Orbital shapes
Each orbital is a hydrogen-like wavefunction (1s, 2s, 2p, 3s, 3p) with the nuclear charge replaced by Slater's effective nuclear charge, Zeff, for that shell. The surfaces are true isosurfaces of the wavefunction: the program evaluates it on a 3D grid and draws the contour that encloses 90% of the probability. Colour shows the sign (phase). Sizes are to scale within an atom, so core orbitals really are that small. Real many-electron orbitals differ in detail, but the shapes, nodes and size trends are the same.
Hybrids and bonds
Hybrids are built as cos β·2s + sin β·2p, with tan²β = 3 (sp³), 2 (sp²) or 1 (sp). In Bonding mode the carbons start out already hybridized; the Hybridization mode shows the 2s and 2p orbitals mixing. In the bonding and hybridization pictures the 2p lobes are deliberately drawn slimmer and longer than real p orbitals (stretched about 1.5× along their axis and narrowed by sharpening the angular part) so p orbitals and hybrids are easy to tell apart from spherical s, and the yellow surfaces mark where in-phase lobes overlap (the product of the two orbitals). Bonding orbitals are the in-phase sum of one orbital on each atom (head-on for σ, side by side for π), and antibonding orbitals are the out-of-phase difference. All of them are drawn at the same contour value as a single carbon 2p orbital, so sizes can be compared. Only the two orbitals that face each other are combined. The other hybrids, which would bond to hydrogen, can be shown faintly, and the C–H bonds themselves are left out to keep the picture one atom to one atom. The overlap S is calculated numerically on the grid.
Hybridization mode
One slider drives everything at once: the hydrogens move in along the bond directions, the central atom's 2s and 2p orbitals mix, and each hybrid merges with an H 1s orbital into a σ bond. The mixing is a smooth rotation of the orbital set (a Cayley-form orthogonal path from the pure 2s and 2p orbitals to the hybrids), so every in-between orbital is still a normalized orbital and the set stays orthogonal. In the Molecular orbitals picture each bond forms directly from an unhybridized 2s or 2p orbital and the H 1s group orbital of matching symmetry, as a two-level mixture whose coupling grows as the hydrogens approach; surfaces there enclose 85% of each MO. In the Hybrid orbitals picture surfaces are drawn at a contour of 0.2 Å−3/2 for carbon, scaled in proportion to Zeff for B and Be (which are more diffuse), and each H 1s (molecular exponent ζ = 1.24) is drawn at a tighter contour, a 0.6 Å sphere, as in textbook figures; the σ bond is drawn as the in-phase sum of the hybrid and that hydrogen. At the true common contour an H 1s sphere is almost as wide as the bond is long and hides the hybrid lobes. Surfaces in the Hybridization and Bonding modes are drawn in a flat, outlined “illustrated” style; in Bonding mode the yellow overlap highlight is drawn on top of the bond so it stays visible. MO energies are schematic: coupling strengths are chosen to give the right ordering, and the hybrid-picture bond energy is set to the average of the bonding MOs so the two pictures have the same total. Bond lengths: C–H 1.09, B–H 1.19, Be–H 1.33 Å. Be valence energies: 2s −9.3, 2p −5.8 eV (Zeff 1.95).
| Element | Valence Zeff | Valence orbital energies |
|---|
| H | 1.00 | 1s −13.6 eV |
| B | 2.60 | 2s −14.0 · 2p −8.3 eV |
| C | 3.25 | 2s −19.4 · 2p −10.7 eV |
| N | 3.90 | 2s −25.6 · 2p −13.2 eV |
| O | 4.55 | 2s −32.4 · 2p −15.9 eV |
| Cl | 6.10 | 3s −25.2 · 3p −13.7 eV |
Core energies are approximate 1s (and Cl 2s/2p) binding energies, drawn compressed below the break. In the MO diagrams, hybrid energies are weighted averages of 2s and 2p. The σ/σ* and π/π* splittings are schematic, chosen to show order and relative size (σ splits more than π), not computed values. Bond lengths: H–H 0.74, C–C 1.54, C=C 1.34, C≡C 1.20 Å. Sources: J. C. Slater,
Phys. Rev. 1930, 36, 57 (shielding rules); valence orbital potential energies as tabulated in G. L. Miessler, P. J. Fischer & D. A. Tarr,
Inorganic Chemistry (Pearson); bond energies are standard average values.