We recently reported a pump-probe method that uses a single laser pulse to introduce parahydrogen (p-H2) into a metal dihydride complex and then follows the time-evolution of the p-H2-derived nuclear spin states by NMR. We present here a theoretical framework to describe the oscillatory behaviour of the resultant hyperpolarised NMR signals using a product operator formalism. We consider the cases where the p-H2-derived protons form part of an AX, AXY, AXYZ or AA′XX′ spin system in the product molecule. We use this framework to predict the patterns for 2D pump-probe NMR spectra, where the indirect dimension represents the evolution during the pump-probe delay and the positions of the cross-peaks depend on the difference in chemical shift of the p-H2-derived protons and the difference in their couplings to other nuclei. The evolution of the NMR signals of the p-H2-derived protons, as well as the transfer of hyperpolarisation to other NMR-active nuclei in the product, is described. The theoretical framework is tested experimentally for a set of ruthenium dihydride complexes representing the different spin systems. Theoretical predictions and experimental results agree to within experimental error for all features of the hyperpolarised 1H and 31P pump-probe NMR spectra. Thus we establish the laser pump, NMR probe approach as a robust way to directly observe and quantitatively analyse the coherent evolution of p-H2-derived spin order over micro-to-millisecond timescales
