Relativistic Effects on Magnetic Resonance Parameters and Other Properties of Inorganic Molecules and Metal Complexes. In: Barysz M, Ishikawa Y, editors. Relativistic Methods for Chemists. Dordrecht: Springer Netherlands; 2010. pp. 521-98. [DOI: 10.1007/978-1-4020-9975-5_12]

Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory

The temperature-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temperature-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as reference. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the density functional approximation and the basis set. Large-scale calculations are routinely possible with the multipole-accelerated resolution of the identity approximation and the seminumerical exchange approximation, as shown for [CeTi₆O₃(OiPr)₉(salicylate)₆].

Hyperion: A New Computational Tool for Relativistic Ab Initio Hyperfine Coupling

Herein we describe Hyperion, a new program for computing relativistic picture-change-corrected magnetic resonance parameters from scalar relativistic active space wave functions, with or without spin–orbit coupling (SOC) included a posteriori. Hyperion also includes a new orbital decomposition method for assisting active space selection for calculations of hyperfine coupling. For benchmarking purposes, we determine hyperfine coupling constants of selected alkali metal, transition metal, and lanthanide atoms, based on complete active space self-consistent field spin–orbit calculations in OpenMolcas. Our results are in excellent agreement with experimental data from atomic spectroscopy as well as theoretical predictions from four-component relativistic calculations.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C₅Me₅)₂Y(μ-S)₂Mo(μ-S)₂Y(C₅Me₅)₂]^-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C₆Cl₅)₄]^- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

Hyperfine Coupling Constants in Local Exact Two-Component Theory

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r²SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C₆Cl₅)₄]^- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)₃]^-, [Lu(NR₂)₃]^-, [Lu(OAr*)₃]^-, and [TbPc₂]^-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth

This study investigates the possibility of detecting relativistic effects and electron correlation in single-crystal X-ray diffraction experiments using the examples of diphenyl mercury (HgPh2) and triphenyl bismuth (BiPh3). In detail, the importance of electron correlation (ECORR), relativistic effects (REL) [distinguishing between total, scalar and spin-orbit (SO) coupling relativistic effects] and picture change error (PCE) on the theoretical electron density, its topology and its Laplacian using infinite order two component (IOTC) wave functions is discussed. This is to develop an understanding of the order of magnitude and shape of these different effects as they manifest in the electron density. Subsequently, the same effects are considered for the theoretical structure factors. It becomes clear that SO and PCE are negligible, but ECORR and scalar REL are important in low- and medium-order reflections on absolute and relative scales-not in the high-order region. As a further step, Hirshfeld atom refinement (HAR) and subsequent X-ray constrained wavefunction (XCW) fitting have been performed for the compound HgPh2 with various relativistic and nonrelativistic wave functions against the experimental structure factors. IOTC calculations of theoretical structure factors and relativistic HAR as well as relativistic XCW fitting are presented for the first time, accounting for both scalar and spin-orbit relativistic effects.

Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules

Using a recently proposed equation for NMR nuclear magnetic shielding for molecules with unpaired electrons [A. Soncini and W. Van den Heuvel, J. Chem. Phys. 138, 021103 (2013)], equations for the temperature (T) dependent isotropic shielding for multiplets with an effective spin S equal to 1/2, 1, 3/2, 2, and 5/2 in terms of electron paramagnetic resonance spin Hamiltonian parameters are derived and then expanded in powers of 1/T. One simplifying assumption used is that a matrix derived from the zero-field splitting (ZFS) tensor and the Zeeman coupling matrix (g-tensor) share the same principal axis system. The influence of the rhombic ZFS parameter E is only investigated for S = 1. Expressions for paramagnetic contact shielding (from the isotropic part of the hyperfine coupling matrix) and pseudo-contact or dipolar shielding (from the anisotropic part of the hyperfine coupling matrix) are considered separately. The leading order is always 1/T. A temperature dependence of the contact shielding as 1/T and of the dipolar shielding as 1/T(2), which is sometimes assumed in the assignment of paramagnetic chemical shifts, is shown to arise only if S ≥ 1 and zero-field splitting is appreciable, and only if the Zeeman coupling matrix is nearly isotropic (Δg = 0). In such situations, an assignment of contact versus dipolar shifts may be possible based only on linear and quadratic fits of measured variable-temperature chemical shifts versus 1/T. Numerical data are provided for nickelocene (S = 1). Even under the assumption of Δg = 0, a different leading order of contact and dipolar shifts in powers of 1/T is not obtained for S = 3/2. When Δg is not very small, dipolar and contact shifts both depend in leading order in 1/T in all cases, with sizable contributions in order 1/T(n) with n = 2 and higher.

Relativistic calculations of magnetic resonance parameters: background and some recent developments

This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.

Perspective: relativistic effects

This perspective article discusses some broadly-known and some less broadly-known consequences of Einstein's special relativity in quantum chemistry, and provides a brief outline of the theoretical methods currently in use, along with a discussion of recent developments and selected applications. The treatment of the electron correlation problem in relativistic quantum chemistry methods, and expanding the reach of the available relativistic methods to calculate all kinds of energy derivative properties, in particular spectroscopic and magnetic properties, requires on-going efforts.