The Rigidity of the (BH₄)-Anion Dispersed in Halides AX, A = Na, K; X = Cl, Br, I, and in MBH₄ with M = Na, K, Rb, Cs

Abstract

The B–H bond length of the borohydride anion (BH₄⁻) in alkali metal borohydrides MBH₄ with M = Na, K, Rb, Cs, and diluted in different alkali halide matrices, was investigated experimentally by infrared spectroscopy (FTIR) and theoretically using first principles calculations. The peak positions in IR absorption spectra of NaBH₄ pressed at 754 MPa in halides NaX and KX with X = Cl, Br, I show significant variations indicating ion exchange effects between the halide and NaBH₄. For NaBH₄ in NaBr, NaI, KBr and KI pellets, the peak positions indicate that BH₄⁻ could be highly diluted in the AX matrix, which renders an isolation of BH₄⁻ in AX (i-BH₄⁻). For NaBH₄ in NaCl and KCl pellets, a solution of BH₄⁻ in AX occurred only after a further thermal treatment up to 450 °C. The observed peak positions are discussed with respect to the lattice parameter (a₀), anion to cation ratio (R = r_A/r_X), standard enthalpy of formation (Δ_fH) and ionic character (I_c) of the halides. A linear relation is obtained between ν₃(i-BH₄⁻) and the short-range lattice energies of AX. Density functional theory (DFT) calculations at generalized gradient approximation (GGA) level were used to calculate the IR vibrational frequencies ν₄, ν₃ and ν₂ + ν₄ for series of compositions Na(BH₄)_0.25X_0.75 with X = Cl, Br, I, and MBH₄. The theoretical and experimental results show the same trends, indicating the rigidity of the B–H bond length and the failure of Badger’s rule.

1. Introduction

Borohydrides, conventionally called tetrahydroborates following the recommendation of the International Union of Pure and Applied Chemistry (IUPAC), have received attention due to their remarkable gravimetric and volumetric hydrogen contents [1]. Here, the anionic part is constituted of the tetrahedral BH₄⁻ group and the negative charge on the borohydride unit is generally compensated by alkali or alkaline earth metals. This has been known since the discovery of NaBH₄ and moreover their potential use as reducing agents [2]. Alkaline solutions of NaBH₄ were already used for producing electrical energy with fuel cell technology in Apollo missions. However, a breakthrough for a viable hydrogen economy is not achieved because of the irreversibility of the hydrogen release, although there were great efforts for better understanding the related problems and also receiving technical solutions for suitable applications of using NaBH₄ as hydrogen storage [3].

Fundamental research shows that the thermal desorption of H₂ occurs at temperatures above 450 °C and leads to a decomposition of the compound into the elements according to MBH₄ => M + B + 2H₂ [1,4]. Some novel approaches could reduce the hydrogen desorption temperature including catalysis [5,6], nano-engineering [7,8] and use of reactive additives [9,10,11]. Some work includes the enclosure and handling of the BH₄⁻ anion in sodalite cages [12,13], or NaBH₄ in an aluminosilicate gel [13,14]. The investigations of the BH₄⁻ anion in the sodalite cages were mainly focused on the thermal control of steps of the effective reaction:

BH4⁻ + 2H₂O = > BO₂⁻ + 4H₂,

(1)

Reaction (1) is supported by previous investigations showing that half of the released hydrogen is supplied by water [3]. In a combined theoretical and experimental approach [15], B(OH)₃/(OH)⁻, B(OH)₄⁻ and BO(OH)₂⁻ could be identified as intermediates in the sodalite cages finally receiving BO₂⁻. In a more recent investigation [16], it was concluded that an initial reaction reveal (BH₃) and (OH)⁻ in separated sodalite cages followed by a fast reaction:

BH₃ + 3H₂O = > B(OH)₃ + 3H₂,

(2)

Thus, the reaction sequence could be in close analogy to conclusions by Mesmer and Jolly [17] that hydrolysis of NaBH₄ in aqueous solutions take place forming initially BH₅ which separates into BH₃ + H₂. Kreevoy and Hutchins [18] attributed it to an acid catalyzed mechanism, where BH₃ rather rapidly reacts with 3H₂O to B(OH)₃ plus 3H₂, and B(OH)₃ reacts successively with (OH)⁻ to B(OH)₄⁻.

