Metric spaces in NMR crystallography

Abstract

ABSTRACT:The anisotropic character of the chemical shift can be measured by experiments that provide shift tensor values and comparing these experimental components, obtained from microcrystalline powders, with 3D nuclear shielding tensor components, calculated with quantum chemistry, yields structural models of the analyzed molecules. The use of a metric tensor for evaluating the mean squared deviations, d 2 , between two or more tensors provides a statistical way to verify the molecular structure governing the theoretical shielding components. The sensitivity of the method is comparable with diffraction methods for the heavier organic atoms (i.e., C, O, N, etc.) but considerably better for the positions of H atoms. Thus, the method is especially powerful for H-bond structure, the position of water molecules in biomolecular species, and other proton important structural features, etc. Unfortunately, the traditional Cartesian tensor components appear as reducible metric representations and lack the orthogonality of irreducible icosahedral and irreducible spherical tensors, both of which are also easy to normalize. Metrics give weighting factors that carry important statistical significance in a structure determination. Details of the mathematical analysis are presented and examples given to illustrate the reason nuclear magnetic resonance are rapidly assuming an important synergistic relationship with diffraction methods (X-ray, neutron scattering, and high energy synchrotron irradiation).

Solid-state 13C NMR investigations of 4,7-dihydro-1H-tricyclopenta[def,jkl,pqr]triphenylene (sumanene) and indeno[1,2,3-cd]fluoranthene: Buckminsterfullerene moieties

“…The error values reported in this paper are calculated using the RMS distance metric analysis approach introduced and discussed in detail by Alderman et al 50 and Grant and Halling. 51 The error between experiment and theory for sumanene is 3.2 ppm. 50,51 The error between experiment and theory for indenofluoranthene is 3.1 ppm.…”

“…51 The error between experiment and theory for sumanene is 3.2 ppm. 50,51 The error between experiment and theory for indenofluoranthene is 3.1 ppm. 50,51 These results show that there is excellent agreement between the theoretical and experimental data.…”

“…50,51 The error between experiment and theory for indenofluoranthene is 3.1 ppm. 50,51 These results show that there is excellent agreement between the theoretical and experimental data. Furthermore, it also suggests that the crystallographic conformations of both sumanene and indenofluoranthene not only are the lowest energy conformations, but they also have the best agreement with NMR data.…”

Cartesian and Spherical Tensors in NMR Hamiltonians

“…( show that among the main three interactions (chemical shift, dipole, and quadrupole interactions) observed in solid state NMR, only chemical shift has non‐zero spherical components of the spin‐part tensor. As the antisymmetric Hamiltonian of the chemical shift tensor is usually neglected, it means that the three components and are not considered . This is the reason why the change of signs in the space‐part tensor components and is seldom mentioned in the literature.…”

From NMR spectra to structure

“…Therefore, this is a good place to discuss what it is the PAS (principal axis system) system. The shielding tensor of nucleus A, is a second‐rank tensor with nine independent components . It can be decomposed into anti‐symmetric and symmetric contributions, .…”