VB Structures Research Articles

On the nature of the chemical bond in valence bond theory.

This Perspective outlines a panoramic description of the nature of the chemical bond according to valence bond theory. It describes single bonds and demonstrates the existence of a "forgotten family" of charge-shift bonds (CSBs) in which the entire/most of the bond energy arises from the resonance between the covalent and ionic structures of the bond. Many of the CSBs are homonuclear bonds. Hypervalent molecules (e.g., XeF2) are CSBs. This Perspective proceeds to describe multiple bonded molecules with an emphasis on C2 and 3O2. C2 has four electron pairs in its valence shell and, hence, 14 covalent structures and 1750 ionic structures. This Perspective outlines an effective methodology of peeling the electronic structure to the minimal and important number of structures: a dominant structure that displays a quadruple bond and two minor structures with π + σ bonds, which stabilize the quadruple bond by resonance. 3O2 is chosen because it is a diradical, which is persistent and life-sustaining. It is shown that the persistence of this diradical is due to the charge-shift bonding of the π-3-electron bonds. This section ends with a discussion of the roles of π vs σ in the geometric preferences of benzene, acetylene, ethene, and their Si-based analogs. Subsequently, this Perspective discusses bonding in clusters of univalent metal atoms, which possess only parallel spins (n+1Mn), and are nevertheless bonded due to the resonance interactions that stabilize the repulsive elementary structure (all spins are up). The bond energy reaches ∼40 kcal/mol for a pair of atoms (in n+1Cun; n ∼ 10-12). The final subsection discusses singlet excited states in ethene, ozone, and SO2. It demonstrates the capability of the breathing-orbital VB method to yield an accurate description of a variety of excited states using merely 10 or few VB structures. Furthermore, the method underscores covalent structures that play a key role in the correct description and bonding of these excited states.

A valence bond perspective of the reaction force formalism

The reaction force formalism represents a convenient approach to analyze the course of a reaction step. From this analysis, the reaction path can be separated in a number of regions that are associated to either structural changes or electronic reorganization. This empirical observation is rationalized in this work on the basis of a simple two-state valence bond correlation diagram. We demonstrate that the ratio between the integrated reaction force and the region of interest ($$w_{\text{ii}}/w_{\text{i}}$$ for the forward reaction and $$w_{\text{iii}}/w_{\text{iv}}$$ for the backward reaction) increases with the ratio between the quantum mechanical resonance energy and the energy required to reach the crossing point at the transition state, we call to this ratio the strength of the resonance. This observation means that the size of the transition region (region ii and iii), that includes the transition state, depends on the strength of the resonance, and the structural zones (region i and iv), that are regions associated with the pure valence bond state curves (no resonance). We propose a simple analytical relationship for $$w_{\text{ii}}/w_{\text{i}}$$ and $$w_{\text{iii}}/w_{\text{iv}}$$ based on three parameters: (i) the quantum mechanical resonance energy, (ii) the energy of the reaction and (iii) the overlap between the VB structures at the transition state. The previous conclusions were supported by a reaction force analysis of a $${\text{S}}_{N}2$$ reactions, $${\text{X}}^{-} + {\text{CH}}_{3}{-}{\text{Y}} \rightarrow {{\text{X}}{-}{\text{CH}}}_{3} + {\text{Y}}^{-} ({\text{X}} = {\text{F}}, {\text{Cl}}, {\text{Br}})$$. The valence bond parameters for these reactions are estimated from empirical considerations. A very good agreement is found between the computed reaction force ratios and the predicted one.

Does the Electronic Structure of Möbius Annulenes Follow Heilbronner's Ideas?

It is shown, on the example of the monocyclic cyclononatetraenyl cation, , that the fully-variational optimization of modern ab initio wavefunctions based on spin-coupled generalized valence bond (SCGVB) theory vindicates, in a surprising level of detail, essential features of Heilbronner's ideas for the electronic structure of Möbius annulenes such as the arrangement of overlapping carbon 2p atomic orbitals along a Möbius strip, leading to a phase inversion between the first and last orbitals. In the SCGVB description, the aromaticity of this Möbius system with eight π electrons follows from the extensive resonance between VB structures.

