The mapping of the conditional pair density onto the electron density

Atomic shell structure from Born probabilities: Comparison to other shell descriptors and persistence in molecules

Real space chemical bonding descriptors, such as the electron localization function or the Laplacian of the electron density, have been widely used in electronic structure theory thanks to their power to provide chemically intuitive spatial images of bonded and non-bonded interactions. This capacity stems from their ability to display the shell structure of atoms and its distortion upon molecular formation. Here, we examine the spatial position of the N electrons of an atom at the maximum of the square of the wavefunction, the so-called Born maximum, as a shell structure descriptor for ground state atoms with Z = 1-36, comparing it to other available indices. The maximization is performed with the help of variational quantum Monte Carlo calculations. We show that many electron effects (mainly Pauli driven) are non-negligible, that Born shells are closer to the nucleus than any other of the examined descriptors, and that these shells are very well preserved in simple molecules.

From the Linnett-Gillespie model to the polarization of the spin valence shells of metals in complexes

In this paper, we present a novel approach to track the origin of the metal complex structure from the topology of the α and β spin densities as an extension of the Linnett-Gillespie model. Usually, the theories that explain the metal-ligand interactions consider the disposition and the relative energies of the empty or occupied set of d orbitals, ignoring the spin contribution explicitly. Our quantum topological approach considers the spatial distribution of the α and β spin valence shells, and the energy interaction between them. We used the properties of the atomic graph, a topological object that summarises the charge concentrations and depletions on the valence shell of an atom in a molecule, and the interacting quantum atoms (IQA) energy partition scheme. Unlike the Linnett-Gillespie model, which is based on electron-electron repulsion, our approach states that the ligands provoke a redistribution of the electron density to maximize the nuclear-electron interactions in each spin valence shell to bypass the concentration of electron-electron interactions, resulting in a polarization pattern which determines the position of the ligands.

Partitioning a Molecule into the Atomic Basins and the Resultant Atomic Charges from Quantum Chemical Topology Analysis of the Kohn-Sham Potential

Quantum chemical topology (QCT) solidifies the chemical basic concepts demonstrating how a molecular system is intrinsically partitioned into its components and what the interaction lines between them are. Here, QCT analysis using a Kohn-Sham one-electron potential (KSpot) in KS equation as a scalar function is initiated and explored, showing KSpot and its resultant electron force lines have novel spatial features which reveal that an atom in a molecule is a spatial basin governed by its nucleus as a 3D-attractor that terminates all the electron force lines defined by the negative gradient of KSpot and that a chemical bond line is just a minimum path of KSpot for the electron motion. Particularly, the atomic charges from this KSpot QCT analysis are moderate and good, having much lower dependence on basis sets chosen for computation. This may provide a platform for the study of molecular structures and properties, intra- and intermolecular electrostatic interaction, energy decomposition, and construction of force field.

The nature of multiple boron-nitrogen bonds studied using electron localization function (ELF), electron density (AIM), and natural bond orbital (NBO) methods

Local nature of the boron-nitrogen (BN) bonding with different formal multiplicities (B≡N, B=N, B-N) have been investigated for 25 experimentally established organoboron molecules in both real and the Hilbert space, using topological analysis of electron localization function (ELF), electron density (AIM), and natural bond orbital (NBO) method. Each BN bond has been represented (ELF) by the bonding disynaptic attractor V(B,N), with the basin electron population between 5.72e and 1.83e, confirming possible existence of all the three bond types. A covalent character of bonding can be associated with the dative mechanism due to the V(B,N) bonding basin formed mainly (91-96%) by the N electron density. Similarly, the NBO method shows 2-center natural orbitals, consisting largely of the hybrids from the N atom. The AIM analysis yields the features typical for shared (H(3,-1)(r) < 0) and closed-shell (∇2ρ(3,-1)(r) > 0) interactions. The delocalization indices, describing electron exchanges between B and N quantum atoms, are smaller than 1.5, even for formally very short triple B≡N bonds. Graphical abstract .

First report of a planar and a quasi-planar Al13+ cluster having localized antiaromatic deltas within an aromatic sea: NICS, ELF, AIM, and AdNDP bonding analysis

A perfectly planar Al₁₃⁺ cluster (CI) and a quasi-planar Al₁₃⁺ cluster (CII) have been found for the first time. Both clusters have a triangular core surrounded by a set of ten Al atoms in the form of a ring. These cationic clusters have substantial aromatic character. The planar CI cluster has local antiaromatic patches within global aromatic sea. It is doubly aromatic having both σ and π aromatic character. The quasi-planar CII cluster is also aromatic but it has more σ-delocalization. Graphical abstract Planar and quasi-planar Al₁₃⁺ clusters with triangular core surrounded by a ring of ten atoms.

