Correlating geminal ²J_Si–O–Si couplings to structure in framework silicates

Abstract

The dependence of a ²⁹Si geminal J coupling across the inter-tetrahedral linkage on local structure was examined using first-principles DFT calculations. The two main influences on ²J_Si–O–Si were found to be a primary dependence on the linkage Si–O–Si angle and a secondary dependence on mean Si–O–Si linkage of the two coupled ²⁹Si nuclei. An analytical expression describing these dependences was proposed and used to develop an approach for relating the correlated pair of ²J_Si–O–Si coupling and mean ²⁹Si isotropic chemical shift to the linkage Si–O–Si angle and the mean Si–O–Si angle of the two coupled ²⁹Si nuclei. An example of this analysis is given using ²⁹Si NMR results from the siliceous zeolite Sigma-2.

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1 Introduction

The isotropic chemical shift of ²⁹Si NMR has long been a valuable probe of structure in silicate materials.^1–4 In changing coordination from SiO₄ to SiO₅ to SiO₆ the chemical shift range of ²⁹Si in silicates varies from approximately −100 to −150 to −200 ppm, respectively.⁵ In the case of tetrahedral coordination, the chemical shift varies over a range of −120 to −70 ppm as the second coordinate sphere changes from fully connected Q⁴ to fully disconnected Q⁰, respectively.⁶ For Q⁴ sites it is well established that variations in the range of −105 ppm to −120 ppm in the isotropic chemical shift of ²⁹Si are correlated to the mean of the Q⁴'s four inter-tetrahedral angles.^7–9

The anisotropy of the ²⁹Si chemical shift is also strongly correlated to changes in the first-coordination sphere. As first noted by Grimmer and coworkers,^10,11 as the Si–O bond length decreases there is an increase in the s-character of the bonding orbital at Si and corresponding increase in shielding along the direction of the shorter bond. This strong dependence of the ²⁹Si chemical shift anisotropy has been exploited not only for distinguishing and quantifying Qⁿ sites,^12–16 but recently has been found to be an effective probe of the modifier cation coordination to the non-bridging oxygen of Q³ sites¹⁷ and useful in the NMR refinement of crystal structures.¹⁸

In contrast to the ²⁹Si chemical shift, accurate measurements of geminal ²J_Si–O–Si couplings across the inter-tetrahedral linkage in silicates have yielded no clear relationships between coupling constant and local structure. While ²J_Si–O–Si couplings in solution have been measured,^19–23 motional averaging makes it difficult to use these measurements to establish empirical relationships. The number of ²J_Si–O–Si measurements in solids have been limited, particularly at ²⁹Si natural abundance levels, 4.67%, and thus far few efforts^24–26 have been made in trying to establish quantitative relationships between ²J_Si–O–Si and structure. Compounding this issue is that commercial computational chemistry packages have only recently reached the level where accurate J-couplings can be calculated through using expensive density function theory (DFT) methods.²⁷ Nevertheless, geometric dependences have been reproduced using modest DFT calculations. The most recent attempts in determining such relationships, using ab initio DFT methods, were in 2009 by Cadars et al.²⁶ and Florian et al.²⁵ Although both groups acknowledged a dependence of ²J_Si–O–Si coupling on Si–O–Si bond angle, Florian et al. suggested a one-to-one mapping, whereas Cadars et al. found that the dependence of the ²J_Si–O–Si coupling is not limited to the Si–O–Si bond angle but is also influenced significantly by the local geometry around the Si–O–Si linkage. Cadars et al. concluded that such dependencies led to a relatively “large scatter” of ²J_Si–O–Si coupling, with respect to Si–O–Si bond angle, and did not attempt to deduce any empirical relationship between the ²J_Si–O–Si coupling and the local structure. To get a sense of that scatter, we plot the ²J_Si–O–Si couplings—calculated here—as a function of Si–O–Si bond angle in Fig. S5 of the ESI.†

Here we re-examine the variation in ²J_Si–O–Si using Q⁴–Q⁴ clusters that are centered on the Si–O–Si linkage extending out to four coordination spheres away from the bridging oxygen between the two coupled silicon. With greater computational resources than were available to Cadars et al. 8 years ago, we were able to increase the level of theory and obtain excellent agreement with known experimental ²J_Si–O–Si couplings. Through systematic structural variations of the cluster, we investigated the influence of (1) the Si–O–Si linkage angle, (2) the Si–O bond distance, (3) the inter-tetrahedral dihedral angle, and (4) the outer Si–O–Si linkage angles of the two coupled ²⁹Si nuclei. Most significantly we show here that after the central Si–O–Si linkage angle it is the outer Si–O–Si linkage angles of the Q⁴–Q⁴ couple that play the next most significant role in determining ²J_Si–O–Si. A smaller dependence on the dihedral angle is observed while a negligible dependence on Si–O distance over relevant length scales is observed. In this study we assume the SiO₄ tetrahedron does not deviate from local tetrahedral symmetry and take the intra-tetrahedral O–Si–O angle as constant. While this is a reasonable approximation in a fully connected Q⁴ network, it is likely that this assumption will break down as the silicate network becomes depolymerized. Further work will be required to understand the dependence of J coupling on local structure in networks having significant distortions around SiO₄ tetrahedra.

