Domain-averaged Fermi Holes Research Articles

Ni Oxidation State and Ligand Saturation Impact on the Capability of Octaazamacrocyclic Complexes to Bind and Reduce CO2.

Two 15-membered octaazamacrocyclic nickel(II) complexes are investigated by theoretical methods to shed light on their affinity forwards binding and reducing CO2. In the first complex 1[NiIIL]0, the octaazamacrocyclic ligand is grossly unsaturated (π-conjugated), while in the second 1[NiIILH]2+ one, the macrocycle is saturated with hydrogens. One and two-electron reductions are described using Mulliken population analysis, quantum theory of atoms in molecules, localized orbitals, and domain averaged fermi holes, including the characterization of the Ni-CCO2 bond and the oxidation state of the central Ni atom. It was found that in the [NiLH] complex, the central atom is reduced to Ni0 and/or NiI and is thus able to bind CO2 via a single σ bond. In addition, the two-electron reduced 3[NiL]2− species also shows an affinity forwards CO2.

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.

Charge density of 4-methyl-3-[(tetrahydro-2H-pyran-2-yl)oxy]thiazole-2(3H)-thione. A comprehensive multipole refinement, maximum entropy method and density functional theory study.

The structure of 4-methyl-3-[(tetrahydro-2H-pyran-2-yl)oxy]thiazole-2(3H)-thione (MTTOTHP) was investigated using X-ray diffraction and computational chemistry methods for determining properties of the nitrogen-oxygen bond, which is the least stable entity upon photochemical excitation. Experimentally measured structure factors have been used to determine and characterize charge density via the multipole model (MM) and the maximum entropy method (MEM). Theoretical investigation of the electron density and the electronic structure has been performed in the finite basis set density functional theory (DFT) framework. Quantum Theory of Atoms In Molecules (QTAIM), deformation densities and Laplacians maps have been used to compare theoretical and experimental results. MM experimental results and predictions from theory differ with respect to the sign and/or magnitude of the Laplacian at the N-O bond critical point (BCP), depending on the treatment of n values of the MM radial functions. Such Laplacian differences in the N-O bond case are discussed with respect to a lack of flexibility in the MM radial functions also reported by Rykounov et al. [Acta Cryst. (2011), B67, 425-436]. BCP Hessian eigenvalues show qualitatively matching results between MM and DFT. In addition, the theoretical analysis used domain-averaged fermi holes (DAFH), natural bond orbital (NBO) analysis and localized (LOC) orbitals to characterize the N-O bond as a single σ bond with marginal π character. Hirshfeld atom refinement (HAR) has been employed to compare to the MM refinement results and/or neutron dataset C-H bond lengths and to crystal or single molecule geometry optimizations, including considerations of anisotropy of H atoms. Our findings help to understand properties of molecules like MTTOTHP as progenitors of free oxygen radicals.

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).

Nature of the Metal‐Ligand Interactions in Complexes M(PH3)2(η2‐L) (M=Ni, Pd, Pt; L=CO2, COS, CS2): A Theoretical Study

AbstractDFT calculations have been carried out for complexes M(0)(PH3)2(η2‐CE1E2), where M=Ni, Pd, Pt, and E1, E2=O, S. The electronic structure of the complexes have been elucidated using various methods, such as quantum theory of atoms in molecules (QTAIM), electron localization function (ELF), domain‐averaged Fermi hole (DAFH), and energy decomposition analysis combined with the natural orbital of chemical valence (EDA‐NOCV). In the carbonyl sulfide complexes the η2‐(S,C) coordination is clearly preferred over η2‐(O,C) which might be attributed to the harder character of oxygen reflected in excess kinetic energy density preventing the formation of bond paths in the vicinity of the metal center and the coordinating oxygen. The nature of the interaction between the metal containing fragment and the ligand is closed‐shell donor‐acceptor, rather than ′classical′ (shared‐shell) covalent type which is dominated by the back‐donating interactions.

Theoretical insights into the nature of PtSn bond: Reevaluating the bonding/back-bonding properties of trichlorostannate with comparison to the cyano ligand.

