客座编辑: Shubin Liu, Professor (刘述斌教授) Research Computing Center
University of North Carolina
Chapel Hill, NC 27599-3420, USA

特刊介绍
Density functional theory (DFT) has been well accepted as the most successful development in theoretical and computational chemistry in last few decades. Nevertheless, for most people, it is only a computational approach. It lacks well behind in offering chemical understanding and conceptualization. A similar situation occurred many years ago in the development of the molecular orbital theory, whose computational methodology matured well ahead of its chemical conceptualization and understanding such as the frontier orbital theory and Woodward–Hoffmann rules.

Using density or its associated quantities to appreciate the molecular structure, bonding, and reactivity can be witnessed by Bader’s Atoms-In-Molecules theory. It is, however, Robert G. Parr of University of North Carolina, USA, who pioneered this field with the establishment of the theoretical framework called Conceptual DFT. Formulization of concepts such as electronegativity, hardness/softness, Fukui function, electrophilicity, and dual descriptor is a few well-known examples of chemical concepts from DFT. More recent developments have also been reported in the literature.

In this special issue, to promote efforts to quantify chemical understanding in terms of DFT, we invite experts from across the world to present their recent results on this topic. This issue is not simply just another occasion to bring together people with the same interest. Rather, it serves as a reminder to our readers—especially students—that newcomers are welcome to contribute and opportunities are abundant.

In Quantum Information Theory (QIT) the classical measures of information content in probability distributions are replaced by the corresponding resultant entropic descriptors containing the nonclassical terms generated by the state phase or its gradient (electronic current). The classical Shannon (S[p]) and Fisher (I[p]) information terms probe the entropic content of incoherent local events of the particle localization, embodied in the probability distribution p, while their nonclassical phase-companions, S[φ] and I[φ], provide relevant coherence information supplements. Thermodynamic-like couplings between the entropic and energetic descriptors of molecular states are shown to be precluded by the principles of quantum mechanics. The maximum of resultant entropy determines the phase-equilibrium state, defined by “thermodynamic” phase related to electronic density, which can be used to describe reactants in hypothetical stages of a bimolecular chemical reaction. Information channels of molecular systems and their entropic bond indices are summarized, the complete-bridge propagations are examined, and sequential cascades involving the complete sets of the atomic-orbital intermediates are interpreted as Markov chains. The QIT description is applied to reactive systems R=A-B, composed of the Acidic (A) and Basic (B) reactants. The electronegativity equalization processes are investigated and implications of the concerted patterns of electronic flows in equilibrium states of the complementarily arranged substrates are investigated. Quantum communications between reactants are explored and the QIT descriptors of the A-B bond multiplicity/composition are extracted.

Multi-scale quantum-mechanical/molecular-mechanical (QM/MM) and large-scale QM simulation provide valuable insight into enzyme mechanism and structure-property relationships. Analysis of the electron density afforded through these methods can enhance our understanding of how the enzyme environment modulates reactivity at the enzyme active site. From this perspective, tools from conceptual density functional theory to interrogate electron densities can provide added insight into enzyme function. We recently introduced the highly parallelizable Fukui shift analysis (FSA) method, which identifies how frontier states of an active site are altered by the presence of an additional QM residue to identify when QM treatment of a residue is essential as a result of quantum-mechanically affecting the behavior of the active site. We now demonstrate and analyze distance and residue dependence of Fukui function shifts in pairs of residues representing different non-covalent interactions. We also show how the interpretation of the Fukui function as a measure of relative nucleophilicity provides insight into enzymes that carry out S_{N}2 methyl transfer. The FSA method represents a promising approach for the systematic, unbiased determination of quantum mechanical effects in enzymes and for other complex systems that necessitate multi-scale modeling.

The evolution of cesium iodide band gap as a function of pressure is studied in the range from 0 to 60 GPa. Within this range, two structural phase transitions occurred, and the band gap was affected by the compression pressure and structural rearrangement. The band gap estimation under pressure, as obtained by the density functional theory methods, successfully reproduced the experimental trend of the optical gap and electrical resistivity, namely, a general decreasing tendency, an early maximum, and a discontinuous peak around 40 GPa.

