{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T10:50:23Z","timestamp":1767869423400,"version":"3.49.0"},"reference-count":86,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T00:00:00Z","timestamp":1767657600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["22203071"],"award-info":[{"award-number":["22203071"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["22303022"],"award-info":[{"award-number":["22303022"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the High-Level Talent Special Support Plan, the Anhui Province Innovation Plan for Science and Technolog","award":["202423r06050002"],"award-info":[{"award-number":["202423r06050002"]}]},{"DOI":"10.13039\/501100000038","name":"NSERC","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001804","name":"Canada Research Chairs","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001804","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the Digital Research Alliance of Canada"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Accurately and efficiently predicting redox potentials and Hammett constants using simple density-based functions derived from information-theoretic approach (ITA) quantities remains an unresolved challenge. In this work, we employ two recently proposed protocols, DL(ITA) (deep learning) and QML(ITA) (quantum machine learning), to a broad range of quinone derivatives with available experimental data. The molecular electrostatic potential (MEP) at the nucleus of the acidic atom and the sum of valence natural atomic orbital (NAO) energies are used within a linear regression (LR) framework to assess the first redox potentials and Hammett parameters of these quinone derivatives. The DL(ITA) protocol enables the construction of a transferable model trained on quinone derivatives that can be applied to both quinone and non-quinone systems. Interestingly, the QML(ITA) model exhibits superior performance compared to the DL(ITA) approach. Moreover, the structure of the QML(ITA) method suggests that it may be readily implemented on real quantum hardware in the near future.<\/jats:p>","DOI":"10.3390\/e28010067","type":"journal-article","created":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T08:42:58Z","timestamp":1767688978000},"page":"67","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Predicting the Redox Potentials and Hammett Parameters of Quinone Derivatives with the Information-Theoretic Approach"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9870-9697","authenticated-orcid":false,"given":"Mingxin","family":"Xu","sequence":"first","affiliation":[{"name":"School of Pharmaceutical Sciences and Yunnan Key Laboratory of Pharmacology for Natural Products, Kunming Medical University, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1250-2764","authenticated-orcid":false,"given":"Yilin","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON L8S 4M1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-9729-2102","authenticated-orcid":false,"given":"Hui","family":"Li","sequence":"additional","affiliation":[{"name":"School of Life Sciences, Yunnan University, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2605-3883","authenticated-orcid":false,"given":"Paul W.","family":"Ayers","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON L8S 4M1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0880-9686","authenticated-orcid":false,"given":"Dandan","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Pharmaceutical Sciences and Yunnan Key Laboratory of Pharmacology for Natural Products, Kunming Medical University, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0966-9782","authenticated-orcid":false,"given":"Qingchun","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Artificial Intelligence, Hefei Comprehensive National Science Center, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0927-4361","authenticated-orcid":false,"given":"Dongbo","family":"Zhao","sequence":"additional","affiliation":[{"name":"Institute of Biomedical Research, Yunnan University, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3577","DOI":"10.1021\/cr100290v","article-title":"Electrochemical Energy Storage for Green Grid","volume":"111","author":"Yang","year":"2011","journal-title":"Chem. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"11533","DOI":"10.1021\/cr500720t","article-title":"Flow Batteries: Current Status and Trends","volume":"115","author":"Soloveichik","year":"2015","journal-title":"Chem. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1073\/pnas.1520211113","article-title":"Redox potential of the terminal quinone electron acceptor QB in photosystem II reveals the mechanism of electron transfer regulation","volume":"113","author":"Kato","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","unstructured":"Bard, A.J., and Faulkner, L.R. (2001). Electrochemical Methods: Fundamentals and Applications, Wiley. [2nd ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sav\u00e9ant, J.