Skip to main content
SpringerLink
Log in

Relativistic Methods in Computational Quantum Chemistry

  • Living reference work entry
  • First Online:
Handbook of Computational Chemistry

  • 715 Accesses

Abstract

In this chapter, we briefly discuss the theoretical foundations of relativistic two-component methods used in quantum chemistry calculations. Specifically, we focus on two groups of methods. These are (i) methods based on the elimination of the small component, such as the zeroth-order regular approximation (ZORA), the first-order regular approximation (FORA), and the normalized elimination of small component (NESC) formalisms, and (ii) approaches that use a unitary transformation to decouple the electronic and positronic states such as the Douglas–Kroll–Hess (DKH) and the infinite-order two-component (IOTC) Hamiltonians. Furthermore, we describe the algebraic approach to IOTC and scrutinize pure algebraic schemes that paved the way to the eXact 2-Component (X2C) Hamiltonians taking advantage of the nonsymmetric algebraic Riccati equation (nARE). Finally, we assess the accuracy of the aforementioned methods in calculating core and valence properties of heavy-element compounds and discuss some challenging examples of computational actinide chemistry.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Notes

  1. 1.

    The atomic unit of speed is defined as \(\frac{a_{0}E_{h}} {\hslash } =\alpha c\), which corresponds to 2. 18769126277 × 106 \(\mathrm{\frac{m} {s} }\).

  2. 2.

    The geometric series is defined as \(\frac{a} {1-x} =\sum _{ k}^{\infty }ax^{k},\) for | x | < 1.

  3. 3.

    Note that V is negative and hence the denominator is even larger than 2c 2.

  4. 4.

    The term unnormalized refers o the normalization property of the large component of the wavefunction, which does not fulfill the normalization condition \(\int \vert \psi ^{L}\vert ^{2}\mathrm{d}\boldsymbol{r}\neq 1 =\int \vert \psi ^{L}\vert ^{2}\mathrm{d}\boldsymbol{r} +\int \vert \psi ^{S}\vert ^{2}\mathrm{d}\boldsymbol{r}\).

References

  • Autschbach, J. (2009). Magnitude of finite-nucleus-size effects in relativistic density functional computations of indirect NMR nuclear spin–spin coupling constants. Chem Phys Chem, 10, 2274.

    CAS  Google Scholar 

  • Autschbach, J. (2012). Perspective: Relativistic effects. Journal of Chemical Physics, 136, 150902.

    Article  Google Scholar 

  • Autschbach, J., Peng, D., & Reiher, M. (2012). Two-component relativistic calculations of electric-field gradients using exact decoupling methods: Spin–orbit and picture-change effects. Journal of Chemical Theory and Computation, 8, 4239.

    Article  CAS  Google Scholar 

  • Balabanov, N. B., & Peterson, K. A. (2005). Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc–Zn. Journal of Chemical Physics, 123, 064107.

    Google Scholar 

  • Barysz, M., & Sadlej, A. J. (2002). Infinite-order two-component theory for relativistic quantum chemistry. Journal of Chemical Physics, 116, 2696.

    Article  CAS  Google Scholar 

  • Barysz, M., Sadlej, A. J., & Snijders, J. G. (1997). Nonsingular two/one-component relativistic Hamiltonians accurate through arbitrary high order in alpha. International Journal of Quantum Chemistry, 65, 225.

    Article  CAS  Google Scholar 

  • Borkowski, M., Żuchowski, P. S., Ciuryło, R., Julienne, P. S., Kędziera, D., Mentel, L., Tecmer, P., Münchow, F., Bruni, C., & Görlitz, A. (2013). Scattering lengths in isotopologues of the RbYb system. Physical Review A, 88, 052708.

    Article  Google Scholar 

  • Bross, D. H., & Peterson, K. A. (2014). Correlation consistent, Douglas–Kroll–Hess relativistic basis sets for the 5p and 6p elements. Theoretica Chimica Acta, 133, 1.

    CAS  Google Scholar 

  • Brown, G. E., & Ravenhall, D. G. (1951). On the interaction of two electrons. Proceedings of the Royal Society of London Series A, 208, 552.