Basic questions still concern details of the stability of the BH₄⁻ anion in various borohydrides. Nakamori et al. [19,20] reported a correlation between the first hydrogen desorption temperature (T_d) of the metal borohydrides and the Pauling electronegativity (χ_P) of the metal cations, while T_d decreases with increasing χ_P of the metal. An apparent linear relationship was obtained between the T_d and χ_P for the borohydrides of Na, Li, Mg, Sc, Zr and Zn indicating that χ_P could be considered as an indicator for approximate estimations of the stability of borohydrides. This relation is applicable for other classes of hydrides as tetra-alanates (AlH₄⁻) [21] and hexa-alanates (AlH₆³⁻) [22], too. Hence, one approach to affect the thermal stability of metal borohydrides is to introduce a metal with higher electronegativity through lattice cation substitution. For example, for the cation substitution of Na (χ_P = 0.93), (The χ_P values are dimensionless and are given according to Pauling’s scale for the elements of the periodic table; χ_P is defined as the power of an atom in a molecule to attract electrons [23]) and Li (χ_P = 0.98) with Zn (χ_P = 1.65) in NaBH₄ and LiBH₄ salts, respectively, by mechanically ball milling the borohydride salts with ZnCl₂ produces mixed-metal of borohydrides NaZn(BH₄)₃, NaZn₂(BH₄)₅ and LiZn₂(BH₄)₅, which are thermally less stable than their unsubstituted patterns [24].

The anion substitution in the alkali metal borohydride is also used to tune the thermodynamics of the borohydrides. The mixed hydride-fluoride Na₃AlH_(6-x)F_x prepared from NaF, Al and H₂ is destabilized relative to the pure hydride Na₃AlH₆ [25]. Likewise, first-principles calculations of the decomposition reaction of LiBH₄ with and without fluorine F⁻ suggested that doping LiBH₄ with F⁻ results in a partial substitution of H by F in the hydride lattice yielding favorable thermodynamics modifications [26]. In a more recent study on ball milled samples of NaBH₄ and NaBF₄, the authors reported a fluorine substitution in the temperature range 200–215 °C yielding NaBH₂F₂. The NaBH₄-NaBF₄ composite decomposes at temperatures by at least 100 °C lower in comparison with NaBH₄ [10].

The anion substitution may involve the whole borohydride unit as the substitution of BH₄⁻ by Cl⁻ in NaBH₄ [27,28], and with Br⁻ [29] and I⁻ [30] in LiBH₄, respectively. Differential Scanning Calorimetry (DSC) measurements of the solid solutions reveal the endothermic peaks of the melt as an indicator of the thermal stability at higher temperatures compared to the unsubstituted borohydride salts, suggesting that hydrogen release is not improved by these substitutions.

The stabilization/destabilization effect upon the lattice substitutions could be attributed to the different types of interactions altered during the ionic exchange. The ionic bonding is effective between the cation element and the BH₄⁻ group, whereas the bonding between the boron B and the hydrogen H atoms within the BH₄⁻ group shows a covalent character [31]. The present work is focused on a better understanding of the interactions between the BH₄⁻ anion and its environment. Room temperature, Raman and infrared spectroscopic studies of alkali borohydrides MBH₄, M = Na, K, Rb and Cs, could be fully assigned by Renaudin et al. [32] for both hydrides and deuterides, including the overtones with Fermi resonance and combination bands of the BH₄⁻ anion, which was in agreement with previous assignments [33,34,35,36]. The data were found to be in perfect agreement with Badger′s rule possessing a linear relationship between vibrational frequencies and the B–D bond length. We show here that for the BH₄⁻ anion highly diluted in halide matrices AX, A = Na, K; X = Cl, Br, I, a simple space argument does not consistently predict extension or contraction of B–H bond distances. Instead, details of the effective potential function are found to be more important. First principles calculations support the conclusion that there are only rather small changes in B–H bond distances although there are large variations in lattice parameter dependent on composition.

2. Materials and Methods

2.1. Experimental Study of BH₄⁻ Anion Isolated in AX Halides

Commercial samples of sodium borohydride NaBH₄ (Merck, ≥98%), potassium borohydride (53.94 g·mol⁻¹, Aldrich, 99.9%), sodium chloride NaCl (Merck, ≥9.5%), sodium bromide NaBr (Fluka, ≥99.0%), sodium iodide NaI (Fluka, ≥99.0%), potassium chloride KCl (Merck, ≥99.5%), potassium bromide KBr (Roth) and potassium iodide KI (Riedel-de Haën, 99–100.5%) were used in this study.

Attenuated total reflectance FTIR-ATR spectra of as-received NaBH₄ and KBH₄ samples were recorded on the Bruker ifs 66 v/S using an ATR accessory with a diamond crystal. The vibrational spectra of borohydrides can be divided into two regions: the bond bending region (with ν₂ and ν₄, the symmetric and asymmetric bending modes, respectively) from 1050 to 1300 cm⁻¹ and the bond stretching region (with ν₁ and ν₃, the symmetric and asymmetric stretching modes, respectively) from 2100 to 2500 cm⁻¹ [32]. The asymmetric modes ν₄ and ν₃ are IR active. The B–H bond bending vibrations occur at about half the bond stretching vibrations implying overtone (2ν₄) and combination (ν₂ + ν₄) with Fermi resonance [35,36].

