How NMR Calculation Works¶

The Physical Picture¶

An external magnetic field, B₀, induces circulating currents in a molecule's electron density. By Lenz's law, these induced currents generate a small local magnetic field that opposes B₀ at each nucleus. The nucleus therefore feels a slightly different effective field than the one actually applied:

B(effective) = B₀ (1 − σ)

where σ is the shielding constant — a property of the local electronic environment around that specific nucleus, not a single number for the whole molecule. Every chemically distinct nucleus has its own shielding tensor.

Why a Tensor, Not Just a Number¶

The induced currents — and therefore the shielding — depend on the orientation of the molecule relative to B₀. A full description needs a 3×3 tensor, not a single value. Two quantities are typically extracted from it (see Isotropic Shielding and Anisotropy):

the isotropic shielding, σ_iso — the average over all orientations, and the value that connects directly to a chemical shift measured on a molecule tumbling freely in solution,
the anisotropy, Δσ — how much the shielding varies with orientation, relevant for solid-state NMR and for understanding the tensor's shape.

The Gauge Origin Problem¶

Computing the induced current requires choosing an origin for the vector potential describing the magnetic field — a gauge origin. In principle, the physical shielding shouldn't depend on this arbitrary mathematical choice. In practice, with a finite (incomplete) basis set, it does: different gauge origins give different, both formally "correct," but numerically inconsistent shielding values.

Gaussian resolves this with Gauge-Including Atomic Orbitals (GIAO): each basis function carries its own, individual gauge origin, tied to the nucleus it's centered on. This removes the artificial gauge dependence and is why the NMR=GIAO keyword names the method explicitly rather than just "NMR". See Gauge-Including Atomic Orbitals (GIAO) for the full derivation.

Getting From the Wavefunction to a Shielding Tensor¶

Formally, the shielding tensor is a second derivative of the energy — with respect to the nuclear magnetic moment and the external field. Computing it requires knowing how the molecular orbitals themselves respond to an applied magnetic field, which is obtained by solving the Coupled-Perturbed Hartree-Fock/DFT (CPHF/CPKS) equations — see Coupled-Perturbed Equations for Shielding.

Converged Ground-State Wavefunction
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Apply GIAO Basis (gauge origin tied to each nucleus)
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Solve Coupled-Perturbed HF/DFT Equations
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Second Derivative of Energy w.r.t. B-field and Nuclear Moment
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Shielding Tensor at Each Nucleus

From Shielding to a Spectrum¶

The raw output of the calculation is a shielding constant, in absolute terms — not directly comparable to an experimental chemical shift, which is always reported relative to a reference compound. That last conversion step is covered in Referencing Chemical Shifts.