The coordinating properties of the trichlorostannate ligand in hydrido platinum trichlorostannato and platinum bis-trichlorostannato complexes, containing various phosphine ligands, have been elucidated by means of charge decomposition analysis, the Ziegler-Rauk Energy Decomposition with the Natural Orbitals for Chemical Valence, Domain-Averaged Fermi Hole, and natural bond orbital methods. Trichlorostannate has been found to be a strong σ-donor and a weak π-acceptor ligand with coordination properties not far from those of the cyano ligand. For back-bonding, the tin-chlorine σ∗ orbitals are mostly responsible. In contrast to previous assumptions, the 5d orbitals of tin play no role at all in the interaction with the platinum center. QTAIM calculations suggest, that the platinum-tin interaction should be interpreted as donor-acceptor, rather than covalent type. Trichlorostannate has been found to have weaker trans influence in comparison to the cyano ligand. © 2017 Wiley Periodicals, Inc.

Insights from domain-averaged Fermi hole (DAFH) analysis and multicenter bond indices into the nature of Be(0) bonding

The nature of the bonding interactions in the recently synthesized first stable neutral complex of so-called zero-valent beryllium (i.e. formally Be(0)) is investigated using the analysis of domain-averaged Fermi holes (DAFHs) and of multicenter bond indices. It is shown that both of these types of analysis, which have previously proved useful for various molecules with nontrivial bonding patterns, basically corroborate the appealing model suggested in the original study to explain the stability of the complex (except for a more realistic specification of the actual valence state of the Be atom). Nevertheless, as well as confirming the anticipated dominance of three-center two-electron π bonding in the central C−Be−C fragment, reinforced by the existence of two donor-acceptor Be−C σ bonds, a more detailed scrutiny of the multicenter bond indices also reveals somewhat unexpected features which suggest also the existence of delocalized 3c-4e σ bonding in the C−Be−C skeleton.

Topology vs. thermodynamics in chemical reactions: the instability of PH5.

The topological approach, based on Bader theory, is compared to the common thermodynamical methodology to study chemical reactivity. It is shown how the former indeed has numerous advantages and provides a more detailed description with respect to the latter about the course of the reaction. The comparison between the two approaches is performed by considering a classical reaction, i.e. the decomposition of PX5 (X = H, F). The topological investigation was supported by using different state-of-the-art topological tools, such as the source function, Espinosa indexes, delocalisation indexes, and domain-averaged Fermi hole analysis. Furthermore, in this work a new topological descriptor, the Bader energy density, PBADER, is introduced and applied to the study case. For the first time since Bader theory was introduced, the distribution of atomic energies in the atomic basins was analysed in detail and used to explain the chemical reactivity a priori.

Reassessing spin-coupled (full generalized valence bond) descriptions of ozone using three-center bond indices

Domain-averaged Fermi hole analysis is carried out for the ground state of O3 at its equilibrium geometry using a complete-active-space self-consistent field CASSCF(18,14) wavefunction, based on a slightly expanded full-valence active space. This initial analysis is augmented with an examination of the corresponding localized natural orbitals (LNOs) and of the numerical values obtained with a new improved definition of three-center bond indices for correlated singlet systems. Much the same pattern of LNOs is observed when using instead a subsequent internally-contracted multiconfiguration-reference configuration interaction construction, which also provides very similar values for the three-center bond indices. This gives us confidence to use such bond indices, alongside relative energies and the electric dipole moment, to assess the relative merits of various combinations of spin-coupled (full generalized valence bond) components with ten active electrons: four π, four σ bonding and the two nonbonding σ electrons associated with the central O atom. These multi-component valence bond descriptions were generated either with or without subsequent orbital reoptimization. The description of the π system which emerges from all of our analysis conforms to a standard model of three-center four-electron π bonding that incorporates a nontrivial degree of (partial) diradical character. Whereas certain combinations of ten-electron spin-coupled components can faithfully reproduce such a picture, none of the individual rival components appears to have sufficient flexibility on its own.

