An electron spin density matrix description of nuclear spin–lattice relaxation in paramagnetic molecules

Abstract

A density matrix treatment applicable to molecular electronic systems is formulated and used to derive an expression for the spin–lattice relaxation rate (T−11) of ligand nuclei in paramagnetic transition-metal complexes. In this way the unpaired spin density is introduced into the nuclear relaxation expression. The two-center dipolar integrals which occur in the expression for T1 are evaluated by a method based on solid spherical harmonic expansions whereby the Hamiltonian operator is reexpressed in terms of coordinates with respect to the orbital’s origin. The theory is used to treat 13C and 1H spin–lattice relaxation in ruthenium acetylacetonate. Good agreement is found between the calculated and experimental ratio TC1/TH1 whereas the Solomon–Bloembergen point-dipole approach yields results significantly in variance with the experimental. The theory is used to treat proton spin relaxation in a representative low-spin Fe(III) complex to demonstrate under what conditions the point-dipole approximation is invalidated.

Calculating NMR paramagnetic relaxation enhancements without adjustable parameters: the spin‐3/2 complex Cr(III)(AcAc)3

“…This allows a more direct and positive examination of the point-dipole assumption than is provided by distance considerations alone. Failure of the point-dipole approximation in metal acetylacetonates has been proposed by Gottlieb et al 35 based on calculations of spin density on the acac ligands. Supporting experimental evidence was also found 36 in the parallel behavior of isotropic hyperfine chemical shifts and ( 1 H/ 13 C) R 1 ratios.…”

Determination of the geometric structure of the metal site in a blue copper protein by paramagnetic NMR

“…In contrast to the longitudinal paramagnetic relaxation, the transverse relaxation, R 2p , is affected both by the dipole interaction and the Fermi contact interaction (37,40,44)…”

“…The integral in Eq. 6 was evaluated, assuming that the unpaired electron spin density, (r), is parameterized by the hydrogen-like Slater-type atomic orbitals mentioned above and that overlap spin population can be neglected (37). Thus…”

“…Two different mechanisms contribute to the interaction, namely the dipolar spin-spin interaction (through space) and the scalar Fermi contact interaction (through bonds). Because of the relatively long electron relaxation time, s , of the blue copper proteins (28, 39), the longitudinal paramagnetic relaxation rate, R 1p , of the protons is, to a good approximation, affected only by the dipolar interaction (37,(40)(41)(42)(43) c,1 …”

“…Thus, the electron-nuclear spinspin interactions derived from paramagnetic NMR relaxation rates are included as restraints in a conventional NMR structure determination, together with interproton distance restraints (nuclear Overhauser enhancements; NOEs) and dihedral angle restraints (scalar nuclear spin-spin couplings) normally used for structure determination of proteins by NMR (36). Because the paramagnetic restraints are derived from the relaxation rates of nuclei close to the metal site, the point dipole approximation of the electron spin fails and a description of the paramagnetic relaxation that takes the electron delocalization into account must be used (37). Therefore, the approach requires knowledge of the distribution of the unpaired electron spin, which can be obtained experimentally by x-ray absorption spectroscopy (16,17), electron nuclear double resonance (38), paramagnetic NMR (28-31) spectroscopy, or theoretically by quantum chemical calculations (5,(25)(26)(27).…”