First-principles calculation of ¹¹B solid-state NMR parameters of boron-rich compounds I: the rhombohedral boron modifications and B₁₂X₂ (X = P, As, O)

Abstract

In the present study, solid-state nuclear magnetic resonance (NMR) spectra under magic angle spinning conditions of the rhombohedral structures α-B and B₁₂P₂ are reported together with the corresponding parameter sets from first principles calculations on α-B B₁₂X₂ (X = P, As, O). With the combination of density functional theory (DFT) and the gauge-including projector-augmented wave (GIPAW) approach as the theoretical tools at hand the computed ¹¹B parameters lead to unambiguous explanation of the measurements. Thereby, we overcome common obstacles of processing recorded NMR spectra of solid-state compounds with several crystallographic positions, in particular non-trivial signal assignments and parameter determination due to peak overlap or even unexpected intensity/area ratios. In fact, we find very good agreement between the theoretical results and measured spectra without applying fitting procedures. Using the Perdew–Burke–Ernzerhof (PBE) functional, the results of the common construction types for pseudopotentials and referencing methods for the chemical shift determination are compared. Suggestions and conclusions from experimental ¹¹B NMR studies on parameters according to the icosahedral positions are critically discussed, for instance the early suspected correlation to chemical shifts is not confirmed. Regarding the electric field gradient (EFG) a detailed explanation for obtaining small deviations amongst all investigated structures of the icosahedral polar sites compared to the equatorial sites is given. Our results show an important link between the exohedral bonding situation of compounds with icosahedral structure elements and the main axis of the EFG and therefore, also measurable quadrupole coupling constants if certain geometrical conditions are fulfilled. Finally, this work also contributes to establishing the number of unique sites measured by solid-state NMR methods within the modification of β-B.

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1 Introduction

With the analysis of the electronic structure of a regular icosahedron of B atoms by Longuet-Higgins and Roberts,¹ research and discussion, and thus also challenges posed by the element B, still remain vivid nowadays for both experimentalists and theoreticians.² The formation of multi-center bonds, for which B is prominent, extends to the icosahedral arrangement from the molecular world^3–6via its rhombohedral modifications α-B (α-rB₁₂)^7–11 and β-B (β-rB₁₀₅)^12,13 to numerous solid phase compounds with rhombohedral structures.^14–28

As an analytical tool nuclear magnetic resonance (NMR) spectroscopy is capable of providing multiple insights into structure–property relations of a compound through parameters that characterize the tensors of chemical shift, quadrupolar coupling and spin–spin interaction.^29–31 In contrast to molecules in solutions, the orientation-dependent, that is the anisotropic, effects play a crucial role in the signal shape of solid-state NMR (SSNMR) spectra and can cumber their processing. One way to minimize a large number of these perturbations is to rotate the solid phase sample sufficiently fast by the so-called magic angle relative to the external magnetic field. Still, issues may remain in spectral processing, especially for measurements of different crystallographic positions. For quadrupolar nuclei with spin I larger than one-half, such as ¹⁰B and ¹¹B, anisotropic second-order interactions cannot be removed under magic angle spinning (MAS) conditions resulting in non-trivial signal assignments and parameter determination.^29,30 This is especially true for systems with multiple symmetry sites. Due to the presence of quadrupolar coupling, peak overlap may arise in the procedure of spectral simulation parameter sets that are ambiguous and hinder a full exploration of meaningful results.^31,32

To overcome this issue experimentalists propose certain assumptions in order to reduce the parameters and, hence, the problem complexity to reproduce a measured spectrum via fit algorithms. These educated guesses to the geometrical or electronic situation for the compound under investigation are sometimes not justified and become even more precarious considering that measured intensity/area ratios may show noticeable deviation from the expected position multiplicities.³³

Regarding the structural element of B₁₂ icosahedra, besides molecular crystals of boranes,^34,35 in boron-rich compounds, mainly rhombohedral phases such as the modification β-rB₁₀₅^36,37 and boron carbide (B₄C)^{32,36,38–43} have been investigated by SSNMR spectroscopy. As has been shown from recent analysis of β-rB₁₀₅ spectra,³⁷ the anisotropy of the chemical shift and the dipolar homo- and heteronuclear coupling can be excluded as dominant mechanisms of signal broadening. Thus, the main interactions influencing recorded signals can be suspected to be the isotropic chemical shift in combination with the quadrupolar coupling interaction.

In this context, it is surprising that there are hitherto no published MAS NMR spectra as well as corresponding parameters of the commonly known α-rB₁₂. Although this modification contains the characteristic structure element of many boron-rich compounds, that is interconnected B₁₂ icosahedra, to the best of the authors' knowledge only one static NMR measurement has been reported,⁴¹ but without quantified statements on characteristic NMR parameters. Note that the mentioned solid phases of β-rB₁₀₅ and B₄C are known for various defect variants and partial occupations.^{12–16,44,45} Therefore, and together with the aforementioned obstacles of spectra processing it is just reasonable to first study and understand compounds that are well-ordered or at least simpler in their structural building units, such as the rhombohedral phases B₁₂X₂ (X = P, As, O), in order to go beyond accumulation of information by conceptual studies. Nonetheless, these missing investigations seem to be caused by the fact that the synthesis and quantification of samples with high phase purity remain a major experimental challenge.

By the aid of the gauge-including projector-augmented wave (GIPAW) method⁴⁶ which allows for an ab initio calculation of the NMR parameters of systems under periodic boundary conditions the above mentioned issues in the obligatory fit process to simulate recorded solid-state spectra can be made feasible.^30,47 With its high accuracy, the range of a parameter set for nuclei at certain atomic positions is significantly reduced as it is shown in the example of the spectral assignments for different ¹⁷O sites in the zeolite Ferrierite.⁴⁸ Regarding ¹¹B NMR, the ionic compounds of simple alkaline (earth) borohydrides⁴⁹ and borates⁵⁰ are particularly noteworthy. The latter study shows on the basis of calculations that certain assumptions about the shielding tensor for simulating recorded spectra are made without justification and, consequently, the parameters obtained from fitting algorithms are proved to be incorrect.

In their pioneering work Mauri et al.⁵¹ have applied GIPAW to determine the most common type of defect in B₄C, that is B₁₁C icosahedra with C–B–C chains. This, however, was achieved by investigating isotropic chemical shifts of ¹³C for several structural arrangements relating them to experimental observations. Considering the mentioned obstacles for quadrupolar nuclear species it becomes clear that a detailed exploration of possible parameters for the B sites in B₄C is cumbersome and even less promising. Accordingly, no concrete chemical shifts of ¹¹B are reported and it is noted with respect to the plotted model spectra that a distinct isotropic shift of B in the C–B–C unit is found in the range of 67 ppm, whereas a signal is observed at 37 ppm in experiments.^32,40

While Mauri et al. suggested further experimental measurements because of discrepancy to the calculation, we believe that this deviation in value is an effect caused by strong quadrupolar coupling. Because the larger the quadrupolar interaction becomes the more one has to consider a high-/upfield shift of the center of gravity for the central transition (−1/2 ↔ 1/2).⁵² This effect implies that the bare observation from the MAS NMR spectrum for an assignment of a peak may lead to the impression of a species being more shielded than the actual isotropic chemical shift, calculated for example by theory, would suggest.⁵³ Although this effect decreases with increasing external magnetic field, the second-order quadrupole interaction is inversely proportional to the applied field, it also plays an important role in recent NMR studies on the reorientation dynamics of molecular glass former trimethoxyboroxine.⁵⁴ Thus, the consideration of both the chemical shift as well as the quadrupole coupling is essential for an elaborated description of ¹⁰B and ¹¹B MAS NMR spectra of boron-rich compounds.²⁹

In fact, with their early study in 1959 Silver and Bray³⁸ already suggested from the experimental quadrupole coupling of 5.58 ± 0.02 the existence of a linear C–B–C unit in B₄C. Later on, Schwarz et al.⁵⁵ confirmed this assignment by first-principles computations of the electric field gradient (EFG). Also, the much smaller values of the icosahedral B atoms predicted by these calculations are found to be consistent with measurements. In general, quantum chemical calculations are mandatory for a sophisticated understanding of connections between the bonding situation and parameters arising from the EFG as had been shown for molecules^34,56,57 but also for periodic systems,^58,59 in particular various borides.^60,61

However, except for the work of Mauri et al.⁵¹ on ¹³C shifts in B₄C, and of Schwarz et al.⁵⁵ on the EFG of α-rB₁₂ and B₄C no theoretical investigation of the SSNMR parameters exists to date of boron-rich compounds with icosahedral structure element, and none reporting on explicit information regarding the chemical ¹¹B shift tensor. One of the main challenges seems to be the prediction on isotropic chemical shifts by GIPAW for B in homonuclear bonds, as the example of molecular crystals with diboron units connecting cyclic systems may suggest.⁶² There, the calculated values consistently diverge by at least 14 ppm compared to experiment, whereas a treatment of the magnetic shielding based on molecular models reduces the error by half. By contrast, our investigation⁶³ showed that chemical ¹¹B shifts of icosahedral molecules with multi-center bonds can be determined by GIPAW. Depending on the type of referencing approach the errors of 1 ppm to 2 ppm correspond very well to those of conventional molecular methods that use localized basis sets. The mimic of inner shell electrons by pseudopotentials (PPs) does not play a significant role due to the error compensation in the calculation of relative shift values. On the other hand, the quadrupole interaction is described by the EFG which depends strongly on the electron density in the vicinity of the nucleus.^47,48 In this case, a careful check on the description of the inner shell electrons might be mandatory when the approximation of pseudoization is considered.

