Relaxation theory of the electronic spin of a complexed paramagnetic metal ion in solution beyond the Redfield limit

The relaxation of the electronic spin S of a paramagnetic metal ion with fully quenched orbital angular momentum in its ground state is investigated in an external magnetic field through a systematic study of the time correlation functions governing the evolution of the statistical operator (density matrix). Let ω 0 be the Larmor angular frequency of S . When the relaxation is induced by a time-fluctuating perturbing Hamiltonian ℏ H 1 ( t ) of time correlation τ c , it is demonstrated that after a transient period the standard Redfield approximation is relevant to calculate the evolution of the populations of the spin states if ∥ H 1 ∥ 2 τ c 2 ∕ ( 1 + ω 0 2 τ c 2 ) ⪡ 1 and that this transient period becomes shorter than τ c at sufficiently high field for a zero-field splitting perturbing Hamiltonian. This property, proven analytically and confirmed by numerical simulation, explains the surprising success of several simple expressions of the longitudinal electronic relaxation rate 1 ∕ T 1 e derived from the Redfield approximation well beyond its expected validity range ∥ H 1 ∥ τ c ⪡ 1 . It has favorable practical consequences on the interpretation of the paramagnetic relaxation enhancement of nuclei used for structural and dynamic studies.