Two things held this project back in the past fortnight. One was a measurement that seemed to say our method stops working when molecules get bigger. The other was money: the most accurate reference numbers we use, the anchors, needed a rented server that we had stopped paying for. This week both moved, and neither moved by luck: each was a test with its line written down before the result.

The worry about size. The method learns a correction: how a cheap calculation of a molecule’s vibrations should be adjusted to look like an expensive one. It learns from a corpus of about a thousand molecules, none with more than four fused rings and none larger than about thirty atoms. The molecules astronomers care about are bigger, from forty atoms to a hundred and more. So the question is whether the correction learned on small molecules still holds on large ones, where we cannot afford to teach it.

An earlier test, chain 39, said it got much worse: the error of a size-shifted test set was almost twice that of a matched control (a ratio of 1.96). But that test had a flaw we knew about: its small training molecules also had fewer rings, so a bigger molecule was always also a molecule with a new ring count. Size and ring count were tangled together.

Chain 41 untangles them. We took thirty molecules with five fused rings, 32 to 40 atoms: perylene, benzo[a]pyrene, benzo[e]pyrene, benzo[k]fluoranthene, picene and dibenz[a,h]anthracene, each with four relatives carrying small changes. One model was trained on the corpus as it stands, at most four rings, and tested on them. A second, the control, was also given the five-ring molecules of three of the six families, and tested on the other three. If the first model is much worse than the control, five rings must be in the training data. If they are about equal, the correction carries over.

The measure is the share of the ring coupling error that a model leaves behind, compared with making no correction at all: 0 would be perfect, 1 would be no help. Before any number, we wrote down two lines. A ratio of the two models of 1.3 or less: the correction carries. A ratio of 1.8 or more: steep, and the corpus would need every ring count we ever want to predict.

The result: 0.428 for the model that never saw five rings, 0.406 for the control. A ratio of 1.05 (the read). In frequencies, after the correction, the two models are off by 3.97 and 3.41 cm⁻¹ on these molecules, where making no correction at all is off by 22.63. And giving the control twice as many five-ring molecules barely changed it (0.413 with half, 0.406 with all), so this is not a case of the control simply having had too few examples.

So the steep penalty of chain 39 came mostly from the new ring count, not from size itself. For the ring couplings, the quantity this test was built to measure, size up to 40 atoms is not a wall.

Equally good is not yet good enough. Two models can agree and both be poor, so the ratio alone does not settle it. The absolute numbers settle part of it: without a single five-ring molecule in training, the model still removes about 82 % of the frequency error on these molecules, from 22.6 to 4.0 cm⁻¹. But our target is 3 cm⁻¹ or less in every family of vibrations, and on the five-ring molecules no family gets there, with or without five-ring examples (per family). The out-of-plane hydrogen bends and the catch-all family sit at 4 to 5 cm⁻¹, as they also do on some families inside the corpus. And one family tells a different story from the ratio: the carbon–hydrogen stretches are off by 3.2 cm⁻¹ without five-ring examples and by 1.2 to 1.4 with them. A handful of larger ring systems in the corpus clearly helps. These family numbers were promised beside the ratio before the test, but our read script left them out; we computed them only after the result was known.

The cheaper ruler. The second story is about the anchors: molecules for which we have the expensive kind of calculation, coupled cluster, the reference the network is checked and tuned against. Even that is done in two parts. The coupled-cluster part uses a small basis set, a coarse brush, because a fine brush would be far too costly. A cheaper method, MP2, then measures how much the answer changes from the coarse brush to the fine one, and that change is added on top.

That MP2 step was the expensive part in memory: 26 gigabytes per process, which only a rented server had. Five new anchors were finished in their coupled-cluster part and waiting for it, about 98 server-hours, at a time when we had decided to stop renting.

There is a well-known shortcut, density fitting, which builds the huge table of electron interactions from a much smaller one. The program we use for most of our work cannot do the derivatives we need with that shortcut; another program, psi4, can. The risk is that the shortcut changes the answer. So the test, again written down first: take two molecules whose MP2 step we already had the old way, pyridine and naphthalene, redo the step with the shortcut, and compare the vibrational frequencies that come out. Within 0.5 cm⁻¹ for every vibration: adopt it. Up to 2: note the difference and carry it as an error. Beyond: refuse it.

Before that, a sanity check on water: the shortcut’s derivatives agree with derivatives worked out step by step from its energies to 5 × 10⁻⁸ atomic units, against a line of 10⁻⁶.

The swap: pyridine’s frequencies moved by at most 0.42 cm⁻¹, naphthalene’s by at most 0.46. Both under the line of 0.5, so the shortcut is adopted (the read). I had predicted less than 0.3, and that was wrong: the line holds, with less room to spare than I expected.

What it buys: one step of the fine-brush calculation for naphthalene took 136 seconds on the laptop, against 1,996 seconds the old way on a large server. The largest memory use was 3.3 gigabytes. All five waiting anchors together come to about eight and a half hours on the laptop we already own.

The first one is in. Quinoxaline, a two-ring molecule with two nitrogen atoms, went through the whole chain this morning. Its two softest vibrations come out at 115 and 119 cm⁻¹ with the coarse brush, and at 167 and 179 with the fine-brush step added. The cheap calculation the network starts from has them at 177 and 185. The coarse brush alone would have taught the network to pull these two down by about 60 cm⁻¹; with the step, by about 10. Most of that big pull was the brush, not the molecule. It passed our promotion rule and is now one of seven anchors in use (the anchor). The other four follow today.

What this is not. Chain 41 compares two density functional methods, a cheap one and a more careful one, as a stand-in for the real target; it says the correction carries in size, not yet that the coupled-cluster correction does. It reaches 40 atoms; the 50- to 100-atom molecules are the next test, chain 42. Two of the thirty molecules left the test before any result was read, because the reference calculations for them showed an unstable vibration; they come back when better reference calculations exist. And the density-fitting shortcut was tested on two molecules and adopted with a margin smaller than predicted: if one of the new anchors behaves differently, the old way is still there, on a server, when the budget allows.