Bonding and Antibonding Orbitals

The bonding/antibonding distinction is not a historical artifact confined to textbook diatomics — it is load-bearing in nearly every corner of modern chemistry and materials science. In transition-metal chemistry, Crystal Field Theory's d-orbital splitting is itself a consequence of metal d orbitals mixing (or failing to mix, by symmetry) with ligand orbitals into bonding, nonbonding, and antibonding combinations; the antibonding $e_g^*$ orbitals in octahedral complexes are precisely why high-spin versus low-spin configurations and Jahn–Teller distortions arise. In organometallic chemistry, back-bonding from filled metal d orbitals into empty ligand π orbitals (as in metal carbonyls, alkene, and N₂ complexes) is a direct antibonding-orbital acceptor phenomenon, and it is exactly why CO's own C≡O bond weakens (lower IR stretching frequency) upon coordination — electron density is being dumped into an antibonding orbital of the ligand. In semiconductor physics and materials chemistry, the valence and conduction bands of a solid are the many-atom generalization of bonding and antibonding MOs: as more atoms are added to a chain or lattice, the bonding combinations spread into the valence band and the antibonding combinations spread into the conduction band, with the band gap tracing directly back to the same $\Delta E_\pm$ splitting derived here for two orbitals. Photovoltaic and LED design, and the entire semiconductor band-structure toolkit, is this two-orbital problem scaled up to $10^{23}$ atoms. In pharmacology and photochemistry, HOMO–LUMO gaps (frontier molecular orbital theory) that govern reactivity, UV-Vis absorption, and photostability of drug molecules are bonding-HOMO/antibonding-LUMO transitions in disguise; photoexcitation routinely promotes an electron from a bonding π orbital to an antibonding π orbital, which is the electronic event underlying sunscreen chemistry, DNA photodamage (thymine dimer formation), and the design of photoswitchable drugs. In spectroscopy, vibrational frequency shifts upon oxidation/reduction or ligand binding (IR, Raman) are routinely interpreted via bond-order changes computed exactly as in the O₂/O₂⁻/O₂²⁻ worked example above — a direct, everyday diagnostic use of the bonding/antibonding framework in analytical and biochemical laboratories.

Ask the tutor: How does adding a third atomic orbital (as in an allyl or NO₂-type system) change the bonding/nonbonding/antibonding pattern compared to the simple two-orbital case derived here?

Start with two 1s orbitals, $\chi_A$ on atom A and $\chi_B$ on atom B, each an exponentially decaying, node-free function centered on its own nucleus. In isolation each has energy $\alpha$ (the atomic orbital energy, formally the Coulomb integral $H_{AA}$). When the atoms are brought together, the electron no longer "belongs" to A or B — the correct one-electron wavefunctions are the two independent linear combinations

The plus combination, $\psi_+$, adds the two atomic wavefunctions in phase. Because $\chi_A$ and $\chi_B$ are both positive in the internuclear region (for s orbitals), their sum builds up extra amplitude exactly where a bond needs it: between the nuclei. This is the bonding orbital, usually called $\sigma_{1s}$, or $\sigma_g$ in a centrosymmetric molecule. It has no node anywhere along the internuclear axis except possibly nodes inherited from the atomic orbitals themselves (1s has none). The minus combination, $\psi_-$, subtracts one atomic orbital from the other. Exactly on the plane bisecting the bond, $\chi_A$ and $\chi_B$ have equal magnitude and opposite sign in this combination, so the wavefunction is forced through zero: a nodal plane perpendicular to the bond axis, sitting between the nuclei. This is the antibonding orbital, $\sigma^*_{1s}$ or $\sigma_u^*$. Electron density is depleted between the nuclei and pushed into the outer lobes, away from the bond region, exposing bare nuclear charge to unshielded repulsion. Energetically, $\psi_+$ lies below $\alpha$ and $\psi_-$ lies above $\alpha$, and — crucially — the antibonding rise is larger than the bonding drop. This asymmetry comes directly from the overlap integral $S = \langle \chi_A|\chi_B\rangle$, which is positive for in-phase 1s orbitals; it appears in the denominator of the secular solution and skews the splitting upward. Filling both $\psi_+$ and $\psi_-$ with one electron each (as in a hypothetical He₂ analysis) therefore gives net repulsion, not a stable bond — this is precisely why He₂ does not exist as a bound ground-state molecule. The same in-phase/out-of-phase logic extends to p orbitals overlapping end-on (giving σ and σ from $p_z$ orbitals directed along the bond) and side-on (giving π and π from $p_x$ or $p_y$ orbitals perpendicular to the bond, each with a nodal plane containing the bond axis in addition to whatever nodes the antibonding combination adds). δ bonds, from face-on d-orbital overlap, carry two nodal planes containing the axis and appear only in metal–metal multiply-bonded systems.

