Using topology for understanding your computational results
-
Ángel Martín Pendás
und
Julia Contreras-García
Abstract
The integration of quantum theory (QT) into chemistry has significantly enhanced computational accuracy, yet challenges remain in translating quantum mechanical results into intuitive chemical concepts. Traditional atomic and molecular models, while empirically effective, lack direct representation in Hilbert space, leading to ambiguities in chemical interpretation. Here, we present a summary of topological methods as a bridge between QT and chemical reasoning, focusing on the quantum theory of atoms in molecules (QTAIM) and the electron localization function (ELF). These approaches provide rigorous frameworks for defining atomic and bonding regions, enabling additive decompositions of quantum mechanical observables. By analyzing critical points of the electron density and other scalar fields, we demonstrate how the QTAIM and the ELF offer complementary insights into molecular bonding. As a case study, we examine the electronic structure of carbon suboxide (C3O2), revealing that a combined QTAIM-ELF approach resolves discrepancies between two bonding descriptions.
Funding source: Emergence-SU
Award Identifier / Grant number: H2Ox S23JR31014
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: Fisciency S23JRAR060
Award Identifier / Grant number: TcPredictor S22JRAR036
Funding source: MICIU
Award Identifier / Grant number: PID2021-122763NB-I00
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None used.
-
Conflict of interest: The author states no conflict of interest.
-
Research funding: JCG thanks ANR TcPredictor S22JRAR036, ANR Fisciency S23JRAR060 and Emergence-SU H2Ox S23JR31014 for funding. AMP thanks grant PID2021-122763NB-I00 funded by the Spanish MICIU (https://dx.doi.org/10.13039/501100011033) and by “ERDF A way of making Europe”, for financial support.
-
Data availability: All data are contained in the manuscript.
References
1. Dalton, J. A New System of Chemical Philosophy. S. Russell, for R. Bickerstaff: Manchester, 1808.10.5479/sil.324338.39088000885681Suche in Google Scholar
2. Lewis, G. N. The Atom and the Molecule. J. Am. Chem. Soc. 1916, 38, 762–785; https://doi.org/10.1021/ja02261a002.Suche in Google Scholar
3. Carleo, G.; Troyer, M. Solving the Quantum Many-Body Problem with Artificial Neural Networks. Science 2017, 355 (6325), 602–606; https://doi.org/10.1126/science.aag2302.Suche in Google Scholar PubMed
4. Primas, H. Chemistry, Quantum Mechanics and Reductionism. Lecture Notes in Chemistry, 1981 edition; Springer: Berlin, Germany, 2013.10.1007/978-3-662-11314-1Suche in Google Scholar
5. Scerri, E.; Fisher, G., Eds. Essays in the Philosophy of Chemistry; Oxford University Press: New York, NY, 2016.10.1093/oso/9780190494599.001.0001Suche in Google Scholar
6. Seifert, V. A. The Chemical Bond is a Real Pattern. Philos. Sci. 2022, 90 (2), 269–287; https://doi.org/10.1017/psa.2022.17.Suche in Google Scholar
7. Mulliken, R. S. Molecular Orbital Theory and the Electronic Structure of Molecules. Nobel Lect. 1966, 188–212.Suche in Google Scholar
8. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. B 1964, 136, 864–871; https://doi.org/10.1103/physrev.136.b864.Suche in Google Scholar
9. Riess, J.; Münch, W. The Theorem of Hohenberg and Kohn for Subdomains of a Quantum System. Theor. Chim. Acta 1981, 58 (4), 295–300; https://doi.org/10.1007/bf02426905.Suche in Google Scholar
10. Mezey, P. A. U. L. G. The Holographic Electron Density Theorem and Quantum Similarity Measures. Mol. Phys. 1999, 96 (2), 169–178; https://doi.org/10.1080/00268979909482950.Suche in Google Scholar
11. Bader, R. F. W.; Henneker, W. H.; Cade, P. E. Molecular Charge Distributions and Chemical Binding. J. Chem. Phys. 1967, 46, 3341–3363; https://doi.org/10.1063/1.1841222.Suche in Google Scholar
12. Martín Pendás, Á.; Contreras-García, J. Topological Approaches to the Chemical Bond; Springer International Publishing: Cham, 2023.10.1007/978-3-031-13666-5Suche in Google Scholar
13. Chérif, F. M. Special Issue: Philosophical Aspects and Implications of the Quantum Theory of Atoms in Molecules (Qtaim). Found. Chem. 2013, 15 (3), 245–251; https://doi.org/10.1007/s10698-013-9194-0.Suche in Google Scholar
14. Berlin, T. Binding Regions in Diatomic Molecules. J. Chem. Phys. 1950, 19, 208–213; https://doi.org/10.1063/1.1748161.Suche in Google Scholar