As the isotropic chemical shift has a well known dependence on the mean Si–O–Si angle, we also show here that a correlation plot of mean ²⁹Si isotropic chemical shift versus²J_Si–O–Si coupling can be mapped into a two dimensional structural correlation of linkage Si–O–Si angle to mean Si–O–Si angle of the two coupled ²⁹Si nuclei.

2 Method

All ab initio calculations were carried out using Gaussian 09²⁸ at the Ohio supercomputing center,²⁹ running HP SL390 G7 two-socket servers with Intel Xeon x5650 (Westmere-EP, 6 core, 2.67 GHz) processors and 48 GB memory. The natural atomic orbital and natural bond orbital analysis was performed using Gaussian NBO version 3.1.³⁰ The ²J_Si–O–Si couplings were calculated using DFT with B3LYP functional on a small O centered SiH₃ terminated cluster, (H₃SiO)₃–Si–O–Si–(OSiH₃)₃, shown in Fig. 1. A perfect tetrahedron angle ∠O–Si–O = 109.5° was imposed about Si atoms in all calculations.

Fig. 1 (H₃SiO)₃–Si–O–Si–(OSiH₃)₃ symmetric cluster used in calculating the ²J_Si–O–Si coupling across Si⁽ⁱ⁾–O–Si^(j).

A locally dense basis set was implemented on this cluster, as suggested by Cadars et al.,²⁶ for accurate ²J_Si–O–Si coupling calculations. This follows an implementation of cc-PV5Z basis set on the two central Si (labeled Si⁽ⁱ⁾ and Si^(j) in Fig. 1), 6-31++G basis set on all H, and 6-31++G* basis set on all O and remaining Si. Single point NMR calculations were run with tight self consistent field (SCF) convergence criteria. The integration grid size was increased to a pruned (99, 590), ‘ultrafine’ grid, although no differences in ²J_Si–O–Si couplings were observed from the pruned (75, 302) ‘fine’ integration grid. All further calculations were, therefore, subjected to a ‘fine’ integration grid. With this setup each calculation took approximately four and a half hours split over 12 cpu cores.

Systematic structural variations of the cluster were performed to investigate the ²J_Si–O–Si coupling dependence on and correlations between the central Si–O–Si bond angle, Ω₀, and (a) Si–O bond distances, d_Si–O, (b) O–Si–Si–O inter-tetrahedral dihedral bond angle, ϕ (index O₁–Si⁽ⁱ⁾–Si^(j)–O₄ in Fig. 1) and (c) outer Si–O–Si bond angles (Ω₁ to Ω₆). With the geometry constrained out to the third coordination sphere of the central linkage oxygen, a geometry optimization of the outermost Si–H bond distances and the remaining dihedral angles was performed once using restricted Hartree–Fock, RHF/6-311G(d) basis set. We found, as did Cadars et al.,²⁶ that variations in these fourth coordination sphere geometries had negligible effect on the predicted J-coupling. We confirmed this finding with several ²J_Si–O–Si coupling calculation starting with RHF/6-311G(d) optimized initial geometries and the results are tabulated in Table S3 of the ESI.† A complete list of all geometrical constraints imposed in this study is tabulated in Tables S1–S5 of the ESI.†

Si–O bond distance, d_Si–O

The ²J_Si–O–Si coupling dependence on and correlation between Ω₀ and d_Si–O was explored by performing a series of ²J_Si–O–Si coupling calculations on the optimized structure by varying Ω₀ from 120° to 180° on a uniform grid for three Si–O bond distances, d_Si–O = 1.58 Å, 1.60 Å and 1.62 Å, respectively, Fig. 2C.

Fig. 2 Dependence and correlation of ²J_Si–O–Si couplings between Si⁽ⁱ⁾–O–Si^(j) bond angle, Ω₀ and (A) , (B) O–Si–Si–O inter tetrahedral dihedral angle, ϕ and (C) Si–O bond distance, d_Si–O. For calculations in (A) d_Si–O = 1.60 Å and Ω_k≠0 = Ω_out varied from 120° to 180°. In (B) d_Si–O = 1.60 Å, Ω_out = 146° and ϕ varied from −60° to +60°. In (C) Ω_out = 146° and d varied from 1.58 Å to 1.62 Å.