Nature of the Three-Electron Bond.

We analyze the properties of 15 3-electron bonds, which include σ-3-electron-bonds, such as dihalide radical anions and di-noble gas radical cations, π-3-electron-bonds as in hydrazine radical cations, and doubly-π-(3e)-bonded species such as O2, FeO+, S2, etc. The primary analytical tool is the breathing-orbital valence-bond (BOVB) method, which enables us to quantify the charge shift resonance energy (RECS) of the three electrons, and the bond dissociation energies (De). BOVB is tested reliable against MRCI calculations. Our findings show that in all 3-electron bonds, none of the VB structures have by themselves any bonding. In fact, in each VB structure, the three electrons maintain Pauli repulsion, while the entire bonding energy arises from resonance due to the charge shift between the two or more constituent VB structures. Hence, 3e-bonds are charge shift bonds (CSBs). The CSB character is probed by calculating the Laplacian (L) of the 3e-bond. Thus, much like the CSBs in electron-pair bonds, such as F2 or the central bond in [1.1.1]propellane, here too L is positive, thus showing the excess kinetic energy of the shared density due to the Pauli repulsion in the 3-electron VB structures. The RECS values for 3-electron bonds are invariably larger than the corresponding bond energies. For the doubly-π-(3e)-bonded species, RECS is very large, exceeding 100 kcal mol-1. As such, it is fitting to conclude that σ- and π-3-electron-bonds find their natural place in the CSB family along with two-electron CSBs, with which they share identical energetic and topological characteristics. Experimental manifestations/tests of 3e-CSBs are proposed.

Ozone and Other 1,3-Dipoles: Toward a Quantitative Measure of Diradical Character.

Ozone and its sulfur-substituted isomers are studied by means of the Breathing Orbital Valence Bond ab initio method, with the objective of estimating their controversial diradical characters. The calculated weights of the various VB structures and their individual diabatic energies are found to be consistent with each other. All 1,3-dipoles can be described in terms of three major VB structures, one diradical and two zwitterionic ones, out of the six structures, forming a complete basis. Ozone has a rather large diradical character, estimated to 44%-49%. SOO and SOS are even more diradicalar, whereas SSO and especially OSO are better described as closed-shell zwitterions. Moreover, the description of 1,3-dipoles, in terms of the three major structures, yields VB weights in full agreement with simple chemical wisdom, i.e., a diradical weight of 33% when the three structures are quasi-degenerate, and a smaller (larger) value when the diradical structure is higher (lower) in energy than the zwitterionic ones. Therefore, the VB-calculated weight of the diradical structure of a molecule qualifies itself as a quantitative measure of diradical character, and not only as an indicator of tendencies. Other definitions of the diradical character, based on molecular orbital/configuration interaction methods, are discussed.

A valence bond model for electron-rich hypervalent species: application to SFn (n=1, 2, 4), PF5 , and ClF3.

Some typical hypervalent molecules, SF4 , PF5 , and ClF3 , as well as precursors SF ((4) Σ(-) state) and SF2 ((3) B1 state), are studied by means of the breathing-orbital valence bond (BOVB) method, chosen for its capability of combining compactness with accuracy of energetics. A unique feature of this study is that for the first time, the method used to gain insight into the bonding modes is the same as that used to calculate the bonding energies, so as to guarantee that the qualitative picture obtained captures the essential physics of the bonding system. The (4) Σ(-) state of SF is shown to be bonded by a three-electron σ bond assisted by strong π back-donation of dynamic nature. The linear (3) B1 state of SF2 , as well as the ground states of SF4 , PF5 and ClF3 , are described in terms of four VB structures that all have significant weights in the range 0.17-0.31, with exceptionally large resonance energies arising from their mixing. It is concluded that the bonding mode of these hypervalent species and isoelectronic ones complies with Coulson's version of the Rundle-Pimentel model, but assisted by charge-shift bonding. The conditions for hypervalence to occur are stated.

