Electronic relaxation of paramagnetic metal ions and NMR relaxivity in solution: Critical analysis of various approaches and application to a Gd(III)-based contrast agent

The time correlation functions (TCFs) G(alphaalpha(t)triple bond(Salpha(t)Salpha(0)) (alpha = x,y,z) of the electronic spin components of a complexed paramagnetic metal ion give information about the time fluctuations of its zero-field splitting (ZFS) Hamiltonian due to the random dynamics of the coordination polyhedron. These TCFs reflect the electronic spin relaxation which plays an essential role in the inner- and outer-sphere paramagnetic relaxation enhancements of the various nuclear spins in solution. When a static ZFS Hamiltonian is allowed by symmetry, its modulation by the random rotational motion of the complex has a great influence on the TCFs. We discuss several attempts to describe this mechanism and show that subtle mathematical pitfalls should be avoided in order to obtain a theoretical framework, within which reliable adjustable parameters can be fitted through the interpretation of nuclear-magnetic relaxation dispersion experimental results. We underline the advantage of the numerical simulation of the TCFs, which avoids the above difficulties and allows one to include the effect of the transient ZFS for all the relative magnitudes of the various terms in the electron-spin Hamiltonian and arbitrary correlation times. This method is applied for various values of the magnetic field taken to be along the z direction. At low field, contrary to previous theoretical expectations, if the transient ZFS has negligible influence, the longitudinal TCF GII(t) triple bond G(zz)(t) has a monoexponential decay with an electronic relaxation time T1e different from 1/(2D(r)), D(r) being the rotational diffusion coefficient of the complex. At intermediate and high field, the simulation results show that GII (t) still has a monoexponential decay with a characteristic time T1e, which is surprisingly well approximated by a simple analytical expression derived from the Redfield perturbation approximation of the time-independent Zeeman Hamiltonian, even in the case of a strong ZFS where this approximation is expected to fail. These results are illustrated for spins S = 1, 3/2, and 5/2 in axial and rhombic symmetries. Finally, the simulation method is applied to the reinterpretation of the water-proton relaxivity profile due to P760-Gd(III), an efficient blood pool contrast agent for magnetic-resonance imaging.