O–Si–Si–O dihedral angle, ϕ

A similar series of calculations were performed on the optimized geometry to investigate the ²J_Si–O–Si coupling dependence on and correlation between Ω₀ and ϕ. This was accomplished by independently varying ϕ and Ω₀ from −60° to +60° and 120° to 180°, respectively, Fig. 2B. All Si–O bond distances were set to 1.6 Å.

Outer Si–O–Si bond angle, Ω_k≠0

A complete systematic exploration of the ²J_Si–O–Si dependence on the six outer Si–O–Si bond angles would have exceeded our computational capabilities. In light of the well established⁸ linear dependence of the isotropic ²⁹Si chemical shift on the mean Si–O–Si bond angle for a Q⁴ tetrahedra we attempted to reduce the dimensionality of the problem by using the mean Si–O–Si bond angle for each Q⁴ tetrahedra, (Si⁽ⁱ⁾ and Si^(j)) involved in the ²J_Si–O–Si coupling, that is,to calculate a double meanThrough systematic variation of Ω₀ (the central linkage angle) and we show (vide infra) that a combined measurement of isotropic ²⁹Si chemical shift and ²J_Si–O–Si can be exploited to determine the local structure around the Q⁴–Q⁴ linkage.

Despite this effort to reduce the dimensionality of this problem from seven to two, there still exist infinite combinations of Ω_k≠0 that lead to the same in eqn (2), with the exception of the singular—and highly unlikely—occurrence of . To ensure a systematic variation of the local structure, we choose to constrain all Ω_k≠0 = Ω_out. A series of ²J_Si–O–Si calculations were performed by independently vary Ω_out and Ω₀ from 120° to 180° on a uniform grid. The calculated ²J_Si–O–Si as a function of and Ω₀ is presented in Fig. 2A. All Si–O bond distances were set to 1.6 Å. Of course, the constraint Ω_k≠0 = Ω_out, implemented only to ensure a systematic local structural variation, is unrealistic even for most crystalline silicates and highly siliceous zeolites, as well as in silica glass and other silica rich disordered materials. To break free from this constraint numerous ²J_Si–O–Si coupling calculations were performed at arbitrary outer Si–O–Si bond angles Ω_k≠0 and ϕ, with values listed in Tables S3 and S4 of the ESI.† These calculations are used to further verify agreement with our proposed ²J_Si–O–Si coupling model.

All additional numerical analysis codes were written in python using NumPy libraries.³¹ The least square analysis was performed using python's LMFIT³² module. The graphics were produced using python's matplotlib library.³³

3 Theory

The J coupling contribution to the nuclear spin Hamiltonian can be written³⁴

Ĥ_J = [small mu, Greek, circumflex]_N·[scr K, script letter K]·[small mu, Greek, circumflex]_N′ = ℏ²γ_Nγ_N′Î_N·[scr K, script letter K]·Î_N′. (3)

The convention is to combine the gyromagnetic ratio constants and the reduced [scr K, script letter K] tensor such that

Ĥ_J = ℏ2πÎ_N·J·Î_N′, (4)

withThis gives a J tensor with dimensions of inverse time.

As we explored the calculated variations in ²J_Si–O–Si with changing cluster structure, we searched for the possible empirical relationships that might characterize the observed correlation of ²J_Si–O–Si to structure. In our calculations we found that the majority of the J-coupling arises from the Fermi contact (FC) contribution, and the remaining contributions from spin–dipolar (SD), paramagnetic spin–orbit (PSO), and diamagnetic spin–orbit (DSO), as illustrated in Fig. S2 of the ESI,† account for less than 10% of the net J-coupling. With this in mind, we looked for guidance in the older literature of J coupling theory and focused on the simple and highly approximate MO theory approach outlined by Pople and Santry,^35–40 which considers only the isotropic Fermi contact contribution and yields the expression

where μ₀ is the magnetic constant, μ_B is the Bohr magneton, s_N(0) and s_N′(0) are the values of the valence s-orbitals of atoms N and N′ at the nuclei, and π_N,N′ is the mutual atom–atom polarizability of Coulson and Longuet-Higgins,⁴¹ given byHere the summation is over occupied (i) and unoccupied (j) molecular orbitals, |ψ_i〉, with energy ε_i and given bywhich are expressed in terms of the valence hybrid type orbitals (HTOs), |ϕ_μ〉, which are given by

|ϕ_μ〉 = a_μ|s〉 + (1 − a²_μ)^1/2|p〉, (9)

where |s〉 and |p〉 are the atomic-type orbitals and a²_μ is the s-character of the HTO. In the summation of eqn (7) the μ and ν index the HTOs on N and N′, respectively.

