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The dichotomy between chemical concepts and numbers after almost 100 years of quantum chemistry: conceptual density functional theory as a case study

  • Paul Geerlings ORCID logo , Bin Wang ORCID logo and Frank De Proft ORCID logo EMAIL logo
Published/Copyright: August 6, 2025
Pure and Applied Chemistry
From the journal Pure and Applied Chemistry Volume 97 Issue 10

Abstract

The dichotomy whether in Quantum Chemistry insight and numbers are to be placed on equal footing is situated in a historical perspective starting from Coulson’s famous quote “Give us insight, not numbers” to Neese’s recent adaptation “Give us insights and numbers”. In parallel, the problem of the chemical interpretation of complex computational results in terms of classical chemical concepts is addressed starting from Mulliken’s quote that “the more accurate the calculations became the more concepts tend to vanish in the air”. Conceptual Density Functional Theory is one of the techniques which avoids the latter issue by its density based approach which can be applied to computational results of any level of sophistication. The crucial role of response functions of the energy E with respect to perturbations in the number of electrons N and/or external potential v(r) is thereby highlighted. However a new confrontation “insight versus numbers” appears. When gradually refining the evaluation of these descriptors, in order to pass from a qualitative to a quantitative level, it turns out from our previous studies that either minor influences show up, or that also fundamental issues may arise hidden in the definition and the physical background of the response function. The derivative discontinuity of the E vs. N curve hereby plays a fundamental role. This issue is documented scrutinizing a series of recent studies on the analytical evaluation of the three second order response functions: the linear response function, the Fukui function and the chemical hardness. They are shown to behave in a fundamentally different way under refinement. For the linear response function, the pure second order v functional derivative of the energy, no fundamental problems arise: when passing to a full analytical evaluation an increasing level of complexity of the equations is observed leading however to a smooth convergence. In the Fukui function case, involving a mixed N and v energy derivative, the issue with the E = E(N) curve and N derivative can be circumvented by directly deriving the electronic chemical potential with respect to v using a Maxwell type relation. Finally for the chemical hardness involving the pure second order N derivative, a fundamental problem arises due to the derivative discontinuity when refining the venerable Parr-Pearson parabolic E = E(N) curve. It forces us to stick to its result identifying the hardness as the (band) gap. On the other hand the analytical expression yields a “condition” that Density Functional Approximations should obey. It is shown how its implementation leads to a straightforward estimate of their delocalization error, on the road for further improvement of DFAs. The inclusion of temperature may be a way out for further refining the chemical hardness and all other response functions involving second or higher order N derivatives, the simplest case being the dual descriptor. Overall this evolution reflects the basic characteristics Löwdin’s accuracy vs. refinement graph.

Keywords: Conceptual density functional theory (CDFT); concepts vs. numbers; quantum chemistry; quantum science and technology; reactivity
Corresponding author: Frank De Proft, Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050, Brussels, Belgium, e-mail:
Article note: A collection of invited papers to celebrate the UN’s proclamation of 2025 as the International Year of Quantum Science and Technology.

Funding source: Chinese Scholarship Council

Award Identifier / Grant number: 202106720017

Funding source: Vrije Universiteit Brussel

Award Identifier / Grant number: SRP73

Acknowledgments

The authors wholeheartedly acknowledge and thank their close collaborators on recent work on the development of the analytical conceptual DFT: Prof. Paul W. Ayers (Department of Chemistry and Chemical Biology, Mc Master University, Canada), Prof. Farnaz Heidar-Zadeh (Department of Chemistry, Queen’s University, Canada), Prof. Shubin Liu (Research Computing Center and Department of Chemistry, University of North Carolina, Chapel Hill, United States) and Prof. Christian Van Alsenoy (Chemistry Department, University of Antwerp, Antwerp, Belgium). The authors are also grateful to Prof. Weitao Yang (Department of Chemistry and Department of Physics, Duke University, United States), for the collaboration on this topic at an earlier stage. BW also wants to thank Prof. Yang for the private conversations about the hardness condition. FDP acknowledges support of the Vrije Universiteit Brussel through a Strategic Research Program awarded to his research group. BW acknowledges the support from Chinese Scholarship Council (No. 202106720017).

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: FDP acknowledges support of the Vrije Universiteit Brussel through a Strategic Research Program awarded to his research group. BW acknowledges the support from Chinese Scholarship Council (No. 202106720017).

  7. Data availability: Not applicable.

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Received: 2025-05-15
Accepted: 2025-07-18
Published Online: 2025-08-06
Published in Print: 2025-10-27

© 2025 IUPAC & De Gruyter

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  9. The quantum chemistry revolution and the instrumental revolution as evidenced by the Nobel Prizes in chemistry
  10. Influence of symmetry on the second-order NLO properties: insights from the few state approximations
  11. The dichotomy between chemical concepts and numbers after almost 100 years of quantum chemistry: conceptual density functional theory as a case study
  12. How ‘de facto variational’ are fully iterative, approximate iterative, and quasiperturbative coupled cluster methods near equilibrium geometries?
  13. Electronic structure of methyl radical photodissociation
  14. Bridging experiment and theory: a computational exploration of UMG-SP3 dynamics
  15. Research Articles
  16. O–Li⋯O and C–Li⋯C lithium bonds in small closed shell and open shell systems as analogues of hydrogen bonds
  17. Metal–ligand bonding and noncovalent interactions of mutated myoglobin proteins: a quantum mechanical study
https://doi.org/10.1515/pac-2025-0525
Keywords for this article
Conceptual density functional theory (CDFT); concepts vs. numbers; quantum chemistry; quantum science and technology; reactivity

Articles in the same Issue

  1. Frontmatter
  2. Review Articles
  3. Minimum energy path methods and reactivity for enzyme reaction mechanisms: a perspective
  4. The quantum revolution in enzymatic chemistry: combining quantum and classical mechanics to understand biochemical processes
  5. A quantum chemical perspective of photoactivated biological functions
  6. Does chemistry need more physics?
  7. Rotational dynamics of ATP synthase: mechanical constraints and energy dissipative channels
  8. Transforming dreams into reality: a fairy-tale wedding of chemistry with quantum mechanics
  9. The quantum chemistry revolution and the instrumental revolution as evidenced by the Nobel Prizes in chemistry
  10. Influence of symmetry on the second-order NLO properties: insights from the few state approximations
  11. The dichotomy between chemical concepts and numbers after almost 100 years of quantum chemistry: conceptual density functional theory as a case study
  12. How ‘de facto variational’ are fully iterative, approximate iterative, and quasiperturbative coupled cluster methods near equilibrium geometries?
  13. Electronic structure of methyl radical photodissociation
  14. Bridging experiment and theory: a computational exploration of UMG-SP3 dynamics
  15. Research Articles
  16. O–Li⋯O and C–Li⋯C lithium bonds in small closed shell and open shell systems as analogues of hydrogen bonds
  17. Metal–ligand bonding and noncovalent interactions of mutated myoglobin proteins: a quantum mechanical study
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