Electronic structure of Li1,2,3 +,0,- and nature of the bonding in Li2,3 +,0,.

The current study of the small lithium molecules Li2 +,0,- and Li3 +,0,- focuses on the nature of the bonding in these molecules as well as their structures and energetics (bond energies, ionization energies, and electron affinities). Valence CASSCF (2s,2p) calculations incorporate nondynamical electron correlation in the calculations, while the corresponding multireference configuration interaction and coupled cluster calculations incorporate dynamical electron correlation. Treatment of nondynamical correlation is critical for properly describing the Li2,3 +,0,- molecules as well as the Li- anion with dynamical correlation, in general, only fine-tuning the predictions. All lithium molecules and ions are bound, with the Li3 + and Li2 + ions being the most strongly bound, followed by Li3 - , Li2 , Li2 - and Li3 . The minimum energy structures of Li3 +,0,- are, respectively, an equilateral triangle, an isosceles triangle, and a linear structure. The results of SCGVB calculations are analyzed to obtain insights into the nature of the bonding in these molecules. An important finding of this work is that interstitial orbitals, a concept first put forward by McAdon and Goddard in 1985, play an essential role in the bonding of all lithium molecules considered here except for Li2 . The interstitial orbitals found in the Li3 +,0 molecules likely give rise to the non-nuclear attractors/maxima observed in these molecules.

Dynamical electron correlation and the chemical bond. II. Recoupled pair bonds in the a4Σ- states of CH and CF.

We extended our studies of the effect of dynamical electron correlation on the covalent bonds in the AH and AF series (A = B-F) to the recoupled pair bonds in the excited a4Σ- states of the CH and CF molecules. Dynamical correlation is energetically less important in the a4Σ- states than in the corresponding X2Π states for both molecules, which is reflected in smaller changes in bond energies (De). Changes in the equilibrium bond distance (Re) and vibrational frequency (ωe), on the other hand, are influenced by the changes in the slope and curvature of the dynamical electron correlation energy as a function of the internuclear distance R, EDEC(R). In the CH(a4Σ-) state, these changes are much smaller than in the CH(X2Π) state, but in the CF(a4Σ-) state, they are larger, reflecting a significant difference in the shapes of EDEC(R) curves. At large R, the shape of EDEC(R) curves for covalent and recoupled pair bonds is similar although different in magnitude. For the CH(a4Σ-) state, EDEC(R) has a minimum at R = Re + 0.72 Å as the orbitals associated with the formation of the recoupled pair bond switch places. EDEC(R) for the CF(a4Σ-) state decreases continuously throughout the bound region of the potential energy curve because the dynamical correlation energy associated with the electrons in the lone pair orbitals is increasing. These results support our earlier conclusion that we still have much to learn about the nature of dynamical electron correlation in molecules.

Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B-F).

Dynamical electron correlation has a major impact on the computed values of molecular properties and the energetics of molecular processes. This study focused on the effect of dynamical electron correlation on the spectroscopic constants (Re, ωe, De), and potential energy curves, ΔE(R), of the covalently bound AH and AF molecules, A = B-F. The changes in the spectroscopic constants (ΔRe, Δωe, ΔDe) caused by dynamical correlation are erratic and, at times, even surprising. These changes can be understood based on the dependence of the dynamical electron correlation energies of the AH and AF molecules as a function of the bond distance, i.e., ΔEDEC(R). At large R, the magnitude of ΔEDEC(R) increases nearly exponentially with decreasing R, but this increase slows as R continues to decrease and, in many cases, even reverses at very short R. The changes in ΔEDEC(R) in the region around Re were as unexpected as they were surprising, e.g., distinct minima and maxima were found in the curves of ΔEDEC(R) for the most polar molecules. The variations in ΔEDEC(R) for R ≲ Re are directly correlated with major changes in the electronic structure of the molecules as revealed by a detailed analysis of the spin-coupled generalized valence bond wave function. The results reported here indicate that we have much to learn about the nature of dynamical electron correlation and its effect on chemical bonds and molecular properties and processes.

New Insights into the Remarkable Difference between CH5- and SiH5.

It has long been known that there is a fundamental difference in the electronic structures of CH5- and SiH5-, two isoelectronic molecules. The former is a saddle point for the SN2 exchange reaction H- + CH4 → [CH5-]‡ → CH4 + H-, while the latter is a stable molecule that is bound relative to SiH4 + H-. SCGVB calculations indicate that this difference is the result of a dramatic difference in the nature of the axial electron pairs in these anions. In SiH5-, the axial pairs represent two stable bonds-a weak recoupled pair bond dyad. In CH5-, the axial electron pairs represent an intermediate transition between the electron pairs in the reactants and those in the products. Furthermore, the axial orbitals at the saddle point in CH5- are highly overlapping, giving rise to strong Pauli repulsion and a high barrier for the SN2 exchange reaction.

