The minimum belongs to the heavy atom's node

A σ orbital made of one s function on each of two identical atoms has a momentum profile along the bond that is exactly zero at q = π/R, because the two contributions are equal and opposite in phase there. Make the atoms different and the zero becomes a floor, whose depth moves with the bond’s polarity — which made the floor look like a measure of polarity in momentum space. The measure then ran into two obstacles in turn. A p function on both atoms fills the floor fifteen times faster than polarity moves it, and carbon monoxide, the molecule the measure was built on, has no single σ bond: three occupied σ orbitals of mixed character, the highest peaking at π/R rather than dipping there.

That essay’s lead was the obvious way out. A hydride has one σ bond. Lithium hydride has exactly one occupied valence orbital, so nothing else can swamp the bond at π/R; hydrogen fluoride has two σ orbitals, but the lower is nearly pure fluorine 2s, which ought to be a smooth background. If the floor is readable anywhere, it should be readable there.

It is not, in either, and the way it fails shows that the minimum the measure was looking for is often not the bond’s at all.

One σ bond, nine-tenths hydrogen #

The calculation is the one that gave carbon monoxide’s weights, cut down to three functions: lithium’s 2s and 2pσ, hydrogen’s 1s, all hydrogenic at Slater’s effective charges (1.30 for lithium, 1 for hydrogen), at the measured length of 1.5949 Å. The diagonal elements are Hoffmann’s valence-state ionisation energies — −5.4 and −3.5 eV for lithium, −13.6 for hydrogen — and the off-diagonal ones follow Wolfsberg and Helmholz with K = 1.75. Each orbital’s profile is built from the momentum amplitudes of its three functions, each carrying a factor of −i for each unit of angular momentum, interfering across the bond through a relative phase of qR, and integrated over the plane of momenta whose component along the bond is q.

The one occupied orbital lies at −13.78 eV and is 91.3 per cent hydrogen, 5.4 per cent lithium 2s and 3.3 per cent lithium 2pσ. It is the hydride ion with a small lithium tail — Li⁺H⁻, the direction chemistry expects — and its profile is the profile of a hydrogen 1s with a little interference on it. It falls monotonically from the origin and reads 10.2 per cent of its peak at π/R. There is no minimum anywhere out to 2π/R.

Two features of the number are worth isolating. Lithium’s p function contributes nothing to it: the same orbital with its 2pσ coefficient set to zero reads 10.22 per cent instead of 10.24, so the quadrature that made p fill carbon monoxide’s floor has almost nothing to work on here. And the reading is not small because the bond is covalent; it is small because hydrogen’s 1s is broad in momentum and lithium’s 2s is narrow, so at π/R the tail of a compact function is all that is left.

So a single σ bond is not sufficient. A minimum needs the two centres to contribute comparably near π/R, and lithium hydride’s bond is too polar for that by a wide margin.

Where the σ electrons of seven hydrides sit #

Lithium hydride is one end of a series, and the series is cheap — the same three functions with the heavy atom changed, at each molecule’s measured length.

From beryllium on there are two occupied σ orbitals, and they separate cleanly by character. The lower is mostly the heavy atom’s 2s with some hydrogen: two-thirds hydrogen in beryllium hydride, falling through 44, 26, 18 and 12 per cent to 7 in hydrogen fluoride. Beryllium hydride’s upper σ holds a single electron; every other orbital counted here holds two. The upper is mostly the heavy atom’s 2pσ, with between 5 and 14 per cent hydrogen throughout. The π electrons — none in lithium, beryllium and boron hydride, one to four from methylidyne to hydrogen fluoride — sit on the heavy atom alone and contribute a smooth one-centre background that this argument leaves out.

That split matters for what follows. The upper σ orbitals are built from a p function on one centre and an s function on the other, and at π/R those two are in quadrature: they cannot cancel there whatever their weights. Any minimum near π/R has to come from the lower orbitals, the ones made of two s functions. And hydrogen fluoride’s upper σ, the orbital the lead hoped would show the bond cleanly, is the clearest case of all.

Its lower σ is 92.9 per cent fluorine 2s and falls monotonically, reading 3.7 per cent of its peak at π/R. Its upper σ, the bond, is 93.0 per cent fluorine 2pσ. A p function along its own axis contributes nothing to the profile at the origin — its momentum amplitude carries a factor of the momentum’s component along the bond — so the bond starts near zero, peaks at 0.625 π/R and still stands at 71 per cent of its peak at π/R. Hydrogen fluoride’s bond reads near its maximum at exactly the momentum where a covalent s bond would read zero. It is the parity rule again, in a molecule where the bond is almost all one p function.

