Skip to main content
SpringerLink
Log in

On Quantum Chemical Topology

  • Chapter
  • First Online:
Applications of Topological Methods in Molecular Chemistry

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 22))

Abstract

Quantum Chemical Topology (QCT) is a branch of theoretical chemistry that uses the language of dynamical systems (e.g. attractor, basin, homeomorphism, gradient path/phase curve, separatrix, critical points) to partition chemical systems and characterise them via associated quantitative properties. This methodology can be applied to a variety of quantum mechanical functions, the oldest and most documented one being the electron density. We define and discuss the topological atom, and justify the name topology. Then we define the quantum atom without reference to the topological atom. Subsequently, it turns out that each topological atom is a quantum atom, a property that enables the construction of a topologically inspired force field called QCTFF. We briefly discuss the four primary energy contributions governing this force field under development, and how the machine learning method kriging captures the variation in these energies due to geometrical change. Finally, in a more philosophical style, we advocate falsification in the area of chemical interpretation by means of quantum mechanical tools, introducing the concept of a non-question.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 349.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 449.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 449.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bader RFW (1991) A quantum-theory of molecular-structure and its applications. Chem Rev 91:893–928

    Article  CAS  Google Scholar 

  2. Popelier PLA (2014) The quantum theory of atoms in molecules. In: Frenking G, Shaik S (eds) The nature of the chemical bond revisited, Chapter 8. Wiley, pp 271–308

    Google Scholar 

  3. Bader RFW (1985) Atoms in molecules. Acc Chem Res 18:9–15

    Article  CAS  Google Scholar 

  4. Bader RFW (1990) Atoms in molecules. A quantum theory. Oxford Univ. Press, Oxford, Great Britain

    Google Scholar 

  5. Popelier PLA (2000) Atoms in molecules. An introduction. Pearson Education, London

    Google Scholar 

  6. Matta CF, Boyd RJ (2007) The quantum theory of atoms in molecules. From solid state to DNA and drug design. Wiley, Weinheim

    Book  Google Scholar 

  7. Popelier PLA (2012) Quantum chemical topology: on descriptors, potentials and fragments. In: Banting L, Clark T (eds) Drug design strategies: computational techniques and applications, vol 20, Chapter 6. Roy Soc Chem, pp 120–163

    Google Scholar 

  8. Popelier PLA, Aicken FM (2003) Atomic properties of amino acids: computed atom types as a guide for future force field design. Chem Phys Chem 4:824–829

    CAS  Google Scholar 

  9. Popelier PLA (2005) Quantum chemical topology: on bonds and potentials. In: Wales DJ (ed) Structure and bonding. intermolecular forces and clusters, vol 115. Springer, Heidelberg, pp 1–56

    Google Scholar 

  10. Popelier PLA, Brémond ÉAG (2009) Geometrically faithful homeomorphisms between the electron density and the bare nuclear potential. Int J Quant Chem 109:2542–2553

    Article  CAS  Google Scholar 

  11. Koritsanszky TS, Coppens P (2001) Chemical applications of X-ray charge-density analysis. Chem Rev 1583–1627

    Google Scholar 

  12. Bader RFW, Essen H (1984) The characterization of atomic interactions. J Chem Phys 80:1943–1960

    Article  CAS  Google Scholar 

  13. Malcolm NOJ, Popelier PLA (2003) An improved algorithm to locate critical points in a 3D scalar field as implemented in the program MORPHY. J Comp Chem 24:437–442

    Article  CAS  Google Scholar 

  14. Popelier PLA (2012) New insights in atom-atom interactions for future drug design. Curr Top Med Chem 12:1924–1934

    Article  CAS  Google Scholar 

  15. Bader RFW, Anderson SG, Duke AJ (1979) Quantum topology of molecular charge distributions. 1. J Am Chem Soc 101:1389–1395

    Article  CAS  Google Scholar 

  16. Collard K, Hall GG (1977) Orthogonal trajectories of the electron density. Int J Quant Chem 12:623–637

    Article  CAS  Google Scholar 

  17. Thom R (1975) Structural stability and morphogenesis; english ed. Benjamin: Reading, MA

    Google Scholar 

  18. Popelier PLA (1996) On the differential geometry of interatomic surfaces. Can J Chem 74:829–838

    Article  CAS  Google Scholar 

  19. Bader RFW, Beddall PM (1972) Virial field relationship for molecular charge distributions and the spatial partitioning of molecular properties. J. Chem. Phys. 56:3320–3329

