On-site repulsion — where it appears
Named by 29 essays across 6 fields — each of them below, with the objects they name alongside it.
What couples two spins
Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.
The hole that is not repulsion
Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.
The smallest many-electron calculation
A Hückel energy comes from a model with one electron in it. Add a single term — a cost for two electrons on the same site — and the problem stops being a matrix of size n and becomes a matrix over configurations. Four sites give thirty-six of them, which is small enough to solve exactly, and the answers correct two things the one-electron model got wrong.
Where molecular orbital theory dissociates
The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.
The insulator band theory cannot see
A half-filled ring of four sites has a degenerate shell and no gap at all in the one-electron picture, which is the definition of a metal at that size. Its exact charge gap is zero when the electrons do not repel and grows without limit when they do — so a material can have a half-filled band and not conduct, and here is the number.
Two kinds of correlation, and only one is small
Correlation energy is defined as a subtraction, and the definition hides that the thing subtracted is not one thing. In the two-site model the local power of the correlation energy in the interaction falls from two to one — and at every interaction strength it equals, exactly, the occupation of the bonding natural orbital.
Two pictures, one plane
Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.
A method that is not additive
Put a molecule next to a copy of itself, far enough away that they do not interact, and the exact energy doubles exactly. A truncated calculation does not — because the truncation forbids both halves being excited at once, which is something the pair can do and neither half can.
A mean field cannot get out of the way
The energy a mean field misses grows without limit as the repulsion rises — 27.82 at U = 32 for four electrons on four sites, against an exact energy of −0.2946, so the error is ninety-four times the answer. The rate it grows at is not an energy at all: it is n²/4N, a count of the coincidences a spread-out density cannot avoid.
Two wrong numbers and a right difference
A mean field gets the total energy of a four-site system wrong by 12.11 and of two two-site systems wrong by 12.49, and 96.9 per cent of that error cancels out of the difference between them. The residue is 0.38 — and the reaction energy it is a residue of is 0.09, so the cancellation improves and the answer gets worse at the same time. Change the pair being compared to a singlet and a triplet and nothing cancels at all: the sign goes.
More bands than there are orbitals
A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.
The correction that was computed somewhere else
Every composite method rests on one assumption: that an expensive correction computed on a small case can be added to a cheap calculation on a large one. Tested on four sites where both answers are exact, it removes 99.74 per cent of the cheap method's error near the reference and −1450 per cent of it further away — at which point the recipe is fifteen times as wrong as the calculation it was improving.
A hundred lines and no way to sort them
A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.
Where the electrons are, without subtracting anything
A correlation energy is an exact energy less a mean-field one, so the answer depends on which mean field was picked — by a factor of 259. The pair distribution says the same thing with no subtraction in it, and two systems matched to the same correlation energy turn out to have electrons in visibly different places.
The warning a cheap calculation gives
A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.
The give-back that turned into a saving
Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.
A contrast with a closed form
Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.
A satellite that never loses its place
The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.
The half of the square a ring of four cannot show
There is a warning that says in advance whether a transferred correction will hold: the mean field's own symmetry breaking, which collapses at a definite site-energy modulation and takes the transfer with it. The other axis of the same square has a threshold too — on the other side — and the ring of four it was all measured on is the one system with no threshold to find.
A sign change is not always a zero
The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.
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.
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.
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.
A cancellation between two separations
The give-back's numerator has one term in it, so its root is the repulsion at which the pair count at one bond returns to its free value. Let the interaction reach two bonds and the numerator has two terms with opposite signs, and its root becomes a cancellation between an enhancement at one separation and a depletion at the next — at a repulsion where neither is at its free value. Which side of the old zero the new root falls on reverses between a reach of a third and a half, because whether two bonds are enhanced or depleted is the interaction's own doing.
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.
Spin below zero
A three-orbital radical read from its singly occupied orbital has no spin on its middle orbital unless its ends are both unequal and coupled — and never negative spin anywhere. Solved exactly with an on-site repulsion, the middle orbital goes negative at once, reaches the allyl radical's measured −0.15 to −0.18 at a repulsion of five and a half to seven electronvolts, and responds to the end–end coupling at first order where the one-determinant leak responds at second. The two have opposite signs, so in a bent, unequal radical the middle orbital's spin falls and then rises, and the unequal ends that let the leak in drain most of the polarisation first.
Six electronvolts was a difference
A three-orbital radical solved with an on-site repulsion reproduced the allyl radical's negative central spin at a repulsion of about six electronvolts — six tenths of what a carbon atom's own charges cost. Add the repulsion between neighbouring sites and the repulsion allyl needs rises almost one for one with it. With the neighbour values π-electron theory uses, it lands within a quarter of an electronvolt of Pariser's 11.13 for carbon. And the band McConnell's constant left can be closed without that constant at all.
The agreement was hydrogen's energy
A three-orbital radical solved exactly with on-site and neighbour repulsion reproduced the allyl radical's central spin at an on-site repulsion of 10.89 eV, a quarter of an electronvolt from Pariser's 11.13 for carbon. Its hopping, 2.91 eV, was never allyl's: it came from a round overlap of 0.25 and hydrogen's energy of −13.6 eV. Computed from carbon, the overlap barely changes — carbon's own 2pπ overlap at 1.39 Å is 0.248 — but the energy does, and with carbon's −11.4 eV the hopping is 2.34 eV, Pariser and Parr's own β. Then the repulsion allyl needs is 9.49 eV with one standard form of the neighbour terms and 12.68 with the other. Pariser's value sits between them, not on either.
A root that moves when nothing else does
Give the ring of six an interaction that falls off as one over the distance and the give-back's numerator has three terms. It still has one root, and at that root every separation's pair change has the opposite sign from the two-bond case. But the root has a weakness the earlier essays never tested: lower the interaction by the same amount at every separation and not one state changes, while the root moves — by as much as the reach of the interaction did, and out of existence altogether if the zero is put at one bond.
Named alongside it
The objects these essays reach for when they reach for this one.
Hubbard modelExact diagonalisationElectron correlationMany-electron wavefunctionsModel limitDouble occupancyCorrelation energyDegeneracyHartree–FockPhotoelectron spectroscopyReference stateConfiguration interaction