Molecular Geometry & Bonding
what the shape decides
Concept

Spectral weight — where it appears

The intensity a spectral line carries, computed as the squared overlap between the state reached and the state prepared. Summed over every line it equals the number of electrons available to be removed, which is the sum rule that says a spectrum has been accounted for completely.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The contrast at three fillings, and the floor two of them reach. The intensity contrast on a ring of 6 against the on-site repulsion, at three fillings. At two electrons it settles on a number well above the factor of two the test needs. At half filling it falls through two and lands on exactly one from U = 64 upward — and every point where it reads exactly one is a point where the cut between fundamental and satellite falls between two lines of identical weight. Those are drawn hollow.

A ratio of exactly one is a tie

Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.

spectra · Photoelectron The same three fillings, on a ring and on a chain. The intensity contrast against the on-site repulsion at two, four and six electrons, for both geometries. Two of the ring's three curves flatten onto exactly one and stay there — the hollow marks, where the rank cut falls between two degenerate lines. The chain's corresponding curve approaches the same value from above without reaching it, because a chain of six has no exactly degenerate removal lines at any repulsion at all.

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

spectra · Photoelectron Six removal lines, each smooth, and the contrast is whichever two sit at the cut. The weight of each of the six strongest removal lines of the half-filled chain of six, followed from U = 16 upward by continuity in energy and labelled by the energy it tends to. Every line is monotone from U = 19. The contrast is the third strongest over the fourth, so it changes whenever two lines exchange those ranks: at U = 28.92 the line tending to +0.45 overtakes the one tending to −1.80 and the two weights are equal, and at U = 157 the fourth and fifth exchange. Lines at ±E, drawn in one colour, converge to one weight.

The limit of one is a parity

A half-filled chain of six has no degenerate removal lines, and its intensity contrast still goes to one — after dipping near U = 32 and rising again to U = 128. Followed line by line, every removal line is smooth and monotone; the dip is an exact tie between two lines at U = 28.916, and the rise ends where two satellites change places. At large repulsion the lines pair up at ±E with equal weights, so the contrast goes to one exactly when the cut falls inside a pair. On a chain of four it does not, and the limit is 1.3125.

wrong · Photoelectron The chain of four's strongest line is a hole's standing wave read against the singlet's ends. For the half-filled chain of four at infinite repulsion, each removal line's weight is a sum over the four sites of two factors. The first is the hole's standing wave squared on each site: the inner levels at ±0.62 put more of the hole on the end sites, the outer levels at ±1.62 more on the middle ones. The second belongs to the spin chain alone: removing an up electron at an end site leaves the stronger spin doublet with weight (2 + √3)/8 = 0.4665, at a middle site 1/4. Their sums are 0.4067 and 0.3098, which are 1/4 + √3/16 ± √15/80, and the limit of the contrast is their ratio, 1.312498.

The limit was two chains at once

At infinite repulsion every removal line of a half-filled chain is a hole's standing wave read against the spin chain the hole moves through, and every hole level carries exactly half an electron. On the chain of four that makes the contrast's limit (20 + 5√3 + √15)/(20 + 5√3 − √15) = 1.3124978 — not 21/16, which it misses in the sixth figure. On chains of eight and twelve, which no exact spectrum had reached, the limit is 1.0458 and 1.0195, and an energy that vanishes in the limit decides it.

wrong · Photoelectron

Named alongside it

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

DegeneracyExact diagonalisationHubbard modelMany-electron wavefunctionsOn-site repulsionPhotoelectron spectroscopySatelliteKoopmans theoremClosed formExchange couplingExtrapolationIonisation energy

All concepts