Covalent and Ionic Contributions to the Bond

The ionic-covalent resonance idea, though first worked out for the simplest possible molecule, underlies how chemists still think about bond polarity across the periodic table. Every time a textbook draws a curved double-headed arrow between a Lewis structure like H–Cl and an ionic structure H⁺Cl⁻ to explain a permanent dipole moment, it is invoking exactly the mechanism derived here, generalized to a heteronuclear pair where the two ionic structures ($\rm H^+Cl^-$ and $\rm H^-Cl^+$) are no longer symmetry-equivalent and so mix with unequal weights — the heteronuclear analog flagged in the limiting-cases table. In modern computational chemistry, the direct descendants of this idea appear as valence bond self-consistent field (VBSCF) and breathing-orbital VB methods, used to obtain chemically transparent, resonance-structure-based descriptions of reaction mechanisms — particularly useful for diagnosing how much ionic (charge-separated) character develops along a reaction coordinate in polar organic mechanisms, SN1/SN2 borderline cases, and proton-transfer reactions in enzyme active sites. The same logic underlies how computational chemists interpret natural bond orbital (NBO) analyses of ab initio wavefunctions, which decompose delocalized molecular orbitals back into Lewis-structure-like covalent and "donor–acceptor" (effectively ionic-resonance) contributions to quantify hyperconjugation and polarization in drug-like molecules. In materials science, ionic-covalent resonance concepts underpin how chemists rationalize the partial ionic character of semiconductor bonds (Si–O, Ga–As), connecting Pauling's electronegativity-based ionicity scale to measured band gaps and piezoelectric response. In biochemistry, the resonance-stabilized peptide bond — where the C–N bond of an amide has partial double-bond character because of resonance with a charge-separated (zwitterionic, O⁻–C=N⁺) structure — is the same ionic-covalent VB logic applied to a polyatomic system, and it directly explains the planarity of the peptide backbone that underlies protein secondary structure. Finally, in spectroscopy and instrumentation, ionic-covalent VB wavefunctions provide the conceptual basis for interpreting charge-transfer excited states and avoided crossings in photodissociation dynamics, where a molecule can switch character from covalent to ionic along a bond-stretching coordinate — the excited-state analog of the ground-state mixing worked out here for H₂.

Ask the tutor: How does the same ionic-covalent mixing idea generalize when there are more than two possible resonance structures, as in benzene or the allyl radical?

Picture the two electrons in H₂ as billiard balls that are not actually confined to their "home" atoms. In the Heitler–London description, electron 1 rides on atom A while electron 2 rides on atom B (or vice versa, with the two arrangements added or subtracted according to the required spin symmetry). This is the covalent picture: each atom keeps one electron, and the bond arises from constructive interference between the "1-on-A, 2-on-B" and "1-on-B, 2-on-A" arrangements — the exchange stabilization that makes the singlet state bind and the triplet state repel. But nothing in the true Hamiltonian forbids both electrons visiting the same atom. Sometimes both electrons will be found near nucleus A, leaving nucleus B as a bare proton momentarily — an H⁻(A)···H⁺(B) configuration. Equally likely by symmetry, both electrons can be near B, giving H⁺(A)···H⁻(B). These are the two ionic structures. They are perfectly legitimate two-electron functions built from the same 1s orbitals; they are simply a different way of putting the two electrons into the two available atomic orbitals. The chemically essential point is that the true ground-state wavefunction of H₂ is not required to be purely covalent or purely ionic — it is whatever linear combination the variational principle selects as lowest in energy. Since the ionic structures are legal trial functions with the correct overall symmetry (space-symmetric, spin-singlet, just like the covalent term), the variation theorem guarantees that allowing them to mix in can only lower the energy relative to pure Heitler–London, never raise it. The physical content of "let them mix" is resonance: the true stationary state is a coherent superposition, not a statistical mixture, of covalent and ionic descriptions. Why should this recover missing binding energy? At the covalent-only level, the bonding electron pair is somewhat too rigidly localized, one electron per center, which overestimates electron-electron repulsion relative to what the electrons could achieve if they were allowed some additional flexibility in how they distribute density. Mixing in the ionic terms gives the wavefunction more variational freedom — more "knobs to turn" — and correlated motion between the electrons improves, lowering the energy. This is the same physical idea, in miniature, that later reappears as configuration interaction in molecular orbital language: the ionic VB structures are the direct analog of doubly-excited configurations mixed into a Hartree–Fock reference. At long range the story must reverse. As R → ∞, moving both electrons onto one atom to form H⁻ + H⁺ costs the ionization energy of hydrogen (13.6 eV) but only gains back the electron affinity of hydrogen (0.75 eV) — a net energetic disaster of roughly 12.8 eV. So the ionic structures must contribute less and less as the atoms separate, vanishing exactly at the dissociation limit. Near the equilibrium bond length, however, the two nuclei are close enough that the extra nuclear attraction the "ionic" electron pair feels from the second nucleus substantially offsets that cost, so a modest ionic admixture survives and helps bind the molecule. This distance-dependence of the ionic weight is precisely what a bond-polarity argument in ordinary organic and inorganic chemistry is really describing.

