Quantum Mechanical Expression for the NMR Parameters

In this section, we are going to derive quantum mechanical expressions for the elements of the nuclear magnetic shielding and reduced indirect nuclear spin-spin coupling tensors, (T and of closed-shell molecules. According to Eqs. (5.40) and (5.78) 

But we also need the second-order perturbation Hamiltonians jj t are bilinear in the external magnetic induction and a nuclear magnetic moment in the case of the shielding tensor and bilinear in two nuclear magnetic moments in the case of the coupling tensor. Inserting, therefore, the sum of the vector potential for an external field A (r)) = B x (r — Rao)-, Eq. (5.19), and for a nuclear magnetic moment 

The latter is often called the orbital diamagnetic (OD) or diamagnetic nuclear spin-electron orbit operator (DSO), whereas the first could be called the 

Exercise 5.8 Derive the second-order perturbation Hamiltonian, I5qs. (5.81) to (5.84), for the vector potential of a nuclear magnetic moment, and an external magnetic induction, Eq. (5.19), by inserting the two vector potentials in the general expression of the molecular Hamiltonian, Eq. (2.101), retaining the second-order terms. 

Taking the derivatives we arrive at the sum-over-states expressions for the nuclear magnetic shielding tensor 

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