A simple method of identifying π orbitals for non-planar systems and a protocol of studying π electronic structure

  • Tian LuEmail author
  • Qinxue Chen
Regular Article
Part of the following topical collections:
  1. Chemical Concepts from Theory and Computation


The characteristic of π electrons has a crucial role in determining various properties of chemical systems, such as reactivity, aromaticity and spectroscopy. There are a large number of methods could be used for investigating π electronic structure, for example, the well-known electron localization function and multicenter bond order. For completely planar systems, the π molecular orbitals can be unambiguously identified and thus studying their π electronic structure is easy. However, for non-planar systems, identification of π orbitals and then analysis of π electrons are often not trivial. In this work, based on localized molecular orbitals (LMOs), we propose a conceptually simple and easy way to automatically identify π orbitals for any kind of systems, which makes subsequent analyses of π electrons straightforward. In addition, we show that the identified π LMOs can also be used to reliably estimate π component of molecular orbitals or other kinds of orbitals. The method proposed in this work has been implemented into our wavefunction analysis code Multiwfn as a key ingredient of standard analysis protocol for π electrons. Application examples given in this article illustrated that this protocol makes analysis of π electronic structure for a wide variety of chemical systems unprecedentedly convenient and reliable.


Orbital localization Electron structure Electron localization function Multiwfn Localized orbital locator Electron density π electron Bond order 



