Symmetry and reactivity of π-systems in electric and magnetic fields: a perspective from conceptual DFT.

The extension of conceptual density-functional theory (conceptual DFT) to include external electromagnetic fields in chemical systems is utilised to investigate the effects of strong magnetic fields on the electronic charge distribution and its consequences on the reactivity of π-systems. Formaldehyde, H2CO, is considered as a prototypical example and current-density-functional theory (current-DFT) calculations are used to evaluate the electric dipole moment together with two principal local conceptual DFT descriptors, the electron density and the Fukui functions, which provide insight into how H2CO behaves chemically in a magnetic field. In particular, the symmetry properties of these quantities are analysed on the basis of group, representation, and corepresentation theories using a recently developed automatic program for symbolic symmetry analysis, QSYM2. This allows us to leverage the simple symmetry constraints on the macroscopic electric dipole moment components to make profound predictions on the more nuanced symmetry transformation properties of the microscopic frontier molecular orbitals (MOs), electron densities, and Fukui functions. This is especially useful for complex-valued MOs in magnetic fields whose detailed symmetry analyses lead us to define the new concepts of modular and phasal symmetry breaking. Through these concepts, the deep connection between the vanishing constraints on the electric dipole moment components and the symmetry of electron densities and Fukui functions can be formalised, and the inability of the magnetic field in all three principal orientations considered to induce asymmetry with respect to the molecular plane of H2CO can be understood from a molecular perspective. Furthermore, the detailed forms of the Fukui functions reveal a remarkable reversal in the direction of the dipole moment along the CO bond in the presence of a parallel or perpendicular magnetic field, the origin of which can be attributed to the mixing between the frontier MOs due to their subduced symmetries in magnetic fields. The findings in this work are also discussed in the wider context of a long-standing debate on the possibility to create enantioselectivity by external fields.

Theoretical Study on the Regioselectivity of Leapfrog B18 and B30 Boron Sheets in Electrophilic and Nucleophilic Reactions Using DFT-Based Reactivity Indices.

A combined computational and interpretational DFT study is performed to investigate the regioselectivity of B18 and B30 leapfrog boron sheets upon reaction with XH3-type electrophiles and nucleophiles (X = N, P, As, B, Al). The M062X, B3LYP, and B3LYP-D3 functionals are used combined with the 6-31+G(d,p) basis. The molecular electrostatic potential (MEP), Fukui functions, and the dual descriptor are employed to predict the local reactivity of B18 and B30. Our results reveal that both clusters are hard and prefer to react with hard bases and acids, such as NH3 and BH3. Further, these leapfrog B6n clusters can play the role of catalysts as they break B-H and Al-H bonds of BH3 and AlH3 in s-BH3-B6n and s-AlH3-B6n complexes, respectively. Leapfrog B6n-XH3 complexes (X = B and Al) can be considered as an interaction between two electron-deficient systems. Therefore, the chemical reactivity between these systems cannot be interpreted in terms of the Hard-Soft-Acid-Base principle.

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies.

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.

Wandering through quantum-mechanochemistry: from concepts to reactivity and switches.

Mechanochemistry has experienced a renaissance in recent years witnessing, at the molecular level, a remarkable interplay between theory and experiment. Molecular mechanochemistry has welcomed a broad spectrum of quantum-chemical methods to evaluate the influence of an external mechanical force on molecular properties. In this contribution, an overview is given on recent work on quantum mechanochemistry in the Brussels Quantum Chemistry group (ALGC). The effect of an external force was scrutinized both in fundamental topics, like reactivity descriptors in Conceptual DFT, and in applied topics, such as designing molecular force probes and tuning the stereoselectivity of certain types of reactions. In the conceptual part, a brief overview of the techniques introducing mechanical forces into a quantum-mechanical description of a molecule is followed by an introduction to conceptual DFT. The evolution of the electronic chemical potential (or electronegativity), chemical hardness and electrophilicity are investigated when a chemical bond in a series of diatomics is put under mechanical stress. Its counterpart, the influence of mechanical stress on bond angles, is analyzed by varying the strain present in alkyne triple bonds by applying a bending force, taking the strain promoted alkyne-azide coupling cycloaddition as an example. The increase of reactivity of the alkyne upon bending is probed by Fukui functions and the local softness. In the applied part, a new molecular force probe is presented based on an intramolecular 6π-electrocyclization in constrained polyenes operating under thermal conditions. A cyclic process is conceived where ring opening and closure are triggered by applying or removing an external pulling force. The efficiency of mechanical activation strongly depends on the magnitude of the applied force and the distance between the pulling points. The idea of pulling point distances as a tool to identify new mechanochemical processes is then tested in [28]hexaphyrins with an intricate equilibrium between Möbius aromatic and Hückel antiaromatic topologies. A mechanical force is shown to trigger the interconversion between the two topologies, using the distance matrix as a guide to select appropriate pulling points. In a final application, the Felkin-Anh model for the addition of nucleophiles to chiral carbonyls under the presence of an external mechanical force is scrutinized. By applying a force for restricting the conformational freedom of the chiral ketone, otherwise inaccessible reaction pathways are promoted on the force-modified potential energy surfaces resulting in a diastereoselectivity different from the force-free reaction.