Density functional theory-based calculations have been carried out to study the bonding and reactivity in RB-AsR (R=H, F, OH, CH_{3}, CMe_{3}, CF_{3}, SiF_{3}, BO) systems. Our calculations demonstrated that all the studied systems adopted bent geometry (∠R-B-As ≈ 180° and ∠B-As-R ≈ 90° or less). The reason for this bending was explained with the help of a valence-orbital model. The potential energy surfaces for three possible isomers of RB-AsR systems were also generated, indicating that the RB-AsR isomer was more stable than R_{2}B-AsR when R=SiF_{3}, CMe_{3}, and H. The B-As bond character was analyzed using natural bond orbital (NBO) and Wiberg bond index (WBI) calculations. The WBI values for B-As bonds in F_{3}SiB-AsSiF_{3} and HB-AsH were 2.254 and 2.209, respectively, indicating that this bond has some triple-bond character in these systems. While the B centers prefer nucleophilic attack, the As centers prefer electrophilic attack.

In this work, the Fukui functions of the two ^{2}P resonance states of Be^{-}, a ^{2}P resonance state of Mg^{-}, and a ^{2}D resonance state of Ca^{-} have been determined. The trajectories of these resonance states, in conjunction with the complex rotation of the Hamiltonian, were used to determine their wave functions. The electron densities, Fukui functions, and values of the hyper-radius < r^{2} > were computed from these wave functions. The Fukui functions have negative regions in the valence shell in addition to the inner shell regions, indicating screening effects of the outer temporary electron. Selected configuration interactions with up to quadruple excitations were used along the trajectories and for computing the final wave function. Based on this data, the densities, Fukui functions, and < r^{2} > were calculated.

The concept of resonance-assisted hydrogen bonds (RAHBs) highlights the synergistic interplay between the π-resonance and hydrogen bonding interactions. This concept has been well-accepted in academia and is widely used in practice. However, it has been argued that the seemingly enhanced intramolecular hydrogen bonding (IMHB) in unsaturated compounds may simply be a result of the constraints imposed by the σ-skeleton framework. Thus, it is crucial to estimate the strength of IMHBs. In this work, we used two approaches to probe the resonance effect and estimate the strength of the IMHBs in the two exemplary cases of the enol forms of acetylacetone and o-hydroxyacetophenone. One approach is the block-localized wavefunction (BLW) method, which is a variant of the ab initio valence bond (VB) theory. Using this approach, it is possible to derive the geometries and energetics with resonance shut down. The other approach is Edmiston’s truncated localized molecular orbital (TLMO) technique, which monitors the energy changes by removing the delocalization tails from localized molecular orbitals. The integrated BLW and TLMO studies confirmed that the hydrogen bonding in these two molecules is indeed enhanced by π-resonance, and that this enhancement is not a result of σ constraints.

As evidenced from recent literature, interest in employing information theory measures for understanding different properties of atomic and molecular systems is increasing tremendously. Following our earlier efforts in this field, we here evaluate the feasibility of using information theory functionals such as Fisher information, Shannon entropy, Onicescu information energy, and Ghosh-Berkowitz-Parr entropy as measures of steric effects for the steric analysis of water nanoclusters. Taking the structural isomers of water hexamers as working models and using information theoretic quantities, we show that the relative energies of water nanoclusters and the computed steric energies are related. We also show the strong effects of steric repulsion on conformational stabilities. At the same time, we have also assessed the usefulness of simultaneously considering the different information theoretic quantities, and achieved more accurate descriptions of the stability of water nanoclusters. In order to consider the effects of cluster size on the obtained results and the extent of applicability of information theoretic quantities, we have also benchmarked larger water nanoclusters with 32 and 64 units. Scrutinizing the obtained data from information theory functionals, we found that Fisher information shows the best overall performance. Our findings underline that the information theoretic quantities, especially Fisher information, can be used as quantitative measures of relative energies and consequently the order of stability of nanoclusters, which affirmed the utility of information theory for investigating various physical and chemical problems.

The density functional theory and its extension to ensembles of excited states can be formalized as thermodynamics. However, these theories are not unique because one of their key quantities, the kinetic energy density, can be defined in several ways. Usually, the everywhere positive gradient form is applied; however, other forms are also acceptable, provided they integrate to the true kinetic energy. Recently, a kinetic energy density of the ground-state theory maximizing the information entropy has been proposed. Here, ensemble kinetic energy density, leading to extremum information entropy, is derived via constrained search. The corresponding ensemble temperature is found to be constant.

