Resonance in Valence Bond Theory

Resonance language remains the everyday vocabulary organic and biochemists use to rationalize reactivity, and it has real quantitative descendants in modern computational chemistry. Curly-arrow mechanisms in every organic reaction — enolate stabilization, amide-bond planarity and rotational barriers, aromatic electrophilic substitution directing effects, the stability of carboxylate and guanidinium cations in amino-acid side chains — are all applications of exactly the resonance-mixing idea developed here, even when no explicit secular determinant is solved by hand. The planarity and restricted rotation of the peptide bond, central to protein secondary structure, is a direct consequence of resonance mixing between the neutral amide structure and a zwitterionic structure with C=N double-bond character, and the resulting partial double-bond character (roughly 40% by typical VB weightings) explains both the observed C–N bond shortening and the roughly 15–20 kcal/mol rotational barrier that keeps peptide backbones locked in trans or cis conformations. In materials and drug design, resonance-stabilized aromatic and heteroaromatic scaffolds (purines, pyrimidines, indoles, imidazoles) owe their chemical robustness and characteristic spectroscopic signatures to exactly the kind of multi-structure delocalization worked out above; medicinal chemists routinely reason about which tautomer or resonance form dominates a binding pocket. Modern ab initio valence bond programs (VB2000, XMVB, and related codes) implement the resonance secular-equation framework computationally, solving genuinely non-orthogonal multi-structure VB problems for real molecules and recovering resonance energies, weights, and reaction mechanisms (including VB-based descriptions of transition states as resonance hybrids of reactant-like and product-like structures) that agree closely with experiment and with Molecular Orbital Theory of Polyatomic Molecules predictions for the same systems. The conceptual throughline — that mixing basis structures variationally always stabilizes the ground state, and that degenerate mixing produces delocalization — also underlies the qualitative reasoning chemists use every day when assigning Hybridization: sp, sp2, and sp3 Orbitals and when comparing the complementary strengths of VB and MO pictures, an explicit comparison developed in Valence Bond versus Molecular Orbital Theory.

Ask the tutor: How would the resonance secular equations change if the mixed VB structures included both covalent and ionic contributions simultaneously, as in a polar aromatic system?

A single VB structure — one specific way of pairing electrons into bonds, exactly as a Lewis structure depicts it — is a legitimate but approximate wavefunction. For a molecule that admits more than one reasonable Lewis structure with the same atomic skeleton (same $\sigma$-framework, same atom positions), each such structure $\Phi_1, \Phi_2, \dots, \Phi_n$ is an antisymmetrized product of orbitals paired according to that particular bonding scheme. None of them, alone, is an eigenfunction of the true molecular Hamiltonian $\hat H$. The variational principle then instructs you to build the best possible wavefunction as a linear combination,

and to choose the coefficients $c_i$ to minimize the energy expectation value. This is formally identical to constructing molecular orbitals as linear combinations of atomic orbitals — the difference is that here the basis functions are entire many-electron structures, each already a fully paired covalent (or ionic) bonding pattern, rather than one-electron atomic orbitals. For allyl radical (three $p$ orbitals, three $\pi$ electrons), there are two covalent-radical resonance structures corresponding to the two ways of placing a double bond and a radical: $\mathrm{CH_2}\!=\!\mathrm{CH}\!-\!\dot{\mathrm{CH}}_2 \leftrightarrow {}^\bullet\mathrm{CH}_2\!-\!\mathrm{CH}\!=\!\mathrm{CH}_2$. These two structures are related by the mirror symmetry of the carbon skeleton and are exactly degenerate. Mixing them is the simplest non-trivial resonance problem and it already displays every essential feature: symmetric and antisymmetric combinations, an avoided crossing in energy, and a net stabilization relative to either localized structure. Benzene is the two-structure problem promoted to six $\pi$ electrons over six $p$ orbitals, with two degenerate Kekulé structures as the dominant basis, augmentable by three higher-energy Dewar (para-bonded) structures for quantitative accuracy. It is essential to be precise about what resonance is not. The molecule does not oscillate between structures in time; there is no experiment, however fast, that would ever catch benzene "as" one Kekulé structure. $\Psi$ is a single, time-independent, stationary quantum state — a genuine superposition, the same sense in which $2p_z$ is a superposition of $m_\ell = +1$ and $-1$ states, not a molecule hopping between them. The individual $\Phi_i$ are mathematical basis functions, useful because chemists already know how to think with Lewis structures; the "percentage contribution" of a structure quantifies its weight in that expansion, not a fraction of time or population.

