The indirect nuclear spin–spin coupling constants of ${\mathrm{C}}_2{\mathrm{H}}_4,$ ${\mathrm{CH}}_2$ NH, ${\mathrm{CH}}_2\mathrm{O,}$ and ${\mathrm{CH}}_2\mathrm{S}$ were investigated by means of correlated ab initio calculations at the level of the second order polarization propagator approximation (SOPPA) and the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes—SOPPA(CCSD) using large basis sets, which are optimized for the calculation of coupling constants. It is found that at the self-consistent-field (SCF) level ${\mathrm{CH}}_2\mathrm{NH}$ and ${\mathrm{CH}}_2\mathrm{S}$ exhibit triplet instabilities whereas ${\mathrm{CH}}_2{\mathrm{CH}}_2$ and ${\mathrm{CH}}_2\mathrm{O}$ show triplet quasi-instabilities, which renders the SCF results meaningless. Our best results deviate between 0.3 and 2.7 Hz from the experimental values. We find that although the one-bond C–H and Y–H couplings as well as the two- and three-bond H–H couplings are dominated by the Fermi contact term, significant contributions of the orbital paramagnetic and sometimes even spin–dipolar terms are observed for the one-bond C–Y and two-bond C–H and Y–H coupling constants. Similarly the changes in the couplings caused by the electronegativity and the lone-pair of Y are mostly due to changes in the Fermi contact (all couplings) and the orbital paramagnetic contribution (C–Y and two-bond Y–H couplings). However, the trend in the changes are neither the same for both terms not for all couplings. In particular, the position of ${\mathrm{CH}}_2\mathrm{S}$ in the series varies indicating that either the electronegativity or the lone pairs are the dominating perturbation. Furthermore, small but optimized Gaussian basis sets for the calculation of indirect nuclear spin–spin coupling constants are presented. They were obtained by contraction of the s- and p-type basis functions for C, N, O, and S and of the s-type basis functions for H of the large uncontracted basis sets. Molecular orbital coefficients of self-consistent-field calculations on ${\mathrm{CH}}_4,$ N${\mathrm{H}}_3,$ ${\mathrm{H}}_2\mathrm{O,}$ ${\mathrm{H}}_2\mathrm{S,}$ and ${\mathrm{H}}_2$ with the uncontracted basis sets were used as contraction coefficients. Applied in the calculation of all coupling constants in ${\mathrm{C}}_2{\mathrm{H}}_4,$ ${\mathrm{CH}}_2$ NH, ${\mathrm{CH}}_2\mathrm{O,}$ and ${\mathrm{CH}}_2\mathrm{S}$ the contraction leads to a maximum basis set error of $\text{∼0.5}$ Hz.

The relativistic Dirac Hamiltonian that describes the motion of electrons in a magnetic field contains only paramagnetic terms (i.e., terms linear in the vector potential A) while the corresponding nonrelativistic Schrödinger Hamiltonian also contains diamagnetic terms (i.e., those from an $A^2$ operator). We demonstrate that all diamagnetic terms relativistically arise from second-order perturbation theory and that they correspond to a “redressing” of the electrons by the magnetic field. If the nonrelativistic limit is taken with a fixed no-pair Hamiltonian (no redressing), the diamagnetic term is missing. The Schrödinger equation is normally obtained by taking the nonrelativistic limit of the Dirac one-electron equation, we show why nonrelativistic use of the $A^2$ operator is also correct in the many-electron case. In nonrelativistic approaches, diamagnetic terms are usually considered in first-order perturbation theory because they can be evaluated as an expectation value over the ground state wave function. The possibility of also using an expectation value expression, instead of a second-order expression, in the relativistic case is investigated. We also introduce and discuss the concept of “magnetically balanced” basis sets in relativistic calculations.

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.

A new approach for calculating relativistic corrections to the nuclear magnetic shieldings is presented. Starting from a full relativistic second order perturbation theory expression a two-component formalism is constructed by transforming matrix elements using the elimination of small component scheme and separating out the contributions from the no-virtual pair and the virtual pair part of the second order corrections to the energy. In this way we avoid a strong simplification used previously in the literature. We arrive at final expressions for the relativistic corrections which are equivalent to those of Fukui et al. [J. Chem Phys. 105, 3175 (1996)] and at some other additional terms correcting both the paramagnetic and the diamagnetic part of the nuclear magnetic shielding. Results for some relativistic corrections to the shieldings of the heavy and light nuclei in HX and ${\mathrm{CH}}_3\mathrm{X}$ $(\mathrm{X}=\mathrm{Br},\mathrm{I})$ at both random phase and second order polarization propagator approach levels are given.

We have calculated the vicinal indirect nuclear spin-spin coupling constants ${}^3{J}_{{}^1{\mathrm{H}}^{\mathrm{1}}\mathrm{H}}$ in the series of molecules ${\mathrm{H}}_{\mathrm{3}}{\mathrm{C–CH}}_{\mathrm{2}}\mathrm{X}$ with X=H, F, Cl, Br, and I at the self-consistent field level and using the second order polarization propagator approximation (SOPPA). We have studied the effect of electron correlation and of the substituents (X=F, Cl, Br, and I) on all four contributions to the coupling constants. But in particular we have investigated the possibility of using locally dense basis sets, i.e., we have carried out calculations with basis sets, where the basis functions on the hydrogen atoms were optimized for the calculation of spin–spin coupling constants whereas on the other atoms smaller, contracted sets of basis functions were used. This changes the results for the couplings by ∼0.3 Hz or 3%. However, the change is almost entirely due to the orbital paramagnetic term and is independent of electron correlation, which enables one to estimate the SOPPA results in the full basis sets. Furthermore we find that the Fermi contact term is the dominant contribution to the vicinal coupling constants, because it is about an order of magnitude larger than the other contributions and because the two orbital angular moment terms almost cancel each other completely. Also the changes in the calculated couplings due to electron correlation are solely due to the Fermi contact term. However, the shifts in the coupling constants caused by the different substituents arise in equal amounts from the Fermi contact and the orbital diamagnetic term, whereas the changes in the orbital paramagnetic term are smaller and are in the opposite direction. In comparison with the experimental data we find very good agreement for ${\mathrm{C}}_{\mathrm{2}}{\mathrm{H}}_6$ and ${\mathrm{C}}_{\mathrm{2}}{\mathrm{H}}_5\mathrm{F}.$ However, the agreement becomes less good with increasing nuclear charge of the substituent X.