New Experimental NMR Shielding Scales Mapped Relativistically from NSR: Theory and Application

Influence of the nuclear charge distribution and electron correlation effects on magnetic shieldings and spin-rotation tensors of linear molecules

The nuclear charge distribution effects (NChDE) on two response properties, the NMR magnetic shielding ( σ ) and the nuclear spin-rotation (SR) constants ( M ), are analyzed. We do it employing point-like and Gaussian-like models for describing the nuclear charge density of three linear molecules: HBr, HI and HAt. According to our results, both properties are sensitive to the NChDE. We show that the NChDE are almost completely relativistic, i.e., they nearly vanish in the non-relativistic limit of both properties. We calculated the NChDE on σ and M , and analyzed the differences between them in terms of a relativistic relation between these two properties. Using that relation we found that the electronic core mechanisms are the main ones for the NChDE on the shielding of nuclei of both, molecules and free atoms. The NChDE are smaller on SR constants than on shieldings. Nevertheless, within the relativistic polarization propagator formalism at the RPA level of approach they are very important for SR constants of nuclei in heavy-atom-containing compounds. Astatine in HAt has the largest influence: M At = -9.95 kHz for a point-like model and -50.10 kHz for a Gaussian-like model. Correlation effects must be included and we do it using different DFT schemes. The PBE0 functional gives results that are closest to experiments for Br and I, though the LDA gives the closest for hydrogen. The value of the SR constant of At is reduced among 350 kHz and 500 kHz from its RPA value, when different and usual functionals are applied. Given that the NChDE on M and σ are mostly relativistic in their origin, these effects are also dependent on electron correlation. They have also a nonvanishing dependence with the Gaunt electron-electron interactions.

Role of Spin-Dependent Terms in the Relationship among Nuclear Spin-Rotation and NMR Magnetic Shielding Tensors

The broadly accepted procedure to obtain the experimental absolute scale of NMR magnetic shieldings, σ, is well-known for nonheavy atom-containing molecules. It was uncovered more than 40 years ago by the works of Ramsey and Flygare. They found a quite accurate relationship among σ and the nuclear spin-rotation constants. Its relativistic extension was very recently proposed, although it has an intrinsic weakness because a new SO-S two-component term needs to be considered. We show how to overcome this problem. We found that (ν_Y^S - ν_Y^atom,S) generalizes the SO-S term, where ν_Y^S = ⟨⟨[((r - r_Y) × α)/(|r - r_Y|³)]; S⁽⁴⁾⟩⟩, r - r_Y is the electron position with respect to the position of nucleus Y, and S⁽⁴⁾ is the four-component total electron spin. When including this new term, one finds that the best of our relativistic Flygare-like models fits quite well with the results of the most accurate method available at the moment. We also show that the difference among the parallel component of σ(Xe) in XeF₂ and σ(Xe) of the free atom is almost completely described by that new term.

Toward an absolute NMR shielding scale using the spin-rotation tensor within a relativistic framework

How can one extend Flygare's rule to the relativistic framework? Three models are proposed here. The best of them shows that σ is related with the spin-rotation tensor, the atomic shielding and a new term coined as σ^SO-S.

Big picture of relativistic molecular quantum mechanics

Any quantum mechanical calculation on electronic structure ought to choose first an appropriate Hamiltonian H and then an Ansatz for parameterizing the wave function Ψ, from which the desired energy/property E(λ) can finally be calculated. Therefore, the very first question is: what is the most accurate many-electron Hamiltonian H? It is shown that such a Hamiltonian i.e. effective quantum electrodynamics (eQED) Hamiltonian, can be obtained naturally by incorporating properly the charge conjugation symmetry when normal ordering the second quantized fermion operators. Taking this eQED Hamiltonian as the basis, various approximate relativistic many-electron Hamiltonians can be obtained based entirely on physical arguments. All these Hamiltonians together form a complete and continuous ‘Hamiltonian ladder’, from which one can pick up the right one according to the target physics and accuracy. As for the many-electron wave function Ψ, the most intriguing questions are as follows. (i) How to do relativistic explicit correlation? (ii) How to handle strong correlation? Both general principles and practical strategies are outlined here to handle these issues. Among the electronic properties E(λ) that sample the electronic wave function nearby the nuclear region, nuclear magnetic resonance (NMR) shielding and nuclear spin-rotation (NSR) coupling constant are especially challenging: they require body-fixed molecular Hamiltonians that treat both the electrons and nuclei as relativistic quantum particles. Nevertheless, they have been formulated rigorously. In particular, a very robust ‘relativistic mapping’ between the two properties has been established, which can translate experimentally measured NSR coupling constants to very accurate absolute NMR shielding scales that otherwise cannot be obtained experimentally. Since the most general and fundamental issues pertinent to all the three components of the quantum mechanical equation HΨ = EΨ (i.e. Hamiltonian H, wave function Ψ, and energy/property E(λ)) have fully been understood, the big picture of relativistic molecular quantum mechanics can now be regarded as established.

Spin-rotation and NMR shielding constants in XF molecules (X = B, Al, Ga, In, and Tl)

Accurate spin-rotation and absolute shielding constants in a series of XF molecules (X = (11)B, (27)Al, (69)Ga, (115)In, and (205)Tl) determined using high-level ab initio coupled-cluster and four-component relativistic density-functional theory (DFT) calculations are presented. The accuracy of the results is established by comparing the relativistically and vibrationally corrected calculated values with available experimental data; for spin-rotation and shielding constants for which no experimental data exist, we provide new and reliable values. For both properties, our results can be considered as reference values against which more approximate quantum-chemical methods can be benchmarked.