An exhaustive search of the literature reveals few MO theory studies considering the geminal ²J_AB coupling between tetrahedrally coordinated atoms. The most relevant and detailed discussion we could find on this topic is a chapter in 1988 by Klessinger and Barfield,⁴² examining the dependence of geminal ¹³C–¹³C coupling constants. In this case they derive the mutual atom–atom polarizability as

where a²₁ and a²₃ are the s-character of the HTOs at carbon C₁ and C₃ along C–C bond directed towards C₂, in a C₁–C₂–C₃ linkage. The integral β_μ,ν is the matrix element of the Hamiltonian operator in the HTO basis set |ϕ_μ〉,

β_μ,ν = 〈ϕ_μ|H|ϕ_ν〉. (11)

The definition of these integrals are given in Klessinger and Barfield.⁴² If all integrals in eqn (10) except are ignored, the mutual atom–atom polarizability term can be approximated⁴² to

π_C1,C3 ∝ a²₁a²₃a⁴₂, (12)

where a²₂ is the s-character of the valence HTO at C₂. The ²J_C1,C3 can then be approximated to

²J_C1,C3 ∝ a²₁a²₃a⁴₂. (13)

While eqn (13) predicts a simple linear correlation of ²J_C1,C3 to the s-character product, we expect this highly approximated correlation to deviate from linearity due to the neglect of the vicinal integrals.⁴² Nevertheless, this approximate model provides a useful starting point for developing an empirical expression for geminal J coupling across two coupled ²⁹Si. On the basis of eqn (13), we propose that ²J_Si–O–Si is approximately given bywhere and are the products of the s-character of the valence HTOs associated with the Si⁽ⁱ⁾–O and Si^(j)–O bonds across Si⁽ⁱ⁾–O–Si^(j) linkage, respectively, as illustrated in Fig. 3.

Fig. 3 Simple illustration of valence HTOs associated with the Si⁽ⁱ⁾–O and Si^(j)–O bonds across Si⁽ⁱ⁾–O–Si^(j) linkage.

4 Results and discussion

4.1 Dependence on local structure

In this subsection, we discuss and examine the contributions to the net ²J_Si–O–Si coupling arising from the variations in the local structure on the basis of the underlying s-characters a²_O, , and at the bridging oxygen and adjacent silicons respectively. These values were determined from the quantum chemistry DFT cluster calculation using Gaussian NBO version 3.1. For clarity, we only present a subset of results from Fig. 2 per structural parameters considered.

Shown in Fig. 4A is the expected dependence of ²J_Si–O–Si coupling on Ω₀ ∈ [120°,180°], for a subset of results from Fig. 2A where the outer Si–O–Si angles are held constant at Ω_k≠0 = 180° and the distances held constant at d_Si–O = 1.6 Å. This dependence of ²J_Si–O–Si coupling on the Si–O–Si linkage angle, Ω₀, has been previously discussed by both Cadars et al.²⁶ and Florian et al.²⁵ In Fig. 4B we see the more intriguing result that the ²J_Si–O–Si coupling has a markedly linear correlation to the product , as predicted by eqn (14). As noted earlier, the slight deviation from linearity observed is not unexpected and is likely attributed to the neglected terms in the mutual atom–atom polarizability term. In Fig. 4C and D we see that the variation in ²J_Si–O–Si primarily arises from variation in a⁴_O,—the s-character product of the two valence HTOs at the bridging oxygen—while there is a minor yet non-negligible variation coming from —the s-character product of the two silicon valence HTOs in the Si–O–Si linkage. The change in a⁴_O is the result of the change in the hybridization of the valence orbitals at the bridging oxygen from sp² (33.33% s-character) at Ω₀ = 120° to sp (50% s-character) at Ω₀ = 180°. A popular approximation for the s-character at the bridging oxygen^9,42 is given by

Here we use the symbol f_O(Ω) to distinguish the approximated s-character at the bridging oxygen from the symbol a²_O for the s-character calculated using quantum chemistry DFT calculations.

Fig. 4 Dependence of ²J_Si–O–Si coupling on (A) Si–O–Si bond angle Ω₀, (B) , (C) a⁴_O, and (D) for the constraints Ω_k≠0 = 180° and d = 1.6 Å.

With the s-character of each sp³ valence HTO on a tetrahedral silicon expected to be 25%, the calculated value of is also as expected at ∼(25%)² = 6.25%. The slight increase in from 6.1% to 6.7% in Fig. 4D with decreasing Ω₀ may seem surprising from a simple hybrid orbital picture—as all intra-tetrahedral angles and Si–O distances are held constant at ∠O−Si−O = 109.5° and d_Si–O = 1.6 Å, respectively, in these calculations. In fact, for this subset of results, even the outer Si–O–Si angles are held fixed at 180°, so it is only the variation of the linkage angle Ω₀ that is responsible for this slight change in . We will examine the origin of this variation shortly when the influence of the outer Si–O–Si angles on ²J_Si–O–Si are considered.

