The wrong shape, fitted as well as it can be

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.

Six 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 Gaussian is the wrong shape

The exact hydrogen 1s and the best sums of one, two, three and six Gaussians, each with its exponents optimised for the energy here rather than taken from a published table. Three of them already sit almost on top of the exact curve over the range where the density is.

The error in the energy against the number of functions, on a logarithmic scale, with the overlaps between each fit and the exact orbital alongside. Every doubling of the basis takes about a factor of four off the error, and every point is above the exact value.

The first few tenths of a bohr. The exact orbital arrives at the nucleus with a corner and a slope of −1; every fit arrives flat. More functions raise the peak towards the right height and none of them produces the corner, because a sum of smooth functions is smooth.

A contraction is a decision made once

The exact hydrogen 1s and the best sums of one, three and six Gaussians. Six of them reproduce the function to better than 99.99 per cent by overlap over the range where the density is, which is why a minimal contracted set is a usable description of an atom — and it is an atom that is being described.

The orbital in momentum space

The same failure in position: every sum of Gaussians arrives at the nucleus flat, where the exact function arrives with a corner and a slope of −Z. In momentum this is the missing far tail, and a fitted function that has too little amplitude at large p has too little kinetic energy from the region near the nucleus.

The measurement a basis was not fitted to

The same fit in position space, where the two failures are the two ends of the curve. The cusp at the origin cannot be reproduced by any sum of Gaussians, because every Gaussian has zero slope there; the tail cannot either, because a Gaussian dies as exp(−ar²) and the exact function dies as exp(−Zr).

The property that gets worse

The relative error in the energy and in four properties, against the number of Gaussians. Only the energy is guaranteed to fall, and it does, at every step. The mean radius does not: it is exact for one function and wrong by more than a per cent for two. The density at the nucleus is still 6.2 per cent wrong at six functions, where the energy is wrong by 0.011.

The functions themselves, against the exact one. The single Gaussian is visibly the wrong shape everywhere: too flat at the nucleus, too fat in the middle, and dead in the tail. It is this function whose mean radius is exact.

The energy alone, which is the quantity every one of these bases was chosen to minimise. This curve is what “converged” usually means, and it is the only curve in this essay with a theorem behind it.

A correction computed at one length

Where all of this starts: a Gaussian has the wrong shape at the nucleus and the wrong shape far out, and a finite sum of them is a compromise. Every error in this essay is a consequence of that compromise being made separately on two centres that then have to describe one molecule.