Nature of the Bonding in the Bifluoride Anion, FHF.

The nature of the bonding in the bifluoride anion, FHF-, has long been a topic of discussion with little resolution. A recent (2021) spectroscopic-theoretical study concluded that the bonds in this molecule represent a "crossover from hydrogen to chemical bonding." Spin-coupled generalized valence bond (SCGVB) theory is an advanced orbital theory that describes a broad range of molecules and molecular processes, and its application has provided valuable insights in the electronic structure of many molecules with "unusual" bonding motifs. SCGVB calculations on the FHF- anion indicate that the bonding in this molecule cannot be attributed to a traditional hydrogen bond or a traditional covalent bond. Instead, the bonds in FHF- represent a new bonding motif-two polarized, delocalized F- anions held together by a positively charged hydrogen atom, i.e., the bonding resembles that for a proton-bound anion pair.

Spin-Coupled Generalized Valence Bond Theory: New Perspectives 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.

A cautionary tale: Problems in the valence-CASSCF description of the ground state (X1Σ+) of BF.

In the full optimized reaction space and valence-complete active space self-consistent field (vCAS) methods, a set of active orbitals is defined as the union of the valence orbitals on the atoms, all possible configurations involving the active orbitals are generated, and the orbitals and configuration coefficients are self-consistently optimized. Such wave functions have tremendous flexibility, which makes these methods incredibly powerful but can also lead to inconsistencies in the description of the electronic structure of molecules. In this paper, the problems that can arise in vCAS calculations are illustrated by calculations on the BH and BF molecules. BH is well described by the full vCAS wave function, which accounts for molecular dissociation and 2s-2p near-degeneracy in the boron atom. The same is not true for the full vCAS wave function for BF. There is mixing of core and active orbitals at short internuclear distances and swapping of core and active orbitals at large internuclear distances. In addition, the virtual 2π orbitals, which were included in the active space to account for the 2s-2p near degeneracy effect, are used instead to describe radial correlation of the electrons in the F2pπ-like pairs. Although the above changes lead to lower vCAS energies, they lead to higher vCAS+1+2 energies as well as irregularities and/or discontinuities in the potential energy curves. All of the above problems can be addressed by using the spin-coupled generalized valence bond-inspired vCAS wave function for BF, which includes only a subset of the atomic valence orbitals in the active space.

The nature of the chemical bond and the role of non-dynamical and dynamical correlation in Be2.

In the spin-coupled generalized valence bond (SCGVB) description of Be2, there is a pair of electrons in highly overlapping "inner" orbitals corresponding to a traditional σ bond, but this bond is compromised by Pauli repulsion arising from its overlap with a second "outer" pair. The presence of this outer pair of electrons leads to a repulsive potential energy curve at long range and a bound, but metastable molecule at short range. To obtain further insights into the nature of the bond in Be2, we determined the non-dynamical and dynamical correlation contributions to the potential energy curve of Be2 using four different choices for the zero-order wave function: Restricted Hartree-Fock (RHF), SCGVB, valence-CASSCF(4,4), and valence-CASSCF(4,8). The SCGVB and valence-CASSCF(4,4) wave functions yield similar breakdowns of the total correlation energy, with non-dynamical correlation being the more important contribution. For the RHF and valence-CASSCF(4,8) wave functions, dynamical correlation is critical, without which the potential energy curve is purely repulsive. High accuracy calculations on the HBen-1Be-BeBen-1H molecule as a function of n (n = 1-6) suggest that the intrinsic strength of a Be-Be σ bond uncompromised by Pauli repulsion is on the order of 62-63 kcal/mol, and its length is 2.13-2.14 Å, ∼60 kcal/mol stronger and ∼0.35 Å shorter than in Be2.

Resolving a puzzling anomaly in the spin-coupled generalized valence bond description of benzene.

