilizations occurring in the M+BH4− clusters. The most favorable type of reorientations utilizes C2 rotations, and, surprisingly, its energy barrier cannot be explained by the global interaction energy. Its contribution is mostly negligible because stabilizations from shortened M···H interactions are neutralized by lengthened counterparts. When such compensation is unrealized, the change is either favorable (Cs) or unfavorable (Na). The former is instrumental for the energy barrier peak at RbBH4, while the latter represents the sole case where the interaction energy is capable of reproducing the energy barrier. Indeed, starting from K, deformation of the alkaline metals transforms the fundamental barrier construction. A distinct reduction in the ionization energy and increase in the electronic polarizability mark the activation of commanding metal deformations. Although unfavorable from an energetic point of view, such enhanced deformations enable the metal cation to form stronger and even partially covalent bonds with approaching hydrogen atoms. Higher energy barriers consequently entail stronger M···H interactions and more deformed metal cations. Similar conclusions apply to the more expensive C3 scenario; however, local symmetry-breaking requires a more complicated model cluster approach, and the nearest M···H contact experiences a more pronounced shortening. The latter causes stronger perturbations of the interacting moieties, which explains the enhanced barrier height.

5. CONCLUSIONS By exploiting the IQA approach, this work provides fundamental insights into the energetic components of BH4− reorientation in MBH4 (M = Li−Cs) crystals. This new knowledge has been deciphered in easily graspable chemical terms, rendering the present analysis certainly useful and likely predictive for future studies. For example, such approach could profitably be applied to understand the driving forces behind 12116


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The Journal of Physical Chemistry C The BH4−BH4− clusters occupy a secondary role, relieving the C2 and C3 reorientational friction through more favorable deformations of the fixed and rotating moiety. This feature is linked to a lengthening of the strongest intermoiety H···H interaction, and according to IQA, this type of interaction remains destabilizing for all computed distances with a binding energy almost independent of the alkaline metal. The latter, nevertheless, regulates the H···H distances and therefore indirectly the reorientational energy gain. Lastly, the reorientational interplay between H···H and M··· H interactions is a striking inversion of the final stages of solidstate hydrogen evolution, where a weakening of the M···H interactions parallels the progressive formation of covalent H··· H bonds.54 Such discrepancies are, however, not in contradiction because they reflect the exploitation of different distance ranges, implicating quite different bond natures and interaction energies.53

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AUTHOR INFORMATION

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S Supporting Information *

IQA details of the complete series of BH4−BH4− clusters and on the formation of Na+BH4− cluster from its isolated monomers. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b00899. Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The Danish National Research Foundation is thanked for partial funding of this work through the Center for Materials Crystallography (CMC, DNRF93).

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Understanding the Reorientational Dynamics of Solid-State MBH4 (M

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