In this study, we show how to generalize Hirshfeld partitioning methods to possibly include non-spherical proatom densities. While this generalization is numerically challenging (requiring global optimization of a large number of parameters), it is conceptually appealing because it allows the proatoms to be pre-polarized, or even promoted, to a state that more closely resembles the atom in a molecule. This method is based on first characterizing the convex set of proatom densities associated with the degenerate ground states of isolated atoms and atomic ions. The preferred orientation of the proatoms' densities are then obtained by minimizing the information–theoretic distance between the promolecular and molecular densities. If contributions from excited states (and not just degenerate ground states) are included in the convex set, this method can describe promoted atoms. While the procedure is intractable in general, if one includes only atomic states that have differing electron-numbers and/or spins, the variational principle becomes a simple convex optimization with a single unique solution.

Tools have been designed obtain information about chemical bonds from quantum mechanical calculations. They work well for solutions of the stationary Schrödinger equation, but it is not clear whether Lewis electron pairs they aim to reproduce survive in time-dependent processes, in spite of the underlying Pauli principle being obeyed in this regime. A simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, is used to study this problem, because it allows presenting the detailed structure of the wave function. It is shown that ⅰ) oscillations persisting after the Hamiltonian stopped changing produce for certain time intervals states where Lewis electron pairs are spatially separated, and ⅱ) methods (like density analysis, or the electron localization function) that are widely used for describing bonding in the stationary case, have limitations in such situations. An exception is provided by the maximum probability domain (the spatial domain that maximizes the probability to find a given number of particles in it). It is conceptually simple, and satisfactorily describes the phenomenon.

Herein we have investigated the interaction between hydrazoic acid (HN_{3}) and a pristine graphyne system based on density functional theory (DFT) method using generalized gradient approximation. The van der Waals dispersion correction is also considered for predicting the possibility of using the graphyne system for detection of hydrazoic acid. Pristine graphyne has a band gap of 0.453 eV, which decreases to 0.424 eV when HN_{3} is adsorbed on graphyne. The electrical conductivity of HN_{3}-adsorbed graphyne is greater than that of its pristine counterpart. Charge transfer analysis reveals that the HN_{3}-adsorbed graphyne system behaves as an n-type semiconductor; however, its pristine analogue acts as an intrinsic semiconductor. Pristine graphyne has zero dipole moment; however, its interaction with HN_{3} increases its dipole moment. The electronic properties of graphyne is significantly influenced by the presence of HN_{3}, leading to the possibility of designing graphyne-based sensors for HN_{3} detection.

In this work it is shown that the kinetic energy and the exchange-correlation energy are mutual dependent on each other. This aspect is first derived in an orbital-free context. It is shown that the total Fermi potential depends on the density only, the individual parts, the Pauli kinetic energy and the exchange-correlation energy, however, are orbital dependent and as such mutually influence each other. The numerical investigation is performed for the orbital-based non-interacting Kohn-Sham system in order to avoid additional effects due to further approximations of the kinetic energy. The numerical influence of the exchange-correlation functional on the non-interacting kinetic energy is shown to be of the order of a few Hartrees. For chemical purposes, however, the energetic performance as a function of the nuclear coordinates is much more important than total energies. Therefore, the effect on the bond dissociation curve was studied exemplarily for the carbon monoxide. The data reveals that, the mutual influence between the exchange-correlation functional and the kinetic energy has a significant influence on bond dissociation energies and bond distances. Therefore, the effect of the exchange-correlation treatment must be considered in the design of orbital-free density functional approximations for the kinetic energy.

By extending the Levy wavefunction constrained search to Fock Space, one can define a wavefunction constrained search for electron densities in systems having noninteger number of electrons. For pure-state v-representable densities, the results are equivalent to what one would obtain with the zero-temperature grand canonical ensemble. In other cases, the wavefunction constrained search in Fock space presents an upper bound to the grand canonical ensemble functional. One advantage of the Fock-space wavefunction constrained search functional over the zero-temperature grand-canonical ensemble constrained search functional is that certain specific excited states (i.e., those that are not ground-state v-representable) are the stationary points of the Fock-space functional. However, a potential disadvantage of the Fock-space constrained search functional is that it is not convex.

