Decay Rate of Correlated Real-Space Delocalization Measures: Insights into Chemical Bonding and Mott Transitions from Hydrogen Chains

Localization and Delocalization in Solids from Electron Distribution Functions

The extent of electron localization and delocalization in molecular and condensed phases has been the subject of intense scrutiny over the years. In Chemistry, where real, instead of momentum space viewpoints are many times closer to intuition, a plethora of localization descriptors exist, including a family of indices invariant under orbital transformations that rely only on an underlying partition of the physical space into meaningful regions. These localization and delocalization indices measure the fluctuation of the electron population contained in such domains, and have been rigorously related to the insulating or conductive character of extended systems. Knowledge of the full electron population probability distribution function is also available in molecules, where it has provided many meaningful results as well as uncovered exotic interaction regimes in excited states. Electron distribution functions (EDFs), which can be seen as real space analogs of Pauling resonance structures, are now reported in periodic systems. In agreement with what is known in finite systems, ionic compounds display narrow EDFs that get wider as covalency sets in. Contrarily to conventional wisdom, most electrons delocalize over their nearest neighbors, even in quasi electron-gas metals like sodium, and it is only in the decay rate of the probability distribution where conductors and insulators can be distinguished.

Understanding Topological Insulators in Real Space

A real space understanding of the Su–Schrieffer–Heeger model of polyacetylene is introduced thanks to delocalization indices defined within the quantum theory of atoms in molecules. This approach enables to go beyond the analysis of electron localization usually enabled by topological insulator indices—such as IPR—enabling to differentiate between trivial and topological insulator phases. The approach is based on analyzing the electron delocalization between second neighbors, thus highlighting the relevance of the sublattices induced by chiral symmetry. Moreover, the second neighbor delocalization index, δi,i+2, also enables to identify the presence of chirality and when it is broken by doping or by eliminating atom pairs (as in the case of odd number of atoms chains). Hints to identify bulk behavior thanks to δ1,3 are also provided. Overall, we present a very simple, orbital invariant visualization tool that should help the analysis of chirality (independently of the crystallinity of the system) as well as spreading the concepts of topological behavior thanks to its relationship with well-known chemical concepts.

Energetics of Electron Pairs in Electrophilic Aromatic Substitutions

The interacting quantum atoms approach (IQA) as applied to the electron-pair exhaustive partition of real space induced by the electron localization function (ELF) is used to examine candidate energetic descriptors to rationalize substituent effects in simple electrophilic aromatic substitutions. It is first shown that inductive and mesomeric effects can be recognized from the decay mode of the aromatic valence bond basin populations with the distance to the substituent, and that the fluctuation of the population of adjacent bonds holds also regioselectivity information. With this, the kinetic energy of the electrons in these aromatic basins, as well as their mutual exchange-correlation energies are proposed as suitable energetic indices containing relevant information about substituent effects. We suggest that these descriptors could be used to build future reactive force fields.

Local spin and open quantum systems: clarifying misconceptions, unifying approaches

The theory of open quantum systems (OQSs) is applied to partition the squared spin operator into fragment (local spin) and interfragment (spin-coupling) contributions in a molecular system.

Exotic Bonding Regimes Uncovered in Excited States

Real-space tools were employed to show that the chemical bonding scenario used routinely to understand ground states lacks the necessary flexibility in excited states. It is shown that, even for two-center, two-electron bonds, the real-space bond orders have exotic values that have never been reported. The nature of these situations was uncovered by using electron-counting techniques that provide an appealing statistical interpretation of bonding descriptors, together with simple physical models. Bond orders greater than one as well as negative bond orders for a single bonding electron pair emerge in situations in which the electrons in the pair show a gregarious (bosonic) instead of the usual lonely (fermionic) behavior. In the first case the gregarious pair is intra-atomic, whereas the coupling is interatomic in the second. A number of examples are used to substantiate these claims.

The Pauli principle and the confinement of electron pairs in a double well: Aspects of electronic bonding under pressure

It has become recently clear that chemical bonding under pressure is still lacking guiding principles for understanding the way electrons reorganize when their volume is constrained. As an example, it has recently been shown that simple metals can become insulators (aka electrides) when submitted to high enough pressures. This has lead to the general believe that "a fundamental yet empirically useful understanding of how pressure alters the chemistry of the elements is lacking" [R. J. Hemley, High Pressure Res. 30, 581 (2010)]. In this paper, we are interested in studying the role that the Pauli principle plays on the localization/delocalization of confined noninteracting electrons. To this end, we have considered the simple case of a 1-dimensional (1-D) double well as a confining potential, and the Electron Localization Function (ELF) has been used to characterize the degree localization/delocalization of the systems of noninteracting electrons. Then, we have systematically studied the topology of the ELF as a function of the double well parameters (barrier eight and wells distance) and of the number of electrons. We have found that the evolution of the ELF distributions has a good correspondence with the evolution of chemical bonding of atomic solids under pressure.

A first step towards quantum energy potentials of electron pairs

A first step towards the construction of a quantum force field for electron pairs in direct space is taken. Making use of topological tools (Interacting Quantum Atoms and the Electron Localisation Function), we have analysed the dependency of electron pairs electrostatic, kinetic and exchange-correlation energies upon bond stretching. Simple correlations were found, and can be explained with elementary models such as the homogeneous electron gas. The resulting energy model is applicable to various bonding regimes: from homopolar to highly polarized and even to non-conventional bonds. Overall, this is a fresh approach for developing real space-based force fields including an exchange-correlation term. It provides the relative weight of each of the contributions, showing that, in common Lewis structures, the exchange correlation contribution between electron pairs is negligible. However, our results reveal that classical approximations progressively fail for delocalised electrons, including lone pairs. This theoretical framework justifies the success of the classic Bond Charge Model (BCM) approach in solid state systems and sets the basis of its limits. Finally, this approach opens the door towards the development of quantitative rigorous energy models based on the ELF topology.

An unexpected bridge between chemical bonding indicators and electrical conductivity through the localization tensor

While the modern theory of the insulating state shows that the conducting or insulating properties of a system can be extracted solely from the ground state properties via the so-called localization tensor (LT), no chemical reading of this important quantity has ever been offered. Here, a remarkable link between the LT and the bond orders as described by the delocalization indices (DIs) of chemical bonding theory is reported. This is achieved through a real space partition of the LT into intra- and interatomic contributions. We show that the convergence or divergence of the LT in the thermodynamic limit, which signals the insulating or conducting nature of an extended system, respectively, can be nailed down to DIs. This allows for the exploitation of traditional chemical intuition to identify essential and spectator atomic groups in determining electrical conductivity. The thermodynamic limit of the LT is controlled by the spatial decay rate of the interatomic DIs, exponential in insulators and power-law in conductors. Computational data of a few selected toy systems corroborate our results.

Real space bond orders are energetic descriptors

Orbital invariant position space techniques are used to show a theoretical link between the conventional concept of bond order and the energetics of chemical interactions.

Introducing DDEC6 atomic population analysis: part 3. Comprehensive method to compute bond orders

A new method to compute accurate bond orders for metallic, covalent, polar-covalent, ionic, multi-centered, aromatic, dative, dispersion, and hydrogen bonding.