Exploring the role of the 3-center–4-electron bond in hypervalent λ3-iodanes using the methodology of domain averaged Fermi holes

Hypervalent iodine compounds, in particular λ(3)-iodanes, have become important reagents in organic synthesis for the electrophilic transfer of substituents to arenes and other nucleophiles. The structure and reactivity of these compounds are usually described based on a 3-center-4-electron bond model, involving the iodine central atom and its two trans substituents. The goal of this computational study is to explore Fermi correlation in view of a more advanced description of bonding in these compounds. For that matter, we apply the analysis of Domain Averaged Fermi Holes (DAFH). The DAFH analysis reveals a relationship between the occurrence of multicenter bonding and structural parameters which cannot be easily observed based on simple MO theory. Whereas for λ(3)-iodanes carrying electron-rich ligands pairing of electrons over three centers is indeed observed, compounds with electron-withdrawing substituents fall into a different category: the pairing of electrons is restricted to extend over two centers only, thus challenging the multicenter bonding picture in this case. Accordingly, a drastic reduction of the DAFH three center bond index is observed. The establishment of the multicenter bond in λ(3)-iodanes is driven by a pseudo Jahn-Teller (PJT) effect, whose extent is tightly coupled to the reactivity of the corresponding compound. The PJT stabilization scales with the degree of s-p hybridization of the central atom, which, in return, depends on the electron-withdrawing power of the ligands in the trans position. The response of the multicenter bond to the iodine "ligand field" can be expressed quantitatively in terms of DAFH bond indices. These show, for example, that the activation of the reacting hypervalent species by means of protonation results in a weaker 3-center-4-electron bond, thus making the reagent more reactive. In this work we explain a number of experimentally known facts concerning the reactivity of these compounds. We also show that the DAFH analysis offers a more complete understanding of hypervalency in λ(3)-iodanes, and that it is a tool to assist the search for novel reagents.

New insights from domain-averaged Fermi holes and bond order analysis into the bonding conundrum in C2

ABSTRACTThe bonding in the ground state of C2 is examined using a combined approach based on the analysis of domain-averaged Fermi holes and of the contributions to covalent bond orders that can be associated with individual localised natural orbitals. The σ system in this molecule turns out to be particularly sensitive, evolving from a description that includes a fairly traditional shared electron pair σ bond, for a range of intermediate nuclear separations, to a somewhat different situation near equilibrium geometry, where non-classical repulsive interactions are particularly important. The various results provide further support for the view that the electronic structure of this molecule sufficiently exceeds the scope of traditional bonding paradigms that attempts to classify the bonding in terms of a classical bond multiplicity are highly questionable.

One-electron images in real space: natural adaptive orbitals.

We introduce a general procedure to construct a set of one-electron functions in chemical bonding theory, which remain physically sound both for correlated and noncorrelated electronic structure descriptions. These functions, which we call natural adaptive orbitals, decompose the n-center bonding indices used in real space theories of the chemical bond into one-electron contributions. For the n = 1 case, they coincide with the domain natural orbitals used in domain-averaged Fermi hole analyses. We examine their interpretation in the two-center case, and show how they behave and evolve in simple cases. Orbital pictures obtained through this technique converge onto the chemist's molecular orbital toolbox if electron correlation may be ignored, and provide new insight if it may not.

Are orbital-resolved shared-electron distribution indices and Cioslowski covalent bond orders useful for molecules?

Values of a delocalisation index that is also known as the shared-electron distribution index can be resolved into contributions from individual domain-averaged Fermi hole functions. In order to assess the utility of this approach, as well as of the partitioning of Cioslowski covalent bond orders into contributions associated with individual localised natural orbitals, results are examined for three molecular test cases: HeH and Be2 are considered for short nuclear separations, for which these systems are demonstrably not bound, and the σ framework of N2 near its equilibrium geometry is examined.

