Overview--Missions 0/6FPS 0

Drag to orbit, wheel or pinch to zoom, click a station (or a button on the right) to focus it. Model units: eV, angstrom, kJ/mol.

Bond Landscape: Orbitals, Polarity and the Energy Hill

One gallery, seven stations: orbitals, the H–H bond, water, ammonia, an ionic or metallic lattice, benzene and a reaction energy hill.

Station

Missions

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom; click a station in the scene or use the buttons above to focus it.
Each station has its own sliders on the right; every number shown is computed from the formulas in "The science".
Reset the view with Overview. Quality Low uses lighter meshes and a lower pixel ratio (also ?q=low).

The science

Morse bond. V(r) = D(1 − e^{−β(r−r_e)})² − D with D = 4.747 eV, r_e = 0.7414 Å, β = 1.943 Å⁻¹ for H₂. The harmonic wavenumber is ν̃ = (β / 2πc) √(2D/μ) (≈ 4397 cm⁻¹ against 4401 measured). The curve is anharmonic: steep on the compressed side, a plateau when stretched (bond broken at about 3 Å).

Polarity. Pauling ionic character f = 1 − exp(−ΔEN²/4) gives a partial charge q = f·e on each H; the dipole of the bent molecule is μ = 2 q d cos(θ/2) (point-charge model: water 1.8 D against 1.85 D measured; at 180° it cancels, like CO₂). The coloured cloud is the electrostatic potential of the point charges, φ = Σ q_i e²/(4πε₀ |r − r_i|), drawn on a density-like surface: red negative, blue positive. Bending energy is a harmonic model around 104.5°; VSEPR: 4 electron domains (2 bonds + 2 lone pairs), the lone pairs squeeze the angle below 109.5°. Ammonia: bond-dipole sum only, the lone pair adds the rest (1.47 D measured).

Ionic lattice. The Madelung constant of rock salt is summed by the Evjen cube method (M = 1.7476). Born–Mayer energy per mole: U(r) = N_A(M e²/4πε₀ r − B/rⁿ), n = 9.1, B set so that the minimum sits at r₀ = 2.82 Å → 766 kJ/mol (Born–Haber experiment: 787). Metallic mode shows a free-electron "sea" that drifts under a field.

Benzene. Hückel π model on a 6-ring: eigenvalues α + xβ with x = 2, 1, 1, −1, −1, −2 (matrix diagonalised numerically). Three localised double bonds give 6|β|, the ring 8|β|: delocalisation energy 2|β| ≈ 150 kJ/mol for |β| ≈ 75 kJ/mol (the slider interpolates, it is not a quantum calculation).

Energy hill. Collinear A + BC → AB + C on a London–Eyring–Polanyi–Sato (LEPS) surface built from three Morse curves with a Sato parameter S that weakens the anti-bonding triplet term (barrier ≈ 0.42 eV for S ≈ 0.135). Trajectories: kinetic energy T = (m/3)(ṙ₁² + ṙ₁ṙ₂ + ṙ₂²), integrated with velocity Verlet. Classical, zero-point energy and tunnelling ignored.

Simplifications. Point charges, model force constants, no electron-density calculation; shapes of the orbital lobes are schematic angular parts. Atom and bond meshes built with Blender, scene by Three.js. Educational use only.