Essential tools of linear algebra for calculating nuclear spin dynamics of chemically exchanging systems

Abstract

In this work, we describe essential tools of linear algebra necessary for calculating the effect of chemical exchange on spin dynamics and polarization transfer in various nuclear magnetic resonance (NMR) experiments. We show how to construct matrix representations of Hamiltonian, relaxation, and chemical exchange superoperators in both Hilbert and Liouville space, as well as demonstrate corresponding codes in Python. Examples of applying the code are given for problems involving chemical exchange between NH${}_3$ and NH${}_4^{+}$ at zero and high magnetic field and polarization transfer from parahydrogen relevant in SABRE (signal amplification by reversible exchange) at low magnetic field (0-20 mT). The presented methodology finds utility for describing the effect of chemical exchange on NMR spectra and can be extended further by taking into account non-linearities in the master equation.