MBH₄/AX pellets (M = Na and K) of 13 mm diameter were produced by exerting a force of 100 kN (equivalent to a pressure of 754 MPa) on a mixture consisting approximately of 0.5 wt% MBH₄ and 99.5 wt% AX for one minute. Fourier transform infrared (FTIR) spectra of NaBH₄ diluted in different halide matrices were taken using the Bruker Vertex 80 v FTIR spectrometer under vacuum. NaBH₄/NaCl and NaBH₄/KCl pellets were further heated under vacuum to 450 °C. IR spectra after the thermal treatment were recorded on the Bruker ifs 66 v/S FTIR spectrometer.

2.2. First Principles Calculations

All calculations were performed with the crystalline orbital program CRYSTAL14 [37,38]. All parameters and functionals were tested in a previous work [15]. We used the PWGGA [39] functional, CRYSTAL standard basis sets for sodium, potassium, boron, hydrogen and chlorine, a TZVP basis set for bromine with an additional f-function and pseudopotential basis sets for rubidium, cesium and iodine (

Table 1. Atomic basis sets used in the CRYSTAL calculations.

Atom	Basis Set
Na	8-511G [40]
K	86-511G [40]
Rb	ECP28MWB [41]
Cs	ECP46MWB [41]
H	5-11G * [42]
B	6-21G * [43]
Cl	86-311G [44]
Br	Pob-TZVP [45]
I	ECP [46]

* related to the nomenclature of the basis set.

).

The truncation criteria in all calculations for bielectronic integrals were set to 10⁻⁹ (overlap and penetration threshold for Coulomb-integrals, overlap threshold for exchange-integrals, pseudo-overlap) and 10⁻¹⁸ (pseudo-overlap), respectively. The SCF (self-consistent-field) convergence threshold on total energy was set to 10⁻⁷ a.u. for geometry optimizations and to 10⁻¹⁰ a.u. for frequency calculations, respectively. A Monkhorst-Pack-net of 8 × 8 × 8 was employed and a maximum trust radius of 0.3 Bohr was used in the optimizations. The Anderson method for accelerating convergence was used.

We used the experimental structures [32,47] as a starting point and fixed the lattice constants in the geometry optimizations, but not the atomic positions. Due to the computational afford, we calculated no supercells for dilution of BH₄⁻ in metal halides but unit cells with one BH₄⁻ and three halides, which is a ratio of 25% of BH₄⁻. The frequencies were calculated with fully optimized atomic positions. The antisymmetric stretching mode was enharmonically corrected [15]. The calculated modes were visualized with the program Jmol [48] and could thereby be assigned to the different mode types.

3. Results and Discussion

3.1. Spectra of NaBH₄/KBH₄ in ATR and Pressed in Various Halides

The ATR spectra of as-received NaBH₄ and KBH₄ powders are shown in Figure 1. The positions of the peak maxima are in good agreement with values reported by Renaudin et al. [32] as collected in

Table 2. ATR frequencies of NaBH₄ and KBH₄ compared to ATR values of Renaudin et al. [32] (values in brackets) in addition to the transmission IR frequencies of NaBH₄ in KBr and NaCl pellets.

		ν4	2ν4	ν3	ν2 + ν4
ATR	NaBH4	1110 (1110)	2217 (2217)	2283 (2284)	2397 (2404)
	KBH4	1111 (1112)	2205 (2208)	2268 (2270)	2375 (2376)
Transmission (pellets)	NaBH4/NaCl	1121	2229	2302	2403
	NaBH4/KBr (<30 s pressed)	1120	2220	2299	2398
	NaBH4/KBr (90 s pressed)	1126	2224	2291	2387

together with the assignment given. It is well known that the bands in the ATR spectra are generally shifted to lower frequencies compared to the transmission spectra due to the dispersion of the refractive index. This is one reason that the IR peak positions obtained using the standard pressed pellet technique, e.g., in KBr halide shows deviations from peak positions determined using ATR.

