Molecular Geometry & Bonding
what the shape decides
Concept

Orbital approximation — where it appears

Assigning each electron its own one-particle function and building the state from the product. It is the assumption that makes the whole orbital vocabulary possible, and it is exactly what correlation is defined as the failure of.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. 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.

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

wrong · Approximation 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 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 The hybrids do not point at the atoms. For each cycloalkane, the angle between its two ring hybrids — fixed by orthogonality once the measured H–C–H angle has said how much s character the hydrogens take — against the angle between its carbons. Cyclopropane's differ by 45.5°, so each hybrid points 22.75° outside the bond it makes; cyclohexane's agree to 0.035°, which is the control.

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

bonding · Hybrids

Named alongside it

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

Closed formHartree–FockMany-electron wavefunctionsModel limitOne-electron modelsOverlap integralthe aufbau principleBasisBasis setBond angleBond lengthBorn–Oppenheimer separation

All concepts