Chemically driven surface effects in polar intermetallic topological insulators A3Bi

Surface electronic spectra, surface and bulk properties as well as the underlying chemical bonding characteristics in topological insulators with complex bonding patterns are considered for the example of cubic, polar intermetallics KNa₂Bi, K₃Bi and Rb₃Bi (with the general formula A₃Bi, A – alkali metal).

The bonding picture in hypervalent XF3 (X = Cl, Br, I, At) fluorides revisited with quantum chemical topology

Hypervalent XF₃ (X = Cl, Br, I, At) fluorides exhibit T-shaped C_2V equilibrium structures with the heavier of them, AtF₃ , also revealing an almost isoenergetic planar D_3h structure. Factors explaining this behavior based on simple "chemical intuition" are currently missing. In this work, we combine non-relativistic (ClF₃ ), scalar-relativistic and two-component (X = Br - At) density functional theory calculations, and bonding analyses based on the electron localization function and the quantum theory of atoms in molecules. Typical signatures of charge-shift bonding have been identified at the bent T-shaped structures of ClF₃ and BrF₃ , while the bonds of the other structures exhibit a dominant ionic character. With the aim of explaining the D_3h structure of AtF₃ , we extend the multipole expansion analysis to the framework of two-component single-reference calculations. This methodological advance enables us to rationalize the relative stability of the T-shaped C_2v and the planar D_3h structures: the Coulomb repulsions between the two lone-pairs of the central atom and between each lone-pair and each fluorine ligand are found significantly larger at the D_3h structures than at the C_2v ones for X = Cl - I, but not with X = At. This comes with the increasing stabilization, along the XF₃ series, of the planar D_3h structure with respect to the global T-shaped C_2v minima. Hence, we show that the careful use of principles that are at the heart of the valence shell electron pair repulsion model provides reasonable justifications for stable planar D_3h structures in AX₃ E₂ systems. © 2017 Wiley Periodicals, Inc.

On the electron density localization in elemental cubic ceramic and FCC transition metals by means of a localized electrons detector

The electron density localization in insulator and semiconductor elemental cubic materials with diamond structure, carbon, silicon, germanium, and tin, and good metallic conductors with face centered cubic structure such as α-Co, Ni, Cu, Rh, Pd, Ag, Ir, Pt, and Au, was studied using a localized electrons detector defined in the local moment representation. Our results clearly show an opposite pattern of the electron density localization for the cubic ceramic and transition metal materials. It was found that, for the elemental ceramic materials, the zone of low electron localization is very small and is mainly localized on the atomic basin edges. On the contrary, for the transition metals, there are low-valued localized electrons detector isocontours defining a zone of highly delocalized electrons that extends throughout the material. We have found that the best conductors are those in which the electron density at this low-value zone is the lowest.

Quantifying solvated electrons' delocalization

The electron delocalization range EDR(r;u_av) (left) captures the spin density (right) of an electron delocalized over u_av = 5.77 Å on the surface of an (H₂O)₂₀⁻ cluster.

Definition of molecular structure: by choice or by appeal to observation?

There are two schools of thought in chemistry: one derived from the valence bond and molecular orbital models of bonding, the other appealing directly to the measurable electron density and the quantum mechanical theorems that determine its behavior, an approach embodied in the quantum theory of atoms in molecules, QTAIM. No one questions the validity of the former approach, and indeed molecular orbital models and QTAIM play complementary roles, the models finding expression in the principles of physics. However, some orbital proponents step beyond the models to impose their personal stamp on their use in interpretive chemistry, by denying the possible existence of a physical basis for the concepts of chemistry. This places them at odds with QTAIM, whose very existence stems from the discovery in the observable topology of the electron density, the definitions of atoms, of the bonding between atoms and hence of molecular structure. Relating these concepts to the electron density provides the necessary link for their ultimate quantum definition. This paper explores in depth the possible causes of the difficulties some have in accepting the quantum basis of structure beginning with the arguments associated with the acceptance of a "bond path" as a criterion for bonding. This identification is based on the finding that all classical structures may be mapped onto molecular graphs consisting of bond paths linking neighboring atoms, a mapping that has no known exceptions and one that is further bolstered by the finding that there are no examples of "missing bond paths". Difficulties arise when the quantum concept is applied to systems that are not amenable to the classical models of bonding. Thus one is faced with the recurring dilemma of science, of having to escape the constraints of a model that requires a change in the existing paradigm, a process that has been in operation since the discovery of new and novel structures necessitated the extension of the Lewis model and the octet rule. The paper reviews all facets of bonding beginning with the work of Pauling and Slater in their accounting for crystal structures, taking note of Pauling's advocating possible bonding between large anions. Many examples of nonbonded or van der Waals interactions are considered from both points of view. The final section deals with the consequences of the realization that bonded quantum atoms that share an interatomic surface do not "overlap". The time has come for entering students of chemistry to be taught that the electron density can be seen, touched, and measured and that the chemical structures they learn are in fact the tracings of "bonds" onto lines of maximum density that link bonded nuclei. Matter, as we perceive it, is bound by the electrostatic force of attraction between the nuclei and the electron density.

