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 A͒ while the corresponding nonrelativistic Schrödinger Hamiltonian also contains diamagnetic terms ͑i.e., those from an A 2 operator͒. 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 redressing͒, 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.

Large long-range indirect nuclear spin coupling constants are of great interest for quantum computers. But they are rarely observed and are usually considered very small, unless the coupled nuclear spins are proximate in space. Looking for counterexamples, we have calculated F-F couplings in four different series of acyclic hydrocarbons (alkanes, conjugated polyenes, conjugated polyynes, and cumulenes) where the coupled fluorine nuclei are separated by up to 11 bonds or 1.4 nm. The calculations were carried out at the level of the second-order polarization propagator approximation using locally dense basis sets. This approach has, in recent years, been shown to be particularly successful in reproducing indirect nuclear spin-spin couplings in organic molecules. We find that the F-F couplings in saturated alkanes diminish very quickly with the number of bonds between the coupled fluorine atoms, whereas in the conjugated polyenes and in particular polyynes the F-F couplings can be transmitted over much longer distances. We predict that the F-F coupling over 9 bonds or 1.1 nm is 12 Hz in (1E,3E,5E,7E)-1,8-difluoroocta-1,3,5,7-tetraene and the coupling over 11 bonds or 1.4 nm is 7 Hz in difluorodecapentayne. Analyzing the four Ramsey contributions, we find that the F-F couplings in the polyenes are dominated by the spin-dipolar term, which is known to be favored by π-electronic systems, whereas in the case of the polyynes the orbital paramagnetic terms make the largest contributions, although the spin-dipolar and the Fermi contact contributions are also significant.

The indirect nuclear spin–spin coupling constants of C2H4, CH2NH, CH2O, and CH2S 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 CH2NH and CH2S exhibit triplet instabilities whereas CH2CH2 and CH2O 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 CH2S 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 CH4, NH3, H2O, H2S, and H2 with the uncontracted basis sets were used as contraction coefficients. Applied in the calculation of all coupling constants in C2H4, CH2NH, CH2O, and CH2S the contraction leads to a maximum basis set error of ∼0.5 Hz.

The nuclear magnetic shieldings of Si, Ge, and Sn in MH(4-n) Y(n) (M = Si, Ge, Sn; Y = F, Cl, Br, I and n = 1-4) molecular systems are highly influenced by the substitution of one or more hydrogens by heavy-halogen atoms. We applied the linear response elimination of small components (LRESC) formalism to calculate those shieldings and learn whether including only a few of the leading relativistic correction terms is sufficient to be able to quantitatively reproduce the full relativistic value. It was observed that the nuclear magnetic shieldings change as the number of heavy halogen substituents and their weights vary, and the pattern of σ(M) generally does not exhibit the normal halogen dependence (NHD) behavior that can be seen in similar molecular systems containing carbon atoms. We also analyzed each relativistic correction afforded by the LRESC method and split them in two: core-dependent and ligand-dependent contributions; we then looked for the electronic mechanisms involved in the different relativistic effects and in the total relativistic value. Based on this analysis, we were able to study the electronic mechanism involved in a recently proposed relativistic effect, the "heavy atom effect on vicinal heavy atom" (HAVHA), in more detail. We found that the main electronic mechanism is the spin-orbit or σ p (T(3)) correction, although other corrections such as σ p (S(1)) and σ p (S(3)) are also important. Finally, we analyzed proton magnetic shieldings and found that, for molecules containing Sn as the central atom, σ(H) decreases as the number of heavy halogen substituents (of the same type: either F, Cl, or Br) increases, albeit at different rates for different halogens. σ(H) only increase as the number of halogen substituents increases if the halogen is iodine.

Articles you may be interested in A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equationa) J. Chem. Phys. 128, 104101 (2008); 10.1063/1.2837472 Relativistic effects on the nuclear magnetic shieldings of rare-gas atoms and halogen in hydrogen halides within relativistic polarization propagator theory J. Chem. Phys. 123, 214108 (2005) 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 ͑1996͔͒ 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 CH 3 X (XϭBr,I) at both random phase and second order polarization propagator approach levels are given.

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We present a relativistic theory for the nuclear spin-spin coupling tensor within the polarization propagator approach using the particle-hole Dirac-Coulomb-Breit Hamiltonian and the full four-component wave function. We give explicit expressions for the coupling tensor in the random-phase approximation, neglecting the Breit interaction. A purely relativistic perturbative electron-nuclear Hamiltonian is used and it is shown how the single relativistic contribution to the coupling tensor reduces to Ramsey's three second-order terms (Fermi contact, spin-dipole, and paramagnetic spin-orbit) in the nonrelativistic limit. The principal propagator becomes complex and the leading property integrals mix atomic orbitals of different parity. The well-known propagator expressions for the coupling tensor in the nonrelativistic limit is obtained neglecting terms of the order c-" ( n 2 1).

Fully relativistic calculations of NMR magnetic shielding on XYH3 (X = C, Si, Ge and Sn; Y = Br, I), XHn (n = 1-4) molecular systems and noble gases performed with a fully relativistic polarization propagator formalism at the RPA level of approach are presented. The rate of convergence (size of basis set and time involved) for calculations with both kinetic balance prescriptions, RKB and UKB, were investigated. Calculations with UKB makes it feasible to obtain reliable results for two or more heavy-atom-containing molecules. For such XYH3 systems, the influence of heavy vicinal halogen atoms on sigma(X) is such that heavy atom effects on heavy atoms (vicinal plus their own effects or HAVHA + HAHA effects) amount to 30.50% for X = Sn and Y = I; being the HAHA effect of the order of 25%. So the vicinal effect alone is of the order of 5.5%. The vicinal heavy atom effect on light atoms (HALA effect) is of the order of 28% for X = C and Y = I. A similar behaviour, but of opposite sign, is observed for sigma(Y) for which sigmaR-NR (I; X = C) (HAHA effect) is around 27% and sigmaR-NR(I; X = Sn) (HAVHA + HAHA effects) is close to 21%. Its electronic origin is paramagnetic for halogen atoms but both dia- and paramagnetic for central atoms. The effect on two bond distant hydrogen atoms is such that the largest variation of sigma(H) within the same family of XYH3 molecules appears for X = Si and Y = I: around 20%. In this case sigma(H; X = Sn, Y = I) = 33.45 ppm and sigma(H; X = Sn, Y = H) = 27.82 ppm.