Why electrons in our atoms line their spins up, what a d orbital is in three and four dimensions, and the spin-paired state that only four dimensions has. Everything on this page is computed as you move it, from the same integrals the paper uses.
An atom's electrons sit in orbitals, grouped into subshells of equal energy: one s orbital, then p, d and f subshells with more orbitals each. Draw each orbital as a box. Two rules decide how electrons go in. Pauli's: a box holds at most two, and then their spins point opposite ways. Hund's first rule: in a part-filled subshell, the lowest-energy arrangement puts as many spins parallel as it can, so electrons spread one to a box before any pair up. In our world every atom's ground state follows it.
Four dimensions has more orbitals in each subshell: 4 p orbitals instead of 3, 9 d instead of 5, 16 f instead of 7. Add electrons one at a time and watch.
Try 4D d with 2 electrons, then 16. Those two fillings, and 4D f with 3 or 29, are where the rule breaks.
An orbital is a wave, and its angular part is a pattern of + and − on the sphere around the nucleus. An s orbital is the same everywhere. A p orbital is + on one side and − on the other. A d orbital changes sign twice around the sphere: four lobes, or a band around the middle. In ours there are exactly five independent such patterns. In four dimensions the sphere around the nucleus is a 3-sphere, and it carries nine. You can't see a 3-sphere whole, so the viewer shows one slice of it at a time, at a chosen value of the fourth coordinate w, the way a 3D creature would meet it. Drag to turn.
Six of the nine 4D d orbitals each live in a plane: xy, xz, xw, yz, yw and zw. Four dimensions has six planes through a point where ours has three, and some pairs of them, xy and zw for one, meet only at the centre. No two planes in our space can do that; any two share a line. Keep that in mind for section 4.
Electrons repel, so the lowest state is the one that keeps them apart. Spin enters through Pauli. Two electrons with parallel spins can never be found at the same point, because their shared wave has to change sign when they swap places and so vanishes where they meet. Two electrons with opposite spins have no such hole and can crowd together. That is the usual reason for Hund's rule: same-spin electrons stay apart by nature, so they repel less. But that is a tendency of the arrangement, not a law. What decides is how far apart each possible two-electron state actually keeps them.
Put two electrons in a d subshell. Their possible states fall into a few families (terms), each with a definite spin: a triplet (spins parallel) or a singlet (spins paired). The chart shows, for each term, how likely the two electrons are to be found a given angle apart, seen from the nucleus. The dashed line is two electrons placed at random; the two thick curves are the lowest triplet and the term that comes lowest overall.
⟨cos²γ⟩ measures how much the two line up along a common axis (0 at right angles, 1 along one line). The repulsion's main angular part charges for exactly that.
Every triplet curve falls to zero at 0°: same-spin electrons never meet. In ours that is enough, and the triplet ³F is also the term that keeps the electrons closest to right angles. In four dimensions a singlet keeps them at right angles better than any triplet can. It is the ten-state term the tables call (2,0)⊕(0,2), and it leans hardest on pairs of orbitals in completely perpendicular planes (xy with zw, xz with yw, xw with yz), the arrangement section 2 said only four dimensions has.
For two d electrons each term's energy is exactly E = F⁰ + a₂F² + a₄F⁴. F⁰ is the average repulsion, the same for every term. F² and F⁴ are its angular parts, the quadrupole and the hexadecapole, which depend on the atom's orbital. a₂ and a₄ depend only on the term; they are the columns of the table above. F² is always the bigger angular part, and a₂ is how much the term lines the electrons up, so the term that keeps them at right angles tends to win. The slider sets F⁴/F²; the buttons put it where real atoms are.
In ours no value of the slider breaks Hund's rule for d²: ³F is always lowest. In four dimensions the singlet wins whenever F⁴/F² is below 5/3, and for real orbitals it is far below, 0.39 for element 26. It isn't the law between charges either: under Gauss's 1/r² the same orbital gives about two-thirds, and the singlet still wins, by more. It is four-dimensional space and its d shell.
Every open d and f subshell of the 4D table, solved the same way with each atom's own orbital. Hund's rule breaks for d² and its mirror d¹⁶ (two missing from a full shell) in all three d rows, and for 4f³ and its mirror 4f²⁹ (three missing). For most of the f block only the highest spins could be checked, so there the answer is a bound. Everywhere else it holds. Hover an element for its numbers.
For the table. The periodic table in the paper was solved assuming Hund's rule. For these elements the true energy of their configuration is lower, by the gap. The table is being re-solved with that correction, and it will be the paper's main table.
For magnetism. In ours a titanium atom, d², carries two parallel spins and is magnetic. Element 26, its counterpart in four dimensions, pairs them: as a free atom it has no net spin. The same goes for 39, 54, 66, 114 and 126.
For light and lasers. What colour an ion absorbs and whether it can lase depends on its lowest states and their spins. On the Bluminium page the d² dopant's ground state is a singlet for the same reason.
Why only four dimensions. In four dimensions every rotation splits into two independent halves, a left-handed and a right-handed turn. A d orbital carries one unit of each, which is what (1,1) means. Two electrons can add their left halves and cancel their right ones, or the other way round: that is (2,0) and (0,2). A three-dimensional rotation can't be split that way, so ours has no such state.
Is it known? We found no earlier treatment of Hund's rules in four or more dimensions (three web searches; a citation-index search is still to do). The computation is the standard one for atomic terms, carried to the 3-sphere; the checks reproduce the known terms of our own ions.
Computed live from the d shell's angular integrals (tools/multiplets.mjs), the term solver js/multiplet.js, the radial integrals of real atoms (papers/p1/hundlaw.mjs) and the survey (tools/hund.mjs, papers/p1/data/hund.json). The paper's section 7 is the written version.
ATOM⁴: the position · how it works · Lumen⁴ · IRIS⁴ · KATABASIS · Four · the arcade