    Article  CAS  Google Scholar 

  • Chang, C., Pelissier, M., & Durand, P. (1986). Regular two-component Pauli-like effective Hamiltonians in Dirac theory. Physica Scripta, 34, 394.

    Article  CAS  Google Scholar 

  • Dirac, P.A.M. (1928a). The quantum theory of the electron. Proceedings of the Royal Society of London Series A, 117, 610.

    Article  Google Scholar 

  • Dirac, P.A.M. (1928b). The quantum theory of the electron part II. Proceedings of the Royal Society of London Series A, 118, 351.

    Article  CAS  Google Scholar 

  • Douglas, N., & Kroll, N. M. (1974). Quantum electrodynamical corrections to fine-structure of helium. Annals of Physics, 82, 89.

    Article  CAS  Google Scholar 

  • Dyall, K. G. (1994). An exact separation of the spinfree and spindependent terms of the dirac–coulomb–breit Hamiltonian. Journal of Chemical Physics, 100, 2118.

    Article  CAS  Google Scholar 

  • Dyall, K. G. (1997). Interfacing relativistic and nonrelativistic methods. I. Normalized elimination of the small component in the modified Dirac equation. Journal of Chemical Physics, 106, 9618.

    Google Scholar 

  • Dyall, K. G. (1998). Interfacing relativistic and nonrelativistic methods. II. Investigation of a low-order approximation. Journal of Chemical Physics, 109, 4201.

    Google Scholar 

  • Dyall, K. G., & Faegri K., Jr. (2007). Introduction to Relativistic Quantum Chemistry. Oxford: Oxford University Press.

    Google Scholar 

  • Dyall, K. G., & van Lenthe, E. (1999). Relativistic regular approximations revisited: An infinite-order relativistic approximation. Journal of Chemical Physics, 111, 1366.

    Article  CAS  Google Scholar 

  • Filatov, M. (2002). Relativistic Ab Initio Theory using the Regular Approximation. Weinheim, Amsterdam. Wiley.

    Google Scholar 

  • Foldy, L. L., & Wouthuysen, S. A. (1950). On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Physical Review, 78, 29.

    Article  Google Scholar 

  • Hess, B. A. (1985). Applicability of the no-pair equation with free-particle projection operators to atomic and molecular structure calculations. Physical Review A, 32, 756.

    Article  CAS  Google Scholar 

  • Hess, B. A., Marian, C. M., Wahlgren, U., & Gropen, O. (1996). A mean-field spin-orbit method applicable to correlated wavefunctions. Chemical Physics Letters, 251, 365.

    Article  CAS  Google Scholar 

  • Heully, J. L., Lindgren, I., Lindroth, E., & Lundqvist, S., Martensson-Pendrill, A. M. (1986). Diagonalisation of the Dirac Hamiltonian as a basis for a relativistic many-body procedure. Journal of Physics B: Atomic and Molecular Physics, 19, 2799.

    Article  CAS  Google Scholar 

  • Hill, J. G. (2013). Gaussian basis sets for molecular applications. International Journal of Quantum Chemistry, 113, 21.

    Article  CAS  Google Scholar 

  • Iliaš, M., & Saue, T. (2007). An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation. Journal of Chemical Physics, 126, 064102.

    Article  Google Scholar 

  • Kędziera, D. (2005). Convergence of approximate two-component Hamiltonians: How far is the Dirac limit. Journal of Chemical Physics, 123, 074109.

    Article  Google Scholar 

  • Kędziera, D. (2006). Solving of the infinite-order two-component method equations. In Recent Progess in Computational Sciences and Engineering (VSP BV-C/O BRILL ACAD PUBL, Leiden, The Netherlands), (Lecture Series on Computer and Computational Sciences, vol. 7A–B, pp. 252–255)

    Google Scholar 

  • Kędziera, D., & Barysz, M. (2004). Two-component relativistic methods for the heaviest elements. Journal of Chemical Physics, 121, 6719.

    Article  Google Scholar 

  • Kędziera, D., & Barysz, M. (2007). Non-iterative approach to the infinite-order two-component (iotc) relativistic theory and the non-symmetric algebraic riccati equation. Chemical Physics Letters, 446, 176.

    Article  Google Scholar 

  • Kędziera, D., & Kaczmarek-Kędziera, A. (2012). Remarks on wave function theory and methods. In J. Leszczynski (Ed.), Handbook of computational chemistry (pp. 55–93). Netherlands: Springer.

