Coupled-Perturbed Equations for Shielding¶
What Needs to Be Solved¶
Computing the shielding tensor (see The Nuclear Magnetic Shielding Tensor) requires knowing how the molecular orbitals respond to a small applied magnetic field — how each orbital changes to first order in B. This response is obtained by solving the Coupled-Perturbed Hartree-Fock (CPHF) equations, or their DFT analogue, Coupled-Perturbed Kohn-Sham (CPKS).
Why "Coupled"¶
Perturbing one orbital with the field changes the effective potential felt by every other orbital (through the Fock/Kohn-Sham operator, which depends on the full electron density). The equation for how orbital i responds therefore depends on how every other orbital responds too — a coupled system of linear equations, solved self-consistently rather than one orbital at a time.
Schematically, for each field component:
where U_i describes the first-order change in orbital i, F₀ is the unperturbed Fock/Kohn-Sham matrix, and the right-hand side involves the field-dependence introduced by the GIAO basis functions themselves.
Where the Shielding Comes From¶
Once these perturbed orbitals are known, the shielding tensor is assembled from them together with the perturbed density — giving the second derivative of the energy defined in The Nuclear Magnetic Shielding Tensor. This is the computational step that actually produces the numbers seen in the Output File: the XX, YX, ZX, ... tensor components, and from them the isotropic value and anisotropy.