Valence Bond versus Molecular Orbital Theory

Modern electronic structure software still descends from both lineages, and the choice of framework is far from academic. Hartree–Fock and Kohn–Sham density functional theory, the workhorses of routine computational chemistry in drug discovery and materials screening, are direct descendants of the molecular orbital picture; their single-determinant (or single-Kohn–Sham-determinant) starting point inherits exactly the same dissociation pathology analyzed here, which is why naive DFT bond-breaking curves for radicals, transition-metal spin states, and homolytic cleavage reactions can be qualitatively wrong unless multireference or broken-symmetry corrections are applied. Anyone modeling a reaction that breaks a covalent bond homolytically — radical polymerization initiation, C–H activation transition states, or photodissociation — needs to know exactly the static-correlation error this article derives, because it silently corrupts single-reference calculations exactly in the bond-breaking region chemists care about most. Valence bond methods, meanwhile, survive both as a pedagogical language (curly arrows, resonance structures, orbital hybridization) and as active modern computational tools: spin-coupled valence bond theory, breathing-orbital VB, and VB self-consistent field methods are used today to dissect reaction mechanisms into explicit covalent and ionic (or diabatic) contributions, particularly useful for charge-transfer states in photochemistry, diradical intermediates in pericyclic reactions, and describing bond-breaking without the artificial symmetry-breaking that plagues restricted MO methods. Multiconfigurational methods such as CASSCF and its variants are, in the language of this article, simply full CI within a chosen active space — they are the rigorous unification point where "VB versus MO" stops being a meaningful distinction, because both limits are contained inside the same configuration expansion, weighted by whatever the variational principle demands at each geometry. Understanding the H₂ case in full — the 50:50 MO ratio, the pure-covalent VB limit, and the CI bridge between them — is the cleanest possible entry point into why every serious quantum chemistry package needs, in one guise or another, a multireference option.

Ask the tutor: Can you show me how the 2×2 VB-CI matrix elements H_cc, H_ii, and H_ci are actually computed from the Coulomb, exchange, and overlap integrals over 1s Slater orbitals?

Consider H₂ with nuclei $a$ and $b$, each carrying a 1s atomic orbital, $\chi_a$ and $\chi_b$, and two electrons labeled 1 and 2. Any acceptable two-electron spatial wavefunction for this problem, expanded in just these two atomic orbitals, must be some combination of exactly four possible electron arrangements: electron 1 near $a$ and electron 2 near $b$ (or vice versa) — the covalent arrangements — and both electrons near the same nucleus, $a$ or $b$ — the ionic arrangements corresponding to H⁻H⁺ and H⁺H⁻. Simple MO theory and simple VB theory are two different, and differently wrong, ways of weighting these four arrangements. The molecular orbital route builds one delocalized bonding orbital,

where $S = \langle \chi_a|\chi_b\rangle$ is the atomic-orbital overlap integral, and then puts both electrons into it with opposite spins: $\Psi_{\text{MO}} = \sigma_g(1)\sigma_g(2)\,[\alpha(1)\beta(2)-\beta(1)\alpha(2)]/\sqrt2$. Because $\sigma_g$ itself is delocalized over both centers, squaring it to build the two-electron density automatically generates covalent and ionic terms in a fixed, geometry-independent ratio. This is the root of the dissociation catastrophe: no matter how far apart the atoms sit, the orbital used to describe each electron is still "half on $a$, half on $b$," so the wavefunction insists on retaining 50% ionic character forever. The valence bond route instead builds the wavefunction directly from the covalent picture: electron 1 in $\chi_a$ and electron 2 in $\chi_b$, plus the electron-exchanged term required by indistinguishability, giving the Heitler–London function

This function never lets both electrons sit on the same nucleus — it is purely covalent by construction — so it dissociates flawlessly to two neutral, spin-paired hydrogen atoms. Its failure is the opposite one: near the equilibrium distance, where quantum mechanics really does want a modest admixture of ionic character to lower the energy further, simple VB has no ionic terms available to mix in, so it underestimates the binding energy and gets a bond length slightly too long.

At the equilibrium bond distance, the exact ratio of ionic to covalent weight is small but nonzero — chemical intuition and experiment agree that H₂ is overwhelmingly covalent but slightly "ionic-resonance-stabilized." Simple MO theory forces a 50:50 ratio that is wildly too ionic; simple VB theory forces a 0% ionic ratio that is too covalent. Both are wrong in exactly the direction their construction predicts, and the size of each error can be computed explicitly, which is the point of the two derivations below.