The objective of this study is to investigate whether DFT calculations with GIPAW are capable of providing SSNMR parameters that can explain our measured MAS NMR spectra of α-rB₁₂ and B₁₂P₂ which are reported for the first time to the best of our knowledge. The used sample of high purity α-rB₁₂ was kindly provided by Prof. Dr Ismail Duman,⁶⁴ while B₁₂P₂ was synthesized in our laboratories according to ref. 17. Both compounds have been characterized by X-ray diffraction (XRD) analysis. The calculated parameter sets of the closely related thermodynamically stable rhombohedral phases B₁₂As₂ and B₁₂O₂ are additionally presented. On this, we compare the two common construction approaches of PPs and find good agreement for chemical shift tensors but differences for the quadrupole parameters. The results are compared to those of other theoretical and experimental work on borides. In detail, the connections of the nuclear quadrupole coupling constants of the sites in α-rB₁₂ to closely related β-rB₁₀₅ and (B₁₂H₁₂)²⁻ in ionic boronates are discussed. This may contribute to the question which crystallographic positions are responsible for measured quadrupole frequencies in β-rB₁₀₅.^36,37 Furthermore, the parameters are discussed in terms of information extracted from the calculated electron density by Bader's analysis.⁶⁵ Other than assumed by static measurements on α-rB₁₂, β-rB₁₀₅ and B₄C,⁴¹ the results do not indicate a trend in the chemical shifts with respect to the B position, that is the linkage to an atom outside the B₁₂ icosahedron. Instead, we obtain and explain relationships between the quadrupole coupling and the bonding situation. The calculations and considerations on structural constraints are in contradiction to the often made assumption of a symmetrically distributed electron density around the intericosahedral bond, which may have an effect on the determined quadrupolar coupling constants and the corresponding frequencies in spectral processing procedures.

2 Methods

2.1 Definitions and theoretical background

The eigenvalues of the calculated absolute shielding tensor σ_xx, σ_yy, and σ_zz are given in parts per million (ppm) and define its isotropic value σ_iso = (σ_xx + σ_yy + σ_zz)/3. For reporting our results we follow the recommendation of ref. 66 and choose |σ_zz − σ_iso| ≥ |σ_xx − σ_iso| ≥ |σ_yy − σ_iso|. The absolute magnetic shielding is related to the chemical shift by δ = (σ_ref − σ_sample)/(1 − σ_ref) which can be approximated in the case of light nuclei to δ ≈ σ_ref − σ_sample (|σ_ref| ≪ 1). Accordingly, there are two main methods in the literature³⁰ for the conversion of calculated isotropic shieldings to chemical shifts that one may generalize to

δ^calc_iso = m·σ^calc_iso + σ^ref_iso, (1)

with the dimensionless parameter m and a certain reference value σ^ref_iso in ppm. A further generalization is obtained by converting all eigenvalues of the shielding tensor according to δ_ii = m·σ_ii + σ^ref which naturally leads to eqn (1). Referring to the SIMPSON/Haeberlen convention⁶⁷ this yields for the chemical shift anisotropy

δ_csa = m·(σ_zz − σ_iso), (2)

and for the asymmetryWe refer to the conservative conversion of calculated shieldings to shifts by setting m = −1 and σ^ref_iso = σ^calc_iso + δ^exp_iso with δ^exp_iso = −15.3 ppm⁶⁸ of (B₁₂H₁₂)²⁻ measured relative to boron trifluoride diethyl etherate BF₃·OEt₂. For conversion by the method of linear regression we choose m = −0.866 and σ^ref_iso = 80.426 ppm. Both reference approaches had been shown to yield accurate ¹¹B shifts of closo-(hetero)dodecaboranes.⁶³

A nucleus with spin I > 1/2 possesses a quadrupole moment Q which couples with the gradient of the electric field in its vicinity. The EFG is a groundstate property and its 3 × 3 tensor components can be directly computed from the charge density ρ(r)⁴⁸

The EFG's eigenvalues are conventionally designated V_xx, V_yy, and V_zz and chosen as |V_zz| ≥ |V_yy| ≥ |V_xx|. Due to Laplace's equation the sum of these components vanishes and only two parameters are required for a complete description, namely the quadrupole coupling constant C_q (in magnitudes of Hz)

where e is the electron charge and h is Planck's constant, as well as the dimensionless quadrupolar asymmetry parameter η_q

SSNMR experiments only give the absolute value of the quadrupole coupling constant |C_q| and it is also convenient to refer to the quadrupolar frequency ν_q

For the two isotopes ¹⁰B (I = 3) and ¹¹B (I = 3/2) we consider the quadrupole moments Q(¹⁰B) = 8.459 × 10⁻³⁰ m² and Q(¹¹B) = 4.059 × 10⁻³⁰ m² reported by Pyykkö.⁶⁹

2.2 Computational details

All presented results of the ab initio electron structure calculations are obtained with the periodic code Quantum ESPRESSO (QE)^70,71 using DFT on the level of the Perdew–Burke–Ernzerhof (PBE) exchange correlation functional.⁷² Description of the core-valence interactions is based on applying suitable PPs with an energy cutoff of 80 Ry (≈1088 eV) for all calculations. Self-consistency was reached within an energy error of less than 0.3 μeV. We sample the Brillouin zone by a Monkhorst–Pack mesh⁷³ of 6 × 6 × 6 for the rhombohedral unit cells. Convergence is assumed due to negligible changes in the absolute values of the presented quantities. Conversion of the crystallographic information files for the plane-wave input of QE was done by the program CIF2CELL.⁷⁴

Based on the experimental determined crystal structures (see Table 1) geometry is optimized under the constraint of constant lattice parameters.⁷⁵ We want to note that boron suboxide is known to occur as non-stoichiometric B₁₂O_2−x (x ≈ 0.5). However, in the present study we assume all positions to be fully occupied and thus, refer to the compound as B₁₂O₂. The applied PPs are generated under the Goedecker–Hartwigsen–Hutter contraction considering scalar relativistics⁷⁶ which had been proven to result in reliable geometries for boron compounds with multi-center bonds.⁶³ We assume convergence when all components of all forces are smaller than 2.5 meV Å⁻¹. Structure parameters before and after relaxation can be found in Table S1 of the ESI.†

Table 1 Details of crystallographic data and refinement of α-rB₁₂ and B₁₂X₂ (X = P, As, O) used in this work

	α-rB12^7	B12P2^17	B12As2^14	B12O2^18
a Primitive cell. b Hexagonal. c No experimental deviations reported. d Rhombohedral.
Composition^a	B12	B12P2	B12As2	B12O2
Crystal system	Rhombohedral
Space group	R3̄m (no. 166)
Lattice parameters^b
a (Å)	4.9179^c	5.9752(10)	6.1490(15)	5.3862(3)
c (Å)	12.5805	11.8230(20)	11.9140(10)	12.3190(10)
Volume (Å)	263.5	365.6	390.1	309.5
Lattice parameters^d
a (Å)	5.0643	5.2376(7)	5.3268(8)	5.1510(3)
α (°)	58.10	69.56(1)	70.50(1)	63.05(1)
Volume (Å)	87.8	121.9	130.0	103.2

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Investigation on the charge distributions of the optimized structures is performed by Bader's quantum theory of atoms in molecules (QTAIM).⁶⁵ Besides the charges the bond critical points (BCPs) are determined for detailed analysis. For this the electron density was recalculated with projector augmented-wave (PAW) PPs in the Kresse–Joubert contraction⁷⁷ including nonlinear core correction scalar-relativistics. The real space grid was chosen to be three times denser in all lattice directions than the default settings of the QE program package. Using the post processing tool the all-electron charge density is reconstructed and analyzed by CRITIC2^78,79 using the Yu-Trinkle integration⁸⁰ to obtain the presented Bader charges.

For the calculation of the presented MAS NMR parameters we used the GIPAW method⁴⁶ in combination with Troullier–Martins normconserving (NC) PPs,⁸¹ for which we proofed reliability and accuracy of chemical ¹¹B shifts within 2.5 ppm or less for compounds with multi-center bonds.⁶³ According to Cuny et al.⁴⁷ one should be aware that nonphysical shape of the pseudowave functions give meaningless results especially for eqn (4) where the charge density distribution close to the nucleus is crucial. Therefore, the presented parameters are additionally evaluated with the aforementioned PAW PPs. The calculated shielding values are determined under consideration of the term σ(G = 0) = −(8π/3)χ with the macroscopic magnetic susceptibility χ,^82,83 assuming spherical shape of the sample.