Another reason for deviations in the peak positions determined in the pressed pellet method is related to a rather uncontrolled anion or cation exchange effect between the probe and the matrix. This is shown in Figure 2 for NaBH₄ in KBr (spectrum b, c). Rather broad and less well-determined peak positions are observed for short mixing and pressing time (30 s, spectrum b). At variance, good mixing and 90 s pressing at the same force (100 kN/13 mm diameter sample = 754 MPa) reveal much sharper and shifted peak positions (spectrum c). The positions of the peak maxima are also given in Table 2 for comparison. The better mixing and longer pressing time obviously leads to better resolved peaks, e.g., for the isotope effect ¹¹B/¹⁰B depicted by the shoulder at higher wavenumbers seen for ν₄ and 2ν₄. This better resolution is explained, however, by a higher degree of solution of the BH₄⁻ anion into the KBr matrix. In this respect, the broader peaks seem to be more representative for NaBH₄. This is supported by inspection of the spectrum obtained for NaBH₄ diluted and pressed into NaCl also shown in Figure 2 (spectrum a). The peaks well coincide with those of NaBH₄ in KBr also taken from the typically broader one. Spectra given for NaBH₄ in NaBr (d) and KI (e) also show rather sharp peaks but at significant different peak positions. This indicates the effect of a sufficient dilution and incorporation of the BH₄⁻ anion in the various metal halide type lattices.

It has been reported that the BH₄⁻ anions are highly diluted in AX by heating discs constituted of AX and traces of borohydride (NaBH₄ or KBH₄) to temperatures between 500 and 600 °C [49]. For our purpose, this effect is demonstrated in Figure 3. Shown are spectra for NaBH₄ (1 mg) diluted into 200 mg of NaCl and KCl (dashed curves) compared to those after heating to 450 °C (red solid lines). For NaBH₄ in NaCl after the thermal treatment, the IR peaks appeared to be sharpened and shifted to higher wavenumbers compared to spectra taken before the heating. For NaBH₄ diluted and pressed with KCl, the spectrum shows peak splitting indicating the formation of two types of solid solution with a higher and smaller BH₄⁻ anion concentration in KCl. However, after thermal treatment, a more homogeneous dilution of BH₄⁻ anions in KCl crystals is obtained.

3.2. BH₄-Frequency Variations Dependences on Halide Parameters

The pellet preparation procedure using a pressure of approximately 754 MPa was sufficient to solve NaBH₄ (a₀ = 6.16 Å) in NaI (a₀ = 6.48 Å), NaBr (a₀ = 5.97 Å), KBr (a₀ = 6.60 Å) and KI (a₀ = 7.06 Å), whereas for NaCl (a₀= 5.64 Å) and KCl (a₀ = 6.29 Å), a subsequent annealing to 450 °C was required to achieve the total solution of BH₄⁻ in the halide lattice. This can be related to the smaller effective ionic radius of Cl⁻ (1.80 Å [28]) compared to BH₄⁻ (2.03 Å [29]). On the other hand, the ionic radii of Br⁻ (1.96 Å [29]) and I⁻ (2.20 Å [30]) are close or significantly larger, respectively. Therefore, a much better exchange is already achieved by the pressure effect. The larger cation radius of K⁺ compared to Na⁺ may enhance the solution effect as could be suggested from the two type of compositions indicated in the NaBH₄ spectrum in KCl compared to NaBH₄ in NaCl (Figure 3).

The obtained IR frequencies of BH₄⁻ in different halide matrices in presumably the high dilution limit, denominated as i-BH₄⁻ (i for isolated) in the following, are summarized in

Table 3. IR frequencies of i-BH₄⁻ in different AX. Structural parameters of AX are also given.

AX	IR Frequencies of i-BH4− in AX				Structural Parameters of AX
	ν4	2ν4	ν3	ν2 + ν4	a0	R = rA/rX
NaCl	1165	2303	2375	2482	5.6402	0.5635
	1166 *	2307 *	2373 *	2486 *
	1167 **	2311 **	2382 **	2493 **
NaBr	1138	2252	2326	2426	5.9738	0.5204
	1136 *	2254 *	2328 *	2427 *
	1135 **	2257 **	2331 **	2430 **
NaI	1112	2208	2281	2364	6.479	0.4636
	1111 *	2202 *	2278 *
KCl	1144	2257	2323	2422	6.290	0.7624
	1144 *	2257 *	2324 *	2422 *
	1146 **	2262 **	2328 **	2428 **
KBr	1127	2226	2293	2390	6.598	0.7041
	1128 *	2226 *	2293 *	2387 *
	1128 **	2229 **	2295 **	2391 **
KI	1107	2191	2258	2350	7.064	0.6273
	1108 *	2191 *	2258 *	2341 *
	1108 **	2192 **	2258 **	2350 **
RbCl	1132 **	2236 **	2303 **	2395 **	6.582	0.8398
RbBr	1118 **	2212 **	2279 **	2369 **	6.8768	0.7755
RbI	1104 **	2181 **	2250 **	2335 **	7.3291	0.6909

* IR values according to ref. [49]. ** IR values according to ref. [50].

and compared to previous works [49,50]. The IR values for BH₄⁻ in RbX matrices are taken from ref. [50]. The cell parameter (a₀) [47] and the radius ratio of the effective ionic radius (NaCl structure, coordination number 6) of the cation to the ionic radius of the anion (R = r_A/r_X) [51] are also given in Table 3.