The whole comparison rests on writing every wavefunction in the same four-term atomic-orbital basis and reading off the covalent and ionic coefficients directly.

where: $\sigma_g = (\chi_a+\chi_b)/\sqrt{2(1+S)}$ is the normalized bonding MO, and the spin singlet factor is suppressed since it is common to both theories. Reading: expanding the square of the sum $(\chi_a+\chi_b)$ over both electrons produces all four arrangements — two covalent, two ionic — with identical coefficients. The MO wavefunction cannot distinguish "ionic" from "covalent" weighting because it was never given the freedom to do so; the ratio is baked in by the algebra of squaring a delocalized orbital.

where: this is the Heitler–London function, containing only the two covalent arrangements with equal weight and no ionic terms at all. Reading: VB theory imposes the opposite constraint — ionic terms are simply absent — which is exact only in the limit $R \to \infty$, where the true wavefunction genuinely has zero ionic weight (moving an electron from one atom to the other costs a finite ionization/affinity energy that becomes infinitely expensive relative to zero coupling at infinite separation).

where: $c_{\text{cov}}$ and $c_{\text{ion}}$ are variationally optimized mixing coefficients found by diagonalizing the $2\times2$ Hamiltonian matrix in the {covalent, ionic} basis, and $\Psi_{\text{ion}}$ is the normalized symmetric combination of the two ionic structures. Reading: this is the full valence bond CI wavefunction — VB theory's route to the exact answer. It contains the same four determinants as $\Psi_{\text{MO}}$ but with a variable, $R$-dependent ratio $c_{\text{ion}}/c_{\text{cov}}$ instead of the fixed ratio 1 that simple MO theory imposes.

where: $\sigma_u$ is the antibonding MO, $c_1,c_2$ are CI coefficients, and the minus sign convention keeps both terms real and the ionic content controllable. Reading: this is MO theory's route to the exact answer — mixing in the doubly-excited configuration $(\sigma_u)^2$ subtracts away ionic character as $c_2/c_1$ grows, precisely canceling the excess ionic weight that plain $(\sigma_g)^2$ carries. At $R\to\infty$, $c_1 \to c_2$ in magnitude and the ionic terms cancel exactly, restoring the correct dissociation limit.

Example 1 — Foundational: confirming the 50:50 MO ratio numerically at a specific, non-symmetric overlap. Take a deliberately non-round overlap value, $S = 0.4637$ (a realistic value near the H₂ equilibrium distance for Slater 1s orbitals with an effective exponent), rather than a convenient round number, to make sure the 50:50 result isn't an artifact of a special choice. The normalization constant is

Expanding $\Psi_{\text{MO}} = N^2[\chi_a(1)\chi_a(2)+\chi_b(1)\chi_b(2)+\chi_a(1)\chi_b(2)+\chi_b(1)\chi_a(2)]$, each of the four bracketed terms carries coefficient $N^2 = 0.34160$ exactly. Grouping,

The ratio is

regardless of the specific numerical value of $S$ chosen — confirming algebraically that the 1:1 covalent–ionic split of simple MO theory does not depend on distance, basis exponent, or any other parameter; it is a structural feature of building the wavefunction from one delocalized product orbital.

Limiting check: setting $S\to 0$ (the large-$R$ limit) gives $N^2 \to 1/2$, covalent weight $\to 1$, ionic weight $\to 1$, ratio still exactly 1 — the pathology persists undiminished all the way to infinite separation, which is precisely the dissociation catastrophe stated in the overview.

Example 2 — Applied: the VB-CI stabilization energy for asymmetric, off-round trial matrix elements. Suppose (units of eV, chosen to be non-symmetric and non-round so the formula is genuinely exercised) that at some intermediate H–H distance the diagonal and off-diagonal VB matrix elements are

(These are illustrative numbers of the right order of magnitude and sign pattern — $H_{\text{ii}} > H_{\text{cc}}$ because the ionic structure is higher in energy — used to exercise the formula, not literature-tabulated H₂ integrals.) Compute the average and half-difference:

The discriminant:

Ground-state VB-CI energy:

Compare to the pure covalent (simple VB) value $H_{\text{cc}} = -31.7$ eV: the ionic admixture lowers the energy by

a genuine, computable stabilization purely from allowing ionic resonance — confirming Step 4 of the second derivation ($E_- < H_{\text{cc}}$) with concrete asymmetric numbers. The mixing angle:

So $\cos^2\theta = 0.9098$ and $\sin^2\theta = 0.0902$: the exact-in-this-basis ground state is about 91.0% covalent, 9.0% ionic — a small but non-negligible ionic weight, exactly the qualitative pattern chemists invoke when they say H₂ is "mostly covalent with a little ionic resonance."