-M. (2006). Elements of Molecular and Biomolecular Electrochemistry, Wiley.","DOI":"10.1002\/0471758078"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1021\/ac60210a007","article-title":"Theory of Stationary Electrode Polarography. Single Scan and Cyclic Methods Applied to Reversible, Irreversible, and Kinetic Systems","volume":"36","author":"Nicholson","year":"1964","journal-title":"Anal. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3373","DOI":"10.1039\/b504286k","article-title":"Spectroelectrochemistry: The Best of Two Worlds","volume":"38","author":"Kaim","year":"2009","journal-title":"Chem. Soc. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1021\/acs.jctc.1c01040","article-title":"Bridging the Experiment-Calculation Divide: Machine Learning Corrections to Redox Potential Calculations in Implicit and Explicit Solvent Models","volume":"18","author":"Hruska","year":"2022","journal-title":"J. Chem. Theory Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"7166","DOI":"10.1021\/acs.jpca.0c05052","article-title":"Benchmark Study of Electrochemical Redox Potentials Calculated with Semiempirical and DFT Methods","volume":"124","author":"Neugebauer","year":"2020","journal-title":"J. Phys. Chem. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3522","DOI":"10.1021\/ar500268y","article-title":"Redox Potentials and Acidity Constants from Density Functional Theory Based Molecular Dynamics","volume":"47","author":"Cheng","year":"2014","journal-title":"Acc. Chem. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7407","DOI":"10.1021\/jp025853n","article-title":"Computing Redox Potentials in Solution:\u2009 Density Functional Theory as A Tool for Rational Design of Redox Agents","volume":"106","author":"Baik","year":"2002","journal-title":"J. Phys. Chem. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"14566","DOI":"10.1021\/jacs.4c01221","article-title":"Accelerating Computation of Acidity Constants and Redox Potentials for Aqueous Organic Redox Flow Batteries by Machine Learning Potential-Based Molecular Dynamics","volume":"146","author":"Wang","year":"2024","journal-title":"J. Am. Chem. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5037","DOI":"10.1021\/jo100735s","article-title":"Accurate Estimation of the One-Electron Reduction Potentials of Various Substituted Quinones in DMSO and CH3CN","volume":"75","author":"Zhu","year":"2010","journal-title":"J. Org. Chem."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"15903","DOI":"10.1021\/jacs.6b05797","article-title":"Quinone 1 e\u2212 and 2 e\u2212\/2 H+ Reduction Potentials: Identification and Analysis of Deviations from Systematic Scaling Relationships","volume":"138","author":"Huynh","year":"2016","journal-title":"J. Am. Chem. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7227","DOI":"10.1021\/ja0421856","article-title":"Quantum-Chemical Predictions of Absolute Standard Redox Potentials of Diverse Organic Molecules and Free Radicals in Acetonitrile","volume":"127","author":"Fu","year":"2005","journal-title":"J. Am. Chem. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6227","DOI":"10.1021\/acs.jctc.4c00602","article-title":"First and Second Reductions in an Aprotic Solvent: Comparing Computational and Experimental One-Electron Reduction Potentials for 345 Quinones","volume":"20","author":"Elhajj","year":"2024","journal-title":"J. Chem. Theory Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3648","DOI":"10.1021\/jp811250r","article-title":"Estimation of Molecular Acidity via Electrostatic Potential at the Nucleus and Valence Natural Atomic Orbitals","volume":"113","author":"Liu","year":"2009","journal-title":"J. Phys. Chem. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14697","DOI":"10.1021\/jp209540p","article-title":"Modeling Molecular Acidity with Electronic Properties and Hammett Constants for Substituted Benzoic Acids","volume":"115","author":"Huang","year":"2011","journal-title":"J. Phys. Chem. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"164107","DOI":"10.1063\/1.3251124","article-title":"Molecular acidity: A quantitative conceptual density functional theory description","volume":"131","author":"Liu","year":"2009","journal-title":"J. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3802","DOI":"10.1021\/acs.jpca.1c02099","article-title":"Quantifications and Applications of Relative Fisher Information in Density Functional Theory","volume":"125","author":"Wang","year":"2021","journal-title":"J. Phys. Chem. A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhao, D., Liu, S., and Chen, D. (2022). A Density Functional Theory and Information-Theoretic Approach Study of Interaction Energy and Polarizability for Base Pairs and Peptides. Pharmaceuticals, 15.","DOI":"10.20944\/preprints202207.0080.v1"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1039\/D2CP04690C","article-title":"Efficient and accurate density-based prediction of macromolecular polarizabilities","volume":"25","author":"Zhao","year":"2023","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6461","DOI":"10.1021\/acs.jctc.3c00646","article-title":"Accurate