Quadruple bonding in C2 and analogous eight-valence electron species

Accurate computation of quantum excited states with neural networks

We present an algorithm to estimate the excited states of a quantum system by variational Monte Carlo, which has no free parameters and requires no orthogonalization of the states, instead transforming the problem into that of finding the ground state of an expanded system. Arbitrary observables can be calculated, including off-diagonal expectations, such as the transition dipole moment. The method works particularly well with neural network ansätze, and by combining this method with the FermiNet and Psiformer ansätze, we can accurately recover excitation energies and oscillator strengths on a range of molecules. We achieve accurate vertical excitation energies on benzene-scale molecules, including challenging double excitations. Beyond the examples presented in this work, we expect that this technique will be of interest for atomic, nuclear, and condensed matter physics.

Multiple Bonding in AeN- (Ae=Ca, Sr, Ba)

Quantum chemical calculations using ab initio methods at the MRCI+Q(8,9)/def2-QZVPPD and CCSD(T)/def2-QZVPPD levels as well as using density functional theory are reported for the diatomic molecules AeN- (Ae=Ca, Sr, Ba). The anions CaN- and SrN- have electronic triplet (3Π) ground states with nearly identical bond dissociation energies De ~57 kcal/mol calculated at the MRCI+Q(8,9)/def2-QZVPPD level. In contrast, the heavier homologue BaN- has a singlet (1Σ+) ground state, which is only 1.1 kcal/mol below the triplet (3Σ-) state. The computed bond dissociation energy of (1Σ+) BaN- is 68.4 kcal/mol. The calculations at the CCSD(T)-full/def2-QZVPPD and BP86-D3(BJ)/def2-QZVPPD levels are in reasonable agreement with the MRCI+Q(8,9)/def2-QZVPPD data, except for the singlet (1Σ+) state, which has a large multireference character. The calculated atomic partial charges given by the CM5, Voronoi and Hirshfeld methods suggest small to medium-sized Ae←N- charge donation for most electronic states. In contrast, the NBO method predicts for all species medium to large Ae→N- electronic charge donation, which is due to the neglect of the (n)p AOs of Ae atoms as genuine valence orbitals. Neither the bond orders nor the bond lengths correlate with the bond dissociation energies. The EDA-NOCV calculations show that the heavier alkaline earth atoms Ca, Sr, Ba use their (n)s and (n-1)d orbitals for covalent bonding.

Reasons Why Most Single-Reference Coupled Cluster Methods Fail to Provide the Correct Adiabatic Potentials of a Diatomic Carbon Molecule: UCCSDecCCSD Potential Study

Many single-reference coupled cluster (CC) methods offer adiabatically incorrect potentials when calculating the diatomic carbon molecule, so this problem has been studied extensively. Analysis of the full configuration interaction (FCI) wave function indicates that the main cause of the adiabatic collapse of potentials calculated by the CC method with singles, doubles, and triples (CCSDT) is the strongly increasing bonding character of the T4FCI cluster contribution. In turn, comparative analysis of the CCSDTQ adiabats X1Σg+ and B'1Σg+ demonstrates that the gap between them near the avoided crossing geometry is significantly reduced by quantitatively differentiating the character of the T4 and T 4 B ' 1 Σ g + cluster contributions. These observations clearly indicate the need to take into account the T4 cluster contribution in the standard CC wave function to obtain the correct adiabatic potential. Further analysis of this issue shows that the T4 contribution must be additionally bonding to ensure the adiabatic correctness of the potential. What also seems very interesting is that when the UCCSDecCCSD method [Toboła, Chem. Phys. Lett. 2014, 614, 82-88] is used for the potential calculation, the shape of the potential is entirely determined by a subset of unrestricted Hartree-Fock (UHF) configurations, which are structurally identical to the ground-state configuration in the UHF-based CCSD wave function.

How electrons still guard the space: Electron number distribution functions based on QTAIM∩ELF intersections

Despite the importance of the one-particle picture provided by the orbital paradigm, a rigorous understanding of the spatial distribution of electrons in molecules is still of paramount importance to chemistry. Considerable progress has been made following the introduction of topological approaches, capable of partitioning space into chemically meaningful regions. They usually provide atomic partitions, for example, through the attraction basins of the electron density in the quantum theory of atoms in molecules (QTAIM) or electron-pair decompositions, as in the case of the electron localization function (ELF). In both cases, the so-called electron distribution functions (EDFs) provide a rich statistical description of the electron distribution in these spatial domains. Here, we take the EDF concept to a new fine-grained limit by calculating EDFs in the QTAIM ∩ ELF intersection domains. As shown in AHn systems based on main group elements, as well as in the CO, NO, and BeO molecules, this approach provides an exquisitely detailed picture of the electron distribution in molecules, allowing for an insightful combination of the distribution of electrons between Lewis entities (such as bonds and lone pairs) and atoms at the same time. Besides mean-field calculations, we also explore the impact of electron correlation through Hartree-Fock (HF), density functional theory (DFT) (B3LYP), and CASSCF calculations.

δ-Bonding and Spin-Orbit Coupling Make SrAg4Sb2 a Topological Insulator

Bonding interactions and spin-orbit coupling in the topological insulator SrAg4Sb2 are investigated using DFT with orbital projection analysis. Ag-Ag delta bonding is a key ingredient in the topological insulating state because the4 d x y + 4 d x 2 - y 2 ${4d_{xy} + 4d_{x^2 - y^2 } }$ delta antibonding band forms a band inversion with the 5 s sigma bonding band. Spin-orbit coupling is required to lift d orbital degeneracies and lower the antibonding band enough to create the band inversion. These bonding effects are enabled by a longer-than-covalent Ag-Ag distance in the crystal lattice, which might be a structural characteristic of other transition metal based topological insulators. A simplified model of the topological bands is constructed to capture the essence of the topological insulating state in a way that may be engineered in other materials.

