On the nature of the chemical bond in valence bond theory

Classical VB Theory (CVBT)

HL

Heitler and London who were the first to calculate the H₂ molecule using VB theory (Ref. 34) with a single VB structure that corresponds to the covalent structure

HAO

a localized hybrid atomic orbital as used in classical VB theory

VBSCF

valence-bond self-consistent field: the classical VB method that uses HAOs for the active VB space; all orbitals are optimized self-consistently under the mean field of the VB structures

BOVB

the breathing orbital VB method, which allows all the VB structures to be different than one another and, hence, to include dynamic correlation effects particular to the structure

L-BOVB

BOVB that uses only HAOs

D-BOVB

BOVB that uses HAOs for the VB treated part and delocalized orbitals for the inactive part

SL-BOVB

BOVB that uses HAOs for the singly occupied atomic orbitals while splitting the doubly occupied HAOs (e.g., in the ionic structures)

SD-BOVB

BOVB that combines features of SL-BOVB and D-BOVB

L-VBSCF, D-VBSCF, S-VBSCF, SL-VBSCF, SD-VBSCF

VBSCF methods with the same treatments of the singly and doubly occupied orbitals as described above for BOVB

VBCISD

a VBSCF calculation followed by configuration interaction using virtual HAOs that are localized on the same atoms as in the VBSCF calculations

VBPT2

a VBSCF calculation followed by a second order perturbation theory

VBSCD

a valence-bond state correlation diagram

CSB

charge-shift bonding, which arises mostly (solely) from the resonance between the respective VB structures, e.g., the covalent–ionic resonance in two-electron bonds

TPB

triplet-pair-bond in monovalent structures

Generalized VB Theory

SCGVB

a VB theory that uses local HAOs that possess small delocalization tails (Coulson–Fischer orbitals) on other atoms

SCGVB(PP)

in molecules like methane (CH₄), one can use a single structure to describe the wave function; this wave function is written as a product of bonds wave functions, which are based on Coulson–Fischer atomic orbitals (having delocalization tails); this wave function is labeled SCGVB(PP), where PP is the abbreviation for a perfect pairing

SCGVB for Many Electron Systems

in molecular species, which contain a few main VB structures (e.g., benzene), the SCGVB wave function is a linear combination of SCGVB product wave functions, each corresponding to a particular VB structure

Molecular Orbital Related Terminology

FCI

refers to a full CI calculation

MCSCF

refers to a multi-configuration self-consistent field calculation

MRCI

refers to a multi-reference CI calculation

NBO Charges

charges derived from Weinhold’s natural bond orbital method

TCSCF-PT2

a two-configuration SCF calculations augmented by second order perturbation theory

The chemical bond is the building block from which chemists construct their molecular universe by breaking bonds and making new ones. There are two main alternative descriptions of bonding: valence-bond (VB) theory and molecular orbital (MO) theory. As has repeatedly been demonstrated and stated,^1–8 the VB and MO descriptions of molecules ultimately converge and constitute therefore two alternative representations⁸ that form a “chemical duality principle”⁹ of the same reality. This Perspective focuses on the description of the bond by VB theory.

Having said that, let us recall that there are additional approaches, which articulate the essence of the bond based on electron-density criteria.^10–18 Density functional theory (DFT), in its most commonly used formulation in chemistry, describes bonding through delocalized Kohn–Sham orbitals, much like in MO theory.¹¹ However, in other electron-density-based approaches, the starting point is the density itself.^12–15 Some of these approaches reconstruct orbitals, e.g., localized bonds,^16,17 while others use the density to reveal a description akin to VB theory with covalent, ionic, and resonance-energy contributions.¹⁸ 

Are bonds “real”? Coulson tried to answer this question in his famous “nightmare speech” that is cited in his obituary.¹⁹ He stated that the concepts that are associated with the bond do not correspond to anything that is measurable. Nevertheless, he continued poetically:

“These concepts make a chemical bond seem so real, so life-like, that I can almost see it. Then I wake with a shock to the realization that a chemical bond does not exist; it is a figment of the imagination that we have invented, and no more real than the square root of −1. I will not say that the known is explained in terms of the unknown, for that is to misconstrue the sense of intellectual adventure. There is no explanation: there is form: there is structure: there is symmetry: there is growth: and there is therefore change and life.” In short, the existence of bond properties is basic to all chemistry.

Against Coulson’s nightmare, there exist recent techniques for bond imaging, e.g., by atomic force microscopy.^20–22 Whether such imaging results in real objects or it just mimics what is already imprinted in the chemical mind is an interesting question. However, even then, there is no doubt that the “bond” is a guiding concept that helps chemists to find their way in the infinitely vast molecular universe.

As such, this Perspective describes a variety of bonding issues, within the VB framework and based on VB concepts. The various topics are outlined at the beginning of Sec. III.

Following the formulation of the purely covalent wavefunction,³⁴ Pauling and Slater developed the classical VB approach in the early 1930s,^52–56 in which all bonds are described by their covalent and ionic structures [Eqs. (1a)–(1d) and Scheme 1(a)]. Since then, the classical VB method has been generalized and refined by van Lenthe and Balint Kurti to the valence-bond self-consistent field (VBSCF) method.^27,28 VBSCF optimizes simultaneously the hybrid atomic orbitals (HAOs) and structural coefficients Scheme [1(a) and Eq. (1d)] within a given basis set; these HAOs are common for all the VB structures. Thus, VBSCF involves static electron-correlation between the electrons of the bond. Subsequently, the method was included along with others into the software TURTLE,²⁹ which is implemented in the GAMESS-UK package.⁵⁷ 

On the other side of the world, Wu and his co-workers developed the XMVB package.⁵⁸ XMVB involves a variety of classical computational VB methods, among which are VBSCF and the breathing-orbital VB (BOVB) method.⁵⁹ In BOVB, the HAOs in the three structures in Scheme 1(a) are allowed to assume individual shapes and sizes that depend on the local charges and electron densities of the A and B fragments in the respective VB structure. BOVB also involves, therefore, dynamic correlation, which is embedded in the VB structures, and accompanies the bond forming and breaking events.

BOVB may be carried out at different levels of sophistication, which further take care of static and dynamic correlation effects.⁶⁰ The basic level is called L-BOVB, where L refers to the practice of using localized orbitals for the inactive doubly occupied orbital shell. D-BOVB is a higher level, which delocalizes the inactive orbitals, thus lowering some of the Pauli repulsion between the bonded atoms. Finally, SD-BOVB is the highest level, in which the doubly occupied orbitals are split into two lobes that are singly occupied and the electrons are singlet paired. The D- and SD-options can also improve the VBSCF method (hence, D-VBSCF and SD-VBSCF). Scheme 2 shows the splitting of the doubly occupied orbitals of the ionic structures.

Incorporation of dynamic correlation can also be achieved by augmenting VBSCF with configuration interaction (CI), including single- and double-excitations from the occupied HAOs into localized virtual HAOs, hence VBCISD.⁶¹ VBCISD is more demanding than VBSCF and BOVB. All three methods involve the same number of VB structures for the bond, as in Scheme 1(a). In addition, the software includes the option to use SCGVB orbitals. XMVB is currently a widely used package, which handles classical VB calculations for molecular species having 20 electrons in 20 HAOs or more in the VB shell. The package is described in two recent perspectives by Chen et al.^30,31

The GVB and SCGVB methods are based on the Coulson–Fischer ansatz, which arose from the early attempts of Coulson³ to bridge MO/MO-CI and the HL-VB. As shown in Scheme 1(b) and Eq. (2), this is a very compact bond-wavefunction, which involves a single VB structure. This structure is formally covalent while incorporating the contribution of the ionic structures via the orbital tails. Even though Coulson referred to these orbitals as “MOs,” this compact electron-pair function became the building block of the GVB and SCVB methods, which were subsequently developed independently by Goddard et al.^35–40 and Gerratt et al.^41–45 for many-electron systems. The methods were then unified under a single program, SCGVB,^47–49 which is incorporated in various packages (MOLPRO, TURTLE, GAUSSIAN) and most recently in DALTON.⁵¹ 

The wavefunctions for molecules with multiple bonds, e.g., benzene, acetylene, and C₂, involve several VB structures.

The number of the covalent structures depends on the number of electrons and the number of non-redundant ways of coupling these electrons to have a given spin state. This number can be found in the branching diagram.^1,62 For example, the 6$π$-electrons of benzene can be coupled to a singlet state in five non-redundant ways. The most well-known singlet spin-coupled forms are the Rumer structures⁶³ in Scheme 3, which involve two Kekulé and three Dewar structures, with the two Kekulé structures being dominant.^6,44,45 The mode of coupling depends on the way we number the electrons (and C atoms), and Scheme 3 shows a chemically meaningful numbering system for $π$-benzene.

