The cost of a ratio decided somewhere else

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.

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.

A contraction is a decision made once

The six optimised primitives, each drawn at its own contraction coefficient, and the function they add up to. The tightest describes the region near the nucleus where the exact orbital has its corner; the most diffuse carries the tail. A contraction is the statement that these six will only ever appear in this ratio.

The energy penalty against the effective nuclear charge, on a logarithmic scale. At the charge the contraction was fitted to it is zero. At 1.238 the fully contracted basis is 0.0283 hartree above what the same six functions could give — seventy-four kilojoules a mole, which is a bond energy — and it climbs from there. One freed coefficient removes most of it; two remove nearly all.

At the effective charge a bond gives an orbital: what the frozen ratio costs, and what each way of relaxing it recovers. One freed coefficient at the diffuse end recovers 78 per cent; one at the tight end recovers 0.9 per cent. Two at the diffuse end recover 98.8 per cent, which is why a basis set has more than one split.

The basis the other atom lent

What a contraction costs. The relevance here is the exponent: the molecule’s best exponent is not the atom’s, and a contracted basis freezes the atom’s. Superposition error and contraction error are two consequences of the same fact — that a basis is attached to atoms and a molecule is not an atom.