It is observed that ν₃(i-BH₄⁻) is similar in NaI and RbBr, 2281 and 2279 cm⁻¹, respectively, while the corresponding a₀ (6.479 Å and 6.877 Å, respectively) and ion radius ratio R (0.4636 and 0.7755, respectively) of the halide matrices are very different. Therefore, the frequencies of ν₃(i-BH₄⁻) cannot be interpreted in terms of only the lattice parameter a₀ or R determining the available space for the substituting BH₄⁻ in AX. At variance, the IR frequencies of i-BH₄⁻ are dependent on the nature of the host matrix, as suggested in previous works for many polyatomic ions isolated in halide matrices [30,50,]. To show this effect more clearly, graphical representations of ν₃(i-BH₄⁻) as a function of the structural parameters, a₀ and R of the corresponding AX, are given in Figure 4a,b, respectively.

Approximately, linear relationships are observed between ν₃(i-BH₄⁻) and a₀(AX) for the homologous series of halides with same cationic element A (A-series, solid red lines) or with the same anionic element X (X-series, dashed blue lines). The linearity between ν₃(i-BH₄⁻) and a₀(AX) applies within each of the defined series implying a decrease of ν₃(i-BH₄⁻) with increasing a₀(AX) (Figure 4a). Furthermore, according to Figure 4b, an increase of ν₃(i-BH₄⁻) is accompanied with an increase of R for the A-series (for a given A-series r_A is constant while r_X is decreasing) but with a decrease of R for the X-series (for a given X-series r_A is increasing while r_X is maintained constant). This indicates that ν₃(i-BH₄⁻) varies in an opposite direction to the ionic radii of the cations (r_A) and the anions (r_X). Smaller cations and anions tend to increase ν₃(i-BH₄⁻). In other words, cations and anions of higher Pauling electronegativity (χ_P) within the A- or X-series exhibit higher ν₃(i-BH₄⁻) since χ_P varies in an opposite direction to that observed for the ionic radius in the periodic table (χ_P increases by going from left to right across a period, and decreases by moving down a group). This still could indicate that increasing the available space for the substituting BH₄⁻ anion along a given A-series with increasing anion size (for e.g., NaCl, NaBr and NaI) decreases ν₃ by increasing the B–H distances. A similar effect occurs with increasing the size of the cation along a given X-series (for e.g., NaI, KI and RbI), an elongation of the B–H distances could be responsible of the decrease of ν₃(i-BH₄⁻). This explanation fails, however, between different series as alluded above for the case of NaI and RbBr which exhibit similar ν₃(i-BH₄⁻) but completely different available anion spaces. The same holds for NaBr and KCl matrices. Therefore, the main factor responsible of frequency variations is not based on the B–H bond distance considerations in these cases.

The linear relationship established between ν₃(i-BH₄⁻) and a0(AX) can be presented empirically according to:

ν₃(i-BH₄⁻) = α.a₀(AX) + β,

(3)

α is the gradient and β a constant obtained from the linear regressions and given in

Table 4. Empirical constants (α and β) of the linear relationship ν₃(i-BH₄⁻) = α.a₀(AX) + β for common cation A- series and common anion X-series.

A-Groups	α (cm−1 Å−1)	β (cm−1)	R2
Na-group	−119.9275	3054.0962	0.9848
K-group	−88.8481	2884.8024	0.9928
Rb-group	−69.1765	2757.6490	0.9956
X-groups
Cl-group	−83.1449	2850.9477	0.9999
Br-group	−56.6220	2668.9545	0.9999
I-group	−35.7190	2511.7233	0.9957

. The R² values (>0.98) are satisfactory and confirm the effectiveness of the correlations.

The absolute value of α of the A-series decreases when descending group 1 of the periodic table, i.e., in the order Na-series > K-series > Rb-series. Similarly, the absolute values of α of the X-series decrease in the order Cl-series > Br-series > I-series. Assuming that the gradient α of the linear regression gives insight into the rate at which changes are taking place, i.e., how much the variations of a₀(AX) would affect the values of ν₃(i-BH₄⁻), obtained results suggest that the dependency of ν₃(i-BH₄⁻) on a₀(AX) diminishes across group 1 and group 17 of the periodic table, i.e., when increasing r_A of A-series and r_X of X-series, respectively. Furthermore, the absolute values of α of A-series are greater than those of the X-series. Since in a given A-series, the anionic element is varying and for a given X-series, the cationic element is changing; the obtained results suggest that the anionic element X of AX has more impact on ν₃(i-BH₄⁻) than the cationic element A.