For simulating the MAS NMR spectra from the calculated parameters we use the DMfit program package.⁸⁴ In detail, we applied the model “Quad 1st” with intensities and line widths under consideration of measurements and under guidance by eye. Central transitions of the recorded spectra are fitted using the “Gaus/Lor” model (see the ESI,† for further information). The ellipsoids associated with the EFG tensor components are visualized by the webbrowser application MagresView.⁸⁵ With this software we also determined the corresponding Euler angles (Table S4 of the ESI†). All crystal structures in this work are drawn using the VESTA package.⁸⁶

2.3 NMR spectroscopy

The reported SSNMR experiments were performed using 2.5 mm rotors on a Bruker Avance 500 MHz solid-state NMR spectrometer (11.7 T) operating at a Lamor frequency ν_l = 160.48 MHz for ¹¹B. Radio frequency pulses were applied at a transverse B₁ field of 125 kHz corresponding to a π/2 pulse width of 2 ms. All spectra were recorded under MAS conditions at 30 kHz with bearing gas at ambient temperature, leading to sample temperatures of approximately 323 K due to frictional heating. ¹¹B spectra were recorded using a rotor-synchronized Hahn-Echo experiment. The background signal originating from the BN-stator was recorded in a separate experiment and subtracted. The spectra were referenced according to IUPAC recommendations for the unified chemical shift scale using the residual protons in D₂O as secondary standard and the substitution method without reshimming the magnet (Ξ(¹¹B) = 32.083974%).⁶⁶

3 Results and discussion

3.1 Structural aspects, bonding situation and QTAIM analysis

α-rB₁₂ and B₁₂X₂ (X = P, As, O) crystallize in the space group R3̄m with an icosahedral arrangement of the B atoms shown in Fig. 1. The B₁₂ building units form cubic closest packing with only slight deviations in the ABC-stacking sequence of layers. Geometrically, the arrangement of the icosahedra can be viewed as being placed on the eight vertices of a rhombohedron with one of the threefold axis along the [111]-direction. This leads to a reduction from point group symmetry I_h to D_3d. Within the rhombohedral unit cell one may, therefore, distinguish between polar B(1) positions on the rhombohedral sites forming equilateral triangles and equatorial B(2) positions of the staggered belt around the B₁₂ icosahedron. Selected distances of the intra- and intericosahedral atomic positions after geometry optimization are given in Table 2.

Fig. 1 Unit cells of α-rB₁₂ and B₁₂X₂ (X = P, As, O): (a) rhombohedral setting; (b) and (c) hexagonal setting. Dotted lines in (a) and (b) show intericosahedral distances of the equatorial B(2) sites in α-rB₁₂; dashed lines in (c) indicate distance of the interstitial X₂ unit in B₁₂X₂.

Table 2 Selected atomic distances (Å) of α-rB₁₂ and B₁₂X₂ (X = P, As, O) of optimized structures

	α-rB12^a	B12P2	B12As2	B12O2
a With X as exohedral RCP from QTAIM analysis. b Both atoms located in the mirror plane σd.
Intraicosahedral
B(1)–B(1)	1.749	1.875	1.904	1.781
B(1)–B(2)	1.805	1.768	1.780	1.779
B(1)–B(2)^b	1.797	1.804	1.813	1.812
B(2)–B(2)	1.780	1.745	1.730	1.749
Average	1.787	1.792	1.801	1.780
Intericosahedral
B(1)–B(1)	1.676	1.734	1.774	1.697
B(2)–B(2)	2.020
Others
B(2)–X	1.167	1.910	2.005	1.503
X–X		2.243	2.406	2.979

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The structural arrangement is well-known for its versatile bonding situation¹ in molecular systems^3–6 as well as in solid-state compounds:^8,9,20–25 In the radial direction 2-electron-2-center (2e2c) bonds are formed while on the polyhedral surface multi-center connectivity occurs like in 2-electron-3-center (2e3c) bonds. For the investigated rhombohedral compounds the six B(1) vertices of one icosahedron connect to two neighboring layers via six 2e2c bonds extended along the rhombohedral diagonal to the polar vertices of six other icosahedra. Being formed outside the B₁₂ unit we refer to them as exohedral bonds, whereas the term endohedral indicates the electronic situation inside a polyhedron. In α-rB₁₂ the equatorial B(2) atoms of one B₁₂ unit give rise to six 2e3c exohedral bonds interconnecting the icosahedra within the same layer. In contrast, in B₁₂X₂ (X = P, As, O) the B(2) sites extend with exohedral 2e2c bonds to an interstitial X₂ unit placed in the octahedral voids of the cubic closed packing. X is tetrahedrally coordinated by three B(2) and one X atom. In the case of B₁₂O₂ the two O atoms do not form bonds with each other, as will be discussed in further detail below. The resulting effect of interstitial intercalation on the crystal structure is an extension of the hexagonal a axis and a reduction in the c direction as shown in Table 1. Inter- and intraicosahedral B(1)–B(1) distances increase while intraicosahedral B(2)–B(2) distances decrease. Compared to α-rB₁₂ the intraicosahedral B(1)–B(2) distances located in the mirror plane σ_d remain rather constant while the others are shortened. All intraicosahedral distances are in the range of 1.745 Å to 1.904 Å showing for each compound minor deviations from the average value⁸⁷ and indicating small distortions of the B₁₂ unit from a regular icosahedron.

The electronic structure of the investigated compounds had been studied and discussed in detail in several works. Our results are in good agreement with those reported in ref. 8–10 and 20–25. Table 3 summarizes the QTAIM analysis of the calculated electron density in a way that gives an overview of the bonding situation important for understanding the later analysis of the EFGs. The charges are consistent with the results of ref. 25 determined by the (linear)-augmented-plane-wave (LAPW) method. Small deviations are mainly caused by geometry relaxation. In α-rB₁₂ there is a small amount of charge transfer from the polar B(1) to the equatorial B(2) site in agreement with the former reports.¹⁰ The participation of the more electronegative interstitial atoms causes the B₁₂ units in B₁₂X₂ to be positively charged. This is in contrast to the expectations based on Wade's rules³ suggesting B₁₂²⁻X2²⁺ but is closer to the real charge distribution. A considerably high amount of charge on the X₂ unit can be compensated with only small charge changes of icosahedral B atoms. Compared to α-rB₁₂ we find main impact of the interstitial atoms on B(2) while charges associated with B(1) atoms show only minor changes.

Table 3 Parameters of QTAIM analysis from PAW calculations of the optimized structures with charge q_PAW (e) compared with the results from ref. 25, and properties of the BCP associated with the exohedral bond at r_BCP with charge density ρ(e Å⁻³), the Laplacian ∇²ρ (e Å⁻⁵) and the distance between atoms and BCP d_X–BCP (Å)

Compound	Site	QTAIM charge		Properties of BCP
		Ref. 25	q PAW	ρ(rBCP)	∇^2ρ(rBCP)	d X–BCP
a LAPW (PW91-LDA) without geometry optimization. b Intraicosahedral BCP to B(2): ρ(rBCP) = 0.799 e Å^−3, ∇^2ρ(rBCP) = −2.739 e Å^−5. c Intraicosahedral BCP to B(2): ρ(rBCP) = 0.857 e Å^−3, ∇^2ρ(rBCP) = −4.263 e Å^−5.
α-rB12	B(1)	0.06	0.09	1.067	−10.120	0.838
	B(2)	−0.06	−0.09	0.531^b	−1.276	1.067
B12P2	B(1)	0.13	0.13	0.974	−7.749	0.867
	B(2)	0.06	0.04	0.932	−7.490	0.691
	P	−0.52	−0.55	0.774	−3.778	1.122
B12As2	B(1)	0.13	0.15	0.914	−6.660	0.887
	B(2)	−0.04	−0.10	0.822^c	−5.279	0.794
	As	−0.27	−0.15	0.625	−1.973	1.203
B12O2	B(1)	0.12	0.12	1.017	−7.798	0.848
	B(2)	0.41	0.41	0.985	12.018	0.487
	O	−1.58	−1.58	0.053	0.285	1.490

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Let us start the discussion of the bonding situation with the connectivity in the structural building elements B₁₂ and X₂. In all compounds BCPs are located near the edges and ring critical points (RCPs) in the center of the triangular faces of the B polyhedra corresponding to endohedral multi-center bonds in accordance to ref. 10 and 25. For all interstitial X₂ units we obtain BCPs with vanishing ellipticity. The increase of the unit cell volume for B₁₂X₂ is a consequence of the enlargement of the interstitial voids which is strongly correlated with the bonding situation of X₂ and the size of X.²⁷ Applying QTAIM analysis we consider a bond formation by a BCP with non-vanishing electron density. Therefore, in B₁₂P₂ and B₁₂As₂ the interstitial atoms are bonded to each other. In contrast, the BCP of the O₂ unit in B₁₂O₂ has negligible electron density with a slight positive Laplacian. In their calculations Li and Ching²³ also found a vanishing charge density distribution along the O–O path and Lee et al.²⁰ reported a small but non-negligible stress on the O₂ dumbbell corresponding to Coulomb repulsion by overlapping charges. Together with conceptual arguments of Slack and Morgan²⁷ and the large atomic distance of 2.979 Å in Table 2 our results confirm that the O atoms are not bound to each other.

For all intericosahedral B(1)–B(1) bonds the BCPs are located at the center of the bonding paths (see also Table 2 and d_X–BCP in Table 3) and show vanishing ellipticities. These bonds can be considered as classic covalent indicated by the negative Laplacian being straight and cylindric. Turning to the exohedral B(2)–X connection we find that the electron density on the BCP increases with a decrease of atom distance from which we conclude bond strengthening. As the electronegativity of the interstitial atom rises the BCP shifts towards the B(2) site. In B₁₂O₂ the positive Laplacian for the B(2)–O bond illustrates an ionic interaction in correspondence to the results from calculations with local basis sets.²³ The B(2) atoms in α-rB₁₂ form exohedral bonds to two neighboring B₁₂ polyhedra with multi-center characteristics.^8–10 We find a longer bonding path than the atomic intericosahedral distance of 2.020 Å, a bending angle B(2)–BCP–B′(2) of 142.6° and an ellipticity of 3.6 indicating a considerably high bond asphericity. The three BCPs of these exohedral bonds form an equilateral triangle with a RCP in its center. With results from analysis by the electron localizability indicator²⁵ and other reports^9,10 we may interpret this RCP as the center of this intericosahedral bond formation. Its distance from the B(2) site is listed together with B(2)–X bond lengths in Table 2.