    Article  CAS  Google Scholar 

  20. Bader RFW, MacDougall PJ, Lau CDH (1984) Bonded and nonbonded charge concentrations and their relation to molecular-geometry and reactivity. J Am Chem Soc 106:1594–1605

    Article  CAS  Google Scholar 

  21. Bader RFW, Gillespie RJ, MacDougall PJ (1988) A physical basis for the VSEPR model of molecular geometry. J Am Chem Soc 110:7329–7336

    Article  CAS  Google Scholar 

  22. Popelier PLA (2000) On the full topology of the Laplacian of the electron density. Coord Chem Rev 197:169–189

    Article  CAS  Google Scholar 

  23. Malcolm NOJ, Popelier PLA (2003) The full topology of the Laplacian of the electron density: scrutinising a physical basis for the VSEPR model. Faraday Discuss 124:353–363

    Article  CAS  Google Scholar 

  24. Tal Y, Bader RFW, Erkku J (1980) Structural homeomorphism between the electron density and the nuclear potential of a molecular system. Phys Rev A 21:1–11

    Article  CAS  Google Scholar 

  25. Becke AD, Edgecombe KE (1990) A simple measure of electron localization in atomic and molecular systems. J Chem Phys 92:5397–5403

    Article  CAS  Google Scholar 

  26. Silvi B, Savin A (1994) Classification of chemical bonds based on topological analysis of electron localization functions. Nature(London) 371:683–686

    Google Scholar 

  27. Polo V, Andres J, Berski S, Domingo LR, Silvi B (2008) Understanding reaction mechanisms in organic chemistry from catastrophe theory applied to the electron localization function topology. J Phys Chem A 112:7128–7136

    Article  CAS  Google Scholar 

  28. Naray-Szabo G, Ferenczy GG (1995) Molecular electrostatics. Chem Rev 95:829–847

    Article  CAS  Google Scholar 

  29. Gadre SR, Kulkarni SA, Shrivastava IH (1992) Molecular electrostatic potentials: a topographical study. J Chem Phys 96:5253–5261

    Article  CAS  Google Scholar 

  30. Gadre SR, Shrivastava IH (1991) Shapes and sizes of molecular aniosn via topographical analysis of electrostatic potential. J Chem Phys 94:4384–4391

    Article  CAS  Google Scholar 

  31. Balanarayan P, Gadre SR (2003) Topography of molecular scalar fields. I. algorithm and poincare-hopf relation. J Chem Phys 119:5037–5043

    Article  CAS  Google Scholar 

  32. Balanarayan P, Kavathekar R, Gadre SR (2007) electrostatic potential topography for exploring electronic reorganizations in 1,3 dipolar cycloadditions. J Phys Chem A 111:2733–2738

    Article  CAS  Google Scholar 

  33. Aray Y, Rodriguez J, Coll S, Rodrıguez-Arias EN, Vega D (2005) Nature of the lewis acid sites on molybdenum and ruthenium sulfides: an electrostatic potential study. J Phys Chem B 109:23564–23570

    Article  CAS  Google Scholar 

  34. Tsirelson VG, Avilov AS, Lepeshov GG, Kulygin AK, Stahn J, Pietsch U, Spence JCH (2001) Quantitative analysis of the electrostatic potential in rock-salt crystals using accurate electron diffraction data. J Phys Chem B 105:5068–5074

    Article  CAS  Google Scholar 

  35. Keith TA, Bader RFW, Aray Y (1996) Structural homeomorphism between the electron density and the virial field. Int J Quant Chem 57:183–198

    Article  CAS  Google Scholar 

  36. Keith TA, Bader RFW (1993) Topological analysis of magnetically induced molecular current distributions. J Chem Phys 99:3669–3682

    Article  CAS  Google Scholar 

  37. Cioslowski J, Liu GH (1999) Topology of electron-electron interactions in atoms and molecules. II. The correlation cage. J Chem Phys 110:1882–1887

