A Diagnostic Analysis of B3LYP Thermochemical Errors: Physical Origins, System Dependence, and Practical Implications

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Hadi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8818002/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract Density functional theory remains a cornerstone of thermochemical modeling; however, its reliability depends critically on the interplay between functional form, basis-set representation, and the underlying physical interactions governing a given system. In this work, a diagnostic assessment of the B3LYP functional is presented, focusing on the identification of systematic error sources rather than on global statistical performance alone. Mean absolute errors reorganized from established benchmark datasets are used as a reference framework to analyze trends associated with molecular size, structural compactness, dispersion interactions, and electron density localization. Global thermochemical statistics show that B3LYP improves substantially over Hartree–Fock methods, yet exhibits only limited and non-monotonic sensitivity to basis-set extension beyond moderate polarization and diffuse functions. A persistent size-dependent error growth is identified for linear hydrocarbons, indicating cumulative deficiencies related to medium-range correlation effects. Branching-sensitive isomerization reactions reveal a pronounced overstabilization of linear isomers, which is significantly—but not completely—mitigated by the inclusion of empirical dispersion corrections, highlighting dispersion as a dominant but non-exclusive error source. In contrast, isomerizations involving nitrogen- and oxygen-containing molecules display strong sensitivity to diffuse basis functions, demonstrating that density localization and polarization effects dominate over dispersion in heteroatom-rich systems. These contrasting behaviors confirm that B3LYP errors arise from multiple, system-dependent physical origins that cannot be resolved by a single correction strategy. Overall, this study establishes a physically motivated diagnostic framework for interpreting B3LYP performance, emphasizing that functional reliability must be evaluated in relation to the dominant interactions of the target chemical system. The resulting classification provides practical guidance for method selection and expectation management in thermochemical applications. DFT B3LYP Thermochemical accuracy Isomerization energies Dispersion corrections Figures Figure 1 Figure 2 Figure 3 Introduction Density functional theory (DFT) has become a central tool in modern computational chemistry, providing a practical balance between accuracy and computational cost that enables routine investigation of thermochemical properties for molecular systems far beyond the reach of high-level wavefunction-based methods [ 1 – 4 ]. Its favourable scaling with system size and its ability to incorporate electron correlation effects at moderate computational expense have made DFT indispensable in studies of organic, biological, and materials systems. Within the broad family of density functionals, hybrid approaches have gained particular prominence due to their partial inclusion of exact exchange, which improves the description of molecular energetics compared to pure generalized gradient approximations (GGAs)[ 5 , 6 ]. Among these, the B3LYP functional has emerged as one of the most widely applied methods in organic thermochemistry. Its enduring popularity does not stem from exceptional accuracy in any single chemical domain, but rather from its perceived robustness and transferability across a diverse range of molecular systems[ 7 – 9 ]. As a result, B3LYP is frequently employed as a default computational choice, often without explicit consideration of its known and well-documented limitations. Extensive benchmarking studies conducted over the past two decades have demonstrated that the apparent reliability of B3LYP is strongly system-dependent[ 7 , 10 , 11 ]. While satisfactory agreement with experimental thermochemical data is commonly reported for small and moderately sized molecules, systematic deviations become increasingly pronounced for larger systems, highly branched hydrocarbons, and molecules dominated by weak intramolecular or medium-range correlation effects[ 10 – 13 ]. Importantly, these deficiencies are often masked when performance is assessed solely through global statistical indicators such as mean absolute errors (MAEs), which average over chemically diverse datasets and obscure physically meaningful trends [ 14 , 15 ]. Attempts to mitigate these shortcomings have primarily focused on technical refinements, including enlargement of one-electron basis sets, empirical adjustment of atomic reference energies, and the incorporation of dispersion corrections into conventional hybrid functionals[ 16 – 19 ]. Although such strategies can reduce numerical errors in selected cases, their effectiveness is highly non-uniform and strongly system-specific. In many instances, numerical improvement does not correspond to a genuine resolution of the underlying physical deficiencies of the exchange–correlation functional, but rather reflects fortuitous error cancellation [ 20 , 21 ]. More fundamentally, conventional benchmarking approaches rarely address a critical conceptual question: whether the observed errors in density functional predictions arise from random numerical artifacts or from systematic, physically grounded deficiencies inherent to the functional form itself[ 22 , 23 ]. Without explicitly distinguishing between these possibilities, method selection remains largely empirical and may fail when extrapolated to chemically complex, larger, or previously unexplored systems. In the present work, we adopt a diagnostic benchmarking perspective aimed at elucidating the physical mechanisms governing the performance and failure of B3LYP-based methods. Rather than introducing new electronic structure calculations, we reorganize established large-scale thermochemical benchmark data according to physically motivated molecular descriptors, including molecular size, structural compactness, degree of branching, heteroatom content, and sensitivity to dispersion-dominated interactions[ 7 , 10 ]. This reclassification reveals distinct and reproducible error patterns that cannot be inferred from global statistical metrics alone. By treating dispersion corrections not merely as empirical improvements but as diagnostic probes, this study distinguishes dispersion-dominated failures from error sources rooted in density localization, basis-set incompleteness, and exchange–correlation imbalance. On this basis, we develop a transferable diagnostic framework that directly links specific molecular features to dominant sources of electronic structure error. This approach provides physically motivated guidance for assessing the applicability limits of B3LYP and establishes a general strategy for interpreting the performance of density functionals beyond conventional mean-error benchmarking. Methodology Reference Dataset and Source of Data The present study is based on a large and well-established thermochemical benchmark dataset originally reported for a chemically diverse collection of neutral, closed-shell organic molecules composed of carbon, hydrogen, nitrogen, and oxygen. The dataset comprises 622 molecular systems with reliable experimental heats of formation, in addition to a carefully selected subset of 34 isomerization reactions that span a broad range of bonding environments, molecular sizes, degrees of branching, and steric compactness. All reference thermochemical data employed in this work originate from experimentally validated sources that have been critically assessed in prior large-scale benchmarking studies. No new experimental measurements or ab initio electronic structure calculations are introduced in the present investigation. Instead, the existing benchmark data are deliberately reused to construct a diagnostic and interpretative framework aimed at identifying systematic, physically grounded error patterns in density functional approximations, rather than at producing improved numerical benchmarks. This distinction is essential: the methodological novelty of the present work lies not in data generation, but in the reorganization and physical interpretation of established results[ 7 , 12 , 24 ]. Computational Levels Considered The electronic structure methods examined in this study include the Hartree–Fock (HF) method, the hybrid density functional