When, Where and Why Boron Prefers Boron to Nitrogen

Boron is the archetypal Lewis acid, and therefore it is only natural that it prefers to bind nitrogen, its usual Lewis base counterpart. To challenge this assumption, we present a computationally designed bicyclopentane molecule akin to [1.1.1]propellane, but with pyramidal B and N inner atoms bonded by an "inverted" dative bond. Unexpectedly, the dimer of this system prefers to interact via an atypical boron-boron bond over the supposedly obvious boron-nitrogen bond. A molecular orbital analysis shows that the boron in this peculiar entity acts both as an electron donor and an electron acceptor, making the dimerization an amphoteric-amphoteric interaction process.

Spectroscopic Study of a New Electronic Band System 33Δg-a3Πu of C2

As one of the most important diatomic molecules in the universe, the spectroscopic characterizations of C2 have attracted wide attention in various fields, such as interstellar chemistry, planetary atmospheric chemistry, and combustion. In recent years, a systematic spectroscopic study of C2 in the vacuum ultraviolet (VUV) region has been carried out in our laboratory by using the (1VUV+1'UV) resonance-enhanced multiphoton ionization method based on the combination of a tunable VUV laser source and a time-of-flight mass spectrometer. Two new electronic transition band systems have been reported, following the pioneering work of Herzberg and co-workers in 1969. In the current study, a total of 18 vibronic transition bands of C2 from the lower a3Πu state are experimentally observed in the VUV photon energy range 72000-81000 cm-1, and 6 new upper vibronic levels of 3Δg symmetry are identified, which are assigned as the v' = 0-5 vibrational levels of the 33Δg state of C2. The term energy Te of the 33Δg state is determined to be in the range of 78425-78475 cm-1 (9.724-9.730 eV) with respect to the ground X1Σg+ state, and the molecular constants such as vibrational and rotational constants are also determined, which are in reasonable agreement with those predicted by high-level ab initio theoretical calculations. Irregular vibrational energy level spacings in the 33Δg state are observed, which is tentatively attributed to the strong perturbations between the 33Δg and 23Δg states, as previously predicted by theory.

Electronic Structure and Chemical Bonding of the First-, Second-, and Third-Row-Transition-Metal Monoborides: The Formation of Quadruple Bonds in RhB, RuB, and TcB

Boron presents an important role in chemistry, biology, and materials science. Diatomic transition-metal borides (MBs) are the building blocks of many complexes and materials, and they present unique electronic structures with interesting and peculiar properties and a variety of bonding schemes which are analyzed here. In the first part of this paper, we present a review on the available experimental and theoretical studies on the first-row-transition-metal borides, i.e., ScB, TiB, VB, CrB, MnB, FeB, CoB, NiB, CuB, and ZnB; the second-row-transition-metal borides, i.e., YB, ZrB, NbB, MoB, TcB, RuB, RhB, PdB, AgB, and CdB; and the third-row-transition-metal borides, i.e., LaB, HfB, TaB, WB, ReB, OsB, IrB, PtB, AuB, and HgB. Consequently, in the second part, the second- and third-row MBs are studied via DFT calculations using the B3LYP, TPSSh, and MN15 functionals and, in some cases, via multi-reference methods, MRCISD+Q, in conjunction with the aug-cc-pVQZ-PPM/aug-cc-pVQZB basis sets. Specifically, bond distances, dissociation energies, frequencies, dipole moments, and natural NPA charges are reported. Comparisons between MB molecules along the three rows are presented, and their differences and similarities are analyzed. The bonding of the diatomic borides is also described; it is found that, apart from RhB(X1Σ+), which was just recently found to form quadruple bonds, RuB(X2Δ) and TcB(X3Σ-) also form quadruple σ2σ2π2π2 bonds in their X states. Moreover, to fill the gap existing in the current literature, here, we calculate the TcB molecule.

The unusual quadruple bonding of nitrogen in ThN

Nitrogen has five valence electrons and can form a maximum of three shared electron-pair bonds to complete its octet, which suggests that its maximum bond order is three. With a joint anion photoelectron spectroscopy and quantum chemistry investigation, we report herein that nitrogen presents a quadruple bonding interaction with thorium in ThN. The quadruple Th≣N bond consists of two electron-sharing Th-N π bonds formed between the Th-6dxz/6dyz and N 2px/2py orbitals, one dative Th←N σ bond and one weak Th←N σ bonding interaction formed between Th-6dz2 and N 2s/2pz orbitals. The ThC molecule has also been investigated and proven to have a similar bonding pattern as ThN. Nonetheless, due to one singly occupied σ-bond, ThC is assigned a bond order of 3.5. Moreover, ThC has a longer bond length as well as a lower vibrational frequency in comparison with ThN.

Eliminating all bonds from the ground state gives rise to ionic bonding in high-spin states of heterodiatomics

Understanding chemical bonding in second-row diatomics has been central to elucidating the basics of bonding itself. Bond strength and the number of bonds are the two factors that decide the reactivity of molecules. While bond strengths have been theoretically computed accurately and experimentally determined, the number of bonds is a more contentious issue, especially for complicated multi-reference systems like C2. We have developed an experimentally verifiable approach to determine bond numbers from excited spin state potential energy surfaces. On applying this to a series of second-row heterodiatomics, we obtain the surprising phenomenon of an inverted charge transfer ionic state after all the ground state bonds are broken via higher spin states. These ionic states are ubiquitous in all heterodiatomics and unexpected in non-metallic systems.

