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Relativistic Effects on Magnetic Resonance Parameters and Other Properties of Inorganic Molecules and Metal Complexes

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Relativistic Methods for Chemists

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 10))

Abstract

This chapter is concerned with relativistic effects on molecular properties other than energy and structure. The chapter has two parts: The first part consists of a brief overview of theoretical formalisms to calculate energy-derivative properties. In the second part benchmark data and selected case studies are presented. The examples emphasize inorganic and organometallic systems but data for some popular benchmark series of molecules with main group atoms are also included. Molecular properties that are discussed in detail this chapter include NMR and EPR parameters, electric field gradients, electronic spectra, and polarizabilities.

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Notes

  1. 1.

    The orders differ by one, however, if “response” refers to a perturbation of a molecular property which is in turn defined as a first-order energy derivative. Then a linear property response is a second-order energy derivative, and so on.

  2. 2.

    However, so-called picture change effects occur which may complicate certain operators. See, e.g., [16, 21, 22].

  3. 3.

    Note however that, depending on the potential and the location, the “small component” can be much larger than the “large” one, e.g., for p1 ∕ 2 orbitals close to the nucleus and therefore the terms “upper” and “lower” component will be used here.

  4. 4.

    However, c ≈ 137 atomic units and the Bohr “velocity” of an electron in the field of a nucleus with charge Z is equal to Z atomic units. Relativistic corrections, which are of order (Z ∕ c)2, cannot be expected to be small for atoms and molecules with heavy nuclei.

  5. 5.

    It is also possible to separate off a spin–orbit part in the Dirac equation and in four-component perturbation theories which leaves two components to be determined.

  6. 6.

    A serious technical limitation of using Eq. (12-24) and its derivatives is that, generally, in an incomplete basis the equation is only valid if the basis set is independent of the perturbation parameters. This is the case for standard AO basis sets and electric fields or nuclear spins, but not for nuclear-position derivatives if the AO basis is nucleus-centered as usual. We will keep using expressions that are valid for complete basis sets but keep in mind that the computation might involve additional terms to account for basis set incompleteness.

  7. 7.

    For magnetic properties, this part is the paramagnetic term and \(\langle {\Psi }^{(0)}\vert \hat{{\mathcal{H}}}^{({\mathcal{F}}_{i},{\mathcal{F}}_{j})}\vert {\Psi }^{(0)}\rangle\) which involves the A 2 contribution is the diamagnetic term.

  8. 8.

    If the momentum representation is chosen one obtains contributions from different electron momenta instead which can be useful, too, but may often be less intuitive for chemical applications.

  9. 9.

    In a computational analysis such changes in bonding tend to show up in significantly increased or reduced matrix elements in the numerator of the SOS expression, Eq. (12-26).

  10. 10.

    This is analogous to the (non) treatment of the nuclear Zeeman operator in NMR computations.

  11. 11.

    The EFG proper is \(-\hat{{V }}_{uv}\). We use the term ‘EFG’ but omit the negative sign here which is consistent with the definition of the quadrupole coupling used here.

  12. 12.

    This is conceptually similar to the magnetic shielding tensor which may also be defined at any point in space by considering a point magnetic dipole μ at a position r X and using its vector potential \({\mbox{ $A$}}^{X} = \frac{1} {{c}^{2}} \frac{\mbox{ $\mu $}\times {\mbox{ $r$}}_{X}} {{r}_{X}^{3}}\) instead of the nuclear vector potential (12-45) in the derivation of the NMR shielding. In NMR the measurement relies on the presence of a nuclear spin and therefore the shielding tensor is measured at the position of the nucleus.

  13. 13.

    The EFG is also vanishing in highly symmetric environments such as tetrahedral or octahedral.

  14. 14.

    According to the discussion of dynamic properties in Section 12.2.2 the only nonvanishing first quasienergy derivative is obtained if the field’s frequency ω is zero, which is a consequence of the time averaging procedure.

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  1. Department of Chemistry, University at Buffalo, State University of New York, Buffalo, NY, 14260-3000, USA

    Jochen Autschbach

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  1. Inst. Chemistry, Dept. Quantum Chemistry, Nicolaus Copernicus University, Gagarin Street 7, Torun, 87-100, Poland

    Maria Barysz

  2. Dept. Chemistry, University of Puerto Rico, San Juan, USA

    Yasuyuki Ishikawa

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Autschbach, J. (2010). Relativistic Effects on Magnetic Resonance Parameters and Other Properties of Inorganic Molecules and Metal Complexes. In: Barysz, M., Ishikawa, Y. (eds) Relativistic Methods for Chemists. Challenges and Advances in Computational Chemistry and Physics, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9975-5_12

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