Molecular Geometry & Bonding
what the shape decides
Concept

Gaussian basis — where it appears

A set of Gaussian functions used to expand molecular orbitals, chosen because their integrals are cheap rather than because they are the right shape. An exponential cusp at the nucleus and an exponential tail both need several Gaussians to imitate.

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

The correction collapses and the error made assembling it grows. At a separation of 2 bohr, two quantities against the number of Gaussians a centre. The trimer's own counterpoise correction falls by a factor of 2717 from one function to six — a bigger basis has less to borrow. The fraction by which summing the pairwise corrections overshoots it rises from 5.3 per cent to 38.0. Improving the calculation makes the assembly proportionally worse.

The correction that gets harder to assemble

Summed across a trimer, pairwise counterpoise corrections come to fifteen per cent more than the trimer's own. Three Gaussians a centre is a small basis, so the natural question was whether the fraction shrinks with a better one or stays put. It does neither. The correction falls by three orders of magnitude and the fraction grows fivefold.

orbitals · Basis Both quantities, and the line they have to be read against. At the closest separation, the counterpoise correction itself and the error a pairwise assembly of it makes, against the number of Gaussians a centre. Both fall — the correction by a factor of 3631, the error by 793 — and the fraction rises by exactly the ratio of those two. The dashed line is a kilocalorie a mole. The only basis where the correction is above it and the error below it is two.

One basis size where it is worth doing

The pairwise assembly's error grows with the basis — 5.25 per cent at one Gaussian a centre, 37.98 at six — while the correction itself falls by a factor of three thousand. Which of the two should a practitioner care about? Drawing a line at a kilocalorie a mole answers it: there is exactly one basis size at which the correction is worth computing and its pairwise assembly is accurate enough to use.

orbitals · Basis The consecutive triples fail for a uniform chain at every length. The share of three decades of accuracy line over which some basis size is usable, against the number of fragments from three to eight, for three arrangements and three ways of assembling the correction. On the uniform chain the consecutive-triple assembly covers less than pairs alone at every length from four, and the shortfall grows. On the heavy-inside chain it covers exactly what every triple covers. On the heavy-outside chain it covers more than every triple from five fragments on.

Length did not rescue the consecutive triples

On four fragments, keeping only the consecutive triples of a counterpoise assembly worked for two arrangements and did worse than pairs for the uniform chain, and a short chain was the obvious excuse. Carried to eight fragments the excuse fails: the uniform chain's consecutive assembly settles at 48 per cent of the line against 68 for pairs. And the heavy-outside chain turns the lesson over — from five fragments every triple together covers less than the consecutive ones alone.

orbitals · Basis Lithium's 2s three ways, in momentum: three nodes, one of them past the fringe. The momentum radial functions of the same three lithium 2s orbitals, weighted by p² as a radial density is, each scaled by its largest magnitude and signed to start negative, against momentum in units of lithium hydride's π/R. Every one has exactly one node — the hydrogenic at 0.624 π/R, the Hartree–Fock at 0.889 and the nodeless Slater function, which has no node in position, at 1.080, just past the fringe.

The node a nodeless orbital has

Lithium hydride's would-be minimum in momentum was traced to the 2s function's node at 0.624 π/R, with a warning that the position belonged to the hydrogenic model and that a nodeless Slater function would have no node at all. Solving the lithium atom properly moves the node to 0.889 π/R; the nodeless function has one too, at 1.080, because a node in momentum is not a node in position but a disagreement between a function's sum and its slope at the nucleus. Lithium hydride's own bond has no minimum with any of the three. What the function decides is whether a minimum can be born at the node at all.

orbitals · Orbital

Named alongside it

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

Basis setBasis set superposition errorFragment methodMany-body expansionModel limitCounterpoiseVariationalEffective nuclear chargeFinite-size effectHartree–FockMomentum orbitalNode

All concepts