Comparing Cosine and Sine Functions for Calculating the Magnetic Shielding of ²⁹Si Nuclei across Siliceous Zeolites

The relationship between the s-fraction of O orbitals, electronic density at the O lone pair, and Si-O-Si angles in the total magnetic shielding σ(²⁹Si) of zeolite materials is typically approximated using the cos(⟨Si-O-Si⟩)/(cos(⟨Si-O-Si⟩) - 1) function, with the ⟨Si-O-Si⟩ being the averaged Si-O-Si angle over the four O neighbors of Si. In this work, an analogous sin(⟨Si-O-Si⟩) dependence is introduced based on the approximation of the absolute O dipoles Q₁(O) versus the Si-O-Si angle in five zeolite structures (BEA, FER, GOO, ZSM-12, and ZSM-57). The models, along with α-quartz, were considered to evaluate the advantages of the effects on both chemical shifts and chemical shift anisotropies, as measured for α-quartz and ZSM-12. The decrease of the O dipole with the Si-O-Si angle is usually interpreted as a variation of the electronic density at the O lone pairs. The dependence of Q₁(O) versus sin(Si-O-Si) demonstrated a similar high level of accuracy for Q₁(O) in the zeolites. The sin(⟨Si-O-Si⟩) function for the magnetic isotropic δ_iso(²⁹Si) chemical shifts was also demonstrated for some of the five structures. The final conclusion regarding a better approximation of the angular dependence of the σ(²⁹Si) chemical shifts between four candidate functions is linked to the distribution of Si-O bond lengths in the zeolites.