Theoretical Chemistry Accounts

, Volume 129, Issue 3–5, pp 423–436 | Cite as

Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations

  • Qiming Sun
  • Wenjian LiuEmail author
  • Werner Kutzelnigg
Regular Article


Various kinetic balances for constructing appropriate basis sets in four-component relativistic calculations are examined in great detail. These include the well-known restricted (RKB) and unrestricted (UKB) kinetic balances, the less-known dual kinetic balance (DKB) as well as the unknown inverse kinetic balance (IKB). The RKB and IKB are complementary to each other: The former is good for positive-energy states, whereas the latter good for negative-energy states. The DKB combines the good of both RKB and IKB and even provides full variational safety. However, such an advantage is largely offset by its complicated nature. The UKB does not offer any particular advantages as well. Overall, the RKB is the simplest ansatz. Although the negative-energy states by a finite RKB basis are in error of O(c 0), there is no objection to using them as intermediates for a sum-over-states formulation of perturbation theory, provided that the magnetic balance is also incorporated in the case of magnetic properties. In particular, the RKB is also an essential ingredient for formulating two-component relativistic theories, while all the others are simply incompatible. As such, the RKB should be regarded as the cornerstone of relativistic electronic structure calculations.


Restricted kinetic balance Unrestricted kinetic balance Inverse kinetic balance Dual kinetic balance Magnetic balance 



The research of this work was supported by grants from the National Natural Science Foundation of China (Project Nos. 20625311, 20773003 and 21033001) and from MOST of China (Project Nos. 2006CB601103 and 2006AA01A119).


