²H NMR Theory of Transition Metal Dihydrides:  Coherent and Incoherent Quantum Dynamics

Abstract

In this paper a simple phenomenological description of the effects of coherent quantum and incoherent mutual exchange of two deuteron nuclei in solid state transition metal complexes on their 2 H NMR spectra is given. This description is based on the quantum-mechanical density matrix formalism developed by Alexander and Binsch. Only the nuclear spin system is treated quantum mechanically. The quantum exchange interaction in NMR is included in the nuclear spin Hamiltonian, and the interaction with the surrounding bath and incoherent exchange processes are treated as phenomenological rate processes described by rate constants. The incoherent exchange corresponds formally to 180° rotations or jumps of the D−D vector around an axis perpendicular to this vector and averages the different quadrupole splitting of the two deuterons. In principle the dideuteron pair will exist in several rovibrational states. However, if the interconversion among these states is fast, the dideuteron exchange can be described by an average exchange coupling or tunnel frequency X 12 and a single average rate constant k 12 of the incoherent exchange. It is shown that the incoherent exchange gives rise to a relaxation of rate −2 k 12 between coherences created between states of different symmetry. The 2 H NMR line shape of a dideuteron pair in the solid state as a function of tunnel and incoherent exchange rate is studied numerically. For single crystals, the effects of coherent and incoherent exchange are strongly different, in particular if the rate constants are on the order of the quadrupole splitting. The spectra of nonoriented powder samples are more similar to each other. Nevertheless, our calculations show that there are still pronounced differences, which should allow the distinction between coherent and incoherent exchange even in nonoriented samples.