The V state of ethylene: valence bond theory takes up the challenge

The ground state and first singlet excited state of ethylene, so-called N and V states, respectively, are studied by means of modern valence bond methods. It is found that extremely compact wave functions, made of three VB structures for the N state and four structures for the V state, provide an N → V transition energy of 8.01 eV, in good agreement with experiment (7.88 eV for the N → V transition energy estimated from experiments). Further improvement to 7.96/7.93 eV is achieved at the variational and diffusion Monte Carlo (MC) levels, respectively, VMC/DMC, using a Jastrow factor coupled with the same compact VB wave function. Furthermore, the measure of the spatial extension of the V state wave function, 19.14 a 0 2 , is in the range of accepted values obtained by large-scale state-of-the-art molecular orbital-based methods. The σ response to the fluctuations of the π electrons in the V state, known to be a crucial feature of the V state, is taken into account using the breathing orbital valence bond method, which allows the VB structures to have different sets of orbitals. Further valence bond calculations in a larger space of configurations, involving explicit participation of the σ response, with 9 VB structures for the N state and 14 for the V state, confirm the results of the minimal structure set, yielding an N → V transition energy of 7.97 eV and a spatial extension of 19.16 a 0 2 for the V state. Both types of valence bond calculations show that the V state of ethylene is not fully ionic as usually assumed, but involving also a symmetry-adapted combination of VB structures each with asymmetric covalent π bonds. The latter VB structures have cumulated weights of 18–26 % and stabilize the V state by about 0.9 eV. It is further shown that these latter VB structures, rather than the commonly considered zwitterionic ones, are the ones responsible for the spatial extension of the V state, known to be ca. 50 % larger than the V state.

Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. II. An efficient algorithm for matrix elements and analytical energy gradients in VBSCF method

In this paper, by applying the reduced density matrix (RDM) approach for nonorthogonal orbitals developed in the first paper of this series, efficient algorithms for matrix elements between VB structures and energy gradients in valence bond self-consistent field (VBSCF) method were presented. Both algorithms scale only as nm(4) for integral transformation and d(2)n(β)(2) for VB matrix elements and 3-RDM evaluation, while the computational costs of other procedures are negligible, where n, m, d, and n(β )are the numbers of variable occupied active orbitals, basis functions, determinants, and active β electrons, respectively. Using tensor properties of the energy gradients with respect to the orbital coefficients presented in the first paper of this series, a partial orthogonal auxiliary orbital set was introduced to reduce the computational cost of VBSCF calculation in which orbitals are flexibly defined. Test calculations on the Diels-Alder reaction of butadiene and ethylene have shown that the novel algorithm is very efficient for VBSCF calculations.

A quadratically convergent VBSCF method

A quadratically convergent valence bond self-consistent field method is described where the simultaneous optimisation of orbitals and the coefficients of the configurations (VB structures) is based on a Newton-Raphson scheme. The applicability of the method is demonstrated in actual calculations. The convergence and efficiency are compared with the Super-CI method. A necessary condition to achieve convergence in the Newton-Raphson method is that the Hessian is positive definite. When this is not the case, a combination of the Super-CI and Newton-Raphson methods is shown to be an optimal choice instead of shifting the eigenvalues of the Hessian to make it positive definite. In the combined method, the first few iterations are performed with the Super-CI method and then the Newton-Raphson scheme is switched on based on an internal indicator. This approach is found computationally a more economical choice than using either the Newton-Raphson or Super-CI method alone to perform a full optimisation of the nonorthogonal orbitals.

How the Conical Intersection Seam Controls Chemical Selectivity in the Photocycloaddition of Ethylene and Benzene

The photocycloaddition reaction of benzene with ethylene has been studied at the CASSCF level, including the characterization of an extended conical intersection seam. We show that the chemical selectivity is, in part, controlled by this extended conical intersection seam and that the shape of the conical intersection seam can be understood in terms of simple VB arguments. Further, the shape and energetics of the asynchronous segment of the conical intersection seam suggest that 1,2 (ortho) and 1,3 (meta) will be the preferred chemical products with similar weight. The 1,4 (para) point on the conical intersection is higher in energy and corresponds to a local maximum on the seam. VB analysis shows that the pairs of VB structures along this asynchronous seam are the same and thus the shape will be determined mainly by steric effects. Synchronous structures on the seam are higher in energy and belong to a different branch of the seam separated by a saddle point on the seam. On S1 we have documented three mechanistic pathways corresponding to transition states (with low barriers) between the reactants and the conical intersection seam: a mixed asynchronous/synchronous [1,2] ortho path, an asynchronous [1,3] meta path, and a synchronous [1,3] meta path.

