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.