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.

Questioning the orbital picture of magnetic spin coupling: a real space alternative.

The prevailing magnetic spin coupling paradigm is based on a one-electron picture, and is therefore orbital dependent and unsatisfactory from a physical point of view. We examine it under a truly invariant real space perspective, focusing on the role of electron delocalization. We show that this view, compatible with orbital thinking, overcomes its limitations. By examining simple model systems we show that it is electron delocalization that drives any singlet-triplet gap, and that delocalization and ionic mixing are two sides of the same reality. It is in the end delocalization, fostered or hindered by the specific structure of a system, that lies behind its preferred magnetic coupling mode. In the case of superexchange-mediated coupling through atomic bridges, we also point out the non-essential role of the bridge's electrons in setting up singlet-triplet preferences. We show that the use of real space thinking allows for tuning singlet-triplet gaps using knobs that are not easily grasped from the orbital standpoint, opening new avenues in the design and control of molecular magnets.

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. An atomic or fragment subsystem is described by a quantum mechanical mixed density operator composed of sectors, characterized by different integer number of electrons that appear with specific probabilities. The OQS fragment spin operators coincide with those defined by Clark and Davidson in their paper on local spins (J. Chem. Phys., 2001, 115, 7382) and are fully consistent with the theory of local operators by Stollhoff and Fulde (J. Chem. Phys., 1980, 73, 4548). OQSs provide a unique way to rationalize the non-zero values of local spins found in closed-shell molecules, a fact that has led to a large number of modified definitions being proposed, which we show suffer from inconsistencies. The OQS viewpoint makes it easy to build models for localized and itinerant spins. These models are used to classify possible local spin arrangements. The role of electron correlation is also studied through the analysis of the Hubbard Hamiltonian in small chains. Local spins result from a game played differently by localized and delocalized electrons. A number of examples exemplifying the ability of the OQS local spin perspective to uncover simple chemical patterns are examined.

Electronegativity equalization: taming an old problem with new tools.

Electronegativity equalization is examined after understanding an atom-in-a-molecule as an open quantum system, characterized by a variable fluctuating number of electrons whose average is set through charge-constrained electronic structure calculations. It is shown that actual results in toy systems can be easily modeled through electron distribution functions, and that by doing so several conflicting interpretations converge onto a common formalism.

Electron-pair bonding in real space. Is the charge-shift family supported?

Charge-shift bonding (CSB) has been introduced as a distinct third family of electron-pair links that adds to the covalent and ionic tradition. However, the full battery of orbital invariant tools provided by modern real space artillery shows that it is difficult to find CSB signatures outside the original valence-bond framework in which CSB was developed. The CSB concept should therefore be further investigated.

Reply to the 'Comment on "Decoding real space bonding descriptors in valence bond language"' by S. Shaik, P. Hiberty and D. Danovich, Phys. Chem. Chem. Phys., 2019, 21, DOI: 10.1039/C8CP07225F.

A recent comment by Hiberty, Danovich and Shaik to our previous communication on interpreting valence bond (VB) concepts in real space raises concerns about the map between quantum chemical topology (QCT) concepts and those of other conceptual frameworks, such as VB theory. We clarify here why some of these discrepancies appear, particularly as resonance structures (RSs) are regarded. As originally shown in our communication, we do not redefine VB structures, but we compare them with their real space equivalent instead.

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.

From quantum fragments to Lewis structures: electron counting in position space.

An electron counting technique that easily provides Lewis structures from real space analyses of general wavefunctions is proposed. We base our approach on reformulating the adaptive natural density partitioning (AdNDP) algorithm proposed by Zubarev and Boldyrev (Phys. Chem. Chem. Phys., 2008, 10, 5207) in position space through the use of domain-averaged cumulant densities, which take into account many-electron correlations. Averages are performed over the basins provided by the quantum theory of atoms in molecules. The decomposition gives rise to a set of n-center, two-electron orbitals which describe the dominant Lewis structures of a molecular system, and is available both for single- and multi-determinant wavefunctions. As shown through several examples, chemically intuitive Lewis descriptions are now available from fully orbital invariant position space descriptors. In this sense, real space methods are now in a position to compete with natural bond order (NBO) orbital procedures without the many biases of the latter.

