Polarization propagators: A powerful theoretical tool for a deeper understanding of NMR spectroscopic parameters

Magnetic molecular spectroscopic properties, like NMR J-coupling and magnetic shielding σ, have been studied by non-relativistic quantum methods since their discovery. When they were found to depend strongly on relativistic effects in molecules containing heavy atoms, this started a new area of intensive research into the development of methods that include such effects. In most cases non-relativistic concepts were extended to the new field though keeping the previous non-relativistic point of view. Quantum mechanics can be formulated by two different formal approaches. Molecular physics and quantum chemistry were developed mostly within the Schrödinger or Heisenberg approaches. The path integral formalism of Feynman is less well known. This may be the reason why propagators are not broadly known in this field of research. Polarization propagators were developed in the early 1970s. Since that time they have been successfully applied to calculate NMR spectroscopic parameters. They are special theoretical devices from which one can do a deep analysis of the electronic mechanisms that underly any molecular response property from basic theoretical elements, like molecular orbitals, electronic excitation energies, coupling pathways, entanglement, contributions within different levels of theory, etc. All this is obtained in a natural way in both regimes: relativistic and non-relativistic. Its relativistic generalization in the early 1990s and the finding of a quantum electrodynamic (QED)-based theory for them, has given us the opportunity to improve our understanding of the physics behind such parameters. In this paper we give a presentation of polarization propagators that start in non-relativistic quantum physics and end up with the introduction of QED effects. The same and powerful basic quantum ideas are applied throughout this review, so that coherence and beauty arise in a natural way. We will give a new understanding that comes from the three levels of theory: non-relativistic, relativistic and QED. We will be cautious to highlight what one can learn when working in any of these regimes, being aware that the same concepts may have different meanings. Starting with a general overview of polarization propagators we develop both the theory and applications in each of the three levels mentioned above. We show how some of the most subtle properties of the NMR spectroscopic parameters can be explained in a simple and beautiful way: the sign of J, the Karplus rule, cooperative effects, diamagnetic and paramagnetic contributions; we also show new rules that appear within the relativistic regime and the way previously valid rules within the non-relativistic theory are broken. We highlight the fact that within the relativistic theory of polarization propagators the whole set of electronic mechanisms that appears within the NR regime and within its quasi-relativistic extension is unified. Furthermore the usual notion of diamagnetism and paramagnetism are now nonsense as such because they arise only after as an approximation from expressions which include a unique type of electronic (electron–positron) mechanism.