The simplest chemical bond: one electron shared by two protons — the hydrogen molecule-ion H₂⁺. Its molecular orbitals are the in-phase and out-of-phase combinations of a 1s orbital on each nucleus, Ψ = CA 1sA ± CB 1sB (LCAO): the bonding Σg1s and antibonding Σu*1s. For one electron the orbitals and their energies Eg(R), Eu(R) are computed exactly (atomic units, a0=1, e²=1, E1s=−½ ha; E includes the nuclear repulsion e²/R). The neutral molecule H₂ reuses the very same orbitals — you just add a second electron. Only the occupancy, bond order, and total energy then change; the orbital shapes and the one-electron levels do not. Drag R and toggle the electron count to see.
The two 1s atomic orbitals (equal energy E1s) mix into a bonding Σg1s below and an antibonding Σu*1s above. These MOs and their energies are fixed by the orbitals — not by how many electrons occupy them: H₂⁺ puts one electron in, H₂ two (the levels don't move). Fill from the bottom (aufbau + Pauli), and the count sets the bond order = ½(nbonding − nantibonding): each electron in a bonding MO is worth half a bond.
ψ(z) — the MO (solid) built from 1sA, 1sB (dashed)
|ψ(z)|² — charge density along the axis
The MO is axially symmetric about the bond (z) axis, so its boundary surface — the smallest region enclosing the chosen fraction of |ψ|² — is a surface of revolution. Bonding Σg: a single lobe wrapping both nuclei (charge shared in the bond). Antibonding Σu*: two lobes of opposite sign (blue ψ>0, red ψ<0) split by the nodal plane between the nuclei. Drag to rotate.
ψ — signed boundary surface (blue: ψ>0, red: ψ<0)
|ψ|² — density boundary surface (same enclosed %)
Binding energy relative to the separated atoms (eV). H₂⁺ Σ_g and H₂ (MO) both develop a well; the extra bonding electron makes the H₂ well deeper and shorter (bond order ½ → 1) — the H₂-vs-H₂⁺ comparison made visible. The occupancy toggle highlights the active species; the Σ_u* antibonding curve is purely repulsive. Two flaws of plain MO are visible: it underbinds (H₂ well sits above the dotted experimental De), and the green H₂ curve rises above the separated atoms at large R — the spurious 50% ionic (H⁺H⁻) dissociation that neglected electron correlation leaves in. The computed H₂ well (≈0.85 Å, 2.70 eV) matches the course's approximate MO result.
binding energy (eV) vs R (a₀) — blue: H₂⁺ Σ_g, red: Σ_u*, green: H₂ MO
the H₂⁺ integrals that build Eg,u = E1s + (J±K)/(1±S): overlap S, Coulomb J, resonance K — K<0 makes the bond
The equilibrium bond length Re and dissociation energy De from the computed curves above, placed next to the measured values — for both the ion (H₂⁺) and the molecule (H₂), so you can compare theory with experiment directly.