Local Spin Analysis and Chemical Bonding

Reply to the 'Comment on "The oxidation state in low-valent beryllium and magnesium compounds"' by S. Pan and G. Frenking, Chem. Sci., 2022, , DOI: 10.1039/D2SC04231B

A recent article by Pan and Frenking challenges our assignment of the oxidation state of low valent group 2 compounds. With this reply, we show that our assignment of Be(+2) and Mg(+2) oxidation states in Be(cAAC^Dip)₂ and Mg(cAAC^Dip)₂ is fully consistent with our data. Some of the arguments exposed by Pan and Frenking were based on visual inspection of our figures, rather than a thorough numerical analysis. We discuss with numerical proof that some of the statements made by the authors concerning our reported data are erroneous. In addition, we provide further evidence that the criterion of the lowest orbital interaction energy in the energy decomposition analysis (EDA) method is unsuitable as a general tool to assess the valence state of the fragments. Other indicators based on natural orbitals for chemical valence (NOCV) deliver a more reliable bonding picture. We also emphasize the importance of using stable wavefunctions for any kind of analysis, including EDA.

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.

Beyond the Classical Electron-Sharing and Dative Bond Picture: Case of the Spin-Polarized Bond

Chemical bonds are traditionally assigned as electron-sharing or donor-acceptor/dative. External criteria such as the nature of the dissociation process, energy partitioning schemes, or quantum chemical topology are invoked to assess the bonding situation. However, for systems with marked multi-reference character, this binary categorization might not be precise enough to render the bonding properties. A third scenario can be foreseen: spin polarized bonds. To illustrate this, the case of a NaBH₃ ^- cluster is presented. According to the analysis NaBH₃ ^- exhibits a strong diradical character and cannot be classified as either electron-sharing or a dative bond. Elaborated upon are the common problems of popular bonding descriptions. Additionally, a simple model, based on the bond order and local spin indicators, which discriminates between all three bonding situations, is provided.

Insight into spin-orbital interaction using MCSCF method: A special analysis of the 1 Σg + electronic state in C2 and the linear polyacetylenic C4 and C6

The symmetry-broken wave function can transform the ¹ Σ_g ⁺ state of C₂ from the classic double bonding to the quadruple bonding, where the transformed wave functions of ϕ _L and ϕ _R are singly occupied by two opposite-spinning electrons. In this article, the effective bond order (EBO) contribution of the fourth bond in C₂ is assessed through the overlap integral between ϕ _L and ϕ _R , namely the value (0.60) is the EBO contribution of the fourth bond in the transformed scheme. Hence, the new EBO is 3.36, which is more equitable than the original EBO (2.15) in the traditional scheme. In addition, the singlet diradical character of the linear polyacetylenic C₄ and C₆ in the ¹ Σ_g ⁺ state is addressed for the first time. No spin-polarized bonding exists in other linear C_2n clusters, because the ionic interaction in the polyacetylenic ¹ Σ_g ⁺ state of C₄ is negligible. Moreover, the coupling energy between α and β single electrons in C₄ is only 4.0 kcal mol^-1 based on the electron spin-flip energy. © 2019 Wiley Periodicals, Inc.

Chemical bonding analysis of excited states using the adaptive natural density partitioning method

A novel approach to chemical bond analysis for excited states has been developed. Using an extended adaptive natural density partitioning method (AdNDP) as implemented in AdNDP 2.0 code, we obtained chemically intuitive bonding patterns for the excited states of H2O, B5+, and C2H4+ molecules. The deformation pathway in the excited states could be easily predicted based on the analysis of the chemical bond pattern. We expect that this new method of chemical bonding analysis would be very helpful for photochemistry, photoelectron spectroscopy, electron spectroscopy and other chemical applications that involved excited states.