B3LYP, dispersion-corrected variants of B3LYP, and the semiempirical PDDG/PM3 approach. For the B3LYP calculations, several commonly used Pople-type basis sets were considered, including 6-31G(d), 6-31G(d,p), and 6–31 + G(d,p), enabling assessment of polarization and diffuse function effects on thermochemical accuracy[ 6 , 7 , 25 ]. Dispersion corrections were incorporated at the energy level to account for long-range correlation effects absent in conventional hybrid functionals [ 7 – 9 ]. These corrections were applied consistently across the entire dataset using parameters reported in the original benchmark studies and were not refitted or reparameterized in the present work. The semiempirical PDDG/PM3 results were included as a low-cost reference point to contextualize the balance between computational efficiency and achievable accuracy[ 10 , 17 , 26 ]. Thermochemical Evaluation Protocol Heats of formation were evaluated following a standardized thermochemical protocol in which molecular total energies were combined with optimized atomic reference energies, thereby minimizing systematic atomic contributions to the total error. This approach enables a more meaningful comparison of relative molecular energetics across different electronic structure methods and reduces artificial size-dependent error amplification. For isomerization reactions, reaction energies were evaluated directly as differences between optimized molecular structures. Zero-point energy and finite-temperature corrections were not explicitly included in the isomerization analysis, as previous studies have demonstrated that such contributions exert only a minor influence on relative isomerization errors compared to dominant electronic structure deficiencies associated with exchange–correlation treatment and electron density distribution [ 27 ]. Diagnostic Reorganization of Benchmark Data A central methodological element of the present study is the diagnostic reorganization of benchmark results. Rather than relying exclusively on global statistical indicators such as mean absolute errors (MAEs), the molecular systems were grouped according to physically motivated descriptors, including: Molecular size and number of heavy atoms Degree of branching and structural compactness Presence of heteroatoms and hydrogen-bonding motifs Sensitivity to dispersion-dominated interactions This classification strategy enables the identification of systematic error patterns and failure modes that are obscured when chemically heterogeneous datasets are analyzed using averaged metrics alone [ 14 , 20 ]. In particular, it allows separation of dispersion-dominated deficiencies from errors arising due to density localization, basis-set incompleteness, and exchange–correlation imbalance. Error Metrics and Comparative Analysis The primary quantitative metric employed to assess method performance is the mean absolute error (MAE) with respect to experimental reference data. MAEs were evaluated for the full dataset as well as for chemically and structurally distinct subsets in order to expose trends associated with molecular size, topology, and interaction regime. In addition to numerical error analysis, reconstructed diagnostic plots were used to visualize systematic behaviors such as cumulative error growth with molecular size and enhanced sensitivity to molecular branching. These graphical representations were derived directly from published benchmark values and are used here strictly as interpretative tools, rather than as sources of new numerical data or independent validation. Methodological Scope and Positioning The methodological scope of the present study is intentionally diagnostic rather than predictive. By relying on established benchmark data, the analysis focuses on uncovering intrinsic limitations of commonly used density functionals and on elucidating the physical origins of their failures, rather than on achieving improved numerical agreement with experiment. While this approach does not replace full ab initio benchmarking based on newly computed datasets, it provides a complementary and conceptually distinct perspective that emphasizes interpretability, transferability, and physics-based guidance for method selection in large-scale organic thermochemistry. The proposed diagnostic framework is therefore intended to support more informed and robust application of density functional methods, particularly in chemically complex or size-extended systems where conventional benchmarking metrics may be misleading. Results and Discussion Global Thermochemical Accuracy and Basis-Set Effects Table 1 and Fig. 1 present the mean absolute errors (MAEs) in the calculated heats of formation obtained using different electronic structure methods and basis sets. The numerical values summarized in Table 1 are reorganized from established benchmark studies and are employed here as a diagnostic reference to analyze systematic accuracy trends, rather than to introduce new performance metrics. As shown in Table 1 and illustrated in Fig. 1 , the Hartree–Fock (HF) method exhibits the largest deviation from experimental values, with an MAE of approximately 3.9 kcal mol⁻¹, reflecting its well-known inability to account for electron correlation effects. The hybrid B3LYP functional yields a clear improvement, reducing the MAE to about 3.1 kcal mol⁻¹ when combined with the 6-31G(d) basis set. The inclusion of additional polarization and diffuse functions leads to only a modest and non-monotonic reduction in the MAE, with values in the narrow range of 2.6–2.7 kcal mol⁻¹ for the 6-31G(d,p) and 6–31 + G(d,p) basis sets. This behavior, evident in both Table 1 and Figure X, indicates that basis-set incompleteness is not the dominant source of error in B3LYP thermochemical predictions for this dataset. Instead, the residual inaccuracies appear to be largely intrinsic to the exchange–correlation functional itself. A more pronounced improvement is observed only when dispersion effects are explicitly included. The dispersion-corrected B3LYP approach achieves an MAE of approximately 2.4 kcal mol⁻¹, representing the best overall performance among the methods considered. This result highlights the significant role of missing medium-range correlation effects in conventional B3LYP calculations, particularly for larger or more compact molecular systems. Notably, the semiempirical PDDG/PM3 method yields an MAE of about 2.8 kcal mol⁻¹, comparable to that of uncorrected B3LYP, as clearly reflected in Figure X. This comparison underscores an important diagnostic point: when assessed solely on global thermochemical averages, uncorrected B3LYP does not offer a decisive accuracy advantage over substantially less expensive semiempirical approaches. Consequently, agreement at the MAE level should not be interpreted as evidence of uniformly reliable electronic structure descriptions across chemically diverse systems[ 8 , 14 , 23 , 28 ]. Table 1 Global thermochemical accuracy of selected electronic structure methods: diagnostic comparison of mean absolute errors (MAE). Method Basis set Dispersion correction MAE (kcal·mol⁻¹) HF 6-31G(d) No 3.9 B3LYP 6-31G(d) No 3.1 B3LYP 6-31G(d,p) No 2.6 B3LYP 6–31 + G(d,p) No 2.7 B3LYP 6–31 + G(d,p) Yes (dispersion) 2.4 PDDG/PM3 — — 2.8 System-Size Dependence and Cumulative Error Growth While global error statistics provide a useful first baseline, they can mask pronounced and chemically systematic size-dependent trends. Figure 2 examines this behavior for linear hydrocarbons, showing the error in the calculated heats of formation as a function of the number of carbon atoms. Two key features emerge clearly. First, the error does not fluctuate randomly around zero; instead, it becomes progressively more negative with increasing chain length, consistent with a cumulative size-dependent bias rather than isolated outliers. Second, the “diagnostic” trend (reorganized from the published benchmark dataset) displays a noticeably reduced slope relative to earlier B3LYP assessments, indicating that improved handling of atomic reference energies can partially suppress—but not eliminate—the size dependence. The persistence of a monotonic error growth even after such reprocessing supports the interpretation that the dominant limitation is intrinsic to the functional’s treatment of exchange–correlation effects in extended nonpolar systems, where medium-range correlation and dispersion-like contributions become increasingly important with molecular size.