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. Taking advantage of the parallelism between the covalent and ionic interaction energies in the interacting quantum atoms (IQA) approach, a real space ionic bond order is defined. Expanding the covalent and ionic interaction energies as a multipolar series we show that the zeroth order terms in the expansion, those dominating the total interaction, are nothing but distance-scaled bond orders. A chemically intuitive picture emerges in which bonding is brought about by the Coulomb attraction of permanently transferred electrons, that give rise to ionic terms, and of the Coulombic attraction of half the shared pairs, which provide the covalent contributions. A set of representative molecules are examined to explore how the zeroth order approximation works. We show that, as expected, the approximation improves with interatomic distance.

Decoding real space bonding descriptors in valence bond language.

Real space bonding descriptors are orbital invariant indices that can be obtained independently of the theoretical framework used to compute a given wavefunction. Here we show how to use them to read in real space some widely used concepts in Valence Bond (VB) theory, such as ionic/covalent characters or covalent-ionic resonance energies. All of these are essential ingredients used when building VB chemical insight. Electron number distribution functions are employed to directly map ionic and covalent weights with real space delocalization indices. We show that covalency, understood as delocalization, emerges in position space from the fluctuation of electron populations. This is mapped in VB to covalent-ionic resonance. The reasons why this is not so in the standard language of non-orthogonal VB are examined. A simple real space ionic character index that maintains the essence of its VB equivalent is defined and examined in simple model systems. The conclusions of this work ease travelling among the sometimes conflicting molecular orbital, real space, and valence bond interpretations in chemical bonding theory.

Decay rate of real space delocalization measures: a comparison between analytical and test systems.

We examine in this contribution the possible relation between the spatial decay rate of real space delocalization measures and the insulating- or metallic-like character of molecular and extended systems. We first show that in simple one-electron models, like the Hückel or tight binding approximations, delocalization indices (DIs) are intimately linked to the first-order reduced density matrix (1RDM), whose decay rate is known to be exponential in gapped systems and algebraic in gapless ones. DIs are shown to behave equivalently, with wild oscillations in gapless 1D, 2D and 3D models that do only persist in one-dimensional real cases, as computed at the Hartree-Fock or Kohn-Sham levels. Oscillations are shown to be directly related to Pauling resonant structures and chemical mesomerism. DIs in insulating-like moieties decay extremely fast. We propose that examining the decay of DIs along different directions in real materials may be used to detect facile and non-facile conductivity channels.

An interacting quantum atoms analysis of the metal-metal bond in [M2(CO)8]n systems.

The metal-metal interaction in policarbonyl metal clusters remains one of the most challenging and controversial issues in metal-organic chemistry, being at heart of a generalized understanding of chemical bonding and of specific applications of these molecules. In this work, the interacting quantum atoms (IQA) approach is used to study the metal-metal interaction in dimetal polycarbonyl dimers, analyzing bridged (Co2(CO)8)), semibridged ([FeCo(CO)8](-)) and unbridged (Co2(CO)8, [Fe2(CO)8](2-)) clusters. In all systems, a delocalized covalent bond is found to occur, involving the metals and the carbonyls, but the global stability of the dimers mainly originates from the Coulombic attraction between the metals and the oxygens.

On the interpretation of domain averaged Fermi hole analyses of correlated wavefunctions.

