Molecular Geometry & Bonding
what the shape decides
Concept

Radial moment — where it appears

An expectation value of a power of the distance from the nucleus, computed from an orbital's radial function. Different powers weight different regions — the tail, the middle, the cusp — so they are separate measurements of size and an approximation can be good at one and poor at another.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Two moments of the n = 2 shell, and only one of them agrees. ⟨1/r⟩ and ⟨1/r²⟩ for each orbital of the n = 2 shell of hydrogen, computed from the radial functions and checked against their closed forms. The first is the same number for every member — which is why they share an energy — and the second differs by a factor of 3 across the shell.

The degeneracy no group predicts

How many orbitals can share an energy is decided by a group before any energy is computed, and the rotation group of a central potential permits one, three, five and seven. Hydrogen's second shell has four orbitals at one energy and its third has nine. The extra degeneracy is not an accident of the arithmetic — it is the signature of a symmetry that has not been named, and it survives only for a potential that goes exactly as one over r.

symmetry · Representation 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 How nearly a broken symmetry survives. A screened potential splits the n = 2 shell and destroys the degeneracy the linear Stark effect depends on. The field needed to overcome the splitting and restore the linear behaviour runs from 4.3e+4 volts a centimetre at a quantum defect of 0.00040 to 2.9e+7 at a defect of 0.208. The dipole between the states is 3.000 throughout, so the field is exactly the splitting divided by twice it.

How nearly a broken symmetry survives

The hydrogen shell's extra symmetry is what makes its Stark effect linear, and a real atom does not have it. Screening splits the shell, and the field needed to overcome the splitting and restore the linear behaviour is a curve — from forty thousand volts a centimetre at a quantum defect of 0.0004 to thirty million at a defect of 0.21.

symmetry · Representation

Named alongside it

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

Closed formConserved quantityDegeneracyExpectation valueModel limitOne-electron modelsVirial theoremApproximationthe aufbau principleBasis setConvergenceEffective nuclear charge

All concepts