[ 8 , 29 ] Branching Sensitivity and Structural Compactness A more detailed structural analysis reveals that molecular branching constitutes a dominant and systematic source of error in B3LYP-based thermochemical calculations. Representative linear-to-branched isomerization reactions, summarized in Table 2 , are reorganized from the benchmark dataset to highlight this effect diagnostically rather than to provide exhaustive statistical coverage. In all examined cases, B3LYP consistently overs tabilizes linear isomers relative to their more compact, branched counterparts, leading to positive isomerization errors that increase sharply with molecular compactness. This behavior is illustrated in Fig. 3 , where branching-sensitive systems exhibit disproportionately large deviations compared to less-compact structures. The rapid growth of the error from the neopentane–pentane transformation to the more compact tetramethylbutane–octane case indicates that the energetic contributions associated with intramolecular dispersion and steric crowding are inadequately captured by the uncorrected functional. Although the inclusion of dispersion corrections significantly reduces the magnitude of the error, substantial residual deviations persist, particularly for highly compact systems. Importantly, this failure mode is not eliminated through conventional error-cancellation strategies such as isodesmic reaction schemes, indicating that the observed imbalance is intrinsic to the functional form rather than methodological in origin. Consequently, branching-sensitive isomerizations serve as a stringent diagnostic probe for assessing the reliability of density functional approximations in systems where compactness and medium-range correlation effects play a critical role.[ 30 , 31 ] Table 2 Representative linear-to-branched isomerization errors illustrating the sensitivity of hydrocarbons to molecular branching. Isomerization reaction Experimental ΔH (kcal·mol⁻¹) B3LYP error (kcal·mol⁻¹) B3LYP + dispersion (kcal·mol⁻¹) Neopentane → Pentane 3.5 + 3.5 + 2.8 Tetramethyl butane → Octane 10.5 + 10.5 + 4.1 Diagnostic Role of Dispersion Corrections The inclusion of empirical dispersion corrections provides direct insight into the physical origin of the errors observed in B3LYP-based thermochemical predictions. As demonstrated by the representative data in Table 2 and the trends illustrated in Fig. 2 , dispersion corrections lead to a substantial reduction in isomerization errors for branching-sensitive hydrocarbons, in some cases exceeding a 50% decrease relative to the uncorrected functional. This behavior confirms that missing dispersion interactions play a key role in destabilizing compact molecular frameworks within conventional B3LYP. Importantly, the improvement introduced by dispersion corrections is selective rather than universal. While errors are markedly reduced for flexible aliphatic systems dominated by intramolecular dispersion, similar corrections yield only limited improvement for rigid polycyclic or bridged structures, where residual deviations remain significant. This contrast indicates that dispersion constitutes a dominant, but not exclusive, source of error and that additional deficiencies—such as imbalance in exchange–correlation treatment—persist in certain classes of molecules. From a diagnostic perspective, dispersion corrections therefore function as a discriminating probe rather than a universal remedy. A pronounced error reduction upon their inclusion identifies dispersion-dominated failure modes, whereas persistent discrepancies point to deeper, non-dispersion-related shortcomings of the functional. This distinction reinforces the value of dispersion corrections as a tool for interpreting, rather than merely improving, density functional performance[ 16 , 32 ]. Heteroatom-Containing Systems and Density Localization Effects Isomerization reactions involving nitrogen- and oxygen-containing molecules exhibit a distinct error pattern compared to hydrocarbon systems. As summarized in Table 3 , reactions that involve changes in N–H or O–H bonding environments display a pronounced sensitivity to the inclusion of diffuse basis functions, with mean absolute errors reduced by more than 1 kcal mol⁻¹ upon basis-set extension. In these systems, the improvement achieved through the use of diffuse and additional polarization functions is significantly greater than that obtained from empirical dispersion corrections. This contrast indicates that the dominant source of error is not missing dispersion interactions, but rather an inadequate description of electron density localization and molecular polarization, particularly in regions associated with heteroatoms and polar bonds. The clear divergence between the behavior of hydrocarbon systems and heteroatom-containing molecules underscores the necessity of distinguishing between different physical origins of functional error. While dispersion corrections effectively diagnose and mitigate failures in compact, nonpolar frameworks, heteroatom-rich systems are primarily limited by basis-set flexibility and the treatment of localized electron density. This distinction reinforces the central premise of the present work: reliable functional assessment requires a physically informed, system-specific diagnostic approach rather than reliance on uniform correction strategies[ 20 , 33 ]. Table 3 Influence of diffuse basis functions on isomerization errors in heteroatom-containing systems. System type MAE with 6-31G(d) (kcal·mol⁻¹) MAE with 6–31 + G(d,p) (kcal·mol⁻¹) Nitrogen-containing molecules 2.5 1.1 Oxygen-containing molecules 2.6 1.7 Failure Mode Classification and Practical Implications The combined evidence from Tables 1 – 3 and the associated figures enables the identification of distinct, physically interpretable failure modes in B3LYP-based thermochemistry, summarized in Table 4 . Size-extended linear hydrocarbons exhibit systematic, cumulative deviations consistent with missing medium-range correlation effects, whereas branching-sensitive isomerizations reveal pronounced overstabilization trends that are strongly reduced—but not eliminated—upon inclusion of dispersion corrections. In contrast, isomerizations involving N–H and O–H bonding environments show marked sensitivity to diffuse basis functions, indicating that density localization and polarization effects dominate over dispersion in these cases. Collectively, these results demonstrate that no single correction strategy provides a universal remedy for B3LYP limitations. Instead, reliability is strongly system-dependent and should be assessed by matching the target chemical problem to its dominant physical interactions, using Table 4 as a practical diagnostic guide for method selection and expectation management[ 21 , 23 ]. Table 4 Diagnostic classification of dominant failure modes in B3LYP calculations. Molecular feature Observed error behavior Dominant physical origin Large linear chains Error increases with size Missing dispersion Branched hydrocarbons Overstabilization of linear isomers Poor mid-range correlation N–H / O–H isomerizations Strong basis-set sensitivity Density localization Polycyclic bridged systems Large residual errors Exchange–correlation imbalance Conclusions This work provides a physically grounded diagnostic assessment of the performance of the B3LYP density functional for thermochemical predictions, with particular emphasis on identifying the dominant origins of systematic error across chemically diverse systems. Rather than relying solely on global statistical metrics, the analysis decomposes B3LYP inaccuracies into distinct, physically interpretable failure modes linked to molecular size, structural compactness, dispersion interactions, and electron density localization. Global thermochemical benchmarks reveal that while B3LYP offers a clear improvement over Hartree–Fock methods, its apparent accuracy is only weakly sensitive to basis-set enlargement beyond moderate polarization and diffuse functions. The limited and non-monotonic improvement observed upon basis-set extension indicates that basis-set incompleteness is not the primary source of residual error for typical thermochemical datasets. Instead, intrinsic limitations of the exchange–correlation functional dominate the overall accuracy. A systematic size-dependent error growth is identified for linear hydrocarbons, demonstrating that B3LYP exhibits cumulative deviations as molecular size increases. Although partial mitigation can be achieved through improved treatment of atomic reference energies, a persistent monotonic trend remains, highlighting the functional’s inadequate description of medium-range correlation