    Article  CAS  Google Scholar 

  38. Pendas AM, Hernandez-Trujillo J (2012) The Ehrenfest force field: topology and consequences for the definition of an atom in a molecule. J Chem Phys 137:134101

    Article  Google Scholar 

  39. Dillen J (2015) The Topology of the Ehrenfest force density revisited. a different perspective based on slater-type orbitals jan. J Comput Chem. doi:10.1002/jcc.23869

    Google Scholar 

  40. Popelier PLA, Kosov DS (2001) Atom-atom partitioning of intramolecular and intermolecular Coulomb energy. J Chem Phys 114:6539–6547

    Article  CAS  Google Scholar 

  41. Blanco MA, Pendas AM, Francisco E (2005) Interacting quantum atoms: a correlated energy decomposition scheme based on the quantum theory of atoms in molecules. J Chem Theor Comput 1:1096–1109

    Article  CAS  Google Scholar 

  42. Pendas AM, Francisco E, Blanco MA, Gatti C (2007) Bond paths as privileged exchange channels. Chem Eur J 13:9362–9371

    Article  CAS  Google Scholar 

  43. Webster B (1990) Chemical bonding theory. Blackwell, Oxford

    Google Scholar 

  44. Reed AE, Curtiss LA, Weinhold F (1988) Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem Rev 88:899–926

    Article  CAS  Google Scholar 

  45. Stone AJ (1981) Distributed multipole analysis, or how to describe a molecular charge-distribution. Chem Phys Lett 83:233–239

    Article  CAS  Google Scholar 

  46. Mulliken RS (1955) Electronic population analysis on LCAO-MO molecular wave functions. I J Chem Phys 23:1833–1840

    Article  CAS  Google Scholar 

  47. Pearson RG (2007) Applying the concepts of density functional theory to simple systems. Int J Quant Chem 108:821–826

    Article  Google Scholar 

  48. Kovacs A, Esterhuysen C, Frenking G (2005) The nature of the chemical bond revisited: an energy-partitioning analysis of nonpolar bonds. Chem Eur J 11:1813–1825

    Article  CAS  Google Scholar 

  49. McWeeny R (1992) Methods of molecular quantum mechanics, 2nd edn. Academic Press, SanDiego

    Google Scholar 

  50. Anderson JSM, Ayers PW, Hernandez JIR (2010) How ambiguous is the local kinetic energy? J Phys Chem A 114:8884–8895

    Article  CAS  Google Scholar 

  51. Fletcher TL, Kandathil SM, Popelier PLA (2014) The prediction of atomic kinetic energies from coordinates of surrounding atoms using kriging machine learning. Theor Chem Acc 133(1499):1–10

    CAS  Google Scholar 

  52. Cohen L (1978) Local kinetic energy in quantum mechanics. J Chem Phys 70:788–799

    Article  Google Scholar 

  53. Bader RFW, Preston HJT (1969) The kinetic energy of molecular charge distributions and molecular stability. Int J Quant Chem 3:327–347

    Article  CAS  Google Scholar 

  54. Nasertayoob P, Shahbazian S (2008) Revisiting the foundations of quantum theory of atoms in molecules (QTAIM): the variational procedure and the zero-flux conditions. Int J Quant Chem 108:1477–1484

    Article  CAS  Google Scholar 

  55. Rafat M, Popelier PLA (2007) Atom-atom partitioning of total (super)molecular energy: the hidden terms of classical force fields. J Comput Chem 28:292–301

    Article  CAS  Google Scholar 

  56. Rafat M, Popelier PLA (2007) Topological atom-atom partitioning of molecular exchange energy and its multipolar convergence. In: Matta CF, Boyd RJ (eds) Quantum theory of atoms in molecules, vol 5. Wiley, Weinheim, pp 121–140

    Google Scholar 

  57. AIMAll: Todd A, Keith TK (2014) Gristmill software, overland Park KS, USA, (aim.tkgristmill.com)

    Google Scholar 

  58. Popelier PLA (2012) Quantum chemical topology: knowledgeable atoms in peptides. AIP Conf Proc 1456:261–268

    Article  CAS  Google Scholar 

  59. Popelier P, Rafat M, Devereux M, Liem SY, Leslie M (2005) Towards a force field via quantum chemical topology. Lect Series Comput Comput Sci 4:1251–1255