  1. 1.
    Yu D, Rong C, Lu T, Chattaraj PK, De Proft F, Liu S (2017) Aromaticity and antiaromaticity of substituted fulvene derivatives: perspectives from the information-theoretic approach in density functional reactivity theory. Phys Chem Chem Phys 19:18635PubMedCrossRefPubMedCentralGoogle Scholar
  2. 2.
    Yu D, Rong C, Lu T, Proft FD, Liu S (2018) Aromaticity study of benzene-fused fulvene derivatives using the information-theoretic approach in density functional reactivity theory. Acta Phys Chim Sin 34:639Google Scholar
  3. 3.
    Yu D, Stuyver T, Rong C, Alonso M, Lu T, De Proft F, Geerlings P, Liu S (2019) Global and local aromaticity of acenes from the information-theoretic approach in density functional reactivity theory. Phys Chem Chem Phys 21:18195PubMedCrossRefPubMedCentralGoogle Scholar
  4. 4.
    Klessinger M, Michl J (1994) Excited states and photochemistry of organic molecules. VCH Publishers Inc, New YorkGoogle Scholar
  5. 5.
    Grimme S (2008) Do special noncovalent π–π stacking interactions really exist? Angew Chem Int Ed 47:3430CrossRefGoogle Scholar
  6. 6.
    Yang Y-F, Liang Y, Liu F, Houk KN (2016) Diels-Alder reactivities of benzene, pyridine, and di-, tri-, and tetrazines: the roles of geometrical distortions and orbital interactions. J Am Chem Soc 138:1660PubMedCrossRefPubMedCentralGoogle Scholar
  7. 7.
    Liu Z, Lu T, Hua S, Yu Y (2019) Aromaticity of Hückel and Möbius topologies involved in conformation conversion of macrocyclic [32]Octaphyrin( refined evidence from multiple visual criteria. J Phys Chem C 123:18593CrossRefGoogle Scholar
  8. 8.
    Lu T, Manzetti S (2014) Wavefunction and reactivity study of benzo[a]pyrene diol epoxide and its enantiomeric forms. Struct Chem 25:1521CrossRefGoogle Scholar
  9. 9.
    Geuenich D, Hess K, Köhler F, Herges R (2005) Anisotropy of the induced current density (ACID), a general method to quantify and visualize electronic delocalization. Chem Rev 105:3758PubMedCrossRefPubMedCentralGoogle Scholar
  10. 10.
    Hirshfeld FL (1977) Bonded-atom fragments for describing molecular charge densities. Theor Chem Acc 44:129CrossRefGoogle Scholar
  11. 11.
    Lu T, Chen F (2012) Atomic dipole moment corrected Hirshfeld population method. J Theor Comput Chem 11:163CrossRefGoogle Scholar
  12. 12.
    Lu T, Chen F (2012) Comparison of computational methods for atomic charges. Acta Phys Chim Sin 28:1Google Scholar
  13. 13.
    Lu T, Chen F (2013) Bond order analysis based on the Laplacian of electron density in fuzzy overlap space. J Phys Chem A 117:3100PubMedCrossRefGoogle Scholar
  14. 14.
    Bader FW (1994) Atoms in molecules: a quantum theory. Oxford University Press, New YorkGoogle Scholar
  15. 15.
    Rong C, Lu T, Liu S (2014) Dissecting molecular descriptors into atomic contributions in density functional reactivity theory. J Chem Phys 140:024109PubMedCrossRefGoogle Scholar
  16. 16.
    Mulliken RS (1955) Electronic population analysis on LCAO-MO molecular wave functions. II. Overlap populations, bond orders, and covalent bond energies. J Chem Phys 23:1841CrossRefGoogle Scholar
  17. 17.
    Becke AD, Edgecombe KE (1990) A simple measure of electron localization in atomic and molecular systems. J Chem Phys 92:5397CrossRefGoogle Scholar
  18. 18.
    Fuentealba P, Chamorro E, Santos JC (2007) Understanding and using the electron localization function. In: Toro-Labbé A (ed) Theoretical aspects of chemical reactivity. Elsevier, Amsterdam, p 57CrossRefGoogle Scholar
  19. 19.
    Lu T, Chen F (2011) Meaning and functional form of the electron localization function. Acta Phys Chim Sin 27:2786Google Scholar
  20. 20.
    Lu T, Chen Q (2018) Revealing molecular electronic structure via analysis of valence electron density. Acta Phys Chim Sin 34:503Google Scholar
  21. 21.
    Manzetti S, Lu T (2013) Alternant conjugated oligomers with tunable and narrow HOMO-LUMO gaps as sustainable nanowires. RSC Adv 3:25881CrossRefGoogle Scholar
  22. 22.
    Manzetti S, Lu T, Behzadi H, Estrafili MD, Thi Le H-L, Vach H (2015) Intriguing properties of unusual silicon nanocrystals. RSC Adv 5:78192CrossRefGoogle Scholar
  23. 23.
    Savin A, Jepsen O, Flad J, Andersen OK, Preuss H, von Schnering HG (1992) Electron localization in solid-state structures of the elements: the diamond structure. Angew Chem Int Ed Engl 31:187CrossRefGoogle Scholar
  24. 24.
    Santos JC, Andres J, Aizman A, Fuentealba P (2004) An aromaticity scale based on the topological analysis of the electron localization function including σ and π contributions. J Chem Theory Comput 1:83CrossRefGoogle Scholar
  25. 25.
    Santos JC, Tiznado W, Contreras R, Fuentealba P (2004) Sigma-Pi separation of the electron localization function and aromaticity. J Chem Phys 120:1670PubMedCrossRefGoogle Scholar
  26. 26.
    Liu S, Rong C, Lu T, Hu H (2018) Identifying strong covalent interactions with pauli energy. J Phys Chem A 122:3087PubMedCrossRefGoogle Scholar
  27. 27.
    Astakhov AA, Tsirelson VG (2014) Spatial localization of electron pairs in molecules using the fisher information density. Chem Phys 435:49CrossRefGoogle Scholar
  28. 28.
    Schmider HL, Becke AD (2000) Chemical content of the kinetic energy density. J Mol Struct (THEOCHEM) 527:51CrossRefGoogle Scholar
  29. 29.
    Tsirelson V, Stash A (2002) Analyzing experimental electron density with the localized-orbital locator. Acta Crystallogr Sect B Struct Sci 58:780CrossRefGoogle Scholar
  30. 30.
    Jacobsen H (2013) Bond descriptors based on kinetic energy densities reveal regions of slow electrons—another look at aromaticity. Chem Phys Lett 582:144CrossRefGoogle Scholar