effects in extended nonpolar systems. Structural compactness emerges as an even more stringent diagnostic. Branching-sensitive isomerization reactions consistently reveal a pronounced overstabilization of linear isomers relative to compact, branched structures. The inclusion of empirical dispersion corrections significantly reduces these errors, confirming the central role of missing dispersion interactions. However, the persistence of residual deviations—particularly in highly compact systems—demonstrates that dispersion alone does not fully account for the observed failures. In contrast, isomerizations involving nitrogen- and oxygen-containing molecules follow a fundamentally different error pattern. For these systems, the dominant sensitivity arises from basis-set flexibility and the accurate description of localized and polarized electron density, rather than from missing dispersion. The pronounced improvement obtained by including diffuse functions underscores the importance of density localization effects in heteroatom-rich environments. Taken together, these findings demonstrate that no single correction strategy provides a universal remedy for the limitations of B3LYP. Functional reliability is strongly system-dependent and must be evaluated by matching the chemical problem of interest to its dominant physical interactions. The diagnostic framework summarized in Table 4 offers a practical guide for method selection and expectation management, enabling more informed and physically consistent use of density functional approximations in thermochemical applications. Declarations Acknowledgements The author would like to thank the Departments of Physics and Medical Physics at the Colleges of Education and Science for their institutional support during the completion of this research. Author Contributions The author is the sole contributor to this work and was responsible for the study design, data analysis, interpretation of results, and manuscript preparation. Data Availability No new datasets were generated during the current study. All data analysed in this work were obtained from previously published benchmark studies. Declarations of Competing Interests The author declares no competing interests. Funding Declaration This research received no external funding References Calais J (1993) Density-functional theory of atoms and molecules. R.G. Parr and W. Yang, Oxford University Press, New York, Oxford, 1989. IX + 333 pp. Price £45.00. Int J Quantum Chem 47:101–101. https://doi.org/10.1002/qua.560470107 W KochM., C. A. Holthausen (2001) Elementary Quantum Chemistry. In: A Chemist’s Guide to Density Functional Theory. Wiley, pp 3–18 Burke K (2012) Perspective on density functional theory. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8818002","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":590731540,"identity":"5e959167-69a1-4ced-8a19-94eccf99ef79","order_by":0,"name":"Hayder M. Hadi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIiWNgGAWjYBACAyCWALMgJLMcmOIBIyK1GJOuJbEBqgUnMGc/Y3jj4x6GaH7p5qebblRYp/fdSGB88LaNQYbvAHYtlj05xpYznjHkzpxzzOx2zpn03Jk3EpgN57Yx8Eji0GJwIMdMmucAQ+6GGwlmt3PbDoMYbNK8QC0GuLScf2Mm/QesJf3b7dx/h9MNbiSw/8ar5QbQFgawlhygLQ2HE4Ba2Jjxa3lWbNlzQCJ35oycsts5x9INZ5552Cw555wEbr+cT95448cBm9x+ifRtt3NqrOX5jicf/PCmzMYeV4gxMHAgogYCDjA2QERwamF/gCZwAIMxCkbBKBgFIxwAANn9Y+sUtAzCAAAAAElFTkSuQmCC","orcid":"","institution":"University of Al-Qadisiyah","correspondingAuthor":true,"prefix":"","firstName":"Hayder","middleName":"M.","lastName":"Hadi","suffix":""}],"badges":[],"createdAt":"2026-02-07 21:08:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8818002/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8818002/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103174441,"identity":"1a78c6e5-cf50-497a-be5d-9befb2f28609","added_by":"auto","created_at":"2026-02-22 15:41:51","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":54745,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of mean absolute errors for different electronic structure methods.\u003c/p\u003e\n\u003cp\u003eDescription: Bar chart comparing MAEs obtained using HF, B3LYP, dispersion-corrected B3LYP, and the semiempirical PDDG/PM3 method. Dispersion-corrected B3LYP provides the lowest overall MAE, while uncorrected B3LYP offers no decisive advantage over the semiempirical approach.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8818002/v1/75c1e8edbd2a7dd4440c1a6f.png"},{"id":103174439,"identity":"af643e01-1b83-465b-8410-934e89337e62","added_by":"auto","created_at":"2026-02-22 15:41:51","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":160156,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSystem-size dependence of B3LYP errors for linear hydrocarbons. \u003c/strong\u003eErrors in ΔHf increase monotonically with chain length, with a reduced but persistent size-dependent trend in the reorganized benchmark data.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8818002/v1/bf9d6925a6491108fb0da65f.png"},{"id":103174440,"identity":"90b371fe-67af-4724-864e-953fa041f3ef","added_by":"auto","created_at":"2026-02-22 15:41:51","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":57348,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eEffect of molecular branching and dispersion correction on isomerization errors.\u003c/strong\u003e\u003cbr\u003e\nUncorrected B3LYP increasingly over stabilizes linear isomers as molecular compactness increases, while dispersion corrections significantly reduce, but do not fully eliminate, this branching-related error.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8818002/v1/6b3dcf85af738fbae114322d.png"},{"id":104834921,"identity":"a8dd4d00-3606-4e90-8496-8af9559a167a","added_by":"auto","created_at":"2026-03-17 17:36:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1098543,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8818002/v1/b47919e9-a032-400e-938a-aa018e2b0a3c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Diagnostic Analysis of B3LYP Thermochemical Errors: Physical Origins, System Dependence, and Practical Implications","fulltext":[{"header":"Introduction","content":"\u003cp\u003eDensity functional theory (DFT) has become a central tool in modern computational chemistry, providing a practical balance between accuracy and computational cost that enables routine investigation of thermochemical properties for molecular systems far beyond the reach of high-level wavefunction-based methods [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Its favourable scaling with system size and its ability to incorporate electron correlation effects at moderate computational expense have made DFT indispensable in studies of organic, biological, and materials systems.\u003c/p\u003e \u003cp\u003eWithin the broad family of density functionals, hybrid approaches have gained particular prominence due to their partial inclusion of exact exchange, which improves the description of molecular energetics compared to pure generalized gradient approximations (GGAs)[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Among these, the B3LYP functional has emerged as one of the most widely applied methods in organic thermochemistry. Its enduring popularity does not stem from exceptional accuracy in any single chemical domain, but rather from its perceived robustness and transferability across a diverse range of molecular systems[\u003cspan additionalcitationids=\"CR8\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. As a result, B3LYP is frequently employed as a default computational choice, often without explicit consideration of its known and well-documented limitations.