An efficient algorithm for complete active space valence bond self-consistent field calculation

This article describes a novel algorithm for the optimization of valence bond self-consistent field (VBSCF) wave function for a complete active space (CAS), so-called VBSCF(CAS). This was achieved by applying the strategies adopted in the optimization of CASSCF wave functions to VBSCF(CAS) wave functions, using an auxiliary orthogonal orbital set that generates the same configuration space as the original nonorthogonal orbital set. Theoretical analyses and test calculations show that the VBSCF(CAS) method shares the same computational scaling as CASSCF. The test calculations show the current capability of VBSCF method, which involves millions of VB structures.

Resonance and Aromaticity: An Ab Initio Valence Bond Approach

Resonance energy is one of the criteria to measure aromaticity. The effect of the use of different orbital models is investigated in the calculated resonance energies of cyclic conjugated hydrocarbons within the framework of the ab initio Valence Bond Self-Consistent Field (VBSCF) method. The VB wave function for each system was constructed using a linear combination of the VB structures (spin functions), which closely resemble the Kekulé valence structures, and two types of orbitals, that is, strictly atomic (local) and delocalized atomic (delocal) p-orbitals, were used to describe the π-system. It is found that the Pauling-Wheland's resonance energy with nonorthogonal structures decreases, while the same with orthogonalized structures and the total mean resonance energy (the sum of the weighted off-diagonal contributions in the Hamiltonian matrix of orthogonalized structures) increase when delocal orbitals are used as compared to local p-orbitals. Analysis of the interactions between the different structures of a system shows that the resonance in the 6π electrons conjugated circuits have the largest contributions to the resonance energy. The VBSCF calculations also show that the extra stability of phenanthrene, a kinked benzenoid, as compared to its linear counterpart, anthracene, is a consequence of the resonance in the π-system rather than the H-H interaction in the bay region as suggested previously. Finally, the empirical parameters for the resonance interactions between different 4n+2 or 4n π electrons conjugated circuits, used in Randić's conjugated circuits theory or Herdon's semi-emprical VB approach, are quantified. These parameters have to be scaled by the structure coefficients (weights) of the contributing structures.

Can Aromaticity Coexist with Diradical Character? An Ab Initio Valence Bond Study of S2N2 and Related 6π‐Electron Four‐Membered Rings E2N2 and E42+ (E=S, Se, Te)

A series of 6π-electron 4-center species, E(2)N(2) and E(4)(2+) (E=S, Se, Te) is studied by means of ab initio valence bond methods with the aims of settling some controversies on 1) the diradical character of these molecules and 2) the radical sites, E or N, of the preferred diradical structure. It was found that for all molecules, the cumulated weights of the two possible diradical structures are always important and close to 50 %, making these molecules comparable to ozone in terms of diradical character. While the two diradical structures are degenerate in the E(4)(2+) dications, they have on the contrary strongly unequal weights in the E(2)N(2) neutral molecules. In these three molecules, the electronic structure is dominated by one diradical structure, in which the radical sites are the two nitrogen atoms, while the other diradical structure is much less important. The ordering of the various VB structures in terms of their calculated weights is confirmed by the relative energies of individual VB structures. In all cases, the major diradical structure (or both diradical structures when they are degenerate) is (are) the lowest one(s), while the covalent VB structures lie higher in energy. The vertical resonance energies are considerable in S(2)N(2) and S(4)(2+), about 80 % of the estimated value for benzene, and diminish as one goes down the periodic table (S→Se→Te). This confirms the aromatic character of these species, as already demonstrated for S(2)N(2) on the basis of magnetic criteria. This and the high weights and stabilities of one or both diradical structures in all systems indicates that aromaticity and diradical character do not exclude each other, contrary to what is usually claimed. Furthermore, it is shown that the diradical structures find their place in a collective electron flow responsible for the ring currents in the π system of these species.