Another example is the triple bond of acetylene with its six electrons, which can be coupled also in five linearly independent ways. However, now the numbering of the HAOs is not intuitively obvious. Scheme 4 shows the application of the Rumer rules, using a numbering system of the AOs, which ends up with one triply bonded structure, three singly bonded structures (two with a $π$ bond and one with a $σ$ bond), and one non-bonded. It is apparent that the triply bonded structure in Scheme 4 is the lowest in energy and, hence, it will dominate the total wave function of acetylene. However, the wave function will involve, as well, finite weights of the singly bonded and non-bonded structures. This is true for both SCGVB^46,49 and classical VB.^6,30–32

Since the VB structures overlap with one another, their weights can be determined in various manners:⁶⁴ (i) the Coulson–Chirgwin⁶⁵ method defines weights that, by analogy to the Mulliken population analysis, divide the overlaps between the VB structures equally between them; (ii) Löwdin’s⁶⁶ method, which orthogonalizes the weights; (iii) Gallup–Norbeck’s inverse weights method, which orthogonalizes the weights using an inverse overlap matrix,⁶⁷ and the method of renormalized weights, which is based on the normalized sum of the squared coefficients.⁵⁸ 

In addition to the Rumer spin-coupling, one can use Kotani⁶⁸ and Serber^69,70 spin wave functions,⁴³ which are orthogonalized, but have different starting points (the Serber method couples high spin configurations, e.g., a triplet and a doublet to a doublet state). While the three types of spin functions, and the corresponding population analyses, will give rise to different weights for the various structures, the total wave functions are mutually transformable⁴³ and the total energies are identical. Rather than quibbling which set of weights is better than the others, one should take the weights with a grain of salt as qualitative indicators of the nature of the multiple bonds.

Each VB method has typical advantages. SCGVB has the advantage of being compact and involving only “covalent-like structures” in which the “ionic structures” are implicitly embedded via the delocalization tails of the HAOs on adjacent atoms. In contrast, the entire structure-set in classical VB involves explicitly ionic structure.⁷¹ Thus, for example, in benzene with six $π$-electrons, there are 175 covalent and ionic structures. C₂ has 14 covalent and 1750 ionic structures, and the calculations are more extensive.

On balance, the compactness in SCGVB is an advantage, but this often comes at the expense of an average quantitative accuracy, which requires further CI corrections using the CASSCF wave function like in the CASVB approach. On the other hand, classical VB is structure-extensive, but the usage of explicit ionic and covalent structures endows the method with flexibility and a quantitative advantage. In some cases, which are discussed later, the explicit covalent–ionic treatment brings about fundamental insight and novel features. For example, when the entire structure-set (1764 structures) for C₂ is used, the VBSCF/6-31G* calculation gives virtually the same results as full-CI/6-31G*.^72,73 The full VBSCF structure-set (with a number given by the Weyl formula⁷⁴) is automatically generated in the XMVB package.⁵⁸ Furthermore, as will be shown later for C₂, one can systematically truncate the ionic structures by limiting their number to mono- and di-ionic structures. Importantly, the application sections also show conceptual insights, wherein the use of explicit covalent and ionic structures also leads to the discovery of forgotten bond families, like the charge-shift bond (CSB) in F₂,⁷⁵ and the discovery of triplet-pair bonds (TPB) (Sec. VI G), and to compact bonding schemes for excited states.^76,77

In discussing the nature of the chemical bond, we shall address the following subtopics:

• Single bonds of main elements and transition metals (TM), with a focus on the “missed” charge-shift bond (CSB) family.^75,78–81

• Why and when do atoms exhibit hypervalency?^82–84 

• New features of the hybridization concept in modern VB theory.^48,85,86

• Descriptions of multiple bonds with a focus on unusual bonding patterns in C₂ and O₂.^{50,72,87–90}

• The geometric and stability outcomes of $π$–$σ$ interplay in multiply bonded molecules, such as benzene,^91–95 etc., and in higher row analogs of doubly and triply bonded molecules.^96,97

• VB modeling of the unusual form of triplet-pair bonds (TPB) in ferromagnetic no-pair clusters.^98,99

• VB descriptions of bonding in excited states.^76,77

The electron-pair bond was formulated in a stroke of genius by Lewis (Fig. 1) in his key 1916 paper.^100,101 Lewis began by noting that most stable chemical compounds possess an even number of electrons, and hence, he envisioned that the chemical bond arises by pairing electrons. Although he speculated in passing about the driving force for the counterintuitive pairing, he did not commit himself to a mechanism whereby this seemingly “electrostatically forbidden” electron-pairing occurs.¹⁰¹ For him the pairing emerged from an observation that most of the stable molecules contain an even number of electrons.

Lewis also needed to bridge between the branch of inorganic chemistry, which involved very polar compounds and featured charged (ionic) species, and discipline of organic chemistry that dealt with generally nonpolar compounds, and where the “structure” was important. As such, on p. 782 of the paper, Lewis explained his motivation: “I believe enough has been said to show how, through simple hypotheses, we may explain the most diverse types of chemical union and how we may construct models which illustrate the continuous transition between the most polar and the most nonpolar of substances.”

Following the publication of the HL wave function for a shared electron pair,³⁴ Pauling^52–54 and Slater^55,56 formulated VB theory using the covalent and ionic structures shown in Scheme 1(a) for an A–B bond. Pauling who was a renowned crystallographer packaged the theory to the experimental community in an effective manner. He used empirical data of bond energies and quantified the electronegativity scale of atoms¹⁰² (pp. 73–80, 97–103). He then used electronegativity differences to devise an effective scheme for predicting bond polarity.

Despite the great progress in computational chemistry, this classification has remained an influential conceptual guide. Nevertheless, one might question the basic assumption used by Pauling to derive Eqs. (3) and (4), namely, that the covalent–ionic resonance of homo-nuclear bonds is negligible and can be ignored. In the early 1990s, when two of us^78,79 started using ab initio VB calculations (with orbital optimization), we were initially convinced that Pauling’s scheme is perfectly reasonable. We tested a few electron pair bonds and met our first surprise—the F–F bond. The result of the calculations is shown for H–H vs F–F in Fig. 2.

Just look at the difference between the two bonds. H–H is a true covalent bond; its covalent structure provides most of the bonding, as well as the “structure” of the bond, namely, its bond length. The covalent–ionic resonance energy is a minor contribution to bonding, ∼6%–9% of the total bond energy.

Since the F–F bonding is provided only by the charge fluctuation of the electron pair in the bond, we named this bond a charge-shift bond (CSB)^79,104 and the corresponding resonance energy as RE_CS. Interestingly, similar to VBSCF the SCGVB method underestimates the bond energy for F–F, ∼15 kcal/mol, compared with BOVB that yields 37 kcal/mol (experimental value: 38 kcal/mol). Thus, even though the SCGVB wave function implicitly contains the ionic contribution,⁴⁰ the mean-field approximation does not endow the ionic and covalent structures with the freedom of being different as in BOVB, and hence, much like VBSCF, SCGVB damps the RE_CS quantity to about two-thirds of its magnitude in BOVB.^75,104 Nevertheless, SCGVB followed by CI reproduces the correct bond energy, but at the expense of losing the visual impact of Fig. 2(b).

Along with F₂, we found a number of such homonuclear bonds, e.g., Cl–Cl, Br–Br, O–O, N–N, the central C–C bond in [1.1.1]propellane,¹⁰⁴ bonds between transition metals (TM), where TM = Mn, Tc, Co, Re,¹⁰⁵ and the coinage Au–Au bond;¹⁰⁶ see Scheme 5.

Table I collects a variety of homo-nuclear bonds, which are characterized based on the corresponding weights of their covalent structures (w_cov), covalent-bonding energy (D_cov), and charge-shift resonance energies (RE_CS) along with the corresponding %RE_CS values [%RE_CS = 100 × (RE_CS/D_e)]. It is seen that entries 1–4 involve covalent bonds, e.g., H–H, C–C (in ethane), in which the bond energy (D_e) is dominated by the intrinsic bonding of the covalent structure (D_cov), while the RE_CS and the %RE_CS values are rather low.

All the homo-nuclear bonds that are CSBs according to Table I are depicted in Scheme 5. Thus, entries 5–10 (Table I) display homo-nuclear bonds, for which the covalency is either repulsive, or small (entry 5), and the bond energy is dominated by RE_CS. Similarly, homo-nuclear bonds between transition metals (entries 11, 14–16) are CSBs,¹⁰⁵ while in coinage metal the Au–Au bond (entry 18) is a CSB.¹⁰⁶ Note that all these CSBs are typified by very large RE_CS quantities that reach 72 kcal/mol for the inverted central C–C bond of [1.1.1]propellane (entry 10), and %RE_CS values that range from 52% to 184%. As such, in CSBs, the individual VB structures are not intrinsically important, and what determines the nature of the bond is the RE_CS between the structures.

Table II shows five bonds that are ionic A⁺X⁻, and their properties resemble those of the respective ionic structures. Thus, in all the bonds, the electrostatic energy (D_ion) of the ionic structure by itself dominates the entire bonding energy (D_e) of the molecule. The charge-shift resonance energies are small. Once again, we witness that the classical bond families, covalent and ionic, resemble their main VB structures; the covalent bonds (Table I) resemble the covalent structure, while the ionic bonds resemble the ionic structure (Table II). In contrast, the CSBs (Table I) do not resemble any of the structures but exist due to the resonance energy between the structures.

Table III lists some hetero-nuclear A–X bonds. The C–H, Si–H, B–H, and Cl–H bonds (entries 1–4) are typified by small RE_CS values and %RE_CS is smaller than 50%; these are polar–covalent bonds. However, the rest of the bonds (entries 6–15) are CSBs (e.g., H–F, C–F, Si–Cl, F–Cl, Cl–Br, etc.). For example, H–F possesses a huge RE_CS of 91 kcal/mol that provides most of the bonding to the molecule. Similarly, all the trigger bonds of explosives (e.g., Me₂N–NO₂, I₂N–I) are CSBs.¹⁰⁷ Once again, the dominant features that distinguish polar–covalent bonds from polar CSBs are the RE_CS and %RE_CS quantities. Furthermore, the dative bonds that we looked at [e.g., H₃N–BX₃ (X = H, F), H₃N–Zn⁺, R–NH₃₊, H₃N–Cr(CO)₅, etc.] are generally CSBs.⁷⁵ 

The CSBs in Tables I and III exhibit clear patterns. It is seen from Table I that in a period of main elements, moving from left to right—toward the electronegative and lone-pair rich atoms—one shifts from a covalent bond (e.g., C–C, entry 4) to CSBs (entries 5–7). The increase in the CSB character is observed also in a period for bonds between transition metals, as the transition metal changes from left to right (e.g., Mn–Mn vs Co–Co). However, going further to groups 11 and 12, the CSB diminishes and Zn–Zn bond (entry 19 in Table I) becomes covalent.

