How chapter 10 first missed the strongest pairs, and what the miss shows: two molecules meet in six ways in our world and ten in 4D, and a crowd of them in far more. Watch the search, and see what it can and can't promise.
Chapter 10 first reported that a pair of Xater molecules binds at 13.3 kcal/mol. A liquid run on a force field fitted to Hartree–Fock pairs then held pairs together harder than that. Searched properly, the pair binds at 26.9: twice as hard, and in a different shape. Nothing was wrong with the arithmetic. The census had only looked in one place. It slid one molecule along a line and tilted it in one plane, two numbers, inside the family of arrangements our water's hydrogen bond makes. That family was our world's intuition, carried into a world where it doesn't hold.
This chapter is about that miss, because it holds a finding. Molecules have many more ways to meet in four dimensions than in three. Any search that stops looking early can miss the answer, and in 4D there is far more room to miss it in.
To place one rigid molecule against another you move it, one number for each direction, and turn it, one number for each plane of turning (chapter 3). In d dimensions that's d moves and d(d − 1)/2 turns. Moving grows like d and turning like d², so as dimensions are added, turning takes over:
| space | moves | turns | a pair: numbers to search | 100 molecules |
|---|---|---|---|---|
| 2D | 2 | 1 | 3 | 297 |
| 3D | 3 | 3 | 6 | 594 |
| 4D | 4 | 6 | 10 | 990 |
| 5D | 5 | 10 | 15 | 1485 |
| 6D | 6 | 15 | 21 | 2079 |
A rigid molecule's pose in d dimensions takes d(d + 1)/2 numbers; a pair's relative pose takes the same, and N molecules take N − 1 times as many. Bendable molecules add their own shapes on top.
Our water pair has six numbers to get right. Xater's has ten, and six of those are turns. A search that checks ten values along each number checks a million arrangements in our world and ten billion in 4D. Nobody searches that way; a descent from a random start follows the slope instead. But it shows how thin a two-number slice is: the census looked at a sheet through a ten-dimensional space.
Two molecules from a random start, sliding down the force field's slope to the nearest minimum, every number at once (BFGS: each step learns how the slope curves). The line under each pair is that descent's energy, and the dashed line is where the census's pair sits. When a descent ends, the molecule takes its basin's colour: arrangements within 0.3 kcal/mol and 0.15 bohr are one basin. A descent that runs out of steps before the slope has gone is counted as unfinished, not as a basin. Turn the 4D tile through w to see the turns you can't see in a slice. race runs whole descents without showing them.
The curve counts the basins found as starts accumulate. If it keeps rising, the search isn't done. Energies are the force field's; where Hartree–Fock has scored a basin (chapter 10) its number is shown beside it.
For a single pair, the landscape is kind. Of 400 random starts on Xater's force field, all 400 slid into the same arrangement, the compact antiparallel pair of chapter 10. Of 400 for our water, 378 found its one linear hydrogen bond, and the rest drifted apart or ran out of steps. Each pair's landscape is a funnel: one minimum, with every slope leading to it. For our water that's the known answer. Of the ten stationary points mapped on the water pair's surface, only the linear pair is a minimum; the others are passes between copies of it.
So the miss wasn't the landscape being hard. Any descent over all ten numbers, started almost anywhere, would have found the deeper pair. The miss was choosing the slice. Our first full search made a related mistake, and it is worth saying plainly. It counted descents that stalled before the bottom as separate minima. At a turn wound past many radians, the numbers describing it get badly conditioned and the descent crawls. Six "minima" for Xater were one minimum reached six times, not quite all the way. The descent now re-centres its turn as it goes and stops only when the slope is gone. Chapter 10's energies stand, because each was rescored and relaxed by Hartree–Fock itself.
A pair is the easy case. Add molecules and the minima multiply. Quench a liquid's arrangement to its nearest minimum, the trick behind settling every start above, and the number of distinct minima grows exponentially with the number of molecules. That is the inherent-structure picture of liquids and glasses. Three molecules already have 12 numbers in our world and 20 in 4D; seven have 36 and 60.
So we counted. Clusters of three to seven molecules were searched from random starts on each world's force field, with 2400 starts for six and 1600 starts for seven, the same in every world; three, four, five had the first count's 500 to 1000. Every start descended until its slope was gone, and the distinct minima were tallied.
Solid lines: minima found, where two are the same if their energies agree within 0.1 kcal/mol and their heavy atoms' distances within 0.1 bohr. The bar under each point runs down to the count when minima within 0.3 kcal/mol and 0.5 bohr are merged. Dashed: the Chao1 estimate of how many there are, from how many were found only once or twice, drawn only where fewer than half were seen just once (beyond that it's a guess bigger than anything measured, and the table says "still climbing"). The force field is fitted to pairs and adds them up; it has no three-body forces.