In view of its use as reactivity theory, Conceptual Density Functional Theory (DFT), introduced by Parr et al., has mainly concentrated up to now on the E = E[N, v] functional. However, different ensemble representations can be used involving other variables also, such as ρ and μ. In this study, these different ensemble representations (E, ?, F, and R) are briefly reviewed. Particular attention is then given to the corresponding second-order (functional) derivatives, and their analogies with the second-order derivatives of thermodynamic state functions U, F, H, and G, which are related to each other via Legendre transformations, just as the DFT functionals (Nalewajski and Parr, 1982). Starting from an analysis of the convexity/concavity of the DFT functionals, for which explicit proofs are discussed for some cases, the positive/negative definiteness of the associated kernels is derived and a detailed comparison is made with the thermodynamic derivatives.The stability conditions in thermodynamics are similar in structure to the convexity/concavity conditions for the DFT functionals. Thus, the DFT functionals are scrutinized based on the convexity/concavity of their two variables, to yield the possibility of establishing a relationship between the three second-order reactivity descriptors derived from the considered functional. Considering two ensemble representations, F and ?, F is eliminated as it has two dependent (extensive) variables, N and ρ. For ?, on the other hand, which is concave for both of its intensive variables (μ and υ), an inequality is derived from its three second-order (functional) derivatives: the global softness, the local softness, and the softness kernel. Combined with the negative value of the diagonal element of the linear response function, this inequality is shown to be compatible with the Berkowitz-Parr relationship, which relates the functional derivatives of ρ with υ, at constant N and μ. This was recently at stake upon quantifying Kohn’s Nearsightedness of Electronic Matter. The analogy of the resulting inequality and the thermodynamic inequality for the G derivatives is highlighted. Potential research paths for this study are briefly addressed; the analogies between finite-temperature DFT response functions and their thermodynamic counterparts and the quest for analogous relationships, as derived in this paper, for DFT functionals that are analogues of entropy-dimensioned thermodynamic functions such as the Massieu function.

Quantitative correlation of several theoretical electrophilicity measures over different families of organic compounds are examined relative to the experimental values of Mayr et al. Notably, the ability to predict these values accurately will help to elucidate the reactivity and selectivity trends observed in charge-transfer reactions. A crucial advantage of this theoretical approach is that it provides this information without the need of experiments, which are often demanding and time-consuming. Here, two different types of electrophilicity measures were analyzed. First, models derived from conceptual density functional theory (c-DFT), including Parr’s original proposal and further generalizations of this index, are investigated. For instance, the approaches of Gázquez et al. and Chamorro et al. are considered, whereby it is possible to distinguish between processes in which a molecule gains or loses electrons. Further, we also explored two novel electrophilicity definitions. On one hand, the potential of environmental perturbations to affect electron incorporation into a system is analyzed in terms of recent developments in c-DFT. These studies highlight the importance of considering the molecular surroundings when a consistent description of chemical reactivity is needed. On the other hand, we test a new definition of electrophilicity that is free from inconsistencies (so-called thermodynamic electrophilicity). This approach is based on Parr’s pioneering insights, though it corrects issues present in the standard working expression for the calculation of electrophilicity. Additionally, we use machine-learning tools (i.e., symbolic regression) to identify the models that best fit the experimental values. In this way, the best possible description of the electrophilicity values in terms of different electronic structure quantities is obtained. Overall, this straightforward approach enables one to obtain good correlations between the theoretical and experimental quantities by using the simple, yet powerful, interpretative advantage of c-DFT methods. In general, we observed that the correlations found at the HF/6-31G(d) level of theory are of semi-quantitative value. To obtain more accurate results, we showed that working with families of compounds with similar functional groups is indispensable.