and Efficient Prediction of Post-Hartree\u2013Fock Polarizabilities of Condensed-Phase Systems","volume":"19","author":"Zhao","year":"2023","journal-title":"J. Chem. Theory Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhao, D., He, X., Ayers, P.W., and Liu, S. (2023). Excited-State Polarizabilities: A Combined Density Functional Theory and Information-Theoretic Approach Study. Molecules, 28.","DOI":"10.20944\/preprints202302.0116.v1"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2655","DOI":"10.1021\/acs.jctc.3c01415","article-title":"Fragment-Based Deep Learning for Simultaneous Prediction of Polarizabilities and NMR Shieldings of Macromolecules and Their Aggregates","volume":"20","author":"Zhao","year":"2024","journal-title":"J. Chem. Theory Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5997","DOI":"10.1021\/acs.jctc.5c00414","article-title":"Multiproperty Deep Learning of the Correlation Energy of Electrons and the Physicochemical Properties of Molecules","volume":"21","author":"Yuan","year":"2025","journal-title":"J. Chem. Theory Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s12039-025-02434-2","article-title":"Can the FCI Energies\/Properties be Predicted with HF\/DFT Densities?","volume":"137","author":"Zhao","year":"2025","journal-title":"J. Chem. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wang, P., Hu, D., Lu, L., Zhao, Y., Chen, J., Ayers, P.W., Liu, S., and Zhao, D. (2025). Predicting the Post-Hartree\u2013Fock Electron Correlation Energy of Complex Systems with the Information-Theoretic Approach. Molecules, 30.","DOI":"10.20944\/preprints202507.2326.v1"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"8879","DOI":"10.1073\/pnas.97.16.8879","article-title":"Information theory, atoms in molecules, and molecular similarity","volume":"97","author":"Nalewajski","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1007\/s00214-006-0121-5","article-title":"Information Theory, the Shape Function, and the Hirshfeld Atom","volume":"115","author":"Ayers","year":"2006","journal-title":"Theor. Chem. Acc."},{"key":"ref_31","unstructured":"Parr, R.G., and Yang, W. (1989). Density Functional Theory of Atoms and Molecules, Oxford University Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"28700","DOI":"10.1039\/D2CP02827A","article-title":"DFT exchange: Sharing perspectives on the workhorse of quantum chemistry and materials science","volume":"24","author":"Teale","year":"2022","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1017\/S0305004100009580","article-title":"Theory of statistical estimation","volume":"22","author":"Fisher","year":"1925","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_35","unstructured":"Kullback, S. (1997). Information Theory and Statistics, Dover Publications."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"141103","DOI":"10.1063\/1.5124244","article-title":"Identity for Kullback-Leibler divergence in density functional reactivity theory","volume":"151","author":"Liu","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.3866\/PKU.WHXB201509183","article-title":"R\u00e9nyi entropy, Tsallis entropy and Onicescu information energy in density functional reactivity theory","volume":"31","author":"Liu","year":"2015","journal-title":"Acta Phys.-Chim. Sin."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"8028","DOI":"10.1073\/pnas.81.24.8028","article-title":"Transcription of ground-state density-functional theory into a local thermodynamics","volume":"81","author":"Ghosh","year":"1984","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"8057","DOI":"10.1021\/acs.jcim.5c01019","article-title":"Quantum-Embedded Graph Neural Network Architecture for Molecular Property Prediction","volume":"65","author":"Lu","year":"2025","journal-title":"J. Chem. Inf. Model."},{"key":"ref_40","unstructured":"Lu, L., Zhao, Y., Xu, M., Yuan, Y., Xu, T., Li, Y., Chattaraj, P.K., Chen, J., Liu, S., and Ayers, P.W. (2025). Quantum Machine Learning via the Information-Theoretic Representation. J. Chem. Theory Comput., submitted."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"148558","DOI":"10.1016\/j.bbabio.2022.148558","article-title":"The aprotic electrochemistry of quinones","volume":"1863","author":"Prince","year":"2022","journal-title":"Biochim. Biophys. Acta (BBA) Bioenerg."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1021\/cr00002a004","article-title":"A survey of Hammett substituent constants and resonance and field parameters","volume":"91","author":"Hansch","year":"1991","journal-title":"Chem. Rev."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5648","DOI":"10.1063\/1.464913","article-title":"Density-functional thermochemistry. III. The role of exact exchange","volume":"98","author":"Becke","year":"1993","journal-title":"J. Chem. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1103\/PhysRevB.37.785","article-title":"Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density","volume":"37","author":"Lee","year":"1988","journal-title":"Phys. Rev. B"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3297","DOI":"10.1039\/b508541a","article-title":"Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy","volume":"7","author":"Weigend","year":"2005","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"B864","DOI":"10.1103\/PhysRev.136.B864","article-title":"Inhomogeneous Electron Gas","volume":"136","author":"Hohenberg","year":"1964","journal-title":"Phys. Rev."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"A1133","DOI":"10.1103\/PhysRev.140.A1133","article-title":"Self-Consistent Equations Including Exchange and Correlation Effects","volume":"140","author":"Kohn","year":"1965","journal-title":"Phys. Rev."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1021\/acsphyschemau.3c00067","article-title":"Harvesting Chemical Understanding with Machine Learning and Quantum Computers","volume":"4","author":"Liu","year":"2024","journal-title":"ACS Phys. Chem. Au"},{"key":"ref_49","first-page":"223","article-title":"Constructing Exact Density Functionals from the Moments of the Electron Density","volume":"34","author":"Ayers","year":"2002","journal-title":"Acta Phys. Chim. Debrecina"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00894-016-3175-x","article-title":"Evaluating frontier orbital energy and HOMO\/LUMO gap with descriptors from density functional reactivity theory","volume":"23","author":"Huang","year":"2017","journal-title":"J. Mol. Model."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3634","DOI":"10.1021\/acs.jpca.6b01197","article-title":"Information Functional Theory: Electronic Properties as Functionals of Information for Atoms and Molecules","volume":"120","author":"Zhou","year":"2016","journal-title":"J. Phys. Chem. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"23103","DOI":"10.1021\/jacs.4c04670","article-title":"When Do Quantum Mechanical Descriptors Help Graph Neural Networks to Predict Chemical Properties?","volume":"146","author":"Li","year":"2024","journal-title":"J. Am. Chem. Soc."},{"key":"ref_53","first-page":"1","article-title":"NEVAE: A Deep Generative Model for Molecular Graphs","volume":"21","author":"Samanta","year":"2020","journal-title":"J. Mach. Learn. Res."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3468","DOI":"10.1021\/acs.jpclett.3c00246","article-title":"Global Optimization: A Soft Computing Perspective","volume":"14","author":"Pal","year":"2023","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1002\/qua.560030308","article-title":"The kinetic energy of molecular charge distributions and molecular stability","volume":"3","author":"Bader","year":"1969","journal-title":"Int. J. Quantum Chem."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1002\/qua.560140813","article-title":"Studies of the energy density functional approach. I. Kinetic energy","volume":"14","author":"Tal","year":"1978","journal-title":"Int. J. Quantum Chem."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1063\/1.437511","article-title":"Local kinetic energy in quantum mechanics","volume":"70","author":"Cohen","year":"1979","journal-title":"J. Chem. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.1063\/1.447257","article-title":"Representable local kinetic energy","volume":"80","author":"Cohen","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/0009-2614(96)00624-0","article-title":"Uniqueness and Asymptotic Behavior of the Local Kinetic Energy","volume":"258","author":"Yang","year":"1996","journal-title":"Chem. Phys. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1002\/qua.989","article-title":"Local kinetic energy and local temperature in the density-functional theory of electronic structure","volume":"90","author":"Ayers","year":"2002","journal-title":"Int. J. Quantum Chem."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"8884","DOI":"10.1021\/jp1029745","article-title":"How Ambiguous Is the Local Kinetic Energy?","volume":"114","author":"Anderson","year":"2010","journal-title":"J. Phys. Chem. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1016\/0009-2614(86)80233-0","article-title":"Exponential approximation for the density matrix and the Wigner distribution","volume":"129","author":"Berkowitz","year":"1986","journal-title":"Chem. Phys. Lett."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.1021\/cr990029p","article-title":"Conceptual Density Functional Theory","volume":"103","author":"Geerlings","year":"2003","journal-title":"Chem. Rev."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Gatti, C., and Macchi, P. (2012). Charge density and chemical reactions: A unified view from conceptual DFT. Modern Charge-Density Analysis, Springer.","DOI":"10.1007\/978-90-481-3836-4"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"590","DOI":"10.3866\/PKU.WHXB20090332","article-title":"Conceptual Density Functional Theory and Some Recent Developments","volume":"25","author":"Liu","year":"2009","journal-title":"Acta Phys.-Chim. Sin."