Going down a column of the Periodic Table, the CSB character in main elements (entries 7–9 in Table I) and transition metals (entries 11–13; 14–16) is sustained albeit with some weakening of this property. In groups 11 and 12, the CSB character increases on going down the column.

For polar bonds (in Table III), the same elements that possess propensities for CSB formation, e.g., F, Cl, Br, O, and N, generate polar CSBs. Thus, the bond family, which we call CSB, exhibits orderly patterns, which depend on the constituent atoms.^75,105 Further down we explain these trends based on the Pauli repulsion pressure that is applied by lone-pairs or filled semi-core orbitals on the electron pair in the covalent structure.¹⁰⁵ 

In retrospect, pure logic could have led one to expect the three families of bonding, as can be gleaned from Scheme 6 where the trinity of families is symbolized by a triangle. Thus, the logic rests on the three variables indicated on the corners of the triangle: there are two types of VB structures, covalent (⁠$Φ$_cov) and ionic (⁠$Φ$_ion) and the resonance interaction between them. As such, one logically expects to find two families where the bonding derives from the respective VB structures, namely, covalent and ionic bonds, and a third family—CSB—wherein the bond energy derives from the covalent–ionic resonance.

This CSB family was overlooked by Pauling who focused only on the resonance energy of hetero-nuclear bonds while dismissing the resonance energy in homo-nuclear bonds. In fact, the total covalent–ionic resonance energy, RE_CS, does not obey the Pauling formulation [in Eq. (1)]. It depends primarily on the sum of the electronegativities of the bond constituent atoms and not on the difference. This is true for $σ$-bonds as well as for single and double $π$-bonds.^{75,97,108,109}

The answer is yes. Consider the C–C, N–N, O–O, and F–F bonds, which are found in Table I to exhibit a transition from a covalent bond (C–C) to CSBs (N–N, O–O, and F–F). Figure 3 shows the %RE_CS determined by BOVB and by TCSCF-PT2.¹¹⁰ It is seen that there is a reasonable correlation between the BOVB and TCSCF-PT2 values. C–C is a covalent bond with %RE_CS < 50, while N–N, O–O, and F–F are CSBs, with %RE_CS values that range between 70% and 180%. Clearly, a high-level MO method reproduces the nature of the bonds in correspondence with the BOVB method.

CSB is further categorized independently by the recent energy decomposition analysis (EDA) scheme devised by Head-Gordon et al.^111,112 as well as by electron density theories: QTAIM,^113–115 electron density tensor theory,^14,15 and electron localization function (ELF).^104,116,117

Figure 4 exemplifies this categorization using QTAIM and ELF for two bonds, one covalent (C–C) and the other CSB (F–F). QTAIM characterizes interatomic interactions via the Laplacian (L) of the electron density at the critical point along the “bond path.”^{14,15,113,114} A repulsive interaction is characterized by a positive L, e.g., L = +0.25 for He⋯He, while a negative L characterizes a stabilizing covalent interaction, e.g., L = −1.39 for H–H. ELF outlines the bonding regions that are separated in a given molecule by the Pauli exclusion rule,^116,117 and hence ELF characterizes the nature of the electron density in the bonds.

Figure 4(a) shows the Laplacian (L) at the bond critical points for the H₃C–CH₃ and F–F bonds. It is seen that the C–C bond has a negative L as expected from a stabilizing covalent interaction in a covalent bond. On the other hand, F–F, which is a CSB, has a positive L, which is in accord with the highly repulsive covalent structure of this bond (Fig. 2). Figure 4(b) shows the ELF properties of the bonding basins for the same bonds, which already, at first glance, appear to be very different. The C–C bond is a kosher covalent bond, which possesses almost two electrons (1.81) in the bonding basin, and a much smaller variance.¹⁰⁴ By contrast, the F–F bond has a dismal population of 0.44 electrons in the bonding basin, and a $σ$² value that is almost equal to the entire population. Therefore, both QTAIM and ELF properties in Fig. 4 are in accord with the VB distinction of the two bonds as covalent (C–C) and a CSB (F–F). More examples can be found in other publications of the group.^75,104,109

This distinction emerges also from other types of analyses: the electron density stress tensor^14,15 and energy decomposition.^111,112 Furthermore, as we shall see later, the distinction between CSB and covalent bonds rests also on experimental manifestations.⁷⁵ 

Main Elements: As we noted above, many of the CSBs are observed among main elements that are compact and/or lone-pair rich, such as F, O, and N, and in some highly strained bonds like the inverted C–C bond of [1.1.1]propellane. Figure 5(a) shows the covalent structures of F–F, while Fig. 5(b) shows the covalent structure of the inverted C–C bond in [1.1.1]propellane. It is seen that the two $σ$-lone pairs of the F atoms apply 3e-Pauli repulsion on the electrons of the covalent structure. This raises the energy of the covalent structure and makes it repulsive as shown in Fig. 2(b). Similarly, in [1.1.1]propellane, the wing-bond-orbitals apply Pauli repulsion on the covalent structure (⁠$Φ$_cov) of the inverted central C–C bond and destabilize it.

Figure 6 shows the compactness index I_c(rel) of the bond orbitals, relative to the free atoms, plotted for a series of bonds against the corresponding %RE_CS.¹²² The I_c(rel) is a measure of the orbital diffuseness in the bond vis-à-vis the free atoms. It is seen that, for the covalent C–C bond in ethane, the compactness index is 0.497, indicating a considerable orbital shrinkage for this covalent bond (for H–H, it is 0.519). However, as we move to the CSBs, the I_c(rel) index gradually increases up to 0.9 for the F–F bond. The bond-orbital shrinkage in F₂ is minor and serves to lower V.

Transition Metals: For TM–TM (TM—transition metal) bonds, the filled semi-core shell applies the overlap repulsion on the valence nd electrons of the bond.¹⁰⁵ For the 3d series, the semi-core shell is 3s²3p⁶ and the valence orbital is 3d. The maximum electron density of the 3d orbital coincides with the maximum of the 3s/3p semi-core orbitals. Hence, the semi-core electrons apply a high Pauli repulsion on the bond orbitals and destabilize the covalent structure, thus making the TM–TM bonds (TM = Mn, Co) CSBs. As we go down the respective columns, the nd orbitals (n = 4, 5) develop radial nodes and experience relativistic masking of the nucleus by the respective ns² subshell. As such, the Pauli repulsion on the bond orbitals decreases and the charge-shift character decreases.

Figure 7 shows the plot for the respective I_C(rel) vs the %RE_CS values for transition metal bonds in groups 7 and 8.^75,105 It is seen that, for each group of complexes, the TM–TM bond in the 3d metals has a major CSB character, and a compactness index, I_C(rel), which is close to 1 or larger than 1 (namely, the bond-orbitals expand rather than contract). Upon moving from group 7 to group 8 TM–TM bonds, the CSB characters increase because the effective nuclear charge causes the metals to shrink from left to right and thereby increases the Pauli repulsion of the filled semi-core shell on the nd bonding orbitals.

Post Transition Metals, Groups 11 and 12: As soon as the period of transition metal reaches ten electrons in the respective nd orbitals, the next atom populates a new valence shell, (n + 1)s. For example, Cu (group 11), which is the first post-transition-metal element in the respective period, has a 3d¹⁰4s¹ configuration and it forms a Cu–Cu bond using the corresponding 4s orbitals. Since the 3d¹⁰ shell is an inner-shell, its Pauli repulsion pressure on the 4s¹-4s¹ covalent structure of the Cu–Cu bond is smaller than the repulsion exerted on the bonds of the preceding transition metals. Hence, the Cu–Cu bond is covalent though still having a borderline CSB character. Moving on to Zn (group 12), the 3d¹⁰ shell further shrinks, and the (Zn–Zn)²⁺ bond in (η⁵-Cp)Zn–Zn(η⁵-Cp) is perfectly covalent.

However, going down the group of the coinage metals to Au–Au, the 6s orbital shrinks due to relativistic effects, while the 5d¹⁰ shell expands, and, hence, the 6s¹-6s¹ covalent structure of the Au–Au bond suffers Pauli repulsion and becomes a CSB but again on the borderline. Moving to (η⁵-Cp)Hg–Hg(η⁵-Cp), the bond becomes covalent.

While the Pauli repulsion is the major driver of CSB, there are other factors, e.g., in $π$-bonds.^97,108 Ideally, a $π$-bond strives to be as short as possible. However, since it comes in combination with a $σ$-bond that prefers to be longer, the $π$-bond gets stretched. As such, its covalent structure gets destabilized, and its kinetic energy rises. Consequently, many $π$-bonds tend to be CSBs or close to that.