Investigating the Linear Response Function under Approximations Following the Coupled-Perturbed Approach for Atoms and Molecules.

The linear response kernel also referred to as linear response function (LRF) in the framework of conceptual density functional theory has gained tremendous success in time-dependent density functional theory. Comparatively less attention has been devoted to the LRF from a chemical reactivity perspective in its time- or frequency-independent context, although it has recently been used to qualitatively describe electron delocalization, (anti-)aromaticity, inductive and mesomeric effects, etc. Despite these successes, which were obtained by approximating the LRF using the independent particle approximation deriving from a coupled-perturbed Kohn-Sham computation, the robustness of this LRF approach needs to be assessed. In this work, we compute the LRF at four levels of approximation (independent particle approximation, random phase approximation, Hartree-Fock approximation, and the (exact) DFT (density functional theory) expression) using functionals from the first four rungs of Jacob's ladder of exchange-correlation energy functionals. To scrutinize the impact of these approximations, new visualization strategies are discussed and systematized. The overall conclusion is that the independent particle approximation yields qualitatively correct results (ergo previous conceptual applications of the LRF are trustworthy), but for quantitative results, LRF expressions including coulomb and exchange(-correlation) terms should be included. With respect to functionals, density-gradient contributions to the exchange-correlation kernel are less than 10% and may be omitted safely where that is preferable computationally.

Mechanochemical Felkin-Anh Model: Achieving Forbidden Reaction Outcomes with Mechanical Force.

Anti-Felkin-Anh diastereoselectivity can be achieved for nucleophilic additions to α-chiral ketones upon stretching the ketone with a mechanical pulling force. Herein, a mechanochemical Felkin-Anh model is proposed for predicting the outcome of a nucleophilic addition to an α-chiral ketone. Essentially, the fully stretched chiral ketone has one substituent shielding each side of the carbonyl, in contrast to the Felkin-Anh model, in which free rotation around a bond is required to achieve the two rotamers of the ketone. Depending on the pulling scenario, either Felkin-Anh or anti-Felkin-Anh diastereoselectivity is obtained. The model is entirely based on the distance between the pulling points, which is maximized in the anti-periplanar arrangement. The major diastereomer is associated with the approach with the least steric interactions. The intuitive model is validated by means of mechanochemical density functional theory calculations. Importantly, the ketone is fully stretched in the sub 1 nN force regime, thus minimizing the risk of undesired homolytic bond rupture. Moreover, the mechanical force is not used for lowering the reaction barriers associated with the nucleophilic addition; instead, it is solely applied for locking the conformation of a molecule and provoking otherwise inaccessible reaction pathways on the force-modified potential energy surface.

External fields in conceptual density functional theory.

The necessity of the recent incorporation of new external variables in the context of conceptual DFT (CDFT) is discussed based on the ever-increasing portfolio of experimental reaction conditions in the endeavor of experimentalists to synthesize new molecules with unprecedented properties. Electric and magnetic fields (ε and B), mechanical forces (F), and confinement are proposed as valuable new variables, extending conventional CDFT and its associated response functions. A finite field approach is used to calculate the evolution of both global and local descriptors in a selected series of atomic and molecular applications, and from it derive new response function involving, with one exception, the first derivative to the field considered. The electric field results, displaying, for example, a case of a field-induced enantioselectivity in the Fukui function, may be instrumental in the recent upsurge of chemistry in oriented external electric fields. The study of atomic electronegativity and hardness in magnetic fields displays a piecewise behavior, associated to configurational jumps upon increasing field strength and reveals an overall compression of their ranges for stronger fields, which may be guiding upon investigating chemistry in extremely high fields like in white dwarfs. The evolution of the electronegativity and hardness of diatomics under mechanical force can elegantly be traced back to differences in their equilibrium distance in the neutral, cationic, and anionic state. The well-known reduction of the polarizability under confinement can be seen as a fore-runner of the increasing hardness of atoms under pressure, presently under investigation. Periodicity showing up in a spontaneous way in the variety of properties is a leitmotiv in this study, as well as the interconnections/analogies between the different response functions.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Properties of the density functional response kernels and its implications on chemistry.

