Theoretical and physical aspects of nuclear shielding

Cynthia J. Jameson and Angel C. De Dios

DOI: 10.1039/9781849730846-00042

1.1 General theory

Several methodologies are being developed for including relativistic effects in the calculation of molecular magnetic properties, in particular the nuclear magnetic shielding tensor. Approaches used can be grouped as follows: four-component or fully relativistic, two-component or quasi-relativistic, and perturbational. Volume 36 of this series gives an overview and newer developments have been discussed in succeeding volumes. In the four-component relativistic treatment of nuclear shielding, the Dirac operator is linear with respect to the vector potential such that the second order energy consists only of a single term, the paramagnetic term, when the first order wave function is directly expanded in the full space of unperturbed states. This standard linear response theory has been used by several groups. Alternative formulations of four-component treatments of nuclear magnetic properties including shielding have been proposed by Kutzelnigg and others by explicitly incorporating the magnetic balance condition between the small and large components of the Dirac spinor in the presence of magnetic fields. But, is it mandatory to resort to a magnetically balanced basis set for calculations of magnetic properties with perturbative fully relativistic methods? Calculations of molecular properties using finite basis sets in relativistic quantum mechanics are contaminated with spurious states. In order to eliminate them, kinetically balanced basis sets were suggested. Kutzelnigg has shown that the exact relativistic wavefunction of the ground state of H-like ions is expandable in a kinetically balanced even-tempered Gaussian basis. Two kinetically balanced bases have previously been investigated, restricted kinetic balanced (RKB) and unrestricted kinetic balanced (UKB). In this reporting period, the application of different kinetic balance prescriptions using the four-component polarization propagator approach in the calculation of nuclear magnetic shielding were investigated. They find that working with relativistic polarization propagators there is no formal requirement to enforce the application of magnetic kinetic balance prescription. On the contrary, the RKB prescription is found to be a necessary condition, but is not enough to obtain reliable results. The kinetic balance prescription ensures that the matrix representation of the kinetic energy operator of the unperturbed system will properly be described in the non-relativistic limit. Both the RKB and the UKB prescriptions were applied to generate small components from large components in the four component basis set. Calculations with both RKB and UKB prescriptions are found to converge to the same value, although UKB was found to be more efficient; i.e., the UKB prescription ensures basis set convergence with quite smaller basis sets than RKB, leading to potentially large time savings. Convergence threshold was defined as the difference within RKB calculations were less than 0.5%. Also UKB is de- fined as converged when the difference between converged RKB and the UKB calculations were less than 0.5%.

The performance of the UKB prescription in such calculations was studied for molecules containing more than one heavy atom in order to examine the electronic effects on the shielding of a heavy atom due to the presence of vicinal heavy atoms. The shieldings of X, Y, and H nuclei in XYH3 molecular systems with X = C, Si, Ge, Sn and Y = Br, I were calculated. Relativistic effects on the shielding of X due to heavy halogen atoms are larger for heavier X nuclei. For example, for substituent Y = I, the difference between Rel and non-rel shielding for X = C in CH3I is 235.57 ppm–198.31 ppm, i.e., (235.57 – 198.31)/198.31 = 18.79%, which the authors refer to as a HALA effect (vicinal heavy atom effect on the shielding of the light atom). For Sn in SnH3I, this difference is 4059.63 ppm – 3111.44 ppm, i.e., (4059.63 – 3111.44)/3111.44 = 30.47%, much larger. The authors refer to the latter as HAVHA + HAHA effects (heavy atom effects on the shielding of the vicinal heavy atom + heavy atom effects on its own shielding). The total shielding for C in CH3I is 235.57 ppm compared to C in CH4 where the total shielding is 195.55 ppm, i.e., (235.57 – 195.55)/ 195.55 = 20.50%. We compare this with the total shielding for Sn in SnH3I is 4059.63 ppm compared to Sn in SnH4 where the total shielding is 4110.16 ppm, i.e., (4059.63 – 4110.16)/4110.16 = 1.23%. For I shielding, the relativistic effects in SnH3I is (6652.74 – 5505.12)/5505.12 = 20.84%. We compare this with the relativistic effects on I shielding in CH3I, (5636.10 – 4429.34)/4429.34 = 27.24%, which is somewhat larger than 20.84%. The hydrogen shieldings exhibited an effect from the two-bond distant heavy atom Br or I; this effect is found to be more pronounced when the central atom is X = Si. The authors found that UKB is much more efficient, and that one can obtain reliable results working with Sadlej basis sets for calculations of paramagnetic components at RPA level. They also found that calculations of the diamagnetic components at PZOA level give a time savings of 50% when compared with RPA calculations of the same. All of these results are at the RPA level. These calculations were all carried out at the RPA level and so did not include electron correlation.