Notice the pattern that recurs throughout VB resonance theory: whenever two basis structures are connected by a nonzero off-diagonal Hamiltonian matrix element (a "resonance integral"), the true eigenstates split into a lower, stabilized combination and a higher, destabilized combination — precisely the two-level mixing familiar from any two-orbital or two-state perturbation problem. What is new here is that the "orbitals" being mixed are entire electron-pairing schemes.

The resonance problem is a linear variation problem in a non-orthogonal basis of $n$ VB structures $\{\Phi_i\}$. Write the trial function and minimize $\langle\Psi|\hat H|\Psi\rangle/\langle\Psi|\Psi\rangle$ with respect to the coefficients $c_i$. This gives the standard generalized eigenvalue problem,

where:

• $H_{ij} = \langle \Phi_i|\hat H|\Phi_j\rangle$ is the Hamiltonian matrix element ("resonance integral" for $i\ne j$, structure energy for $i=j$),
• $S_{ij} = \langle \Phi_i|\Phi_j\rangle$ is the overlap between the (generally non-orthogonal) structures,
• $E$ is a variational estimate of an energy eigenvalue,
• $c_j$ are the coefficients of each structure in the resonance hybrid, obtained after solving.

Nontrivial solutions require the secular determinant to vanish,

a polynomial of degree $n$ in $E$ whose $n$ roots are the variational energies of the $n$ mixed states; the lowest root is the ground-state resonance energy estimate. Reading this equation: it is the exact same structure as the LCAO secular determinant, except the basis functions are many-electron VB structures rather than one-electron atomic orbitals, and $S_{ij}$ is generally not the identity matrix because two different electron-pairing schemes on the same atoms are not orthogonal. For the common and instructive special case of two degenerate, equally weighted structures ($H_{11}=H_{22}\equiv\alpha$, $H_{12}=H_{21}\equiv\beta$, $S_{11}=S_{22}=1$, $S_{12}=S_{21}\equiv S$), the $2\times 2$ determinant gives two roots,

Reading this: because $\beta$ is negative (a stabilizing resonance integral) and $S>0$ for real bonding structures, $E_-$ (the symmetric combination, $c_1=c_2$) lies below $\alpha$ and $E_+$ (the antisymmetric combination, $c_1=-c_2$) lies above it. The resonance energy relative to an isolated (non-resonating) structure is

Reading this: the stabilization is driven by the resonance integral $\beta$ but is moderated by the overlap $S$ between the two structures — larger $|\beta|$ (better geometric/orbital alignment for delocalization) and larger overlap both increase the resonance energy, though $S$ appears in the denominator too, so the net dependence is not simply linear in either. For $n$ degenerate covalent structures with equal pairwise interaction $\beta$ between every pair (an idealization useful for symmetric rings), the ground-state energy of the fully symmetric combination is

where:

• $n$ is the number of mixed degenerate structures,
• $\alpha$ is the energy of a single (unmixed) localized structure,
• $\beta$ is the (negative) resonance integral between any pair of structures.

Reading this: adding more mutually resonating, energy-degenerate structures multiplies the stabilization, which is part of why aromatic and polyene delocalization problems reward writing out all reasonable resonance structures, not just the two most obvious Kekulé forms.