A band of shares, and the real orbitals outside it #

So the question becomes narrower: for each hydride, at what share of hydrogen would a bond made of the heavy atom’s 2s and hydrogen’s 1s have a minimum at all? That is a sweep with one parameter. Hold the two functions and the length fixed, take the bonding combination at every mixing angle from one degree to eighty-nine, and record whether the profile has an interior minimum below 1.9 π/R and where.

The band narrows and moves as the heavy atom gets heavier. For lithium a minimum exists at any hydrogen share up to 0.45; for beryllium and boron up to about 0.36; for carbon between 0.14 and 0.38; for nitrogen between 0.34 and 0.42; for oxygen only between 0.49 and 0.51; for fluorine between 0.53 and 0.64. The light hydrides allow a minimum over a wide range of polarity, the heavy ones only near equal sharing.

And the computed orbitals almost all fall outside. Lithium hydride’s bond is at 0.91, twice the band’s upper edge. The lower σ orbitals of beryllium, boron, nitrogen, oxygen and fluorine hydride sit at 0.67, 0.44, 0.18, 0.12 and 0.07 — each on the wrong side of its own band, and the heavier the atom the further from equal sharing the orbital is while its band closes in on it. Methylidyne’s lower σ, at 0.26 hydrogen, is the only one inside, and it is the one occupied σ orbital of the thirteen with a minimum: at 0.862 π/R, 2.5 per cent of its peak. Its small 2pσ admixture moves that minimum a little below where the pure 2s–1s bond at the same share puts it.

The upper orbitals of beryllium and boron hydride sit inside their bands by share, at 0.05 and 0.12 hydrogen, and have no minimum. The band is the two-s-function bond’s; those orbitals are half and two-thirds 2pσ, and the band does not describe them.

Not the fringe: the node #

The band’s width is not the most interesting thing about the sweep. Where the minimum sits is.

The fringe argument says the minimum should be at π/R: that is where the two centres’ contributions are in antiphase. For oxygen and fluorine it is — at 1.000 and between 1.000 and 1.038 π/R. For lithium it is not remotely there. Lithium hydride’s minimum, at any share that has one, sits between 0.525 and 0.600 π/R. Beryllium’s is between 0.66 and 0.75, boron’s between 0.79 and 0.94.

What those three have in common is visible on the horizontal axis. A hydrogenic 2s function has one radial node in position and, as the momentum picture of the same orbital showed, one in momentum: its transform is proportional to $(4p^2 - Z^2)/(4p^2 + Z^2)^3$ and vanishes at p = Z/2. For lithium’s 2s at an effective charge of 1.30 that is 0.650 bohr⁻¹, which at lithium hydride’s length is 0.624 π/R. Beryllium’s node is at 0.787 π/R and boron’s at 0.964. In all three the minimum sits just below the node — every minimum the sweep finds is within a fifth of the node’s position below it — and nowhere near the fringe.

From carbon on the node moves above π/R — 1.095, 1.215, 1.327 and 1.434 for carbon to fluorine — and the minimum stops following it and settles onto the fringe. Carbon and nitrogen are in between, their minima drawn towards π/R from below.

So there are two features competing to be the minimum: the bond’s fringe at π/R and the heavy atom’s own momentum node. When the node is lower, it wins.

Why a node makes a minimum #

The mechanism is in the essay that counted momentum nodes, and it needs only one line.

For a single s function the directional profile is the momentum density integrated over a plane, which is an integral from |q| upwards of the density times the momentum. Its slope is therefore minus 2π q times the density at q — and that is zero exactly where the momentum function has a node. The node appears in the profile as a horizontal tangent, a flat place in a falling curve rather than a dip. On lithium’s 2s alone the flat place is at 0.624 π/R, and the curve still falls on both sides of it.

Add anything to it and a flat point becomes a stationary pair. The interference term with hydrogen’s 1s varies slowly through the node, and a small slowly varying addition to a curve with zero slope gives it a small minimum beside the flat point and a small maximum on its other side. At 2 per cent hydrogen the minimum is already there, at 0.56 π/R; at 25 per cent it has deepened and moved to 0.525. And because the single function’s slope vanishes quadratically at the node — the density has a double zero there — the minimum and its partner maximum separate as the square root of the admixture rather than in proportion to it: at under one per cent hydrogen the minimum is already 0.04 π/R below the node. That is why the band of shares reaches all the way down to almost no hydrogen for the light atoms: the minimum is the 2s function’s own, and any hydrogen at all reveals it.

Read this way, the light hydrides’ minimum carries information about the heavy atom’s radial function and its effective charge rather than about the bond. Its position follows Z/2, not π/R, so it moves with the effective charge — which a molecule is free to change when the bond forms — and not with the bond length. Its existence requires a node, which a nodeless Slater-type 2s would not have.

