The second-order functional structure of the chemical potential is examined within conceptual density functional theory by treating μ[N, v(r)] as a functional of the electron number and the external potential. In close analogy with the nonperturbative functional expansion of the total energy introduced by Liu and Parr, the chemical potential admits an explicit functional form up to second order that does not rely on a Taylor expansion about a reference state. This representation organizes established first- and second-order response descriptors into a unified response-theoretical framework, where the linear terms recover the standard hardness and Fukui function contributions, while quadratic terms encode nonlinear charge effects and nonlocal response through the hyperhardness, charge sensitivity, and response kernel. The resulting formulation clarifies the hierarchical organization of response properties associated with the chemical potential and provides a complementary perspective on electronegativity equalization within conceptual density functional theory.

alpha,beta-Unsaturated carbonyl compounds are frequently associated with Michael-type electrophilic reactivity toward biological nucleophiles. However, structurally similar electrophiles within the same chemical class may exhibit markedly different mutagenic responses, making mechanistically interpretable mutagenicity prediction challenging. In this study, we examined a curated dataset of 34 alpha,beta-unsaturated acids, a chemically coherent subclass of Michael acceptors, to evaluate whether electronically derived reactivity descriptors could discriminate mutagenic from non-mutagenic compounds. Descriptors derived from conceptual density functional theory (CDFT) and electrotopological state (E-State) indices were reduced using correlation filtering, ANOVA, and recursive feature elimination. Among the evaluated classifiers, Logistic Regression provided the most consistent balance between predictive agreement and interpretability (kappa = 0.72). The final two-descriptor model, based on the global electrophilicity index (omega) and the beta-carbon E-State descriptor (SC beta), indicates that higher electrophilicity and increased electronic activation at the beta-carbon are associated with a greater probability of a mutagenic Ames response within this restricted chemical domain. The proposed model is intended as a category-specific binary hazard classifier for alpha,beta-unsaturated acids rather than as a generalized mutagenicity predictor.

This review analyses the chemical bonding and reactivity implications of recent findings concerning the rigorous application of the bonding evolution theory (BET) to describe electron rearrangements in cycloaddition reactions in the ground and electronically excited states. Computational evidence shows that characterizing bond formations and scissions using parametric polynomials derived from catastrophe theory (CT) is critical for gaining further insights into chemical bonding and reactivity. However, most of the applications of BET conduct this association without considering the robust mathematical basis supporting BET. Consequently, misinterpretations and incorrect results have arisen due to the inherent ambiguity of such oversight. The proper use of BET involves calculating the Hessian matrix at potentially degenerate critical points of the electron localization function (ELF) and measuring their relative distances along a reactive coordinate. This methodical approach tailors key CT concepts into the original BET framework, thereby recovering the rigor of the latter. The systematic application of these steps has led to various unexpected outcomes, including the correlation between electron density symmetry and CT’s polynomials, the interplay between these polynomials and the heterolytic/homolytic character of bond breakages, and a CT-based model for scaling bond polarity. These discoveries underscore the relevance of adhering to concepts and highlight that rationalizing electron reorganizations using CT’s functions is far from a technical subtlety.

The transmetalation step in the ligand-free Suzuki-Miyaura reaction remains a topic of debate, especially regarding the roles of the base and the order of bond-breaking and bond-forming processes. In this study, we investigate the carbonate-assisted boronate pathway and demonstrate that aryl transfer proceeds through an electronically asynchronous mechanism, in which B-C bond cleavage precedes Pd-C bond formation despite a smooth and featureless free-energy profile. Free-energy calculations combined with intrinsic reaction coordinate analysis reveal a kinetically accessible and highly exergonic pathway. However, analysis of the electron localization function (ELF) shows that the reaction involves a series of discrete electronic events that are not directly visible in the energy profile. Pretransmetalation involves stepwise eta 2-aryl coordination, where Pd-Cortho interaction occurs before Pd-Cipso bonding. Transmetalation happens through B-Cipso bond cleavage followed by Pd-Cipso bond formation in separate steps, revealing that aryl transfer occurs through electronically asynchronous steps. These findings reveal that key bond-reorganization events in palladium-catalyzed cross-coupling can proceed through electronically distinct stages even in the absence of detectable energetic signatures, highlighting a fundamental limitation of energy-based mechanistic analysis and establishing ELF topology as a powerful tool to uncover hidden electronic processes in organometallic reactivity.

