Theoretical Chemistry Accounts

, 132:1298 | Cite as

The one-electron picture in the Piris natural orbital functional 5 (PNOF5)

  • Mario Piris
  • Jon M. Matxain
  • Xabier Lopez
  • Jesus M. Ugalde
Regular Article
Part of the following topical collections:
  1. 8th Congress on Electronic Structure: Principles and Applications (ESPA 2012)

Abstract

The natural orbital functional theory provides two complementary representations of the one-electron picture in molecules, namely, the natural orbital (NO) representation and the canonical orbital (CO) representation. The former arises directly from the optimization process solving the corresponding Euler equations, whereas the latter is attained from the diagonalization of the matrix of Lagrange multipliers obtained in the NO representation. In general, the one-particle reduced-density matrix (1-RDM) and the Lagrangian cannot be simultaneously brought to the diagonal form, except for the special Hartree-Fock case. The 1-RDM is diagonal in the NO representation, but not the Lagrangian, which is only a Hermitian matrix. Conversely, in the CO representation, the Lagrangian is diagonal, but not the 1-RDM. Combining both representations we have the whole picture concerning the occupation numbers and the orbital energies. The Piris natural orbital functional 5 leads generally to the localization of the molecular orbitals in the NO representation. Accordingly, it provides an orbital picture that agrees closely with the empirical valence shell electron pair repulsion theory and the Bent’s rule, along with the theoretical valence bond method. On the other hand, the equivalent CO representation can afford delocalized molecular orbitals adapted to the symmetry of the molecule. We show by means of the extended Koopmans’ theorem that the one-particle energies associated with the COs can yield reasonable principal ionization potentials when the 1-RDM remains close to the diagonal form. The relationship between NOs and COs is illustrated by several examples, showing that both orbital representations complement each other.

Keywords

Molecular orbitals Orbital energies One-particle reduced-density matrix Natural orbital functional PNOF5 