Similarly, in hetero-polar bonds, there are factors that affect the energy gap between the covalent and ionic structures. For example, in H–F, Pauli repulsion raises the energy of the covalent structure, while, in the Si–Cl and Ge–Cl, positive charge concentration on Si and Ge lowers the energy of the ionic structures, compared with their carbon analog C–Cl.^75,123 Figure 8 shows charge distributions in the cations of Me₃C–Cl vs Me₃Si–Cl (Me = CH₃). It is seen that the positive charge is delocalized in Me₃C⁺ while in Me₃Si⁺ it resides entirely on Si where it reaches +2 (due to the electron withdrawal by the more electronegative Me groups). Consequently, the Me₃Si⁺Cl⁻ VB structure has a deeper and tighter energy minimum and it approaches the covalent structure much more than in Me₃C⁺Cl⁻, thus maximizing the RE_CS. For example, RE_CS(Me₃Si–Cl) ∼ 60 kcal/mol vs RE_CS(Me₃C–Cl) ∼ 40 kcal/mol.^123,124

A useful concept must also have experimental manifestations or at the very least a connection to experimental patterns. Many of these patterns have been discussed in a recent treatment of bonding.⁷⁵ For example, the emergence of hypervalency, e.g., in XeF₂ is an outcome of the CSB character of its constituent electron-pair bond (see more in Sec. III D). The resistance of Si–X bonds to undergo solvolysis, despite their high ionic character, is due to the stickiness of Si⁺X⁻ interactions compared with the carbon analogs.^75,125–128 As such, Si–X bonds conserve the RE_CS in condensed phases. In solution, this stickiness results in an energy barrier >60 kcal/mol for Si–X bond heterolysis.^123,124 The role of amines as the potent nucleophile, e.g., in Michael addition to enones, was postulated to be rooted in the large RE_CS of the dative R₃N⁺–C bond.^75,129 Similarly, the conjecture of Patil and Bhanage¹³⁰ links the trends in the key physico-chemical properties of ionic liquids to the CSB character of the protonated N–H bond.

Other matches between the VB predictions and experimental findings concern the Laplacians of various bonds. Thus, the findings that bonds such as O–O and Mn–Mn possess positive experimental Laplacians are in accord with the CSB characters of these bonds.^75,105 Especially intriguing is the experimental characterization by Messerschmidt et al.¹³¹ of two differently signed Laplacians for the C–C bonds in [1.1.1]propellane. Figure 9 shows the experimentally determined Laplacians for a [1.1.1]propellane derivative, as well as computed values by Jenkins et al. using ab initio methods,¹⁴ which match DFT results⁸⁰ and calculations done for this Perspective. The different signs are in full agreement with the characterizations of these bond types by VB theory, namely, that the wing C–C bonds are normal covalent bonds with L < 0, while the inverted C–C bond is a CSB with L > 0.^80,104,132 The same bond-dichotomy appears in a variety of small-ring propellanes.¹³³ Furthermore, it is predicted that ring substitution can shift the bond character from covalent to CSB.¹³³ 

In the remainder of this section, we focus on a special problem, which is related to the impact of ionic structures on chemical reactivity, and which at the same time offers a potential method for determining the RE_CS quantity of a variety of bonds.

The intriguing reactivity puzzle deals with the theoretical and experimental findings in simple exchange reactions of F vs H. In the early 1970s, Schaefer et al. predicted^134,135 that the barrier for F exchange (H· + F–H → H–F + ·H) is astonishingly large, compared with the smaller barrier for the corresponding H exchange (F· + H–F → F–H + ·F). In 1978, Polanyi confirmed this prediction using the molecular beam experiment for the reactions between D and H–F.¹³⁶ 

To comprehend these results, Hiberty et al.¹³⁷ performed VB calculations of the reactions for the various hydrogen halides (H–X; X = F, Cl, Br). VB calculations enable one to compute the reaction barrier as well as the purely covalent energy barrier (⁠$Δ$E_cov^‡), in which we used only the covalent structures, for reactants, products, and transition states (e.g., HFH^‡ and FHF^‡). Table IV displays these results for the two reactions with H–F, which we recall is a CSB.

Inspection of Table IV reveals that the energy barrier for the F-exchange reaction is approximately twice the size of the barrier for the corresponding H-exchange, precisely as reported by Schaefer et al. and confirmed by Polanyi et al. However, inspection of the covalent-only barrier, $Δ$E_cov^‡, reveals similar barriers, and if at all, the barrier is somewhat larger for the H-exchange reaction. This necessarily means that the barrier differences in the first line in Table IV  are determined by the different ionic–covalent mixing in the TS. Indeed, as the RE_CS(TS) data reveal, the charge-shift resonance energy for HFH^‡ is 29 kcal/mol lower compared with the same quantity for FHF^‡.

To understand the root cause of the barrier difference between the H- and F-exchange reactions, we refer to Scheme 7 that draws the two lowest ionic structures for the two reactions.

It is seen that in the F-exchange reaction [in panel (a)] the ionic structure is destabilized by Pauli repulsion due to 3e-interactions between F:⁻ and the right-hand side H· radical (the mirror image ionic structure suffers repulsion on the left-hand side). As such, the energy of these ionic structures is raised and the structures mix less into the covalent TS, thus endowing the TS with a smaller RE_CS. In contrast, the ionic structure of the H-exchange reaction [in panel (b)] has no Pauli repulsion, and as such this ionic structure is low in energy, and it mixes more strongly into the covalent TS, yielding a larger RE_CS quantity.

The implication of this relation may be far reaching. It provides, in principle, a way to quantify the RE_CS of bonds based on the barrier difference for the two isomeric reactions, which can be easily devised and studied.

As demonstrated earlier, covalent bonds and CSBs have unique ways to obey the virial ratio [Eq. (6)], and they exhibit distinct bonding patterns. Furthermore, the covalent → CSB transitions in a period and a column of the Periodic Table are orderly and can be explained based on fundamental properties: the Pauli repulsion in the covalent structure and other physical properties (e.g., the ionic–covalent energy gap, relativistic effect, inner-shell’s proximity to the valence orbital). Additionally, CSBs have distinct experimental manifestations, and the CS-resonance energy of a bond can be quantified.

Many elements possess the ability to form more bonds than expected based upon the octet rule,¹⁰⁰ which limits the number of electron pairs around an atom to four. A typical example is xenon, which already has an octet in its valence shell, but is nevertheless able to bind two fluorine atoms by forming a linear three-center four-electron (3c-4e) bond. Other atoms, such as sulfur, phosphorus, chlorine, and krypton, as well as elements below them in the Periodic Table, also have this property, which is referred to as hypervalency.¹³⁸ Indeed, these atoms form molecules, such as XeF₂, XeCl₂, KrF₂, RnF₂, ClF₃, SF₄, PCl₅, and so on. By contrast, first-row analogs of the family, such as O, N, F, and Ne, strictly comply with the octet rule, thus exhibiting the so-called¹³⁹ “the first-row anomaly” (the stable F₃⁻ anion is an exception, which can be formally considered hypervalent—see later).

A tentative explanation for the hypervalency of P and S has first been proposed by Pauling, in terms of an expanded octet model, through promotion of electrons into vacant high-lying d orbitals, leading to sp³d hybridization (pp. 145–153 in Ref. 102). This model was later ruled out when accurate quantum calculations became available.¹⁴⁰ 

Pimentel and Rundle presented an alternative MO model, which does not involve d-orbital participation, that relies only on the three MOs that are involved in the 3c-4e system. Since the two occupied MOs of 3c-4e systems are strongly bonding and nonbonding, respectively, some stability might be expected for the hypercoordinated compound. The Rundle–Pimentel model served to rationalize and predict many structures, but was also considered by Munzarová and Hoffmann¹⁴¹ to be oversimplified, since the model would predict that all 3c-4e systems should be stable, hence failing to explain the above-mentioned “first-row anomaly.” Moreover, why F₃⁻ is stable whereas the isoelectronic H₃⁻ hypercoordinated anion is a transition state? Then, to be entirely successful and predictive, the Rundle–Pimentel model should involve a missing factor. As demonstrated by Braïda and Hiberty,⁸²  this missing factor is the presence or absence of charge-shift bonding (CSB). This will be illustrated below using XeF₂ as an archetype of the hypervalent molecule.

This model shows that only two factors can possibly account for the stability of a hypercoordinated compound: (a) either significant bonding energy of at least one of the four VB structures R₁–R₄ of Scheme 8 or (b) a large resonance energy that results from their mixing.⁸⁴ The ab initio VB calculations show that none of these VB structures is bonded by itself,⁸² and the lowest-lying one (R₃ in Scheme 8), the F:⁻ Xe²⁺ F:⁻ ionic structure, is still ∼79 kcal/mol above the dissociation limit, Xe + 2F·. On the other hand, the charge-shift resonance energy (RE_CS) arising from the mixing of R₁–R₄ amounts to 128 kcal/mol, thus fully accounting for the stability of XeF₂.

What is the origin of such a large resonance energy? It has already been our past experience that large resonance energies are observed in hypercoordinated compounds when the corresponding normal-valent species is bound by charge-shift bonding.^83,84,137 Indeed, the single bond in F–Xe⁺ is a typical charge-shift bond, with a resonance energy, RE_CS = 69 kcal/mol, which is larger than the total bonding energy of the molecule, ∼40 kcal/mol.⁸² Importantly, in other rare gas–fluorine complexes, RgF₂, the RE_CS values of the F–Rg⁺ bonds are in the order NeF⁺ < ArF⁺ < KrF⁺ < XeF⁺. Accordingly, NeF₂ and ArF₂ are unstable, whereas KrF₂ and XeF₂ are stable (the latter more than the former), showing that the charge-shift bonding character of diatomic RgF⁺ is critical for the stability of the three-atom cluster RgF₂.⁸² Then, there is a correlation between the charge-shift bonding (CSB) character of the normal-valent compounds and the stability of the hypercoordinated ones. Applying this rule, we now have the key to understanding why F₃⁻ is stable whereas H₃⁻ is not; F–F is a CSB, whereas H–H is a classical covalent bond.