The photocyclization of 3H-azepines exhibits a striking substituent-dependent preference toward the formation of the 2-azabicyclo[3.2.0]hepta-2,6-diene isomer, in agreement with experimental findings, while 6-substituted analogues remain unobserved. Through time-dependent density functional theory (TD-DFT) and bonding evolution theory (BET) analyses, we reveal how subtle electronic effects govern the excited-state reaction landscape. The nature of the substituent at position 2 dictates the topology of the excited S-1 state. For electron-donating groups (-NH2 and -OEt), the excitation is mainly localized over the olefinic bonds of the ring, whereas electron-withdrawing groups (-CN and -CF3) localize it on the imine bond. Donor-substituted systems access two minimum-energy conical intersections (MECI-1 and MECI-2) on the excited-state surface: deactivation through MECI-1 regenerates the 3H-azepine reactant, while the MECI-2 pathway leads to a strained intermediate that efficiently evolves into the experimentally observed bicyclic product, overcoming low energy barriers (<8 kcal mol(-1)). In contrast, electron-withdrawing substituents only provide access to the MECI-1 pathway, preventing the formation of the bicyclic product. This study provides a unified mechanistic rationale for the regioselectivity of 3H-azepine photocyclization, and illustrates how substituent electronics sculpt the excited-state reactivity landscape of heterocyclic systems.

This mini-review discusses recent advances in the rigorous application of Bonding Evolution Theory (BET) to elucidate electron rearrangements in cycloaddition reactions occurring in both ground and electronically excited states. Computational studies reveal that describing bond formation and cleavage through parametric polynomials derived from the Catastrophe Theory (CT) provides a deeper and more coherent understanding of chemical bonding and reactivity. However, several existing BET applications have adopted CT concepts without fully incorporating the mathematical rigor on which BET is based, resulting in conceptual ambiguities and inaccurate interpretations. A proper implementation of BET requires evaluating the Hessian matrix at potentially degenerate critical points (CPs) of the Electron Localization Function (ELF) and assessing their relative evolution along the reaction coordinate. This systematic protocol integrates key CT principles within BET’s original framework, restoring its formal consistency. The resulting analyses have revealed correlations between electron-density symmetry and CT polynomials, relationships between these polynomials and the homolytic or heterolytic character of bond dissociation, and the development of a CT-based model for scaling bond polarity. These findings demonstrate that incorporating CT-derived functions into BET is not merely a formal refinement but a fundamental step toward achieving a more rigorous and predictive understanding of electron rearrangements in cycloadditions.

Quantitative structure–activity relationship (QSAR) modeling is a central component of molecular informatics and ligand-based drug discovery, enabling the analysis and interpretation of structure–activity relationships from molecular descriptors. In this study, mechanistically interpretable binary QSAR models were developed to classify the activity of 89 serotonin 5-HT1A receptor ligands by integrating Conceptual Density Functional Theory (CDFT) descriptors with no-code machine learning workflows. Neutral and protonated molecular forms were considered under both vacuum and aqueous conditions, and electronic reactivity indices (η, μ, ω, and ΔNmax) were computed at the GFN1-xTB level of theory. Model construction employed the RandomTree algorithm within an OECD-consistent framework, using Ki values as biological endpoints. Across all decision trees and validation splits, the electrophilicity index of protonated molecules in aqueous solvent repeatedly emerged as the root node, suggesting a potential contribution of electrostatic complementarity under physiologically relevant conditions. Chemical hardness of neutral molecules in aqueous phase was identified as a secondary descriptor associated with activity classification, reflecting stability-related effects. External validation yielded a Cohen’s Kappa value of 0.49, with precision, recall, and F1-score above 0.80, and AUC–ROC values between 0.69 and 0.73, indicating moderate but consistent exploratory classification performance within the studied applicability domain. These results suggest that explicitly accounting for protonation and solvation may improve the mechanistic interpretability and exploratory classification behavior of CDFT-based QSAR models for protonatable 5-HT1A ligands. The proposed no-code workflow provides a transparent and reproducible exploratory molecular informatics framework for exploratory and mechanistically interpretable ligand-based modeling in early-stage drug discovery.