Frequency Upshift in BO2 and CO2+ upon Electronic Excitation: A Twin-State Model Rationalization

The twin-state model, previously shown to provide a simple physical rationalization for the frequency upshift (exaltation) of the Kekulé mode in benzene upon S(0) to S(1) electronic excitation, is extended to the case of the BO(2) radical and the CO(2)(+) radical cation. In the case of BO(2)/CO(2)(+), the ground and excited states are degenerate, yet the model applies to the degenerate two-state system as well. In contrast with a pseudo-Jahn-Teller model, the twin-state one can predict which frequency is exalted and also which pair of electronic states are coupled, thus explaining the specificity of the phenomenon. The frequency exaltation is a spectroscopic manifestation of the resonance between the pair of VB structures describing twin states. In analogy with the case of benzene, it is predicted that the ν(3) asymmetric stretch fundamental will be a dominant peak in the two-photon absorption spectrum of BO(2) and CO(2)(+).

Bonding Conundrums in the C2 Molecule: A Valence Bond Study.

The ab initio VB study for the electronic structure of the C2 molecule in the ground state is presented in this work. VB calculations involving 78 chemically relevant VB structures can predict the bonding energy of C2 quite well. Sequentially, a VBCIS calculation provides spectroscopic parameters that are very close to full CI calculated values in the same basis set. Furthermore, the analysis of the bonding scheme shows that a triply bonded structure is the major one in terms of weights, and the lowest in energy at the equilibrium distance. The second structure in terms of weights is an ethylene-like structure, displaying a σ + π double bond. The structure with two suspended π bonds but no σ bond contributes only marginally to the ground state. This ordering of weights for the VB structures describing the C2 molecule is shown to be consistent with the shape of the molecular orbitals and with the multireference character of the ground state. With the triply bonded bonding scheme, the natures of the π and σ bonds are investigated, and then the corresponding "in situ" bond strengths are estimated. The contribution of the covalent-ionic resonance energy to π and σ bonding is revealed and discussed.

Valence Bond Perturbation Theory. A Valence Bond Method That Incorporates Perturbation Theory

A post-VBSCF method, called valence bond second-order perturbation theory (VBPT2), is developed in this paper and is shown to be (i) economical and (ii) at par with more sophisticated VB and MO-based methods. The VBPT2 method starts with VBSCF using a minimal structure set. Subsequently, the Møller-Plesset (MP) partition of the zeroth-order Hamiltonian is obtained by introducing a generalized Fock matrix constructed from the VBSCF density matrix. The first-order wave function is expressed in terms of singly and doubly excited VB structures, which are generated by replacing occupied orbitals by virtual orbitals, the latter being defined as orthogonal to the occupied orbitals. The VBPT2 method retains the simplicity of a VB presentation by condensing contributions from the excited structures into the minimal number of fundamental structures that are involved in the VBSCF calculation. The method is tested by calculating the bond energies of H(2), F(2), N(2), O(2), the barrier of identity hydrogen abstraction reaction, the atomization energy and a potential energy curve for the water molecule and the structural weights and covalent-ionic resonance energy of F(2). It is shown that the VBPT2 method gives results in good agreement with those of the VBCI method and molecular-orbital based methods such as MRPT and MRCI at the same truncation levels. However, the computational effort is greatly reduced, compared to that of VBCI. Future potential directions for the development of the VBPT2 method are outlined.

Ab initio valence bond calculations. IX. Ionization potentials of ethylene, allyl radicall, trans-, and cis-butadiene

π-electron ab initio valence bond calculations on the ground and ionized states of ethylene, allyl radical, trans-, and cis-butadiene have been performed using the 6-31G basis set after contraction. Vertical ionization potentials have been computed both using the π core of the molecule and that of the cation: the VB results and those obtained by means of the Koopmans' theorem and the ŠF method have been compared with the experimental data. The wave functions have been interpreted in terms of VB structures by means of a population analysis.

