Spin resolved electron number distribution functions: How spins couple in real space

How electrons still guard the space: Electron number distribution functions based on QTAIM∩ELF intersections

Despite the importance of the one-particle picture provided by the orbital paradigm, a rigorous understanding of the spatial distribution of electrons in molecules is still of paramount importance to chemistry. Considerable progress has been made following the introduction of topological approaches, capable of partitioning space into chemically meaningful regions. They usually provide atomic partitions, for example, through the attraction basins of the electron density in the quantum theory of atoms in molecules (QTAIM) or electron-pair decompositions, as in the case of the electron localization function (ELF). In both cases, the so-called electron distribution functions (EDFs) provide a rich statistical description of the electron distribution in these spatial domains. Here, we take the EDF concept to a new fine-grained limit by calculating EDFs in the QTAIM ∩ ELF intersection domains. As shown in AHn systems based on main group elements, as well as in the CO, NO, and BeO molecules, this approach provides an exquisitely detailed picture of the electron distribution in molecules, allowing for an insightful combination of the distribution of electrons between Lewis entities (such as bonds and lone pairs) and atoms at the same time. Besides mean-field calculations, we also explore the impact of electron correlation through Hartree-Fock (HF), density functional theory (DFT) (B3LYP), and CASSCF calculations.

Persistence of atoms in molecules: there is room beyond electron densities

Evidence that the electronic structure of atoms persists in molecules to a much greater extent than has been usually admitted is presented. This is achieved by resorting to N-electron real-space descriptors instead of one- or at most two-particle projections like the electron or exchange-correlation densities. Here, the 3N-dimensional maxima of the square of the wavefunction, the so-called Born maxima, are used. Since this technique is relatively unknown to the crystallographic community, a case-based approach is taken, revisiting first the Born maxima of atoms in their ground state and then some of their excited states. It is shown how they survive in molecules and that, beyond any doubt, the distribution of electrons around an atom in a molecule can be recognized as that of its isolated, in many cases excited, counterpart, relating this fact with the concept of energetic promotion. Several other cases that exemplify the applicability of the technique to solve chemical bonding conflicts and to introduce predictability in real-space analyses are also examined.

The role of references and the elusive nature of the chemical bond

Chemical bonding theory is of utmost importance to chemistry, and a standard paradigm in which quantum mechanical interference drives the kinetic energy lowering of two approaching fragments has emerged. Here we report that both internal and external reference biases remain in this model, leaving plenty of unexplored territory. We show how the former biases affect the notion of wavefunction interference, which is purportedly recognized as the most basic bonding mechanism. The latter influence how bonding models are chosen. We demonstrate that the use of real space analyses are as reference-less as possible, advocating for their use. Delocalisation emerges as the reference-less equivalent to interference and the ultimate root of bonding. Atoms (or fragments) in molecules should be understood as a statistical mixture of components differing in electron number, spin, etc.

The theory of chemical bonding relies on arbitrary references. Here the authors report a fundamental study on the chemical bond showing that considering the binding fragments as objects in real space enables to eliminate inherent biases.

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.

Lewis Structures from Open Quantum Systems Natural Orbitals: Real Space Adaptive Natural Density Partitioning

Building chemical models from state-of-the-art electronic structure calculations is not an easy task, since the high-dimensional information contained in the wave function needs to be compressed and read in terms of the accepted chemical language. We have already shown ( Phys. Chem. Chem. Phys. 2018, 20, 21368) how to access Lewis structures from general wave functions in real space by reformulating the adaptive natural density partitioning (AdNDP) method proposed by Zubarev and Boldyrev ( Phys. Chem. Chem. Phys. 2008, 10, 5207). This provides intuitive Lewis descriptions from fully orbital invariant position space descriptors but depends on not immediately accessible higher order cumulant density matrices. By using an open quantum systems (OQS) perspective, we here show that the rigorously defined OQS fragment natural orbitals can be used to build a consistent real space adaptive natural density partitioning based only on spatial information and the system's one-particle density matrix. We show that this rs-AdNDP approach is a cheap, efficient, and robust technique that immerses electron counting arguments fully in the real space realm.

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.

