Molecular Geometry & Bonding
what the shape decides
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 7 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 The upper separation sits where the first spin-raising multiplet does. Every five-centre doublet, every six-centre triplet and the three stars, by the energy of the lowest multiplet with a larger spin than the ground state, in coupling units, and the coupling at which the fit's two nuisance parameters separate for the second time. On logarithmic axes a constant product is a line of slope minus one; the line drawn puts that multiplet 359 cm⁻¹ above the ground, and the forty-four clusters lie within a factor of one and a half of it.

The excitation that raises the spin

A susceptibility fit's upper separation — the coupling at which its monomer fraction and temperature-independent term decouple for the second time — was placed by a band that broke, and the lead it left was that at fixed ground spin the first excitation ordered it. Every connected cluster of four, five and six spin-½ centres answers that. Among twenty five-centre doublets the first excitation ranks the separations at 0.36 and fewer bonds does better; the first excitation that raises the total spin ranks them at 0.86, and its product with the separation stays within a factor of one and a half where the separation itself spans three and a half. The separation sits where the first higher-spin multiplet is about 355 cm⁻¹ above the ground.

applied · Magnetism The window's top moves every separation with it. The median of the upper separation times the first spin-raising gap, over the forty-one clusters, for windows starting at 20 K and ending between 150 and 500 K. The dashed line is the reading that it sits 1.7 kT above the ground at the window's top; the dotted line is strict proportion through the 300 K value. The medians are 189, 243, 354, 463, 569 cm⁻¹: the earlier prediction of about 240 at 200 K and 475 at 400 K comes out 243 and 463. The rise is slightly slower than proportion — as the top temperature to the power 0.92.

The window sets the constant

A susceptibility fit's upper separation, multiplied by the gap to the first spin-raising multiplet, came to about 355 cm⁻¹ across forty-one clusters, and that was read as a position: the raising multiplet sitting 1.7 kT above the ground at the top of a window ending at 300 K. The reading predicted about 240 at a 200 K top and 475 at 400 K. Re-read under nine windows, the forty-one clusters give 243 and 463, and the raising gap collapses them equally well under every one. But the constant is not quite a multiple of kT at the top — it drifts from 1.81 to 1.64 as the top rises — and the drift is the window's width: windows that are scaled copies of one another give the same constant to a per cent, because the model has no energy scale of its own.

applied · Magnetism How the temperatures are spaced moves the constant more than how wide the window is. The median of the upper separation times the spin-raising gap, in units of kT at the window's top, for windows of span 7.5, 15 and 25 with forty temperatures spaced three ways: evenly in the temperature, in its logarithm and in its inverse. At a span of fifteen they give 1.70, 1.49, 1.36. From span 7.5 to 25 even spacing falls by 9.5 per cent and logarithmic by 14.0: the logarithmic design leans more, not less.

The spacing sets the constant

A susceptibility fit's upper separation, in units of kT at the window's top, was found to depend on the window's span — 1.81 at a span of 7.5, 1.64 at 25 — and every window read had forty temperatures evenly spaced, so a wider window was also a sparser one at its cold end. Holding the span and changing the design separates the two. The number of temperatures barely matters: twenty, forty and eighty converge on one value from above, and a design with the census's own step reproduces the forty-point constant at both ends of the range. How the temperatures are spaced matters a great deal. Spaced evenly in their logarithm, as many measurements are, the constant is twelve per cent lower and leans more, not less; spaced evenly in their inverse, lower again.

applied · Magnetism

Named alongside it

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

Exchange couplingMagnetic susceptibilityModel limitSpin stateUnderdeterminationTemperatureBasis setBipartiteContractionConvergenceEigenvalueExact diagonalisation

All concepts