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.

From Density Functional Theory to Conceptual Density Functional Theory and Biosystems.

The position of conceptual density functional theory (CDFT) in the history of density functional theory (DFT) is sketched followed by a chronological report on the introduction of the various DFT descriptors such as the electronegativity, hardness, softness, Fukui function, local version of softness and hardness, dual descriptor, linear response function, and softness kernel. Through a perturbational approach they can all be characterized as response functions, reflecting the intrinsic reactivity of an atom or molecule upon perturbation by a different system, including recent extensions by external fields. Derived descriptors such as the electrophilicity or generalized philicity, derived from the nature of the energy vs. N behavior, complete this picture. These descriptors can be used as such or in the context of principles such as Sanderson's electronegativity equalization principle, Pearson's hard and soft acids and bases principle, the maximum hardness, and more recently, the minimum electrophilicity principle. CDFT has known an ever-growing use in various subdisciplines of chemistry: from organic to inorganic chemistry, from polymer to materials chemistry, and from catalysis to nanotechnology. The increasing size of the systems under study has been coped with thanks to methodological evolutions but also through the impressive evolution in software and hardware. In this flow, biosystems entered the application portfolio in the past twenty years with studies varying (among others) from enzymatic catalysis to biological activity and/or the toxicity of organic molecules and to computational peptidology. On the basis of this evolution, one can expect that "the best is yet to come".

Extending conceptual DFT to include external variables: the influence of magnetic fields.

An extension of conceptual DFT to include the influence of an external magnetic field is proposed in the context of a program set up to cope with the ever increasing variability of reaction conditions and concomitant reactivity. The two simplest global reactivity descriptors, the electronic chemical potential (µ) and the hardness (η), are considered for the main group atoms H-Kr using current density-functional theory. The magnetic field strength, |B|, is varied between 0.0 and 1.0 B 0 = he -1 a 0 -2 ≈ 2.3505 × 105 T, encompassing the Coulomb and intermediate regimes. The carbon atom is studied as an exemplar system to gain insight into the behaviour of the neutral, cationic and anionic species under these conditions. Their electronic configurations change with increasing |B|, leading to a piecewise behaviour of the ionization energy (I) and electron affinity (A) values as a function of |B|. This results in complex behaviour of properties such as the electronegativity χ = -1/2(I + A) = -µ and hardness η = 1/2(I - A). This raises an interesting question: to what extent are atomic properties periodic in the presence of a magnetic field? In the Coulomb regime, close to |B| = 0, we find the familiar periodicity of the atomic properties, and make the connections to response functions central to conceptual DFT. However, as the field increases in the intermediate regime configurational changes of the atomic species lead to discontinuous changes in their properties; fundamentally changing their behaviour, which is illustrated by constructing a periodic table of χ and η values at |B| = 0.5 B 0. These values tend to increase for groups 1-2 and decrease for groups 16-18, leading to a narrower range overall and suggesting substantial changes in the chemistry of the main group elements. Changes within each group are also examined as a function of |B|. These are more complex to interpret due to the larger number of configurations accessible to heavier elements at high field. This is illustrated for group 17 where Cl and Br have qualitatively different configurations to their lighter cogener at |B| = 0.5 B 0. The insight into periodic trends in strong magnetic fields may provide a crucial starting point for predicting chemical reactivity under these exotic conditions.

A Combined Experimental/Quantum-Chemical Study of Tetrel, Pnictogen, and Chalcogen Bonds of Linear Triatomic Molecules.