In Fig. 5A is the variation of ²J_Si–O–Si with the two central linkage Si–O bond distances, d_Si–O ∈ [1.58 Å, 1.62 Å], for a subset of results from Fig. 2C where the central linkage angle was fixed at Ω₀ = 180° and the outer Si–O–Si angles and outer distances are held constant at Ω_k≠0 = 146° and d_Si–O = 1.6 Å, respectively. The ²J_Si–O–Si coupling remains relatively constant around 21.5 Hz over this range of central linkage Si–O bond distances, with a slight decrease from ∼22 to ∼21 Hz with increasing bond length. This relative independence of ²J_Si–O–Si on the central linkage Si–O bond distances is consistent with previous observations by Florian et al.²⁵ In Fig. 5B we find again that the ²J_Si–O–Si coupling has a linear correlation to the product , and in Fig. 5C we see no observable dependence of ²J_Si–O–Si coupling on the s-character product a⁴_O at the bridging oxygen. This is because, for this subset of calculations, the hybridization at the bridging oxygen was locked to sp through the constraint Ω₀ = 180°. Since the s-character at the bridging oxygen predominantly depends on the Si–O–Si bond angle, as noted in eqn (15), the change in Si–O bond distance, d_Si–O, shows no observable change in its s-character. In Fig. 5D we find that the minor variation in ²J_Si–O–Si coupling is dominated by the change in . While a decrease in the a²_Si with increasing Si–O bond distance has been previously reported by Grimmer and coworkers^10,11 it was in the context of correlated changes in the intra-tetrahedral angle ∠O–Si–O. With the intra-tetrahedral angles in this calculation fixed at ∠O–Si–O = 109.5°, it seems that changes in Si–O length alone lead to minor variations in a²_Si—although these result in relatively insignificant variations in ²J_Si–O–Si.

Fig. 5 Dependence of ²J_Si–O–Si coupling on (A) Si–O bond distance, d_Si–O, (B) , (C) a⁴_O, and (D) for the constraints Ω₀ = 180° and Ω_k≠0 = 146°.

In Fig. 6A is the variation of ²J_Si–O–Si with the inter-tetrahedral dihedral angle ϕ ∈ [−60°,60°] for a subset of results from Fig. 2B where the central linkage angle was fixed at Ω₀ = 180°, and the outer Si–O–Si angles and outer distances are held constant at Ω_k≠0 = 146° and d_Si–O = 1.6 Å, respectively. A periodic modulation of the form cos 3ϕ for ²J_Si–O–Si over a range of 1.6 Hz is observed due to the local three fold symmetry of the (H₃SiO)₃–Si–O–Si–(OSiH₃)₃ cluster when rotating about ϕ. There is no observable variation in the hybrid orbital s-character products in Fig. 6B–D as expected, since this variation is associated with the vicinal integral terms of the mutual atom–atom polarizability expansion which are completely neglected in eqn (14).

Fig. 6 Dependence of ²J_Si–O–Si coupling on (A) O–Si–Si–O inter tetrahedral dihedral angle, ϕ, (B) , (C) a⁴_O, and (D) for the constraints Ω_k≠0 = 146°, Ω₀ = 180° and d = 1.6 Å.

In Fig. 7A is the variation of ²J_Si–O–Si with , as calculated by eqn (2), for a subset of results from Fig. 2A subjected to the constraint Ω₀ = 180°, Ω_k≠0 = Ω_out and d = 1.6 Å where Ω_out varied from 120° to 180°. The variation in Fig. 7A is about 25% of that shown in Fig. 4A and is found to be the second most dominant dependence of ²J_Si–O–Si coupling. As Ω₀ is fixed, it is the change in the outer Si–O–Si bond angles, Ω_out, that leads to this variation in ²J_Si–O–Si. In Fig. 7B we again find the markedly linear correlation of ²J_Si–O–Si coupling with as predicted by eqn (14). As might be expected, no observable dependence of ²J_Si–O–Si coupling on a⁴_O is observed in Fig. 7C since the hybridization at the bridging oxygen was fixed to sp with the constraint Ω₀ = 180°.

Fig. 7 Dependence of ²J_Si–O–Si coupling on (A) the average Si–O–Si bond angles, , (B) , (C) a⁴_O, and (D) for the constraints Ω₀ = 180° and d = 1.6 Å.

