Conceptual Density Functional Theory

We extend the framework of conceptual density functional theory (DFT) to include fourth-order energy derivatives. We present the series of quantum reactivity descriptors at this order for the canonical and the grand canonical ensembles. After introducing a direct computational methodology, we investigate and describe the quartic curvature λ and the third-order Fukui function f(3)(r). These higher-order descriptors capture nonlinear aspects of electronic reactivity that go beyond the conventional concepts of chemical potential, hardness, or Fukui functions. The global descriptor λ, obtained via finite-difference approximations of frontier orbital energies, quantifies the curvature of the chemical hardness and offers insight into the electronic (in)stability of molecular systems under charge perturbations. The local descriptor f(3)(r), derived as the third-order response of the electron density, reveals spatially resolved regions of charge-transfer sensitivity. We apply these descriptors to a series of push-pull organic molecules and open-shell systems, demonstrating their ability to highlight differences in internal charge transfer character and electronic delocalization. The results support the use of fourth-order conceptual DFT tools as chemically meaningful indicators of reactivity beyond the harmonic regime.

Conceptual Density Functional Theory and Some Recent Developments

Taylor-Expansion Monte Carlo Simulations of Classical Fluids in the Canonical and Grand Canonical Ensemble

The chemical hardness concept and its realization within the conceptual density functional theory is approached with innovative perspectives, such as the electronegativity and hardness equalization of atoms in molecules connected with the softness kernel, in order to examine the structure-reactivity equalization ansatz between the electronic sharing index and the charge transfer either in the additive or geometrical mean picture of bonding. On the other hand, the maximum hardness principle presents a relation with the chemical stability of the hardness concept. In light of the inverse relation between hardness and polarizability, the minimum polarizability principle has been proposed. Additionally, this review includes important applications of the chemical hardness concept to solid-state chemistry. The mentioned applications support the validity of the electronic structure principles regarding chemical hardness and polarizability in solid-state chemistry.

Metal hexafluorides hydrolyze at ambient temperature to deposit compounds having fluorine-to-oxygen ratios that depend upon the identity of the metal. Uranium-hexafluoride hydrolysis, for example, deposits uranyl fluoride (UO2F2), whereas molybdenum hexafluoride (MoF6) and tungsten hexafluoride deposit trioxides. Here, we pursue general strategies enabling the prediction of depositing compounds resulting from multi-step gas-phase reactions. To compare among the three metal-hexafluoride hydrolyses, we first investigate the mechanism of MoF6 hydrolysis using hybrid density functional theory (DFT). Intermediates are then validated by performing anharmonic vibrational simulations and comparing with infrared spectra [McNamara et al., Phys. Chem. Chem. Phys. 25, 2990 (2023)]. Conceptual DFT, which is leveraged here to quantitatively evaluate site-specific electrophilicity and nucleophilicity metrics, is found to reliably predict qualitative deposition propensities for each intermediate. In addition to the nucleophilic potential of the oxygen ligands, several other contributing characteristics are discussed, including amphoterism, polyvalency, fluxionality, steric hindrance, dipolar strength, and solubility. To investigate the structure and composition of pre-nucleation clusters, an automated workflow is presented for the simulation of particle growth. The workflow entails a conformer search at the density functional tight-binding level, structural refinement at the hybrid DFT level, and computation of a composite free-energy profile. Such profiles can be used to estimate particle nucleation kinetics. Droplet formation is also considered, which helps to rationalize the different UO2F2 particle morphologies observed under varying levels of humidity. Development of predictive methods for simulating physical and chemical deposition processes is important for the advancement of material manufacturing involving coatings and thin films.

We have studied the predictions of the canonical ensemble concerning the occupation probability for a model with an infinite uniform ladder of degenerate levels. Results show that in the high-degeneracy or high-temperature limit, the canonical ensemble occupation probability would be the same as that under the grand canonical ensemble (given by the Fermi-Dirac distribution). The difference between them is of the order of O(1/g) where g is the degeneracy of the levels. In the low-temperature limit, this difference depends on the corresponding Fermi energy. If the number of electrons is such that the Fermi energy is equal to the energy of a highly degenerate level, then this difference is small. The maximum difference occurs when the Fermi energy is in between two levels. These results are applicable to 1D perfect mesoscopic rings.

Fluctuation formulas for the external potential v(r) are introduced in a modified Legendre-transformed representation of the density functional theory of electronic structure (isomorphic ensemble). A new (nuclear/geometric) reactivity index h(r), having the same status as the electronic Fukui function in the canonical ensemble, is thereby identified, h(r)=(1/N)[δμ/δσ(r)]N,T=(1/kT) [〈μ⋅v(r)〉−〈μ〉〈v(r)〉], where μ is the electronic chemical potential, σ is the shape factor of the electron density distribution, N is the number of electrons, 〈...〉 denotes the ensemble average of a quantity, and 〈v(r)〉 is the ensemble averaged external potential. This new local quantity is shown to be an inverse of the local softness, and to provide a useful definition of a local hardness.

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few.