Few methods allow for a physically sound analysis of chemical bonds in cases where electron correlation may be a relevant factor. The domain averaged Fermi hole (DAFH) analysis, a tool firstly proposed by Robert Ponec in the 1990's to provide interpretations of the chemical bonding existing between two fragments Ω and Ω' that divide the real space exhaustively, is one of them. This method allows for a partition of the delocalization index or bond order between Ω and Ω' into one electron contributions, but the chemical interpretation of its parameters has been firmly established only for single determinant wavefunctions. In this paper we report a general interpretation based on the concept of excluded density that is also valid for correlated descriptions. Both analytical models and actual computations on a set of simple molecules (H2, N2, LiH, and CO) are discussed, and a classification of the possible DAFH situations is presented. Our results show that this kind of analysis may reveal several correlated assisted bonding patterns that might be difficult to detect using other methods. In agreement with previous knowledge, we find that the effective bond order in covalent links decreases due to localization of electrons driven by Coulomb correlation.

Chemical Interactions and Spin Structure in (O2)4: Implications for the ε-O2 Phase.

The chemical interactions and spin structure of (O2)4 in its ground singlet state are analyzed by means of Quantum Chemical Topology descriptors. The energetic contributions of the Interacting Quantum Atoms approach are used to obtain information about the class of interactions displayed along the dissociation path of (O2)4. The exchange-correlation contribution to the binding energy is non-negligible for the O2-O2 interactions at intermolecular distances close to those found for the pressure induced ε phase of solid (O2) and this strengthening of the intermolecular bonding is built up from a simultaneous weakening of the intramolecular bond. This result is of interest in connection with the observed softening of the IR vibron frequency in the lower pressure range of the ε phase. The spin structure in the real space along the dissociation process is interpreted with the help of the so-called electron number distribution functions. At large distances, the four triplet O2 molecules are arranged in a way consistent with an antiferromagnetic structure, whereas at short distances, a significant spin redistribution is driven by the exchange process and it involves a propensity toward a null magnetic moment per molecule. Such probability behavior can be related with the magnetic evolution of solid oxygen across the δ → ε phase transition. Additional calculations of (O2)4 excited states support the conclusion that the relative stabilization and magnetic features of the ground singlet state are due to the onset of the new intermolecular bonds, and not to an exclusive modification of the electronic character within the O2 molecules.

The Ehrenfest force field: Topology and consequences for the definition of an atom in a molecule.

The Ehrenfest force is the force acting on the electrons in a molecule due to the presence of the other electrons and the nuclei. There is an associated force field in three-dimensional space that is obtained by the integration of the corresponding Hermitian quantum force operator over the spin coordinates of all of the electrons and the space coordinates of all of the electrons but one. This paper analyzes the topology induced by this vector field and its consequences for the definition of molecular structure and of an atom in a molecule. Its phase portrait reveals: that the nuclei are attractors of the Ehrenfest force, the existence of separatrices yielding a dense partitioning of three-dimensional space into disjoint regions, and field lines connecting the attractors through these separatrices. From the numerical point of view, when the Ehrenfest force field is obtained as minus the divergence of the kinetic stress tensor, the induced topology was found to be highly sensitive to choice of gaussian basis sets at long range. Even the use of large split valence and highly uncontracted basis sets can yield spurious critical points that may alter the number of attraction basins. Nevertheless, at short distances from the nuclei, in general, the partitioning of three-dimensional space with the Ehrenfest force field coincides with that induced by the gradient field of the electron density. However, exceptions are found in molecules where the electron density yields results in conflict with chemical intuition. In these cases, the molecular graphs of the Ehrenfest force field reveal the expected atomic connectivities. This discrepancy between the definition of an atom in a molecule between the two vector fields casts some doubts on the physical meaning of the integration of Ehrenfest forces over the basins of the electron density.

Performance of the density matrix functional theory in the quantum theory of atoms in molecules.

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.

Restoring orbital thinking from real space descriptions: bonding in classical and non-classical transition metal carbonyls.