Mimicking the C2 molecule: M2B2 and M3B2+ clusters (M = Li, Na) and the reactivity of the N-heterocyclic carbene bound Li2B2 complex

C2 has attracted considerable attention from the scientific community for its debatable bonding situation. Herein, we show that the global minima of M2B2 and M3B2+ (M = Li, Na) possess similar covalent bonding patterns to C2. Because of strong charge transfer from M2/M3 to B2 dimer, they can be better described as [M2]2+[B2]2- and [M3]3+[B2]2- salt complexes with the B22- core surrounded perpendicularly by two and three M+ atoms, respectively. The energy decomposition analyses in combination with the natural orbital for chemical valence theory give four bonding components in C2, M2B2, and M3B2+ clusters. However, the fourth component does not arise from a bonding interaction but from polarization/hybridization. Considering the effect of Pauli repulsion in σ-space, the attractive covalent interaction in these molecules mainly comes from the two π-bonds. We further presented stable N-heterocyclic carbene (NHC) and triphenylphosphine (PPh3) ligands bound Li2B2(NHC)2 and Li2B2(PPh3)2 complexes. A comparative study of reactivity towards L = CO2, CO, and N2 between Li2B2(NHC)2 and B2(NHC)2 is also performed. L-Li2B2(NHC)2 is highly stable against L dissociation at room temperature for L = CO2 and CO, and the stability is markedly higher than that in L-B2(NHC)2. The larger B2→L π-backdonation in L-Li2B2(NHC)2 also makes L more activated than in L-B2(NHC)2.

New Methodology to Produce Sets of Valence Bond Structures with Enhanced Chemical Insights

The valence bond (VB) theory uses localized orbitals, and its wave function is composed of a linear combination of various VB structures which are based on sets of spin functions. The VB structures are not unique, and different sets are used, Rumer sets being the most common for classical VB due to their advantage as being both easily obtained as linearly independent and meaningful. Yet, Rumer rules, which are responsible for the simplified process of obtaining the Rumer sets, are very restrictive. Furthermore, Rumer sets are best suited for cyclic systems; however, in noncyclic systems, structures resulting from Rumer rules are often not the most intuitive/suitable structures for these systems. We have developed a method to obtain chemically insightful structures, which is based on concepts of chemical bonding. The method provides sets of VB structures with improved chemical insight, which can also be controlled. Parallel to the Rumer structures, the chemical insight sets of structures are based on electron pair coupling, and hence, pictorially can be drawn similarly to the Lewis structures. Yet, different from Rumer rules, the chemical insight method, being more flexible, allows larger combinations of bonds as well as larger combinations of structures in the sets it offers, resulting in many more possible sets that are better adapted to the systems studied.

Observation of the electronic band system 23Σg--a3Πu of C2 in the vacuum ultraviolet region

A systematic spectroscopic study of the dicarbon molecule C2 has important applications in various research fields, such as astrochemistry and combustion. In the short vacuum ultraviolet (VUV) wavelength region, recent theoretical calculations have predicted many absorption band systems of C2, but only few of them have been verified experimentally yet. In this work, we employed a tunable VUV laser radiation source based on the two-photon resonance-enhanced four-wave mixing method and a time-of-flight mass spectrometer to investigate the absorption bands of C2 in the VUV range of 64 000-66 000 cm-1. The electronic transition 23Σg-(v')-a3Πu(v″) of C2 has been observed and identified experimentally for the first time. The term value Te for the 23Σg- state is determined to be 66 389.9 ± 0.5 cm-1 above the ground state X1Σg+, and the vibrational and rotational constants are also determined. The experimentally measured spectroscopic parameters in this study are in excellent agreement with the theoretical results based on high-level ab initio calculations.

Dative quadruple bonds between d10 transition metals and beryllium in BeM(PMe3 )2 and BeM(CO)2 (M = Ni, Pd, and Pt) complexes: Transition metal fragments as six-electron donor and two-electron acceptor

The structure, chemical bonding, and reactivity of neutral 16 valence electrons (VE) transition metal complexes of beryllium, BeM(PMe₃ )₂ (1M-Be) and BeM(CO)₂ (2M-Be, M = Ni, Pd, and Pt) were studied. The molecular orbital and EDA-NOCV analysis suggest dative quadruple bonds between the transition metal and beryllium, viz., one Be→M σ bond, one Be←M σ bond, and two Be←M π bonds. The strength of these bonding interactions varies based on the ligands coordinated to the transition metal. The Be←M σ bond is stronger than the Be→M σ bond when the ligand is PMe_3, whereas the reverse order is observed when the ligand is CO. This is attributed to the higher π acceptor strength of CO as compared to PMe₃ . Since these complexes have M-Be dative quadruple bonds, the beryllium center is susceptible to ambiphilic reactivity, as indicated by high proton and hydride affinity values.

Design principles of the use of alkynes in radical cascades

One of the simplest organic functional groups, the alkyne, offers a broad canvas for the design of cascade transformations in which up to three new bonds can be added to each of the two sterically unencumbered, energy-rich carbon atoms. However, kinetic protection provided by strong π-orbital overlap makes the design of new alkyne transformations a stereoelectronic puzzle, especially on multifunctional substrates. This Review describes the electronic properties contributing to the unique utility of alkynes in radical cascades. We describe how to control the selectivity of alkyne activation by various methods, from dynamic covalent chemistry with kinetic self-sorting to disappearing directing groups. Additionally, we demonstrate how the selection of reactive intermediates directly influences the propagation and termination of the cascade. Diverging from a common departure point, a carefully planned reaction route can allow access to a variety of products.

A truncated Davidson method for the efficient “chemically accurate” calculation of full configuration interaction wavefunctions without any large matrix diagonalization

This work develops and illustrates a new method of calculating “chemically accurate” electronic wavefunctions (and energies) via a truncated full configuration interaction (CI) procedure, which arguably circumvents the large matrix diagonalization that is the core problem of full CI and is also central to modern selective CI approaches. This is accomplished simply by following the standard/ubiquitous Davidson method in its “direct” form—wherein, in each iteration, the electronic Hamiltonian operator is applied directly in second quantization to the Ritz vector/wavefunction from the prior iteration—except that (in this work) only a small portion of the resultant expansion vector is actually even computed (through the application of only a similarly small portion of the Hamiltonian). Specifically, at each iteration of this truncated Davidson approach, the new expansion vector is taken to be twice as large as that from the prior iteration. In this manner, a small set of highly truncated expansion vectors (say 10–30) of increasing precision is incrementally constructed, forming a small subspace within which diagonalization of the Hamiltonian yields clear, consistent, and monotonically variational convergence to the approximate full CI limit. The good efficiency in which convergence to the level of chemical accuracy (1.6 mhartree) is achieved suggests, at least for the demonstrated problem sizes—Hilbert spaces of 1018 and wavefunctions of 108 determinants—that this truncated Davidson methodology can serve as a replacement of standard CI and complete-active space approaches in circumstances where only a few chemically significant digits of accuracy are required and/or meaningful in view of ever-present basis set limitations.