Highly Hazardous Pesticides (HHPs) pose severe risks to human health and the environment, making it essential to understand their molecular stability and degradation pathways. In this study, the Quantum Theory of Atoms in Molecules (QTAIM) was applied to four representative organophosphate pesticides, allowing the identification of electronically weak bonds as intrinsic sites of lability. These findings are consistent with reported hydrolytic, oxidative, enzymatic, and microbial degradation routes. Importantly, QTAIM descriptors proved largely insensitive to solvation, confirming their intrinsic character within the molecular electronic structure. To complement QTAIM, conceptual DFT (Density Functional Theory) reactivity indices were analyzed, revealing that solvent effects induce more noticeable variations in global and local descriptors than in topological parameters. In addition, a Topological Analysis of the Fukui Function (TAFF) was performed, which mapped nucleophilic, electrophilic, and radical susceptibilities directly onto QTAIM basins. The TAFF analysis confirmed that bonds identified as weak by QTAIM (notably P–O, P–S, and P–N linkages) also coincide with the most reactive sites, thereby reinforcing their mechanistic role in degradation pathways. This integrated framework highlights the robustness of QTAIM, the sensitivity of global and local reactivity descriptors to solvation revealed by conceptual DFT, and the complementary insights provided by TAFF, contributing to risk assessment, remediation strategies, and the rational design of safer pesticides.

The catalytic activity of a heterogeneous catalyst is routinely described by the Sabatier principle using molecule/proton adsorption energy ΔGH as the thermodynamic descriptor. This descriptor, however, fails in quantitatively tracking the reaction rate due to the lack of charge transfer information during a reaction. Herein, we use density functional theory (DFT) calculations combining both canonical and grand-canonical ensembles to report a positive correlation between reaction kinetics and interfacial charge transfer for the hydrogen evolution reaction (HER). The Sabatier relationship for the HER catalysts exhibits a large dispersion in the canonical calculation but an improved linearity under the grand-canonical ensemble, indicating that surface charges are indeed important. This charge effect can be well captured by the Mulliken electronegativity χ of a catalyst because the reaction barrier shows a linear dependence on χ for a wide range of catalysts. Specifically, a catalyst with lower electronegativity exhibits a lower barrier and thus a faster reaction rate. This electronegativity principle is reaction route and pH independent and is founded on the thermoneutral requirement on ΔGH.

In this work, we make use of a model chemistry within Density Functional Theory (DFT) recently presented, which is called CHIH-DFT, to calculate the molecular structure of 3,4-diphenyl-4-(4-hydroxyphenyl)-1,2,5-thiadiazoline 1,1-dioxide (Tphenol), as well to predict its infrared (IR), ultraviolet (UV-Vis) and fluorescence (Fluo) spectra, the dipole moment and polarizability, the free energy of solvation in different solvents as an indication of solubility, and the chemical reactivity parameters that arise from Conceptual DFT. The calculated values are compared with the available experimental data for this molecule as a means of validation of our proposed model chemistry.

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.

The structures, electronic properties, bonding characters and UV–Vis spectra of [Formula: see text] ([Formula: see text]–[Formula: see text]) transition-metal phthalocyanines (TMPcs) molecules have been studied with different density function theory (DFT) methods. The calculated structural parameters agree well with previous experimental or theoretical values. Natural Population Analysis (NPA) charge revealed that [Formula: see text]–[Formula: see text] hybridizations occur when [Formula: see text] TM atoms are involved in chemical bondings. The spin magnetic moments of TMPcs are mainly from the contribution of [Formula: see text] electrons. Conceptual density functional theory (CDFT) results indicate that [Formula: see text] TMPcs molecules are willing to accept further electrons. The TM–N chemical bonds show very weak covalent nature, and are consistent with bond order analyses. Time-dependent density function theory (TD-DFT) calculations were carried out to simulate the UV–Vis spectra, and corresponding electronic transfers for dominant peaks were also obtained.

In this work we studied three dyes which are proposed for potential photovoltaic applications and named Dye7, Dye7-2t and Dye7-3t. The Density Functional Theory (DFT) was utilized, using the M05-2X hybrid meta-GGA functional and the 6–31+G(d,p) basis set. This level of calculation was used to find the optimized molecular structure and to predict the main molecular vibrations, the absorption and emission spectra, the molecular orbitals energies, dipole moment, isotropic polarizability and the chemical reactivity parameters that arise from Conceptual DFT. Also, the pKa values were calculated with the semi-empirical PM6 method.

The antiradical properties and possible mechanisms of action of the tautomers of curcumin, caffeic acid phenethyl ester (CAPE), and chicoric acid (CA) have been studied within density functional theory (DFT). We calculated global chemical reactivity descriptors from conceptual DFT, pKa, bioavailability, and toxicity to evaluate the antiradical properties and characterize these species. Our final level of theory is the M06-2X functional with the 6-31 + G* basis set; we selected this level after performing a benchmark calibration and validation among different levels. Solvent effects were modeled via the continuum solvation model based on density (SMD). We used water and pentyl ethanoate as solvents to simulate the physiological conditions. The free radical scavenger capacity was analyzed for three possible oxidative stress mechanisms: single electron transfer (SET), hydrogen atom transfer (HAT), and xanthine oxidase (XO) inhibition. The results indicate that neutral curcumin, CA, and CAPE behave as antireductants. The most favorable mechanism turns out to be HAT, where CA and CAPE stand out. In conclusion, our DFT study strongly indicates that neutral curcumin, CAPE, and CA would very likely perform well as antiradical drugs with recommended therapeutic use, supported by their non-toxic nature.