Isotropic Shielding and Anisotropy¶
Diagonalizing the Tensor¶
The raw shielding tensor reported for each atom (see Output File) is a general 3×3 matrix, not necessarily symmetric in the Cartesian frame it's printed in. Diagonalizing it gives three principal values, σ₁₁, σ₂₂, σ₃₃ (Gaussian reports these as Eigenvalues), corresponding to the shielding along the tensor's three principal axes.
For example, for atom 1 (carbon) in the formic acid dimer output:
Isotropic Shielding¶
The isotropic shielding is simply the average of the three principal values:
For atom 1: (−64.5819 + 38.5983 + 99.7689) / 3 = 24.5951 ppm, matching the Isotropic = value Gaussian prints directly. This is the physically relevant quantity for a molecule tumbling freely in solution, since rapid molecular tumbling averages out the orientation-dependent part of the tensor, leaving only this isotropic component to be observed.
Anisotropy¶
The anisotropy describes how much the shielding varies with orientation, how far the tensor is from being perfectly isotropic. Gaussian computes it as:
using the convention that eigenvalues are ordered so σ₃₃ is the one furthest from the other two. For atom 1: 99.7689 − (−64.5819 + 38.5983)/2 = 99.7689 − (−12.9918) = 112.7607 ppm, matching the reported value.
A large anisotropy (as seen for the bridging oxygen, atom 5, at 523.07 ppm) means the local electronic environment responds very differently depending on the molecule's orientation relative to the field, relevant for solid-state NMR, where molecules are fixed in orientation rather than tumbling freely.