    Chapter  Google Scholar 

  • Kello, V., & Sadlej, A. J. (1997). Picture change and calculations of expectation values in approximate relativistic theories. International Journal of Quantum Chemistry, 68, 159.

    Article  Google Scholar 

  • Liu, W. (2010). Ideas of relativistic quantum chemistry. Molecular Physics, 108, 1679.

    Article  CAS  Google Scholar 

  • Liu, W., & Peng, D. (2009). Exact two-component Hamiltonians revisited. Journal of Chemical Physics, 131, 031104.

    Article  Google Scholar 

  • Mastalerz, R., Reiher, M. (2008). Relativistic electronic structure theory for molecular spectroscopy. In M. Quack & F. Merkt (Eds.), Handbook of High-Resolution Spectroscopy (pp. 405–442). Wiley

    Google Scholar 

  • Nakajima, T., & Hirao, K. (2000). The higher-order Douglas–Kroll transformation. Journal of Chemical Physics, 113, 7786.

    Article  CAS  Google Scholar 

  • Noro, T., Sekiya, M., Koga, T. (2012). Segmented contracted basis sets for atoms H through Xe: Sapporo-(DK)-nZP sets (n = D, T, Q). Theoretica Chimica Acta, 131, 1124.

    Google Scholar 

  • Noro, T., Sekiya, M., & Koga, T. (2013). Sapporo-(DKH3)-nZP (n = D, T, Q) sets for the sixth period s-, d-, and p-block atoms. Theoretica Chimica Acta, 132, 1363.

    Google Scholar 

  • Noro, T., Sekiya, M., Koga, T., & Saito, S. L. (2009). Relativistic contracted Gaussian-type basis functions for atoms K through Xe. Chemical Physics Letters, 481, 229.

    Article  CAS  Google Scholar 

  • Pantazis, D. A., & Neese, F. (2009). All-electron scalar relativistic basis sets for the lanthanides. Journal of Chemical Theory and Computation, 5, 2229.

    Article  CAS  Google Scholar 

  • Pantazis, D. A., & Neese, F. (2011). All-electron scalar relativistic basis sets for the actinides. Journal of Chemical Theory and Computation, 7, 677.

    Article  CAS  Google Scholar 

  • Pantazis, D. A., & Neese, F. (2012). All-electron scalar relativistic basis sets for the 6p elements. Theoretica Chimica Acta, 131, 1292.

    Google Scholar 

  • Pantazis, D. A., & Neese, F. (2014). All-electron basis sets for heavy elements. WIREs Computational Molecular Science, 4, 363.

    Article  CAS  Google Scholar 

  • Pantazis, D. A., Chen, X. Y., Landis, C. R., & Neese, F. (2008). All-electronscalar relativistic basis sets for third-row transition metal atoms. Journal of Chemical Theory and Computation 4, 908.

    Article  CAS  Google Scholar 

  • Peng, D., Middendorf, N., Weigend, F., & Reiher, M. (2013). An efficient implementation of two-component relativistic exact-decoupling methods for large molecules. Journal of Chemical Physics, 138, 184105.

    Article  Google Scholar 

  • Peng, D., & Reiher, M. (2012). Exact decoupling of the relativistic Fock operator. Theoretical Chemistry Accounts, 131, 1081.

    Article  Google Scholar 

  • Peterson, K. A. (2015). Correlation consistent basis sets for actinides. I. the Th and U atoms. Journal of Chemical Physics, 142, 074105.

    Google Scholar 

  • Peterson, K. A., & Dyall, K. G. (2015). Gaussian basis sets for lanthanide and actinide elements: Strategies for their development and use. In M. Dolg (Ed.), Computational Methods in Lanthanide and Actinide Chemistry (pp. 195–216). Wiley.

    Google Scholar 

  • Prascher, B. P., Woon, D. E., Peterson, K. A., Dunning, T. H., Jr., & Wilson, A. K. (2011). Gaussian basis sets for use in correlated molecular calculations. VII. Valence, core-valence, and scalar relativistic basis sets for Li, Be, Na, and Mg. Theoretica Chimica Acta, 128, 69.