Limiting check: if $H_{\text{ci}} \to 0$ (structures decoupled — e.g., orbitals on infinitely separated atoms), then $\tan 2\theta \to 0$, so $\theta \to 0$, $E_- \to H_{\text{cc}}$: the ionic admixture vanishes and VB-CI collapses back exactly to simple Heitler–London, as it must at dissociation.

Example 3 — Exam/competition level: showing MO-CI reproduces the VB-CI ground state exactly (same numbers, both routes). This example demonstrates the central claim of the article — that full CI on the MO side and full CI on the VB side give the same wavefunction — using the same non-symmetric matrix elements framework, now built from the MO configurations $(\sigma_g)^2$ and $(\sigma_u)^2$. The two-configuration MO wavefunction is

Expand each configuration in atomic orbitals. From the earlier derivation,

By the same method with $\sigma_u = M(\chi_a-\chi_b)$, $M^2 = 1/[2(1-S)]$,

So the combination, written schematically in un-normalized covalent/ionic amplitudes (dropping the constants $N^2,M^2$ for the moment and tracking only relative signs and weights):

Matching coefficients to the VB-CI ground state $\Psi_- \propto \cos\theta\,\Psi_{\text{cov}} + \sin\theta\,\Psi_{\text{ion}}$ from Example 2 ($\cos\theta = 0.9539$, $\sin\theta = 0.3003$, taking square roots of the population figures found there), the condition for the two routes to agree is

Solving for the ratio $c_2/c_1$ (with $S=0.4637$ as before, so $N^2 = 0.3416$ and $M^2 = 1/[2(1-0.4637)] = 1/1.0726 = 0.9323$):

Let $x = c_2/c_1$. Then

So a modest admixture of the doubly-excited $(\sigma_u)^2$ configuration, with $c_2/c_1 = -0.1908$ (about 19% of the amplitude of the ground configuration, entering with the sign that removes ionic character since $(\sigma_u)^2$ carries a minus sign on the covalent terms), reproduces exactly the same 91:9 covalent:ionic population that the VB route obtained by explicitly mixing in an ionic structure with $\tan 2\theta = 0.6992$. This is the demonstration promised throughout: MO-CI with one excited determinant and VB-CI with one ionic structure are two parametrizations of the identical two-configuration wavefunction.

Limiting check: at $R\to\infty$, $S\to 0$ so $N^2\to M^2\to 1/2$, and the physical requirement that ionic character vanish entirely forces $c_1N^2 = c_2M^2$, i.e., $c_2/c_1 \to 1$ — the well-known result that full CI on the MO side needs equal weighting of $(\sigma_g)^2$ and $(\sigma_u)^2$ at dissociation to cancel the ionic terms completely, recovering the correct Heitler–London limit from the MO side.

Walter Heitler and Fritz London, working together in Zürich in 1927, produced the calculation that both launched valence bond theory and, in a real sense, launched quantum chemistry as a discipline. Their treatment of H₂ was the first application of the new quantum mechanics to a genuine chemical bond, and it succeeded in explaining, from first principles, why two neutral hydrogen atoms attract each other at all — a question classical physics could not answer since two neutral, spherically symmetric atoms have no obvious source of attraction. Heitler and London showed the attraction was a purely quantum mechanical exchange effect tied to the symmetry requirements of the two-electron wavefunction under spin pairing. The molecular orbital alternative grew up alongside it through the work of Robert Mulliken and Friedrich Hund in the late 1920s and early 1930s, framed around delocalized orbitals built from linear combinations of atomic orbitals and filled according to an aufbau-like scheme. For several decades the two camps were genuinely in competition — Linus Pauling championed and extended VB theory into a general chemical language (hybridization, resonance, electronegativity), while Mulliken's MO approach proved better suited to spectroscopy, symmetry classification, and eventually to computational implementation via Roothaan's equations. The reconciliation described in this article — that both are truncations of the same exact expansion — became fully explicit only once configuration interaction and multiconfigurational self-consistent field methods matured in the mid-20th century, showing that "VB versus MO" was never really a disagreement about the physics, only about which basis to truncate first.