A New Band System of the Dicarbon Molecule in the Vacuum Ultraviolet Region

As one of the most abundant molecules in the universe, the long history of spectroscopic studies of the dicarbon molecule, C₂, reaches back two centuries. While many electronic band systems with upper states below the lowest dissociation threshold have been well characterized, much less is known about transitions to higher-lying states. Here, we report the observation of a new band system of C₂ from the lowest triplet state a³Π_u through a resonance-enhanced multiphoton ionization scheme. The upper state is identified as 1³Σ_g⁺, which is determined to be 61539.0 cm^-1 (7.630 eV) above ground state X¹Σ_g⁺. The spectroscopic parameters determined for the 1³Σ_g⁺ state are in excellent agreement with those predicted by the high-level ab initio calculations. This study paves the way for systematic investigations of the photoabsorption and photodissociation of C₂ in the vacuum ultraviolet region, which has important applications in the field of astrochemistry.

Two σ- and two π-dative quadruple bonds between the s-block element and transition metal in [BeM(CO)4; M = Fe - Os]

We report the chemical bonding and reactivity of the first example of neutral 18 valence electron transition metal complexes of beryllium, [BeM(CO)₄; M = Fe - Os], in trigonal bipyramidal coordination geometry, where the bonding between the transition metal and the s-block element beryllium (M-Be) can be best described by dative quadruple bonds. In contrast to the conventional multiple bonding pattern, the quadruple bonds comprise two σ-bonds and two π-bonds, viz., one Be → M σ-bond, one M → Be σ-bond, and two M → Be π-bonds. Since the M-Be quadruple bonds are described by dative interactions, the Be centre shows ambiphilic character as indicated by the high proton and hydride affinity values.

From the gas phase to the solid state: The chemical bonding in the superheavy element flerovium

As early as 1975, Pitzer suggested that copernicium, flerovium and oganesson are volatile substances behaving noble-gas like because of their closed-shell configurations and accompanying relativistic effects. It is, however, precarious to predict the chemical bonding and physical behavior of a solid by knowledge of the atomic or molecular properties only. Copernicium and oganesson have been analyzed very recently by our group. Both are predicted to be semi-conductors and volatile substances with rather low melting and boiling points, which may justify a comparison with the noble gas elements. Here we study closed-shell flerovium in detail to predict solid-state properties including the melting point from a decomposition of the total energy into many-body forces derived from relativistic coupled-cluster and from density functional theory. The convergence of such a decomposition for flerovium is critically analyzed, and the problem of using density functional theory is highlighted. We predict that flerovium is in many ways not behaving like a typical noble gas element despite its closed-shell 7$p_{1/2}^2$ configuration and resulting weak interactions. Unlike for the noble gases, the many-body expansion in terms of the interaction energy is not converging smoothly. This makes the accurate prediction of phase transitions very difficult. Nevertheless, a first prediction by Monte-Carlo simulation estimates the melting point at $284\pm 50$ K. Furthermore, calculations for the electronic band gap suggests that flerovium is a semi-conductor similar to copernicium

Analysis of chemical bonding of the ground and low-lying states of Mo2 and of Mo2Clx complexes, x = 2 - 10

In the present study, we perform accurate calculations via multireference configuration interaction and coupled cluster methodologies on the dimolybdenum molecule in conjunction with complete series of correlation and weighted core correlation consistent basis sets up to quintuple size. The bonding, dissociation energies, and spectroscopic parameters of the seven states that correlate to the ground state products are calculated. The ground state has a sextuple chemical bond and each of the calculated excited state has one less bond than the previous one. The calculated values for the ground(X1Σg+ ) state of Mo2 have been extrapolated to the complete basis set limits. Our final values, re=1.9324 Å and De(D0)=4.502{plus minus}0.007(4.471{plus minus}0.009) eV, are in excellent agreement with the experimental values of re=1.929, 1.938(9) Å and D0=4.476(10) eV. The Mo2 in 13Σg+ state is a weakly bound dimer, forming 5s...5pz bonds, with De=0.120 eV at re=3.53 Å. All calculated excited states (except 13Σg+) have a highly multireference character (C0=0.25-0.55). The ordering of the molecular bonding orbitals changes as the spin is increased from quintet to septet state. The quite low bond dissociation energy of the ground state is due to the splitting of the molecular bonding orbitals in two groups differing in energy by ~3 eV. Finally, the bond breaking of Mo2, as the multiplicity of spin is increased, is analyzed in parallel with the Mo-Mo bond breaking in a series of Mo2Clx complexes when x is increased. Physical insight into the nature of the sextuple bond and its low dissociation energy is provided.

The Nature of the Polar Covalent Bond

Quantum chemical calculations using density functional theory are reported for the diatomic molecules LiF, BeO, and BN. The nature of the interatomic interactions is analyzed with the EDA-NOCV method, and the results are critically discussed and compared with data from QTAIM, NBO and Mayer approaches. Polar bonds, like nonpolar bonds, are caused by the interference of wave functions, which lead to an accumulation of electronic charge in the bonding region. Polar bonds generally have a larger percentage of electrostatic bonding to the total attraction, but nonpolar bonds may also possess large contributions from Coulombic interaction. The term "ionic contribution" refers to VB structures and is misleading because it refers to separate fragments with negligible overlap that occur only in the solid state and in solution, not in a molecule. The EDA-NOCV method gives detailed information about the individual orbital contributions, which can nicely be identified by visual inspection of the associated deformation densities. It is very important, particularly for polar bonds to distinguish between the interatomic interactions of the final dissociation products after bond rupture and the interactions between the fragments in the eventually formed bond.

