Molecular Geometry & Bonding
what the shape decides
Concept

Variational principle — where it appears

The statement that the energy computed from any trial wavefunction is at or above the true ground-state energy. It turns approximation into optimisation, since a lower number is a better function; it promises nothing about any other property, which can be worse in the lower-energy wavefunction.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The two halves change places. What fraction of the counterpoise correction belongs to the lighter of two unlike atoms, against their separation. At a bonding distance it is 1.26 per cent — essentially the whole correction is the heavier atom's — and by 9.0 bohr it is 67. The two change places at 6.29 bohr. A symmetric pair's share is exactly a half everywhere, which is what makes one frozen number a complete description there and nowhere else.

A correction that is two functions

A symmetric pair's counterpoise correction splits exactly in half, at every separation, to the last digit — which is why one frozen number describes it. Give the two atoms different charges and the split runs from 0.03 per cent to 67, changing places at 6.29 bohr: the correction a single number was standing in for is two functions of different shapes.

orbitals · Basis One half is a curve and the other is not. The two halves of a counterpoise correction for an unequal pair, against separation, on a logarithmic axis. The heavier centre's falls smoothly over two decades; the lighter centre's scatters over more than one decade between neighbouring points. It is not a rough function — it is a difference of two energies of order a hartree whose difference is a millionth, and the solver does not have seven figures to spare.

The half that cannot be computed

How many points does each half of a counterpoise correction need to interpolate? The natural expectation is two different numbers. The answer is that the question is not yet askable: the lighter centre's half is a difference of two energies agreeing to six figures, its second differences are seven per cent of its own value, and no interpolation of it means anything. The third thing worth checking — the symmetric-pair check — works perfectly.

orbitals · Basis The trimer's correction, and the sum of its pairs. The counterpoise correction of a three-fragment system computed directly — each fragment's energy alone less its energy in the whole trimer's basis — against the sum of the three pairwise corrections, on a logarithmic axis. The sum is the larger everywhere the difference is above the solver's noise: 30.4 per cent at 1.6 bohr and nothing by six.

The assembly that counts one share twice

A counterpoise correction is divided unequally between its two centres, and the first place that matters is a three-fragment system, where the pairwise corrections are added up and the assembly must double-count one share and undercount another. It does: the heavy centre is over-corrected by seventeen per cent and the light ones under-corrected by two and a half, and the two do not cancel.

orbitals · Basis A ring of 6 as the neighbour repulsion is turned up. At an on-site repulsion of 8, three quantities against the nearest-neighbour repulsion: the alternating structure factor, the double occupancy, and the nearest-neighbour opposite-spin pair distribution. The rise is steepest at V = 4.5, which is 0.563 times the on-site repulsion. Far past it the ring is charge ordered — nearly every electron paired on alternate sites, which is what a double occupancy approaching a half means.

The give-back that turned into a saving

Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.

beyond · Correlation

Named alongside it

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

Reference stateApproximationBasisBasis set superpositionClosed formModel limitConvergenceNormalisationBond lengthConventionCorrelation energyCounterpoise

All concepts