The quantitative content behind the picture above is a 2×2 eigenvalue problem. Writing the trial MO as $\psi = c_A\chi_A + c_B\chi_B$, the variational method requires solving the secular equations built from three integrals:

where: $\alpha$ is the Coulomb integral (roughly the atomic orbital energy in the molecular environment, always negative); $\beta$ is the resonance (exchange) integral governing the strength of interaction (negative, and its magnitude grows with overlap); $S$ is the overlap integral, $0 \le S \le 1$ for normalized real orbitals with the same phase convention. The resulting orbital energies, from solving the 2×2 secular determinant, are

where: $E_+$ (bonding) takes the upper signs, $E_-$ (antibonding) takes the lower signs; both $\alpha$ and $\beta$ are negative numbers, so $E_+ < \alpha < E_-$. Reading it: because $\beta<0$, the numerator $\alpha+\beta$ (bonding) is more negative than $\alpha$ alone — stabilization. For the antibonding root the numerator $\alpha - \beta$ is less negative (destabilization), and additionally the denominator $(1-S)$ is smaller than $(1+S)$, which inflates $|E_- - \alpha|$ relative to $|E_+-\alpha|$. This is the algebraic origin of the asymmetric splitting seen in the figure above: antibonding destabilizes more than bonding stabilizes, by roughly $2S|\beta|$ in the simplest expansion. A second essential relationship is bond order, which converts orbital occupancy into a bond-strength index:

where: $n_{\text{bonding}}$ and $n_{\text{antibonding}}$ are the numbers of electrons in bonding and antibonding MOs respectively (nonbonding electrons are excluded from the count entirely). Reading it: each pair of electrons in a bonding MO contributes +1 to bond order; each pair in an antibonding MO cancels one unit. A bond order of zero (equal bonding and antibonding occupation) means no net stabilization — the molecule does not form, which is exactly the He₂ case.

Example 1 — Foundational: H₂⁺, the simplest two-center problem. H₂⁺ has exactly one electron and two protons, so it is the cleanest test of the bonding/antibonding split with no electron–electron repulsion to complicate matters. Using tabulated values appropriate to the equilibrium bond length $R = 106$ pm ($2.00\ a_0$), the relevant integrals (in units of the hydrogen atom ionization energy, $E_h = 27.2$ eV, or more conveniently in eV) are approximately

(these are the standard textbook LCAO integrals for H₂⁺ at its equilibrium separation; exact numerical values differ slightly between texts because $\alpha$ includes a nuclear-attraction correction beyond the bare 1s energy, but the qualitative pattern is robust). Apply the exact formula:

Wait — check the physical ordering: since $\beta<0$, $\alpha+\beta$ is more negative than $\alpha-\beta$; and dividing by the smaller denominator $(1-S)=0.41$ versus $(1+S)=1.59$ inflates the antibonding magnitude far more. Redo cleanly with signed arithmetic:

Because $-14.7 > -14.9$, this arithmetic shows $E_+$ actually sits above $\alpha$, which contradicts the bonding requirement. The resolution is that $\alpha$ itself, as tabulated in this convention, already includes the nuclear-attraction integral $j = \langle\chi_A|(-1/r_B)|\chi_A\rangle$, and the correct comparison energy is the isolated 1s energy, $-13.6$ eV, not the dressed $\alpha=-14.9$ eV. Comparing to the true reference:

This value is more negative than $E_+$, which is unphysical for an antibonding level; the discrepancy signals that the illustrative numbers above are only qualitatively representative, not a literal reproduction of the converged H₂⁺ integrals. The point that survives this arithmetic exercise, and the one to take away, is structural rather than numerical: whatever the precise integrals, the algebra of Step 4 above guarantees $\Delta E_- > \Delta E_+$ in magnitude whenever $S>0$, so the antibonding root is always pushed farther from the atomic reference than the bonding root is pulled down. The experimentally measured H₂⁺ bond dissociation energy, $2.65$ eV, with equilibrium bond length $106$ pm, is the correct benchmark: it confirms $\sigma_{1s}$ is bound by roughly this much relative to H + H⁺, consistent with a single electron occupying the bonding combination.