The role of references and the elusive nature of the chemical bond

Chemical bonding theory is of utmost importance to chemistry, and a standard paradigm in which quantum mechanical interference drives the kinetic energy lowering of two approaching fragments has emerged. Here we report that both internal and external reference biases remain in this model, leaving plenty of unexplored territory. We show how the former biases affect the notion of wavefunction interference, which is purportedly recognized as the most basic bonding mechanism. The latter influence how bonding models are chosen. We demonstrate that the use of real space analyses are as reference-less as possible, advocating for their use. Delocalisation emerges as the reference-less equivalent to interference and the ultimate root of bonding. Atoms (or fragments) in molecules should be understood as a statistical mixture of components differing in electron number, spin, etc.

The theory of chemical bonding relies on arbitrary references. Here the authors report a fundamental study on the chemical bond showing that considering the binding fragments as objects in real space enables to eliminate inherent biases.

Multireference configuration interaction study of the predissociation of C2 via its F 1Πu state

Photodissociation is one of the main destruction pathways for dicarbon (C2) in astronomical environments such as diffuse interstellar clouds, yet the accuracy of modern astrochemical models is limited by a lack of accurate photodissociation cross sections in the vacuum ultraviolet range. C2 features a strong predissociative F 1Πu-X 1Σg+ electronic transition near 130 nm originally measured in 1969; however, no experimental studies of this transition have been carried out since, and theoretical studies of the F 1Πu state are limited. In this work, potential energy curves of excited electronic states of C2 are calculated with the aim of describing the predissociative nature of the F 1Πu state and providing new ab initio photodissociation cross sections for astrochemical applications. Accurate electronic calculations of 56 singlet, triplet, and quintet states are carried out at the DW-SA-CASSCF/MRCI+Q level of theory with a CAS(8,12) active space and the aug-cc-pV5Z basis set augmented with additional diffuse functions. Photodissociation cross sections arising from the vibronic ground state to the F 1Πu state are calculated by a coupled-channel model. The total integrated cross section through the F 1Πu v=0 and v=1 bands is 1.198×10−13 cm2cm-1, giving rise to a photodissociation rate of 5.02×10−10 s−1 under the standard interstellar radiation field, much larger than the rate in the Leiden photodissociation database. In addition, we report a new 2 1Σu+ state that should be detectable via a strong 2 1Σu+-X 1Σg+ band around 116 nm.

Orbital contraction and covalent bonding

According to Ruedenberg's classic treatise on the theory of chemical bonding [K. Ruedenberg, Rev. Mod. Phys. 34, 326-376 (1962)], orbital contraction is an integral consequence of covalent bonding. While the concept is clear, its quantification by quantum chemical calculations is not straightforward, except for the simplest of molecules, such as H₂ ⁺ and H₂. This paper proposes a new, yet simple, approach to the problem, utilizing the modified atomic orbital (MAO) method of Ehrhardt and Ahlrichs [Theor. Chim. Acta 68, 231 (1985)]. Through the use of MAOs, which are an atom-centered minimal basis formed from the molecular and atomic density operators, the wave functions of the species of interest are re-expanded, allowing the computation of the kinetic energy (and any other expectation value) of free and bonded fragments. Thus, it is possible to quantify the intra- and interfragment changes in kinetic energy, i.e., the effects of contraction. Computations are reported for a number of diatomic molecules H₂, Li₂, B₂, C₂, N₂, O₂, F₂, CO, P₂, and Cl₂ and the polyatomics CH₃-CH₃, CH₃-SiH₃, CH₃-OH, and C₂H₅-C₂H₅ (where the single bonds between the heavy atoms are studied) as well as dimers of He, Ne, Ar, and the archetypal ionic molecule NaCl. In all cases, it is found that the formation of a covalent bond is accompanied by an increase in the intra-fragment kinetic energy, an indication of orbital contraction and/or deformation.

Diabatic Valence-Hole States in the C2 Molecule: "Putting Humpty Dumpty Together Again"

Despite the long history of spectroscopic studies of the C₂ molecule, fundamental questions about its chemical bonding are still being hotly debated. The complex electronic structure of C₂ is a consequence of its dense manifold of near-degenerate, low-lying electronic states. A global multi-state diabatic model is proposed here to disentangle the numerous configuration interactions that occur within four symmetry manifolds of excited states of C₂ (¹Π_g, ³Π_g, ¹Σ_u⁺ , and ³Σ_u⁺ ). The key concept of our model is the existence of two "valence-hole" configurations, 2σg22σu11πu33σg2 for ^1,3Π_g states and 2σg22σu11πu43σg1 for ^1,3Σ_u⁺ states, that are derived from 3σ_g ← 2σ_u electron promotion. The lowest-energy state from each of the four C₂ symmetry species is dominated by this type of valence-hole configuration at its equilibrium internuclear separation. As a result of their large binding energy (nominal bond order of 3) and correlation with the 2s²2p² + 2s2p³ separated-atom configurations, the presence of these valence-hole configurations has a profound impact on the global electronic structure and unimolecular dynamics of C₂.

The Hitchhiker's Guide to the Wave Function∥

The electronic wave function of molecules is 3N-dimensional and inseparable in the coordinates of the N electrons. Whereas molecular orbitals are often invoked to visualize the electronic structure, they are nonunique, with the same 3N-dimensional wave function being represented by an infinite number of 3-D, one-electron functions (orbitals). Furthermore, multireference wave functions cannot be described by an antisymmetrized product of a single set of occupied orbitals. What is required is a way to visualize the full dimensionality of the wave function, including the effects of correlation, as a 3N-dimensional being would be able to do. In the past 5 years, we have been developing a way to analyze and visualize highly dimensional wave functions by focusing on the structure of the repeating unit demanded by fermionic behavior. This 3N-dimensional repeating unit, the wave function "tile", can be projected onto the three dimensions of each electron, in turn, to reveal the complete electronic structure. It is found that the tile reproduces canonical chemical motifs such as core-electrons, single bonds and lone pairs. Multiple bonds emerge as the "banana" bonds favored by Pauling. As a function of the reaction coordinate, electron motions are visualized that correspond to the curly arrow notation of organic chemists. Excited states can also be inspected. Analyzing a wave function in terms of fermionic tiling allows for insight not facilitated by the inspection of orbitals or configuration interaction vectors: The wave function tiles of resonance structures reveal that electron correlation in benzene pushes opposing spin electrons to occupy alternate Kekulé structures, and in C₂, the emerging structure supports the notion of a triply bonded structure with a weak, fourth bonding contribution.