The interrelations established between ν₃(i-BH₄⁻), a₀(AX) and R(AX) can be extended and discussed in term of other parameters of AX as the negative value of the standard enthalpy of formation (Δ_fH, or exothermicity = −Δ_fH), the fractional ionic character of the bond for AX (I_c) and the electronegativity χ_P of A and X, respectively. It has been reported that -Δ_fH increase with increase of I_c of AX. Moreover, both parameters increase when R increases within a given A-series (r_X is changing) or X-series (r_A is changing) with the increase being deeper for A-series [51]. By combining these correlations with those deduced from Figure 4, it can be deduced that for the same cationic A-series, ν₃(i-BH₄⁻) is positively correlated to -Δ_fH and I_c of AX, i.e., an increase of ν₃(i-BH₄⁻) is accompanied with an increase of -Δ_fH and I_c of AX, respectively (Figure 5). For the homologous series with the same anionic element, an inverse correlation lies between ν₃(i-BH₄⁻) and -Δ_fH and I_c of AX, i.e., an increase of ν₃(i-BH₄⁻) is accompanied with a decrease of -Δ_fH and I_c of AX, respectively.

BH₄⁻ anion isolated in AX replaces the anion X⁻ at individual sites in the alkali halide lattice. Several complex empirical functions have been developed to carry out a quantitative evaluation of the frequencies shift of an isolated ion in different surroundings. In a general way, the potential function (V′) governing the vibration of a molecular ion substituted in NaCl-type cube can be regarded as the sum of the potential function of the free ion (V_F) and the interaction energy (V) arising from the presence of the environment expressed in terms of the interaction energy (U) (Equation (4)) [52].

V’ = V_F + V,

(4)

The perturbing potential function (V, expressed in terms of the interaction energy) has two distinct effects on the considered ion: (i) the second derivatives of the perturbing potential with respect to appropriate symmetry coordinates which represent additional contributions to the effective force constants of the impurity ion and (ii) the first derivatives of the perturbing potential represent forces which modify the equilibrium internuclear spacing within the ion [52]. Accordingly, the frequency shift of the $i$th normal mode of the solute is given by:

$$\frac{\Delta \nu i}{\nu i} = \frac{1}{2ki}\left(\frac{{\partial }^{2}V}{\partial S_i^2} - \frac{3a_{iii}}{a_{ii}}\frac{\partial V}{\partial S_i} - {\sum }_r\frac{a_{iir}}{a_{rr}}\frac{\partial V}{\partial S_r}\right),$$

(5)

where k$i$ is the force constant of the $i$th mode, $Si$ are the normal co-ordinates and $a$ are the potential constants [53]. For a diatomic ion described by a Morse potential of the form $V_F = D{\left(1 - e^{ - \beta \xi }\right)}^2$ the equation is reduced to:

$$\frac{\Delta \nu }{\nu } = \frac{1}{2k}\left(\frac{{\partial }^{2}V}{\partial {\xi }^2} + 3\beta \frac{\partial V}{\partial \xi }\right),$$

(6)

where $\xi = r - r_e$ is the change in the internuclear spacing [53,54]. The perturbing potential can be explicitly expressed based on the classical theory of ionic crystals given by Born and Mayer as the sum of a polarization term (V_p) arising from static or dynamic polarization of the isolated ion or the surrounding lattice, a columbic term (V_c) describing the charge–charge interaction between the solute ion and the supporting alkali halide lattice and a repulsive short-range term (V_s) due to the overlap of electron density of the isolated ion with its surrounding ions of the lattice [52,54]. In most cases where the terms of Equation (4) or its equivalent have been evaluated, it was found that the first derivative term (force multiplied by anharmonicity) dominated the frequency shift calculations [53]. For example, for the cyanide ion CN⁻ isolated in various alkali halides [54], the authors observed that the frequency shifts calculations are sensitive to very small changes in the position of the nearest neighbor ions and are dominated by the repulsive term shortening of the CN bond.

An approximately linear relation is obtained between ν₃(i-BH₄⁻) and the short-range lattice energies of AX (Figure 6). The short lattice energy of AX can be considered as the contribution of the repulsive, dipole–dipole and dipole–quadrupole interaction energies given by the Born–Mayer equation AX. These values are tabulated in

Table 5. Contribution of the short-range forces of the alkali halides as given by the Born–Mayer equation (in kg.cal.mol⁻¹ [56]).