We build the wavefunction from the same two 1s atomic orbitals, $1s_A$ and $1s_B$, used in Heitler–London, but now form all four ways of placing two electrons on two orbitals that give the correct singlet spin symmetry and space symmetry ($A \leftrightarrow B$ interchange). The covalent (Heitler–London) spatial function, already familiar, is

where: $1s_A(i)$ places electron $i$ in the 1s orbital on nucleus A; the sum enforces indistinguishability and exchange symmetry appropriate to the singlet (space-symmetric) state.

Reading: one electron is always on A, the other always on B — no double occupancy of either atom is permitted. The two ionic spatial functions are

where: the first places both electrons on A (an H⁻ ion at A, bare proton at B), the second places both on B. Since A and B are chemically identical nuclei, the symmetric combination that respects the $A \leftrightarrow B$ molecular symmetry is the one that matters:

where: this single ionic structure already includes both charge-transfer directions with equal weight, as required by the D∞h symmetry of the homonuclear molecule. The improved trial wavefunction — the ionic–covalent VB wavefunction — is the variational mixture

where: $N$ is an overall normalization constant and $\lambda$ is a single real mixing parameter to be determined by minimizing the energy — this is the "resonance" between covalent and ionic descriptions. Multiplying out, $\Psi$ contains four one-electron-per-orbital terms, and one can equivalently write it as a linear combination of the four determinant-type spatial products:

where: setting $\lambda = 0$ recovers pure Heitler–London exactly; setting $\lambda = 1$ recovers (up to normalization) the fully delocalized product that is closely related to the simple LCAO-MO description of H₂ — this identity is the bridge to molecular orbital theory. The full multiplicative spin singlet function is $\Psi(1,2)\,\theta_{0,0}(1,2)$ with $\theta_{0,0} = \frac{1}{\sqrt2}\big[\alpha(1)\beta(2)-\alpha(2)\beta(1)\big]$, exactly as in Heitler–London; only the spatial part changes here, so we suppress spin bookkeeping below and work entirely with spatial integrals. To find the optimal $\lambda$ we treat this as a two-basis-function variational problem: the two "structures" $\psi_{\rm cov}$ and $\psi_{\rm ion}$ span a two-dimensional trial space, and we solve the associated generalized eigenvalue (secular) problem exactly, then identify $\lambda$ from the resulting eigenvector.

The determinant approach above is exact within the two-structure basis; it is useful, however, to see the same physics from a slightly different and more transparent angle — perturbation-theory reasoning that shows why the ionic admixture must vanish at large R and why it survives at bonding distances, without solving the full quadratic.

Example 1 — Foundational: writing the explicit six-term wavefunction and checking the R→∞ limit. Write out $\Psi = N[\psi_{\rm cov} + \lambda\,\psi_{\rm ion}]$ in full for general $\lambda$, and confirm that setting $\lambda = 0$ recovers Heitler–London while the physical requirement $\lambda(\infty)=0$ is consistent with molecular dissociation into neutral atoms. Expanding term by term,

Setting $\lambda=0$ leaves exactly

which is precisely the Heitler–London spatial function — confirming this construction is a genuine, verifiable generalization rather than a different theory. Now check the long-range limit physically: as $R\to\infty$, the orbitals $1s_A$ and $1s_B$ no longer overlap at all, so any term like $1s_A(1)1s_A(2)$ describes an electron distribution entirely localized at A while $1s_B(1)1s_B(2)$ localizes entirely at B. If $\lambda$ were finite in this limit, the wavefunction would describe a statistical mixture of "both electrons frozen on A, bare proton at B" and "both electrons frozen on B, bare proton at A" — i.e., a 50/50 mixture of H⁻ + H⁺ and H⁺ + H⁻, which is energetically absurd by 12.8 eV per the calculation above. Since the variational principle forbids any wavefunction from paying an unnecessary 12.8 eV energy penalty when a $\lambda=0$ alternative exists at the same $R$, the optimal $\lambda(R)$ is forced to zero as $R\to\infty$. This is the required limiting-case check: as R → ∞, λ(R) → 0 and $\Psi \to \psi_{\rm cov}$, recovering two neutral H atoms exactly as physically demanded.