How the claims are checked #

The generalised construction reproduces the old one. Given carbon monoxide’s functions and its 5σ coefficients, the profile built here — any list of s and pσ functions on either centre — agrees with the construction those essays used to a part in 10¹⁰ at three momenta. The hydrogen 1s is the only thing new.

Every occupied orbital integrates to one along the whole line of momenta, to a part in a thousand, which is the check that the angular-momentum phases and the overlaps agree: lithium hydride’s overlaps are −0.376 between lithium 2s and hydrogen 1s, its outer lobe being the negative one, and +0.554 between lithium 2pσ and hydrogen 1s.

The node is located twice. Numerically, by bisection on the transform the profiles are built from, and against the closed form Z/2; the two agree to 10⁻⁶ for all seven atoms. And the flatness is checked directly: on lithium’s 2s alone the profile’s slope at the node is less than a thousandth of its slope at half that momentum.

The refusal is the homonuclear limit run through the same code. Two identical 1s functions at equal coefficients must cancel at π/R to 10⁻¹², and a lithium 2s against a hydrogen 1s at equal coefficients must not — two radial functions that are not proportional cannot cancel on a whole plane.

The model’s edges #

Hydrogenic functions at Slater’s charges. Every number here is that model’s, and the node is where it is most exposed. A Slater-type 2s has no radial node at all and so no momentum node; a Hartree–Fock 2s has one, from its orthogonality to the 1s core, but not at Z/2. What survives any choice of function is the mechanism — a momentum node is a flat point in the profile, and a flat point near π/R competes with the fringe — and what does not is the position, 0.624 π/R for lithium.

Extended Hückel, not a self-consistent calculation. Hoffmann’s parameters and the Wolfsberg–Helmholz rule put lithium hydride’s bond at 91 per cent hydrogen, which is more ionic than a self-consistent field would make it. The verdict on lithium hydride is robust to that — the band ends at 0.45 and the orbital would have to lose half its hydrogen — but the verdicts on methylidyne and nitrogen hydride, whose orbitals sit near their bands’ edges, are not.

Valence only, and σ only. Lithium’s 1s core is omitted; it is compact in position and so broad and featureless in momentum, adding a smooth background to any measured profile. The π electrons are omitted for the same reason. Both would raise the profile at π/R and neither could create a minimum, so leaving them out favours the measure and it still fails.

Directional, one determinant, and at a fixed length. Every profile is of a single occupied orbital, and the fringe’s zero belongs to a single determinant: correlation fills it linearly, and nothing here asks what correlation does to the node’s flat point. A gas-phase measurement averages over orientations, which the parity argument showed is enough to remove the fringe’s zero even for a homonuclear bond.

The reading that survives #

The polarity reading began as a claim about one object — the depth of a fringe minimum at π/R — and each test since has found a different object occupying its place: a floor from mismatched radial functions, a p function’s contribution in quadrature, an orbital that peaks rather than dips, and now, in the molecules simplest enough to have one bond, a minimum that belongs to the heavy atom’s own radial node. The habit it leaves is worth keeping beyond momentum space: when a feature turns up where the theory put it, check what else in the system could put a feature there. A node at 0.624 π/R and a fringe at 1.000 are well separated in lithium hydride; at boron they are four per cent apart, and a measurement would have no way to tell which one it was seeing.

Who measured what #

Lithium hydride, as the simplest ionic solid, has had its Compton profile measured and calculated many times since the 1970s, and the reciprocal form factor — the profile’s Fourier transform, in which the interference appears at the bond length rather than at π/R — was developed in the late 1970s, notably by Weyrich and co-workers, as a way to read bonding from such data. Lithium hydride’s ionicity and its hydride-ion character are standard. The 2s momentum node at Z/2 is a textbook property of hydrogenic functions.

What is computed here is the hydride series in the extended Hückel model, the band of hydrogen shares in which a 2s–1s bond has a minimum, its position against the heavy atom’s node, and the finding that of thirteen occupied σ orbitals only methylidyne’s lower one has a minimum.

Still open: a crystal, and a node that is not hydrogenic #

The nearer question is the function. A Hartree–Fock lithium 2s, orthogonalised to its 1s, has its momentum node somewhere other than 0.65 bohr⁻¹, and whether that node falls below or above lithium hydride’s fringe decides whether the minimum a better model would predict is the node’s or the fringe’s. It is a single transform of a tabulated function, and it would test the one claim here most tied to the model.

The obvious question is the solid. Lithium hydride’s Compton profile was measured on crystals, where each hydride ion has six lithium neighbours at 2.04 Å rather than one at 1.59, and the interference that appears at π/R for a molecule appears at the lattice’s own reciprocal vectors. Whether the lithium 2s node or a lattice fringe sets the first minimum of the crystal’s directional profile is the same competition with a different set of fringes, and the crystal is the system where a measurement exists to compare against.