  1. 1.
    Kim YK (1967) Phys Rev 154:17CrossRefGoogle Scholar
  2. 2.
    Schwarz WHE, Wallmeier H (1982) Mol Phys 46:1045CrossRefGoogle Scholar
  3. 3.
    Schwarz WHE, Wechsel-Trakowski E (1982) Chem Phys Lett 85:94CrossRefGoogle Scholar
  4. 4.
    Kutzelnigg W (1984) Int J Quantum Chem 25:107CrossRefGoogle Scholar
  5. 5.
    Wallmeier H (1984) Phys Rev A 29:2933CrossRefGoogle Scholar
  6. 6.
    Talman JD (1986) Phys Rev Lett 57:1091CrossRefGoogle Scholar
  7. 7.
    Hill RN, Krauthauser C (1994) Phys Rev Lett 72:2151CrossRefGoogle Scholar
  8. 8.
    Stanton RE, Havriliak S (1984) J Chem Phys 81:1910CrossRefGoogle Scholar
  9. 9.
    Dyall KG, Fægri K Jr (1990) Chem Phys Lett 174:25 (The acronyms RKB and UKB were proposed therein for the first time)Google Scholar
  10. 10.
    Ishikawa Y, Binning RC Jr, Sando KM (1983) Chem Phys Lett 101:111CrossRefGoogle Scholar
  11. 11.
    Heully JL, Lindgren I, Lindroth E, Lundquist S, Mårtensen-Pendrill AM (1986) J Phys B 19:2799CrossRefGoogle Scholar
  12. 12.
    Kutzelnigg W (1996) Phys Rev A 54:1183CrossRefGoogle Scholar
  13. 13.
    Kutzelnigg W (1997) Chem Phys 225:203CrossRefGoogle Scholar
  14. 14.
    Kutzelnigg W (1999) J Chem Phys 110:8283CrossRefGoogle Scholar
  15. 15.
    Kutzelnigg W (1989) Z Phys D 11:15CrossRefGoogle Scholar
  16. 16.
    Kutzelnigg W (2007) J Chem Phys 126:201103CrossRefGoogle Scholar
  17. 17.
    Kutzelnigg W (1994) Int J Quantum Chem 51:447CrossRefGoogle Scholar
  18. 18.
    Klahn BK, Bingel WA (1977) Theor Chim Acta 44:27CrossRefGoogle Scholar
  19. 19.
    Kutzelnigg W, Liu W (2006) Mol Phys 104:2225CrossRefGoogle Scholar
  20. 20.
    Kutzelnigg W, Liu W (2005) J Chem Phys 123:241102CrossRefGoogle Scholar
  21. 21.
    Dyall KG (1997) J Chem Phys 106:9618CrossRefGoogle Scholar
  22. 22.
    Dyall KG (2001) J Chem Phys 115:9136CrossRefGoogle Scholar
  23. 23.
    Dyall KG (2002) J Comput Chem 23:786CrossRefGoogle Scholar
  24. 24.
    Liu W, Peng D (2006) J Chem Phys 125:044102; 125:149901(E)Google Scholar
  25. 25.
    Peng D, Liu W, Xiao Y, Cheng L (2007) J Chem Phys 127:104106CrossRefGoogle Scholar
  26. 26.
    Liu W, Kutzelnigg W (2007) J Chem Phys 126:114107CrossRefGoogle Scholar
  27. 27.
    Kutzelnigg W, Liu W (2006) J Chem Phys 125:107102CrossRefGoogle Scholar
  28. 28.
    Liu W (2007) Progr Chem 19:833Google Scholar
  29. 29.
    Liu W, Peng D (2009) J Chem Phys 131:031104CrossRefGoogle Scholar
  30. 30.
    Iliaš M, Saue T (2007) J Chem Phys 126:064102CrossRefGoogle Scholar
  31. 31.
    Sikkema J, Visscher L, Saue T, Iliaš M (2009) J Chem Phys 131:124116CrossRefGoogle Scholar
  32. 32.
    Liu W (2010) Mol Phys 108:1679CrossRefGoogle Scholar
  33. 33.
    Lee YS, Mclean AD (1982) J Chem Phys 76:735CrossRefGoogle Scholar
  34. 34.
    Shabaev VM, Tupitsyn II, Yerokhin VA, Plunien G, Soff G (2004) Phys Rev Lett 93:130405CrossRefGoogle Scholar
  35. 35.
    Dyall KG (1994) J Chem Phys 100:2118CrossRefGoogle Scholar
  36. 36.
    Dyall KG, Grant IP, Wilson S (1984) J Phys B 17:483CrossRefGoogle Scholar
  37. 37.
    Foldy LL, Wouthuysen SA (1950) Phys Rev 78:29CrossRefGoogle Scholar
  38. 38.
    Dyall KG, Fægri K Jr (2007) Introduction to relativistic quantum chemistry. Oxford University Press, New YorkGoogle Scholar
  39. 39.
    Xiao Y, Peng D, Liu W (2007) J Chem Phys 126:081101CrossRefGoogle Scholar
  40. 40.
    Liu W, Hong G, Li L, Xu G (1996) Chin Sci Bull 41:651Google Scholar
  41. 41.
    Hong W, Liu G, Dai D, Li L, Dolg M (1997) Theor Chem Acc 96:75Google Scholar
  42. 42.
    Liu W, Wang F, Li L (2003) J Theor Comput Chem 2:257CrossRefGoogle Scholar
  43. 43.
    Liu W, Wang F, Li L (2004) Recent advances in relativistic molecular theory, recent advances in computational chemistry, vol 5. In: Hirao K, Ishikawa Y (eds). World Scientific, Singapore, p 257Google Scholar
  44. 44.
    Liu W, Wang F, Li L (2004) Encyclopedia of computational chemistry (electronic edition). In: von Ragué Schleyer P, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF III, Schreiner PR (eds) Wiley, ChichesterGoogle Scholar
  45. 45.
    The interchanged eigenvectors should be \(({\bf -B}^D, {\bf A}^D) . ^T\) if the imaginary unit in Eq. 36. is not extracted explicitlyGoogle Scholar
  46. 46.
    Fægri K Jr (2001) Theor Chem Acc 105:252Google Scholar
  47. 47.
    Ramsey NF (1950) Phys Rev 78:699CrossRefGoogle Scholar
  48. 48.
    Kutzelnigg W (2003) Phys Rev A 67:032109CrossRefGoogle Scholar
  49. 49.
    Xiao Y, Liu W, Cheng L, Peng D (2007) J Chem Phys 126:214101CrossRefGoogle Scholar
  50. 50.
    Kutzelnigg W, Liu W (2009) J Chem Phys 131:044129CrossRefGoogle Scholar
  51. 51.
    Cheng L, Xiao Y, Liu W (2009) J Chem Phys 130:144102CrossRefGoogle Scholar
  52. 52.
    Cheng L, Xiao Y, Liu W (2009) J Chem Phys 131:244113CrossRefGoogle Scholar
  53. 53.
    Sun Q, Liu W, Xiao Y, Cheng L (2009) J Chem Phys 131:081101CrossRefGoogle Scholar
  54. 54.
    Komorovsky S, Repisky M, Malkina OL, Malkin VG, Ondík IM, Kaupp M (2008) J Chem Phys 128:104101CrossRefGoogle Scholar
  55. 55.
    Repiský M, Komorovský S, Malkina OL, Malkin VG (2009) Chem Phys 356:236CrossRefGoogle Scholar
  56. 56.
    Komorovsky S, Repisky M, Malkina OL, Malkin VG (2010) J Chem Phys 132:154101CrossRefGoogle Scholar
  57. 57.
    Hamaya S, Fukui H (2010) Bull Chem Soc Jpn 83:635CrossRefGoogle Scholar
  58. 58.
    Pecul M, Saue T, Ruud K, Rizzo A (2004) J Chem Phys 121:3051CrossRefGoogle Scholar
  59. 59.
    Maldonado A, Aucar G (2007) J Chem Phys 127:154115CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Qiming Sun
    • 1
    • 2
    • 3
    • 4
    • 5
  • Wenjian Liu
    • 1
    • 2
    • 3
    • 4
    • 5
    Email author
  • Werner Kutzelnigg
    • 6
  1. 1.Beijing National Laboratory for Molecular SciencesPeking UniversityBeijingPeople’s Republic of China
  2. 2.Institute of Theoretical and Computational ChemistryPeking UniversityBeijingPeople’s Republic of China
  3. 3.State Key Laboratory of Rare Earth Materials Chemistry and ApplicationsPeking UniversityBeijingPeople’s Republic of China
  4. 4.College of Chemistry and Molecular EngineeringPeking UniversityBeijingPeople’s Republic of China
  5. 5.Center for Computational Science and EngineeringPeking UniversityBeijingPeople’s Republic of China
  6. 6.Lehrstuhl für Theoretische ChemieRuhr-Universität BochumBochumGermany

Personalised recommendations