Valence-bond description of chemical reactions on Born–Oppenheimer molecular dynamics trajectories

The nature of chemical bonds on dynamic paths was investigated using the complete active space valence-bond (CASVB) method and the Born-Oppenheimer dynamics. To extract the chemical bond picture during reactions, a scheme to collect contributions from several VB (resonance) structures into a small numbers of indices was introduced. In this scheme, a tree diagram for the VB structures is constructed with the numbers of the ionic bonds treated as generation. A pair of VB structures is related to each other if one VB structure is transferred into the other by changing a covalent bond to an ionic bond. The former and latter VB structures are named parent and child structures, respectively. The weights of the bond pictures are computed as the sum of the CASVB occupation numbers running from the top generation to the bottom along the descent of the VB structures. Thus, a number of CASVB occupation numbers are collected into a small number of indices, and a clear bond picture may be obtained from the CASVB wave function. The scheme was applied to the hydrogen exchange reaction H(2)+F-->H+HF and the Diels-Alder reaction C(5)H(6)(cyclopentadiene)+CH(2)=CH(2)(ethylene)-->C(7)H(10)(norbornene). In both the reactions, the scheme gave a clear picture for the Born-Oppenheimer dynamics trajectories. The reconstruction of the bonds during reactions was well described by following the temporal changes in weight.

Heterolytic Bond Dissociation in Water: Why Is It So Easy for C4H9Cl But Not for C3H9SiCl?

The recently developed (Song, L.; Wu, W.; Zhang, Q.; Shaik, S. J. Phys. Chem. A 2004, 108, 6017-6024) valence bond method coupled to a polarized continuum model (VBPCM) is used to address the long standing conundrum of the heterolytic dissociation of the C-Cl and Si-Cl bonds, respectively, in tertiary-butyl chloride and trimethylsilyl chloride in condensed phases. The method is used here to compare the bond dissociation in the gas phase and in aqueous solution. In addition to the ground state reaction profile, VB theory also provides the energies of the purely covalent and purely ionic VB structures as a function of the reaction coordinate. Accordingly, the C-Cl and Si-Cl bonds are shown to be of different natures. In the gas phase, the resonance energy arising from covalent-ionic mixing at equilibrium geometry amounts to 42 kcal/mol for tertiary-butyl chloride, whereas the same quantity for trimethylsilyl chloride is significantly higher at 62 kcal/mol. With such a high value, the root cause of the Si-Cl bonding is the covalent-ionic resonance energy, and this bond belongs to the category of charge-shift bonds (Shaik, S.; Danovich, D.; Silvi, B.; Lauvergnat, D.; Hiberty, P. C. Chem.- Eur. J. 2005, 11, 6358). This difference between the C-Cl and Si-Cl bonds carries over to the solvated phase and impacts the heterolytic cleavages of the two bonds. For both molecules, solvation lowers the ionic curve below the covalent one, and hence the bond dissociation in the solvent generates the two ions, Me3E+ Cl- (E = C, Si). In both cases, the root cause of the barrier is the loss of the covalent-ionic resonance energy. In the heterolysis reaction of Si-Cl, the covalent-ionic resonance energy remains large and fully contributes to the dissociation energy, thereby leading to a high barrier for heterolytic cleavage, and thus prohibiting the generation of ions. By contrast, the covalent-ionic resonance energy is smaller for the C-Cl bond and only partially contributes to the barrier for heterolysis, which is consequently small, leading readily to ions that are commonly observed in the classical SN1 mechanism. Thus, the reluctance of R3Si-X molecules to undergo heterolysis in condensed phases and more generally the rarity of free silicenium ions under these conditions are experimental manifestations of the charge-shift character of the Si-Cl bond.

A new formulation of the phase change approach in the theory of conical intersections

We have reformulated the phase change approach to the theory of conical intersections suggested by Zilberg and Haas [S. Zilberg, Y. Haas, Adv. Chem. Phys. 124 (2002) 433]. We use the fact that the phase change loop variable ϕ in polar co-ordinates in the space of nuclear co-ordinates is related to the transformation θ = ϕ 2 of the diabatic VB states defined at the apex of the cone in a conical intersection. Thus one can use the basis transformation generated by θ to yield a sequence of orthogonal pairs of VB structures (anchors) at any value of ϕ in the loop in nuclear configuration space. In this way the branching plane concept becomes related to VB wavefunctions based on the correspondence between the two angles. The theory is developed in detail using a simple three orbital three electron example. Examples are presented for four and six orbitals. In the later case, our results differ from those of Zilberg and Haas.