We investigate the mechanical and vibrational properties of three quasiplanar two-dimensional (2D) mono-layers (Be2C, Be5C2, and B4C3) containing hypercoordinate carbon centers. Density functional theory was employed to determine the elastic response, Young's modulus, Poisson's ratio, and dynamical stability. B4C3 exhibits the highest in-plane stiffness (<^>249 N/m), nearly isotropic elasticity, and large cohesive energy, consistent with high mechanical resilience. Be5C2 displays strong elastic anisotropy and a negative Poisson's ratio (nu = -0.36), indicative of auxetic behavior. Be2C shows intermediate stiffness and moderate flexibility. Phonon dispersions confirm that all three monolayers are dynamically stable. Electron localization function and crystal orbital Hamilton population analyses reveal extensive multicenter sigma networks complemented by modest pi delocalization (<^>10-15%), which facilitates uniform stress redistribution under deformation. These findings connect the topology of multicenter bonding with macroscopic elastic response, offering design guidelines for lightweight and flexible 2D materials.

Conventional bonding theory associates molecular stability with electronic energy lowering relative to separated fragments. While successful for ordinary molecules, this picture may fail when additional quantum particles contribute to stability. Here we construct a multicomponent conceptual density-functional theory (MC-CDFT) as the natural extension of conceptual DFT to multicomponent quantum systems. Starting from a constrained-search formulation of the multicomponent energy functional, we generalize conceptual DFT from scalar electronic descriptors to species-resolved vectors, matrices, and response kernels that explicitly encode intercomponent coupling. Within this framework, the bonding criterion is governed by the curvature of the total multicomponent energy functional, rather than by the sign of any isolated subsystem contribution. A bound minimum may therefore arise even when the electronic contribution alone is destabilizing, provided that intercomponent coupling compensates for this energetic penalty. Recent Quantum Monte Carlo results for the e+:Be2 complex illustrate this mechanism: although the electronic contribution remains repulsive at all internuclear separations, the total multicomponent system remains bound [R. Porras-Roldan et al., Chem. Sci., 2025, 16, 22322-22332]. The present formulation provides a general conceptual framework for analyzing stability in multicomponent quantum systems in which more than one particle species contributes to bonding.

