NMR chemical shift as analytical derivative of the Helmholtz free energy

Abstract

We present a theory for the temperature-dependent nuclear magnetic shielding tensor of molecules with arbitrary electronic structure. The theory is a generalization of Ramsey's theory for closed-shell molecules. The shielding tensor is defined as a second derivative of the Helmholtz free energy of the electron system in equilibrium with the applied magnetic field and the nuclear magnetic moments. This derivative is analytically evaluated and expressed as a sum over states formula. Special consideration is given to a system with an isolated degenerate ground state for which the size of the degeneracy and the composition of the wave functions are arbitrary. In this case, the paramagnetic part of the shielding tensor is expressed in terms of the g and A tensors of the electron paramagnetic resonance spin Hamiltonian of the degenerate state. As an illustration of the proposed theory, we provide an explicit formula for the paramagnetic shift of the central lanthanide ion in endofullerenes Ln@C(60), with Ln = Ce(3+), Nd(3+), Sm(3+), Dy(3+), Er(3+), and Yb(3+), where the ground state can be a strongly spin-orbit coupled icosahedral sextet for which the paramagnetic shift cannot be described by previous theories.

Communication: Paramagnetic NMR chemical shift in a spin state subject to zero-field splitting

“…In this Communication we present an alternative derivation of this formula, based on the general theory of NMR chemical shift we have recently developed. 5,6 Interestingly, we obtain a result that differs from that proposed in Ref. 4.…”

“…Despite the increasingly central role played by paramagnetic NMR in the elucidation of the structure of metallo-proteins, 1 and in the investigation of the spin dynamics in novel magnetic materials, 2 only quite recently rigorous theories have been developed for the ab initio calculation of the paramagnetic NMR chemical shift. [3][4][5][6] Of particular relevance in this respect is the work of Moon and Patchkovskii, 3 that of Pennanen and Vaara, 4 and that of Van den Heuvel and Soncini. 5,6 Moon and Patchkovskii 3 derived an expression for the paramagnetic shielding tensor of a spin doublet state in terms of its gand A-tensors.…”

“…[3][4][5][6] Of particular relevance in this respect is the work of Moon and Patchkovskii, 3 that of Pennanen and Vaara, 4 and that of Van den Heuvel and Soncini. 5,6 Moon and Patchkovskii 3 derived an expression for the paramagnetic shielding tensor of a spin doublet state in terms of its gand A-tensors. This treatment was extended by Pennanen and Vaara 4 to arbitrary spin states, in the limit of weak spinorbit coupling, and later generalised by us to a theory that is valid for arbitrary strength of spin-orbit coupling, and arbitrary size of the degenerate manifold.…”

“…This treatment was extended by Pennanen and Vaara 4 to arbitrary spin states, in the limit of weak spinorbit coupling, and later generalised by us to a theory that is valid for arbitrary strength of spin-orbit coupling, and arbitrary size of the degenerate manifold. 5,6 In Ref. 4 the authors also proposed a general formula for the paramagnetic shielding tensor of a spin state subject to zero-field splitting (see Eq.…”

“…9 A general formula for the shielding tensor was proposed in Ref. 6, to which the reader is referred for more details:…”