Linear triatomic molecules (CO2, N2O, and OCS) are scrutinized for their propensity to form perpendicular tetrel (CO2 and OCS) or pnictogen (N2O) bonds with Lewis bases (dimethyl ether and trimethyl amine) as compared with their tendency to form end-on chalcogen bonds. Comparison of the IR spectra of the complexes with the corresponding monomers in cryogenic solutions in liquid argon enables to determine the stoichiometry and the nature of the complexes. In the present cases, perpendicular tetrel and pnictogen 1:1 complexes are identified mainly on the basis of the lifting of the degenerate ν 2 bending mode with the appearance of both a blue and a red shift. Van 't Hoff plots of equilibrium constants as a function of temperature lead to complexation enthalpies that, when converted to complexation energies, form the first series of experimental complexation energies on sp1 tetrel bonds in the literature, directly comparable to quantum-chemically obtained values. Their order of magnitude corresponds with what can be expected on the basis of experimental work on halogen and chalcogen bonds and previous computational work on tetrel bonds. Both the order of magnitude and sequence are in fair agreement with both CCSD(T) and DFA calculations, certainly when taking into account the small differences in complexation energies of the different complexes (often not more than a few kJ mol-1) and the experimental error. It should, however, be noted that the OCS chalcogen complexes are not identified experimentally, most probably owing to entropic effects. For a given Lewis base, the stability sequence of the complexes is first successfully interpreted via a classical electrostatic quadrupole-dipole moment model, highlighting the importance of the magnitude and sign of the quadrupole moment of the Lewis acid. This approach is validated by a subsequent analysis of the molecular electrostatic potential, scrutinizing the σ and π holes, as well as the evolution in preference for chalcogen versus tetrel bonds when passing to "higher" chalcogens in agreement with the evolution of the quadrupole moment. The energy decomposition analysis gives further support to the importance/dominance of electrostatic effects, as it turns out to be the largest attractive term in all cases considered, followed by the orbital interaction and the dispersion term. The natural orbitals for chemical valence highlight the sequence of charge transfer in the orbital interaction term, which is dominated by an electron-donating effect of the N or O lone-pair(s) of the base to the central atom of the triatomics, with its value being lower than in the case of comparable halogen bonding situations. The effect is appreciably larger for TMA, in line with its much higher basicity than DME, explaining the comparable complexation energies for DME and TMA despite the much larger dipole moment for DME.

Designing Force Probes Based on Reversible 6π-Electrocyclizations in Polyenes Using Quantum Chemical Calculations.

The conjugated π-system in polyenes can be interrupted by electrocyclic ring-closure reactions. In this work, this 6π-electrocylization is shown by means of density functional calculations to be reversible by the application of an external mechanical pulling force at the terminal ends of the interrupted polyene chain. The test systems were constrained in a fused ring system, thus locking the orientation of three π-bonds and generally promoting 6π-electrocyclic ring-closure reactions. For several systems, the forward reaction is exergonic and the corresponding reaction barrier is comparable to those reported in the literature. The reverse reaction is triggered by an external pulling force of 2 nN (nano-Newton) or less and also becomes exergonic in all investigated polyenes under these force conditions. Moreover, it proceeds via a low reaction barrier when a pulling force of 2 nN is active, indicating that the mechanical force is an efficient stimulus for triggering ring-opening reactions. Analysis of the strain energy induced by this mechanical force confirms an optimal activation of the corresponding C-C σ-bond that breaks upon ring opening when the pulling positions are located on the polyene chain.

Extending conceptual DFT to include additional variables: oriented external electric field.

The extension of the E = E[N, v] functional for exploring chemical reactivity in a conceptual DFT context to include external electric fields is discussed. Concentrating on the case of a homogeneous field the corresponding response functions are identified and integrated, together with the conventional response functions such as permanent dipole moment and polarizability, in an extended response function tree associated with the E = E[N, v, ε] functional. In a case study on the dihalogens F2, Cl2, Br2, I2 the sensitivity of condensed atomic charges (∂q/∂ε) is linked to the polarizability of the halogen atoms. The non-integrated (∂ρ(r)/∂ε) response function, directly related to the field induced density change, is at the basis of these features. It reveals symmetry breaking for a perpendicular field, not detectable in its atom condensed counterpart, and accounts for the induced dipole moment directly related to the molecular polarizability. The much higher sensitivity of the electronic chemical potential/electronegativity as compared to the chemical hardness is highlighted. The response of the condensed Fukui functions to a parallel electric field increases when going down in the periodic table and is interpreted in terms of the extension of the outer contours in the non-condensed Fukui function. In the case of a perpendicular field the (∂f(r)/∂ε) response function hints at stereoselectivity with a preferential side of attack which is not retrieved in its condensed form. In an application the nucleophilic attack on the carbonyl group in H2CO is discussed. Similar to the dihalogens, stereoselectivity is displayed in the Fukui function for nucleophilic attack (f+) in the case of a perpendicular electric field, and opposite to the one that would arise based on the induced density. Disentangling the expression for the evolution of the Fukui function in the presence of an electric field reveals that this difference can be traced back to local differences in the polarization or induced density between the anionic and the neutral system. This difference may be exploited, e.g. for an appropriately substituted H2CO, to generate enantioselectivity.

Mechanochemically Triggered Topology Changes in Expanded Porphyrins.