Chemical reactivity towards electron transfer is captured by the Fukui function. However, this is not well defined when the system or its ions have degenerate or pseudo-degenerate ground states. In such a case, the first-order chemical response is not independent of the perturbation and the correct response has to be computed using the mathematical formalism of perturbation theory for degenerate states. Spatial pseudo-degeneracy is ubiquitous in nanostructures with high symmetry and totally extended systems. Given the size of these systems, using degenerate-state perturbation theory is impractical because it requires the calculation of many excited states. Here we present an alternative to compute the chemical response of extended systems using models of local softness in terms of the local density of states. The local softness is approximately equal to the density of states at the Fermi level. However, such approximation leaves out the contribution of inner states. In order to include and weight the contribution of the states around the Fermi level, a model inspired by the long-range behavior of the local softness is presented. Single wall capped carbon nanotubes (SWCCNT) illustrate the limitation of the frontier orbital theory in extended systems. Thus, we have used a C_{360} SWCCNT to test the proposed model and how it compares with available models based on the local density of states. Interestingly, a simple Hückel approximation captures the main features of chemical response of these systems. Our results suggest that density-of-states models of the softness along simple tight binding Hamiltonians could be used to explore the chemical reactivity of more complex system, such a surfaces and nanoparticles.

The addition of electrons to form gas-phase multiply charged anions (MCAs) normally requires sophisticated experiments or calculations.In this work, the factors stabilizing the MCAs, the maximum electron uptake of gas-phase molecules, X, and the electronic stability of MCAs X^{Q}^{-}, are discussed. The drawbacks encountered when applying computational and/or conceptual density functional theory (DFT) to MCAs are highlighted. We develop and test a different model based on the valence-state concept. As in DFT, the electronic energy, E(N, v_{ex}), is a continuous function of the average electron number, N, and the external potential, v_{ex}, of the nuclei. The valence-state-parabola is a second-order polynomial that allows extending E(N, v_{ex}) to dianions and higher MCAs. The model expresses the maximum electron acceptance, Q_{max}, and the higher electron affinities, A_{Q}, as simple functions of the first electron affinity, A_{1}, and the ionization energy, I, of the “ancestor” system. Thus, the maximum electron acceptance is Q_{max, calc} = 1 + 12A_{1}/7(I -A_{1}). The ground-state parabola model of the conceptual DFT yields approximately half of this value, and it is termed Q_{max, GS} = ${}^{1}\!\!\diagup\!\!{}_{2}\; $ + A_{1}/(I -A_{1}). A large variety of molecules are evaluated including fullerenes, metal clusters, super-pnictogens, super-halogens (OF_{3}), super-alkali species (OLi_{3}), and neutral or charged transition-metal complexes, AB_{m}L_{n}^{0/+/-}. The calculated second electron affinity A_{2, calc} = A_{1}-(7/12)(I -A_{1}) is linearly correlated to the literature references A_{2, lit} with a correlation coefficient R = 0.998. A_{2} or A_{3} values are predicted for further 24 species. The appearance sizes, n_{ap}^{3-}, of triply charged anionic clusters and fullerenes are calculated in agreement with the literature.

A new definition of the dual descriptor, namely, the thermodynamic dual descriptor, is developed within the grand canonical potential formalism. This new definition is formulated to describe the same physical phenomenon as the original definition proposed by Morell, Grand, and Toro-Labbé (J. Phys. Chem. A 2005, 109, 205), which is characterized by a second-order response of the electron density towards an electron flux. To formulate the new definition, we performed two successive partial derivatives of the average electron density, one with respect to the average number of electrons, and the other with respect to the chemical potential of the electron reservoir. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures up to temperatures of chemical interest, one finds that the thermodynamic dual descriptor can be expressed as ∆f_{T}(r) = (β/2)C[f^{+}(r)-f^{-}(r)], where β = 1/k_{B}T, C is a global quantity that depends on the temperature and global electronic properties of the molecule (the first ionization potential and the electron affinity), C = 1 for systems with zero fractional charge, and C = C_{ω} > 0 (albeit very close to zero) for systems with nonzero fractional charge, , and f^{+}(r) and f^{-}(r) are the nucleophilic and electrophilic Fukui functions, respectively. The quantity within the square brackets is the original definition of the dual descriptor. As the local terms (the ones containing regioselectivity information) are equal to those of the dual descriptor, ∆f_{T}(r) has the same regioselectivity information, multiplied by the global quantity (β/2)C. This implies that the regioselectivity information contained in the original dual descriptor is preserved at all temperatures different from zero, and for any value of C > 0. One of the most important features of this new definition is that it avoids the undesired Dirac delta behavior observed when the second order partial derivative of the average density is taken with respect to the average number of electrons, using the exact density dependence of the average number of electrons.

