Hybrid orbitals: shape from superposition
Why can carbon form four bonds pointing towards a tetrahedron? Start with spherical s and two-lobed p orbitals, then build directional orbitals on the same atom.
01Add a wave, not an electron
An orbital describes an electron’s possible spatial distribution, not a path it travels along. Its wave amplitude ψ can be positive or negative. The probability density is |ψ|² and is never negative. Same-sign contributions reinforce; opposite signs cancel.
Final formula at 100%. During construction, the p coefficient is multiplied by t and the whole orbital is renormalized. p with a direction label means the corresponding combination of pₓ, pᵧ and p𝓏.
02One input basis, an equally large output basis
We are choosing a new set of orbitals to describe the same space. No extra orbital or electron is created. Select “All orbitals” above to see each independent orbital surface together; this is an overlay, not another sum of their amplitudes. A higher density threshold separates the main lobes; the small back lobes fall below that threshold.
sp · 180°
1 s + 1 p → 2 hybrid orbitals, pointing in opposite directions. Two p orbitals remain unhybridized. In the usual model of ethyne, they allow two perpendicular π bonds.
sp² · 120°
1 s + 2 p → 3 coplanar hybrid orbitals. One p orbital remains perpendicular to the plane. In ethene, side-on overlap of those p orbitals supplies the π bond.
sp³ · 109.47°
1 s + 3 p → 4 tetrahedral hybrid orbitals. Methane uses four equivalent bonding directions. Real bond angles can depart from this ideal when lone pairs or different substituents are present.
Model scope: normalized Gaussian s/p functions illustrate angular shape and interference; atomic radial nodes are omitted. The surfaces have equal |ψ|², not a sharp electron boundary. Hybridization is a bonding model, not an ab initio prediction of molecular geometry.
Check your understanding: does sp³ create four new electrons?
No. Four original orbitals are replaced by four linear combinations. Electron number is unchanged; occupation depends on the atom and its bonding environment.