Molecular Geometry & Bonding
what the shape decides
Concept

Momentum orbital — where it appears

The Fourier transform of an orbital — the same one-electron state written as a function of momentum rather than of position. Its squared modulus is the density a Compton scattering measurement reaches, and its shape need not resemble the position orbital's: a distribution stretched along a bond in position is flattened along it in momentum.

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

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 A bond is a cosine in momentum space. The two factors a two-centre bonding orbital's momentum density is made of. One is the atomic momentum density, unchanged by the bond and falling as the eighth power of the momentum. The other is cos²(q·R/2), the interference between the two centres, whose period is fixed by the bond length and by nothing else. Everything that distinguishes a bond from two atoms in this picture is that cosine — and it has zeros where the atomic factor has none.

Oblate in the picture nobody draws

Every drawing of a σ bond shows a density stretched along the bond, and the position-space calculation agrees: the second moment along the axis is twice the one across it. In momentum the same orbital is flattened in the same direction, because the interference between the two centres cuts the distribution off at π over the bond length — and that cut-off is a zero a measurement could find.

orbitals · Orbital An antibonding occupation fills the zero and moves the minimum. The profile along the bond near π/R, per electron, on a logarithmic scale, for five antibonding occupations of the same two orbitals. With nothing in the antibonding orbital the profile is exactly zero at π/R = 1.573. Two hundredths of an electron leave a minimum at 1.608, which reads the separation as 1.954 bohr instead of 1.997. At 0.104 the minimum becomes a flat shoulder, and the Heitler–London bond, at 0.236, has none.

The zero belongs to one determinant

A bonding orbital's momentum profile along the bond is exactly zero at π/R, and that zero reads a bond length with nothing fitted. It is a property of putting every electron into that one orbital. Any antibonding occupation fills it in linearly and drags the minimum outward, a tenth of an electron erases it, and the valence-bond wavefunction built from the same two functions never has one at any separation.

orbitals · Orbital The zero at π/R follows parity, not bonding. The bonding and antibonding combinations of three atomic functions on nitrogen, each along the bond and each normalised to its own largest value, against momentum in units of π/R. For 2s the bonding combination is zero at π/R and the antibonding one is not. For 2p along the bond it is the other way round: the σ bond carries a sine, is zero at the origin and near its largest at π/R, and the antibonding combination carries the cosine. For 2p across the bond the π bond carries the cosine again. The factor is a cosine exactly when the orbital's inversion parity matches the atomic function's.

The zero is a parity, not a bond

A hydrogen-like σ bond has a momentum profile along its axis that vanishes at π/R, and it is natural to read that zero as a bond's signature. Built from 2p functions pointing along the axis, the σ bond carries a sine instead and sits at 83 per cent of its peak there. Which factor an orbital carries is decided by whether its inversion parity matches its atom's, and bonding has nothing to do with it.

orbitals · Orbital

Named alongside it

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

Closed formEffective nuclear chargeMolecular orbitalNodeOverlap integralProbability densityWavefunctionBond lengthClosed-shell configurationsConfiguration interactionElectron correlationExpectation value

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