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An Introduction to Relativistic Quantum Chemistry

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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

Chemistry is governed by the shell structure of the atoms. This holds in particular concerning the periodic system of chemical elements. Non-relativistic quantum chemistry describes the motion of electrons and nuclei and their mutual interactions to a first approximation. It reproduces a large fraction of chemistry of the more important lighter elements sufficiently well. A significant amount of chemical insight can already be gained from the analysis of the atomic one-electron orbitals. However, while valence electrons have ‘non-relativistically small’ energies, they become ‘relativistically fast’ in the neighborhood of heavy nuclei. The importance of relativistic effects in the atomic valence shells increases approximately as Z2. Relativity significantly changes the chemical trends at the bottom of the periodic table. The relativistic effects of the valence electrons can be classified as direct and indirect ones. The direct ones are due to the increase of the effective mass with velocity, to the change of the electric nuclear attraction of a spinning electron, and to the magnetic spin-orbit coupling. The indirect effects on the valence electrons are due to the relativistic changes of nuclear shielding and Pauli repulsion by the inner orbitals. The changes of the radial, the angular, and the quaternionic phase behavior of the relativistic atomic valence orbitals modify the atomic bonding properties, the energetics, the structure and properties of the molecules.

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Notes

  1. 1.

    In principle there are no local contributions to property expectation-values in holistic quantum mechanics. However, choosing a specific integral representation, discussion of local contributions to this integral can still give physical insight, see below Section 1.3 and Footnote 11.

  2. 2.

    We use the English modification of Galileo Galelei’s name.

  3. 3.

    In his famous, often-cited dictum, [134] expressed four ideas: First – The fundamental physical laws for a ‘mathematical theory of chemistry’ are completely known. (It is not clear whether Dirac realized that a ‘mathematical theory of chemistry’ comprises only a fraction of chemistry. Another question is which proper chemists, computational chemists, theoretical chemists or philosophers of chemistry are aware of this significant aspect.) Second – the relativistic theory is incomplete. (This is still valid, but the left-over problems are mainly relevant to elementary particle physics and cosmology. Most chemically relevant problems of the physical theory of relativity are now solved.) Third – relativity is irrelevant to chemistry. (Now, we know it better.) Fourth – The two main and permanent problems of theoretical chemistry are to develop computational approaches to calculate reasonably accurate observable values, and to derive interpretational tools for a physical understanding of the complex chemical processes in matter.

  4. 4.

    This comprises the approximation of static and dynamic two-electron correlations, the neglect of non-adiabatic electron-nuclear couplings, the neglect of relativistic electron dynamics, or the neglect of environmental perturbations by 3K cosmic background radiation, or perturbations by the condensed phase surroundings.

  5. 5.

    One should distinguish between an expansion which can give, in principle, the correct value, and an approximation consisting only of the first term of the expansion. There are the lowest-order BO approximation and the BO expansion. There are the Hartree, Hartree–Fock and Kohn–Sham orbital approximations and the post-independent-particle expansions.

  6. 6.

    The IUPAC recommends lanthanoids and actinoids instead of former lanthanides and actinides. ‘-ides’ are usually anionic compounds such as halides, sulfides, etc. See http://en.wikipedia.org/wiki/Lanthanoid.

  7. 7.

    Let ‘superheavy’ be defined as Z > 100.

  8. 8.

    This means without accounting for any quantum interference effects.

  9. 9.

    0 4 and 1 4 mean the 4 ×4 zero- and unit-matrices.

  10. 10.

    For Z larger than about 165, the electronic 1s level dives into the positronic continuum of the extended nucleus, with spontaneous electron–positron pair production.

  11. 11.

    Of course, quantum mechanics is a holistic theory without physically defined local contributions to an observable expectation value. However, in a concrete calculation of a physical value, or in a specific explanation of the physical mechanism, one applies one specific formula chosen from a gauge-invariant set. This then gives one of the many complementary, internally consistent pictures of physical realty. For instance, we here choose \(+1/{2}_{\ \mathrm{a}}{ \int }^{\mathrm{e}}\mathrm{d}r \cdot \vert \nabla \Psi {(r)\vert }^{2}\) with positive definite integrand for local contributions of the kinetic energy T. It gives a somewhat different picture than \(-1/{2}_{\ \mathrm{a}}{ \int }^{\mathrm{e}}\mathrm{d}r \cdot ({\Psi }^{{_\ast}}\cdot {\nabla }^{2}\Psi )\). For the total energy we choose \({}_{\mathrm{a}}{ \int }^{\mathrm{e}}\mathrm{d}r \cdot ({\Psi }^{{_\ast}}\cdot \hat{\mathrm{H}}\Psi ) = E {\cdot }_{\mathrm{a}}{ \int }^{\mathrm{e}}\mathrm{d}r \cdot \rho (r)\). In the present Figures, a means the nuclear position, r = 0, and e corresponds to the upper integration limit plotted on the abscissa.

  12. 12.

    This has sometimes been called the ‘sausage phenomenon’, reminding of the carnival song ‘everything has an end, only the sausage (and the molecular potential curve) has two ends.

  13. 13.

    We here define the orbital energy of a many-electron open-shell atom as the difference of the configuration average energies of the neutral and the ionized species.

  14. 14.

    These few elements are the only ones, where the ubiquitous so-called (n + l, n) text book rule holds!

  15. 15.

    A (separated) shell is stable, because its energy is low and therefore filled up. Chemists sometimes apply the inverted pseudo-logics: A shell becomes stable by filling it up.

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  1. W. H. Eugen Schwarz

    Present address: Theoretical Chemistry Group, University, D-57068, Siegen, Germany

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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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Schwarz, W.H.E. (2010). An Introduction to Relativistic Quantum Chemistry. 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_1

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