Concept

Conditioning — where it appears

How strongly a calculation's answer responds to small changes in its inputs. It is read from the spread of a problem's singular values, and a badly conditioned fit leaves some combination of its parameters almost undetermined however well the data agree with the model.

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

The cliff, and the slope leading to it. The smallest eigenvalue of the overlap matrix, and what an extra function is worth, as that function is brought towards one already in the basis. Both fall together, and the energy stops improving long before the matrix stops being invertible.

A function that is already there

Adding a function to a basis set can only lower the energy, so a bigger basis is a better one. What that leaves out is that the same act makes the functions less independent: an optimised basis's smallest overlap eigenvalue halves with every function added, and a function placed on top of one already present buys less than a millionth of what a well-placed one buys while driving that eigenvalue to 4×10⁻¹⁰ — past which the calculation is refused outright.

orbitals · Basis

Sixteen clusters, and the couplings at which each one's sign changes. The sign of the exponent for every cluster across couplings from 2 to 1000 cm⁻¹, read at a 20–300 K window, with each change of sign marked. No cluster keeps one sign across the range. The clusters with a ground spin change sign twice below 150 cm⁻¹, at couplings that fall as the ground spin rises; the singlets first change sign above 300 cm⁻¹. The spin rule — positive for a singlet ground state, negative otherwise — is right on all sixteen at once only between 20.1 and 47.0 cm⁻¹, and the working point every earlier reading used, 50 cm⁻¹, is just outside it.

The sign rule holds between two poles

The sign of a susceptibility fit's exponent was put down to the parity of the spin count, then to the spin of the ground state, and every cluster tried had the two aligned. A star of four spins is even with a ground spin of one, and its exponent is −6.84. But swept across the coupling, every cluster's sign changes, through poles the ground spin places, and the spin rule is right on all sixteen clusters only between 20.1 and 47.0 cm⁻¹.

applied · Magnetism $Where the nuisance parameters separate hardly depends on purity. For each of the seven clusters with a ground spin, the two couplings below 150 cm⁻¹ at which the monomer fraction and the temperature-independent term are uncorrelated, against the monomer fraction from half a per cent to sixteen, on logarithmic axes. Every cluster keeps both at every fraction. For ground spins of one and above they move by under three per cent; for the three doublets by about a tenth. A circle marks a pole of the third direction's exponent there and a square a zero.$

Purity renames the poles

A susceptibility fit's sign poles were located at one monomer impurity, two per cent, and a real sample's impurity is rarely known. Swept from half a per cent to sixteen, the couplings where the fit's nuisance parameters separate barely move for ground spins of one and above and move a tenth for doublets. But the same separation stops being a pole and becomes a zero of the exponent at a few per cent, and above six to eight per cent every singlet acquires separations of its own.

applied · Magnetism

Stars drift above the band and K₂,ₙ graphs fall below it. The upper separation coupling times (S + ½) against ground spin for every cluster computed, with the seven that defined the band shaded between 114.7 and 129.3 cm⁻¹. The stars, from four centres to eight, run 129.3, 114.7, 117.3, 127.2, 140.3 — down and then steadily up. The K₂,ₙ graphs, from K₂,₃ to K₂,₆, run 114.8, 124.0, 100.9, 90.0 — up and then steadily down. K₃,₅ sits at 113.9. Seven clusters of up to six centres happened to lie where the two families cross.

Five more clusters break the band

Seven spin clusters put their upper separation coupling times the ground spin plus a half inside a band from 115 to 129 cm⁻¹, which looked like a law of where a susceptibility fit's nuisance parameters decouple. Five clusters built to test it — two larger stars and three larger bipartite graphs — land inside it once. Stars drift above and bipartite graphs below, and neither the shape of the curve nor the first excitation places the separation instead.

applied · Magnetism

Named alongside it

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

Exchange coupling Magnetic susceptibility Model limit Spin state Underdetermination Basis set Bipartite Contraction Convergence Eigenvalue Exact diagonalisation Gaussian

All concepts