NMR Consequences of the Nucleus–Electron Spin Interactions

GIACOMO PARIGI AND CLAUDIO LUCHINAT

1.1 The Effect of Paramagnetism on NMR Spectra

The presence of unpaired electrons in molecules makes them paramagnetic and largely affects their NMR spectra. The main effects related to the presence of a paramagnetic center, i.e., of atoms or ions with unpaired electrons, are (see Figure 1.1):

(1) the NMR shifts are perturbed, so that the shifts measured for the paramagnetic molecule (d) and for a diamagnetic analogue (ddia) (i.e., for the same molecule without the paramagnetic center or with the paramagnetic metal ion substituted by a diamagnetic one) differ. These differences are called hyperfine shifts (dhs):

d = ddia + dhs (1.1)

(2) the nuclear relaxation rates of the paramagnetic molecule (R1 longitudinal; R2 transverse) are increased with respect to those of a diamagnetic analogue (Rdia1 and Rdia2). This difference is called paramagnetic relaxation enhancement (R1M and R2M):

[MATHEMATICAL EXPRESSION OMITTED] (1.2)

(3) the probabilities for the different orientations of the paramagnetic molecule in a magnetic field are not the same, so a partial self-orientation occurs. This partial self-orientation is responsible for the occurrence of residual dipolar couplings (?vrdc), analogously to what happens when partial molecular orientation is driven by external devices. The presence of paramagnetic residual dipolar couplings affects the J-coupling between nuclei:

[MATHEMATICAL EXPRESSION OMITTED] (1.3)

This chapter deals with the description of these effects, and of their physical origin, which is related to the so-called hyperfine coupling. In eqn (1.1)–(1.3) other minor effects, namely residual anisotropic chemical shifts, cross relaxation terms, and dynamic frequency shifts, respectively, have been neglected (see later).

1.1.1 The Hyperfine Coupling

The interaction between the magnetic moment associated to a nuclear spin I and the magnetic moment associated to an unpaired electron spin S is called hyperfine coupling and is traditionally described by the Hamiltonian

Hhc = S x A x I (1.4)

The energy corresponding to this interaction fluctuates with time (and thus this interaction represents a source of relaxation for the nuclear spin, see Section 1.4) as a result of the molecular motions and of changes in the electron spin state. The other terms describing the contributions to the energy of the system are:

(i)the Zeeman interaction between the applied magnetic field B0 and the nuclear magnetic moment µI=h?II, where h is Planck's constant divided by 2p and ?I is the nuclear magnetogyric ratio,

(ii)the Zeeman interaction between B0 and an 'effective' electron magnetic moment taking into account both the electron spin and the orbital magnetic moments. This is parameterised using the g tensor, with the effective electron magnetic moment given by µS=-µBLD[g x S, where µB is the electron Bohr magneton. Therefore, the g tensor couples the 'effective' electron spin to B0, thus accounting for the presence of the anisotropic orbital contributions to the electron magnetic moment. As a result, the latter depends on the orientation of the molecule with respect to the magnetic field direction,

(iii) the zero-field splitting (ZFS) interaction, due to the spin–orbit coupling, for systems with more than one unpaired electron (S>1/2), and

(iv)all other diamagnetic interactions present in the molecule.

The whole Hamiltonian describing a paramagnetic system is thus composed of the terms

[MATHEMATICAL EXPRESSION OMITTED] (1.5)

where D is the ZFS tensor.

The A tensor in Hhc is composed of two terms: the isotropic Fermi-contact term and the anisotropic electron–nucleus dipole–dipole interaction term

A = AFC1 + ADIP (1.6)

where AFC is a scalar and ADIP is a traceless symmetric tensor. The scalar term, called the contact coupling constant, is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.7)

where µ0 is the permeability of the vacuum, ge is the free-electron g factor and p is the electron spin density at the nucleus (normalized to one electron). Anisotropic orbital contributions are excluded from AFC because the coupling responsible for the contact interaction is only with the unpaired electron spin density residing on the nucleus (and thus in s orbitals).

In the point-dipole approximation (see Section 1.2.2), the dipolar term S · ADIP · I should describe the energy related to the dipolar magnetic field generated by the magnetic moments, which, in the classical approach, is

[MATHEMATICAL EXPRESSION OMITTED] (1.8)

where the effective electron magnetic moment is considered to take into account the presence of the orbital contributions. Therefore,

[MATHEMATICAL EXPRESSION OMITTED] (1.9)

where the superscript T indicates the matrix transpose, and r is the vector connecting the positions of the nuclear and electron spins. Of note, the components of the g tensor are not the experimental EPR values obtained for the ground state, but the molecular g values along the main directions of the tensor.

In quantum chemistry computations, a different formulation for the ADIP tensor is used (see Section 1.6).

1.1.2 The Curie Spin

The tiny difference in population between the nuclear spin states in the presence of a magnetic field, at room temperature, is responsible for the NMR signal originated by that nucleus. In fact, this difference causes an induced average magnetic moment, which is proportional to the observed NMR signal (see Section 8.1.1). The larger the magnetic field (and the smaller the temperature), the larger the difference in spin population, and thus the induced average magnetic moment.

The difference in population between the different spin states is much larger for electrons than for protons, due to their larger...