AX	Repulsive	Dipole-Dipole	Dipole-Quadrupole	Short Range
NaCl	−23.5	5.2	0.1	−18.2
NaBr	−20.6	5.5	0.1	−15
NaI	−17.1	6.3	0.1	−10.7
KCl	−21.5	7.1	0.1	−14.3
KBr	−18.6	6.9	0.1	−11.6
KI	−15.9	7.1	0.1	−8.7
RbCl	−19.9	7.9	0.1	−11.9
RbBr	−17.6	7.9	0.1	−9.6
RbI	−15.4	7.9	0.1	−7.4

and make clear evidence that the repulsive forces constitute the major part. Similar linear correlation was reported for the nitrate anion NO₃⁻ embedded in different halide matrices [55]. We also obtained the same linear dependency for CN⁻ in different halides basing on the IR frequencies of reference [54].

The correlations between ν₃(i-BH₄⁻) and χ_P(A) and χ_P(X) already discussed for the different series, can provide information about the stability of the borohydrides. Cations and anions of higher χ_P within the A- or X-series exhibit higher ν₃(i-BH₄⁻). A more pronounced charge transfer to BH₄⁻ anion results in stronger B–H bonds stabilizing the borohydride [1]. A more electronegative cation destabilizes the borohydride by attracting the electron density and reducing the electron density in the B–H bonds [19]. For MBH₄ salts, the thermal stability increases in the order NaBH₄ < KBH₄ < RbBH₄ < CsBH₄ [57], whereas the χ_P of the cation increases in the opposite way. Orimo et al. [57] found that for lithium, sodium and potassium borohydrides, the Raman active stretching ν₁ and bending ν₂ modes decrease in the order (ν₂ (LiBH₄) = 1295 cm⁻¹ and ν_2′ (LiBH₄) = 1305 cm⁻¹) > (ν₂ (NaBH₄) = 1280 cm⁻¹) > (ν₂ (KBH₄) = 1240 cm⁻¹) and (ν₁ (NaBH₄) = 2325 cm⁻¹) > (ν₁ (KBH₄) = 2305 cm⁻¹), whereas the melting temperature (T_m) varies in the opposite direction, T_m (LiBH₄) < T_m (NaBH₄) < T_m (KBH₄). The value of ν₁ of LiBH₄ was reported at 2293 cm⁻¹ and did not follow the relation due to the difference of the crystal system. LiBH₄ adopts an orthorhombic structure (space group Pcmn) at room temperature, whereas the other alkali metal (Na, K, Rb and Cs) borohydrides crystallize in a rock salt, face-centered-cubic structure. According to the ATR spectra of Renaudin et al. [32], ν₃ is found to decrease in the order NaBH₄ > KBH₄ > RbBH₄ > CsBH₄. This indicates an inverse correlation between T_d of the borohydride and ν₃ of BH₄⁻ anion. Based on these results, it can be said that in a given X-group where A is changing, an increase of ν₃(i-BH₄⁻) would be accompanied with a decrease of T_d. At the same time, a₀ (AX), R (AX), I_c (AX) and -ΔH (AX) decrease.

DSC data reveal stabilization with increasing chloride-substitution of NaBH₄ [28]. Sieverts measurements of the hexagonal Li(BH₄)_0.5Br_0.5 indicate similar H₂ release as for the unsubstituted pattern LiBH₄ [29]. This indicates that the substitution, with a more electronegative anion, localizes the negative charge on the B-H bond, stabilizing, thus, the borohydride. Therefore, it could be deduced that for the A-series where the anionic element X is varying, an increase of ν₃(i-BH₄⁻) is accompanied with an increase of T_d. At the same time, a₀ (AX) decreases, R (AX), I_c (AX) and -Δ_fH (AX) are increased. Further experiments are still needed to demonstrate the role of the electronegativity of the anion and the thermal stability of the borohydride.