In an earlier study of benzene, Small and Head-Gordon found that the spin-coupled generalized valence bond (SCGVB) wave function for the π system predicted a distorted (non-D6h ) geometry, one with alternating CC bond lengths. However, the variations in the energy were very small and the predictions were made using a very small basis set (STO-3G). We re-examined this prediction using a much larger basis set (aug-cc-pVTZ) to determine the dependence of the energy of benzene on the distortion angle, ΔθCXC (ΔθCXC = 0° corresponds to the D6h structure). We also found a distorted geometry with the optimum ΔθCXC being 0.31° with an energy 0.040 kcal mol-1 lower than that for the D6h structure. In the optimum geometry, adjacent CC bond lengths are 1.3861 Å and 1.4004 Å. Analysis of the SCGVB wave function led us to conclude that the cause of the unusual non-D6h geometry predicted by the SCGVB calculations seems to be a result of the interaction between the Kekulé and Dewar components of the full SCGVB wave function. The addition of doubly ionic configurations to the SCGVB wave function leads to the prediction of a D6h geometry for benzene and a dependence on ΔθCXC essentially the same as that predicted by the complete active space self-consistent field wave function.

Orbital Hybridization in Modern Valence Bond Wave Functions: Methane, Ethylene, and Acetylene.

The concept of hybrid orbitals is one of the key theoretical concepts used by chemists to explain the structures and other properties of molecules. Recent work found that the hybrid orbitals from modern ab initio valence bond wave functions differ significantly from traditional hybrid orbitals. We report a detailed analysis of the orbitals of methane, ethylene, and acetylene from spin-coupled generalized valence bond (SCGVB) wave functions, a variationally optimized valence bond wave function that places no constraints on the orbitals and spin function. The carbon-centered orbitals in the SCGVB wave functions are found to be 2s-2p hybrid orbitals largely localized on the carbon atom and pointed directly at the hydrogen atoms to which they are bonded. However, the SCGVB orbitals for methane, ethylene, and acetylene differ markedly from the sp3, sp2, and sp hybrid orbitals traditionally associated with these molecules. It is now clear that the orbitals in modern valence bond wave functions do not follow the hybridization rules of traditional valence bond theory. These findings imply that, in modern valence bond theories, other factors are responsible for the structures and properties of molecules that are traditionally attributed to orbital hybridization.

Spin-Coupled Generalized Valence Bond Description of Group 14 Species: The Carbon, Silicon and Germanium Hydrides, XH n ( n = 1-4).

Although elements in the same group in the Periodic Table tend to behave similarly, the differences in the simplest Group 14 hydrides-XH n (X = C, Si, Ge; n = 1-4)-are as pronounced as their similarities. Spin-coupled generalized valence bond (SCGVB) as well as coupled cluster [CCSD(T)] calculations are reported for all of the molecules in the CH n/SiH n/GeH n series to gain insights into the factors underlying these differences. It is found that the relative weakness of the recoupled pair bonds of SiH and GeH gives rise to the observed differences in the ground state multiplicities, molecular structures, and bond energies of SiH n and GeH n. A number of factors that influence the strength of the recoupled pair bonds in CH, SiH, and GeH were examined. Two factors were identified as potential contributors to the decrease in the strengths of these bonds from CH to SiH and GeH: (i) a decrease in the overlap between the orbitals involved in the bond and (ii) an increase in Pauli repulsion between the electrons in the two lobe orbitals centered on the X atoms. Finally, an analysis of the hybridization of the SCGVB orbitals in XH4 indicates that they are closer to sp hybrids than sp3 hybrids, which implies that the underlying cause of the tetrahedral structure of the XH4 molecules is not a direct result of the hybridization of the X atom orbitals.

Fundamental Aspects of Recoupled Pair Bonds. III. The Frustrated Recoupled Pair Bond in Oxygen Monofluoride.

In a previous paper in this series, we discussed the formation of recoupled pair bonds in the a4Σ- states of CF and SF in which the recoupling process was essentially complete at the equilibrium geometry of the molecule. In this paper, we examine the a4Σ- state of oxygen monofluoride (OF), which could also have a recoupled pair bond. Unlike the other two molecules, generalized valence bond calculations predict that the recoupling in OF is woefully incomplete at Re and the resulting potential energy curve for the OF(a4Σ-) state is purely repulsive; the binding energy, ≈11 kcal/mol, is entirely due to dynamical correlation. A number of factors account for these differences, but the nature of the dominant correlation effect in the oxygen 2p lone pair as well as the spatial extent of the 2p orbital are paramount.

Generalized Valence Bond Description of Chalcogen-Nitrogen Compounds. III. Why the NO-OH and NS-OH Bonds Are So Different.

Crabtree et al. recently reported the microwave spectrum of nitrosyl-O-hydroxide (trans-NOOH), an isomer of nitrous acid, and found that this molecule has the longest O-O bond ever observed: 1.9149 Å ± 0.0005 Å. This is in marked contrast to the structure of the valence isoelectronic trans-NSOH molecule, which has a normal NS-OH bond length and strength. Generalized valence bond calculations show that the long bond in trans-NOOH is the result of a weak through-pair interaction that singlet couples the spins of the electrons in singly occupied orbitals on the hydroxyl radical and nitrogen atom, an interaction that is enhanced by the intervening lone pair of the oxygen atom in NO. The NS-OH bond is the result of the formation of a stable recoupled pair bond dyad, which accounts for both its length and strength.