Example 1 — Foundational: solving the allyl two-structure secular problem numerically. Allyl radical has two degenerate covalent VB structures, $\Phi_1$ (radical on C1, double bond C2=C3) and $\Phi_2$ (double bond C1=C2, radical on C3). Suppose parametrized calculations give $\alpha = -1.00\ \text{eV}$ (the $\pi$-energy of either isolated structure, relative to separated atoms), $\beta = -0.60\ \text{eV}$ (resonance integral between the two structures), and overlap $S=0.20$. Apply the two-structure formulas derived above directly:

The resonance-stabilized ground state lies at $E_-=-1.333$ eV, and the resonance energy relative to a single localized structure is

This matches, in order of magnitude, the experimentally inferred allyl resonance stabilization (roughly 12–15 kcal/mol from radical thermochemistry; the discrepancy simply reflects that $\alpha,\beta,S$ here are illustrative parameters, not a full self-consistent VB calculation). The physically important content is qualitative and quantitative-in-form: because $\Phi_1$ and $\Phi_2$ are exactly degenerate ($H_{11}=H_{22}$), they mix with equal weight $c_1=c_2$, the radical spin density is predicted to be equally shared between C1 and C3, and the true ground state is the symmetric combination lying below either parent structure by an amount set by $\beta$ and $S$. Limiting case check: if the coupling vanished ($\beta\to 0$), $E_-\to\alpha/(1+S)\cdot$... more carefully, with $\beta=0$ both roots collapse to $E_\pm = \alpha/(1\pm S)$, which for $S\ne0$ still differ only because of the nonorthogonality — but the physically meaningful resonance stabilization $\Delta E_{\text{res}} = (\alpha S-\beta)/(1+S)$ becomes $\alpha S/(1+S)$, a small residual overlap artifact rather than a real chemical effect; the true "no resonance" limit is $\beta\to 0$ and $S\to 0$ simultaneously, giving $\Delta E_{\text{res}}\to 0$ exactly, confirming the formula behaves correctly when there is genuinely nothing to mix.

Example 2 — Applied: comparing resonance energies of benzene, naphthalene, and a non-aromatic polyene using per-electron scaling. A simple, widely used approximation (in the spirit of Pauling's original resonance-energy tabulations) takes the total $\pi$-resonance energy of a benzenoid ring system as roughly proportional to the number of independent Kekulé structures $K$ available, via

where $\gamma$ is an empirical scaling constant fit so that benzene ($K=2$) reproduces $36.0$ kcal/mol:

Naphthalene has $K=3$ Kekulé structures. The model predicts

The experimental resonance energy of naphthalene (from hydrogenation/combustion analysis) is about $61$ kcal/mol, so this crude logarithmic model is within about $7\%$ — good enough to illustrate the qualitative trend (more resonance structures, greater stabilization, though sublinearly) without claiming rigorous accuracy. Contrast this with 1,3,5-hexatriene, an open-chain non-cyclic triene with $K=1$ (only one reasonable covalent structure, since there is no ring closure to generate a second, degenerate bonding pattern): the model correctly predicts $\Delta E_{\text{res}} \approx 51.9\ln(1) = 0$, consistent with the experimental observation that open-chain polyenes show only modest conjugative stabilization (a few kcal/mol from adjacent-double-bond conjugation) and nothing resembling the "extra" aromatic stabilization of a closed, degenerately resonating ring. Limiting case check: as $K\to 1$ for any system (only one possible Lewis structure, e.g. ethylene or a simple isolated double bond), $\ln K\to 0$ and the predicted resonance energy vanishes identically, matching the physical expectation that a molecule with a unique, unambiguous Lewis structure has no resonance stabilization to speak of.