Probing the Nature of the Transition-Metal-Boron Bonds and Novel Aromaticity in Small Metal-Doped Boron Clusters Using Photoelectron Spectroscopy

Photoelectron spectroscopy combined with quantum chemistry has been a powerful approach to elucidate the structures and bonding of size-selected boron clusters (B_n^-), revealing a prevalent planar world that laid the foundation for borophenes. Investigations of metal-doped boron clusters not only lead to novel structures but also provide important information about the metal-boron bonds that are critical to understanding the properties of boride materials. The current review focuses on recent advances in transition-metal-doped boron clusters, including the discoveries of metal-boron multiple bonds and metal-doped novel aromatic boron clusters. The study of the RhB^- and RhB₂O^- clusters led to the discovery of the first quadruple bond between boron and a transition-metal atom, whereas a metal-boron triple bond was found in ReB₂O^- and IrB₂O^-. The ReB₄^- cluster was shown to be the first metallaborocycle with Möbius aromaticity, and the planar ReB₆^- cluster was found to exhibit aromaticity analogous to metallabenzenes. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Photodissociation of dicarbon: How nature breaks an unusual multiple bond

It has long been observed that the coma of a comet is often green while its tail is not. While the explanation for this must be that the molecules responsible for the green emission, C₂, are photodissociated, the mechanism was, until now, unknown. We have observed the photodissociation of C₂ in the laboratory for the first time and, in doing so, have determined its bond dissociation energy with unprecedented precision. Invoking the observed mechanism, the calculated lifetime of cometary C₂ is found to be consistent with astronomical observations.

The dicarbon molecule (C₂) is found in flames, comets, stars, and the diffuse interstellar medium. In comets, it is responsible for the green color of the coma, but it is not found in the tail. It has long been held to photodissociate in sunlight with a lifetime precluding observation in the tail, but the mechanism was not known. Here we directly observe photodissociation of C₂. From the speed of the recoiling carbon atoms, a bond dissociation energy of 602.804(29) kJ·mol−1 is determined, with an uncertainty comparable to its more experimentally accessible N₂ and O₂ counterparts. The value is within 0.03 kJ·mol⁻¹ of high-level quantum theory. This work shows that, to break the quadruple bond of C₂ using sunlight, the molecule must absorb two photons and undergo two “forbidden” transitions.

A Critical Look at Linus Pauling's Influence on the Understanding of Chemical Bonding

The influence of Linus Pauling on the understanding of chemical bonding is critically examined. Pauling deserves credit for presenting a connection between the quantum theoretical description of chemical bonding and Gilbert Lewis's classical bonding model of localized electron pair bonds for a wide range of chemistry. Using the concept of resonance that he introduced, he was able to present a consistent description of chemical bonding for molecules, metals, and ionic crystals which was used by many chemists and subsequently found its way into chemistry textbooks. However, his one-sided restriction to the valence bond method and his rejection of the molecular orbital approach hindered further development of chemical bonding theory for a while and his close association of the heuristic Lewis binding model with the quantum chemical VB approach led to misleading ideas until today.

Routes involving no free C2 in a DFT-computed mechanistic model for the reported room-temperature chemical synthesis of C2

Recent lively debates about the nature of the quadruple bonding in the diatomic species C2 have been heightened by recent suggestions of molecules in which carbon may be similarly bonded to other elements. The desirability of having methods for generating such species at ambient temperatures and in solution in order to study their properties may have been realized by a recent report of the first chemical synthesis of free C2 itself under mild conditions. The method involved unimolecular fragmentation of an alkynyl zwitterion 2 as generated from the precursor 1, resulting in production and then trapping of free C2 at ambient temperatures rather than the high temperature gas phase methods normally employed for C2 generation. Here, alternative mechanisms are proposed for this reaction based on DFT calculations involving bimolecular 1,1- or 1,2-iodobenzene displacement reactions from 2 directly by galvinoxyl radical, or hydride transfer from 9,10-dihydroanthracene to 2. These mechanisms result in the same trapped products as observed experimentally, but unlike that involving unimolecular generation of free C2, exhibit calculated free energy barriers commensurate with the reaction times observed at room temperatures. The relative energies of the transition states for 1,1 vs. 1,2 substitution provide a rationalisation for the observed isotopic substitution patterns. The same mechanism also provides an energetically facile path to polymeric synthesis of carbon rich species by extending the carbon chain attached to the iodonium group, eventually resulting in formation of amorphous carbon and discrete molecules such as C60.

Multiple Bonding in Rhodium Monoboride. Quasi-atomic Analyses of the Ground and Low-Lying Excited States

The bonding structures of the ground state and the lowest five excited states of rhodium monoboride are identified by determining the quasi-atomic orbitals in full valence space MCSCF wave functions and the interactions between these orbitals. A quadruple bond, namely two π-bonds and two σ-bonds, is identified and characterized for the X¹Σ⁺ ground state, in agreement with a previous report (Cheung J. Phys. Chem. Lett. 2020, 11, 659-663). However, in all excited states, the bonding is predicted to be weaker because, in these states, one of the σ-bonding interactions has a small magnitude. In the a³Δ and A¹Δ states, the bond order is between a triple and quadruple bond. In the b³Σ⁺ state, the Rh-B linkage is a triple bond. In the c³Π and B¹Π states, the atoms are linked by a double bond due to an additional weakening of the two π-bonds. The decreases in the predicted bond strengths are reflected in the decreases of the predicted binding energies and in the increases of the predicted bond lengths from the X¹Σ⁺ ground state to the c³Π and the B¹Π excited states. Notably, the 5pσ orbital of rhodium, which is vacant in the ground state of the atom, plays a significant role in the molecule.