Example 2 — Applied: estimating λ(Rₑ) from a simplified two-level secular problem with realistic numbers. To see the mixing mechanism with concrete (non-round) numbers, consider a deliberately simplified two-state model at $R = R_e = 1.40\ a_0$ (0.74 Å) in which we approximate the covalent and ionic structures as orthonormal ($S_{cc}=S_{ii}=1$, $S_{ci}=0$) — a simplification made purely for arithmetic transparency, not a claim about the true overlap. Suppose representative (illustrative, asymmetric) matrix elements at this distance are

These are chosen to be self-consistent with the qualitative facts established above: $H_{cc}$ (covalent) lies well below $H_{ii}$ (ionic), reflecting the ~12 eV intrinsic energy cost of charge separation, while $H_{ci}$ is a modest but nonzero coupling. With orthonormal basis functions the secular determinant collapses to the standard symmetric two-level form,

Expanding,

Substituting numbers: $H_{cc}+H_{ii} = -1.835$ hartree, and $H_{cc}H_{ii}-H_{ci}^2 = (-1.115)(-0.720) - (-0.130)^2 = 0.8028 - 0.0169 = 0.7859$ hartree².

giving $E_- = -1.1540$ hartree (lower, physically relevant root) and $E_+ = -0.6811$ hartree. Compare $E_-=-1.1540$ hartree to the pure-covalent value $H_{cc}=-1.115$ hartree: mixing lowers the energy by

a substantial extra stabilization purely from allowing ionic character, consistent with the qualitative claim that ionic–covalent mixing meaningfully improves on Heitler–London. Now extract $\lambda$ from the eigenvector equation $(H_{cc}-E_-)c_1 + H_{ci}c_2 = 0$:

So in this illustrative (orthonormal-basis) model, $\lambda \approx 0.30$, giving an ionic probability weight of

Real (non-orthogonal-basis) calculations at $R_e$ give somewhat smaller ionic weights, typically 2–6% depending on the orbital exponent used for $1s_A,1s_B$, because the neglected overlap $S_{ci}$ in this toy model is not actually zero and modifies the coupling; but the qualitative conclusion — a small, clearly nonzero ionic contribution at $R_e$, and a real energy-lowering of order 1 eV relative to Heitler–London alone — is exactly the physically correct trend and the right order of magnitude.

Example 3 — Exam/competition level: showing λ must be an odd/symmetric function under nuclear exchange, and using this to eliminate one apparent extra parameter. A student proposes to generalize the theory by allowing two independent mixing coefficients, writing

with $\lambda_A \ne \lambda_B$ in general. Show, using only the symmetry of the homonuclear molecule, that the variationally optimal solution must in fact have $\lambda_A = \lambda_B$, so that the single-parameter form used throughout this article, built from $\psi_{\rm ion}=\psi_{A^-B^+}+\psi_{A^+B^-}$, loses no generality. The molecular point group of H₂ (D∞h) contains the operation that exchanges nuclei A and B (in addition to inversion through the bond midpoint, which for a homonuclear diatomic has the same effect on labels). This exchange operation, call it $\hat P_{AB}$, is a genuine symmetry of the full electronic Hamiltonian $\hat H$: swapping the labels "A" and "B" of two identical protons cannot change any physical prediction, so $\hat P_{AB}\hat H \hat P_{AB}^{-1} = \hat H$, i.e., $\hat H$ commutes with $\hat P_{AB}$. Because $\hat P_{AB}$ commutes with $\hat H$, the true ground-state eigenfunction can always be chosen to be a simultaneous eigenfunction of $\hat P_{AB}$ (assuming no accidental degeneracy, which is the generic and physically correct case for the nondegenerate ground state). Applying $\hat P_{AB}$ to the trial function:

since $\hat P_{AB}$ swaps $\psi_{A^-B^+} \leftrightarrow \psi_{A^+B^-}$ (it literally swaps which nucleus is negatively/positively "ionic") while leaving $\psi_{\rm cov}$ invariant (relabeling $A\leftrightarrow B$ inside $\psi_{\rm cov}$ just reorders the two already-symmetric terms). For $\Psi_{\rm proposed}$ to be an eigenfunction of $\hat P_{AB}$ with eigenvalue $+1$ (the ground state of a homonuclear diatomic is symmetric under nuclear exchange, exactly as the Heitler–London singlet already is), we need

which, comparing coefficients of $\psi_{A^-B^+}$ and $\psi_{A^+B^-}$ term by term, requires

Therefore the variational minimum over the enlarged two-parameter space is guaranteed, by symmetry alone (no energy minimization arithmetic needed), to lie on the line $\lambda_A=\lambda_B\equiv\lambda$ — precisely recovering the single-parameter ansatz $\Psi = N[\psi_{\rm cov}+\lambda(\psi_{A^-B^+}+\psi_{A^+B^-})]$ used throughout. This is a clean, exam-style illustration of how molecular symmetry reduces the number of independent variational parameters before any energy is computed — a labor-saving trick used constantly in more elaborate resonance-structure VB calculations.

The ionic-covalent resonance idea in valence bond theory is inseparable from Linus Pauling, who took Heitler and London's 1927 covalent-only treatment of H₂ and, in a series of papers from 1928 through the early 1930s, systematically generalized it: adding ionic structures, formalizing the idea of resonance among multiple structures, and building the entire conceptual vocabulary — hybridization, resonance energy, electronegativity, percent ionic character — that organic and inorganic chemists still use today. His 1939 book The Nature of the Chemical Bond turned these mathematical results into the working language of structural chemistry, and it is largely through Pauling's exposition that "resonance between a covalent structure and an ionic structure" became a standard phrase rather than a technical curiosity buried in a physics journal.