Molecular Geometry & Bonding
what the shape decides
Concept

Virial theorem — where it appears

The relation that the potential energy of a Coulomb system in a stationary state is exactly twice the kinetic energy in magnitude and opposite in sign. It holds identically at any variational minimum with respect to a scaling of the orbitals, so it is satisfied by bad wavefunctions and good ones alike.

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 A correction computed at 3 bohr and used everywhere. Three binding curves for H₂⁺ in 2 Gaussians a centre: uncorrected, properly counterpoise corrected at every separation, and corrected once at 3 bohr with that value subtracted throughout. The frozen curve is the uncorrected one shifted down by a constant, so its minimum sits at 2.2270 bohr — exactly where the uncorrected minimum is, and 4.2 millibohr from where the full correction puts it. The depth moves and the structure does not.

A correction computed at one length

The counterpoise correction is expensive, so it is evaluated once at a reference geometry and subtracted across a whole potential surface. A constant does not move a minimum — so a frozen correction returns the uncorrected bond length exactly, at every reference geometry and in every basis, and everything the correction does to a structure is the part that has just been thrown away.

orbitals · Basis

Named alongside it

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

Basis setClosed formExpectation valueModel limitRadial momentApproximationthe aufbau principleBond lengthBorn–Oppenheimer separationConserved quantityConvergenceDegeneracy

All concepts