In general the results from QTAIM analysis show for the polar B(1) sites that the bonding distance corresponds to the electron density on the BCP and, thus, to the bond strength. In fact, the values of ρ(r_BCP) shown in Table 3 are the highest of all BCPs found for a respective site. Exceptions are the BCPs associated with bonds of the B(2) atoms of α-rB₁₂ and B₁₂As₂ where maximum values of ρ(r_BCP) occur for the endohedral bonds to another B(2) atom. Therefore, one should differentiate for the equatorial B(2) sites: The strongest bond formation is exohedral to the interstitial atoms in B₁₂O₂ and B₁₂P₂, whereas it is endohedral to B(2) atoms in α-rB₁₂ and B₁₂As₂. On the basis of this result the term “inverted molecular solid” introduced by Emin¹⁹ should be used with caution as it does not generally hold true as in the case of B₁₂As₂.

3.2 Calculated NMR parameters and measured MAS NMR spectra

The calculated ¹¹B NMR parameters are given in Table 4 where the results of PPs under NC and PAW construction are compared. For conversion of the magnetic shieldings to chemical shifts we show the results of referencing via linear regression as well as by applying the conservative approach selecting (B₁₂H₁₂)²⁻ as a reference substance. The differences in values for the PPs of NC type increase the more deshielded the ¹¹B site is. Note that the chemical shift asymmetry η_cs is not influenced by the choice of reference method. We found that the macroscopic correction with the magnetic susceptibility assuming spherical system⁸² is essential for the here reported values of δ_iso and also influences η_cs.‡ Shift parameters determined by NC and PAW with the conservative referencing method deviate from each other in a similar way to that reported by studies on more shielded ¹¹B species in borohydrides.⁴⁹ An exception is the asymmetry η_cs of B(1) and the anisotropy δ_csa of B(2) in B₁₂As₂ which is related to the construction of the PPs. For NC, the values of linear regression differ from the conservative ones the more deshielded the B site is, in particular caused by the applied parameters of our preceding work.⁶³ In total, all δ_iso agree in the range of 2 ppm regardless of the referencing method or PP type used which is comparable to the accuracy of computations for experimental values of structurally related molecules.

Table 4 Chemical shift parameters δ_iso (ppm), δ_csa (ppm) and η_cs together with the quadrupolar coupling constants |C_q| (kHz) and asymmetry parameter η_q calculated with NC and PAW type PPs; expected shift for the center of gravity of the central transition δ_cg (ppm) according to PAW values and the shift from the central transition fit of the recorded spectra δ_ctf (ppm)

Compound	Site	NC							PAW						Exp
		Linear regression^a		Conservative		η cs	|Cq|	η q	Conservative		η cs	|Cq|	η q	δ cg
		δ iso	δ ^regcsa	δ iso	δ ^regcsa				δ iso	δ csa					δ ctf
a Referencing according to ref. 63, δ^regcsa/δ^refcsa = 0.866 is assumed (see methods for details). b δ cg = δiso + δq with δq = −Cq^2(1 + ηq^2/3)/(40νl).^52,54 c For details of the central transition fit see the ESI.
α-rB12	B(1)	2.6	−7.7	5.5	−8.9	0.01	1706	0.20	4.5	−8.6	0.07	1651	0.20	3.6	5.6
	B(2)	5.0	15.4	8.2	17.7	0.59	887	0.38	7.4	17.5	0.54	838	0.38	7.2	−11.9
B12P2	B(1)	−2.2	−6.0	−0.1	−6.9	0.49	1611	0.20	−0.8	−7.0	0.53	1483	0.28	−1.5	−8.1
	B(2)	−9.3	17.0	−8.3	19.6	0.38	85	0.57	−8.9	19.0	0.37	251	0.47	−8.9
B12As2	B(1)	−0.9	4.5	1.3	5.2	0.77	1466	0.26	1.5	5.0	0.57	1384	0.29	0.9
	B(2)	−6.5	−10.6	−5.0	−12.3	0.95	328	0.77	−5.1	14.3	0.92	203	0.30	−5.1
B12O2	B(1)	−14.9	−10.9	−14.8	−12.6	0.11	1405	0.13	−13.7	−13.0	0.12	1416	0.10	−14.3
	B(2)	−0.3	12.0	2.1	13.9	0.84	1260	0.17	3.4	14.4	0.78	1430	0.10	2.8

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The calculated δ_iso values are in the same range as that found for experimental MAS NMR spectra of B₄C³² within 10 ppm to −15 ppm but indicate overall more deshielded ¹¹B species than icosahedral closo-borane molecules.⁶⁸ In all cases the interstitial X₂ unit in B₁₂X₂ leads to more shielded B sites compared to α-rB₁₂. However, this increase in shielding correlates neither with the position nor with the charge situation: Relative to polar B(1) sites, the B(2) site in α-rB₁₂ and B₁₂O₂ shows smaller shielding, whereas the opposite is true for the remaining compounds. The nature of the exohedral bonds is, therefore, not clearly related to the differences in chemical shifts as proposed from experimental study.⁴¹ As for their molecular counterparts there is no simple relation between charge and chemical shift which may be a consequence of the distinct covalent bonding character in these icosahedral compounds.⁶³

Usually NMR is not sign sensitive for the coupling constant C_q³¹ and so only absolute values are reported in Table 4. Comparing NC and PAW we observe noteworthy deviations for the B(2) site in B₁₂P₂ and B₁₂As₂ between the parameters |C_q| and η_q describing the quadrupolar interaction. This becomes more evident with the investigation of the principal axes of the EFG tensor in the following section as we obtain a change in the sign of V_zz. Furthermore, the orientations of the EFG main axes change relative to those of the barely changing chemical shift tensors which leads to different Euler angles (see Table S4 in the ESI†). These discrepancies do not change with a finer k-grid sampling or an increase of the plane wave cutoff. Being a “core” property the EFG is highly sensitive to the electronic wave functions and density in the vicinity of the nucleus under investigation (see eqn (4)) and so nonphysical forms of the pseudowave functions for inner shell contributions may lead to meaningless results.⁴⁷ Since the PAW method works with the full reconstruction of the all-electron wave functions, potentials and density⁷⁷ we rely on those values for further discussion.

For all compounds the |C_q| values of B(1) are in the range of about 1450 kHz while we find high fluctuations for B(2). Besides small structural deviations clearly the influence of the interstitial unit causes this effect for which the polar positions seem to be rather unaffected. The magnitude of the calculated quadrupole coupling constants of ¹¹B in Table 4 agrees well with observations reported for other compounds containing icosahedral boron sites such as 450 ± 250 and 1300 ± 100 kHz,³⁸ 361, 380 and 513 kHz⁴¹ or 840 kHz⁴² for B₄C and 1240 kHz⁴² for rhombohedral silicon boride α-SiB_3−x.⁸⁸ It is interesting that for α-rB₁₂ the |C_q| values correspond to B(1) for measurements in carborane B₁₀C₂H₁₂ of 1430 ± 20 kHz³⁴ and on the other hand for B(2) to constants measured in closo-dodecaborates M₂[B₁₂H₁₂] (M = K, Rb, Cs, NH₄, N(CH₃)₄) in the range of 650 kHz to 700 kHz.³⁵ This will become more clear in the next section where the orientation and magnitude of the principal axes of the EFG tensor are analyzed in more detail.

The B(1) sites connecting the B₁₂ icosahedra show higher coupling constants than for other structural arrangements such as octahedra in YB₆ with an experimental value of 1200 kHz⁶⁰ or B₁₂ cuboctahedra in TB₁₂ (T = Y, Zr, Lu) with about 1100 kHz.⁶¹ The presented |C_q| of B(2) is in the range of those obtained in YT′B₄ (T′ = Mo, W, Re)³³ consisting of two dimensional boron networks. Turning to the asymmetry η_q in Table 4 the polar site is in general lower than or equal to that for the equatorial one indicating a more symmetric distribution of the electron density around the largest principal axis V_zz of the EFG tensor. In contrast, values of η_q ≈ 0.95 for the aforementioned dodecaboride TB₁₂ imply a highly asymmetric electronic distribution.⁶¹

Compared to GIPAW calculations and measured parameters of borate crystals and glasses in ref. 50 and references therein the here investigated boron-rich compounds are in general more shielded with chemical shift anisotropies δ_csa of the same magnitude of 10 ppm. The quadrupolar coupling constants |C_q| of the polar B(1) sites are smaller than for trigonal BO₃ units while those for the equatorial B(2) sites in B₁₂P₂ and B₁₂As₂ are comparable to tetrahedral BO₄ formation.

Fig. 2 and 3 show our measured MAS NMR spectra of α-rB₁₂ and B₁₂P₂ together with contributions from the two different B sites. For the simulations we used the theoretical PAW parameters and adjusted the remaining magnitudes as described in the methods section. To the knowledge of the authors, only static samples of α-rB₁₂ were measured⁴¹ which are unsuitable for comparisons of our calculations or recorded spectra. It is difficult to extract reasonable parameters from the measured ¹¹B spectra alone which is also known from reports^32,41,42 on B₄C. Even with the expectation to obtain contributions of the B(1) and B(2) site it seems less appropriate to apply conventional fitting procedures due to the abundance of possible value combinations for parameter assignment. This is even more severe considering ref. 33 which reports that the intensity ratios of side band patterns show a noticeable deviation from expected signal area ratios in the ¹¹B MAS NMR spectrum of YWB₄ (at 14.1 T under 20 kHz spinning), and the same is true in the region of the central transition. Nonetheless, we report shifts δ_ctf in Table 4 obtained from a central transition fit of the recorded spectra (see the ESI† for details). Comparisons to the theoretical expected shift for the center of gravity δ_cg show good agreement and indicate that individual site contributions of α-rB₁₂ and B₁₂P₂ are not resolved separately. The shoulder at −11.9 ppm in the spectrum of α-rB₁₂ will be addressed in more detail below.

