The Nuclear Magnetic Shielding Tensor¶
Definition¶
For a nucleus N in an external magnetic field B, the effective field it experiences is reduced by the induced response of the surrounding electrons:
where σ_N is the nuclear magnetic shielding tensor at nucleus N — a 3×3 matrix, since the induced field need not be parallel to the applied one, and its magnitude generally depends on the molecule's orientation relative to B.
As a Second Derivative of the Energy¶
Formally, each component of the shielding tensor is a mixed second derivative of the molecular energy, with respect to the external magnetic field B and the nuclear magnetic moment μ_N:
This is why an NMR calculation is fundamentally a different kind of task from a geometry optimization or an energy calculation: it doesn't evaluate the energy itself, but a specific second derivative of it with respect to two perturbations (a field and a nuclear moment) that never actually appear in the unperturbed Hamiltonian used to optimize the geometry.
Why This Requires Perturbation Theory¶
Since B and μ_N aren't present in the ordinary electronic Hamiltonian, evaluating this derivative means asking how the wavefunction would change if a small magnetic field were applied — a perturbation theory problem, not a single-point energy evaluation. That response is what the Coupled-Perturbed HF/DFT equations actually solve for.