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NMR

This page gives hints on how to calculate NMR chemical shieldings with the ABINIT package.

Introduction

A key observable measured in NMR spectroscopy is the chemical shielding, which is usually thought of as the shielding of the external magnetic field caused by the electrons in the sample, the effect of which is to reduce slightly the Zeeman splitting of the energy levels of the nuclear magnetic dipole from that of a bare nucleus [Slichter1978]. More precisely, the external magnetic field induces an orbital electronic current, which itself generates a small secondary magnetic field opposite to the bare field. One approach to computing this effect in a DFT context is provided by the Gauge Including Projector Augmented Wave formalism (GIPAW) to compute the induced current, and from that, the effective induced field [Pickard2001].

From an energetic perspective, though, chemical shielding is just the effect on the total energy of both a nuclear magnetic dipole and an external magnetic field:

\[ \sigma_{ij} = \frac{\partial^2 E}{\partial \mathbf{m}_i\partial \mathbf{B}_j} \]

for nuclear dipole \(\mathbf{m}\) and magnetic field \(\mathbf{B}\). From this perspective, the shielding is either the induced field acting on the bare dipole, or the induced dipole acting on the bare field [Thonhauser2009]. The latter approach is implemented in ABINIT, where the first order energy

\[ E^{(1)} = \frac{\partial E}{\partial\mathbf{B}_j} \]

is computed in the presence of a small nuclear magnetic dipole [Zwanziger2023].

Execution

While not a true response function, \(E^{(1)}\) turns out to dependn on both the ground state wavefunctions and the DDK wavefunctions, \(|\partial u_{n\mathbf{k}}/\partial k\rangle\). Thus, to compute the effect in ABINIT, first, ground state wavefunctions are computed in the presence of a small nuclear magnetic dipole moment. The moment is described by the variable nucdipmom, which is input as a set of 3-vectors, one for each atom in the unit cell. Most of these will be zero, and typically just “1 0 0” or “0 1 0” or “0 0 1” will be input for the single atom one wishes to study. The triple of numbers are the Cartesian directions, so to compute the full spatial dependence, three separate calculations will be carried out.

Once the ground state is computed with a dipole on the atom and in the direction of interest, a DDK calculation is carried out, with rfddk 1 and rfdir 1 1 1, again with a dipole imposed with nucdipmom as in ground state. For this calculation in addition, set orbmag 2 to initiate computation of the orbital magnetization and hence shielding at the end of the DDK calculation.

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