As mentioned above, various alternative formulations of four-component treatments of NMR properties have been proposed by Kutzelnigg and others, which explicitly incorporate the magnetic balance condition between the small and large components of the Dirac spinor in the presence of magnetic fields. These methods have achieved the same goal of capturing the diamagnetic part of the shielding in a natural manner, i.e., without using negative energy states in the computation or interpretation of the diamagnetic part. The essence of these methods is that the contributions of negative energy states are reduced to order c -2 or higher so as to guarantee the correct non-relativistic limit even with a finite basis, in sharp contrast to those approaches without the magnetic balance. In this reporting period, Cheng et al. showed that these variants of approaches using magnetic balance can all be recast into one unified form by employing the unique degrees of freedom offered by the generic ansatz of orbital decomposition. That is, each scheme can be understood as a particular decomposition of the first order wave function Ψ10 into a known magnetic term Ψ10m and a residual Ψ10r. The latter is to be expanded in a RKB basis. Compared to the standard linear response theory where Ψ10m = 0, the basis set error should be reduced for the subset of basis functions that overlaps strongly with Ψ10m. The various schemes previously proposed for incorporating the magnetic balance dependence are then shown to be equivalent to this new approach and therefore can be combined with any level of theory for electron correlation. These authors then adopt relativistic spin density functional theory in the Dirac-Kohn-Sham (DKS) framework and set up the coupled and uncoupled-perturbed DKS equations and mixed second order energy. A direct consequence is that the heavy demand on the basis set is greatly reduced such that the standard basis sets optimized for electronic structure calculations are found to be sufficient for reliable NMR paramaters, particularly when a finite nuclear charge distribution is used. Extremely compact negative energy states (additional steep s and p functions in the basis set become necessary) when highly accurate absolute NMR shieldings are sought. Such effects of highly compact negative energy states essentially cancel out for relative shieldings. Although the variant approaches including magnetic balance dependence have been shown to be equivalent in this paper, the authors prefer the external field-dependent unitary transformation (EFUT) at operator level because of its explicit operator form and the absence of "unphysical" one-centered contributions from occupied orbitals to the paramagnetic term. The calculated results are analyzed to elucidate the various contributions to the paramagnetic terms, using the two simple systems: the one-electron Rn85+ ion and the Rn neutral atom as an example of a multi-electron system. The common gauge origin was chosen at the NMR nucleus. Applications to multi-centered molecular systems were not explored in this paper.

The weak interaction responsible for the existence of nuclear β-decay does not conserve parity, the symmetry with respect to inversion of the coordinates of all particles at the origin. A weak interaction correction to the electromagnetic interaction in chiral molecules leads to the theoretical prediction of a slight deviation between the wave function of a right-handed molecule and the mirror image of the wave function of the corresponding left-handed molecule. Consequently, it would also lead to slightly different electronic energies of the left- and right-handed chiral molecules. Tiny PV differences between the electronic energies of the left- and right-handed chiral molecules leads, in principle, to differences in NMR resonance frequencies. Barra et al. have estimated PV NMR frequency splittings between enantiomers and predicted splittings of the order of millihertz for Tl, Pt or Pb nuclei. Ab initio calculations of PV NMR shielding tensors have been presented by Lazzeretti and co-workers at the HartreeFock level and also by Laubender and Berger at the Hartree-Fock and at CCSD level. Density functional theory calculations were carried out by Weijo et al. Four-component relativistic treatment of PV NMR shielding tensors were presented by Bast et al. but they found what was later discovered as spurious Z power dependence of the PV contributions to the shielding tensor in H2X2 molecules, X = O, S, Se, Te, Po. Nahrwold and Berger have established a Z3 scaling and Z5 scaling, respectively, for the paramagnetic and the spin-orbit contributions to the relativistic isotropic PV NMR shielding in this series of molecules. Using a two-component quasi-relativistic ZORA DFT approach to the calculation of PV NMR shielding tensors, they predict, for certain conformations of HPoPoH, a PV NMR frequency splitting of 10 mHz between the two mirror image structures in a magnetic field of 11.7 T. Molecular magnetic properties may be a possible tool for observing parity-violating (PV) effects. This effect is not expected to be size extensive because the part of the PV operator relevant to the leading order effect on shielding does not contain a sum over nuclei, in contrast to the operator used in PV energy calculations. The expected lack of molecular size dependence of the PV contributions to shielding has been confirmed by non-relativistic DFT calculations on polysilylene chains of increasing lengths.