Fig. 2 Experimental ¹¹B MAS NMR spectrum of α-rB₁₂ measured at 11.7 T with 30 kHz spinning frequency and simulation with parameters from PAW calculation in Table 4 together with site contributions. δ_cg is used in the simulation to set the shift position. Intensities and line widths adjusted by eye for the best fit of central transition shown in (c) (see Table S3 in the ESI† for further details).

Fig. 3 Experimental ¹¹B MAS NMR spectrum of B₁₂P₂ measured at 11.7 T with 30 kHz spinning frequency and simulation with parameters from PAW calculation in Table 4 together with site contributions. δ_cg is used in the simulation to set the shift position. Intensities and line widths adjusted by eye for the best fit of central transition shown in (c) (see Table S3 in the ESI† for further details).

The parameters arising from the EFG tensor may have a dominant influence not only on the signal shape but also, more importantly, on the center of gravity for observed signals/peaks being shifted high-/upfield.^52,53 This effect turns out to play a significant role in the experimental observed ¹¹B shift for the atom in the C–B–C chain in B₄C. This can be convincingly demonstrated by inserting values of the experimental setup of Simeone et al.³² (ν_l = 96.216 MHz, 300 MHz spectrometer) into the equation given in the footnote of Table 4. Together with the measured C^exp_q = 5580 kHz ± 20 kHz³⁸ or theoretical C^theo_q = 5722 kHz⁵⁵ one receives δ^exp_q = −27.0 ± 0.2 or δ^theo_q = −28.4 ppm.⁸⁹ Mauri et al.⁵¹ mentioned theoretical δ_iso of 67.3 ppm and 66.1 ppm for the chain center depending on the considered structure model with the remark on discrepancies to the measurement of Simeone et al. with a small signal at 37 ppm. But for δ_cg the resulting range can be estimated from 40.5 ppm to 37.7 ppm. Therefore, the calculated δ_iso is neither unreasonable nor are the measurements deficient, it is just that the shift of the center of gravity on the central transition was not considered for the computed values. For the here investigated cases of α-rB₁₂ and B₁₂X₂ (X = P, As, O) the C_q leads to δ_q ≤ 1 ppm. However, we choose them to set the shift positions in the simulation and contributions in Fig. 2 and 3. Note that δ_cg (given in ppm) is related to the NMR experiment because it depends apart from the Larmor frequency ν_l also on the used carrier frequency.

By using the parameter sets from PAW calculations in Table 4 a reasonable assignment of contributions from the polar and equatorial sites in the spectra of α-rB₁₂ and B₁₂P₂ is possible because they can explain the observed central transition and side band patterns. The theoretical results suggest that the difference of the chemical shifts for the B(1) and B(2) sites in α-rB₁₂ and B₁₂P₂ is smaller than their line broadening and, hence, can be expected to be not resolved in the experimental MAS NMR spectra of Fig. 2 and 3. Therefore, the strong shoulder obtained for α-rB₁₂ cannot be explained by our calculated values and might not be related to the theoretical assumption of an infinite expanded periodic structure. Indeed, powder XRD measurements and scanning electron microscopy (SEM) reveal a particle size of ≈30 nm. So, a significant share of the B₁₂ icosahedra is located on the surface of the particle with a different electronic surrounding (≈10% assuming a diameter of 5 Å for a B₁₂ icosahedron). Furthermore, there may occur a reconstruction of the (111) surface of α-rB₁₂ with single bridging interstitial B atoms.¹¹ Our suggestion is also supported by studies³⁷ on the structurally closely related β-rB₁₀₅ showing that the two apparent peaks of the static sample⁴¹ get removed under MAS measurement to result in only one peak around 14.2 ppm. In this regard it is important to note that the results in Table 4 are based on modeling well-ordered crystal structures. This might not be always the case and even hard to afford for experimental samples as for compound B₁₂O₂ where we assumed all sites to be completely occupied throughout this work although it occurs as non-stoichiometric B₁₂O_2−x.²⁷

Our results also provide insight into the problem of establishing the number of unique sites detected by NMR experiments of the other B modification β-rB₁₀₅.^36,37 The idealized crystal structure shown in Fig. 4a may be described² by placing B₈₄ polyhedra in Fig. 4b on the corners of the rhombohedral unit cell. As illustrated, each of the 12 atoms of the central icosahedron bonds to a pentagonal cap resulting in the arrangement of a “super-icosahedron”. The pentagonal caps in polar positions are linked to polar caps of other B₈₄ units while the equatorial ones are attached to B₁₀–B–B₁₀ (B₂₁) chains visualized with condensed icosahedra in Fig. 4c. Each pentagonal cap within B₈₄ is connected to another one and forms the half of an icosahedron. Therefore, all 84 atoms are sixfold coordinated intericosahedral positions comparable to the polar B(1) site in α-rB₁₂. In the following we refer to the numbering of ref. 12 for the distinguishable sites in β-rB₁₀₅. Three different B₈₄ units are connected via the B₁₀ unit which is formed by the four B sites B11, B12, B13 and B14 (see Fig. 4c). These atoms have no exohedral 2e2c bond following the icosahedral fivefold main axis. B13 is partially occupied and coordinates twice to itself and to the solitary B15 site. The central B14 links with 3 × 3 = 9 bonds to B11, B12 and (with partial 0.75) to B13. B12 has two bonds to B13 (partial), 2 × 2 = 4 bonds to outer B sites (B3 and B4) of two different B₈₄ units and one bond each to the inner atoms of the B11 and B14 sites. B11 is connected twice to itself, with one bond each to the inner B12 and B14 sites as well as overall 2 × 2 = 4 bonds to outer B sites (B4 and B10) of two different B₈₄ units. The bonding situation of these inner atoms, that is within the B₁₀ unit, thus shows a certain similarity to the equatorial B(2) atoms in α-rB₁₂, since they do not have exohedral 2e2c bonds, but are exclusively involved in multi-center bonds. In the idealized structure shown in Fig. 4a the local symmetry of the B14 site is C_3v while the B15 site is located in a trigonal-antiprismatic (distorted octahedral) D_3d arrangement.

Fig. 4 Unit cell of idealized β-rB₁₀₅ and structure elements: (a) rhombohedral setting; (b) B₈₄ unit; and (c) B₂₁ configuration with condensed icosahedra formed by two B₁₀ units (upper part) coordinating the central B atom (B15 site) trigonal-antiprismatic. The shown numbering of B sites and partial occupation refer to ref. 12. Note that partially occupied positions (B13 site) of the B₁₀ unit are shown as full spheres in (a). Colors are meant to support the understanding of the crystal structure in terms of Samson polyhedra (see also ref. 2).

Turning to the NMR experiments, Turner et al.³⁷ excluded a dominant influence of chemical shift anisotropy as well as homo- and heteronuclear dipolar broadening mechanisms on the spectral shape in β-rB₁₀₅ which is in accordance to our theoretical findings for α-rB₁₂ in Table 4. In fact, second-order quadrupolar interactions with varying magnitudes are identified as main contribution in the ¹⁰B wide-line spectrum of static polycrystalline β-rB₁₀₅ samples which is in accordance with older investigations on ¹¹B.³⁶ The corresponding frequencies ν_q are obtained by comparison of simulated spectra with the recordings under the assumption of η_q = 0 and are associated with three distinguishable groups. Note that for β-rB₁₀₅ an assignment to the corresponding sites is difficult by experimental observation alone for there are at least 17 distinguishable atom positions reported in recent studies.^12,13 Therefore, the most recent investigation³⁷ reported ν_q for the sake of comparison while in the former one³⁶ used presumptions based on intensity ratios. The converted values of frequencies determined via |C_q| from PAW for α-rB₁₂ (see Table 4) are listed together with the observed groups for β-rB₁₀₅ in Table 5; we tabulate the data for both isotopes for benefit of discussion. The data sets of both experimental studies agree well with each other for group I and are in the same range of value as the polar B(1) site in α-rB₁₂. Thus, our first-principles calculations confirm the suggestion of Hynes and Alexander³⁶ to assign the largest ν_q to the sixfold coordinated B nuclei of the intericosahedric linkage forming the B₈₄ unit depicted in Fig. 4b.⁹⁰

Table 5 Quadrupolar frequencies ν_q (kHz) for ¹⁰B and ¹¹B of the two rhombohedral boron modifications. For α-rB₁₂eqn (7) is used to convert the PAW calculated |C_q| values in Table 4 while for β-rB₁₀₅ the results from NMR experiments are listed suggesting three distinguishable groups for signal contribution. The actual measured data are emphasized and reevaluated for the different nuclear species

α-rB12			β-rB105^a
This work			Group	Ref. 36		Ref. 37
Site	^10B	^11B		^10B	^11B	^10B	^11B
a Measured data for ^10B and ^11B in ref. 36 and 37, respectively. b Occurence only proposed.
B(1)	165	826	I	136 ± 14	680 ± 70	142	710
B(2)	84	420	II	26 ± 6	130 ± 30	84	420
			III	0^b		0

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Notwithstanding this, the following three observations lead us to a suggestion for signal assignment for the measured groups that differs from the one given by Hynes and Alexander:³⁶ The experimental frequencies obtained for group II clearly differ from each other (i). In addition, only ref. 37 reports on the evidence of group III with the remark that the choice of η_q indirectly influences ν_q obtained by the spectra simulation process, whereas in ref. 36 the occurrence of ν_q ≈ 0 kHz is simply expected from symmetry considerations for the B sites B14 and B15 (ii). On the other hand, the agreement of values for the equatorial B(2) position in α-rB₁₂ with those of Turner et al.³⁷ is striking, although no fivefold coordinated atomic positions are obtained in the idealized structure of β-rB₁₀₅ (iii).