\u003c/p\u003e \u003cp\u003eExtensive benchmarking studies conducted over the past two decades have demonstrated that the apparent reliability of B3LYP is strongly system-dependent[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. While satisfactory agreement with experimental thermochemical data is commonly reported for small and moderately sized molecules, systematic deviations become increasingly pronounced for larger systems, highly branched hydrocarbons, and molecules dominated by weak intramolecular or medium-range correlation effects[\u003cspan additionalcitationids=\"CR11 CR12\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Importantly, these deficiencies are often masked when performance is assessed solely through global statistical indicators such as mean absolute errors (MAEs), which average over chemically diverse datasets and obscure physically meaningful trends [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAttempts to mitigate these shortcomings have primarily focused on technical refinements, including enlargement of one-electron basis sets, empirical adjustment of atomic reference energies, and the incorporation of dispersion corrections into conventional hybrid functionals[\u003cspan additionalcitationids=\"CR17 CR18\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Although such strategies can reduce numerical errors in selected cases, their effectiveness is highly non-uniform and strongly system-specific. In many instances, numerical improvement does not correspond to a genuine resolution of the underlying physical deficiencies of the exchange\u0026ndash;correlation functional, but rather reflects fortuitous error cancellation [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eMore fundamentally, conventional benchmarking approaches rarely address a critical conceptual question: whether the observed errors in density functional predictions arise from random numerical artifacts or from systematic, physically grounded deficiencies inherent to the functional form itself[\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Without explicitly distinguishing between these possibilities, method selection remains largely empirical and may fail when extrapolated to chemically complex, larger, or previously unexplored systems.\u003c/p\u003e \u003cp\u003eIn the present work, we adopt a diagnostic benchmarking perspective aimed at elucidating the physical mechanisms governing the performance and failure of B3LYP-based methods. Rather than introducing new electronic structure calculations, we reorganize established large-scale thermochemical benchmark data according to physically motivated molecular descriptors, including molecular size, structural compactness, degree of branching, heteroatom content, and sensitivity to dispersion-dominated interactions[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. This reclassification reveals distinct and reproducible error patterns that cannot be inferred from global statistical metrics alone.\u003c/p\u003e \u003cp\u003eBy treating dispersion corrections not merely as empirical improvements but as diagnostic probes, this study distinguishes dispersion-dominated failures from error sources rooted in density localization, basis-set incompleteness, and exchange\u0026ndash;correlation imbalance. On this basis, we develop a transferable diagnostic framework that directly links specific molecular features to dominant sources of electronic structure error. This approach provides physically motivated guidance for assessing the applicability limits of B3LYP and establishes a general strategy for interpreting the performance of density functionals beyond conventional mean-error benchmarking.\u003c/p\u003e"},{"header":"Methodology","content":"\u003cp\u003e \u003cb\u003eReference Dataset and Source of Data\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe present study is based on a large and well-established thermochemical benchmark dataset originally reported for a chemically diverse collection of neutral, closed-shell organic molecules composed of carbon, hydrogen, nitrogen, and oxygen. The dataset comprises 622 molecular systems with reliable experimental heats of formation, in addition to a carefully selected subset of 34 isomerization reactions that span a broad range of bonding environments, molecular sizes, degrees of branching, and steric compactness.\u003c/p\u003e \u003cp\u003eAll reference thermochemical data employed in this work originate from experimentally validated sources that have been critically assessed in prior large-scale benchmarking studies. No new experimental measurements or ab initio electronic structure calculations are introduced in the present investigation. Instead, the existing benchmark data are deliberately reused to construct a diagnostic and interpretative framework aimed at identifying systematic, physically grounded error patterns in density functional approximations, rather than at producing improved numerical benchmarks.\u003c/p\u003e \u003cp\u003eThis distinction is essential: the methodological novelty of the present work lies not in data generation, but in the reorganization and physical interpretation of established results[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e].\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eComputational Levels Considered\u003c/h2\u003e \u003cp\u003eThe electronic structure methods examined in this study include the Hartree\u0026ndash;Fock (HF) method, the hybrid density functional B3LYP, dispersion-corrected variants of B3LYP, and the semiempirical PDDG/PM3 approach. For the B3LYP calculations, several commonly used Pople-type basis sets were considered, including 6-31G(d), 6-31G(d,p), and 6\u0026ndash;31\u0026thinsp;+\u0026thinsp;G(d,p), enabling assessment of polarization and diffuse function effects on thermochemical accuracy[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eDispersion corrections were incorporated at the energy level to account for long-range correlation effects absent in conventional hybrid functionals [\u003cspan additionalcitationids=\"CR8\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. These corrections were applied consistently across the entire dataset using parameters reported in the original benchmark studies and were not refitted or reparameterized in the present work. The semiempirical PDDG/PM3 results were included as a low-cost reference point to contextualize the balance between computational efficiency and achievable accuracy[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eThermochemical Evaluation Protocol\u003c/h3\u003e\n\u003cp\u003eHeats of formation were evaluated following a standardized thermochemical protocol in which molecular total energies were combined with optimized atomic reference energies, thereby minimizing systematic atomic contributions to the total error. This approach enables a more meaningful comparison of relative molecular energetics across different electronic structure methods and reduces artificial size-dependent error amplification.\u003c/p\u003e \u003cp\u003eFor isomerization reactions, reaction energies were evaluated directly as differences between optimized molecular structures. Zero-point energy and finite-temperature corrections were not explicitly included in the isomerization analysis, as previous studies have demonstrated that such contributions exert only a minor influence on relative isomerization errors compared to dominant electronic structure deficiencies associated with exchange\u0026ndash;correlation treatment and electron density distribution [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e].\u003c/p\u003e\n\u003ch3\u003eDiagnostic Reorganization of Benchmark Data\u003c/h3\u003e\n\u003cp\u003eA central methodological element of the present study is the diagnostic reorganization of benchmark results. Rather than relying exclusively on global statistical indicators such as mean absolute errors (MAEs), the molecular systems were grouped according to physically motivated descriptors, including:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eMolecular size and number of heavy atoms\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eDegree of branching and structural compactness\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ePresence of heteroatoms and hydrogen-bonding motifs\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eSensitivity to dispersion-dominated interactions\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThis classification strategy enables the identification of systematic error patterns and failure modes that are obscured when chemically heterogeneous datasets are analyzed using averaged metrics alone [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. In particular, it allows separation of dispersion-dominated deficiencies from errors arising due to density localization, basis-set incompleteness, and exchange\u0026ndash;correlation imbalance.\u003c/p\u003e\n\u003ch3\u003eError Metrics and Comparative Analysis\u003c/h3\u003e\n\u003cp\u003eThe primary quantitative metric employed to assess method performance is the mean absolute error (MAE) with respect to experimental reference data. MAEs were evaluated for the full dataset as well as for chemically and structurally distinct subsets in order to expose trends associated with molecular size, topology, and interaction regime.