    Google Scholar 

  60. Popelier PLA (2012) A generic force field based on quantum chemical topology. In: Gatti C, Macchi P (eds) Modern charge-density analysis, vol 14. Springer, Germany, pp 505–526

    Google Scholar 

  61. Cardamone S, Hughes TJ, Popelier PLA (2014) Multipolar electrostatics. Phys Chem Chem Phys 16:10367–10387

    Article  CAS  Google Scholar 

  62. Ivanov MV, Talipov MR, Timerghazin QD (2015) Genetic algorithm optimization of point charges in force field development: challenges and insights. J Phys Chem A 119:1422–1434

    Article  CAS  Google Scholar 

  63. Rafat M, Popelier PLA (2006) A convergent multipole expansion for 1,3 and 1,4 Coulomb interactions. J Chem Phys 124(144102):1–7

    Google Scholar 

  64. Rafat M, Popelier PLA (2007) Long range behaviour of high-rank topological multipole moments. J Comput Chem 28:832–838

    Article  CAS  Google Scholar 

  65. Popelier PLA, Joubert L, Kosov DS (2001) Convergence of the electrostatic interaction based on topological atoms. J Phys Chem A 105:8254–8261

    Article  CAS  Google Scholar 

  66. Yuan Y, Mills MJL, Popelier PLA (2014) Multipolar electrostatics for proteins: atom-atom electrostatic energies in crambin. J Comp Chem 35:343–359

    Article  CAS  Google Scholar 

  67. Solano CJF, Pendás AM, Francisco E, Blanco MA, Popelier PLA (2010) Convergence of the multipole expansion for 1,2 Coulomb interactions: the modified multipole shifting algorithm. J Chem Phys 132:194110

    Article  CAS  Google Scholar 

  68. Pendas AM, Francisco E, Blanco MA (2006) Binding energies of first row diatomics in the light of the interacting quantum atoms approach. J Phys Chem A 110:12864–12869

    Article  CAS  Google Scholar 

  69. Angyan JG, Loos M, Mayer I (1994) Covalent bond orders and atomic valence indices in the topological theory of atoms in molecules. J Phys Chem 98:5244–5248

    Article  CAS  Google Scholar 

  70. Garcia-Revilla M, Francisco E, Popelier PLA, Martin-Pendas AM (2013) Domain-averaged exchange correlation energies as a physical underpinning for chemical graphs. Chem Phys Chem 14:1211–1218

    CAS  Google Scholar 

  71. Chávez-Calvillo R, García-Revilla M, Francisco E, Martín Pendás A, Rocha-Rinza T (2015) Dynamical correlation within the Interacting quantum atoms method through coupled cluster theory. Comput Theor Chem 1053:90–95

    Google Scholar 

  72. Grimme S (2004) Accurate description of van der Waals complexes by density functional theory including empirical corrections. J Comput Chem 25:1463–1473

    Article  CAS  Google Scholar 

  73. Fletcher T, Davie SJ, Popelier PLA (2014) Prediction of intramolecular polarization of aromatic amino acids using kriging machine learning. J Chem Theory Comput 10:3708–3719

    Google Scholar 

  74. Grimme S, Antony J, Ehrlich S, Krieg H (2010) A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J Chem Phys 132:154104–154122

    Article  Google Scholar 

  75. Badenhoop JK, Weinhold F (1997) Natural bond orbital analysis of steric interactions. J Chem Phys 107:5406–5421

    Article  CAS  Google Scholar 

  76. Verstraelen T, Ayers PW, Van Speybroeck V, Waroquier M (2013) Hirshfeld-E partitioning: AIM charges with an improved trade-off between robustness and accurate electrostatics. J Chem Theory Comput 9:2221–2225

    Google Scholar 

  77. Hirshfeld FL (1977) Bonded-atom fragments for describing molecular charge densities. Theor Chem Acta 44:129–138

    Article  CAS  Google Scholar 

  78. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, Cambridge

    Google Scholar 

  79. Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersran. J Chem Metall Mining Soc South Africa 52:119–139