According to Hynes and Alexander³⁶ the atoms of the B11, B12 and B13 sites are assigned to the frequency group characterized with 130 ± 30 kHz (for ¹¹B). Because a higher coordination number leads in general to a decrease of |C_q| and, hence, to ν_q as it is exemplarily shown for ²³Na in several idealized environments in ref. 57 and 58. On the other hand, it is postulated that the central atoms of the B₁₀ and B₂₁ units on the B14 and B15 sites should lead to ν_q = 0 kHz. But in fact, investigating the geometrical arrangements of both centers in the idealized structure (B13 site fully occupied) shows that the conditions for which the EFG contributions add up to zero are not satisfied considering C_3v or D_3d symmetry, respectively.§ So it is much more likely that the corresponding ν_q is very small but not vanishing. This is also consistent with the recent observation³⁷ that within the spectra simulations the signal features associated with a certain asymmetry parameter η_q can also be obtained by small variations in the quadrupolar frequency but with a slightly different η_q and, thereby, a supposed ν_q = 0 kHz is the consequence of assuming η_q = 0. As shown in tab:nmr and in the discussion of the next section this assumption is, however, not supported by theoretical calculations in general for B atoms in icosahedral arrangements. For these reasons, we argue that the associated group III of ref. 37 actually corresponds to groups II and III given in ref. 36. With the previous discussion on the B11, B12, B13 and B14 sites of the B₁₀ unit forming multi-center bonds we conclude that these should be assigned to group II with 84 kHz measured in ref. 37. The contribution of group III might be related by the B15 site but a definite conclusion by comparison with α-rB₁₂ only is not possible.

In addition, the interstitial B atoms on the partially occupied positions must also be taken into account in order to fully understand the experimental NMR spectrum. A relatively homogeneous electron density distribution can be assumed due to the delocalization of the electrons over all boron polyhedra, but the EFGs of B atoms close to the defect can be expected to differ significantly. Since the NMR is a local probe, ordered models have to be set up taking symmetry reduction into account. Our preliminary calculations on β-rB₁₀₅ clearly indicate the trends for the fully occupied sites discussed here. A sophisticated way to model fractional site occupancy is provided by the SOD (site-occupancy disorder) program⁹² considering not only the energetic but also entropic contribution of a certain atomic configuration. Such inclusion of reasonable order variants is required for deeper exploration of the quadrupolar interactions in β-rB₁₀₅.

3.3 Analysis of the EFG

As the EFG is related to the charge distribution in the vicinity of the probe nucleus it follows from symmetry considerations that if the nuclear site is located on the rotational axis in an environment of C_nv symmetry the principal axis V_zz coincides with this axis. This is reported for example for B atoms in C_2v site symmetry of dodecaborides with cuboctahedral B₁₂ units where the exohedral bond is formed along the rotation axis.⁶¹ For the here investigated structures the B₁₂ symmetry of I_h reduces to D_3d and consequently the local environment of the nucleus changes from C_5v to C_s. In this case, the remaining restriction for the orientation of two EFG principal axes is to lie within the plane σ_d causing the third axis to be exclusively perpendicular to σ_d. Based on this analysis, it becomes clear why values of η_q in Table 4 are not zero, that is the reduction of symmetry from I_h to D_3d forces that the small orthogonal axes V_xx and V_yy cannot occur with the same magnitude. An a priori designation of η_q = 0 for icosahedral positions commonly made in experimental studies^37,38,42,43 is, therefore, not justified from symmetry restriction. In fact, this misconception can even lead to deviations from the actual |C_q| (and ν_q) values by applied simulation procedures to recorded spectra.^36,41

It is illustrative to visualize the principal axes of the EFG tensor in their magnitude and orientation as ellipsoids as shown for α-rB₁₂, B₁₂P₂ and B₁₂As₂ in Fig. 5. The large EFG axis V_zz coincides with the longest ellipsoid axis and the value of η_q influences the shape. The more symmetric or asymmetric the electron density is distributed around V_zz the more prolate (η_q = 0) or oblate (η_q = 1) the ellipsoid appears.

Fig. 5 Ellipsoides associated with the magnitude and orientation of principal axes of the EFG tensor determined by PAW: (a) for B(1) and B(2) sites in α-rB₁₂; (b) for B(2) in B₁₂P₂; and (c) for B(2) in B₁₂As₂. The orientation for B(2) in B₁₂O₂ is similar to the one in B₁₂P₂ (see Θ_Vzz and Φ_Vzz in the table). For the sake of appearance ellipsoid sizes are rescaled and do not correspond in relative magnitude to each compound.

At first sight one would expect the highest gradients of the electric field running radial from the icosahedral center due to the dominance of the 2e2c exohedral bond formation^1,19 as well as by the observation that the B₁₂ polyhedron contains almost no electronic charge.²⁵ And, thus, it might seem plausible that a pseudo fivefold axis of an icosahedral atom arrangement leads to the large principal axis pointing along the exohedral bond as in Fig. 5a and b. However, this is not true in general, as shown for carboranes,³⁴ and can be even misleading as in the case of the equatorial B(2) site in B₁₂As₂ (see Fig. 5c).

More detailed information on the orientation of the EFG tensor's principal axes is provided by Table 6 with angles between V_zz and the radial axis of the B₁₂ unit pointing along its center of mass Θ_Vzz, as well as between V_zz and the exohedral bond denoted by Φ_Vzz. Additionally, we refer to the averaged apex angles θ_av and (see Fig. 6a for their definition) as a measure of deformation for the B₁₂ polyhedra compared to the regular icosahedron after structure optimization. Their root-mean-square deviations can be found together with the kink angles of the exohedral bonds in Table S5 of the ESI.† The compound (B₁₂H₁₂)²⁻ is added for comparison as a closely related structure of I_h symmetry using the geometry reported in ref. 63. Overall, we observe small angles between the V_zz axis and the bonding partner Θ_Vzz except for the B(2) site of B₁₂As₂ where also the apex angles are expanded the most compared to (B₁₂H₁₂)²⁻. In fact, the angle to the radial axis Φ_Vzz is smaller than Θ_Vzz for most cases. As one expects for (B₁₂H₁₂)²⁻ the exohedral bond coincides with the large principal axis of the EFG. Note also how the B₁₂ unit in α-rB₁₂ shows the smallest deviations from the I_h symmetry with the V_zz axis pointing almost in the radial direction for both the 2e2c and 2e3c bond formations.

Table 6 Angles (°) related to the orientation of the large V_zz axis (PAW) to the exohedral bonding partner Θ_Vzz, radial to the icosahedral center of mass Φ_Vzz, and averaged apex angles related to geometry θ_av and (see Fig. 6a for definition). Δ[small theta, Greek, macron] denotes the absolute mean deviation from the magic angle θ_mag (≈54.74°) defined as |(θ_av + )/2 − θ_mag|. All values according to optimized structures

Compound	Site	EFG related		Geometry related
		Θ V zz	Φ V zz	θ av		Δ[small theta, Greek, macron]
a Related to the center of the intericosahedral 2e3c bond. b Perpendicular to σd, see Fig. 5c. c All θ′ > θmag.
α-rB12	B(1)	7.66	0.47	58.43	52.77	0.86
	B(2)	9.73^a	0.98 ^a	58.14	52.46	0.56
B12P2	B(1)	5.44	4.44	56.34	50.56	1.29
	B(2)	5.21	1.45	60.22	54.75	2.75
B12As2	B(1)	5.84	5.90	55.67	49.87	1.97
	B(2)		90.00^b	60.89	55.52^c	3.47
B12O2	B(1)	5.40	7.22	57.31	51.58	0.29
	B(2)	5.55	3.52	59.26	53.69	1.74
(B12H12)^2−		0.02	0.02	58.28	52.62	0.71

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Fig. 6 Idealized arrangement and qualitative curvature of resulting V_zz values using the point charge model: (a) construction to calculate v_zz(θ) for an icosahedral model system with apex angles θ and θ′. M denotes the midpoint of the edge; (b) v_zz(θ) (colored) at point P for different distance ratios ω_Y (see the Appendix for details). Gray lines indicate contributions by a charge located at Y with q_Y/q_ring > 0 (solid) or q_Y/q_ring < 0 (dashed).

For simplicity and a better qualitative understanding of the meaning of the EFG components, we may analyze the V_zz values along the icosahedral diagonal by considering the point charge model (PCM). This method had been applied to understand and construct quite unexpected geometrical arrangements for molecules⁹³ and solid phases^91,94 where the EFG vanishes besides cubic symmetry. For instance, a charge distribution symmetric to the rotational axis of a cone with an opening angle twice the magic angle θ_mag, satisfying (3cos² θ_mag − 1) = 0, leads to an overall null tensor at the apex point of the cone. In fact, considering an in-depth treatment of second-order quadrupole effects, multiple magic angles can be derived not only from terms containing the Legendre polynomial of second-order P₂(cos ϑ) but also from such fourth-order P₄(cos ϑ).⁹⁵ Here we only refer to the former one with θ_mag ≈ 54.74°.