Clearly, the origin of the ²J_Si–O–Si dependence on the outer angles, Ω_out comes from the variation in as seen in Fig. 7D. Why would the or increase as the outer angles, Ω_out, increase? This is the same question alluded to earlier with respect to Fig. 4D. The logic is as follows. We expect the sum of the s-characters for the four HTO around each silicon to be constant. So, as the s-character of one HTO deviates from 25%, the s-characters of the other HTO compensate to maintain the constant sum. Hence, we expect the s-character of the HTO that is part of the central linkage to increase as the average s-character of the other three HTOs decreases. Close examination of the variation in as the function of the Si–O–Si tetrahedral angle, Ω₀, and average Si–O–Si bond angle, 〈Ω〉_i, also shown in Fig. S3 of the ESI,† reveals an approximate proportionality given by

In regard to Fig. 4D, since the outer Si–O–Si bond angles are held constant, Ω_k≠0 = 180°, the individual s-characters at Si⁽ⁱ⁾ and Si^(j) and, therefore, the product, , decreases as Ω₀ increases.

The highly approximate J-coupling model in eqn (14) predicts a linear correlation of ²J_Si–O–Si coupling with respect to the s-character product, . From the ab initio calculations, shown in Fig. 8, however, we find that the ²J_Si–O–Si coupling is better described by a quadratic in the s-character product, , with R² = 0.99164. As mentioned earlier, this slight curvature is not unexpected, since the model in eqn (14) neglects all the vicinal integrals from the mutual atom–atom polarizability term. Further discussion on the effect of vicinal integrals can be found in the chapter by Klessinger and Barfield.⁴²

Fig. 8 ² J _Si–O–Si coupling as a function of at Si⁽ⁱ⁾, Si^(j) and O across Si⁽ⁱ⁾–O–Si^(j) linkage. The points in gray and black corresponds to the systematic (Ω_k≠0 = Ω_out) and arbitrary structural variations, respectively.

Fig. 8 shows the ²J_Si–O–Si couplings in two different colors, gray and black. The gray dots correspond to couplings evaluated by systematic variation of the local structure, also shown in Fig. 2, and are provided in the Tables S1 and S2 of the ESI.† The black dots correspond to couplings evaluated at arbitrary Ω_k and ϕ and are listed in Tables S3 and S4 of the ESI.† Most notably a consistent trend in ²J_Si–O–Si is observed with respect to the s-character product, , for both systematic and arbitrary structural variation.

On the basis of the results and discussion presented here we can now construct an expression relating ²J_Si–O–Si to local structure. The most straightforward approach would be a substitution of eqn (15) and (16) into eqn (14). We found, however, that such an approach leads to an excessive number of coefficients for calibrating the relationship. Instead we found that the same relationship can be expressed with fewer coefficients using

For this expression the coefficients m₁ = 0.778 ± 0.004 Hz per °, m₂ = 0.0058 ± 0.0005 Hz per ° and J₀ = −8.3 ± 0.1 Hz were determined by least square minimization of the objective functionNote there is a strong cross correlation coefficient ρ_m1,J0 = −0.918 between m₁ and J₀ in this determination. A comparison of ab initio²J_Si–O–Si couplings with respect to the ²J_Si–O–Si coupling model in eqn (17) is presented in Fig. 9A. Again, the points in gray and black correspond to systematic and arbitrary variation of the local structure, respectively. Excellent agreement between the ²J_Si–O–Si coupling model and ²J_Si–O–Si coupling from ab initio calculation is observed with R² = 0.99518. Note that in this approach the slight dependence on Si–O bond distance, d_Si–O, has been neglected.

Fig. 9 (A) Plot comparing prediction of ²J_Si–O–Si coupling model, eqn (17), and ²J_Si–O–Si coupling from ab initio calculations. Gray and black dots correspond to ²J_Si–O–Si couplings evaluated by systematic and arbitrary structural variations, respectively. The calculated Pearson correlation coefficient is R² = 0.99518. (B) Plot comparing prediction of ²J_Si–O–Si coupling model, eqn (19) and ²J_Si–O–Si coupling from ab initio calculations. The calculated Pearson correlation coefficient is R² = 0.99016.

Given that m₂ ≪ m₁ we find that the ²J_Si–O–Si coupling model of eqn (17) can be simplified by dropping the m₂ cos 3ϕ term to obtain

A comparison of ab initio²J_Si–O–Si couplings with respect to the ²J_Si–O–Si coupling model in eqn (19) is presented in Fig. 9B. Neglecting the ϕ dependence leads to slightly greater scatter which is more noticeable at the higher couplings and a small drop of linear correlation coefficient to R² = 0.99016. Given this agreement all further analysis will use the ²J_Si–O–Si coupling model of eqn (19).

4.2 Mapping to local structure

Even with our approximate model for ²J_Si–O–Si in eqn (19) being only a function of Ω₀ and there is no unique mapping of a single ²J_Si–O–Si coupling back to local structure. Fortunately, the ²⁹Si isotropic chemical shift of a Q⁴ site has a well established correlation^7,9,43 to the mean inter-tetrahedral angle, 〈Ω〉, of a given Q⁴, of which, the linear correlation⁸

δ_CS = a_δ〈Ω〉 + b_δ, (20)

is the simplest, while still giving a reasonably accurate correlation in the relevant range of 〈Ω〉 ∈ [140°,160°] as detailed further in the ESI.† The data for a number of crystalline silicas and siliceous framework silicates taken from the literature^9,44,45 is shown in Fig. 10, along with a fit to eqn (20) with coefficients a_δ and b_δ provided in Table 1.