On the connection between probability density analysis, QTAIM, and VB theory

Classification of bonds is essential for understanding and predicting the reactivity of chemical compounds. This classification mainly manifests in the bond order and the contribution of different Lewis resonance structures. Here, we outline a first principles approach to obtain these orders and contributions for arbitrary wave functions in a manner that is both, related to the quantum theory of atoms in molecules and consistent with valence bond theory insight: the Lewis structures arise naturally as attractors of the all-electron probability density |Ψ|2. Doing so, we introduce a valence bond weight definition that does not collapse in the basis set limit.

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

The concerns posed by S. Shaik, P. Hiberty and D. Danovich regarding the mapping between quantum chemical topology (QCT) and valence bond (VB) concepts are discussed and clarified. We stress that we do not redefine the VB concept of the resonance structure but that we compare it with its QCT equivalent in real space.

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

We study in this contribution the spatial decay rate of real-space localization and delocalization indices in correlated systems. To that end, we examine Hubbard and quantum chemical models of simple cyclic hydrogen chains, showing that all descriptors of delocalization converge quickly toward the infinite chain limits. It is then shown that the localization index may be understood as a generalization of the standard order parameter in Mott insulator transitions and that the origin of the enigmatic sigmoidal profile of delocalization indices in chemical bond-breaking processes lies in the nonlinear mapping between intersite distances and correlation parameters. Although the long-range asymptotic decay of delocalization indices is exponential, we show that as the correlation parameter decreases quantum mechanical interference sets in and a switch to an oscillating pattern, related to core chemical concepts such as resonance or mesomerism, appears.

Curly arrows meet electron density transfers in chemical reaction mechanisms: from electron localization function (ELF) analysis to valence-shell electron-pair repulsion (VSEPR) inspired interpretation

The displacement of the nuclei along the reaction path provides an explanatory interpretation of the electron density transfers making possible to understand chemical reactions.

Excluding hyperconjugation from the Z conformational preference and investigating its origin: formic acid and beyond

A scheme indicating that the preference for the Z conformer in proteins is chemically equivalent to that of amides. Other compounds, such as carboxylic acids, also exhibit the same conformational trend.

Revisiting the carbonyl n → π* electronic excitation through topological eyes: expanding, enriching and enhancing the chemical language using electron number distribution functions and domain averaged Fermi holes

Interpretations of the S₀ → S₁ transition in formaldehyde arising from the DAFH analysis.

A uniform approach to the description of multicenter bonding

A novel method for investigating the multicenter bonding patterns in molecular systems by means of the so-called Electron Density of Delocalized Bonds (EDDB) is introduced and discussed.

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.

Domain-averaged exchange-correlation energies as a physical underpinning for chemical graphs

A novel solution to the problem of assigning a molecular graph to a collection of nuclei (i.e. how to draw a molecular structure) is presented. Molecules are universally understood as a set of nuclei linked by bonds, but establishing which nuclei are bonded and which are not is still an empirical matter. Our approach borrows techniques from quantum chemical topology, which showed for the first time the construction of chemical graphs from wave functions, shifting the focus on energetics. This new focus resolves issues surrounding previous topological analyses, in which domain-averaged exchange-correlation energies (V(xc)), quantities defined in real space between each possible atom pair, hold the key. Exponential decay of V(xc) in non-metallic systems as the intercenter distance increases guarantees a well-defined hierarchy for all possible V(xc) values in a molecule. Herein, we show that extracting the set of atom pairs that display the largest V(xc) values in the hierarchy is equivalent to retrieving the molecular graph itself. Notably, domain-averaged exchange-correlation energies are transferable, and they can be used to calculate bond strengths. Fine-grained details resulted to be related to simple stereoelectronic effects. These ideas are demonstrated in a set of simple pilot molecules.

Nature of Chemical Interactions from the Profiles of Electron Delocalization Indices

We analyze the behavior of the profiles of delocalization indices (DIs) between relevant pairs of atoms along reaction coordinates for a set of model chemical processes. A relationship between the topology of the DI and the nature of the underlying chemical change is reported. As shown, exponential shapes correspond to the traditional category of repulsive/nonbonded interactions, while sigmoidal profiles signal the formation/breaking of chemical links.