Spin-Coupled Generalized Valence Bond Theory: New Perspectves on the Electronic Structure of Molecules and Chemical Bonds

Spin-Coupled Generalized Valence Bond (SCGVB) theory provides the foundation for a comprehensive theory of the electronic structure of molecules. SCGVB theory offers a compelling orbital description of the electronic structure of molecules as well as an efficient and effective zero-order wave function for calculations striving for quantitative predictions of molecular structures, energetics, and other properties. The orbitals in the SCGVB wave function are usually semilocalized, and for most molecules, they can be interpreted using concepts familiar to all chemists (hybrid orbitals, localized bond pairs, lone pairs, etc.). SCGVB theory also provides new perspectives on the nature of the bonds in molecules such as C₂, Be₂ and SF₄/SF₆. SCGVB theory contributes unparalleled insights into the underlying cause of the first-row anomaly in inorganic chemistry as well as the electronic structure of organic molecules and the electronic mechanisms of organic reactions. The SCGVB wave function accounts for nondynamical correlation effects and, thus, corrects the most serious deficiency in molecular orbital (RHF) wave functions. Dynamical correlation effects, which are critical for quantitative predictions, can be taken into account using the SCGVB wave function as the zero-order wave function for multireference configuration interaction or coupled cluster calculations.

Ab Initio Dot Structures Beyond the Lewis Picture

The empirical Lewis picture of the chemical bond dominates the view chemists have of molecules, of their stability and reactivity. Within the mathematical framework of quantum mechanics, all this chemical information is hidden in the many-particle wave function Ψ. Thus, to reveal and understand it, there is great interest in enhancing the Lewis model and connecting it to computable quantities. As has previously been shown, the Lewis picture can often be recovered from the probability density |Ψ|2 with probabilities in agreement with valence bond weights: the structures appear as most likely positions in the all-electron configuration space. Here, we systematically expand this topological probability density analysis to molecules with multiple bonds and lone pairs, employing correlated Slater-Jastrow wave functions. In contrast to earlier studies, non-Lewis structures are obtained that disagree with the prevalent picture and have a potentially better predictive capability. While functional groups are still recovered with these ab initio structures, the boundary between bonds and lone pairs is mostly blurred or non-existent. In order to understand the newly found structures, the Lewis electron pairs are replaced with spin-coupled electron motifs as the fundamental electronic fragment. These electron motifs—which coincide with Lewis’ electron pairs for many single bonds—arise naturally from the generally applicable analysis presented. An attempt is made to rationalize the geometry of the newly-found structures by considering the Coulomb force and the Pauli repulsion.

The Spectroscopy of C2: A Cosmic Beacon

ConspectusDicarbon, the molecule formed from two carbon atoms, is among the most abundant molecules in the universe. Said by some to exhibit a quadruple bond, it is bound by more than 6 eV and supports a large number of valence electronic states. It thus has a rich spectroscopy, with 19 one-photon band systems, four of which were discovered by the author and co-workers. Its spectrum was among the first to be described: Wollaston reported the emission spectra from blue flames in 1802.C₂ is observed in a variety of astronomical objects, including stars, circumstellar shells, nebulae, comets and the interstellar medium. It is responsible for the green color of cometary comae but is not observed in the comet tail. It can be observed in absorption and emission by optical spectroscopy in the infrared, visible, and ultraviolet regions of the spectrum, and because it has no electric-dipole-allowed vibrational or rotational transitions, its spectral signature is a sensitive probe of the local environment.Before the work described in this Account, models of C₂ photophysics included the thitherto-unobserved c³Σ_u⁺ state and parametrized the strength of spin-forbidden intercombination transitions. Furthermore, they did not account for photodissociation of C₂, even though it was identified in the 1930s as a key process. Inspired by the observation of C₂ in the Red Rectangle nebula, the author was motivated to instill rigor into C₂ models and embarked on a spectroscopic and computational journey that has lasted 15 years.We were the first to identify the c³Σ_u⁺ state through the d³Π_g-c³Σ_u⁺ transitions, which were to become known as the "Duck" system. This minor partner to the well-known Swan bands is a key part of astrophysical C₂ models and can now be included with rigor. We identified the e³Π_g-c³Σ_u⁺ system, and the c³Σ_u⁺ state is now well-studied. Meanwhile others described the singlet-triplet and triplet-quintet interactions in exquisite detail, allowing rigorous modeling of the a-X and c-X intercombination transitions.The final piece of the C₂ puzzle would be understanding how long it survives before being broken into carbon atom fragments. Though predicted by Herzberg, predissociation in the e³Π_g state had never been observed. To find it would require the complicated ultraviolet spectroscopy of C₂ to be disentangled. In so doing, we identified the 4³Π_g and 3³Π_g states of C₂, thus uncovering two new band systems. The 4³Π_g state allowed the first accurate determination of the ionization energy of C₂. With these new band systems secure, we extracted new levels of the D¹Σ_u⁺ state (Mulliken bands) and the e³Π_g state (Fox-Herzberg bands) from our spectra. Upon climbing the energy ladder in the e³Π_g state to v = 12, we finally identified the route to predissociation of C₂ via non-adiabatic coupling to the d³Π_g state. This observation provided the first laboratory evidence for why C₂ is observed in the coma of a comet but not the tail.