Example 3 — Competition-level: five-structure VB treatment of benzene including Dewar structures, solved as a full secular determinant. Take benzene's $\pi$ system with the two Kekulé structures $\Phi_1,\Phi_2$ (degenerate, energy $\alpha$) and three Dewar structures $\Phi_3,\Phi_4,\Phi_5$ (degenerate among themselves by symmetry, energy $\alpha+\delta$ with $\delta>0$ since Dewar structures place a bond across the ring and are less stable). Use the (commonly quoted, Pauling-style) simplified coupling scheme with unit overlaps neglected for tractability ($S_{ij}\approx0$ for $i\ne j$, a rough but standard simplification for a first quantitative pass):

• Kekulé–Kekulé coupling: $H_{12}=\beta_{KK}$
• Kekulé–Dewar coupling: $H_{1,3}=H_{1,4}=H_{1,5}=H_{2,3}=H_{2,4}=H_{2,5}=\beta_{KD}$
• Dewar–Dewar coupling: $H_{34}=H_{35}=H_{45}=\beta_{DD}$

Take illustrative numbers (in units of $|\beta_0|$, a reference resonance integral): $\alpha=0$ (energy zero defined at an isolated Kekulé structure), $\delta = 0.40|\beta_0|$, $\beta_{KK}=-1.00|\beta_0|$, $\beta_{KD}=-0.60|\beta_0|$, $\beta_{DD}=-0.10|\beta_0|$. By the full ring symmetry ($D_{6h}$), the lowest eigenvector must be the totally symmetric combination: equal weight on both Kekulé structures and equal weight on all three Dewar structures. Write $\Psi = a(\Phi_1+\Phi_2) + b(\Phi_3+\Phi_4+\Phi_5)$ and substitute into the two independent secular equations obtained by symmetry reduction:

Plugging in numbers with $\alpha=0$:

The determinant condition is

Expand:

so the secular equation becomes

Solve the quadratic:

The lower root, $E_{\text{ground}} = -1.988\,|\beta_0|$, is the ground-state $\pi$-energy in this five-structure basis. Compare to the two-Kekulé-only result (setting $\beta_{KD}=\beta_{DD}=0$, i.e. dropping the Dewar structures entirely): the $2\times2$ block alone gives $E_-=\alpha+\beta_{KK}=-1.00|\beta_0|$ (using the $S=0$ simplification of the earlier formula). Including the three Dewar structures nearly doubles the calculated stabilization, from $-1.00|\beta_0|$ to $-1.988|\beta_0|$ — concretely demonstrating, with real numbers, the qualitative statement made in the second derivation: the two "obvious" Kekulé structures are necessary but not sufficient, and quantitative VB resonance-energy calculations for benzene must include the higher-energy Dewar structures to approach the experimental $36$ kcal/mol. Limiting case check: if the Kekulé–Dewar coupling $\beta_{KD}\to 0$ (Dewar structures decoupled entirely from Kekulé structures), the secular determinant factors, and the lower root reduces exactly to $E=\alpha+\beta_{KK}=-1.00|\beta_0|$, the pure two-structure Kekulé result — confirming the five-structure formula correctly collapses to the simpler two-structure case when the extra structures are switched off.

Linus Pauling built resonance theory in the early 1930s as the organizing framework of his valence bond program, synthesizing the quantum mechanics of Heitler and London's 1927 hydrogen-molecule treatment with the practical bonding intuition chemists already used. Pauling's insight was that a molecule's true wavefunction could be constructed as a superposition of the various "reasonable" Lewis structures a chemist might draw, and that the mathematics of that superposition — secular determinants exactly analogous to those governing atomic term splitting — explained numerically why benzene, carboxylate ions, and countless other species were more stable, more symmetric, and had different bond lengths than any single classical structure would predict. He introduced the term "resonance" deliberately evoking the mechanical phenomenon of coupled oscillators, emphasizing (against frequent misreading even in his own era) that it was a mathematical superposition and not a literal oscillation of the molecule between forms. His 1939 book The Nature of the Chemical Bond systematized resonance alongside electronegativity, hybridization, and bond-length–bond-order relationships into the dominant qualitative bonding language of twentieth-century chemistry, work recognized with the 1954 Nobel Prize in Chemistry.