The Compton profile, exact and fitted

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.

The measurement a basis was not fitted to

The exact profile and three fitted ones. Every curve here integrates to exactly one electron, so the differences between them are shape and nothing else. The one-Gaussian fit is visibly the wrong shape; the three-Gaussian fit looks right at this scale and is not.

Three fitted bases, with the error in the energy and the error in the profile in adjacent columns. The energy error falls by a factor of 1,389 from one Gaussian to six; the profile error falls by 43. The last column is the ratio between the two, and it grows.

Two errors for one basis, at five sizes, in a single table. The energy error falls by a factor of four for every doubling of the basis and the profile error falls by much less — so the two are not two views of one convergence, and a basis chosen by the first is not chosen by the second.

The property that gets worse

The momentum-space quantity, for comparison: the Compton profile, which is a measurement of the momentum distribution and is dominated by exactly the regions of space the fit is worst in. Its error at six functions is twenty-five times the energy’s.