Theoretical Chemistry Accounts

, Volume 126, Issue 3–4, pp 165–175 | Cite as

A fundamental connection between symmetry and spatial localization properties of basis sets

Regular Article


The problem of the compatibility between symmetry and localization properties of basis sets is addressed here. It is shown that both concepts are closely related from a fundamental point of view through the notion of invariance extent. This quantity is a functional that depends on the symmetry group and the basis set choices, and it is shown that all basis sets adapted in a general way to symmetry, i.e. induced from irreducible bases of the subgroups, are stationary points of it. In particular, the usual irreducible bases of the full group display a maximal invariance extent, while those symmetry-adapted basis sets that display a minimal value of this quantity feature in most cases the same symmetry properties as localized functions obtained by means of the Boys scheme. The most relevant conclusions are illustrated by means of simple molecular and periodic examples.


  1. 1.
    Wannier GH (1937) The structure of electronic excitation levels in insulating crystals. Phys Rev 52:191–197CrossRefGoogle Scholar
  2. 2.
    des Cloizeaux J (1964) Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys Rev A 135:685–697CrossRefGoogle Scholar
  3. 3.
    des Cloizeaux J (1964) Analytic properties of n-dimensional energy bands and Wannier functions. Phys Rev A 135:698–707CrossRefGoogle Scholar
  4. 4.
    von Boehm J, Calais JL (1979) Variational procedure for symmetry-adapted Wannier functions. J Phys C 12:3661–3675CrossRefGoogle Scholar
  5. 5.
    Evarestov RA, Smirnov VP (1997) Site symmetry in crystals: theory and applications. Springer series in solid-state sciences, 2nd enlarged edn. Springer, Berlin (ISSN 0171-1873)Google Scholar
  6. 6.
    Smirnov VP, Evarestov RA, Usvyat DE (2002) Wannier-type atomic functions and chemical bonding in crystals. Int J Quantum Chem 88:642–651CrossRefGoogle Scholar
  7. 7.
    Smirnov VP, Usvyat DE (2001) Variational method for the generation of localized Wannier functions on the basis of Bloch functions. Phys Rev B 64:245108CrossRefGoogle Scholar
  8. 8.
    Dovesi R (1986) On the role of symmetry in the ab initio Hartree-Fock linear combination of atomic orbitals treatment of periodic systems. Int J Quantum Chem 29:1755CrossRefGoogle Scholar
  9. 9.
    Pisani C, Dovesi R, Roetti C (1988) Hartree-Fock ab initio treatment of crystalline solids. Lecture notes in chemistry series, vol 48. Springer, BerlinGoogle Scholar
  10. 10.
    Pisani C, Busso M, Capecchi G, Casassa S, Dovesi R, Maschio L, Zicovich-Wilson CM, Schütz M (2005) Local-MP2 electron correlation method for non conducting crystals. J Chem Phys 122:094113CrossRefGoogle Scholar
  11. 11.
    Foster JM, Boys SF (1960) Canonical configurational interaction procedure. Rev Mod Phys 32:300–302CrossRefGoogle Scholar
  12. 12.
    Pipek J, Mezey PG (1989) A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions. J Chem Phys 90(9):4916–4926CrossRefGoogle Scholar
  13. 13.
    Zicovich-Wilson CM (2008) Two points of view to look at symmetry. J Phys Conf Ser 107:012030CrossRefGoogle Scholar
  14. 14.
    Zicovich-Wilson CM, Dovesi R, Saunders VR (2001) A general method to obtain well localized Wannier functions for composite energy bands in LCAO periodic calculations. J Chem Phys 115:9708–9719CrossRefGoogle Scholar
  15. 15.
    Casassa S, Zicovich-Wilson CM, Pisani C (2006) Symmetry-adapted localized Wannier functions suitable for periodic local correlation methods. Theor Chem Acc 116:726–733. doi:10.1007/s00214-006-0119-z
  16. 16.
    Scheurer P, Schwarz WHE (2000) Continuous degeneracy of sets of localized orbitals. Int J Quantum Chem 76:428–433CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad Autónoma del Estado de MorelosCuernavaca (MOR)Mexico
  2. 2.Dipartimento di Chimica IFM, Centre of Excellence NIS (Nanostructured Interfaces and Surfaces)Università di TorinoTurinItaly

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