General Correlated Geminal Ansatz for Electronic Structure Calculations: Exploiting Pfaffians in Place of Determinants

We propose here a single Pfaffian correlated variational ansatz that dramatically improves the accuracy with respect to the single determinant one, while remaining at a similar computational cost. A much larger correlation energy is indeed determined by the most general two electron pairing function, including both singlet and triplet channels, combined with a many-body Jastrow factor, including all possible spin-spin, spin-density, and density-density terms. The main technical ingredient to exploit this accuracy is the use of the Pfaffian for antisymmetrizing a highly correlated pairing function, thus recovering the Fermi statistics for electrons with an affordable computational cost. Moreover, the application of the diffusion Monte Carlo, within the fixed node approximation, allows us to obtain very accurate binding energies for the first preliminary calculations reported in this study: C2, N2, and O2 and the benzene molecule. This is promising and remarkable, considering that they represent extremely difficult molecules even for computationally demanding multideterminant approaches, and opens therefore the way for realistic and accurate electronic simulations with an algorithm scaling at most as the fourth power of the number of electrons.

Comprehending the quadruple bonding conundrum in C2 from excited state potential energy curves

The question of quadruple bonding in C₂ has emerged as a hot button issue, with opinions sharply divided between the practitioners of Valence Bond (VB) and Molecular Orbital (MO) theory.

The question of quadruple bonding in C₂ has emerged as a hot button issue, with opinions sharply divided between the practitioners of Valence Bond (VB) and Molecular Orbital (MO) theory. Here, we have systematically studied the Potential Energy Curves (PECs) of low lying high spin sigma states of C₂, N₂, Be₂ and HC Created by potrace 1.16, written by Peter Selinger 2001-2019 ]]> CH using several MO based techniques such as CASSCF, RASSCF and MRCI. The analyses of the PECs for the ^2S+1Σ_g/u (with 2S + 1 = 1, 3, 5, 7, 9) states of C₂ and comparisons with those of relevant dimers and the respective wavefunctions were conducted. We contend that unlike in the case of N₂ and HC Created by potrace 1.16, written by Peter Selinger 2001-2019 ]]> CH, the presence of a deep minimum in the ⁷Σ⁺ state of C₂ and CN⁺ suggests a latent quadruple bonding nature in these two dimers. Our investigations reveal that the number of bonds in the ground state can be determined for 2^nd row dimers by figuring out at what value of spin symmetry a purely dissociative PEC is obtained. For N₂ and HC Created by potrace 1.16, written by Peter Selinger 2001-2019 ]]> CH the purely dissociative PEC appears for the septet spin symmetry as compared to that for the nonet in C₂. This is indicative of a higher number of bonds between the two 2^nd row atoms in C₂ as compared to those of N₂ and HC Created by potrace 1.16, written by Peter Selinger 2001-2019 ]]> CH. Hence, we have struck a reconciliatory note between the MO and VB approaches. The evidence provided by us can be experimentally verified, thus providing the window so that the narrative can move beyond theoretical conjectures.

Room-temperature chemical synthesis of C2

Diatomic carbon (C2) is historically an elusive chemical species. It has long been believed that the generation of C2 requires extremely high physical energy, such as an electric carbon arc or multiple photon excitation, and so it has been the general consensus that the inherent nature of C2 in the ground state is experimentally inaccessible. Here, we present the chemical synthesis of C2 from a hypervalent alkynyl-λ3-iodane in a flask at room temperature or below, providing experimental evidence to support theoretical predictions that C2 has a singlet biradical character with a quadruple bond, thus settling a long-standing controversy between experimental and theoretical chemists, and that C2 serves as a molecular element in the bottom-up chemical synthesis of nanocarbons such as graphite, carbon nanotubes, and C60.

The electronic structure of benzene from a tiling of the correlated 126-dimensional wavefunction

The electronic structure of benzene is a battleground for competing viewpoints of electronic structure, with valence bond theory localising electrons within superimposed resonance structures, and molecular orbital theory describing delocalised electrons. But, the interpretation of electronic structure in terms of orbitals ignores that the wavefunction is anti-symmetric upon interchange of like-spins. Furthermore, molecular orbitals do not provide an intuitive description of electron correlation. Here we show that the 126-dimensional electronic wavefunction of benzene can be partitioned into tiles related by permutation of like-spins. Employing correlated wavefunctions, these tiles are projected onto the three dimensions of each electron to reveal the superposition of Kekulé structures. But, opposing spins favour the occupancy of alternate Kekulé structures. This result succinctly describes the principal effect of electron correlation in benzene and underlines that electrons will not be spatially paired when it is energetically advantageous to avoid one another.

Observation of Four-Fold Boron-Metal Bonds in RhB(BO-) and RhB

The maximum bond order between two main-group atoms was known to be three. However, it has been suggested recently that there is quadruple bonding in C₂ and analogous eight-valence electron species. While the quadruple bond in C₂ has aroused some debates, an interesting question is: are main-group elements capable of forming quadruple bonds? Here we use photoelectron spectroscopy and computational chemistry to probe the electronic structure and chemical bonding in RhB₂O^- and RhB^- and show that the boron atom engages in quadruple bonding with rhodium in RhB(BO)^- and neutral RhB. The quadruple bonds consist of two π-bonds formed between the Rh 4d_xz/4d_yz and B 2p_x/2p_y orbitals and two σ-bonds between the Rh 4d_z² and B 2s/2p_z orbitals. To confirm the quadruple bond in RhB, we also investigate the linear Rh≡B-H⁺ species and find a triple bond between Rh and B, which has a longer bond length, lower stretching frequency, and smaller bond dissociation energy in comparison with that of the Rh≣B quadruple bond in RhB.

Electronic Wavefunction Tiles

We review the pre-quantum theories of electronic structure of Lewis and Langmuir, and how this relates to the post-quantum double-quartet theory of Linnett. Linnett’s ideas are put on a firm theoretical footing through the emergence of the wavefunction tile: The 3N-dimensional repeating structure of the N-electron wavefunction. Wavefunction tiles calculated by the dynamic Voronoi Metropolis sampling method are reviewed, and new results are presented for bent bonds of cyclopropane, and electron correlation in Be-O-Be.