Both worlds' minima multiply, and 4D's much faster. From two molecules to seven, our water's minima go 1, 5, 5–7, 8–9, 37–45, 138–152; Xater's go 1, 2–3, 8–10, 20–25, 148–268, 342–765, and element 9's XH₃'s 1, 2, 14–16, 64–98, 348–1022, 540–1505. Where two numbers are given, the true count lies between them. The larger tells apart arrangements of the same energy by their shape, which can split one soft minimum in two. The smaller merges any two within a test three times looser, which can merge two different ones. From four molecules on, even 4D's smaller counts are above our water's larger ones. The best arrangement gets harder to find as well. Of the starts for six molecules, 8% found our water's best, 6% Xater's and 2% XH₃'s; for seven, 2%, 0.9% and one start in 1600.
None of this is the whole count. Only our water's six has nearly settled: the second half of its starts added 4 new minima. Everywhere else the discovery curve is still climbing. For seven molecules the second half of the starts added 28 new minima for our water, 266 for Xater and 733 for XH₃, where 1425 of its 1505 minima were seen just once. There the best one found, seen only once, is not shown to be the best there is. The clusters also hold together far harder in 4D: the best of seven binds at 5.1 kcal/mol per molecule for our water and 34.1 for Xater. These are the force field's energies, not Hartree–Fock's, and the force field has no three-body forces.
For a liquid this hardly matters: a liquid visits a great many minima, and the simulation samples them rather than finding one. It matters for the solid. Which crystal a 4D water freezes into is a search for one best arrangement among very many, and on this evidence 4D's search space is far larger than ours. That's where the extra turns will bite.
It may also shape the freezing itself. A liquid with more minima to visit has more entropy, and a crystal freezes where its energy gain pays for that entropy (T = ΔH/ΔS), so more minima push the freezing point down, and many nearly equal crystals make freezing slow, grainy and glassy. But 4D's clusters bind several times harder, which pushes it up. The prediction, for the liquid simulation to test: a freezing point low for its cohesion, and a liquid that would rather be a glass.
A search gives one guarantee. The energy it finds is one that some arrangement really has, so the true best can only be lower. Every binding energy in these chapters is a floor: more searching can only make it stronger. Doubling Xater's pair didn't break the census's conclusion that element 9's XH₃ and Xater tie as the water. It made both stronger.
What a search can't promise is that it's finished. That has to be argued separately, by these checks:
On a bigger machine (the compute runner), the first three checks were run.
Known answers: two found, one missed. For our ammonia pair the full search finds the best arrangement the census's scan had, 2.15 kcal/mol with the nitrogens 3.54 Å apart. (The real ammonia pair is famously floppy, its hydrogen bond barely holding a shape. Hartree–Fock on a small basis gives a definite one.) For our water it finds the linear hydrogen bond and stops 0.10 kcal/mol short of the census's relaxed pair. For hydrogen fluoride it fails. The force field's best arrangement is a straight chain, F–H···F–H. Hartree–Fock scores it at 3.77 kcal/mol and its relaxation, by small steps along each pose number, never leaves the line. The census's bent pair is deeper, 4.05, and the published structure is bent. The energy was within 7%, but the shape was wrong. That's the failure the force field's blind spots predict: the search lands where the force field points.
Hartree–Fock without the force field: nothing deeper. Two molecules were placed at random in close contact and scored by Hartree–Fock itself: 600 placements for our water, 2,400 for Xater, 1,568 for XH₃. None came out deeper than the best pair. The deepest reached 3.5 of water's 4.3 kcal/mol, 24.2 of Xater's 26.9, and 20.1 of XH₃'s 27.5. On the attractive placements the force field is off by 0.5 kcal/mol (rms) for our water and 2.6 and 2.7 for the 4D pair. On single placements it misses up to 12 kcal/mol of the pull, so it does have blind spots in 4D. None hid a deeper pair among these placements.
Chapter 10 re-baked with the corrected descent: it stands. The best pairs come out at 26.94 and 27.54 kcal/mol, as before. The force field's six "distinct" minima per molecule, which the faulty descent had reported, collapse to one each: the funnel.
So the 4D best pairs survive every check that could have beaten them. But hydrogen fluoride shows the guarantee is weaker than we'd like. Random placements are thin in ten dimensions, and a search follows its force field. Next: relax Hartree–Fock from the deepest random placements, and give the force field the lone pairs' directions. Still open: longer cluster searches where the counts haven't settled, and the other searches in this position.
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