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s00214-020-2546-7","article-title":"Conceptual density functional theory: Status, prospects, issues","volume":"139","author":"Geerlings","year":"2020","journal-title":"Theor. Chem. Acc."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"244103","DOI":"10.1063\/1.2747247","article-title":"Steric effect: A quantitative description from density functional theory","volume":"126","author":"Liu","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1408","DOI":"10.1039\/C7CP07678A","article-title":"Steric charge","volume":"20","author":"Liu","year":"2018","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1496","DOI":"10.1039\/C6CP06376D","article-title":"Electronic forces as descriptors of nucleophilic and electrophilic regioselectivity and stereoselectivity","volume":"19","author":"Liu","year":"2017","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"e1461","DOI":"10.1002\/wcms.1461","article-title":"Information-Theoretic approach in density functional theory and its recent applications to chemical problems","volume":"10","author":"Rong","year":"2020","journal-title":"Wiley Interdiscip. Rev. Comput. Mol. Sci."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"11191","DOI":"10.1021\/acs.jpclett.2c03165","article-title":"Development and Applications of the Density-Based Theory of Chemical Reactivity","volume":"13","author":"Rong","year":"2022","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1021\/acs.jpca.3c07997","article-title":"Some Recent Advances in Density-Based Reactivity Theory","volume":"128","author":"He","year":"2024","journal-title":"J. Phys. Chem. A"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"8690","DOI":"10.1021\/acs.jpclett.0c02144","article-title":"Homochirality Originates from the Handedness of Helices","volume":"11","author":"Liu","year":"2020","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"8720","DOI":"10.1021\/acs.jpclett.1c02433","article-title":"Principle of Chirality Hierarchy in Three-Blade Propeller Systems","volume":"12","author":"Liu","year":"2021","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF00549096","article-title":"Bonded-atom fragments for describing molecular charge densities","volume":"44","author":"Hirshfeld","year":"1977","journal-title":"Theor. Chim. Acta"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"4219","DOI":"10.1021\/acs.jpca.7b08966","article-title":"Information-Theoretic Ap-proaches to Atoms-in-Molecules: Hirshfeld Family of Partitioning Schemes","volume":"122","author":"Ayers","year":"2018","journal-title":"J. Phys. Chem. A"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1063\/1.454033","article-title":"A multicenter numerical integration scheme for polyatomic molecules","volume":"88","author":"Becke","year":"1988","journal-title":"J. Chem. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Bader, R.F.W. (1990). Atoms in Molecules: A Quantum Theory, Clarendon Press.","DOI":"10.1093\/oso\/9780198551683.001.0001"},{"key":"ref_79","unstructured":"Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J.R., Scalmani, G., Barone, V., Petersson, G.A., and Nakatsuji, H. (2016). Gaussian 16, Version Rev. C.01, Gaussian, Inc."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1002\/jcc.22885","article-title":"Multiwfn: A multifunctional wavefunction analyzer","volume":"33","author":"Lu","year":"2012","journal-title":"J. Comput. Chem."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"082503","DOI":"10.1063\/5.0216272","article-title":"A comprehensive electron wavefunction analysis toolbox for chemists","volume":"161","author":"Lu","year":"2024","journal-title":"Multiwfn. J. Chem. Phys."},{"key":"ref_82","first-page":"8024","article-title":"PyTorch: An Imperative Style, High-Performance Deep Learning Library","volume":"32","author":"Paszke","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_83","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv."},{"key":"ref_84","first-page":"2825","article-title":"Scikit-learn: Machine Learning in Python","volume":"12","author":"Pedregosa","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_85","unstructured":"PyVQNet Development Team (2025, September 11). PyVQNet (Version 2.17.1) [Python Package]. Python Package Index. Available online: https:\/\/pypi.org\/project\/pyvqnet\/."},{"key":"ref_86","unstructured":"Bian, H., Jia, Z., Dou, M., Fang, Y., Li, L., Zhao, Y., Wang, H., Zhou, Z., Wang, W., and Zhu, W. (2023). VQNet 2.0: A New Generation Machine Learning Framework that Unifies Classical and Quantum. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/67\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T05:14:48Z","timestamp":1767849288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/67"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,6]]},"references-count":86,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010067"],"URL":"https:\/\/doi.org\/10.3390\/e28010067","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,6]]}}}