Why do molecules have shapes? VSEPR theory
The valence electrons around the central atom A form pairs: bonding pairs (X — they join A to another atom) and lone pairs (E — they belong to A alone). The pairs carry the same charge, so they repel and move as far apart as possible. The total number of pairs, the steric number, sets the geometry, while lone pairs are “invisible” in the molecule's shape — only the atom positions show. A double or triple bond counts as one pair.
AXnEm ⟶ steric number = n + m · Erep = Σi<j 1rij → minimum
The builder looks for the minimum of this energy for points on a sphere (gradient descent). Lone pairs repel a little more strongly, so in a trigonal bipyramid they take the equatorial positions (they have fewer neighbors at 90° there). In reality this stronger repulsion squeezes the angles: NH3 has 106.7° and H2O 104.5° instead of the ideal 109.5°. Click a cell in the table to build that shape.
Measuring distance and angle
Every atom has coordinates (x, y, z) in picometers (1 pm = 10−12 m). The distance is the Pythagorean theorem in 3D:
d = √(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2
The angle at the middle atom B comes from the dot product of the vectors u = A − B and v = C − B:
cos θ = u · v|u| · |v|
A triple bond is the shortest: C≡C is 120 pm, C=C 134 pm and C–C 154 pm.
Polarity
The atom with the higher electronegativity χ pulls the shared electrons more strongly and gets a partial charge δ−, its partner δ+. Here we count bonds with Δχ ≥ 0.4 as polar. You will find the χ values in the periodic table.
μ = q · d · μtotal = Σ μi · 1 D ≈ 3.336 · 10−30 C·m
Bond moments add up like vectors. In CO2, BF3 or SF6 symmetry makes the sum zero — the molecule is nonpolar even though its bonds are polar. Arrows are drawn the chemists' way: from δ+ to δ−. Simplification: the direction of the resultant comes from the sum of the bonds (without lone pairs), and the arrow length matches the measured dipole moment when it is known.
How is the 3D image made?
The renderer rotates the coordinates with a rotation matrix and then projects them in perspective — farther atoms look smaller:
x′ = f · xf − z y′ = f · yf − z
Atoms and bonds are sorted from farthest to nearest and painted in order (the painter's algorithm). The spheres are shaded with a radial gradient, with a highlight from light at the upper left.
Two kinds of models
Ball-and-stick shows angles and bond types well (double, triple, delocalized — dashed). The space-filling model draws atoms at the scale of their van der Waals radii — roughly how the molecule “looks” to its neighbors. For NaCl we use ionic radii (Na⁺ 102 pm, Cl⁻ 181 pm).
The colors follow the CPK convention: H white, C gray, N blue, O red, F and Cl green, S yellow, P orange. The geometries of small molecules come from gas-phase measurements; the rest were computed with a force field (described next to this).
Where do the 3D coordinates come from? A force field
We stored most molecules as a graph (who is bonded to whom — SMILES notation), and a simple molecular mechanics model worked out the shape. Atoms are balls, bonds are springs with an ideal length, and angles and twists have their favorite values:
E = Σ kr(r − r0)2 + Σ kθ(θ − θ0)2 + Σ kφ[1 − cos(nφ − φ0)] + Erep.
The computer moves the atoms downhill on the energy landscape (the FIRE algorithm) until the forces vanish. The angles come from hybridization (sp³ 109.5°, sp² 120°, sp 180°), benzene rings come out flat, and cyclohexane and glucose in the chair form. It is a simplified model: accurate to about 1–3 pm and a few degrees. In the editor the same algorithm runs live in your browser.
Chirality and the CIP rules
A carbon atom with four different substituents is a chirality center: the molecule and its mirror image are two different enantiomers, like your left and right hands. To name them, we rank the substituents by the Cahn–Ingold–Prelog rules: higher atomic number = higher priority; on a tie we compare the next “spheres” of atoms, and a double bond counts as two single ones.
1 → 2 → 3 clockwise (4 at the back) ⟹ R · counterclockwise ⟹ S
In the program the sign of the scalar triple product of the vectors from the center to substituents 1, 2, 3 decides: v1 · (v2 × v3) < 0 means R.
Balancing equations is algebra
Each element gives one equation: the number of its atoms on the left = the number on the right. For CH4 + O2 → CO2 + H2O with unknowns a, b, c, d:
C: a = c · H: 4a = 2d · O: 2b = 2c + d ⟹ 1 : 2 : 1 : 2
The program writes this as a matrix A·x = 0, solves it by Gaussian elimination on fractions and looks for the smallest positive whole numbers. If there are several independent solutions, it is really several reactions at once.
Forces between molecules and bond energies
A hydrogen bond (about 20 kJ/mol) forms when an H atom bonded to F, O or N attracts a lone pair on a neighboring molecule. London forces come from momentary dipoles in electron clouds; they are weak and fall off quickly with distance (E ∝ −1/r6), but grow with the size of the molecule.
ΔH ≈ Σ E(bonds broken) − Σ E(bonds formed)
Breaking bonds takes energy, forming them releases it. Average energies from tables give an estimate; exact values are measured in a calorimeter.