References

  1. 1.
    Bent HA (1961) An Appraisal of Valence-bond Structures and Hybridization in Compounds of the First-row elements. Chem Rev 61(3):275–311CrossRefGoogle Scholar
  2. 2.
    Brueckner KA, Wada W (1955) Nuclear saturation and two-body: self-consistent solutions and the effects of the exclusion principle. Phys Rev 103:1008–1016CrossRefGoogle Scholar
  3. 3.
    Coulson CA, Longuet-Higgins HC (1947) The electronic structure of conjugated systems. I. General theory. Proc R Soc Lond A191:39–60CrossRefGoogle Scholar
  4. 4.
    Day OW, Smith DW, Garrod C (1974) A generalization of the hartree-fock one-particle potential. Int J Quantum Chem Symp 8(S8):501–509CrossRefGoogle Scholar
  5. 5.
    Day OW, Smith DW, Morrison RC (1975) Extension of Koopmans’ theorem. II. Accurate ionization energies from correlated wavefunctions for closed-shell atoms. J Chem Phys 62(1):115–119CrossRefGoogle Scholar
  6. 6.
    Donnelly RA (1979) On fundamental difference between energy functionals based on first- and second-order density matrices. J Chem Phys 71(7):2874–2879CrossRefGoogle Scholar
  7. 7.
    Dunning Jr. TH (1989) Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J Chem Phys 90(2):1007–1023CrossRefGoogle Scholar
  8. 8.
    Fock VA (1930) Näherungsmethode zur Lösung des quantenmechanischen Mehrkörper- problems. Z Phys 61(1–2):126–148CrossRefGoogle Scholar
  9. 9.
    Gilbert TL (1975) Hohenberg-Kohn theorem for nonlocal external potentials. Phys Rev B 12(6):2111–2120CrossRefGoogle Scholar
  10. 10.
    Gillespie RJ, Nyholm RS (1957) The valence-shell electron-pair repulsion theory. Quart Rev Chem Soc 11:339–80CrossRefGoogle Scholar
  11. 11.
    Goedecker S, Umrigar CJ (2000) Natural Orbital Functional Theory. In: Cioslowski J (ed.) Many-electron densities and reduced density matrices, pp 165–181. Kluwer, New YorkCrossRefGoogle Scholar
  12. 12.
    Hartree DR (1928) The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and Methods. Proc. Camb Phil. Soc. 24(1):89–110CrossRefGoogle Scholar
  13. 13.
    Heitler W, London F (1927) Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z Phys 44(6–7):455–472CrossRefGoogle Scholar
  14. 14.
    Hund F (1928) Zur Deutung der Molekelspektren. IV. Z Phys 51(11–12):759–795CrossRefGoogle Scholar
  15. 15.
    Jimenez-Hoyos CA, Henderson TM, Scuseria GE (2011) Generalized Hartree-Fock Description of Molecular Dissociation. J Chem Theory Comput 7:2667–2674CrossRefGoogle Scholar
  16. 16.
    Johnson III RD (ed.) (2011) NIST computational chemistry comparison and benchmark database, NIST standard reference database Num 101, Release 15b. http://cccbdb.nist.gov/
  17. 17.
    Kohn W, Sham L (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140(4A):A1133–A1138CrossRefGoogle Scholar
  18. 18.
    Koopmans TA (1934) Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms. Physica 1(1–6):104–113CrossRefGoogle Scholar
  19. 19.
    Kutzelnigg W, Mukherjee D (1999) Cumulant expansion of the reduced density matrices. J Chem Phys 110(6):2800–2809CrossRefGoogle Scholar
  20. 20.
    Leiva P, Piris M (2006) Calculation of vertical ionization potentials with the Piris natural orbital functional. J Mol Struct Theochem 770(1–3):45–49CrossRefGoogle Scholar
  21. 21.
    Levy M (1979) Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc Natl Acad Sci USA 76(12):6062–6065CrossRefGoogle Scholar
  22. 22.
    Levy M (1987) Density matrices and density functionals. In: Erdahl R, Smith VHJ (eds.) Density matrices and density functionals. Reidel, Dordrecht, pp 479–498CrossRefGoogle Scholar
  23. 23.
    Linderberg J, Öhrn Y (2004) Propagators in quantum chemistry, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  24. 24.
    Lopez X, Ruiperez F, Piris M, Matxain JM, Matito E, Ugalde JM (2012) Performance of PNOF5 natural orbital functional for radical formation reactions: hydrogen atom abstraction and C–C and O–O homolytic bond cleavage in selected molecules. J Chem Theory Comput 8:2646–2652CrossRefGoogle Scholar
  25. 25.
    Lowdin PO (1955) Quantum theory of many-particle systems. I. Phys Rev 97(6):1490–1508CrossRefGoogle Scholar
  26. 26.
    Matxain JM, Piris M, Mercero JM, Lopez X, Ugalde JM (2012) sp3 Hybrid orbitals and ionization energies of methane from PNOF5. Chem Phys Lett 531:272–274CrossRefGoogle Scholar
  27. 27.
    Matxain JM, Piris M, Ruipérez F, Lopez X, Ugalde JM (2011) Homolytic molecular dissociation in natural orbital functional theory. Phys Chem Chem Phys 13(45):20129–20135CrossRefGoogle Scholar
  28. 28.
    Matxain JM, Piris M, Uranga J, Lopez X, Merino G, Ugalde JM (2012) Nature of chemical bonds by means of NOFT. Chem Phys Chem 13:2297–2303CrossRefGoogle Scholar
  29. 29.
    Mazziotti DA (1998) Approximate solution for electron correlation through the use of Schwinger probes. Chem Phys Lett 289:419–427CrossRefGoogle Scholar
  30. 30.
    Mazziotti DA (2007) Variational two-electron reduced-density-matrix theory. In: Mazziotti DA (ed.) Reduced-density-matrix mechanics: with applications to many-electron atoms and molecules, 1 edn., Chap 3. Wiley, Hoboken, pp 21–59CrossRefGoogle Scholar