A combined strategy that unifies our interacting quantum atoms approach (IQA), a chemically intuitive energetic perspective within the quantum theory of atoms in molecules (QTAIM), the domain natural orbitals obtained by the diagonalization of the charge-weighted domain-averaged Fermi hole (DAFH), and the statistical analyses of chemical bonding provided by the electron number distribution functions (EDF) is presented. As shown, it allows for recovering traditional orbital images from the orbital invariant descriptions of QTAIM. It does also provide bonding indices (like bond orders) and bond energetics, all in a per orbital basis, still invariant manner, using a single unified framework. The procedure is applied to show how the Dewar, Chatt, and Ducanson model of bonding in simple transition metal carbonyls may be recovered in the real space. The balance between the number of σ-donated and π-backdonated electrons is negative in classical compounds and positive in non-classical ones. The energetic strength of backdonation is, however, smaller than that of donation. Our technique surpasses conventional orbital models by providing physically sound, quantitative energetics of chemical bonds (or interactions) together with effective one-electron pictures, all for arbitrary wavefunctions.

A connection between domain-averaged Fermi hole orbitals and electron number distribution functions in real space.

We show in this article how for single-determinant wave functions the one-electron functions derived from the diagonalization of the Fermi hole, averaged over an arbitrary domain Omega of real space, and expressed in terms of the occupied canonical orbitals, describe coarse-grained statistically independent electrons. With these domain-averaged Fermi hole (DAFH) orbitals, the full electron number distribution function (EDF) is given by a simple product of one-electron events. This useful property follows from the simultaneous orthogonality of the DAFH orbitals in Omega, Omega(')=R(3)-Omega, and R(3). We also show how the interfragment (shared electron) delocalization index, delta(Omega,Omega(')), transforms into a sum of one-electron DAFH contributions. Description of chemical bonding in terms of DAFH orbitals provides a vivid picture relating bonding and delocalization in real space. DAFH and EDF analyses are performed on several test systems to illustrate the close relationship between both concepts. Finally, these analyses clearly prove how DAFH orbitals well localized in Omega or Omega(') can be simply ignored in computing the EDFs and/or delta(Omega,Omega(')), and thus do not contribute to the chemical bonding between the two fragments.

Bases for Understanding Polymerization under Pressure: The Practical Case of CO2.

We present a novel quantitative strategy for monitoring chemical bonding transformations in solids from the topology of their electronic structure. Developed in the context of the electron localization function formalism, it provides an unambiguous characterization of long-range interactions and bond formation. Charge flux between electron localization regions is found to hold the key for identifying the nature of the interaction between the chemically meaningful entities in the solid (valence shells, lone pairs, molecules, etc.). Because of the wide range of interesting properties that high pressure induces in molecular solids, we illustrate the potentialities of our strategy to unveil controversial questions involved in the bond reorganization along the polymerization of CO2. Our study confirms that the topology of the bonding network in the pseudopolymeric phases points toward the incipient formation of the new bonds in the higher pressure polymers. This transformation is identified as a synchronic weakening of the intramolecular (C==O) double bond and the birth of a new intermolecular C--O bond controlled by the oxygen lone pairs. Overall, the relationship that this type of analysis establishes between different polymorphs of the phase diagram could be further exploited for the prediction of the coordination of high pressure phases, opening new avenues for experimental synthesis and structure indexation.

Toward Understanding the Photochemistry of Photoactive Yellow Protein: A CASPT2/CASSCF and Quantum Theory of Atoms in Molecules Combined Study of a Model Chromophore in Vacuo.

Photochemical processes that take place in biological molecules have become an increasingly important research topic for both experimentalists and theoreticians. In this work, we report the reaction mechanism of a model of the photoactive yellow protein (PYP) chromophore in vacuo. The results obtained here, using a strategy based on the simultaneous use of the minimum energy path concept and the quantum theory of atoms in molecules applied to this excited state process, suggest a possible way in which the protein could increase the efficiency of the reaction. The role played by other electronic states of the same and different spin multiplicities in the reaction process is also analyzed, with special emphasis on that played by the lowest-lying triplet state. The possibility of a more complex than expected reaction mechanism is finally discussed, with some suggestions on the possible roles of the protein.