Molecular Geometry & Bonding
what the shape decides
Concept

the aufbau principle — where it appears

The rule that electrons fill the lowest orbitals first, in a stated order. The order is a property of the whole many-electron system rather than of any one atom's orbitals, which is why it is not the order the same orbitals empty in.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The radial function of 3s. 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.

The radial distribution across the periodic table

A 4s orbital is bigger than a 3d by every measure of size except the one that decides which fills first. Penetration is a feature of a small inner peak, and the periodic table's shape depends on it.

orbitals · Orbital 4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

wrong · Approximation 4s and 3d from Sc to Zn. The mean radius of the 4s and 3d orbitals across the elements Sc to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

orbitals · Orbital 2s and 2p from B to Ne. The mean radius of the 2s and 2p orbitals across the elements B to Ne, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What the screening model cannot see

Slater's rules put the 2s and the 2p in one group, so they give both orbitals exactly the same effective nuclear charge at every element from boron to neon. The two are separated by several electronvolts in all six, and the model has no term that could produce it.

orbitals · Orbital 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

Named alongside it

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

Effective nuclear chargeOne-electron modelsPenetrationShieldingMany-electron wavefunctionsQuantum numbersRadial distributionRadial nodeApproximationConserved quantityContour levelDegeneracy

All concepts