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

The theory of chemical bonding is underdeveloped in electronic excited states, even in small molecules. Fortunately, real space tools may be used to offer rich images of simple excitation processes, as is shown in this work. The statistics of electron populations, through a fruitful combination of electron distribution functions (EDFs) and domain averaged Fermi holes (DAFHs), was used to enlighten our chemical knowledge of a paradigmatic process: the n → π* excitation in formaldehyde. Interestingly, our results are perfectly compatible with an alternative perception of the electronic transition: the rotation of one averaged-electron in the oxygen lone pair. This topological model does not require inter-orbital jumps to explain the final electron distribution and, in our humble opinion, this fact makes it, to some extent, more realistic. Finally, other far-reaching conclusions emerge smoothly from our analysis: (i) the σ link may contribute less to the total bond order (as measured by the delocalization index) of a polar double bond than the π one; (ii) populating an antibonding orbital does not necessarily imply decreasing the bond order of its corresponding bond.

Why is the bond multiplicity in C2 so elusive?

We reexamine the full GVB (or spin-coupled) description of the bonding in the ground state of C2 at its equilibrium geometry, prompted by recent controversy as to whether or not this system exhibits four bonds. We show that two apparently different interpretations (namely a single σ bond plus antiferromagnetic coupling of high spin units on the two atoms as opposed to a conventional set of covalent σ and π bonds) do in fact have a high overlap. Neither description is adequate. Further insights into the nature of the bonding in this system emerge from an analysis of spin correlation matrix elements. We also analyze domain-averaged Fermi holes and values of the QTAIM-generalized Wiberg–Mayer index. It proves important to assess the significance of the various numerical results by means of direct comparison with analogous calculations for HCCH. We suggest that an enhanced weight for triplet spin-coupling modes, relative to HCCH, increases the difficulty of describing the electronic structure of C2 in terms of conventional bonding models that feature a whole number of two-center two-electron bonds.

Resonance assisted electron transport in oligophenyl conductors

High electron delocalization is responsible of the unusual electric conductance displayed by some π-conjugated oligophenyl chains. Recently, it has been observed that oligophenyl chains connected via methylene linkers to gold electrodes are significantly better conductors than equivalent chains with a direct ring-metal contact. In recent theoretical studies it has been shown how a simple resonance model may explain the different conductance of these chains, connecting this electric response property with chemical concepts such as aromaticity and bond order. In this work, we validate the previous qualitative model by analysing the effect of an external electric perturbation over domain averaged Fermi hole (DAFH) densities, bond orders and atomic charges obtained with the Hirshfeld iterative atomic partitioning of the molecular electron densities. The variations in the DAFH plots reflect perfectly the role played by resonance on the electrode–electrode electron transfer. Whereas charge transfer is mainly due to π electrons in oligophenyl chains connected through methylene linkers, following a resonance assisted mechanism, it occurs via σ electrons with a direct ring-metal contact and π electrons polarize in counter-field direction. Calculation of the variations experienced by the bond orders and atomic charges in the molecular junction upon the external electric perturbation confirms the bonding reorganization predicted by the resonance model.

Structure and bonding in binuclear metal carbonyls. Classical paradigms vs. insights from modern theoretical calculations

The nature of the bonding interactions in a series of bridged and unbridged binuclear metal carbonyls has been analyzed using the visual insights provided by the analysis of domain averaged Fermi holes (DAFH). Picture of the bonding emerging from this analysis has been confronted with the predictions anticipated in each particular case by the traditional bonding paradigms exemplified by the 18-electron rule and isolobal analogy. Such a confrontation has shown that the frequently observed inconsistencies, especially in what concerns the eventual existence and/or multiplicity of direct metal–metal bonds, are due to simplistic interpretation of 18-electron rule that primarily relies on formal electron count and qualitative consideration of metal–metal distances without taking into account the actual bonding capabilities of individual M(CO)n fragments. The systematic scrutiny of the results of DAFH analysis for the wide series of studied carbonyls allowed to reveal the shortcomings of the original simplistic interpretation and to propose a slight reinterpretation of 18-electron rule that makes its predictions consistent with the insights of moderns theoretical tools.

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.

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.