Fig. 10 Linear correlation between average Si–O–Si bond angle, 〈Ω〉, about the Si tetrahedron and ²⁹Si isotropic chemical shift, δ_CS. Twelve ²⁹Si isotropic chemical shift sites in Tridymite were taken from Kitchin et al.⁴⁵ and average Si–O–Si bond angles from Baur.⁴⁴ The remaining were obtained from Engelhardt and Radeglia⁹ and references within.

Table 1 Final coefficients for eqn (22) after calibration with the results of Fig. 10 and eqn (23) after calibration with Sigma-2

Coefficient	Value	Coefficient	Value
a δ	−0.6148 ppm per °	b δ	−19.297 ppm
a j	107.88°	b j	223.49°
c j	0.00002487°	d j	53.01
m 1	0.778 Hz per °	J 0	−7.5 Hz

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The ²J_Si–O–Si coupling across ²⁹Si⁽ⁱ⁾–O–²⁹Si^(j) linkage involves two ²⁹Si isotropic chemical shifts, δ_CS,i and δ_CS,j, associated with ²⁹Si⁽ⁱ⁾ and ²⁹Si^(j) respectively. Using the linear correlation of eqn (20), the average ²⁹Si isotropic chemical shift is given by

With a simple inversion we haveInverting eqn (19) for Ω₀ giveswhere the coefficient a_j, b_j, c_j and d_j are listed in Table 1. Details on the solution for Ω₀ are given in Appendix A. To calibrate eqn (23) against previous experimental measurements we use the ²⁹Si INADEQUATE NMR results of Cadars et al.²⁶ on polycrystalline highly siliceous zeolite Sigma-2, whose structure, shown in Fig. 11A, was predetermined using single crystal X-ray analysis. There are four observable ²J_Si–O–Si couplings in Sigma-2. The observed ²J_Si–O–Si coupling and the mean ²⁹Si isotropic chemical shift, , of the two coupled nuclei corresponding to the four ²⁹Si–O–²⁹Si pairs from this measurement are listed in Table 2. Here, the mean ²⁹Si isotropic chemical shift, , was determined as half the corresponding ²⁹Si double quantum frequency of the INADEQUATE spectra. The X-ray determined Si–O–Si bond angle, Ω₀ and , for these pairs are also listed in Table 2. We choose to calibrate eqn (23) by only varying J₀ to obtain agreement between model and experiment. This was accomplished by performing a least square minimization of the objective functionwhere Ω₀(X-ray) is the Si–O–Si bond angle inferred from single crystal X-ray analysis, and Ω₀ is the Si–O–Si bond angle calculated using eqn (23) with m₁ = 0.778 Hz per ° held constant. With this approach we obtain best agreement with J₀ = −7.5 ± 0.6 Hz.

Fig. 11 (A) Schematic of highly siliceous zeolite Sigma-2 adapted from Cadars et al.²⁶ (B) A correlation plot of and J coupling for Sigma-2 with superimposed calibrated and Ω₀ grid. (C) Comparison of single crystal X-ray vs. NMR determined and Ω₀ for the four different ²⁹Si pairs in Sigma-2. Superimposed is the and J coupling grid. A good agreement between the two results is observed for pair 4-1, 2-3 and 1-3.

Table 2 Observed ²J_Si–O–Si couplings (column 3) and ²⁹Si average isotropic chemical shift, , (column 2) for the ²⁹Si pairs (column 1) in highly siliceous zeolite Sigma-2.²⁶ Listed along column 4 and 5, is Ω₀ and obtained from the crystal structure determined by single crystal X-ray analysis. Listed in column 6 and 7, is the Ω₀ and calculated using eqn (22) and (23)

^29Si pair	Experimental		Calculated
	NMR		Via X-ray		Via NMR
	/ppm	J/Hz	Ω 0/°	/°	Ω 0/°	/°
1-3	−118.0	23.5	172.8	158.2	176.0	160.5
2-3	−116.8	16.5	153.5	155.6	152.0	158.6
4-2	−111.25	10.0	148.7	148.9	141.5	149.5
4-1	−112.4	6.3	137.2	151.6	134.0	151.4

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A plot of measured ²J_Si–O–Si coupling vs. from Sigma-2 is presented in Fig. 11B. Overlaid on top is a grid map of calibrated and Ω₀. In Fig. 11C, the calculated and Ω₀ (filled circles) along with X-ray determined and Ω₀ (open circles) are presented. Overlaid on top is the grid map of J coupling and . Agreement to within ∼3° between Ω₀ from the X-ray and NMR measurements is observed for pairs 4-1, 2-3 and 1-3, while there is a mismatch of ∼7° for the 4-2 pair. With only the limited data from Sigma-2, it is clear that additional experimental efforts in refinement of the calibration of eqn (23) would be helpful. Such efforts are, in fact, currently in progress in our laboratory on highly silicious zeolites using the recently developed PIETA method⁴⁶ for rapid and sensitive 2D J NMR spectroscopy.