\u003c/p\u003e \u003cp\u003eIn addition to numerical error analysis, reconstructed diagnostic plots were used to visualize systematic behaviors such as cumulative error growth with molecular size and enhanced sensitivity to molecular branching. These graphical representations were derived directly from published benchmark values and are used here strictly as interpretative tools, rather than as sources of new numerical data or independent validation.\u003c/p\u003e\n\u003ch3\u003eMethodological Scope and Positioning\u003c/h3\u003e\n\u003cp\u003eThe methodological scope of the present study is intentionally diagnostic rather than predictive. By relying on established benchmark data, the analysis focuses on uncovering intrinsic limitations of commonly used density functionals and on elucidating the physical origins of their failures, rather than on achieving improved numerical agreement with experiment.\u003c/p\u003e \u003cp\u003eWhile this approach does not replace full ab initio benchmarking based on newly computed datasets, it provides a complementary and conceptually distinct perspective that emphasizes interpretability, transferability, and physics-based guidance for method selection in large-scale organic thermochemistry. The proposed diagnostic framework is therefore intended to support more informed and robust application of density functional methods, particularly in chemically complex or size-extended systems where conventional benchmarking metrics may be misleading.\u003c/p\u003e"},{"header":"Results and Discussion","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003eGlobal Thermochemical Accuracy and Basis-Set Effects\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e present the mean absolute errors (MAEs) in the calculated heats of formation obtained using different electronic structure methods and basis sets. The numerical values summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e are reorganized from established benchmark studies and are employed here as a diagnostic reference to analyze systematic accuracy trends, rather than to introduce new performance metrics.\u003c/p\u003e \u003cp\u003eAs shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the Hartree\u0026ndash;Fock (HF) method exhibits the largest deviation from experimental values, with an MAE of approximately 3.9 kcal mol⁻\u0026sup1;, reflecting its well-known inability to account for electron correlation effects. The hybrid B3LYP functional yields a clear improvement, reducing the MAE to about 3.1 kcal mol⁻\u0026sup1; when combined with the 6-31G(d) basis set.\u003c/p\u003e \u003cp\u003eThe inclusion of additional polarization and diffuse functions leads to only a modest and non-monotonic reduction in the MAE, with values in the narrow range of 2.6\u0026ndash;2.7 kcal mol⁻\u0026sup1; for the 6-31G(d,p) and 6\u0026ndash;31\u0026thinsp;+\u0026thinsp;G(d,p) basis sets. This behavior, evident in both Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Figure X, indicates that basis-set incompleteness is not the dominant source of error in B3LYP thermochemical predictions for this dataset. Instead, the residual inaccuracies appear to be largely intrinsic to the exchange\u0026ndash;correlation functional itself.\u003c/p\u003e \u003cp\u003eA more pronounced improvement is observed only when dispersion effects are explicitly included. The dispersion-corrected B3LYP approach achieves an MAE of approximately 2.4 kcal mol⁻\u0026sup1;, representing the best overall performance among the methods considered. This result highlights the significant role of missing medium-range correlation effects in conventional B3LYP calculations, particularly for larger or more compact molecular systems.\u003c/p\u003e \u003cp\u003eNotably, the semiempirical PDDG/PM3 method yields an MAE of about 2.8 kcal mol⁻\u0026sup1;, comparable to that of uncorrected B3LYP, as clearly reflected in Figure X. This comparison underscores an important diagnostic point: when assessed solely on global thermochemical averages, uncorrected B3LYP does not offer a decisive accuracy advantage over substantially less expensive semiempirical approaches. Consequently, agreement at the MAE level should not be interpreted as evidence of uniformly reliable electronic structure descriptions across chemically diverse systems[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGlobal thermochemical accuracy of selected electronic structure methods: diagnostic comparison of mean absolute errors (MAE).\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBasis set\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDispersion correction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6-31G(d)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB3LYP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6-31G(d)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB3LYP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6-31G(d,p)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB3LYP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6\u0026ndash;31\u0026thinsp;+\u0026thinsp;G(d,p)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB3LYP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6\u0026ndash;31\u0026thinsp;+\u0026thinsp;G(d,p)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYes (dispersion)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePDDG/PM3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026mdash;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026mdash;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003eSystem-Size Dependence and Cumulative Error Growth\u003c/h3\u003e\n\u003cp\u003eWhile global error statistics provide a useful first baseline, they can mask pronounced and chemically systematic size-dependent trends. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e examines this behavior for linear hydrocarbons, showing the error in the calculated heats of formation as a function of the number of carbon atoms. Two key features emerge clearly. First, the error does not fluctuate randomly around zero; instead, it becomes progressively more negative with increasing chain length, consistent with a cumulative size-dependent bias rather than isolated outliers.\u003c/p\u003e \u003cp\u003eSecond, the \u0026ldquo;diagnostic\u0026rdquo; trend (reorganized from the published benchmark dataset) displays a noticeably reduced slope relative to earlier B3LYP assessments, indicating that improved handling of atomic reference energies can partially suppress\u0026mdash;but not eliminate\u0026mdash;the size dependence. The persistence of a monotonic error growth even after such reprocessing supports the interpretation that the dominant limitation is intrinsic to the functional\u0026rsquo;s treatment of exchange\u0026ndash;correlation effects in extended nonpolar systems, where medium-range correlation and dispersion-like contributions become increasingly important with molecular size.[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eBranching Sensitivity and Structural Compactness\u003c/h2\u003e \u003cp\u003eA more detailed structural analysis reveals that molecular branching constitutes a dominant and systematic source of error in B3LYP-based thermochemical calculations. Representative linear-to-branched isomerization reactions, summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, are reorganized from the benchmark dataset to highlight this effect diagnostically rather than to provide exhaustive statistical coverage. In all examined cases, B3LYP consistently overs tabilizes linear isomers relative to their more compact, branched counterparts, leading to positive isomerization errors that increase sharply with molecular compactness.\u003c/p\u003e \u003cp\u003eThis behavior is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, where branching-sensitive systems exhibit disproportionately large deviations compared to less-compact structures. The rapid growth of the error from the neopentane\u0026ndash;pentane transformation to the more compact tetramethylbutane\u0026ndash;octane case indicates that the energetic contributions associated with intramolecular dispersion and steric crowding are inadequately captured by the uncorrected functional. Although the inclusion of dispersion corrections significantly reduces the magnitude of the error, substantial residual deviations persist, particularly for highly compact systems.\u003c/p\u003e \u003cp\u003eImportantly, this failure mode is not eliminated through conventional error-cancellation strategies such as isodesmic reaction schemes, indicating that the observed imbalance is intrinsic to the functional form rather than methodological in origin. Consequently, branching-sensitive isomerizations serve as a stringent diagnostic probe for assessing the reliability of density functional approximations in systems where compactness and medium-range correlation effects play a critical role.