Grid-free powder averages: on the applications of the Fokker-Planck equation to solid state NMR

TL;DR

This paper introduces a novel Fokker-Planck formalism for solid state NMR that simplifies powder averaging by eliminating spherical grids, enabling efficient simulations with tensor train techniques and low correlation basis sets.

Contribution

It presents a grid-free Fokker-Planck approach for magic angle spinning NMR, integrating relaxation and diffusion, and demonstrating scalable simulation methods.

Findings

Elimination of spherical quadrature grids in powder averaging.

Use of tensor train techniques to mitigate increased matrix size.

Accurate simulations with low correlation order basis sets.

Abstract

We demonstrate that Fokker-Planck equations in which spatial coordinates are treated on the same conceptual level as spin coordinates yield a convenient formalism for treating magic angle spinning NMR experiments. In particular, time dependence disappears from the background Hamiltonian (sample spinning is treated as an interaction), spherical quadrature grids are avoided completely (coordinate distributions are a part of the formalism) and relaxation theory with any linear diffusion operator is easily adopted from the Stochastic Liouville Equation theory. The proposed formalism contains Floquet theory as a special case. The elimination of the spherical averaging grid comes at the cost of increased matrix dimensions, but we show that this can be mitigated by the use of state space restriction and tensor train techniques. It is also demonstrated that low correlation order basis sets apparently give accurate answers in powder-averaged MAS simulations, meaning that polynomially scaling simulation algorithms do exist for a large class of solid state NMR experiments.