The Quadruple Bonding in C2 Reproduces the Properties of the Molecule

Ever since Lewis depicted the triple bond for acetylene, triple bonding has been considered as the highest limit of multiple bonding for main elements. Here we show that C2 is bonded by a quadruple bond that can be distinctly characterized by valence-bond (VB) calculations. We demonstrate that the quadruply-bonded structure determines the key observables of the molecule, and accounts by itself for about 90% of the molecule's bond dissociation energy, and for its bond lengths and its force constant. The quadruply-bonded structure is made of two strong π bonds, one strong σ bond and a weaker fourth σ-type bond, the bond strength of which is estimated as 17-21 kcal mol(-1). Alternative VB structures with double bonds; either two π bonds or one π bond and one σ bond lie at 129.5 and 106.1 kcal mol(-1), respectively, above the quadruply-bonded structure, and they collapse to the latter structure given freedom to improve their double bonding by dative σ bonding. The usefulness of the quadruply-bonded model is underscored by "predicting" the properties of the (3)Σ+u state. C2's very high reactivity is rooted in its fourth weak bond. Thus, carbon and first-row main elements are open to quadruple bonding!

The Bond Order of C2 from a Strictly N-Representable Natural Orbital Energy Functional Perspective

The bond order of the ground electronic state of the carbon dimer has been analyzed in the light of natural orbital functional theory calculations carried out with an approximate, albeit strictly N-representable, energy functional. Three distinct solutions have been found from the Euler equations of the minimization of the energy functional with respect to the natural orbitals and their occupation numbers, which expand upon increasing values of the internuclear coordinate. In the close vicinity of the minimum energy region, two of the solutions compete around a discontinuity point. The former, corresponding to the absolute minimum energy, features two valence natural orbitals of each of the following symmetries, σ, σ*, π and π*, and has three bonding interactions and one antibonding interaction, which is very suggestive of a bond order large than two but smaller than three. The latter, features one σ-σ* linked pair of natural orbitals and three degenerate pseudo-bonding like orbitals, paired each with one triply degenerate pseudo-antibonding orbital, which points to a bond order larger than three. When correlation effects, other than Hartree-Fock for example, between the paired natural orbitals are accounted for, this second solution vanishes yielding a smooth continuous dissociation curve. Comparison of the vibrational energies and electron ionization energies, calculated on this curve, with their corresponding experimental marks, lend further support to a bond order for C2 intermediate between acetylene and ethylene.

The Chemical Bond in C2

Quantum chemical calculations using the complete active space of the valence orbitals have been carried out for Hn CCHn (n=0-3) and N2. The quadratic force constants and the stretching potentials of Hn CCHn have been calculated at the CASSCF/cc-pVTZ level. The bond dissociation energies of the C-C bonds of C2 and HC≡CH were computed using explicitly correlated CASPT2-F12/cc-pVTZ-F12 wave functions. The bond dissociation energies and the force constants suggest that C2 has a weaker C-C bond than acetylene. The analysis of the CASSCF wavefunctions in conjunction with the effective bond orders of the multiple bonds shows that there are four bonding components in C2, while there are only three in acetylene and in N2. The bonding components in C2 consist of two weakly bonding σ bonds and two electron-sharing π bonds. The bonding situation in C2 can be described with the σ bonds in Be2 that are enforced by two π bonds. There is no single Lewis structure that adequately depicts the bonding situation in C2. The assignment of quadruple bonding in C2 is misleading, because the bond is weaker than the triple bond in HC≡CH.

Puzzles in bonding and spectroscopy: the case of dicarbon

The unstable molecule C₂ has been of interest since its identification as the source of the "Swan band" features observable in the spectra offlames, carbon arcs, white dwarf stars, and comets, and it continues to serve as a focal point for experimental and theoretical discovery. Recent spectroscopic work has identified a quintet state of the molecule for the first time, while new insights into the bond order of C₂ in its ground state have been provided by sophisticated computational methods based on valence bond theory. This article gives a review of spectroscopic and computational work on C₂ including both historical background and the most recent discoveries.