This study meticulously examines the criteria for assigning electron rearrangements along the intrinsic reaction coordinate (IRC) leading to bond formation and breaking processes during the pyrolytic isomerization of cubane (CUB) to 1,3,5,7 cyclooctatetraene (COT) from both thermochemical and bonding perspectives. Notably, no cusp type function was detected in the initial thermal conversion step of CUB to bicyclo[4.2.0]octa 2,4,7 triene (BOT Contrary to previous reports, all relevant fluxes of the pairing density must be described in terms of fold unfolding. The transannular ring opening in the second step highlights characteristics indicative of a cusp type catastrophe, facilitating a direct comparison with fold features. This fact underscores the critical role of density symmetry persistence near topographical events in determining the type of bifurcation. A fold cusp unified model for scaling the polarity of chemical bonds is proposed , integrating ubiquitous reaction classes such as isomerization, bimolecular nucleophilic substitution, and cycloaddition. The analysis reveals that bond polarity index (BPI) values within the [0, 10 5 au interval correlate with cusp unfolding, whereas fold spans over a broader [10 3 , au spectrum. These insights emphasize that the cusp polynomial is suitable for describing chemical processes involving symmetric electron density distributions, particularly those involving homolytic bond cleavages; in contrast, fold characterizes most chemical events Methods G eometry optimization and frequency calculations were conducted using various DFT functionals . In line with recent finding s concerning the rigorous application of BET , the characterization of bond formations and scissions via unfoldings was carried out by carefully monitoring the determinant of the Hessian matrix at all potentially degenerate CPs and their relative distance . The computed gas phase activation enthalpies strongly align with experimental values, stressing the adequacy of the chosen levels of theory in describ ing the ELF topography along the IRC. The BPI was determined using the methodology proposed by Allen and collaborators

This review analyses the chemical bonding and reactivity implications of recent findings concerning the rigorous application of the bonding evolution theory (BET) to describe electron rearrangements in cycloaddition reactions in the ground and electronically excited states. Computational evidence shows that characterizing formations and scissions of chemical bonds through parametric polynomials derived from catastrophe theory (CT) is critical to gaining further insights into chemical bonding and reactivity. However, most of the insofar application of BET conducts this association without considering the robust mathematical basis supporting BET. Consequently, misinterpretations and incorrect results have arisen because of the inherent ambiguity of such an oversight. The proper utilization of BET involves the calculation of the Hessian matrix at potentially degenerate critical points of the electron localization function (ELF) and measuring their relative distance along a reactive coordinate. This methodical approach is tailored to incorporate key CT concepts into the original BET framework, thereby recovering the rigor of the latter. The systematic application of these steps has led to various unexpected outcomes, including the correlation between electron density symmetry and CT’s polynomials, the interplay between these polynomials and the heterolytic/homolytic character of bond breakages, and a CT-based model for scaling bond polarity. These discoveries underscore the relevance of adhering to concepts and highlight that rationalizing electron reorganizations using CT’s functions is far from a technical subtlety.

Polycyclic aromatic hydrocarbons (PAHs) are persistent pollutants with well-known genotoxic and mutagenic effects, posing risks to ecosystems and human health. Their hydrophobic nature promotes accumulation in soils and aquatic environments, increasing exposure risks. Upon metabolic activation, PAHs generate reactive species that form DNA adducts, driving their mutagenic potential. This study presents an OECD-compliant methodology that integrates conceptual density functional theory (CDFT) calculations at the GFN2-xTB level with machine learning models to predict PAH mutagenicity. Using quantum chemical descriptors of procarcinogens and radical cation metabolites alongside Ames test data, key electronic properties linked to mutagenicity were identified. Feature selection consistently highlighted radical cation descriptors as key indicators of metabolic activation pathways. Machine learning models - including SPAARC, Random Tree, and JCHAID - achieved validation accuracies exceeding 89 %, with minimal false-negative rates, ensuring conservative predictions for environmental risk assessment. The PSL and CDP electrophilicity frameworks proved particularly effective in modeling DNA damage-related processes. This no-code, freeware-based methodology provides a scalable and cost-effective tool for assessing mutagenic risks in environmentally relevant conditions. The findings reinforce the importance of metabolic activation, validate the radical cation as a reliable proxy for this process, and demonstrate the predictive value of electronic properties in QSAR modeling. These insights support advances in environmental toxicology and contribute to improved strategies for regulatory risk assessment.