Let us consider Fig. 6a with the B atom under investigation at point P and where the z-axis coincides with the fivefold rotation axis as do the direction of the exohedral bond due to a geometrical arrangement of I_h symmetry. Since the EFG components scale with the inverse cubic distance (eqn (4)) we may assume that its contributions at point P are only given by the charged ring with radius r representing the endohedral bonds and a charge placed at point Y along the exohedral bond. There are many combinations of parameters to consider but we may write the resultant value of V_zz solely in terms of the apex angle θ and the ring charge to cubic radius ratio q_ring/r³. The corresponding curvature and its contributions are shown in Fig. 6b for the charge ratio |q_Y/q_ring| = 0.1. Reasonable distances between P and Y are indicated by the dimensionless constant ω_Y = OY/a with a as the edge length of the icosahedron. As shown in the Appendix, ω_Y increases with the distance PY. The two characteristic apex angles of the regular icosahedron (≈52.62°) and θ_ico (≈58.28°) are also illustrated in Fig. 6b for further discussion. It is important to mention that V_zz becomes negative in the marked region for charges on the ring and Y with the same sign. If we assume that the contribution of q_ring rises from electrons on the icosahedral surface this accordingly implies negative charge on Y causing a negative value of V_zz which is consistent with the more sophisticated explanation in ref. 57. As it is evident from Fig. 6b the ring contribution v_ring is zero for θ_mag and, hence, the contribution of the charge at Y is significant, this holds also true for angles θ in the interval ≤ θ ≤ θ_ico. The former theoretical and experimental investigations^5,6,25,26 showed that the electron density is only significant on the icosahedron surface as well as on the exohedral bonds and vanishes rapidly elsewhere. Correspondingly, relaxing the assumption of point charges to a continuous charge density evenly spread on the surface leads to an endohedral contribution that nearly vanishes. And for this reason, EFG components V_zz pointing along the radial direction mainly reflect the charge distribution along the exohedral bonds. It is crucial to clarify that this is not true for vertex positions of other boron polyhedra as for example for octahedra in hexaborides^55,96 because the geometry of the regular icosahedron is such that and θ_ico are close to θ_mag with ( + θ_ico)/2 ≈ θ_mag. This assists the understanding why |C_q| values in Table 4 for the B(1) atoms are approximately constant in the range of 1450 kHz and can be expected to barely change in other structurally related compounds. Furthermore, we may conceptualize with this geometric interpretation that for the B(2) site in B₁₂As₂ the V_zz axis is perpendicular to the B(2)–As bond as we observe such an increase in angles θ and θ′ in Table 6, that the influence of the endohedral bonds has a more significant impact on the EFG components compared to all other positions of the investigated compounds. In order to further illustrate this we additionally list in Table 6 the absolute mean deviation from the magic angle as Δ[small theta, Greek, macron] = |(θ_av + θ_av′)/2 − θ_mag|. Especially for B₁₂As₂ and the B(2) site in B₁₂P₂ the highest deviations occur indicating for geometrical reasons a growing influence of the endohedral bonding situation on the orientation of the V_zz axis.

Table 7 shows the full set of EFG components related to the parameters given in Table 4 for both types of used PPs together with those calculated for the (B₁₂H₁₂)²⁻ molecule. The magnitude of values is in the range of other computational investigations.^60,61 For other borides and for α-rB₁₂ our results agree well with calculated V_zz values by Schwarz et al.⁵⁵ for B(1) and B(2) with −15.4 × 10²⁰ and 8.8 × 10²⁰ V m², respectively.

Table 7 Comparison of the EFG tensor components (10²⁰ V m²) determined by the NC and PAW PPs

Compound	Site	NC			PAW
		V xx	V yy	V zz	V xx	V yy	V zz
a V zz axis perpendicular to σd, see Fig. 5c. b V zz axis along the B(2)–X direction. c V xx = Vyy.
α-rB12	B(1)	6.9	10.4	−17.4	6.7	10.1	−16.8
	B(2)	−2.8	−6.2	9.0	−2.7	−5.9	8.5
B12P2	B(1)	6.5	9.9	−16.4	5.5	9.6	−15.1
	B(2)	−0.2	−0.7	0.9^a	0.7	1.9	−2.6^b
B12As2	B(1)	5.5	9.4	−14.9	5.0	9.1	−14.1
	B(2)	−0.4	−3.0	3.3^b	0.7	1.3	−2.1^a
B12O2	B(1)	6.2	8.1	−14.3	6.5	8.0	−14.5
	B(2)	−5.3	−7.5	12.8	−6.6	−8.0	14.6
(B12H12)^2−		4.3^c		−8.5	4.1^c		−8.1

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For now, we exclusively refer to the PAW values given in Table 7. The results of both PPs are discussed and compared below. From eqn (4) it follows that within the principal axis system of the EFG tensor. One may decompose V_zz into a lattice V^lat_zz and spherical part V^ll_zz related to non-vanishing terms of angular momenta.⁵⁹ Defining an anisotropy count Δn_pz = (n_px + n_py)/2 − n_pz with partial charges n_pi in the corresponding p orbitals in a muffin tin sphere, it was shown for borides that the contribution is much larger than V^lat_zz, V^sd_zz and V^pf_zz.^60,61 In the orbital picture one may, thus, neglect in good approximation the contributions by s orbitals and terms of mixtures with higher angular momenta as also discussed in the early work of Townes and Dailey⁵⁶ for molecular systems. In fact, compared to hexaborides the factor shows an increase by more than 50% for the icosahedral sites in α-rB₁₂ and B₄C.⁵⁵ This is a consequence of the observation that 80% of the EFG components stem from contributions within a sphere of 0.69 Å around the B nuclei. Note that this is in the distance range d_X–BCP of the B sites to the exohedral BCP in Table 3. Therefore, the rather large positive value of V_zz for the B(2) site of α-rB₁₂ can be explained by the depopulation of the p_z orbital donating electron density to the intericosahedral 2e3c bond. The density on the BCP ρ(r_BCP) is the smallest value compared to all other BCPs and indicates the weakest bond in this compound which is in accordance with Lee et al.⁸ In B₁₂O₂ the O atom clearly dominates the bonding situation of the B(2) site as the interstitial bond is the shortest occurring amongst all compounds investigated here (see Table 2). The value ρ(r_BCP) is the highest of all exohedral B(2) bonds which is in total agreement with the orientation of the large principal axis. Consequently, the positive V_zz value reflects the large shift of the BCP towards the B(2) atom with d_B(2)–BCP = 0.487 Å and the strong ionic bond character confirmed by the positive Laplacian ∇²ρ(r_BCP) = 12.018 e Å⁻⁵. Regarding B₁₂P₂ and B₁₂As₂ there are intraicosahedral distances smaller than the interstitial bond lengths. Further analysis of ρ(r_BCP) shows that the largest value of the equatorial B(2) is given by the exohedral bond in B₁₂P₂, whereas we obtain the maximal ρ(r_BCP) for the endohedral B(2)–B(2) bond in B₁₂As₂ (see also Table 3). In the case of B₁₂P₂ we interpret the negative value of V_zz caused by depopulation of the p_z orbital by bond polarisation of the more electronegative P atom. In contrast to B₁₂O₂, this bond is covalent, in particular there is much less depopulation, which leads to a more moderate decrease of electron density compared to endohedral B(1)–B(2) or B(2)–B(2) bonds. Consequently, the charge anisotropy Δn_pz tends to zero and causes small negative V_zz. In B₁₂As₂ we obtain the exohedral B(2)–As bond to be weaker than the B(2)–P interaction. This is in accordance with increasing bond distance and the decrease of electronegativity difference from P to As as compared to B. Additionally, the intraicosahedral B(2)–B(2) distance in B₁₂As₂ is shortened compared to the one in B₁₂P₂ culminating to domination of this endohedral bond formation. This is in contrast to all other sites investigated in this work and leads to the large principal axis V_zz being perpendicular to σ_d as shown in Fig. 5c.

The two V_zz for the B(2) site in α-rB₁₂ and the B atoms in (B₁₂H₁₂)²⁻ show different signs but lead to similar |C_q| values. From the former analysis it is clear that the resulting similarity in the quadrupolar constant |C_q| is coincidental because it arises from two completely different electronic situations: In α-rB₁₂ the lack of electron population along the V_zz axis leads to a positive value, whereas the negative sign for (B₁₂H₁₂)²⁻ corresponds to the covalent exohedral B–H formation. Compared to the homonuclear B(1)–B(1) bonds the more positive value of −8.1 × 10²⁰ V m² can be explained by the bond polarisation caused from the more electronegative H atom depopulating the p_z orbital in the vicinity of the B nucleus.

Turning to the applied PP types, NC and PAW yield similar components except for the equatorial B(2) sites in B₁₂P₂ and B₁₂As₂ where we obtain different orientations for the large principal axis V_zz. This can be explained by the inadequate description of the electron density close to the nucleus under investigation by PPs relying on the NC construction. The represented charge density distribution in the vicinity of the B(2) sites differ and accordingly the EFG components become sensitive to the theoretical description by the selected PP. For these sites we also obtain the largest Δ[small theta, Greek, macron] in Table 6 meaning that the correct modelling of the endohedral bonding situation becomes more crucial the more distorted the B₁₂ unit is.