    Google Scholar 

  • Pyykkö, P. (1988). Relativistic effects in structural chemistry. Chemical Reviews, 88, 563.

    Article  Google Scholar 

  • Reiher, M. (2006). Douglas–Kroll–Hess theory: A relativistic electrons-only theory for chemistry. Theoretical Chemistry Accounts, 116, 241.

    Article  CAS  Google Scholar 

  • Reiher, M. (2012). Relativistic Douglas-Kroll-Hess theory. WIREs Computational Molecular Science, 2, 139.

    Article  CAS  Google Scholar 

  • Reiher, M., & Wolf, A. (2004a). Exact decoupling of the Dirac Hamiltonian. I. General theory. Journal of Chemical Physics, 121, 2037.

    Google Scholar 

  • Reiher, M., & Wolf, A. (2004b). Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas-Kroll-Hess transformation up to arbitrary order. Journal of Chemical Physics, 121, 10945.

    Google Scholar 

  • Reiher, M., & Wolf, A. (2014). Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science. Weinheim, Amsterdam. Wiley.

    Book  Google Scholar 

  • Roos, B. O., Lindh, R., Malmqvist, P. A., Veryazov, V., Widmark, P. O. (2004). Main group atoms and dimers studied with a new relativistic ANO basis set. Journal of Physical Chemistry A, 108, 2851.

    Article  CAS  Google Scholar 

  • Roos, B. O., Lindh, R., Malmqvist, P. A., Veryazov, V., & Widmark, P. O. (2005a). New relativistic ANO basis sets for transition metal atoms. Journal of Physical Chemistry A, 109, 6575.

    Article  CAS  Google Scholar 

  • Roos, B. O., Lindh, R., Malmqvist, P. A., Veryazov, V., & Widmark, P. O. (2005b). New relativistic ANO basis sets for actinide atoms. Chemical Physics Letters, 409, 295.

    Article  CAS  Google Scholar 

  • Roos, B. O., Lindh, R., Malmqvist, P. A., Veryazov, V., Widmark, P. O., & Borin, A. C. (2008). New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3. Journal of Physical Chemistry A, 112, 11431.

    Article  CAS  Google Scholar 

  • Saue, T. (2011). Relativistic Hamiltonians for chemistry: A primer. ChemPhysChem, 3, 3077.

    Article  Google Scholar 

  • Schimmelpfennig, B., Maron, L., Wahlgren, U., Teichteil, C., Fagerli, H., & Gropen, O. (1998a). On the combination of ECP-based CI calculations with all-electron spin-orbit mean-field integrals. Chemical Physics Letters, 286, 261.

    Article  CAS  Google Scholar 

  • Schimmelpfennig, B., Maron, L., Wahlgren, U., Teichteil, C., Fagerli, H., & Gropen, O. (1998b). On the combination of ECP-based CI calculations with all-electron spin-orbit mean-field integrals. Chemical Physics Letters, 286, 267.

    Article  CAS  Google Scholar 

  • Schwerdtfeger, P. (2002). Relativistic Electronic Structure Theory. Part I. Fundamentals. Weinheim, Amsterdam. Elsevier.

    Google Scholar 

  • Sekiya, M., Noro, T., Koga, T., Saito, S. L. (2010). Relativistic correlating basis sets for 57la and 89ac. Journal of Computational Chemistry, 31, 497.

    CAS  Google Scholar 

  • Sikkema, J., Visscher, L., Saue, T., & Ilias, M. (2009). The molecular mean-field approach for correlated relativistic calculations. Journal of Chemical Physics, 131, 124116.

    Article  Google Scholar 

  • Tecmer, P., Gomes, A.S.P., Ekström, U., & Visscher, L. (2011). Electronic spectroscopy of UO2+2, NUO+ and NUN: An evaluation of time-dependent density functional theory for actinides. Physical Chemistry Chemical Physics, 13, 6249.

    Article  CAS  Google Scholar 

  • Tecmer, P., Gomes, A.S.P., Knecht, S., & Visscher, L. (2014). Communication: Relativistic Fock-space coupled cluster study of small building blocks of larger uranium complexes. Journal of Chemical Physics, 141, 041107.