This work computationally explores the photochemistry of 3H-azepines, revealing a remarkable se- lectivity toward the formation of the 2-azabicyclo[3.2.0]hepta-2,6-diene isomer, in agreement with experimental findings, while the formation of 6-substituted analogues is not observed. Using TD- DFT-based electronic structure calculations, the reaction and deactivation mechanisms responsible for this selectivity were dissected. The results demonstrate that the nature of the substituent at position 2 plays a decisive role in the behavior of the excited S1 state. For electron-donating groups (–NH2 , –OEt), the excitation is mainly localized over the olefinic bonds of the ring, whereas electron- withdrawing groups (–CN, –CF3 ) localize it on the imine bond. Donor-substituted systems access two minimum-energy conical intersections (MECI1 and MECI2) on the excited-state surface: deactivation through MECI1 regenerates the 3H-azepine reactant, while the MECI2 pathway leads to a strained intermediate that efficiently evolves into the experimentally observed bicyclic product, overcoming low energy barriers (< 8 kcal·mol−1 ). In contrast, electron-withdrawing substituents only provide access to the MECI1 pathway, preventing the formation of the bicyclic product. This study provides a fundamental mechanistic rationale for the photocyclization selectivity of 3H-azepines, establishing a direct connection between substituent electronic effects, the topology of the excited-state energy surfaces, and the non-radiative deactivation pathways.

Nitroaromatic compounds (NAs) are widely used in industrial applications but pose significant genotoxic risks, necessitating accurate mutagenicity prediction for chemical safety assessments. This study integrates conceptual density functional theory (CDFT) descriptors with explainable no-code machine learning (ML) models to predict NA mutagenicity based on Ames test results. Following OECD QSAR guidelines, feature selection and model development were performed using decision-tree-based algorithms (Random Tree, JCHAID*, SPAARC) and multilayer perceptrons (MLPs). These models exhibited high predictive accuracy (internal: >80%, kappa = 0.21-0.37; external: similar to 90%, kappa = 0.41-0.62) with strong interpretability. The study also explores the role of metabolic activation and aqueous-phase descriptors, evaluating a novel electronic analog to LogP (LogQP) to assess hydrophobicity-mutagenicity relationships. Results demonstrate that aqueous-phase electronic properties and electrophilicity descriptors outperform vacuum-based methods in mutagenicity prediction. The combination of CDFT descriptors with shallow ML models proves to be a robust, interpretable, and accessible framework for predictive toxicology. This approach enhances chemical risk assessment and bridges computational chemistry with toxicology for regulatory applications.

This study synergizes machine learning (ML) with conceptual density functional theory (CDFT) to develop OECD-compliant predictive models for the mutagenic activity of aromatic amines (AAs) with a fully No-Code methodology using a comprehensive data set of 251 AAs, Leave-One-Out-Cross-Validation (LOOCV), and three distinct data splits. Our research employs the GFN2-xTB method, known for its robustness and speed, to compute descriptors for procarcinogens and their activated metabolites in vacuum and aqueous phases. We evaluate the effectiveness of different theoretical definitions of electrophilicity within CDFT, namely, PSL, GCV, and CDP schemes, and the newly introduced Log QP descriptor to approximate Log P information. SPAARC, RandomTree, and JCHAID* ML methods were used to build explainable predictive models with highly robust internal validation (Avg. Correct Classifications = 76% and Avg. Kappa = 0.29) and external validation (Avg. Correct Classifications = 79% and Avg. Kappa = 0.33) metrics, and the results were compared to those of a two hidden layer Multilayer Perceptron. The results indicate that the second CDP definition for the electrophilicity in both vacuum and aqueous phases and also the newly presented Log QP descriptors are the most important ones for predicting the mutagenic activity of AA (namely omega+Vac CDP2+, omega+Aq CDP2+, and LogQP1+Vac, respectively). The results indicate that metabolic activation, aqueous solvent properties, and the CDP electrophilicity schemes and Log QP should be considered when building predictive models for the mutagenic activity of AA. This study offers a replicable, No-Code approach to QSAR research, making high-level ML and CDFT applications accessible to a broader audience. Future work will expand these methods to other compound families, enhancing predictive capabilities in the study of mutagenic activities and other biological phenomena.