Molecular Geometry & Bonding
what the shape decides
Concept

Radial node — where it appears

A spherical surface on which an orbital's wavefunction changes sign, of which an orbital has n−l−1. Each one cuts the drawn surface into an extra shell, so a 2s picture is a shell inside a shell.

Named by 9 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.

Nodes

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

orbitals · Orbital 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 1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction. The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

One level is not one comparison

A plate of orbitals drawn at a single contour value looks like a comparison and is not one. At the level that encloses ninety per cent of a 1s, a 2s encloses four per cent, a 3s under one, and a 3d has no surface at all — its wavefunction never reaches that value anywhere in space.

orbitals · Contour 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 Overlap does not always fall as the atoms are pulled apart. The overlap integral of two pairs of orbitals against the separation between their centres, with the turning point and the sign change of each found by bisection on the integral itself. A pair with a radial node in it does not fall monotonically and does not keep one sign.

Closer is not more overlap

Two 1s orbitals overlap more the closer they are, and every curve drawn from that pair says the same thing. Put a radial node into one of them and the rule fails: a 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr, and less again beyond — and two 2p orbitals head-on change sign at 5.06 bohr and are more strongly coupled at eight bohr than at four.

bonding · Overlap The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

wrong · Overlap The same count in both pictures, and no rule between them. Every hydrogenic orbital with a radial node, drawn twice: its position nodes on the left axis and its momentum nodes on the right, at the same nuclear charge. The counts are identical and exact — n − l − 1 in each — because the two radial functions are polynomials of the same degree. The positions are unrelated: a node three quarters of the way out in one picture is not three quarters of the way out, or anywhere in particular, in the other.

The nodes in the other variable

An orbital has n − l − 1 radial nodes, and it has exactly that many in momentum too — the two radial functions are polynomials of the same degree. Nothing pairs one node with another: they are zeros of two different classical families. What is exact is the product over all of them, which is a ratio of factorials and does not depend on the nuclear charge at all.

orbitals · Orbital Lithium hydride's one σ bond, and the minimum it would need to be covalent to have. The momentum profile along the bond of lithium hydride's occupied σ orbital, from an extended Hückel calculation over lithium 2s and 2pσ and hydrogen 1s, each curve scaled to its own peak, against momentum in units of π/R. As computed the orbital is 91% hydrogen and its profile falls monotonically, reading 10% of its peak at π/R. The same lithium 2s and hydrogen 1s with a quarter of the orbital on hydrogen do have a minimum — at 0.525 π/R, not at π/R, just below lithium's 2s node in momentum at 0.624 π/R.

The minimum belongs to the heavy atom's node

Carbon monoxide had no single σ bond for the momentum-space polarity reading to read, so the next place to look was a hydride, where there is one. Lithium hydride's computed σ orbital is ninety-one per cent hydrogen and its profile falls straight through π/R without a minimum. Across all seven first-row hydrides one occupied orbital of thirteen has a minimum at all, and where a lithium-to-boron hydride could have one, it would not be the bond's fringe: it would be the heavy atom's 2s node in momentum, a flat point that a little hydrogen tips over.

orbitals · Orbital 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.

NodeEffective nuclear chargeModel limitAntibondingClosed formMomentum orbitalQuantum numbersRadial distributionWavefunctionApproximationthe aufbau principleBasis set

All concepts