Molecular Geometry & Bonding
what the shape decides
Figure

Where two He atoms stop, with no contact distance put in

The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.
Where two He atoms stop, with no contact distance put in. The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.

One of the figures on overlap: Two functions on two centres, integrated — including the integrals symmetry requires to vanish, which come out at arithmetic noise.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

The radius that was tabulated

The two terms for helium, and their sum. The repulsion comes out of overlap integrals of hydrogenic 1s functions at helium’s effective charge; the attraction is a measured polarisability and a measured ionisation energy divided by the sixth power of the separation. Where they balance is 3.140 ångström, against a tabulated contact of 2.80, and no length was put in.

The computed contact distances of three noble gases against twice their tabulated van der Waals radii. The worst disagreement is 12.1 per cent, and the three atoms span a factor of two in polarisability and a factor of eight in the number of valence electrons.

The same balance for argon. The repulsion at the contact distance is 2.19 meV against helium’s 0.29, and the attraction is 7.83 against 0.81 — both an order of magnitude larger, on an atom whose contact is only a quarter further out, because both terms fall very steeply and the ratio between them is what sets the distance rather than the size of either. That is the same insensitivity an overlap curve’s shape has to the absolute size of the functions.

A size a confound cannot supply

Every pair’s well, with the measured separations marked on the axis. The minima are in the right region, which is what makes the comparison worth making.

The two terms for one pair, and their sum. The minimum is where the balance puts the ions; the dashed line is where they are.

Three computed contact distances against three tabulated ones, with nothing fitted between them. The computation puts two atoms in a well and reads off where they stop; the table was assembled from crystal structures. They agree closely enough that the computed radius can stand in for the tabulated one, which is what the ranking above needs and is the only thing it needs.

Every figure · Every orbital, by what it encloses · All essays