Molecular Geometry & Bonding
what the shape decides
Concept

Conserved quantity — where it appears

An operator that commutes with the Hamiltonian, whose eigenvalues therefore label states. Every degeneracy larger than the geometric symmetry requires is the signature of one that has not yet been found.

Named by 3 essays across one field — 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 A field splits the n = 2 shell into whole numbers. The eigenvalues of z inside the shell, which are the shifts a uniform field produces to first order. There are three distinct ones and each is a whole number times (3/2)n, so the splitting is proportional to the field itself rather than to its square — which is what no other atom does.

The symmetry that is not a rotation

Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.

symmetry · Representation 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.

DegeneracyOne-electron modelsClosed formExpectation valueIrreducible representationsMatrix elementModel limitQuantum numbersRadial momentSymmetry operationAngular momentumthe aufbau principle

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