3.3. BH₄-Frequency Variations Dependences on Fundamental Parameter Sets in DFT

The experimental IR frequencies of i-BH₄⁻ in NaX (Cl, Br, I) are shown in Figure 7a as a function of the lattice parameter a₀ of the halides: ν₄ is represented by 2*ν₄ and obtained by simply doubling the values of ν₄ (* denotes the multiplication), together with the three other peaks, namely 2ν_4F− and ν_3F+ (the subscript _F− and _F+ are used to denote the Fermi resonance effect) and the combination mode ν₂ + ν₄. Values given by Renaudin et al. [32] for compositions MBH₄, M = N, K, Rb, Cs are also included for comparison (the values are taken as obtained by ATR-method for 2ν_4F− and ν_3F+ and ν₂ + ν₄; for 2*ν₄ we used the Raman values of v₄). All frequencies increase with decreasing lattice parameter as discussed above. It is observed that the frequency-dependence on the lattice parameter is weaker for MBH₄ compared to i-BH₄⁻ in NaX. The overtone frequency 2*ν₄ comes in all cases well between 2ν_4F− and ν_3F+. This visualizes the Fermi resonance effect characteristic of the BH₄⁻ anion which occurs between ν₃ and the overtone and combination modes. The Fermi resonance observed in the Raman and IR spectra of the tetrahedral BH₄⁻ occurs between the stretching and overtone and combination modes and redistributes the intensities and changes the frequencies [35,36]. As a result, the overtone 2ν₄ and combination ν₂ + ν₄ modes appear with much greater intensity than would have been expected and the IR frequencies are shifted from the values at which they would otherwise have appeared (2ν4F− < 2*ν₄ < ν_3F+).

The same trends are obtained from DFT calculations at generalized gradient approximation (GGA) level (Figure 7b). It has to be noted that the calculations could not be completed for i-BH₄⁻ directly, i.e., in the highly diluted limit. Instead the calculations were carried out for compositions Na(BH₄)_0.25X_0.75 with X = Cl, Br, I. The Fermi resonance effect cannot be considered since CRYSTAL does not allow for incorporation of anharmonic effects. Thus, simple approximations of overtones and combination bands had to be made. Therefore, the theoretical values for 2 × ν₄ and ν₃, as well as the combination mode ν₂ + ν₄ (by simple addition of the harmonic values) are shown related on the lattice parameter values as obtained with DFT. A quasi-linear relationship between lattice parameter and composition Na(BH₄)_1-xX_x with x = 0, 0.25, 0.5, 1 and X = Cl, Br or I, was obtained, where the parameter values are in excellent agreement with experiment for the endmembers. Moreover, the frequency variation dependent on lattice parameter is more pronounced for NaX, X = Cl, Br, I, compared to the series MBH₄ with M = Na, K, Rb, Cs. The absolute values of calculated and measured frequencies and lattice parameters are close. Different from the expectations, the theoretically optimized structures show only slight variations in B–H bond distances. The changes are about one order of magnitude smaller compared to the data given by Renaudin et al. [32]. Rather small variation were also obtained by others using ab initio methods [58]. We therefore conclude that the B–H bond variation is not responsible for the observed frequency changes, but rather the long-distance tail of the B–H potential which is affected by the nearest and next-nearest neighbors.

4. Conclusions

The nearest-neighbor coordination has significant influence on the B–H bonding strength of the BH₄⁻ anion predicting the variation of the IR frequencies in different halides. The BH₄⁻ anion isolated in AX crystal (i-BH₄⁻/AX) yields very sharp IR absorption bands. The IR frequencies of i-BH₄⁻ are found to depend on the nature of the AX matrix. Systematic variations are observed between the asymmetrical stretching ν₃(i-BH₄⁻) and different structural parameters of AX. The correlations established are applicable to homologous series of AX possessing the same cationic A element (A-series: Na-, K-, Rb-series) or the same anionic X element (X-series: Cl-, Br and I-series). For example, when the cationic element A of AX is kept constant for the A-series of AX (A-series: Na-, K- and Rb-series), ν₃(i-BH₄⁻) decreases linearly with increasing cell parameter a₀ of AX. A similar quasi-linear trend is observed for the respective anionic X-series. The correlation between ν₃(i-BH₄⁻) and a₀(AX) for the homologous series is almost linear and can be written as ν₃(i-BH₄⁻) = α.a₀ (AX) + β. The absolute value of the slope decreases when increasing the ionic radii of the cation A or the anion X of A-groups and X-groups, respectively. Moreover, higher values are obtained for the A-groups compared to the X-groups. This indicates that smaller cations and anions produces deeper changes in ν₃(i-BH₄⁻) and that ν₃(i-BH₄⁻) is strongly affected by the nature of X rather than A. The correlation study on the ν₃(i-BH₄⁻) is extended in terms of the ratio of the effective ionic radius of the cation to the ionic radius of the anion (R = r_A/r_X), the Pauling electronegativity (χ_P) of A and X, respectively, the ionic character (I_c) and the standard enthalpy of formation of AX (Δ_fH). An approximately linear relationship is observed between ν₃(i-BH₄⁻) and the contribution of the short-range interactions energies of AX pointing the strong impact of the repulsive forces between BH₄⁻ and its nearest neighbors on ν₃(i-BH₄⁻). DFT calculations obtain a reasonable agreement with experimental values, without showing significant variations of the B–H bond length.