Latent harmony in dicarbon between VB and MO theories through orthogonal hybridization of 3σg and 2σu

Besides the classic double bond scheme, several novel schemes have been proposed to describe the nature of the chemical bond in dicarbon (C₂), including a quadruple bond and a singlet diradical state. The results from a symmetry-broken CASSCF(8,8)/aug-cc-pVTZ study present a harmony between MO and VB theories, based on the orthogonal hybridization of the 3σ_g and 2σ_u orbitals together with the other six pristine valence orbitals. This scheme achieves the same bonding energy, R_C-C, ω_e and one electron density as that from the eight pristine valence orbitals. A quadruple bond scheme, identical to Prof. Shaik's result from VB theory, is achieved with the 4^th bond energy in the range of 12.8-27.6 kcal mol^-1. Meanwhile, the weight of a singlet open-shell configuration is the highest among all the possible configurations.

Symmetry laws improve electronegativity equalization by orders of magnitude and call for a paradigm shift in conceptual density functional theory

The strict Wigner-Witmer symmetry constraints on chemical bonding are shown to determine the accuracy of electronegativity equalization (ENE) to a high degree. Bonding models employing the electronic chemical potential, μ, as the negative of the ground-state electronegativity, χ(GS), frequently collide with the Wigner-Witmer laws in molecule formation. The violations are presented as the root of the substantially disturbing lack of chemical potential equalization (CPE) in diatomic molecules. For the operational chemical potential, μ(op), the relative deviations from CPE fall between -31% ≤ δμ(op) ≤ +70%. Conceptual density functional theory (cDFT) cannot claim to have operationally (not to mention, rigorously) proven and unified the CPE and ENE principles. The solution to this limitation of cDFT and the symmetry violations is found in substituting μ(op) (i) by Mulliken's valence-state electronegativity, χ(M), for atoms and (ii) its new generalization, the valence-pair-affinity, α(VP), for diatomic molecules. Mulliken's χ(M) is equalized into the α(VP) of the bond, and the accuracy of ENE is orders of magnitude better than that of CPE using μ(op). A paradigm shift replacing the dominance of ground states by emphasizing valence states seems to be in order for conceptual DFT.

How much tetraradical character is present in the Si₆Ge₉ cluster?

This study discusses in detail the supposedly tetraradicaloid nature of a spirobis(pentagerma[1.1.1]propellane) derivative recently reported by Ito et al. (J. Am. Chem. Soc., 2013, 135, 6770). The electronic structure properties of the Si6Ge9 cluster are computationally explored by means of the composition of the ground state wavefunction, excitation energies to low-lying singlet, triplet and quintet states, and magnetic couplings between radical centers. Two main conclusions can be extracted from the obtained results regarding the radical character of spriobis(pentagerma[1.1.1]propellane): (i) the ground state of the Si6Ge9 cluster presents a rather small amount of effective unpaired electrons, which might be related to its chemical stability and (ii) there is in fact a perceptible tetraradical character within the small overall radical nature of the molecule. The proposed description do not contradict the conclusions drawn by the introductory work of Ito et al., but it provides a more detailed and precise interpretation of radical character of the molecule.

The nature of the fourth bond in the ground state of C2: the quadruple bond conundrum

Does, or doesn't C2 break the glass ceiling of triple bonding? This work provides an overview on the bonding conundrum in C2 and on the recent discussions regarding our proposal that it possesses a quadruple bond. As such, we focus herein on the main point of contention, the 4th bond of C2, and discuss the main views. We present new data and an overview of the nature of the 4th bond--its proposed antiferromagnetically coupled nature, its strength, and a derivation of its bond energy from experimentally based thermochemical data. We address the bond-order conundrum of C2 arising from generalized VB (GVB) calculations by comparing it to HC≡CH, and showing that the two molecules behave very similarly, and C2 is in no way an exception. We analyse the root cause of the deviation of C2 from the Badger Rule, and demonstrate that the reason for the smaller force constant (FC) of C2 relative to HC≡CH has nothing to do with the bond energies, or with the number of bonds in the two molecules. The FC is determined primarily by the bond length, which is set by the balance between the bond length preferences of the σ- versus π-bonds in the two molecules. This interplay in the case of C2 clearly shows the fingerprints of the 4th bond. Our discussion resolves the points of contention and shows that the arguments used to dismiss the quadruple bond nature of C2 are not well founded.

Quantum entanglement in carbon–carbon, carbon–phosphorus and silicon–silicon bonds

We present a quantum entanglement analysis to dissect the bond orders in polyatomic molecules.