In the inverse Faraday effect, a shift proportional to the intensity of a laser beam will occur in the NMR spectral lines of the nuclei in a sample that has been exposed to circularly polarized light (CPL). The shift is of opposite signs for the left and right CPL, in which case, a rapid switching between right and left will cause the resonance line to be split. This effect, laser-induced NMR splittings, has not yet been observed experimentally but has been proposed as a possible new probe of electronic structure of molecules. It has been predicted that the splittings would be larger for systems with large electric dipole polarizabilities. Predictions of this splitting for hydrocarbons of increasing complexity, from ethene (C2H4), benzene (C6H6), coronene (C24H12), fullerene (C60) and circumcoronene (C54H18) have been carried out using third order time dependent perturbation theory. The ratio of the splitting to the intensity of the laser beam has been found to increase with the system size in this series. Traditional energy-optimized basis sets could not be used in the calculations because of the combined electric dipole (valence) and orbital magnetic hyperfine (core) character. The authors employed compact completeness-optimized basis sets instead.

1.2 Ab initio and DFT calculations

Relativistic calculations of NMR properties of RgH+ ion (where Rg = Ne, Ar, Kr, Xe), 195Pt shielding in platinum complexes, and 207Pb shielding in solid ionic lead(II) halides have been reported in this review period. For the Rg nucleus in the RgH+ ions, the following methods were used and results compared with each other: non-relativistic uncorrelated method (HF), relativistic uncorrelated methods, four component Dirac Hartree-Fock method (DHF) and two-component zeroth order regular approach (ZORA), non-relativistic correlated methods using second order polar- ization propagator approach SOPPA(CCSD), SOPPA(MP2), respectively coupled cluster singles and doubles or second order Møller-Plesset, and CCSD. The difference between the DHF and HF calculations show that for the Rg nucleus the relativistic effect on shielding scales with atomic number as Z3.4. There are small effects on the shielding of the proton in RgH+ ion, 18.1, 5.2, 1.0 and 0.0 ppm in going from Xe to Ne, behaving rather similarly to the isoelectronic HX halides. The methods including electron correlation produce smaller values of shielding compared to uncorrelated calculations. The authors suggest that the decrease of shielding is caused by orbital decontraction which implies that the average electro-electron distance is larger when correlation effect is taken into account. On the other hand, the relativistic effect produces higher values of shielding arising from the contraction of the orbitals. The opposite effects of relativistic and electron correlation contributions has been noted previously for Xe shielding.

195Pt shieldings in complexes with moderately strongly binding ligands were calculated by Autschbach and Zheng using two-component relativistic density functional theory using zeroth order regular approximation (ZORA).30 In this work, the Vosko-Wilk-Nusair (VWN) local density functional was used along with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation functional. A comparative analysis of localized orbital contributions permits the separate effects of spin-orbit coupling versus paramagnetic shielding from core orbitals, bonds, and lone pairs to be considered, leading one to ascribe intuitive and chemically meaningful contributions to the Pt chemical shifts of Pt and other heavy nuclei. The authors provide a tutorial on the results of the angular momentum operators applied to real d functions which give non-vanishing matrix elements for the paramagnetic term of the shielding; these permit a physically intuitive analysis of the contributions of 5d lone pairs to Pt shielding using only orbital rotation pictures. The arguments apply to any nd atomic orbitals centered on the metal nucleus in transition metal complexes and date back to 1964,31 before computerized calculations of electronic structure. The negative isotropic chemical shifts of Pt(II) complexes relative to [PtCl6]2- is easily predicted by this type of analysis, as are the individual components of the 195Pt chemical shift tensors. The calculated results including the spin-orbit coupling terms provide a reasonable agreement within the chemical shift range of about 3000 ppm for the Pt complexes included in this work and the authors demonstrate the value of analyses of the calculated relativistic shieldings in terms of natural bond orbitals (NBO) and natural localized molecular orbitals (NLMO) to provide an intuitive breakdown of the total computed tensor in terms of the molecule's bonds, lone pairs, core shells, etc. for each of its principal components.