Further VB calculations have been performed on other F-A-F hypervalent species (A = PF₃, SF₂, ClF).⁸³ In all cases, these compounds were found to be charge-shift-bonded, with RE_CS values being always considerably larger than the dissociation energies to A + 2F·. Interestingly, in all cases, including XeF₂, the weights of the four structures analogous to R₁–R₄ Scheme (8) are significant and of the same orders of magnitude, a favorable condition for maximizing the resonance energy. This implies that both the first and second ionization potentials for the central group A must be low in stable hypervalent species.

Thus, the general model for hypervalency in electron-rich systems appears to be the VB version of the Rundle–Pimentel model, coupled with the presence of charge-shift bonding. This latter feature implies the following conditions for the existence of hypervalency: (1) the central group and its ligands form charge-shift bonding in normal-valent species (i.e., being electronegative and bearing lone pairs) and (2) the central atom has low ionization potentials, generally for both the first and the second ionizations. The lack of these features explains the many exceptions to the traditional MO-based Rundle–Pimentel model and the so-called “first-row anomaly.”

Using classical VB theory, the hypervalent molecules are highly ionic and stabilized by large RE_CS. Another VB model of hypervalency, called “recoupled pair bonding” (RCP),^49,143–145 was introduced in 2009 by Woon and Dunning based on SCGVB calculations. The model is based on the facility of electronic promotion in the normal-valent molecule. This model was first applied to SF_n compounds, and later to other hypervalent molecules among which are ClF_n and PF_n.^144,145 The principle of this model is illustrated with the addition of two fluorine atoms to SF₂, leading to the hypervalent SF₄ molecule. Thus, whereas the X¹A₁ ground state of SF₂ is a bent singlet state, there exists a low-lying excited state, the a³B₁ triplet state (hereafter referred to as FSF*), which displays a linear 3c-4e bond, linking together a $π$ lone pair of the sulfur atom (3p_z) and odd electrons on the F atoms Scheme (9). According to the RCP model, polarization of the 3p_z orbital of S results in a pair of highly overlapping (S = 0.862) 3p lobe orbitals (3p_z+, 3p_z−); one (3p_z+) directed toward the right-hand F atom (3p_z+), while the other (3p_z−) toward the left-hand one. As such, two S–F bonds are formed, leaving two free valences (3p_x and 3p_y) that can bind further two more F atoms and form a stable SF₄ molecule.

What is the relationship between the RCP model and the charge-shift assisted Coulson–Rundle–Pimentel outlined above? In classical VB terms, the hypervalent FSF* compound in Scheme 9 can be considered a concise bonding scheme of the resonating combination F⋯S⁺:F⁻ ↔ F· S: ·F ↔ F:⁻ S⁺⋯F, to which F:⁻ S²⁺ F:⁻ is added for completeness. The VB calculations of Braïda and Hiberty⁸³ show that these four VB structures are critically important and display comparable weights, and that their mixing is associated with a very large charge-shift resonance energy. Thus, the RCP model is compatible with the classical VB model of the Coulson–Rundle–Pimentel 4e-3c bonding.

Part of chemical wisdom, hybridization is the concept of blending atomic orbitals so as to create new hybrid atomic orbitals (HAOs) that are suitable for the pairing of electrons and forming local and spatially directed covalent bonds. Pauling and Slater developed this concept in a series of papers, published in 1931,^{52–56,146,147} with the aim of constructing a general quantum chemical theory for polyatomic molecules. The notion of orbital hybridization proved insightful and has been used to discuss molecular geometries and bond angles in a variety of molecules, ranging from organic and transition metal compounds all the way to solids (Ref. 102, pp. 108–144).

Nevertheless, the concept of hybridization is regarded with some caution⁸⁶ because the initial formulation used a fixed recipe with orthogonal hybrids. Thus, tetrahedral sp³ hybrids involve each 25%s and 75%p contributions and maintain angles of 109.5°. Trigonal hybrids (angles of 120°) were denoted as sp², and collinear (angles of 180°) hybrids as sp. Furthermore, hybrid orbitals have been continuously criticized as being illegitimate orbitals, in scientific papers and textbooks,⁷ and this criticism seems to persist, despite proper rebuttals.¹⁴⁸ 

The SCGVB study of CH₄ by Penotti et al.¹⁴⁹ is a straightforward answer to the above criticisms. This study showed that hybridization in CH₄ results naturally when the molecule is calculated by releasing all constraints, such as geometry, double occupancy of orbitals, orthogonality, and spin-coupling between orbitals. Thus, the most general single-configuration wave function that emerges involves a set of four equivalent tetrahedral hybrid orbitals, each being singlet-coupled with the 1s atomic orbital (AO) of a hydrogen atom. As such, hybrid orbitals are variationally optimized objects of ab initio calculations that arise without any preconceived assumptions.

Let us continue in this vein and demonstrate that this holds true for all hybridization modes,⁸⁶ which result from the classical VB description of bonds, without any preconceived idea of hybridization, and without an assumption that hybrids have to be orthogonal. At the same time, we try to get deeper insight and find out the constitution of hybrids and the root cause of this constitution. As we demonstrate, overlapping hybrids are energetically more economical, since they reduce the expensive promotion energy and maximize the bonding.

It is apparent that, at one extreme of orthogonal hybrids (S = 0), λ = 1, and the p/s ratio is exactly one, leading to traditional orthogonal sp hybrids (e.g., in BeH₂), which require full investment of the respective promotion energy and as such lead to intrinsically weak bonds. At the other extreme when λ = 0 in Eq. (13), one gets S = 1 and p/s = 0. In such a nonphysical case, both HAOs would be the pure 2s AO, and there is no need for promotion energy investment. However, now Be–H bonding is no more possible, since the single 2s orbital is doubly occupied and can only be involved in ionic bonding. The variational procedure determines the best λ coefficient that allows the hybrids to maintain some intermediate overlap (1 > S > 0). This will bring about an optimal bond strength, which is a compromise of the fractional promotion-energy vs the intrinsic bond strength of the hybrids having this λ coefficient.

By analogy to the foregoing arguments, it can be seen, from Eqs. (14)–(16), that if the hybrids are constrained to be mutually orthogonal (S = 0), their p/s ratios are exactly 1, 2, and 3, respectively, for linear, trigonal, and tetrahedral cases, as expressed by the sp, sp², and sp³ notations. On the other hand, if the hybrids are allowed to overlap (0 < S < 1), the p/s ratio would be smaller than for corresponding orthogonal hybrids. It follows that the atomic valence state constructed with overlapping hybrids will lie in-between the ground state and the valence state that possesses orthogonal hybrids: the larger the overlap, the lower the promotion energy required for reaching the valence state.

Let us consider the above typical hybridization types: tetrahedral for CH₄, BH₄⁻, and NH₄⁺; trigonal for BH₃ and CH₃⁺; and linear for BeH₂, C₂H₂, CH₂²⁺, and C₂. In all these cases, we use ab initio VBSCF theory,^27,28 in which both coefficients and orbitals are simultaneously optimized, and the orbitals are pure single-centered AOs/HAOs. As we argued in the preceding sections, in classical VB theory, each A–H bond is described as a combination of a covalent structure A·–·H and two ionic ones A⁺:H⁻ and A:⁻ H⁺. Thus, the total number of VB structures necessary for accurately describing a Lewis structure is 3ⁿ, n being the number of two-electron bonds in the molecule. As such, a complete VBSCF calculation involves 81 VB structures for AH₄, 27 for AH₃ and C₂H₂, 9 for AH₂, and 81 for C₂.

The four optimized orbitals of CH₄ in the full VBSCF calculations are displayed in Fig. 11. There are four tetrahedral hybrids on carbon (h₁–h₄) and four 1s AOs of the hydrogen atoms (s₁–s₄). Figure 11 shows that these variationally determined four hybrids, free of any a priori constraints or preconceptions, point exactly along the tetrahedral directions and toward the respective hydrogen atoms to which they are linked.

The orbitals of BH₄⁻ and NH₄⁺ have the same qualitative shapes as those of CH₄. The overlaps between orbitals h_i-s_i are displayed in Table V for the three molecules. It is seen that the overlaps between a given hybrid and the 1s AO of the corresponding hydrogen atom within an A–H bond are quite large (entries 1–4), as expected for bonded atoms. These overlaps are also similar for the different molecules in the range S = 0.63–0.69. As expected, these overlaps slightly decrease in the series B⁻, C, and N⁺, as the central atom becomes increasingly more electronegative, since the larger the atomic electronegativity the less diffuse are the atomic or hybrid orbitals.

The hybrid–hybrid (h_i-h_j) overlaps, in entries 5–10, are virtually constant in a given column. This shows that although no constraints were imposed to get identical hybrids, the emerging hybrids are, nevertheless, identical to one another, within the limits of computational accuracy. Second, the overlaps between hybrids vary significantly among the molecules and exhibit a sharp increase from BH₄⁻ (S = 0.060) toward NH₄⁺ (S = 0.225). Such a trend cannot be explained by the diffuseness/compactness of the AOs of B⁻, C, and N⁺, which would predict the opposite effect. There must therefore be a different and a dominant effect (the magnitude of promotion energy) that is responsible for this trend, as is discussed later.

