Molecular Geometry & Bonding
what the shape decides
Concept

Most probable radius — where it appears

The distance at which an electron is most likely to be found, which is the maximum of the radial distribution rather than of the wavefunction. It differs from the mean radius, and both differ from the radius of a contour.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The radial function of 1s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Where the electron is

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.

orbitals · Orbital Four measures of size for 6 orbitals. The most probable radius, the mean radius, the root-mean-square radius and the radius of the sphere holding ninety per cent of the density, for 1s, 2s, 2pz, 3s, 3dz2, 4s. All four are computed from the same radial function, all four are correct, and they are not the same number.

How big is an orbital

Four measures of size, all computed from the same radial function, all correct, and spanning a factor of two and a half for a 1s orbital. The one that governs chemistry is a fifth, and it is not a measure of size at all.

orbitals · Contour One of these is guaranteed to improve, and it is not the one anybody measures. The relative error in the energy and in four properties of the fitted function, against the number of Gaussians. The energy falls at every step, because that is what the variational principle promises. The mean radius is exact for the single-function basis and 332 thousand times worse for the two-function one, and the density at the nucleus is still 6.2 per cent wrong where the energy is wrong by 0.011 per cent.

The property that gets worse

One Gaussian fitted to hydrogen gets the mean radius exactly right — 1.500000, against an exact 1.5. Two Gaussians get it wrong by 1.4 per cent, which is three hundred and thirty-two thousand times further out, while the energy improves fivefold. The variational principle bounds one number and says nothing whatever about any other, and the sequence of errors in everything else need not even be monotone.

orbitals · Basis

Named alongside it

The objects these essays reach for when they reach for this one.

Expectation valueProbability densityContour levelEnclosed probabilityNodeRadial distributionWavefunctionApproximationBasis setClosed formConvergenceGaussian

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