[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eRepresentative linear-to-branched isomerization errors illustrating the sensitivity of hydrocarbons to molecular branching.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIsomerization reaction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eExperimental ΔH (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eB3LYP error (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eB3LYP\u0026thinsp;+\u0026thinsp;dispersion (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNeopentane \u0026rarr; Pentane\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e+\u0026thinsp;3.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;2.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTetramethyl butane \u0026rarr; Octane\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e10.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e+\u0026thinsp;10.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;4.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eDiagnostic Role of Dispersion Corrections\u003c/h2\u003e \u003cp\u003eThe inclusion of empirical dispersion corrections provides direct insight into the physical origin of the errors observed in B3LYP-based thermochemical predictions. As demonstrated by the representative data in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and the trends illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, dispersion corrections lead to a substantial reduction in isomerization errors for branching-sensitive hydrocarbons, in some cases exceeding a 50% decrease relative to the uncorrected functional. This behavior confirms that missing dispersion interactions play a key role in destabilizing compact molecular frameworks within conventional B3LYP.\u003c/p\u003e \u003cp\u003eImportantly, the improvement introduced by dispersion corrections is selective rather than universal. While errors are markedly reduced for flexible aliphatic systems dominated by intramolecular dispersion, similar corrections yield only limited improvement for rigid polycyclic or bridged structures, where residual deviations remain significant. This contrast indicates that dispersion constitutes a dominant, but not exclusive, source of error and that additional deficiencies\u0026mdash;such as imbalance in exchange\u0026ndash;correlation treatment\u0026mdash;persist in certain classes of molecules.\u003c/p\u003e \u003cp\u003eFrom a diagnostic perspective, dispersion corrections therefore function as a discriminating probe rather than a universal remedy. A pronounced error reduction upon their inclusion identifies dispersion-dominated failure modes, whereas persistent discrepancies point to deeper, non-dispersion-related shortcomings of the functional. This distinction reinforces the value of dispersion corrections as a tool for interpreting, rather than merely improving, density functional performance[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003eHeteroatom-Containing Systems and Density Localization Effects\u003c/h2\u003e \u003cp\u003eIsomerization reactions involving nitrogen- and oxygen-containing molecules exhibit a distinct error pattern compared to hydrocarbon systems. As summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, reactions that involve changes in N\u0026ndash;H or O\u0026ndash;H bonding environments display a pronounced sensitivity to the inclusion of diffuse basis functions, with mean absolute errors reduced by more than 1 kcal mol⁻\u0026sup1; upon basis-set extension.\u003c/p\u003e \u003cp\u003eIn these systems, the improvement achieved through the use of diffuse and additional polarization functions is significantly greater than that obtained from empirical dispersion corrections. This contrast indicates that the dominant source of error is not missing dispersion interactions, but rather an inadequate description of electron density localization and molecular polarization, particularly in regions associated with heteroatoms and polar bonds.\u003c/p\u003e \u003cp\u003eThe clear divergence between the behavior of hydrocarbon systems and heteroatom-containing molecules underscores the necessity of distinguishing between different physical origins of functional error. While dispersion corrections effectively diagnose and mitigate failures in compact, nonpolar frameworks, heteroatom-rich systems are primarily limited by basis-set flexibility and the treatment of localized electron density. This distinction reinforces the central premise of the present work: reliable functional assessment requires a physically informed, system-specific diagnostic approach rather than reliance on uniform correction strategies[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eInfluence of diffuse basis functions on isomerization errors in heteroatom-containing systems.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSystem type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMAE with 6-31G(d) (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMAE with 6\u0026ndash;31\u0026thinsp;+\u0026thinsp;G(d,p) (kcal\u0026middot;mol⁻\u0026sup1;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNitrogen-containing molecules\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e1.1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOxygen-containing molecules\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e1.7\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003eFailure Mode Classification and Practical Implications\u003c/h2\u003e \u003cp\u003eThe combined evidence from Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and the associated figures enables the identification of distinct, physically interpretable failure modes in B3LYP-based thermochemistry, summarized in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Size-extended linear hydrocarbons exhibit systematic, cumulative deviations consistent with missing medium-range correlation effects, whereas branching-sensitive isomerizations reveal pronounced overstabilization trends that are strongly reduced\u0026mdash;but not eliminated\u0026mdash;upon inclusion of dispersion corrections. In contrast, isomerizations involving N\u0026ndash;H and O\u0026ndash;H bonding environments show marked sensitivity to diffuse basis functions, indicating that density localization and polarization effects dominate over dispersion in these cases.\u003c/p\u003e \u003cp\u003eCollectively, these results demonstrate that no single correction strategy provides a universal remedy for B3LYP limitations. Instead, reliability is strongly system-dependent and should be assessed by matching the target chemical problem to its dominant physical interactions, using Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e as a practical diagnostic guide for method selection and expectation management[\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDiagnostic classification of dominant failure modes in B3LYP calculations.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMolecular feature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eObserved error behavior\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDominant physical origin\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLarge linear chains\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eError increases with size\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMissing dispersion\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBranched hydrocarbons\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOverstabilization of linear isomers\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePoor mid-range correlation\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eN\u0026ndash;H / O\u0026ndash;H isomerizations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStrong basis-set sensitivity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDensity localization\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePolycyclic bridged systems\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLarge residual errors\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eExchange\u0026ndash;correlation imbalance\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusions","content":"\u003cp\u003eThis work provides a physically grounded diagnostic assessment of the performance of the B3LYP density functional for thermochemical predictions, with particular emphasis on identifying the dominant origins of systematic error across chemically diverse systems. Rather than relying solely on global statistical metrics, the analysis decomposes B3LYP inaccuracies into distinct, physically interpretable failure modes linked to molecular size, structural compactness, dispersion interactions, and electron density localization.