The kernel energy method (KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries. KEM breaks a molecule into smaller subsets, called kernels, for the purposes of calculation. The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy. A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals. In this study, the kernel expansion for the density matrix is examined in the context of density functional theory (DFT) Kohn-Sham (KS) calculations. A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined, and is then converted into a normalized projector by using the Clinton algorithm. Such normalized projectors are factorizable into linear combination of atomic orbitals (LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis. Both straightforward KEM energies and energies from a normalized, idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion. Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules. In the case of the proof-of-concept system, calculations were performed using the STO-3G and 6-31G(d, p) bases over a range of atomic separations, some very far from equilibrium. The water cluster was calculated in the 6-31G(d, p) basis at an equilibrium geometry. The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases. In the case of the water cluster, the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result. The KS density matrices of this study are applicable to quantum crystallography.

A global and local charge transfer partitioning model, based on the cornerstone theory developed by Robert G. Parr and Robert G. Pearson, which introduces two charge transfer channels (one for accepting electrons (electrophilic) and another for donating (nucleophilic)), is applied to the reaction of a set of indoles with 4, 6-dinitrobenzofuroxan. The global analysis indicates that the prevalent electron transfer mechanism in the reaction is a nucleophilic one on the indoles, i.e., the indoles under consideration transfer electrons to 4, 6-dinitrobenzofuroxan. Evaluating the reactivity descriptors with exchange-correlation functionals including exact exchange (global hybrids) yields slightly better correlations than those obtained with generalized gradient-approximated functionals; however, the trends are preserved. Comparing the trend obtained with the number of electrons donated by the indoles, and predicted by the partitioning model, with that observed experimentally based on the measured rate constants, we propose that the number of electrons transferred through this channel can be used as a nucleophilicity scale to order the reactivity of indoles towards 4, 6-dinitrobenzofuroxan. This approach to obtain reactivity scales has the advantage of depending on the intrinsic properties of the two reacting species; therefore, it opens the possibility that the same group of molecules may show different reactivity trends depending on the species with which they are reacting. The local model allows systematic incorporation of the reactive atoms based on the their decreasing condensed Fukui functions, and the correlations obtained by increasing the number of reactive atoms participating in the local analysis of the transferred nucleophilic charge improve, reaching an optimal correlation, which in the present case indicates keeping three atoms from the indoles and two from 4, 6-dinitrobenzofuroxan. The atoms selected by this procedure provide valuable information about the local reactivity of the indoles. We further show that this information about the most reactive atoms on each reactant, combined with the spatial distribution of the nucleophilic and electrophilic Fukui functions of both reactants, allows one to propose non-trivial candidates of starting geometries for the search of the transition state structures present in these reactions.

In this perspective, we review the chemical information encoded in electron density and other ingredients used in semilocal functionals. This information is usually looked at from the functional point of view: the exchange density or the enhancement factor are discussed in terms of the reduced density gradient. However, what parts of a molecule do these 3D functions represent? We look at these quantities in real space, aiming to understand the electronic structure information they encode and provide an insight from the quantum chemical topology (QCT). Generalized gradient approximations (GGAs) provide information about the presence of chemical interactions, whereas meta-GGAs can differentiate between the different bonding types. By merging these two techniques, we show new insight into the failures of semilocal functionals owing to three main errors: fractional charges, fractional spins, and non-covalent interactions. We build on simple models. We also analyze the delocalization error in hydrogen chains, showing the ability of QCT to reveal the delocalization error introduced by semilocal functionals. Then, we show how the analysis of localization can help understand the fractional spin error in alkali atoms, and how it can be used to correct it. Finally, we show that the poor description of GGAs of isodesmic reactions in alkanes is due to 1, 3-interactions.

Chemical concepts such as structure, bonding, reactivity, etc. have been widely used in the literature and text books to appreciate molecular properties and chemical transformations. Even though modern theoretical and computational chemistry is well established from the perspective of accuracy and complexity, how to quantify these concepts is a still unresolved task. Conceptual density functional theory and its related recent developments provide unique opportunities to tackle this problem. In this Special Issue, 27 contributions from top investigators over the world are collected to highlight the state-of-art research on this topic, which not only showcases the status of where we are now but also unveils a number to possible future directions to be pursued.