Furthermore, this shows that the orientation of the principal axes can have major influence on η_q. Regarding B₁₂As₂, the different oriented V_zz axis for NC and PAW also changes the orientation and magnitude of the two small principal axes V_xx and V_yy. This in turn modifies the ratio of (V_xx − V_yy)/V_zz causing η_q in Table 4 to result in significant different values for B(2) of 0.77 and 0.29. Concomitantly, we want to hint on the sensitivity of η_q with regard to the component value V_zz. The smaller the calculated V_zz values are the more significant the relative deviations in V_xx and V_yy become. For example, in B₁₂P₂ and B₁₂O₂ the absolute values of |V_yy − V_xx| for the B(2) site are comparable but result in very different η_q in Table 4. In summary, the reliability of theoretical values is critical for small |C_q| and explains rather large deviations between NC and PAW in η_q for the B(2) sites in Table 4. Therefore, we recommend a careful check of calculated “core” properties by the PAW construction when the approximation with pseudoization of inner shell electrons is applied, even for the 1s² configuration.

With this we conclude that the main principal axis of the EFG tensor indicates under particular symmetry restrictions the strongest or weakest bond formation in the investigated compounds. Besides the electronegativity difference to exohedral bonding partners, geometric aspects of the B₁₂ unit, in particular its distortion, are also mandatory for an in-depth understanding of the V_zz values for the various B sites in boron-rich borides with icosahedral structure elements. Both the electronic and the geometric structures do not influence the quadrupolar interaction independently but need to be considered in conjunction with each other.

4 Conclusions

In this paper we calculated the quadrupolar interaction and, for the first time, chemical shift parameters of the rhombohedral phases α-rB₁₂ and B₁₂X₂ (X = P, As, O) where B₁₂ icosahedra are the main structure element. The GIPAW approach is capable of providing parameter sets which show a good bridging between XRD and NMR models for our recorded MAS spectra of α-rB₁₂ and B₁₂P₂. The shift parameters are in good agreement for both established construction schemes of PPs with exceptions for δ_csa and η_cs in B₁₂As₂. Despite the pseudoization of the inner shell electrons by NC conditions, the PAW construction allows an all electron treatment. This becomes more significant when quadrupolar parameters are considered which are directly related to the EFG tensor. Consequently, the measured MAS NMR spectra and especially their side band patterns should be addressed with results from PAW. For the systems at hand, the calculated coupling constants are in general too small to obtain second-order quadrupolar splittings as had been proposed by other studies.^38,41 Furthermore, the determined |C_q| values of α-rB₁₂ enabled us to draw a connection to observed quadupolar frequencies ν_q of β-rB₁₀₅, which can be divided into three distinct groups. We suggest an assignment based on the results of our calculations and discuss the indirect influence of the assumption η_q = 0 for values of |C_q| or ν_q yielded by the usual fitting procedure of measured spectra. The reduction of symmetry from I_h to D_3d of the B₁₂ units excludes this commonly a priori applied postulation for η_q in general which might be only justified for ionic cases or at least strong bond polarisations where V_xx − V_yy values are of smaller magnitude than V_zz as for instance in B₁₂O₂. We want to point out that a guessed value of η_q also implies an expectation of orientation for the V_zz axis and the symmetry of its environment. Accordingly, the distribution of the electron density in the vicinity of the nucleus and possibly also the predominant bond formation is assumed. However, this is not always clear without quantum chemical calculations or XRD measurements and can lead to inaccurate results for |C_q| or ν_q obtained by simulations of experimental spectra. Further investigations on the here studied compounds by more sophisticated NMR techniques such as multimagnetic field or multiquantum MAS measurements will help for more detailed insight into experimental values. In this context one should not forget that XRD produces average structural models over long atomic distances whereas both DFT calculations and NMR measurements are based on the local environment of atoms.

The analysis of the polyhedral arrangement of the B atoms by distances and apex angles indicates small distortions in the order of α-rB₁₂, B₁₂O₂, B₁₂P₂ and B₁₂As₂ from a regular icosahedron. Applying QTAIM analysis we observe charges and bond formation that are in general agreement with other investigations on the electronic structures. However, these charges indicate no simple relation to the calculated isotropic chemical shifts which might be a consequence of the covalent character of the studied compounds as this is also obtained for the more shielded ¹¹B species in icosahedral closo-dodecaheteroboranes.⁶³ Thus, our results do not confirm the presumption stated by Lee et al.⁴¹ that measured differences in chemical shifts of the B atoms relate to the bonds they form outside the icosahedron. Interpreting the charge density on a BCP ρ(r_BCP) to be correlated with bond strength, the polar B(1) sites exclusively form strongest bonds exohedral to other B₁₂ units. This holds also true for the equatorial B(2) sites of B₁₂P₂ and B₁₂O₂ connected to the interstitial unit X₂, whereas we find for α-rB₁₂ but more surprisingly for B₁₂As₂ highest ρ(r_BCP) for endohedral B(2)–B(2) bonds.

Investigating the principal axes of the EFG tensor we obtain a good approximation that V_zz is oriented along the exohedral bond except for B(2) of B₁₂As₂ where it is perpendicular to the mirror plane σ_d. A general inspection of the apex angles and θ_ico of the regular icosahedron in combination with the PCM reveals that the most significant contribution for V_zz can be expected from the electronic situation within the nuclear vicinity along the exohedral bond. Therefore, the sign and magnitude of V_zz is nearly constant for the B(1) sites (|C_q| ≈ 1450 kHz) and strongly correlated with the electron density on the exohedral bonding path of the B(2) sites. This effect is closely related to the special icosahedral geometry for which ( + θ_ico)/2 ≈ θ_mag is well fulfilled and consequently the influence by endohedral bonding partners is of minor significance. Note that other regular polyhedra such as the cube or octahedron do not meet this spatial condition. Accordingly, we find for the B(2) sites in B₁₂P₂ and B₁₂As₂ not only strong distortions for the apex angles but also that the description of V_zz in orientation and magnitude depends critically on the type of PP. With PAW, the V_zz axis of B(2) in B₁₂P₂ lines up with the exohedral B(2)–P bond whereas in B₁₂As₂ it points along the direction of the endohedral B(2)–B(2) bond. Thus, the orientation of the large principal axis of the EFG tensor indicates under particular symmetry restrictions the strongest or, in the case of the B(2) site of α-rB₁₂, the weakest bond formation for the here investigated compounds.

Altogether GIPAW may be utilized not only to study structural changes with theoretical modeling and compare to NMR experiments, but to address the obstacles for ¹¹B measurements on solid-state compounds, in particular peak overlap and possibly uncorrelated site to actual peak intensity/area ratio. Thus, calculated parameters may clarify and elucidate which site is measured and to which certain extent they contribute to signals in a variety of complex spectra. With this at hand, one is able to expand to even more complicated systems and investigate certain structural or electronic influences and open the possibility to understand more complex NMR spectra and conceptualize results on theoretical foundation.

Conflicts of interest

There are no conflicts to declare.

Appendix: qualitative understanding by the point charge model

For the qualitative understanding of the EFG we consider Fig. 6 showing V_zz along the fivefold rotation axis of the icosahedron assuming the PCM. For a derivation let us assume the electric field E⃑ on a given point r⃑ caused by a set of point charges q_i located at r⃑_i in atomic units (E_h/(ea₀)) as

Since the EFG scales with the inverse cube of distance we only consider for the electric field in point P (five evenly spaced) charges on a ring summing up to q_ring and the charge q_Y on point Y. Choosing the center of the ring as the origin O the contributions for the EFG component V_zz = V_ring(z) + V_Y(z) in P along the z-axis can be obtained by differentiation of eqn (8) yielding⁹³

In general there is a very wide combination of possible parameters to analyze the behaviour of the EFG. However, we may reduce this complexity by expressing V_zz in terms of q_ring/r³ as v_zz(θ) = v_ring(θ) + v_Y(θ) leading to sole dependence on the apex angle θ

where we used the definition of the cotangents cot(θ) = cos(θ)/sin(θ), the charge ratio q̄_Y = q_Y/q_ring as well as the geometrical characteristic ratios ζ_Y and ω_Y. Note that this expression is related to atomic units, in particular it holds 4πε₀ = 1.

Table 8 shows geometrical quantities of a regular icosahedron with edge length a. Due to the fivefold rotation axis the golden ratio ϕ plays an important role in icosahedral geometry. Assuming that the electronic charge is located on the midpoint of all bonds a reasonable estimation is given by with d_1/2 as half of the exohedral bonding distance. Accordingly, the depicted values of ω_Y in Fig. 6 correspond to the cases of B(2) in Table 2 for α-rB₁₂ and B₁₂As₂. For reasons of appearance q̄_Y = 0.1 is selected.

Table 8 Overview of quantities of the regular icosahedron with edge length a in terms of the golden ratio ϕ used for the qualitative analysis of v_zz(θ) in Fig. 6b

Quantity	Value	Meaning
ϕ		Golden ratio
r ico		Radius of circumcircle of a regular pentagon
R ico		Radius of circumsphere
θ ico		Apex angle between diagonal and edge
		Apex angle between diagonal and centroid of triangular face
ζ Y		Ratio a/rico

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Acknowledgements

The authors thank Prof. Dr Ismail Duman for a high purity sample of α-rB₁₂, and Dr Anke Schaub for the MAS NMR measurements.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp04061d

‡ Without consideration of χ the shifts δ_iso are deshielded by about 10 ppm to 15 ppm, whereas only minor changes occur for δ_csa; for details see Table S2 in the ESI.†

§ For this, we refer to expressions (3) and (4) of ref. 91.

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