The cliff, and the slope leading to it

One of the figures on a basis is not a thing: The same electrons written two ways with the density unchanged, and four electronegativity scales that disagree.

Four 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.

A function that is already there

Optimised bases of one to six Gaussians, with the energy each reaches and the smallest eigenvalue of its own overlap matrix. The energy falls towards the exact −½ hartree. The eigenvalue falls towards zero, halving roughly with each function added.

A fifth Gaussian added to an optimised four-function basis, at a multiple of one of the existing exponents, with the multiple brought steadily towards one. The smallest overlap eigenvalue and the energy the extra function buys fall together, and the energy stops improving long before the matrix stops being invertible.

The same in numbers. At three times an existing exponent the extra function is worth 1.9×10⁻⁵ hartree; at 1.01 times it is worth 3.4×10⁻⁹, and the smallest eigenvalue is already at 4.6×10⁻⁶. Past 1.00003 the set is refused.

The basis the other atom lent

The smallest eigenvalue of the overlap matrix as functions are added: how nearly linearly dependent a basis has become. This is the quantity behind the shape of the artefact — where two centres’ functions are nearly the same functions, the far centre has nothing to lend, and the borrowing has to happen where the sets are still distinguishable.

A correction computed at one length

The other thing that happens when two sets of basis functions are put beside each other: they stop being independent. Near-linear dependence and superposition error are the same phenomenon measured two ways, and both grow with how much basis is on the neighbouring centre.

A correction that is two functions

Where all of this comes from: how nearly linearly dependent a set of functions becomes as it grows. A basis that could describe each atom completely would have no correction to split.