No figure on this site is a drawing that was made once and saved. Each one is a function:
it takes parameters and returns SVG, so the same generator produces the p4 plate and the
p6m plate without either being redrawn.
That is the reason the collection can keep growing without the illustrations drifting apart.
A generator is written once, checked once, and every essay that calls it inherits the same
line weights, the same colour roles, and the same behaviour in dark mode. There are
15 of them so far.
angle-tableThe angles each arrangement gives Every distinct angle subtended at the centre, for each number of sites, measured off the minimised arrangement. Five is the case with three distinct angles, which is the arithmetic signature of sites that are not all equivalent. sites shape angles measured 2 linear 180.0°×1 3 trigonal planar 120.0°×3 4 tetrahedral 109.5°×6 5 trigonal bipyramidal 90.0°×6 120.0°×3 180.0°×1 6 octahedral 90.0°×12 180.0°×3 each row is a separate minimisation
contour-choice
enclosed
five-sitesFive sites are not five of a kind The minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey. axial — 2 sites neighbours at 90°, 90°, 90°, 180° equatorial — 3 sites neighbours at 90°, 90°, 120°, 120° the two are not equivalent neighbour angles measured, not assumed 5 sites
hybridsp3 hybrids The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates. sp3 109.471° between every pair 4 hybrids worst off-diagonal 0e+0 an orthogonal transformation of the atomic orbitals one electron
hybrid-matrixThe sp3 transformation Each row is one hybrid, written in the basis of the atomic orbitals it is made from. The rows are orthonormal, so the matrix is a rotation — and a rotation of a basis changes no observable quantity whatever. s pₓ p_y p_z hybrid 1 0.5000 0.5000 0.5000 0.5000 hybrid 2 0.5000 0.5000 -0.5000 -0.5000 hybrid 3 0.5000 -0.5000 0.5000 -0.5000 hybrid 4 0.5000 -0.5000 -0.5000 0.5000 every row normalised and every pair orthogonal to 0e+0 a change of basis, and nothing more sp3
mo-levelsSplitting goes with overlap For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero. 1s-1s σ overlap S = 0.3900 2pz-2pz σ overlap S = 0.4585 2px-2px π overlap S = 0.7529 1s-2px symmetry forbids it S = 0 exactly no splitting overlaps computed at 2.8 bohr, levels in proportion one electron
moleculewater — C2v The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves. H H O C2v principal axis C2 2 mirror planes no inversion centre may be polar cannot be chiral group recovered from the coordinates 3 atoms
orbital
orbital-set
overlap
overlap-curve
polar-chiralWhat the group settles For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds. molecule group may be polar may be chiral water C2v yes no 2 σ carbon dioxide D∞h no no has i ammonia C3v yes no 3 σ methane Td no no 6 σ boron trifluoride D3h no no 4 σ hydrogen peroxide C2 yes yes — bromochlorofluoromethane C1 yes yes — both columns derived from the symbol, not from the bonds
radial
vsepr4 sites, minimised The arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed. tetrahedral 109.47° × 6 repulsion minimised, angles measured off the result 4 sites