A hitherto unexplored class of molecules for molecular force probe applications are expanded porphyrins. This work proves that mechanical force is an effective stimulus to trigger the interconversion between Hückel and Möbius topologies in [28]hexaphyrin, making these expanded porphyrins suitable to act as conformational mechanophores operating at mild (sub-1 nN) force conditions. A straightforward approach based on distance matrices is proposed for the selection of pulling scenarios that promote either the planar Hückel topology or the three lowest lying Möbius topologies. This approach is supported by quantum mechanochemical calculations. Force distribution analyses reveal that [28]hexaphyrin selectively allocates the external mechanical energy to molecular regions that trigger Hückel-Möbius interconversions, explaining why certain pulling scenarios favor the Hückel two-sided topology and others favor Möbius single-sided topologies. The meso-substitution pattern on [28]hexaphyrin determines whether the energy difference between the different topologies can be overcome by mechanical activation.

Alkaline Earth Metals Activate N2 and CO in Cubic Complexes Just Like Transition Metals: A Conceptual Density Functional Theory and Energy Decomposition Analysis Study.

Following the recent discovery of stable octa-coordinated alkaline earth metals with N2 and CO, the role of group II metals in the catalytic reduction of these ligands by means of density functional theory (DFT) calculations and conceptual DFT-based reactivity indices is investigated. Cubic group IV and octahedral group VI transition metal complexes as well as the free ligands are computed for reference. The outer and most accessible atoms of N2 and CO become much more nucleophilic and electrophilic in all complexes, relevant for N2 fixation, as probed by the Fukui function and local softness. Within one row of the periodic table, the alkaline earth complexes often show the strongest activation. On the contrary, the electrostatic character is found to be virtually unaffected by complexation. Trends in the soft frontier orbital and hard electrostatic character are in agreement with calculated proton affinities and energy decomposition analyses of the protonated structures, demonstrating the dominance of the soft (HOMO-LUMO) orbital interactions.

How Do Local Reactivity Descriptors Shape the Potential Energy Surface Associated with Chemical Reactions? The Valence Bond Delocalization Perspective.

How do local reactivity descriptors, such as the Fukui function and the local spin density distribution, shape the potential energy surface (PES) associated with chemical reactions and thus govern reactivity trends and regioselective preferences? This is the question that is addressed here through a qualitative valence bond (VB) analysis. We demonstrate that common density functional theory (DFT)-based local reactivity descriptors can essentially be regarded-in one way or another-as indirect measures of delocalization, i.e., resonance stabilization, of the reactants within VB theory. The inherent connection between (spatial) delocalization and (energetic) resonance stabilization embedded in VB theory provides a natural and elegant framework for analyzing and comprehending the impact of individual local reactivity descriptors on the global PES. Our analysis provides new insights into the role played by local reactivity descriptors and illustrates under which conditions they can sometimes fail to predict reactivity trends and regioselective preferences, e.g., in the case of ambident reactivity. This treatment constitutes a first step toward a unification of VB theory and conceptual DFT.

Switching between Hückel and Möbius aromaticity: a density functional theory and information-theoretic approach study.

Benziporphyrins are versatile macrocycles exhibiting aromaticity switching behaviors. The existence of both Hückel and Möbius (anti)aromaticity has been reported in these systems, whose validity is respectively governed by the [4n + 2] and [4n] π-electron rule on the macrocyclic pathway. Despite the experimental evidence on the floppiness of benziporphyrins, the switching mechanism between Hückel and Möbius structures is still not clear, as well as the factors influencing the stability of the different π-conjugation topologies. For these reasons, we performed a systematic study on A,D-di-p-benzihexaphyrins(1.1.1.1.1.1) with two redox states corresponding to [28] and [30] π-electron conjugation pathways. Whereas benzi[28]hexaphyrin obeys Möbius aromaticity, benzi[30]hexaphyrin follows Hückel aromaticity. The dynamic interconversion between Möbius and Hückel aromaticity is investigated through the rotation of a phenylene ring, which acts as the topology selector. Further analyses of the energy profiles using energy decomposition and information-theoretic approaches provide new insights into conformational stability, aromaticity and antiaromaticity for these species. Strong and opposite cross correlations between aromaticity indexes and information-theoretic quantities were found for the two macrocyclic systems with opposite global aromaticity and antiaromaticity behaviors. These results indicate that Hückel and Möbius aromaticity and antiaromaticity, though qualitatively different, are closely related and can be interchanged, and information-theoretic quantities provide a novel understanding about their relevance. Our present results should provide in-depth insights to appreciate the nature and origin about Möbius (anti)aromaticity and its close relationship with Hückel (anti)aromaticity.