\u003c/p\u003e \u003cp\u003eGlobal thermochemical benchmarks reveal that while B3LYP offers a clear improvement over Hartree\u0026ndash;Fock methods, its apparent accuracy is only weakly sensitive to basis-set enlargement beyond moderate polarization and diffuse functions. The limited and non-monotonic improvement observed upon basis-set extension indicates that basis-set incompleteness is not the primary source of residual error for typical thermochemical datasets. Instead, intrinsic limitations of the exchange\u0026ndash;correlation functional dominate the overall accuracy.\u003c/p\u003e \u003cp\u003eA systematic size-dependent error growth is identified for linear hydrocarbons, demonstrating that B3LYP exhibits cumulative deviations as molecular size increases. Although partial mitigation can be achieved through improved treatment of atomic reference energies, a persistent monotonic trend remains, highlighting the functional\u0026rsquo;s inadequate description of medium-range correlation effects in extended nonpolar systems.\u003c/p\u003e \u003cp\u003eStructural compactness emerges as an even more stringent diagnostic. Branching-sensitive isomerization reactions consistently reveal a pronounced overstabilization of linear isomers relative to compact, branched structures. The inclusion of empirical dispersion corrections significantly reduces these errors, confirming the central role of missing dispersion interactions. However, the persistence of residual deviations\u0026mdash;particularly in highly compact systems\u0026mdash;demonstrates that dispersion alone does not fully account for the observed failures.\u003c/p\u003e \u003cp\u003eIn contrast, isomerizations involving nitrogen- and oxygen-containing molecules follow a fundamentally different error pattern. For these systems, the dominant sensitivity arises from basis-set flexibility and the accurate description of localized and polarized electron density, rather than from missing dispersion. The pronounced improvement obtained by including diffuse functions underscores the importance of density localization effects in heteroatom-rich environments.\u003c/p\u003e \u003cp\u003eTaken together, these findings demonstrate that no single correction strategy provides a universal remedy for the limitations of B3LYP. Functional reliability is strongly system-dependent and must be evaluated by matching the chemical problem of interest to its dominant physical interactions. The diagnostic framework summarized in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e offers a practical guide for method selection and expectation management, enabling more informed and physically consistent use of density functional approximations in thermochemical applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author would like to thank the Departments of Physics and Medical Physics at the Colleges of Education and Science for their institutional support during the completion of this research.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author is the sole contributor to this work and was responsible for the study design, data analysis, interpretation of results, and manuscript preparation.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo new datasets were generated during the current study. All data analysed in this work were obtained from previously published benchmark studies.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclarations of Competing Interests \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research received no external funding\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eCalais J (1993) Density-functional theory of atoms and molecules. R.G. Parr and W. Yang, Oxford University Press, New York, Oxford, 1989. IX\u0026thinsp;+\u0026thinsp;333 pp. Price \u0026pound;45.00. 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Phys Rev Lett 100:146401. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/PhysRevLett.100.146401\u003c/span\u003e\u003cspan address=\"10.1103/PhysRevLett.100.146401\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-molecular-modeling","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmmo","sideBox":"Learn more about [Journal of Molecular Modeling](https://www.springer.com/journal/894)","snPcode":"894","submissionUrl":"https://submission.nature.com/new-submission/894/3","title":"Journal of Molecular Modeling","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"DFT, B3LYP, Thermochemical accuracy, Isomerization energies, Dispersion corrections","lastPublishedDoi":"10.21203/rs.3.rs-8818002/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8818002/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDensity functional theory remains a cornerstone of thermochemical modeling; however, its reliability depends critically on the interplay between functional form, basis-set representation, and the underlying physical interactions governing a given system. In this work, a diagnostic assessment of the B3LYP functional is presented, focusing on the identification of systematic error sources rather than on global statistical performance alone. Mean absolute errors reorganized from established benchmark datasets are used as a reference framework to analyze trends associated with molecular size, structural compactness, dispersion interactions, and electron density localization.\u003c/p\u003e\n\u003cp\u003eGlobal thermochemical statistics show that B3LYP improves substantially over Hartree–Fock methods, yet exhibits only limited and non-monotonic sensitivity to basis-set extension beyond moderate polarization and diffuse functions. A persistent size-dependent error growth is identified for linear hydrocarbons, indicating cumulative deficiencies related to medium-range correlation effects. Branching-sensitive isomerization reactions reveal a pronounced overstabilization of linear isomers, which is significantly—but not completely—mitigated by the inclusion of empirical dispersion corrections, highlighting dispersion as a dominant but non-exclusive error source.\u003c/p\u003e\n\u003cp\u003eIn contrast, isomerizations involving nitrogen- and oxygen-containing molecules display strong sensitivity to diffuse basis functions, demonstrating that density localization and polarization effects dominate over dispersion in heteroatom-rich systems. These contrasting behaviors confirm that B3LYP errors arise from multiple, system-dependent physical origins that cannot be resolved by a single correction strategy.\u003c/p\u003e\n\u003cp\u003eOverall, this study establishes a physically motivated diagnostic framework for interpreting B3LYP performance, emphasizing that functional reliability must be evaluated in relation to the dominant interactions of the target chemical system. The resulting classification provides practical guidance for method selection and expectation management in thermochemical applications.\u003c/p\u003e","manuscriptTitle":"A Diagnostic Analysis of B3LYP Thermochemical Errors: Physical Origins, System Dependence, and Practical Implications","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-22 15:41:46","doi":"10.21203/rs.3.rs-8818002/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-13T08:14:00+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-13T01:17:36+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-13T01:15:58+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Molecular Modeling","date":"2026-02-07T20:55:57+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-molecular-modeling","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmmo","sideBox":"Learn more about [Journal of Molecular Modeling](https://www.springer.com/journal/894)","snPcode":"894","submissionUrl":"https://submission.nature.com/new-submission/894/3","title":"Journal of Molecular Modeling","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"62870c76-5b5c-4296-a9ce-78e1cc351714","owner":[],"postedDate":"February 22nd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T10:54:14+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-22 15:41:46","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8818002","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8818002","identity":"rs-8818002","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}