  31. 31.
    Giambiagi M, de Giambiagi M, Mundim K (1990) Definition of a multicenter bond index. Struct Chem 1:423CrossRefGoogle Scholar
  32. 32.
    Kar T, Sánchez Marcos E (1992) Three-center four-electron bonds and their indices. Chem Phys Lett 192:14CrossRefGoogle Scholar
  33. 33.
    Ponec R, Mayer I (1997) Investigation of some properties of multicenter bond indices. J Phys Chem A 101:1738CrossRefGoogle Scholar
  34. 34.
    Yu D, Rong C, Lu T, De Proft F, Liu S (2018) Baird’s rule in substituted fulvene derivatives: an information-theoretic study on triplet-state aromaticity and antiaromaticity. ACS Omega 3:18370PubMedPubMedCentralCrossRefGoogle Scholar
  35. 35.
    Matito E (2016) An electronic aromaticity index for large rings. Phys Chem Chem Phys 18:11839PubMedCrossRefPubMedCentralGoogle Scholar
  36. 36.
    Mayer I (1983) Charge, bond order and valence in the AB initio SCF theory. Chem Phys Lett 97:270CrossRefGoogle Scholar
  37. 37.
    Matito E, Poater J, Solà M, Duran M, Salvador P (2005) Comparison of the AIM delocalization index and the mayer and fuzzy atom bond orders. J Phys Chem A 109:9904PubMedCrossRefPubMedCentralGoogle Scholar
  38. 38.
    Lu T, Chen F (2012) Multiwfn: a multifunctional wavefunction analyzer. J Comput Chem 33:580PubMedPubMedCentralCrossRefGoogle Scholar
  39. 39.
    Lu T, Chen F (2011) Calculation of molecular orbital composition. Acta Chim Sin 69:2393Google Scholar
  40. 40.
    Ros P, Schuit GCA (1966) Molecular orbital calculations on copper chloride complexes. Theor Chem Acc 4:1CrossRefGoogle Scholar
  41. 41.
    Szabo A, Ostlund NS (1989) Modern quantum chemistry. Dover Publications, New YorkGoogle Scholar
  42. 42.
    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 90:4916CrossRefGoogle Scholar
  43. 43.
    Edmiston C, Ruedenberg K (1963) Localized atomic and molecular orbitals. Rev Mod Phys 35:457CrossRefGoogle Scholar
  44. 44.
    Reed AE, Schleyer PVR (1990) Chemical bonding in hypervalent molecules the dominance of ionic bonding and negative hyperconjugation over d-orbital participation. J Am Chem Soc 112:1434CrossRefGoogle Scholar
  45. 45.
    Jensen F (2007) Introduction to computational chemistry. Wiley, West SussexGoogle Scholar
  46. 46.
    Foster JM, Boys SF (1960) Canonical configurational interaction procedure. Rev Mod Phys 32:300CrossRefGoogle Scholar
  47. 47.
    Martin RL (2003) Natural transition orbitals. J Chem Phys 118:4775CrossRefGoogle Scholar
  48. 48.
    Weinhold F (1998) Natural bond orbital methods. In: Schleyer PVR (ed) Encyclopedia of computational chemistry, vol 2. Wiley, West Sussex, p 1792Google Scholar
  49. 49.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Petersson GA, Nakatsuji H, Li X, Caricato M, Marenich AV, Bloino J, Janesko BG, Gomperts R, Mennucci B, Hratchian HP, Ortiz JV, Izmaylov AF, Sonnenberg JL, Ding WF, Lipparini F, Egidi F, Goings J, Peng B, Petrone A, Henderson T, Ranasinghe D, Zakrzewski VG, Gao J, Rega N, Zheng G, Liang W, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Throssell K, Montgomery JA Jr, Peralta JE, Ogliaro F, Bearpark MJ, Heyd JJ, Brothers EN, Kudin KN, Staroverov VN, Keith TA, Kobayashi R, Normand J, Raghavachari K, Rendell AP, Burant JC, Iyengar SS, Tomasi J, Cossi M, Millam JM, Klene M, Adamo C, Cammi R, Ochterski JW, Martin RL, Morokuma K, Farkas O, Foresman JB, Fox DJ (2016) Gaussian 16. Wallingford, CTGoogle Scholar
  50. 50.
    Hariharan PC, Pople JA (1973) The influence of polarization functions on molecular orbital hydrogenation energies. Theor Chem Acc 28:213CrossRefGoogle Scholar
  51. 51.
    Stephens PJ, Devlin FJ, Chabalowski CF, Frisch MJ (1994) Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J Phys Chem 98:11623CrossRefGoogle Scholar
  52. 52.
    Humphrey W, Dalke A, Schulten K (1996) VMD: visual molecular dynamics. J Mol Graph 14:33PubMedCrossRefGoogle Scholar
  53. 53.
    Deng Y, Yu D, Cao X, Liu L, Rong C, Lu T, Liu S (2018) Structure, aromaticity and reactivity of corannulene and its analogues: a conceptual density functional theory and density functional reactivity theory study. Mol Phys 116:956CrossRefGoogle Scholar
  54. 54.
    Perdew JP (1986) Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys Rev B 33:8822CrossRefGoogle Scholar
  55. 55.
    Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38:3098CrossRefGoogle Scholar
  56. 56.
    Andrae D, Häußermann U, Dolg M, Stoll H, Preuß H (1990) Energy-adjustedab initio pseudopotentials for the second and third row transition elements. Theor Chim Acta 77:123CrossRefGoogle Scholar
  57. 57.
    Sedlak R, Janowski T, Pitoňák M, Řezáč J, Pulay P, Hobza P (2013) Accuracy of quantum chemical methods for large noncovalent complexes. J Chem Theory Comput 9:3364PubMedPubMedCentralCrossRefGoogle Scholar
  58. 58.
    Johnson ER, Keinan S, Mori-Sánchez P, Contreras-García J, Cohen AJ, Yang W (2010) Revealing noncovalent interactions. J Am Chem Soc 132:6498PubMedPubMedCentralCrossRefGoogle Scholar
  59. 59.
    For manual corresponding to Multiwfn Version 3.7, see section 4.100.22 on how to perform analyses similar to this work. The steps of realizing topology analysis of ELF-π is illustrated in section 4.5.3. The way of rendering isosurface maps by VMD program based on the data calculated by Multiwfn is introduced in section 4.A.14. The manual is freely available at Accessed on 30 Sep 2019

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Beijing Kein Research Center for Natural SciencesBeijingPeople’s Republic of China

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