Analytic Models of Domain-Averaged Fermi Holes: A New Tool for the Study of the Nature of Chemical Bonds

Comparison of DAFH and FALDI-like approaches

Two complementary methodologies for extracting useful insights into electronic structure and bonding from contemporary wavefunctions are compared. The first of these, known as the analysis of domain-averaged Fermi holes (DAFH), mostly provides visually appealing descriptions of the role and the extent of electron sharing in chemical bonding. The second one, known as the fragment, atom, localized, delocalized and interatomic (FALDI) charge density decomposition scheme, uses the partitioning of certain localization and delocalization indices to focus on highly visual contributions associated with individual domains and with pairs of domains, respectively. Four variants of a FALDI-like approach are investigated here in some detail, mostly to establish which of them are the most reliable and the most informative. In addition to ‘full’ calculations that use the correlated pair density, the consequences for the DAFH and FALDI-like procedures of using instead a popular one-electron approximation are explored. Additionally, the geometry dependence of the degree of acceptability of the errors that this introduces for delocalization indices is assessed for different formal bond multiplicities. The familiar molecular test systems employed for these various linked investigations are the breaking of the bonds in H2 and in N2, as well as the nature of the bonding in B2H6, as a simple example of multicenter bonding. One of the key outcomes of this study is a clear understanding of how DAFH analysis and a particular variant of FALDI-like analysis could be most profitably deployed to extract complementary insights into more complex and/or controversial bonding situations.

The influence of correlation on (de)localization indices from a valence bond perspective

When going beyond the Hartree-Fock level to correlated methods, one observes a significant reduction in the delocalization index. This is commonly interpreted as a weakening of electron sharing due to electron correlation, although this is rather counter-intuitive to the concomitant energy lowering. In this study, we use an analytical valence bond model and full CI calculations to show that this reduction in the delocalization index actually goes hand in hand with increased covalent contributions at the expense of ionic contributions. This suggests that we should be careful in formulating interpretations of these results in (de)localization indices. Graphical Abstract Variation of the localization Δ(Ω_A, Ω_A) and delocalization index Δ(Ω_A, Ω_B) as a function of the parameter ω. By adjusting this parameter ω from [Formula: see text] to 0, we can gradually change the underlying wave function from a Hartree-Fock to a Heitler-London description.

Theoretical investigations of the chemical bonding in MM'O2 clusters (M, M' = Be, Mg, Ca)

Motivated by the known stability of the somewhat unusual Be₂O₂ rhombus, which features a short Be-Be distance but no direct metal-metal bonding, we investigate the nature of the bonding interactions in the analogous clusters MM'O₂ (M, M' = Be, Mg, Ca). CCSD/cc-pVTZ and CCSD(T)/cc-pVQZ calculations, amongst others, are used to determine optimized geometries and the dissociation energies for splitting the MM'O₂ clusters into metal oxide monomers. The primary tools used to investigate the chemical bonding are the analysis of domain-averaged Fermi holes, including the generation of localized natural orbitals, and the calculation of appropriate two- and three-center bond indices. Insights emerging from these various analyses concur with earlier studies of M₂O₂ rhombic clusters in that direct metal-metal bonding was not observed in the MM'O₂ rings whereas weak three-center (3c) bonding was detected in the MOM' moieties. In general terms, these mixed MM'O₂ clusters exhibit features that are intermediate between those of M₂O₂ and M'₂O₂, and the differences between the M and M' atoms appear to have little impact on the overall degree of 3c MOM' bonding. Graphical abstract Bonding situation in MM'O₂ clusters (M, M' = Be, Mg, Ca).

Electron-Pair Distribution in Chemical Bond Formation

The chemical formation process has been studied from relaxation holes, Δh(u), resulting from the difference between the radial intracule density and the nonrelaxed counterpart, which is obtained from atomic radial intracule densities and the pair density constructed from the overlap of the atomic densities. Δh(u) plots show that the internal reorganization of electron pairs prior to bond formation and the covalent bond formation from electrons in separate atoms are completely recognizable processes from the shape of the relaxation hole, Δh(u). The magnitude of Δh(u), the shape of Δh(u) ∀ u < R_eq, and the distance between the minimum and the maximum in Δh(u) provide further information about the nature of the chemical bond formed. A computational affordable approach to calculate the radial intracule density from approximate pair densities has been also suggested, paving the way to study electron-pair distributions in larger systems.

Domain-averaged Fermi-hole analysis for solids

The domain-averaged Fermi hole (DAFH) orbitals provide highly visual representation of bonding in terms of orbital-like functions with attributed occupation numbers. It was successfully applied on many molecular systems including those with non-trivial bonding patterns. This article reports for the first time the extension of the DAFH analysis to the realm of extended periodic systems. Simple analytical model of DAFH orbital for single-band solids is introduced which allows to rationalize typical features that DAFH orbitals for extended systems may possess. In particular, a connection between Wannier and DAFH orbitals has been analyzed. The analysis of DAFH orbitals on the basis of DFT calculations is applied to hydrogen lattices of different dimensions as well as to the solids diamond, graphite, Na, Cu and NaCl. In case of hydrogen lattices, remarkable similarity is found between the DAFH orbitals evaluated with both the analytical approach and DFT. In case of the selected ionic and covalent solids the DAFH orbitals deliver bonding descriptions, which are compatible with classical orbital interpretation. For metals the DAFH analysis shows essential multicenter nature of bonding.