The last three entries of Table V show total energies for the 81-structure VBSCF calculations, Hartree–Fock (HF) calculations (no electron correlation), and 1764-structure full-valence-CASSCF (complete static electron correlation). It is apparent that the 81-structure VBSCF wave function is in all cases much lower than HF and close to full-valence-CASSCF, recovering ∼92% of the full static electron correlation. The description of AH₄ molecule by a single hybridized Lewis structure displaying overlapping hybrids is therefore shown to be quite accurate and reliable.

In order to avoid repeating the procedure for trigonal hybrids and linear hybrids, we simply state the results in Tables VI and VII.⁸⁶ 

Borane and methyl cation CH₃⁺ are isoelectronic and, hence, may, in principle, possess trigonal hybridizations. Describing each A–H bond (A = B, C⁺) by one covalent component and two ionic ones produces a set of 27 VB structures in the VBSCF calculation. The resulting optimized HAOs of BH₃, indeed, have the expected shapes of three coplanar hybrids making angles of 120° and each pointing toward their respective hydrogen atoms. The orbitals of CH₃⁺ have analogous shapes.⁸⁶ 

Table VI shows the properties of these hybrids. The last two entries of the table show that the 27-structure VBSCF calculation retrieves ∼95% recovery of the static electronic correlation in the CASSCF calculation. Again, the overlaps between each hybrid and a 1s AO of hydrogen within an A–H bond (entries 1–3) are large and slightly decreasing as the central atom becomes more electronegative, whereas the overlaps between hybrids (entries 4–6) are much smaller and display the opposite tendency.

The two bonds in BeH₂ and CH₂²⁺ require a nine-structure VBSCF calculation for a complete VB description.⁸⁶ On the other hand, the $σ$ system of C₂H₂ involves three bonds with 27-structures each (leaving aside the $π$ bonds that are treated as doubly occupied molecular orbitals). The emerging hybrids are all collinear and have similar appearances.⁸⁶ 

The overlaps are displayed in Table VII. In light of the preceding results, we expect to find that the orbital overlaps for the A–H bonds (A = Be, C, C²⁺, entries 1–4) are decreasing in the series, following the order of increasing electronegativities from Be to C²⁺. By contrast, the hybrid–hybrid overlaps between the HAOs of atom A (entries 5–7) increase in the same series, in line with the above findings for the tetrahedral and trigonal hybridization types. The last overlap entry of Table III shows that the overlap between the two inward HAOs h_in(1)–h_in(2) that form the $σ$ C–C bond in C₂H_2. This overlap is seen to be large, which is not surprising given that this $σ$ bond is strongly compressed by the two $π$ bonds in acetylene. The VBSCF energies recover 99% of the static correlation energies in CASSCF.⁸⁶ 

Equations (14)–(16) show that, if the HAOs are constrained to be mutually orthogonal, their np/ns ratios are exactly 1, 2, and 3 for linear, trigonal, and tetrahedral hybridization types, respectively. In such cases, the energy expense required to reach the valence state from the ground state is the full promotion energy,¹⁵⁰ given by the experimental transition energy (T_e) from the ground state to the high-spin state of the central atom, i.e., T_e(¹S → ³P) for linear hybrids, T_e(²P → ⁴P) for trigonal ones, and T_e(³P → ⁵S) for tetrahedral ones.

These transition energies are collected in Table VIII for atoms and cations and are seen to follow the order of electronegativities. Thus, the transition energy increases steeply as one goes from left to right of the Periodic Table and from an atom to its cation. This is in line with the well-known fact that the 2s → 2p orbital energy gap increases from left to right of the Periodic Table.

Orthogonal hybridization is therefore costly and it creates substantially weakened bonds. For example, in CH₄, the orthogonal hybridization would have reduced the atomization energy of the molecule by 96.4 kcal/mol, which is the full price for the ³P → ⁵S promotion of carbon. Since such a price is highly unfavorable for the molecular stability, overlapping hybrids are formed and reduce the promotion-energy penalty. As revealed from Eqs. (14)–(16), the p/s ratio of the overlapping hybrids is invariably proportional to (1 − S)/(1 + S) and it decreases as the overlap S increases. Indeed, VBSCF calculations reveal that hybrids are formed naturally from the variational VB procedure and they maintain substantial overlaps in order to optimize bonding by reducing the promotion energy. There are only overlapping hybrids!

Our further application to C₂⁸⁶ shows that this molecule restricts the promotion energy per C to about half of the full amount required for orthogonal hybrids, in order to form a quadruple bond with two $π$ bonds and two $σ$ bonds (see also in Sec. VI C 2).

Summary of Hybridization Trends in VB Theory: The above trends produced by the calculations reveal a clear pattern: The VB hybrids are variationally optimized objects, they keep strictly the angles of the classical hybridization, but at the same time they overlap and thereby minimize the requisite promotion energy and increase bonding. The hybrid–hybrid overlap increases with the increase of electronegativity of the central atom (e.g., NH₄⁺ > CH₄ > BH₄⁻), while, for a given electronegativity, the hybrid–hybrid overlap decreases as the number of equivalent bonds increases (e.g., CH₄ < CH₃⁺, CH₂²⁺). Thus, for example, the hybridizations are sp^1.76 (for CH₄), sp^0.94 (for CH₃⁺), and sp^0.40 (for CH₂⁺). Similar conclusions were reported recently by Xu and Dunning,⁴⁸ who used a projection methodology to deduce the hybridization obtained from SCGVB calculations that generate overlapping hybrids. A similar conclusion was reached for the hybridization in CH₄ already in the 1950s by van Vleck and Sherman.¹ 

Goddard et al. have treated a variety of multiply bonded molecules, Cr_2, HC≡CH, N≡N, P≡P, As≡As, and C₂, by means of SCGVB.^{39,40,43,45–50,88,151,152} The authors of this Perspective and their collaborators have treated $π$-only bonding in X = Y and X≡Y molecules (X, Y are either identical or different), where X and Y are fragments of main element atoms, e.g., H₂C, H₂Si, H₂Ge, HC, HSi, HGe, ….^96,97,108 As such, this section is restricted to the description of the unusual bonding patterns in C₂ and O₂.^72,87,89

C₂ is chosen because it has been a source of stimulation and controversy regarding its bond multiplicity.^{17,43,50,88,153–164} Moreover, with its singlet-paired eight valence electrons, C₂ should be described by a large number of VB structures, which may prevent a clear description of the bonding. In view of these issues, we have chosen to address here the C₂ problem because we deem it important to conceptualize electronically complex molecules and facilitate their descriptions. As this section shows, the key properties of C₂ are captured by the quadruply bonded form of this molecule.⁸⁷ 

The second molecule, O₂, is chosen here because its existence confronts us with an existential riddle: How come we breathe ³O₂ and do not catch fire? As we shall show, this has nothing to do with the triplet spin state of ³O₂, but rather with the huge resonance energy, which is brought about by the charge-shift resonance of its two 3-electron $π$-bonds,^89,90 and which makes the molecule persistent against many available reactions, e.g., its own dimerization/trimerization, or reactions with hydrocarbons, H₂, and so on.^89,165

C₂ is one of the most strongly bonded diatomic molecules, and one that has stirred up stimulation and controversy about its bond multiplicity. Since the ground state of this molecule, ¹$Σg+$⁠, possesses four valence–electron pairs, the possibility exists, at least formally, that the nature of these pairs express quadruple bonding between the two atoms. As we discussed in Sec. II, all wave functions of multiple-bonded molecules involve contributions of several structures to the total bonding of the molecules. For example, the 6$π$-electrons of benzene are described by five covalent structures: two Kekulé and three Dewar types. Nevertheless, we feel very comfortable to drop the minor Dewar structures and represent benzene as a resonating mixture of two $π$-Kekulé structures. C₂ with its eight valence-electrons requires 14 structures in SCGVB, or 14 covalent structures and 1750 ionic classical VB structures. One may therefore wonder if, like the benzene case, here too one can easily assign for C₂ bond multiplicity in a meaningful manner.

We demonstrate herein⁸⁷ that the conclusions of SCGVB and classical VB are compatible. In each method, the full wave function of C₂(¹$Σg+$⁠) is a blend of two bonding forms (the major is quadruple-bonded, and the minor is doubly bonded). Nevertheless, we find that what counts as a basis for a lucid description is whether the major structure with the maximal bonding can describe reasonably well the key properties of a multiply bonded molecule. Our foregoing analysis of C₂⁸⁷ shows that the quadruply bonded structure reproduces the key properties of this molecule.

According to the Weyl formula,⁷⁴ C₂ involves 1764 VB structures, of which 14 are covalent and 1750 ionic structures of mono-, di-, and higher-ionic ranks. A full-space VBSCF/6-31G* calculation^58,72,87 reproduces the bonding energy (D_e) as in the full-CI (FCI/6-31G*) calculations of Sherrill and Piecuch⁷³ This result is shown in Table IX along with others (see later, entry 6). Our goal is to define the most economical level of calculations, which is quantitatively reasonable and still providing lucid insight into bonding in C₂. This is done in steps of gradual peeling of VB structures based on their rank of bonding.

The 14 covalent structures are arranged in groups labeled A–D. The first structure is the quadruply bonded $Φ$_A, followed by six doubly bonded ones (one with two $π$ bonds-$Φ$_B; two with $σ$_in and $π$ bonds-$Φ$_C1,C2; and one $Φ$_D14 with a double $σ$ bond—$σ$_in and $σ$_o). In addition, there are four singly bonded structures and three non-bonded ones. The entire set of covalent structures is shown in the supplementary material. The lowest energy covalent structures are the quadruply and doubly bonded structures, which are depicted in Fig. 12, along with the designations of the paired orbitals: $π$_x, $π$_y, $σ$_in, and $σ$_o. The two $σ$ bonds correspond to the inward and outward exo-$σ$-orbitals.