15. Koch, D.; Pavanello, M.; Shao, X.; Ihara, M.; Ayers, P. W.; Matta, C. F.; Jenkins, S.; Manzhos, S. The Analysis of Electron Densities: From Basics to Emergent Applications. Chem. Rev. 2024, 124 (22), 12661–12737; https://doi.org/10.1021/acs.chemrev.4c00297.Suche in Google Scholar PubMed
16. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford Science Publications: Oxford, UK, 1990.10.1093/oso/9780198551683.001.0001Suche in Google Scholar
17. Becke, A. D.; Edgecombe, K. E. A Simple Measure of Electron Localization in Atomic and Molecular Systems. J. Chem. Phys. 1990, 92, 5397–5403; https://doi.org/10.1063/1.458517.Suche in Google Scholar
18. Hirsch, M. W.; Smale, S.; Devaney, R. L. Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd ed.; Elsevier Academic Press: San Diego, 2004.Suche in Google Scholar
19. Coppens, P. Electron Density from X-Ray Diffraction. Annu. Rev. Phys. Chem. 1992, 43, 663–692; https://doi.org/10.1146/annurev.physchem.43.1.663.Suche in Google Scholar
20. Kato, W. A. On the Eigenfunctions of Many-Particle Systems in Quantum Mechanics. Commun. Pure Appl. Math. 1957, 10, 151–177; https://doi.org/10.1002/cpa.3160100201.Suche in Google Scholar
21. Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T. “schrödinger Inequalities” and Asymptotic Behavior of the Electron Density of Atoms and Molecules. Phys. Rev. A 1977, 16 (5), 1782–1785; https://doi.org/10.1103/physreva.16.1782.Suche in Google Scholar
22. Keith, T. A. The AIMAll Program, 2015. Avalaible from: http://aim.tkgristmill.com.Suche in Google Scholar
23. Menéndez-Herrero, M.; Francisco, E.; Martín Pendás, Á. Linnett is Back: Chemical Bonding through the Lens of Born Maxima. J. Chem. Theory Comput. 2025, 21 (5), 2448–2461; https://doi.org/10.1021/acs.jctc.4c01785.Suche in Google Scholar PubMed
24. Bader, R. F. W.; Stephens, M. E. Spatial Localization of the Electronic Pair and Number Distributions in Molecules. J. Am. Chem. Soc. 1975, 97, 7391–7399; https://doi.org/10.1021/ja00859a001.Suche in Google Scholar
25. Francisco, E.; Martín Pendás, A.; Blanco, M. A. Edf: Computing Electron Number Probability Distribution Functions in Real Space from Molecular Wave Functions. Comput. Phys. Commun. 2008, 178 (8), 621–634; https://doi.org/10.1016/j.cpc.2007.11.015.Suche in Google Scholar
26. Silvi, P. D. B.; Jolly, L. H.; D’Arco, P. Pseudopotential Periodic Hartree-Fock Study of the Cristobalite to Stishovite Phase Transition. Theochem. J. Mol. Struct. 1992, 92, 1–9; https://doi.org/10.1016/0166-1280(92)87031-t.Suche in Google Scholar
27. Weizsäcker, C. F. V. Zur theorie der kernmassen. Zeitschrift für Physik 1935, 96 (7–8), 431–458; https://doi.org/10.1007/bf01337700.Suche in Google Scholar
28. Kohout, M.; Savin, A. Atomic Shell Structure and Electron Numbers. Int. J. Quantum Chem. 1996, 60, 875–882; https://doi.org/10.1002/(sici)1097-461x(1996)60:4<875::aid-qua10>3.0.co;2-4.10.1002/(SICI)1097-461X(1996)60:4<875::AID-QUA10>3.0.CO;2-4Suche in Google Scholar
29. Silvi, B.; Savin, A. Classification of Chemical Bonds Based on Topological Analysis of Electron Localization Functions. Nature 1994, 371, 683–686; https://doi.org/10.1038/371683a0.Suche in Google Scholar
30. Silvi, B. The Synaptic Order: A Key Concept to Understand Multicenter Bonding. J. Molec. Struct. 2002, 614, 3–10; https://doi.org/10.1016/s0022-2860(02)00231-4.Suche in Google Scholar
31. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular Geometry; Allyn & Bacon: Boston, 1991.Suche in Google Scholar
32. Michalski, M.; Gordon, A. J.; Berski, S. Topological Analysis of the Electron Localisation Function (Elf) Applied to the Electronic Structure of Oxaziridine: the Nature of N–O Bond. Struct. Chem. 2019, 30 (6), 2181–2189; https://doi.org/10.1007/s11224-019-01407-9.Suche in Google Scholar
33. Diels, O.; Wolf, B. Ueber das kohlensuboxyd. i. Berichte der deutschen chemischen Gesellschaft 1906, 39 (1), 689–697; https://doi.org/10.1002/cber.190603901103.Suche in Google Scholar
34. Jensen, P.; Johns, J. W. C. The Infrared Spectrum of Carbon Suboxide in the V6 Fundamental Region: Experimental Observation and Semirigid Bender Analysis. J. Mol. Spectrosc. 1986, 118 (1), 248–266; https://doi.org/10.1016/0022-2852(86)90239-0.Suche in Google Scholar
35. Zhao, L.; Chai, C.; Petz, W.; Frenking, G. Carbones and Carbon Atom as Ligands in Transition Metal Complexes. Molecules 2020, 25 (21), 4943; https://doi.org/10.3390/molecules25214943.Suche in Google Scholar PubMed PubMed Central
Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/pac-2025-0466).
© 2025 IUPAC & De Gruyter
