Illustrated essays on molecular shape and bonding: orbitals drawn at a contour that encloses a stated probability, overlaps that symmetry forbids coming out exactly zero, and what a point group settles on its own. https://www.molecular-geometry.com/ 2026-08-04T21:12:05.469Z https://www.molecular-geometry.com/essays/what-an-orbital-is/ 2026-08-04T21:12:05.469Z

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.

https://www.molecular-geometry.com/essays/vsepr-computed/ 2026-08-04T21:12:05.469Z

The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down.

https://www.molecular-geometry.com/essays/overlap-decides/ 2026-08-04T21:12:05.469Z

Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it.

https://www.molecular-geometry.com/essays/point-groups-from-coordinates/ 2026-08-04T21:12:05.469Z

A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.

https://www.molecular-geometry.com/essays/hypervalency-without-d-orbitals/ 2026-08-04T21:12:05.469Z

Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre four-electron, and the textbook account has been known to be wrong for fifty years.

https://www.molecular-geometry.com/essays/hybridisation-does-not-explain/ 2026-08-04T21:12:05.469Z

Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.

https://www.molecular-geometry.com/essays/say-what-it-encloses/ 2026-08-04T21:12:05.469Z

An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith.

https://www.molecular-geometry.com/essays/five-sites-are-not-alike/ 2026-08-04T21:12:05.469Z

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.

https://www.molecular-geometry.com/essays/exactly-zero/ 2026-08-04T21:12:05.469Z

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

https://www.molecular-geometry.com/essays/symmetry-forbids-a-dipole/ 2026-08-04T21:12:05.469Z

Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise.

https://www.molecular-geometry.com/essays/delocalisation/ 2026-08-04T21:12:05.469Z

Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.

https://www.molecular-geometry.com/essays/orbitals-are-not-where-the-electron-is/ 2026-08-04T21:12:05.469Z

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

https://www.molecular-geometry.com/essays/nodes/ 2026-08-04T21:12:05.469Z

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

https://www.molecular-geometry.com/essays/why-water-is-bent/ 2026-08-04T21:12:05.469Z

The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.

https://www.molecular-geometry.com/essays/hybrids-are-a-basis/ 2026-08-04T21:12:05.469Z

An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

https://www.molecular-geometry.com/essays/chirality-is-a-symmetry-statement/ 2026-08-04T21:12:05.469Z

A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several.

https://www.molecular-geometry.com/essays/the-dipole-is-not-a-sum-of-bonds/ 2026-08-04T21:12:05.469Z

Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

https://www.molecular-geometry.com/essays/where-the-electron-is/ 2026-08-04T21:12:05.469Z

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.

https://www.molecular-geometry.com/essays/mo-and-valence-bond/ 2026-08-04T21:12:05.469Z

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.