Example 2 — Applied: bond order and magnetism across the O₂/O₂⁺/O₂⁻/O₂²⁻ series. Using the standard period-2 MO filling order for O₂ ($\sigma_{2s}$, $\sigma_{2s}^*$, $\sigma_{2p}$, $\pi_{2p}$ (×2, degenerate), $\pi_{2p}^*$ (×2, degenerate), $\sigma_{2p}^*$), each species differs only in how many electrons occupy the antibonding $\pi^*$ pair, since all lower orbitals stay filled. Neutral O₂ has 12 valence electrons to place: 2 in $\sigma_{2s}$, 2 in $\sigma_{2s}^*$, 2 in $\sigma_{2p}$, 4 in the $\pi_{2p}$ pair, and the remaining 2 electrons go into the doubly degenerate $\pi_{2p}^*$ pair — by Hund's rule, one electron in each of the two π* orbitals, unpaired.

with two unpaired electrons — this is the textbook triumph of MO theory, correctly predicting that O₂ is paramagnetic, something Lewis structures cannot explain. Removing one electron to form O₂⁺ empties one π* electron:

shorter, stronger bond than neutral O₂ (observed: $112$ pm in O₂⁺ vs $121$ pm in O₂), consistent with removing an antibonding electron strengthening the bond. Adding one electron to form O₂⁻ (superoxide) puts three electrons in π*:

longer/weaker bond (observed $\approx 128$ pm), one unpaired electron (paramagnetic, radical anion — consistent with the reactivity of superoxide in biology). Adding two electrons to form O₂²⁻ (peroxide) fully fills π*:

single bond, diamagnetic, longest O–O distance in the series ($\approx 149$ pm, close to an O–O single bond in H₂O₂). This entire trend — bond length shortening monotonically as antibonding occupancy drops from 6→5→4→3 electrons — is a direct, checkable consequence of the antibonding orbitals' destabilizing, bond-weakening nodal structure.

Example 3 — Exam/competition level: symmetry-forbidden mixing and the σ/π crossover in period-2 homonuclear diatomics. A classic subtlety: for Li₂ through N₂, photoelectron spectroscopy and computed orbital orderings show $\sigma_{2p}$ lying above $\pi_{2p}$, the reverse of the naive energy ordering expected from a pure two-orbital 2p–2p treatment. Explain this using bonding/antibonding symmetry arguments, and predict where the crossover occurs. The resolution is that $\sigma_{2s}^*$ and $\sigma_{2p}$ share the same symmetry label ($\sigma_g$ for homonuclear diatomics, since both are symmetric under the full point group's $C_2$ rotation and inversion), so nothing in group theory forbids them from mixing — and when two orbitals of the same symmetry are close in energy, they must be treated with a full secular equation, not a naive two-orbital-per-symmetry picture. For light atoms (Li through N), the 2s and 2p atomic orbital energies are close enough ($\Delta E_{2s,2p}$ small for low nuclear charge) that this s–p mixing is strong: it pushes the $\sigma_{2s}$ orbital further down and the $\sigma_{2p}$ orbital further up, past the $\pi_{2p}$ level, giving the order

for B₂, C₂, N₂. For O₂, F₂, Ne₂, the 2s–2p energy gap has widened enough (increasing effective nuclear charge across the period) that s–p mixing becomes negligible, restoring the naive order with $\sigma_{2p}$ below $\pi_{2p}$:

The diagnostic consequence, testable without any computation, is magnetism: because $\sigma_{2p}$ sits above $\pi_{2p}$ for B₂ and C₂, filling electrons into the low-lying degenerate $\pi_{2p}$ pair before $\sigma_{2p}$ forces B₂ (6 valence e⁻ beyond the $\sigma_{2s}/\sigma_{2s}^*$ pair, i.e. 2 electrons in the frontier levels) to have one electron in each of the degenerate π orbitals — B₂ is experimentally paramagnetic, which is only explained correctly by the s–p-mixed ordering. Using the un-mixed ("late") ordering would incorrectly fill both frontier electrons into a nondegenerate $\sigma_{2p}$ and predict (wrongly) a diamagnetic B₂. This example shows that symmetry labels (which orbitals can mix) combined with quantitative near-degeneracy (whether they do mix strongly) are both required to get bonding/antibonding orderings right — pure nodal counting is necessary but not sufficient.

For the heteronuclear case, the qualitative picture needs one refinement: when $\alpha_A \ne \alpha_B$ (different electronegativities), solving the same secular determinant with unequal diagonal terms gives coefficients that are no longer equal in magnitude. The bonding MO has a larger coefficient on the more electronegative atom (lower $\alpha$, deeper AO), and the antibonding MO has its larger coefficient on the less electronegative atom. This is exactly the polar-covalent-bond picture: HF's bonding σ orbital is roughly 80% fluorine 2p character and 20% hydrogen 1s, consistent with the large electronegativity gap, while the antibonding σ* is mostly hydrogen 1s in character. The bonding/antibonding split survives the asymmetry, but "50:50 sharing" does not.

The mathematical machinery of bonding/antibonding orbitals grew out of the marriage between quantum mechanics and chemical valence theory in the late 1920s and 1930s, primarily through the independent efforts of Friedrich Hund and Robert Mulliken, who built the molecular orbital approach as a rival and complement to Heitler–London valence bond theory.