Overall, the results presented here are extremely promising and open the door to new opportunities to more fully exploit ²J_Si–O–Si couplings as quantitative probes of structure in silicates. With only a ∼1 Hz change from the ab initio derived value of J₀ = −8.3 ± 0.1 Hz, to the Sigma-2 calibrated value of J₀ = −7.5 ± 0.6 Hz, the proposed correlated models of ²⁹Si chemical shift and ²J_Si–O–Si coupling provide an acceptable model for the quantitative interpretation of the ²J_Si–O–Si coupling. It will be interesting to see if a similar analysis can be applied with other geminal J couplings across a bridging oxygen, such as a ³¹P–O–³¹P or ²⁷Al–O–²⁹Si linkage.

5 Summary

While both scalar J-couplings and homonuclear dipolar coupling are used qualitatively to establish connectivities between Si sites, it has been primarily homonuclear ²⁹Si–²⁹Si dipolar couplings through measurements of double quantum buildup curves that have provided some of the most useful quantitative details in the structure refinement of many siliceous zeolites and NMR crystallographic structural studies of meso- and microporous silicate materials.^18,47,48 Here we have examined whether ²J_Si−O−Si, the geminal coupling across a Si–O–Si linkage, can be turned into a more quantitative probe of the local structure in silicate networks.

Using high level density function theory (DFT) methods, we have found that the two main influences on the ²J_Si–O–Si couplings are a primary dependence on the linkage Si–O–Si angle and a secondary dependence on mean Si–O–Si linkage of the two coupled ²⁹Si nuclei. We show that the simple and highly approximate MO theory approach outlined by Pople and Santry^35–40 can provide key insights when developing approximate models for geminal J-couplings based on results from high level density function theory (DFT) methods.^25,26 Exploiting a well established correlation between ²⁹Si isotropic chemical shift and the mean Si–O–Si angle of a Q⁴ site, we have developed an approach where a correlation plot of ²J_Si–O–Si to mean ²⁹Si isotropic chemical shift can be mapped into a 2D correlation of linkage Si–O–Si bond angle, Ω₀, to mean Si–O–Si bond angle of the two coupled ²⁹Si. Using available experimental ²J_Si–O–Si couplings from Sigma-2,²⁶ we found that only a minor adjustment of one ab initio derived coefficient in our ²J_Si–O–Si model was needed to bring our model in line with experimental results.

Conflicts of interest

There are no conflicts to declare.

Appendix

A Inversion of ²J_Si–O–Si to Ω₀

For inversion of eqn (19) with respect to Ω₀, a close form solution can be obtained by first recasting eqn (19) into the form of a cubic equationwhere λ = cos Ω₀, andThe roots of eqn (25) arewhere we have definedThe number of real and complex roots can be determined from the sign of the discriminant, D(x) in eqn (28d):

• If D(x) > 0, one root is real and two are complex conjugate.

• If D(x) = 0, all roots are real with at least two equal.

• If D(x) < 0, all roots are real and unequal.

For the given problem, the discriminant D(x), eqn (28d), is positive if x > −6.75. For given J₀ < 0 Hz and m₁ > 0° Hz⁻¹, x is always positive and so is the discriminant. Therefore, there exists only one real root of eqn (25) given by eqn (27a). Thus, the inversion of eqn (19) with respect to Ω₀ is

To further simplify this result, Ω₀(x) was approximated to a function g(x) that closely resembles Ω₀(x) within the relevant range of 120° to 180°. For Ω₀(x) ∈ [120°,180°], λ₁(x) maps to a range λ₁(x) ∈ [−0.5, −1.0] which further maps to a range x ∈ [1/18,1/4]. Within the relevant range x ∈ [1/18,1/4], Ω₀(x) can be approximated as

g(x) = a_j + b_jx + c_j exp{d_jx}, (30)

where the coefficients are listed in Table 1. The function g(x) provides a good approximation of Ω₀(x) with |g(x) − Ω₀(x)| < 0.5° within the range Ω₀(x) ∈ [120°, 176°]. Deviations from 176° and onwards to a maximum of 3.7° at Ω₀(x) = 180° is significant although can be neglected because of its low probability. A comparison of Ω₀(x) and g(x) as a function of x is provided in the ESI.†

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. CHE-1506870.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp06486a

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