In addition, Fig. 12 places these lowest energy covalent structures on an energy scale, which clearly shows that the quadruple bond, $Φ$_A(cov), has the lowest energy by at least 133.3 kcal/mol relative to the doubly $π$-bonded $Φ$_B(cov) and more so for the higher lying doubly bonded structures. This energy scale already provides us a clue that the double $σ$-bond in $Φ$_D14(cov) (146.2 kcal/mol above $Φ$_A(cov)) has a bonding energy, which is close to the double $π$-bond in $Φ$_B(cov).

In addition, VBSCF calculations using pairwise combinations at the equilibrium distance, e.g., A with B, A with C1 and C2, and A with D14, show that, in each case, the weight of structure A is dominant, ranging from 0.77 to >0.99 (for the different ways of calculating the weights^58,64). In addition, the stabilization due to this mixing is largest (25.1 kcal/mol) for C1,2, quite small for D14 (2.8 kcal/mol), and negligible for B (0.1 kcal/mol). The same conclusion is obtained by mixing all the five covalent structures together. Structure A remains dominant (e.g., the inverse weight is $ω$_A = 0.917), and its net stabilization energy by mixing with others is dominated by structures C1,2. As such, based on the covalent-only structures, C₂ is predominantly quadruply bonded at the equilibrium distance. This conclusion persists after the addition of multiple ionic structures.

Each covalent bond in Fig. 12 possesses two more ionic structures. In this manner, the full set of covalent and ionic structures, which describe the quadruply bonded structure contains 3⁴ = 81 VB structures, hence $Φ$_A(81). This structure-set involves mono-ionic, di-ionic, and higher ionic structures; the latter are high in energy. Therefore, by peeling off the structure-set of $Φ$_A(81) to a bare minimum, we retain for each bond-pair only its two mono-ionic structures and create thereby eight mono-ionic structures and 12 lowest lying di-ionic structures Schemes (10 and 11). The so-resulting peeled total structure-set for the quadruple bond contains the 21 VB structures, $Φ$_A(21). This minimal set $Φ$_A(21) is only 1.2 kcal/mol higher than $Φ$_A,81.

Additionally, in order to calculate D_e values, we must add those VB structures that describe in the asymptote two carbon atoms in their ³P states. These six structures are shown in Scheme 12. By adding these structures to $Φ$_A(21) and $Φ$_A(81), we generate $Φ$_A(27) and $Φ$_A(87) that describe the quadruply bonded structure taken all the way to the dissociation limit.

Table IX shows the VBSCF wave functions, their D_e values, and bond multiplicities, alongside bond energies from other methods. When the bond-multiplicity is not discussed in a given source, we use the qualifier “unspecified.” The D_e values for the wavefunctions of the quadruply bonded structure, $Φ$_A, are shown in entries 1–4. By comparing entries 1–4, we see that the truncated set $Φ$_A(27) in entry 2 gives rise to almost the same D_e value as $Φ$_A(87) (entry 1). We can therefore focus on the truncated wave function and discard the one with the fuller set (⁠$Φ$_A(87)). The D_e value of $Φ$_A(27) (entry 2) is 82% of the corresponding value for the full-VBSCF wave function (⁠$Φ$_full(1764) in entry 5), which itself is compatible with the full-CI (FCI) value in entry 6, due to Sherrill.⁷³ 

The D_e value for $Φ$_A(27) can be further improved as seen in entries 3 and 4 by adding local dynamic correlation to the four bonds by means of VBCISD (a similar effect to BOVB) and quantum Monte Carlo with a Jastrow factor.⁸⁷ These improved D_e values, respectively, 129.3 and 134.9 kcal/mol, are 94% and 98% of the full-VBSCF result (entry 5), which itself is virtually identical to the FCI value at the same basis set (entry 6).⁷³ 

Focusing on VBSCF-only results, we can further add layers of VBSCF structures to $Φ$_A(27) and thereby improve the D_e value. The principle is simple: every new covalent structure (from $Φ$_B, $Φ$_C1, $Φ$_C2, $Φ$_D14, and the remaining covalent structures), which is added to $Φ$_A(27) comes with a corresponding set of ionic structures. If this addition gives rise to an improved D_e value, we continue to add structure-sets up to a point where we approach the D_e value for $Φ$_full(1764).

In a past publication,⁸⁷ we created the 20 ionic structures for the four electron pairs of each of these covalent structures. However, here we use a minimalistic approach and add ionic structures only for the covalent bonds in the respective structure. Since we have two bonds in B, C1,2, and D14, the number of covalent and ionic structures in the doubly bonded set is 3² = 9. Precisely, as we found in the covalent-only structures, here too, adding $Φ$_B(9) and $Φ$_D14(9) has a very small effect on D_e, compared with $Φ$_A(27) (Table IX)_. Thus, the results for $Φ$_A,C1,C2(45) in entry 7 and $Φ$_{A,B,C1C,2 (54)} in entry 8 have identical D_e values, 126.9 kcal/mol, which are 92% of the results for $Φ$_full(1764) and FCI (entries 5 and 6, respectively). Adding the nine structures for D14 generates in entry 9 $Φ$_A-D14(61), which exhibits D_e = 129.0 kcal/mol (93.5% of the FCI value), adding the singly bonded covalent structures leads in entry 10 to $Φ$_{cov-bonded(91)} with D_e = 132.5 kcal/mol, which is 96% of the FCI value. It is clear that we can easily reach the FCI value with a few more structures. As such, the principle is clear: 45–91 VB structures (with well-defined bond multiplicities) give rise to D_e values that are more than 92%–96% of the full VBSCF or FCI values. Analyzing the weights, we find that $Φ$_{cov-bonded(91)} is dominated by the quadruply bonded structure with $ω$_A = 0.89 (see Table X). As such, C₂ is basically a quadruply bonded molecule, much as benzene is described by two Kekulé structures, and N₂ by a triply bonded structure, though their respective wave functions contain contributions of other VB structures with lower bond multiplicities.

A comparison of the VBSCF values to those obtained with SCGVB in Table IX shows that the quadruply bonded wave function, $Φ$_A(27) in entry 2, gives rise to a D_e value that is better than the quadruply bonded $Φ$_SCGVB(PP) structure (entry 11), which implicitly includes the same number of covalent and ionic structures as in $Φ$_A(87) in entry 1. Furthermore, the quadruply bonded $Φ$_A(27) in entry 2 leads to an identical D_e value as the one obtained from the total $Φ$_SCGVB,14 wave function that includes 14 spin-coupled structures (entry 12) and all the 14 covalent structures, and implicitly many, if not all, of their ionic structures. This better quantitative performance of VBSCF is expected since the VBSCF optimizes the individual ionic structures, while $Φ$_SCGVB(PP) and $Φ$_SCGVB,14 do not. Nevertheless, we have verified in the past⁸⁷ that $Φ$_SCGVB,14 includes a mixture of quadruply bonded (⁠$Φ$_SCGVB(PP))_, doubly bonded and single-bonded structures. Furthermore, $Φ$_SCGVB,14 and $Φ$_GVB (entry 13) can be improved systematically by CI.^87,166,167 As such, the VBSCF and SCGVB wave functions are qualitatively similar, albeit their quantitative performances are different.

Table X shows the group weights (inverse-weights⁶⁷) for the VBSCF $Φ$_{cov-bonded(91)} wave function. It is seen that the weight of the group A wave function $Φ$_A(27) is 0.89, while the other weights are very small. The $σ$ + $π$ doubly bonded group of structures $Φ$_C(18) leads among the doubly bonded structures, but its weight is small, 0.06. Other weights (see the supplementary material) do not change this picture in any major way.

In conclusion, the quadruply bonded structure dominates the wave function for C₂, having inverse weights around ∼0.9 and bond energies of ∼113/135 kcal/mol (entries 1–5 in Table IX), while the doubly bonded structures are less important, both in terms of their weight and energy contributions to the D_e value. Among the latter structures, the two (⁠$Φ$_C1 and $Φ$_C2) with the $σ$ + $π$ double bonds are more important than others and contribute 14 kcal/mol to the D_e value (see Fig. 13). At longer R_CC distances, the doubly bonded structures cross below the quadruply bonded one and eventually define the dissociation limit. This is found in both the classical VB⁸⁷ and the SCGVB⁸⁸ computations.

Table XI collects computational data for C₂ of equilibrium bond distances (R_CC), force constants (k_CC), and bond energies (D_e) at various levels.⁸⁷ The differences vis-à-vis experimental data are due to the use of a modest basis set (6-31G*) in the FCI calculations. The FCI level serves as a benchmark for the full-VBSCF calculations that reproduce the FCI data.

It is seen that the single quadruply bonded covalent structure (entry 1) produces R_CC and k_CC values that are reasonably close to the full-VBSCF calculations (entry 4) and to the corresponding multi-reference CI (MRCI) and coupled-cluster single double triple [CCSD(T)] calculations (entries 5 and 6), as well as to the experimental values (entry 7). The quadruply bonded wavefunction $Φ$_A(27) (entry 2) reproduces the R_CC and k_CC values within 92% of the full-VBSCF calculations and the corresponding D_e to within 82% of the corresponding full-VBSCF and FCI values. Adding local dynamic correlation to the four bonds in entry 2a using VBCISD improves the D_e match to within 94% of the benchmark FCI data. The $Φ$_A,C1,C2(45) wave function (entry 3), which includes the main stabilizing effect of mixing the $σ$ + $π$ doubly bonded structures, gets closer to the data of the full-VBSCF and MRCI (FCI) calculations. Finally, the wave function $Φ$_{bonded-cov(91)} (entry 4) that includes the blocks of all the bonded-covalent structures reproduces the data of the full-VBSCF and MRCI (FCI) calculations (entries 5–7).