Variations in the Nature of Triple Bonds: The N2, HCN, and HC2H Series.

The inertness of molecular nitrogen and the reactivity of acetylene suggest there are significant variations in the nature of triple bonds. To understand these differences, we performed generalized valence bond as well as more accurate electronic structure calculations on three molecules with putative triple bonds: N2, HCN, and HC2H. The calculations predict that the triple bond in HC2H is quite different from the triple bond in N2, with HCN being an intermediate case but closer to N2 than HC2H. The triple bond in N2 is a traditional triple bond with the spins of the electrons in the bonding orbital pairs predominantly singlet coupled in the GVB wave function (92%). In HC2H, however, there is a substantial amount of residual CH(a(4)Σ(-)) fragment coupling in the triple bond at its equilibrium geometry with the contribution of the perfect pairing spin function dropping to 82% (77% in a full valence GVB calculation). This difference in the nature of the triple bond in N2 and HC2H may well be responsible for the differences in the reactivities of N2 and HC2H.

Insights into the Electronic Structure of Ozone and Sulfur Dioxide from Generalized Valence Bond Theory: Addition of Hydrogen Atoms.

Ozone (O3) and sulfur dioxide (SO2) are valence isoelectronic species, yet their properties and reactivities differ dramatically. In particular, O3 is highly reactive, whereas SO2 is chemically relatively stable. In this paper, we investigate serial addition of hydrogen atoms to both the terminal atoms of O3 and SO2 and to the central atom of these species. It is well-known that the terminal atoms of O3 are much more amenable to bond formation than those of SO2. We show that the differences in the electronic structure of the π systems in the parent triatomic species account for the differences in the addition of hydrogen atoms to the terminal atoms of O3 and SO2. Further, we find that the π system in SO2, which is a recoupled pair bond dyad, facilitates the addition of hydrogen atoms to the sulfur atom, resulting in stable HSO2 and H2SO2 species.

Insights into the Electronic Structure of Molecules from Generalized Valence Bond Theory.

In this article we describe the unique insights into the electronic structure of molecules provided by generalized valence bond (GVB) theory. We consider selected prototypical hydrocarbons as well as a number of hypervalent molecules and a set of first- and second-row valence isoelectronic species. The GVB wave function is obtained by variationally optimizing the orbitals and spin coupling in the valence bond wave function. The GVB wave function is a generalization of the Hartree-Fock (HF) wave function, lifting the double occupancy restriction on a subset of the HF orbitals as well as the associated orthogonality and spin coupling constraints. The GVB wave function includes a major fraction (if not all) of the nondynamical correlation energy of a molecule. Because of this, GVB theory properly describes bond formation and can answer one of the most compelling questions in chemistry: How are atoms changed by molecular formation? We show that GVB theory provides a unified description of the nature of the bonding in all of the above molecular species as well as contributing new insights into the well-known, but poorly understood, first-row anomaly.

Generalized valence bond description of the ground states (X(1)Σg(+)) of homonuclear pnictogen diatomic molecules: N2, P2, and As2.

The ground state, X1Σg+, of N2 is a textbook example of a molecule with a triple bond consisting of one σ and two π bonds. This assignment, which is usually rationalized using molecular orbital (MO) theory, implicitly assumes that the spins of the three pairs of electrons involved in the bonds are singlet-coupled (perfect pairing). However, for a six-electron singlet state, there are five distinct ways to couple the electron spins. The generalized valence bond (GVB) wave function lifts this restriction, including all of the five spin functions for the six electrons involved in the bond. For N2, we find that the perfect pairing spin function is indeed dominant at Re but that it becomes progressively less so from N2 to P2 and As2. Although the perfect pairing spin function is still the most important spin function in P2, the importance of a quasi-atomic spin function, which singlet couples the spins of the electrons in the σ orbitals while high spin coupling those of the electrons in the π orbitals on each center, has significantly increased relative to N2 and, in As2, the perfect pairing and quasi-atomic spin couplings are on essentially the same footing. This change in the spin coupling of the electrons in the bonding orbitals down the periodic table may contribute to the rather dramatic decrease in the strengths of the Pn2 bonds from N2 to As2 as well as in the increase in their chemical reactivity and should be taken into account in more detailed analyses of the bond energies in these species. We also compare the spin coupling in N2 with that in C2, where the quasi-atomic spin coupling dominants around Re.