Molecular Geometry & Bonding
what the shape decides
Concept

Shielding — where it appears

The reduction in the nuclear charge an electron feels because of the others between it and the nucleus. It is modelled by fitted rules that reproduce orbital sizes well and cannot see the ordering effects penetration produces.

Named by 8 essays across 2 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.

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 4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, 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.

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

wrong · Approximation 4s and 3d from Sc to Zn. The mean radius of the 4s and 3d orbitals across the elements Sc to Zn, 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 an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

orbitals · Orbital 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 One isovalue, many fractions — and one fraction, many isovalues. What each of 4 conventional isovalues encloses, for 7 orbitals of hydrogen, and in the last column the level each one needs to enclose 90 per cent. At 0.02 atomic units the fractions run from 0.4 to 96.2 per cent, and the levels in the last column differ by a factor of 32. A plate of orbitals drawn at one value is not a comparison of sizes. 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; 3pz at 90% of its density, |ψ| = 3.03e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

The isovalue nobody chose

Every program that draws an orbital asks for a number, and almost every user accepts the default. At 0.02 atomic units that default encloses 96 per cent of hydrogen's 1s, 52 per cent of its 2s and four tenths of one per cent of its 4s — three pictures drawn to one rule, meaning three completely different things.

orbitals · Contour Where two He atoms stop, with no contact distance put in. The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.

The radius that was tabulated

A van der Waals radius is a fitted number that every structural argument in chemistry uses. Computing it instead — a repulsion taken from computed overlap integrals, an attraction taken from two measured scalars, and no length anywhere — puts helium's contact at 3.140 ångström against a tabulated 2.80, neon's at 3.226 against 3.08 and argon's at 3.996 against 3.76. And the surface two atoms actually stop at encloses 99.4 per cent of the density, not ninety.

orbitals · Contour The fraction a table encloses is not one number. For each ion, the fraction of its own electron density that lies inside its tabulated radius. Along each isoelectronic series the answer runs over eight percentage points, where three neutral atoms at their contact distances spanned three tenths of one. And it peaks at the neutral rather than falling through it, because a van der Waals radius is fitted to the distance between two atoms that are not bonded and an ionic radius is one term of a sum fitted to the distance between two that are.

The surface a table draws

Three noble gases stop at a surface enclosing between 99.38 and 99.75 per cent of their density — a near-constant, and an argument that a contour is a real boundary. Charge the atoms and it collapses. Across ten electrons the tabulated radius encloses anything from 94.4 to 99.98 per cent, it peaks at the neutral rather than trending through it, and radii built at a fixed enclosure do not add up to a single measured separation.

orbitals · Contour One size scale, computed for every atom at once. Each atom's valence shell at the effective charge Slater's rules give it, with the mean radius of a hydrogenic orbital of that shell and that charge, and its root-mean-square radius. Both are closed forms in the principal and angular quantum numbers and the charge, so both are on one scale for every atom by construction — which is the difficulty a tabulated radius has and this does not. Neither quantity is anywhere in the cubic, which knows three energies and no length.

A size the fit was not made from

Every input tested against the chemical capacity so far has been inside the cubic that produced it, so the search was constrained to fail. A size is not: a hydrogenic orbital at the effective charge Slater's rules give an atom's valence shell has an exact radius, on one scale for every atom, and the cubic knows three energies and no length at all. It fails the test by sixteen times the floor.

wrong · Electronegativity

Named alongside it

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

Effective nuclear chargeRadial distributionthe aufbau principleContour levelModel limitPenetrationConventionEnclosed probabilityMany-electron wavefunctionsOne-electron modelsQuantum numbersClosed-shell configurations

All concepts