The data in Table XI do not exhibit an increase of k_CC nor a concomitant decrease of R_CC with the increase in the bond energy (D_e). For example, a comparison of $Φ$_A(27) (entry 2) to $Φ$_A,C1,C2(45), $Φ$_{(Q,D,S)bonds(91)} and $Φ$_full(1764) (entries 3–5) shows that as structures of lesser bond-multiplicity are mixed into the quadruple bond, the D_e increases but the bond gets somewhat longer and its force constant a bit lower. There is no contradiction here, but just a simple indication that the main effect of raising the D_e is due to the resonance stabilization, which is brought about primarily by the mixing of the $σ$ + $π$ double-bonded structures (C1, C2) into the quadruply bonded structure (cf. Fig. 13). As expected from resonance mixing, this effect also lengthens the bond distance (entries 3 and 4) and lowers the force constant vis-à-vis the corresponding values for $Φ$_A(27) (entry 2).

Note that, while this resonance mixing stabilizes the molecule, as seen in Fig. 13, and changes R_CC and k_CC parameters in Table XI, the molecules are, nevertheless, still described by a dominant quadruple bond structure.

One of the components of the quadruple bond is the double-$σ$-bond Scheme 13. This is a unique double bond because it has an exo-component. Nevertheless, we already addressed this issue in this section and in previous publications^86,87 and concluded that the double $σ$-bond makes a significant contribution to the total C₂ bonding.

As such, the total promotion energy for creating four such hybrids in C₂ is 92.4 kcal/mol. In turn, the strength of the double-$σ$-bond can be estimated from the fact that the doubly $π$-bonded structure-set is situated above the quadruply bonded structure set $Φ$_A(27) by 153.7 kcal/mol. Roughly speaking, this is the cost of breaking (vertically) the two $σ$-bonds. This value is very similar to the in situ bond energies of 156.6 and 147.3 kcal/mol determined before,^86,87 and either value is significantly larger than the requisite total molecular-promotion energy, 92.4 kcal/mol. As such, the $σ$-bonding makes a substantial net contribution to the total bonding of C₂. Furthermore, the promotion energy lowers the bond-dissociation energy of the molecule to 146.7 kcal/mol, while the “strength” of the bond, relative to the promoted state (⁵I, Scheme 13), is 239.1 kcal/mol.

One remaining issue is the strength of exo-bond, $σ$_o Scheme 13. In previous studies, we determined this value from theoretical calculations^72,153,168 as well as from experimental bond dissociation energy (BDE) of the two C–H bonds of acetylene to yield C₂. Scheme 14 shows the formation of C₂ from acetylene in two successive C–H bond breaking steps. The difference between the first and second BDE values is, in principle, the bond energy of the exo-$σ$-bond (D_$σ$o), unless there are significant structural relaxation/rehybridization or radical delocalization in HCC•. As we mentioned earlier, these relaxation/rehybridization/delocalization energies are negligibly small.¹⁶⁸ As such, D(⁠$σ$_o) is 15–17.9 kcal/mol using experimental or computational data (the higher value is determined from CCSDT/complete basis set (CBS) calculations,^{166,169–172} including an estimate from the singlet → triplet excitation of the bond).⁷² 

One obvious lesson is the unusual $σ$-bonding of C₂, which includes an exo-bond that is rather weak (∼17.9 kcal/mol). With such a bond, the C₂ molecule will be highly reactive⁸⁷ in a variety of reactions and would thereby pose a serious challenge to be “isolated.”

A second lesson is the stimulation that such an unusual electronic structure has caused. Our publications have been followed by some debates and claims that C₂ does not have a quadruple bond. However, equally so, other theoretical publications supported by and large the same electronic structure as we did, from different electronic structure analyses.^{17,156,158,160,161} Soon after, the double $σ$-bonding was reported to be essential also in B₂.¹⁶² 

Furthermore, already at the outset,⁷² the bonding in C₂ served as a model for other isoelectronic species (e.g., CN⁺, N₂⁺⁺, BN⁻), and it stimulated the community to find similar transition metal (TM) complexes exhibit quadruple bonds to main elements, , , and molecules.^173–175 The C₂ work⁷² served to stimulate a group of experimentalists¹⁶³ to attempt the arduous room temperature isolation of C₂, which has subsequently run into some difficulties,¹⁶⁴ and elicited further collaboration to clear out the difficulties. All in all, such a stimulating debate on bonding is good for chemistry.

Finally, the analysis of multi-bonded molecules is challenging because any multiple bond from triple onwards contains a few different contributions of different bond multiplicities. This creates a challenge to devise a meaningful simplification, which we think shows clearly the quadruple bond dominance in C₂. The systematic method we presented above may be helpful in this respect for challenging molecules like Cr₂, which has a quintuple or sextuple bond⁴⁰ and for derivatives of ligand supported Cr₂ and other transition metals.^176,177

Why then this diradical (↑•OO•↑) still exists intact? All the more it is an abundant constituent of our atmosphere (∼21%), whereas most other diradicals and radicals are extremely reactive. Is it the spin-state issue that is responsible for the persistence of ³O₂? Most likely not, because spin–orbit coupling for reactions of ³O₂ is sufficient to bypass the spin–flip issue. The root cause of the persistency of the ↑·OO·↑ diradical appears to be the high barrier for most of its reactions irrespective of their exothermicity. Thus, Filatov et al.¹⁶⁵ showed that the barrier for H-abstraction by O₂ from H₂ is very high (exceeding 60 kcal/mol), and the reaction step is highly endothermic (58 kcal/mol), even though the overall reaction [Eq. (18)] is exothermic.

This charge-shift resonance energy, which exceeds the iconic resonance energy of benzene, was thermodynamically estimated as 100.6 kcal/mol from G4 calculations of $Δ$H(298.15 K) by Borden et al.⁹⁰ On the other hand, the resonance energy of a single O–O $π$-(3e) bond was estimated by the same authors as 31.6 kcal/mol. Thus, the bonding energy of the double-$π$-(3e) bond of dioxygen can be considered to arise from two single $π$-(3e) bonds, augmented by a cooperativity factor of 37–38 kcal/mol.⁹⁰ 

This, of course, affects the reactivity of the molecule. As an example, if ³O₂ is involved in a reaction of hydrogen abstraction from another molecule, the double-$π$-(3e) bond, with its huge resonance energy, is reduced to a single $π$-(3e) bond, and the loss of resonance energy is as high as ∼69 kcal/mol, thus accounting for the large endothermicity and high barrier for this reaction¹⁶⁵ and, more generally, for the kinetic persistence of this diradical (↑·OO·↑) and its ability to sustain life on our globe.

This result, as well the other ones discussed in Sec. III A, illustrates the importance of resonance energy as a fundamental concept in chemistry and the importance of being able to estimate it. In addition, the fair agreement between the BOVB calculations⁸⁹ and thermochemical estimate⁹⁰ shows that the CS-resonance energy is, in principle, a quasi-observable that affects the chemical behavior of the molecules.

The $π$–$σ$ model for doubly and multiply bonded molecules, such as ethene, acetylene, and benzene, is a dominant feature in our chemical education. For example, benzene and ethene can be conceptualized in terms of sp² hybridization, while acetylene in terms of a sp hybridization (note that even though the hybridization involves overlapping hybrids, we use here the traditional nomenclature, where sp refers to linear hybridization, while sp² to trigonal hybrids). Using these hybridization modes leads to straightforward predictions of the respective geometries as being planar and linear. The clarity of this model attests also to the conceptual utility of the $π$–$σ$ bonding description in these molecules and their many derivatives. At the same time, this attractive picture raises questions about the interplay between the $π$ and $σ$ components in benzene, ethene, and acetylene: which one of these two components controls the geometry and stability of these molecules and their derivatives?

As we shall show, the answers to these questions are not always intuitively obvious. For example, for benzene and other aromatic molecules,^91–95 we demonstrated that the $π$ electronic component strives to achieve a geometry with alternating C–C bond lengths whereas the $σ$-frame resists this tendency and determines the perfect hexagonal symmetry of benzene.

How does the $π$–$σ$ interplay manifest in ethene and acetylene, and the higher-row analogs of these three iconic molecules? It turns out that higher row analogous molecules with, e.g., Si, Ge, etc., raise additional questions since these molecules adapt geometries that violate the foregoing simple perspective. For example, the sila-analogs of ethene and acetylene^96,97 and the respective derivatives^179–188 are trans-bent and/or twisted. Similarly, hexasila-benzene (Si₆H₆) is either puckered¹⁸⁹ or dubiously planar with a frequency of 10–61 cm⁻¹ for ring puckering, and in any event, it is not the most stable isomer of Si₆H₆.¹⁹⁰ Can these trends be understood in terms of the interplay between the bonding features of the $π$ and $σ$ components? This is the main theme in this subsection.

The $π$-electronic component of benzene is perfectly delocalized in the D_6h symmetric geometry of the molecule. But is this property due to the inherent propensity of the 6$π$-electrons to prefer a delocalized state, or is it a consequence of the $σ$-frame that forces identical $σ$_C–C bonds and a D_6h symmetric structure?