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 10 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 One zero becomes three, because the average couples them. What the interference factor becomes when the molecule is averaged over every orientation, one curve per angular kind. A directional profile along the bond carries cos(qR) for every orbital, so every cosine combination is exactly zero at q = π/R. The average replaces it by j₀(pR) for an s combination, j₀ − 2j₂ for a pσ one and j₀ + j₂ for a pπ one — because the cross term multiplies the interference by the orbital's own angular density and the average of the product is not the product of the averages. Their first zeros are at 0.66π, π and 1.43π.

The pair that cancels only at zero overlap

A directional Compton profile along a bond is exactly zero at π/R for every cosine combination, which makes the depth there a count of parity mismatches. A gas measurement averages over orientations, and the question was whether the count survives. It does not, for a reason nothing in the directional picture shows: the bonding and antibonding combinations of one atomic function are normalised by 2 + 2S and 2 − 2S, so a filled pair leaves a residue proportional to its overlap — and in nitrogen that residue is the largest single term, from a pair whose imbalance is zero.

orbitals · Orbital The minimum fills in, and it fills in before any charge moves. The σ profile along the bond for three pairs of atoms, each with equal coefficients on the two atoms — no charge transfer at all. The homonuclear pair has an exact zero at π/R because its two atomic contributions are identical and cancel. The other two do not, because their radial functions differ: the cancellation at π/R needs cₐ gₐ = cᵇ gᵇ everywhere, and two Slater functions of different exponents are nowhere proportional.

A floor no charge transfer explains

A σ bond between two identical atoms has a momentum profile that is exactly zero at π/R, because the two atomic contributions are identical and cancel. Two different atoms cannot cancel there at any coefficients, so the depth of the minimum becomes a measure of polarity — and it is one, monotone in the charge imbalance in the direction chemistry moves charge. What it is not is a measure with a zero: it has a floor set by how far the two radial functions are from proportional, and B–N's floor is deeper than C–O's although C–O is the more polar bond.

orbitals · Orbital A p function fills the polar bond's minimum, and a pair of them fills it nine times over. Carbon monoxide's σ profile along the bond at equal coefficients, for three bases: one s function on each atom, a quarter p mixed into carbon's, and a quarter p on both atoms in the sense that maximises their overlap. The minimum at π/R is 1.35% of the peak with s alone, 3.63% with p on carbon and 12.53% with p on both. The p function proposed to reduce the floor raises it, and the pair raises it most.

The third function fills the floor

A polar σ bond built from one s function per atom has a momentum profile whose minimum at π/R never reaches zero — for carbon monoxide, 1.35 per cent of the peak. The natural hope was that a p function mixed into the diffuse atom would reshape it towards the compact one and lower that floor. It cannot. At π/R an s function on one atom and a p function on the other are exactly out of phase, so a p function on one atom cancels nothing and only adds itself. With p on both atoms in the sense that makes the bond, the pair adds fifteen times as strongly as the s pair cancels: a quarter p takes the minimum to 12.5 per cent. That is eight times the whole range through which polarity moves the s-only bond.

orbitals · Orbital Carbon monoxide's three σ orbitals, and what each does at π/R. The momentum profile along the bond of each occupied σ orbital of carbon monoxide, with weights from an extended Hückel calculation over carbon and oxygen 2s and 2pσ, each scaled to its own peak, against momentum in units of π/R. At π/R the lowest, 3σ, reads 2.52% of its peak; the middle, 4σ, 8.40%; and the highest, 5σ, 85.3% — its profile peaks at 0.82 π/R and has no minimum near π/R at all.

The lone pair sits where the floor was

A polar σ bond's momentum profile has a minimum at π/R whose floor was proposed as a measure of polarity, and the essay that swept the p weight could not say whether carbon monoxide's own σ bond sat near the s-only floor of 1.35 per cent or the quarter-p value of 12.5. With the weights computed — carbon and oxygen 2s and 2pσ, extended Hückel — the answer is that carbon monoxide has no such bond. Its lowest σ orbital reads 2.5 per cent, its middle one 8.4 per cent of its own peak, and its highest, the orbital with the lone pair's sense, has no minimum at π/R at all: it peaks there. The six σ electrons together have no minimum anywhere, and the polar molecule reads lower at π/R than nitrogen does.

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.

Molecular orbitalModel limitOverlap integralBasis setClosed formNodePolarityEffective nuclear chargeOverlapParity (g and u)Probability densityRadial node

All concepts