    Google Scholar 

  80. Kandathil SM, Fletcher TL, Yuan Y, Knowles J, Popelier PLA (2013) Accuracy and tractability of a Kriging model of intramolecular polarizable multipolar electrostatics and its application to histidine. J Comput Chem 34:1850–1861

    Article  CAS  Google Scholar 

  81. Mills MJL, Popelier PLA (2011) Intramolecular polarisable multipolar electrostatics from the machine learning method Kriging. Comput Theor Chem 975:42–51

    Article  CAS  Google Scholar 

  82. Mills MJL, Popelier PLA (2012) Polarisable multipolar electrostatics from the machine learning method Kriging: an application to alanine. Theor Chem Acc 131:1137–1153

    Article  Google Scholar 

  83. Mills MJL, Hawe GI, Handley CM, Popelier PLA (2013) Unified approach to multipolar polarisation and charge transfer for ions: microhydrated Na+. Phys Chem Chem Phys 15:18249–18261

    Article  CAS  Google Scholar 

  84. Handley CM, Hawe GI, Kell DB, Popelier PLA (2009) Optimal construction of a fast and accurate polarisable water potential based on multipole moments trained by machine learning. Phys Chem Chem Phys 11:6365–6376

    Article  CAS  Google Scholar 

  85. Yuan Y, Mills MJL, Popelier PLA (2014) Multipolar electrostatics based on the Kriging machine learning method: an application to serine. J Mol Model 20:2172–2186

    Article  Google Scholar 

  86. Hughes TJ, Kandathil SM, Popelier PLA (2015) Accurate prediction of polarised high order electrostatic interactions for hydrogen bonded complexes using the machine learning method kriging. Spectrochim Acta A 136:32–41

    Article  CAS  Google Scholar 

  87. Bader RFW (2009) Bond paths are not chemical bonds. J Phys Chem A 113:10391–10396

    Article  CAS  Google Scholar 

  88. Grimme S, Mück-Lichtenfeld C, Erker G, Kehr G, Wang H, Beckers H, Willner H (2009) When do interacting atoms form a chemical bond? spectroscopic measurements and theoretical analyses of dideuteriophenanthrene. Angew Chem Int Ed 48:2592–2595

    Article  CAS  Google Scholar 

  89. Pendas AM, Blanco MA, Francisco E (2009) Steric repulsions, rotation barriers, and stereoelectronic effects: a real space perspective. J Comput Chem 30:98–109

    Article  CAS  Google Scholar 

  90. Ayers PL, Boyd RJ, Bultinck P, Caffarel M, Carbó-Dorca R, Causá M, Cioslowski J, Contreras-Garcia J, Cooper DL, Coppens P, Gatti C, Grabowsky S, Lazzeretti P, Macchi P, Pendás AM, Popelier PLA, Ruedenberg K, Rzepa H, Savin A, Sax A, Schwarz WEH, Shahbazian S, Silvi S, Solà M, Tsirelson V (2015) Six questions on topology in theoretical chemistry. Comput Theor Chem 1053:2–16

    Article  CAS  Google Scholar 

Download references

Acknowledgments

I thank the EPSRC for the award of an Established Career Fellowship.

Author information

Authors and Affiliations

  1. Manchester Institute of Biotechnology (MIB), 131 Princess Street, Manchester, M1 7DN, UK

    Paul L A Popelier

  2. School of Chemistry, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    Paul L A Popelier

Authors
  1. Paul L A Popelier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul L A Popelier .

Editor information

Editors and Affiliations

  1. LCC-CNRS, Toulouse, France

    Remi Chauvin

  2. LCC-CNRS, Toulouse, France

    Christine Lepetit

  3. Théorique, UMR 761, Laboratoire de Chimie, Paris, France

    Bernard Silvi

  4. CNRS/UPM, MONARIS UMR 8233 (LADIR + LM2N), Paris, France

    Esmail Alikhani

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Popelier, P.L.A. (2016). On Quantum Chemical Topology. In: Chauvin, R., Lepetit, C., Silvi, B., Alikhani, E. (eds) Applications of Topological Methods in Molecular Chemistry. Challenges and Advances in Computational Chemistry and Physics, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-29022-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation