HETERONUCLEAR COUPLING MECHANISMS FOR TWO-SPIN SYSTEMS
Spin (I) is a subtle property of elementary and composite particles (bosons, fermions / protons, neutrons, etc.). While spin possesses a few qualities that relate to an object rotating about a center of mass, it is appropriate to consider it a fundamental property, in the vein of mass or charge. The number of protons and neutrons that comprise an atomic nucleus defines whether spin is zero or an integer/half-integer value (I=1, 2, 3... or I=1/2, 3/2...).
Nuclei with non-zero values for spin can exist in one of two distinct spin states, α or β (often called “spin-up” and “spin-down”). These states refer to the orientation of the magnetic moment that results from the intrinsic composition of subatomic particles in the nucleus. In a loose sense, α and β states relate to charges that are equal in magnitude but opposite in sign; they are often thought of as bar magnets with north and south poles. At thermal equilibrium and in the absence of an external magnetic field, these states are nearly equivalent in energy. In a bulk sample, the Boltzmann distribution confirms that α and β are almost equal in population (with a difference of 1.5 x 10-6 %). Nuclei thus have no “default” spin orientation, as evidenced by this random distribution of spin states. However, the presence of another electromagnetic field—specifically, another non-zero spin particle—can have an observable influence on nuclear spin.
Within a molecule or within a bulk sample, interactions between the electromagnetic forces associated with nuclei and electrons result in a number of discrete phenomena. Electromagnetic fields, or more appropriately, spin states in close proximity can interact either constructively or destructively. This interaction is often referred to as polarization or perturbation of spin, alluding to the resulting energy difference (either positive or negative) when two particles with non-zero spin interact. This “spin coupling” is both observable and manifold in effect. Among the most prominent outcomes of spin interaction are dipolar coupling, scalar coupling, and the nuclear Overhauser effect (NOE).
Dipolar coupling is a direct interaction and is strongest in the case of two nuclei adjacent to one another in a bonding condition. The orientation of their “north and south poles” can lead to a higher overall energy if both nuclei are in the α spin state. The interaction results in a lower energy system if the nuclei are of opposite spin (α and β), and a system of minimal energy results when both nuclei are spin α. Dipolar coupling is a strong interaction at the molecular level; energies are typically on the order of 30,000 Hz for hydrogen-hydrogen coupling. Luckily for the sake of other interactions, which would be unobservable due to their fundamentally lower energies, dipolar coupling tends to sum to zero in most conditions. Isotropic tumbling in a fluid solution—simply, the random kinetic motion of individual molecules—averages out the effects of dipolar coupling. This averaging exposes the more subtle and vastly more interesting interatomic perturbations that arise from scalar coupling and the NOE.
The energies associated with nuclear Overhauser interactions are the smallest of these three internuclear phenomena. The NOE is a result of through-space polarization between nuclei with non-zero spin. This interaction does not depend on any bonding structure between atoms; it is a conceptually simple matter of the spatial location of two (or more) atoms with respect to one another. The perturbation that occurs when non-zero spin nuclei are near each other in space is similar to the result of dipolar coupling between adjacent atoms, though the energy involved is much lower in magnitude. Regardless, the nuclear Overhauser effect produces a coupling of nuclear spin that leads to either a higher- or lower-energy coupled system, depending on the resulting spin states (α or β) of the nuclei involved.
Energetically, scalar coupling exists somewhere between dipolar coupling and the perturbation caused by the NOE. Hydrogen-hydrogen coupling tends to be on the scale of 2-3 Hz (about a ten thousandth of the energetic transitions in dipolar coupling). It revolves around the same principle of spin polarization, but here it is an indirect interaction. Unlike dipolar coupling or the NOE, scalar coupling relies on the presence of bonding electrons. Like nuclei with non-zero values of spin, electrons can occupy either α or β states. Thus, the existence of either the α or β state in the first nucleus ("A") affects the spin states of the bonding electrons associated with the system. This perturbation in turn influences the spin state of the adjacent nucleus ("X"). Although these are time-independent assertions, this through-bond polarization of spin can be thought of as a "chain reaction". The two-spin, one-bond system is a common point of discussion (often referred to as an "AX" coupling system), and one-bond coupling is a very apparent interaction. The energies associated with multiple-bond coupling decrease dramatically with distance, but long-range couplings through 2 or 3 bonds can be observed. In high-resolution conditions, coupling through 5 bonds has been witnessed. One-bond coupling, however, produces the highest-energy transitions in the realm of scalar coupling.