  31. 31.
    Mazziotti DA (2008) Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function: Generalizations and applications. Phys Rev Lett 101:253002CrossRefGoogle Scholar
  32. 32.
    Mazziotti DA (2012) Two-electron reduced density matrix as the basic variable in many-electron quantum chemistry and physics. Chem Rev 112(8):244–262CrossRefGoogle Scholar
  33. 33.
    Morrell MM, Parr RG, Levy M (1975) Calculation of I.P. from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density. J Chem Phys 62(2):549–554CrossRefGoogle Scholar
  34. 34.
    Mulliken RS (1928) The assignment of quantum numbers for electrons in molecules. I. Phys Rev 32:186–222CrossRefGoogle Scholar
  35. 35.
    Ortiz JV (1999) Toward an exact one-electron picture of chemical bonding. Adv Quantum Chem 35:33–52CrossRefGoogle Scholar
  36. 36.
    Ortiz JV (2004) Brueckner orbitals, Dyson orbitals, and correlation potentials. Int J Quantum Chem 100(6):1131–1135CrossRefGoogle Scholar
  37. 37.
    Parr R, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, OxfordGoogle Scholar
  38. 38.
    Pernal K, Cioslowski J (2005) Ionization potentials from the extended Koopmans’ theorem applied to density matrix functional theory. Chem Phys Lett 412(1-3):71–75CrossRefGoogle Scholar
  39. 39.
  40. 40.
    Piris M (1999) A generalized self-consistent-field procedure in the improved BCS theory. J Math Chem 25:47–54CrossRefGoogle Scholar
  41. 41.
    Piris M (2006) A new approach for the two-electron cumulant in natural orbital. Int J Quantum Chem 106:1093–1104CrossRefGoogle Scholar
  42. 42.
    Piris M (2007) Natural orbital functional theory. In: Mazziotti DA (ed.) Reduced-density-matrix mechanics: with applications to many-electron atoms and molecules, chap 14. Wiley, Hoboken, pp 387–427Google Scholar
  43. 43.
    Piris M (2012) A natural orbital functional based on an explicit approach of the two-electron cumulant. Int J Quantum Chem. doi:10.1002/qua.24020
  44. 44.
    Piris M, Lopez X, Ruipérez F, Matxain JM, Ugalde JM (2011) A natural orbital functional for multiconfigurational states. J Chem Phys 134(16):164102CrossRefGoogle Scholar
  45. 45.
    Piris M, Lopez X, Ugalde JM (2007) Dispersion interactions within the Piris natural orbital functional theory: the helium dimer. J Chem Phys 126(21):214103CrossRefGoogle Scholar
  46. 46.
    Piris M, Matxain JM, Lopez X, Ugalde JM (2009) Spin conserving natural orbital functional theory. J Chem Phys 131:021,102CrossRefGoogle Scholar
  47. 47.
    Piris M, Matxain JM, Lopez X, Ugalde JM (2010) Communication: The role of the positivity N-representability conditions in natural orbital functional theory. J Chem Phys 133:111101CrossRefGoogle Scholar
  48. 48.
    Piris M, Matxain JM, Lopez X, Ugalde JM (2010) Communications: accurate description of atoms and molecules by natural orbital functional theory. J Chem Phys 132:031103CrossRefGoogle Scholar
  49. 49.
    Piris M, Matxain JM, Lopez X, Ugalde JM (2012) The extended Koopmans’ theorem: vertical ionization potentials from natural orbital functional theory. J Chem Phys 136(17):174116CrossRefGoogle Scholar
  50. 50.
    Piris M, Montero LA, Cruz N (1997) The BCS approach to electron correlation in the density matrix formalism. J Chem Phys 107(1):180–187CrossRefGoogle Scholar
  51. 51.
    Piris M, Otto P (2003) One-particle density matrix functional for correlation in molecular systems. Int J Quantum Chem 94(6):317–323CrossRefGoogle Scholar
  52. 52.
    Piris M, Otto P (2005) Natural orbital functional for correlation in polymers. Int J Quantum Chem 102(1):90–97CrossRefGoogle Scholar
  53. 53.
    Piris M, Ugalde JM (2009) Iterative diagonalization for orbital optimization in natural orbital functional theory. J Comput Chem 30:2078–2086CrossRefGoogle Scholar
  54. 54.
    Sand AM, Schwerdtfeger Ca, Mazziotti DA (2012) Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene. J Chem Phys 136(3):034112CrossRefGoogle Scholar
  55. 55.
    Smith DW, Day OW (1975) Extension of Koopmans theorem. I. Derivation. J Chem Phys 62(1):113–114CrossRefGoogle Scholar
  56. 56.
    Valone SM (1980) Consequences of extending 1 matrix energy functionals pure-state representable to all ensemble representable 1 matrices. J Chem Phys 73(3):1344–1349CrossRefGoogle Scholar
  57. 57.
    Woon D, Dunning Jr. TH (1993) Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J Chem Phys 98(2):1358–1371CrossRefGoogle Scholar
  58. 58.
    Wu W, Su P, Shaik S, Hiberty PC (2011) Classical valence bond approach by modern method. Chem Rev 111(11):7557–7593CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mario Piris
    • 1
    • 2
  • Jon M. Matxain
    • 1
  • Xabier Lopez
    • 1
  • Jesus M. Ugalde
    • 1
  1. 1.Kimika Fakultatea, Donostia International Physics Center (DIPC) Euskal Herriko Unibertsitatea (UPV/EHU)DonostiaSpain
  2. 2.IKERBASQUEBasque Foundation for ScienceBilbaoSpain

Personalised recommendations