Electron localization function at the correlated level

The electron localization function (ELF) has been proven so far a valuable tool to determine the location of electron pairs. Because of that, the ELF has been widely used to understand the nature of the chemical bonding and to discuss the mechanism of chemical reactions. Up to now, most applications of the ELF have been performed with monodeterminantal methods and only few attempts to calculate this function for correlated wave functions have been carried out. Here, a formulation of ELF valid for mono- and multiconfigurational wave functions is given and compared with previous recently reported approaches. The method described does not require the use of the homogeneous electron gas to define the ELF, at variance with the ELF definition given by Becke. The effect of the electron correlation in the ELF, introduced by means of configuration interaction with singles and doubles calculations, is discussed in the light of the results derived from a set of atomic and molecular systems.

Anatomy of bond formation. Domain-averaged fermi holes as a tool for the study of the nature of the chemical bonding in Li(2), Li(4), and F(2)

Domain-averaged Fermi hole (DAFH) analysis represents a relatively new strategy for extracting useful new insights into electronic structure and bonding from correlated wave functions. We analyze a full-valence CASSCF description of the Li4 rhombus, in order to discern the role played by the domains of the non-nuclear attractors in the sharing of the valence electrons. Similarly we examine the electron reorganization that accompanies the bond dissociation process in the Li2 molecule, which also features such a non-nuclear attractor for a significant range of nuclear separations. Full-CI wave functions for H2, for a wide range of bond lengths, are used to determine how robust are the DAFH descriptions from full-valence CASSCF wave functions to the incorporation of dynamical electron correlation. Comparisons are made, for H2 and Li4, with a much cheaper strategy in which restricted Kohn-Sham orbitals from B3LYP calculations are inserted into a simplified DAFH expression which applies at the restricted Hartree-Fock level. We also investigate the breaking of the relatively weak F-F bond in F2, in order to determine the extent to which the DAFH analysis of such a system differs from that of a more conventional homopolar bond, such as the one in H2.

Forces in molecules

Chemistry is determined by the electrostatic forces acting within a collection of nuclei and electrons. The attraction of the nuclei for the electrons is the only attractive force in a molecule and is the force responsible for the bonding between atoms. This is the attractive force acting on the electrons in the Ehrenfest force and on the nuclei in the Feynman force, one that is countered by the repulsion between the electrons in the former and by the repulsion between the nuclei in the latter. The virial theorem relates these forces to the energy changes resulting from interactions between atoms. All bonding, as signified by the presence of a bond path, has a common origin in terms of the mechanics determined by the Ehrenfest, Feynman and virial theorems. This paper is concerned in particular with the mechanics of interaction encountered in what are classically described as 'nonbonded interactions'--are atoms that 'touch' bonded or repelling one another?

Chemical interpretation of molecular electron density distributions

In this study, the two small molecules HS(CH)(CH(2)), 1, and F(CH)(4)F, 2, are presented, which yield different chemical interpretations when one and the same density is interpreted either by means of Natural Bond Orbital and subsequent Natural Resonance Theory application or by the Quantum Theory of Atoms In Molecules. The first exhibits a S-C bond in the orbital based approach, whereas the density based Quantum Theory of Atoms In Molecules detects no corresponding bond. In F(CH)(4)F a F...F bond is detected in the density based approach, whereas in the orbital based approach no corresponding bond is found. Geometrical reasons for the presence of unexpected and the absence of expected bond critical points are discussed.

Spatial Shape of Electron Delocalization: Structure of the Laplacian of the Negative Exchange−Correlation Density

The Laplacian of the negative exchange-correlation density (with respect to coordinate r(2)), nabla<(r)2>(2)[-Gamma(sigma1)(sigma2)(XC) (r(1),r(2))] = nabla(r)2(2)X(sigma1)(sigma2)(r(1),r(2)), is proposed as an instrument for the analysis of electron delocalization in real space. It determines local concentrations in the amount of electrons that are delocalized from a reference point r(1) over space. Integration of the reference coordinate r(1) over an atomic basin Omega(n) gives the function nabla(2)X(sigma1)(sigma2)(Omega(n);r), which contains detailed information about the spatial shape of the delocalization that originates from an atom in a molecule. Its isosurface representations are richly structured and resemble molecular orbitals in their complexity and partly also in their shape. The sum over all nabla(2)X(sigma1)(sigma2)(Omega(n);r) functions of a molecule equals the Laplacian of the electron density nabla(2)rho(r), for which it provides a meaningful partitioning into atomic contributions.

The Electronic Structure of Alkali Aurides. A Four-Component Dirac−Kohn−Sham Study

Spectroscopic constants, including dissociation energies, harmonic and anharmonic vibrational frequencies, and dipole moments, are calculated for the complete alkali auride series (LiAu, NaAu, KAu, RbAu, CsAu). The four-component formulation of relativistic density functional theory has been employed in this study, using the G-spinor basis sets implemented recently in the program BERTHA. The performance of four standard nonrelativistic density functionals employed is investigated by comparing the results with the best available theoretical and experimental data. The present work provides the first theoretical predictions on the molecular properties of RbAu. The intermetallic bond that occurs in the alkali auride series is highly polar and is characterized by a large charge transfer from the alkali metals to gold. The extent of this electron transfer has been investigated using several different charge analysis methods, enabling us to reach some general conclusions on their relative performance. We further report a detailed analysis of the topological properties of relativistic electron density in the bonding region, discussing the features of this approach which characterize the nature of the chemical bond. We have also computed the fully relativistic density for the alkali halides MBr and MI (M = Li, Na, K, Rb, and Cs). The comparative study shows that, on the basis of several topological properties and the variation in bond lengths, the gold atom behaves similarly to a halogen intermediate between Br and I.

Chemical bonding: From Lewis to atoms in molecules

The Lewis electron pair concept and its role in bonding are recovered in the properties of the electron pair density and in the topology of the Laplacian of the electron density. These properties provide a bridge with the quantum mechanical description of bonding determined by the Feynman, Ehrenfest, and virial theorems, bonding being a consequence of the electrostatic forces acting within a molecular system.

Mutual information and correlation measures in atomic systems

Mutual information is introduced as an electron correlation measure and examined for isoelectronic series and neutral atoms. We show that it possesses the required characteristics of a correlation measure and is superior to the behavior of the radial correlation coefficient in the neon series. A local mutual information, and related local quantities, are used to examine the local contributions to Fermi correlation, and to demonstrate and to interpret the intimate relationship between correlation and localization.

Some Electronic Correlation Effects in the Topological Analysis of the Laplacian of the Electronic Charge Density in C-n-Butonium Cations

In this work, we present a topological study of the Laplacian of the electronic density using a 6-311++G basis set, at Hartree-Fock (HF) and second-order Møller-Plesset (MP2) (full-electron and frozen-core) levels of theory, for the carbocations 2-C-n-butonium generated upon the insertion of a proton into the secondary C-C bond during the protonation of n-butane. The charge concentration, CC, critical points of the Laplacian distribution at each valence shell, VS, of carbon atoms, and the charge concentration closer to hydrogen atoms are studied. Also, the bonding critical points of the electronic density are analyzed. We analyze some effects that Coulomb correlation has on topological features of the electronic distribution. It is shown that they are mainly reflected in a decreasing of the charge concentrations at the VS and in a contraction of the VS to the nuclei. They are more pronounced over C-C bonds than in C-H bonds. The sensitivity of some parameters derived from this topological analysis to the correlation effect of core electrons and subtle effects related to hyperconjugative interactions are shown. Some consequences of different schemes (double and triple split-valence basis set with diffuse and polarization functions) in the definition of subtle VS charge concentrations at 3c-2e bond paths are presented. It is also demonstrated here how the facts that allow us to understand the MP2 stability order found in the carbocationic species 2-C-n-butonium > 1-C-n-butonium > 2-H-n-butonium > 1-H-n-butonium are similarly depicted at correlated and uncorrelated levels of calculation.