    Article  Google Scholar 

  • Tecmer, P., Boguslawski, K., Legeza, Ö., & Reiher, M. (2014). Unravelling the quantum-entanglement effect of noble gas coordination on the spin ground state of CUO. Physical Chemistry Chemical Physics, 16, 719.

    Article  CAS  Google Scholar 

  • Van Lenthe, E., & Baerends, E. J. (2003). Optimized Slater type basis sets for the elements 1-118. Journal of Computational Chemistry, 24, 1142.

    Article  Google Scholar 

  • van Lenthe, E., Baerends, E. J., & Snijders, J. G. (1993). Relativistic regular twocomponent Hamiltonians. Journal of Chemical Physics, 99, 4597.

    Article  Google Scholar 

  • van Lenthe, E., Baerends, E. J., & Snijders, J. G. (1994). Relativistic total energy using regular approximations. Journal of Chemical Physics, 101, 9783.

    Article  Google Scholar 

  • van Lenthe, E., van Leeuwen, R., Baerends, E. J., & Snijders, J. G. (1996). Relativistic regular two-component Hamiltonians. International Journal of Quantum Chemistry, 57, 281.

    Article  Google Scholar 

  • van Lenthe, E., Ehlers, A., & Baerends, E. J. (1999). Geometry optimizations in the zero order regular approximation for relativistic effects. Journal of Chemical Physics, 110, 8943.

    Article  Google Scholar 

  • van Wüllen, C. (2004). Journal of Chemical Physics, 120, 7307.

    Article  Google Scholar 

  • Visscher, L. (2013). In P. O. Widmark (Ed.), Lecture Notes in Quantum Chemistry: European Summerschool in Quantum Chemistry (pp. 5–43). Springer.

    Google Scholar 

  • Visscher, L., & Dyall, K. (1997). Atomic electronic structure calculations using different nuclear charge distributions. Atomic Data and Nuclear Data Tables, 67, 207.

    Article  CAS  Google Scholar 

  • Visser, O., Aerts, P., Hegarty, D., & Nieuwpoort, W. (1987). The use of Gaussian nuclear charge distributions for the calculation of relativistic electronic wavefunctions using basis set expansions. Chemical Physics Letters, 134, 34.

    Article  CAS  Google Scholar 

  • Wolf, A., Reiher, M., & Hess, B. A. (2002). In P. Schwerdtfeger (Eds.), Relativistic Quantum Chemistry (Theoretical and Computational Chemistry, pp. 622–663). Elsevier.

    Google Scholar 

  • Wolf, A., Reiher, M., & Hess, B. A. (2002). The generalized Douglas–Kroll transformation. Journal of Chemical Physics, 117, 9215.

    Article  CAS  Google Scholar 

  • Zou, W., Filatov, M., & Cremer, D. (2011). An improved algorithm for the normalized elimination of the small-component method. Theoretica Chimica Acta, 130, 633.

    CAS  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the financial support from the National Science Center Poland, Grants No. DEC-2012/07/B/ST4/01347 and No. DEC-2013/11/B/ST4/00771. K.B. acknowledges financial support from a SONATA BIS grant of the National Science Centre, Poland (no. 2015/18/E/ST4/00584). We had many helpful discussions with Prof. Markus Reiher and Prof. Lucas Visscher.

Author information

Authors and Affiliations

  1. Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Torun, Grudziadzka 5, 87-100, Torun, Poland

    Paweł Tecmer & Katharina Boguslawski

  2. Department of Chemistry and Photochemistry of Polymers, Faculty of Chemistry, Nicolaus Copernicus University in Torun, Torun, Poland

    Dariusz Kędziera

Authors
  1. Paweł Tecmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katharina Boguslawski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dariusz Kędziera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paweł Tecmer .

Editor information

Editors and Affiliations

  1. Jackson State University, Jackson, Mississippi, USA

    Jerzy Leszczynski

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Dordrecht

About this entry

Cite this entry

Tecmer, P., Boguslawski, K., Kędziera, D. (2016). Relativistic Methods in Computational Quantum Chemistry. In: Leszczynski, J. (eds) Handbook of Computational Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6169-8_42-1

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-6169-8_42-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Online ISBN: 978-94-007-6169-8

  • eBook Packages: Springer Reference Chemistry and Mat. ScienceReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation