Are Multicentre Bond Indices and Related Quantities Reliable Predictors of Excited-State Aromaticity?

Systematic scrutiny is carried out of the ability of multicentre bond indices and the NOEL-based similarity index dAB to serve as excited-state aromaticity criteria. These indices were calculated using state-optimized complete active-space self-consistent field wavefunctions for several low-lying singlet and triplet states of the paradigmatic molecules of benzene and square cyclobutadiene and the inorganic ring S2N2. The comparison of the excited-state indices with aromaticity trends for individual excited states suggested by the values of magnetic aromaticity criteria show that whereas the indices work well for aromaticity reversals between the ground singlet and first triplet electronic states, addressed by Baird's rule, there are no straightforward parallels between the two sets of data for singlet excited states. The problems experienced while applying multicentre bond indices and dAB to singlet excited states are explained by the loss of the information inherently present in wavefunctions and/or pair densities when calculating the first-order density matrix.

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).

Electronic Structure and Bonding Situation in M2O2 (M = Be, Mg, Ca) Rhombic Clusters.

Quantum chemical calculations using ab initio methods at the CCSD(T) level and density functional theory have been carried out for the title molecules. The electronic structures of the molecules were analyzed with a variety of charge and energy decomposition methods. The equilibrium geometries of the M2O2 rhombic clusters exhibit very short distances between the transannular metal atoms M = Be, Mg, Ca. The calculated distances are close to standard values between double and triple bonds, but there are no chemical M-M bonds. The metal atoms M carry large positive partial charges, which are even bigger than in diatomic MO. The valence electrons of M are essentially shifted toward oxygen in M2O2, which makes it possible that there is practically no electronic charge in the region between the metal atoms. The bond dissociation energies for fragmentation of M2O2 into two metal oxides MO are very large. The metal-oxide bonds in the rhombic clusters are shorter and stronger than in diatomic MO. A detailed analysis of the electronic structure suggests that there is no significant direct M-M interaction in the M2O2 rhombic clusters, albeit weak three-center M-O-M bonding.

On the mechanism of dihydrogen activation by frustrated Lewis pairs. Insights from the analysis of domain averaged Fermi holes and generalized population analysis.

The electron reorganization responsible for the facilitation of heterolytic splitting of H-H bond by frustrated Lewis pair (FLP) catalysts has been studied using the analysis of domain averaged Fermi holes and generalized population analysis. The analysis of electron structures of the species along the reaction path has revealed that the anticipated synchronicity of previously considered electron shifts of electron pair of the σHH bond to a vacant orbital on B and from the lone pair on the basic N site to an antibonding σHH* orbital is associated with the build up of extensive delocalized bonding that can conveniently be characterized in terms of multicenter bond indices. In addition, the detailed scrutiny of the IRC-dependence of the 2-center bond indices of the disappearing H-H bond resulted in the proposal of a simple heuristic measure of the efficiency of the FLP catalysts. Attention was also paid to the evaluation of the presumed facilitating effect on the dissociation of the H-H bond of the electric field in the cavity of FLP catalyst. It has been shown that the strength of this field does not reach the critical values required for the efficient facilitation of the splitting of the H-H bond.

Bonding quandary in the [Cu3S2]3+ core: insights from the analysis of domain averaged fermi holes and the local spin.

The electronic structure of the trinuclear symmetric complex [(tmedaCu)3S2 ](3+), whose Cu3S2 core represents a model of the active site of metalloenzymes involved in biological processes, has been in recent years the subject of vigorous debate. The complex exists as an open-shell triplet, and discussions concerned the question whether there is a direct S-S bond in the [Cu3S2](3+) core, whose answer is closely related to the problem of the formal oxidation state of Cu atoms. In order to contribute to the elucidation of the serious differences in the conclusions of earlier studies, we report in this study the detailed comprehensive analysis of the electronic structure of the [Cu3S2](3+) core using the methodologies that are specifically designed to address three particular aspects of the bonding in the core of the above complex, namely, the presence and/or absence of direct S-S bond, the existence and the nature of spin-spin interactions among the atoms in the core, and the formal oxidation state of Cu atoms in the core. Using such a combined approach, it was possible to conclude that the picture of bonding consistently indicates the existence of a weak direct two-center-three-electron (2c-3e) S-S bond, but at the same time, the observed lack of any significant local spin in the core of the complex is at odds with the suggested existence of antiferromagnetic coupling among the Cu and S atoms, so that the peculiarities of the bonding in the complex seem to be due to extensive delocalization of the unpaired spin in the [Cu3S2](3+) core. Finally, a scrutiny of the effective atomic hybrids and their occupations points to a predominant formal Cu(II) oxidation state, with a weak contribution of partial Cu(I) character induced mainly by the partial flow of electrons from S to Cu atoms and high delocalization of the unpaired spin in the [Cu3S2](3+) core.

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.

Changes in charge distribution, molecular volume, accessible surface area and electronic structure along the reaction coordinate for a carbocationic triple shift rearrangement of relevance to diterpene biosynthesis.

The nature of the recently described "triple shift" rearrangement of a biologically relevant carbocation (computed in the absence of a surrounding enzyme) is characterized by examining the evolution of charge distribution, molecular volume, accessible surface area, and multicenter bonding indices along its reaction coordinate. Implications for interaction of the rearranging carbocation with a terpene synthase active site are discussed.

Bond indices in solids: extended analytical model.

The analytical model suggested some time ago for the calculation of bond indices in infinite periodical structures was reconsidered and extended so as to provide not only realistic estimate of the extent of electron sharing localized among individual pairs of the atoms in the lattice but also to detect the eventual presence of multicenter bonding in metallic solids.

Theoretical insights into the nature of nickel-carbon dioxide interactions in Ni(PH3)2(η2-CO2).

DFT calculations were carried out for the Ni(0) complex Ni(PH(3))(2)(η(2)-CO(2)), which is a model compound for the well-known Ni(0) carbon dioxide complexes containing various tertiary phosphane ligands. The electronic structure of the complex was elucidated using domain-averaged Fermi hole (DAFH), quantum theory of atoms in molecules (QTAIM), electron localization function (ELF), charge decomposition analysis (CDA), and natural bond orbital (NBO) methods. The carbon dioxide ligand in the complex reveals an unexpected coordination behavior. Apart from the expected π-donation interaction, the C-O σ bond takes also part in the electron donation. Moreover, the back-donation is slightly influenced by the phosphorus atom adjacent to the noncoordinated O of carbon dioxide as it transfers electron density directly to carbon. This unconventional way of back-donation may also explain the bent character of the Ni-C bond path. Due to excess kinetic energy density, no bond critical point was found between the coordinating oxygen and the nickel center. A strong relationship has been found between the DAFH and the NBO methods, which can provide additional information for the interpretation of DAFH eigenvectors.

Bonding analysis of the [C(2)O(4)](2+) intermediate formed in the reaction of CO(2)(2+) with neutral CO(2).

The bonding patterns of the [C(2)O(4)](2+) dication formed upon interaction of CO(2)(2+) with neutral CO(2) are investigated using the analysis of domain-averaged Fermi holes (DAFHs). The DAFH approach provides an explanation for the previously observed "asymmetry" of the energy deposition in the pair of CO(2)(+) monocations formed in the thermal reaction CO(2)(2+) + CO(2) --> [C(2)O(4)](2+) --> 2 CO(2)(+), specifically that the CO(2)(+) monocation formed from the dication dissociates far more readily than the CO(2)(+) monocation formed from the neutral molecule. The bonding pattern is consistent with a description of intermediate [C(2)O(4)](2+) as a complex between the triplet ground state of CO(2)(2+) with the singlet ground state of neutral CO(2), which can, among other pathways, smoothly proceed to a nondegenerate pair of (4)CO(2)(+) + (2)CO(2)(+) where the former stems from the dication and the latter stems from the neutral reactant. Hence the "electronic history" of the components is retained in the [C(2)O(4)](2+) intermediate. In addition, dissociation of (4)CO(2)(+) is discussed based on CCSD and CASSCF calculations. Equilibrium geometries for the ground electronic states of CO(2)(0/+/2+) and some other relevant structures of CO(2)(+) are determined using the MRCI method.

Influence of atoms-in-molecules methods on shared-electron distribution indices and domain-averaged Fermi holes.

The effect of using different methods to obtain atoms in molecules (AIMs) on the shared-electron distribution indices (SEDIs) and domain-averaged Fermi holes (DAFHs) is examined using a test set of diatomic molecules. Use of Bader's binary AIM model gives significantly different SEDIs as a function of internuclear distances than do self-consistent Hirshfeld-based AIM models. DAFH eigenvectors remain very similar for all AIM among the different methods. The corresponding eigenvalues are found to differ significantly, although the sums of complementary eigenvalues show only relatively small changes. The choice of a binary division of molecular space into AIM domains is found to lead to extra structure in SEDI curves, but this probably has limited chemical significance.

Relativistic Effects on Metal-Metal Bonding: Comparison of the Performance of ECP and Scalar DKH Description on the Picture of Metal-Metal Bonding in Re2Cl8(2.).

This paper reports a systematic comparison of the performance of alternative methods of including relativistic effects on the nature of metal-metal bonding in the Re2Cl8(2-) anion. The comparison involved the description using a scalar relativistic Douglas-Kroll-Hess (DKH2) Hamiltonian with all-electron basis sets and the relativistic effective core potential (ECP) basis sets. The impact of the above methods on the picture of the bonding was analyzed using the so-called domain averaged Fermi holes (DAFH). Besides comparing the impact on the picture of the bonding of the two above methods, the focus was also put on the systematic comparison of the "exact" AIM generalized form of DAFH analysis with the approximate Mulliken-like approach used in an earlier DAFH study of ReRe bonding. It has been shown that in contrast to descriptions using ECP basis sets where the differences in the picture of the bonding emerging from the approximate and "exact" DAFH analysis are only marginal, the approximate DAFH approach has been found to dramatically fail in the case of all-electron basis sets required for the description in terms of the Douglas-Kroll-Hess (DKH2) Hamiltonian.

Do the structural changes defined by the electron density topology necessarily affect the picture of the bonding?

The analysis of domain averaged Fermi holes (DAFH) was applied to the elucidation of the nature of the bonding interactions in supported metal carbonyls, where multicenter bonding of the bridging ligands and direct metal-metal bonds are considered as possible alternatives. The main focus is directed on the detailed scrutiny of the possible impact of the changes in the topology of electron density induced by the systematic variation of the geometry of the studied carbonyls on the picture of the bonding provided by the visual description in terms of DAFH analysis. It has been shown that irrespective of the dramatic changes in the topology of electron density exemplified by the existence and/or the lack of direct metal-metal bond path, the DAFH picture of the bonding remains practically unaffected and in all cases consistently suggests that the bonding of the bridging ligands exhibits the typical features of delocalized 3c-2e bonding.

Domain averaged Fermi hole analysis for open-shell systems.

The Article reports the extension of the new original methodology for the analysis and visualization of the bonding interactions, known as the analysis of domain averaged Fermi holes (DAFH), to open-shell systems. The proposed generalization is based on straightforward reformulation of the original approach within the framework of unrestricted Hartree-Fock (UHF) and/or Kohn-Sham (UKS) levels of the theory. The application of the new methodology is demonstrated on the detailed analysis of the picture of the bonding in several simple systems involving the doublet state of radical cation NH(3)((+)) and the triplet ground state of the O(2) molecule.

Structure and bonding in binuclear metal carbonyls from the analysis of domain averaged fermi holes. 2. Fe2(CO)8(2-) and Fe2(CO)8.

The nature of the bonding interactions in individual isomeric structures of the above carbonyls was studied using the analysis of domain averaged Fermi holes (DAFH). The main focus was directed on the confrontation of the picture of the bonding resulting from this analysis with the predictions of empirical 18-electron rule. This rule assumes, namely, the presence of direct metal-metal bond(s) for both carbonyls, but the detailed insights provided by the DAFH analysis show that the straightforward association of metal-metal bond with the favorable electron count only is too simplistic, and provided the actual structure of individual isomeric species is not taken into account, the predictions of this rule may fail. This is, e.g., the case of the C(2v) isomer of the carbonylate anion [Fe2(CO)8](2-) where the DAFH analysis denies the existence of direct metal-metal bond similarly as in the case the isoelectronic Co2(CO)8. Similar discrepancies between the predictions of the 18-electron rule and DAFH analysis were found also in the case of the C(2v) isomer of the neutral Fe2(CO)8 carbonyl, where the DAFH analysis detects the presence of a single bent Fe-Fe bond rather than the double bond anticipated by the 18-electron rule.

A one-electron approximation to domain-averaged Fermi hole analysis.

In general, full domain-averaged Fermi hole (DAFH) analysis for correlated wavefunctions requires explicit use of the correlated pair density, but such a quantity is not always readily available. We propose instead a simple one-electron approximation, which we call pseudo-DAFH or pDAFH, and which requires instead only the natural orbitals (and their occupation numbers). From comparisons of the DAFH and pDAFH modes of analysis for the bond dissociation processes in H2, N2 and LiH, as well as for the electronic structure of more complex bonding patterns, such as in CH2Li2 and Li4, we conclude that pDAFH analysis could indeed prove to be very useful when the correlated pair density is not available. Detailed comparisons are also presented of values of the shared-electron distribution index (SEDI), a proposed one-electron approximation to it (pSEDI) and a generalized Wiberg index.

Analytic models of domain-averaged Fermi holes: a new tool for the study of the nature of chemical bonds.

Simple analytical models are introduced that significantly enhance the ability to understand and rationalise the nature of bonding interactions depicted by domain-averaged Fermi hole (DAFH) analysis. The examples presented show that besides shedding new light on the role of electron-sharing in ordinary two-centre two-electron (2c-2e) chemical bonds that are well represented by the classical Lewis model, the proposed approach also provides interesting new insights into the nature of bonding interactions that go beyond the traditional Lewis paradigm. This is, for example, the case of 3c-2e multicentre bonding, but a straightforward extension of the approach also reveals for direct metal-metal bonding the existence of a completely new type of bonding interaction that involves the mutual exchange of electrons between the lone pairs on adjacent metal atoms.

Structure and bonding in binuclear metal carbonyls from the analysis of domain averaged Fermi holes. I. Fe2(CO)9 and Co2(CO)8.

The nature of the bonding in the above carbonyls was studied using the analysis of domain averaged Fermi holes (DAFH). The results straightforwardly confirm the conclusions of earlier theoretical studies in which the existence of direct metal-metal bond, anticipated for the above carbonyls on the basis of 18-electron rule, was questioned. In addition to indicating the lack of direct metal-metal bond, the DAFH analysis also allowed to characterize the nature of the electron pairs involved in the bonding of the bridging ligands. The analysis has shown that because the number of available electron pairs is not sufficient for the formation of ordinary localized 2c-2e bonds between terminal M(CO)(3) fragments and the bridging ligands, the bonding in both carbonyls exhibits typical features of electron deficiency and one bonding electron pair is effectively involved in multicenter 3c-2e bonding. Because of the symmetry of the complexes the bridging ligands are not distinguishable and all M-C-M bridges have a partial 3c-2e nature via resonance of the localized structures.

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.

Anatomy of bond formation. Bond length dependence of the extent of electron sharing in chemical bonds from the analysis of domain-averaged Fermi holes.

We demonstrate that domain-average Fermi hole (DAFH) analysis, which has previously been used at the Hartree-Fock level, remains useful after the proper introduction of electron correlation. We perform a systematic investigation of the variation of the picture of bonding with increasing bond length in simple diatomic molecules such as N2 and LiH. Alongside values of a shared-electron distribution index (SEDI), this analysis provides further insight into the geometry dependence of the extent of electron sharing in polar and non-polar systems. We also use DAFH analysis, with correlated wave functions, to evaluate the (potential) multicentre bonding in the electron-deficient and electron-rich molecules CH2Li2 and CH2N2, respectively.