Singlet and Triplet States in Two-Electron Systems

Two electrons in an atom carry two independent labels — where they are (space) and which way they spin. Quantum mechanics does not let those labels be chosen freely. The total electron wavefunction must be antisymmetric under exchange, and that single constraint chains the spatial symmetry to the spin symmetry: the two are locked together like the two sides of a ledger that must balance to minus one. Singlet and Triplet States built the four spin states of two spin-½ particles — one antisymmetric singlet and three symmetric triplet states. This article cashes that construction for the specific system that matters most: two electrons, where the antisymmetry requirement turns a spin choice into a spatial verdict. The verdict is a pairing rule. Because the singlet spin state is antisymmetric, it must ride on a symmetric spatial wavefunction; because the triplet spin state is symmetric, it must ride on an antisymmetric one. That is not a preference or an energetic tendency — it is forced by Fermi statistics, and it holds even though the Coulomb Hamiltonian never mentions spin. The consequence is a spatial correlation with no classical analogue: triplet electrons are pushed apart by an exchange "hole," singlet electrons are pulled together, and the energy difference between the two — set entirely by how far the electrons sit from each other — is the exchange splitting that organizes atomic spectra. We prove the pairing table by evaluating exchange eigenvalues explicitly, exhibit the spatial consequence with a computed number for two electrons in a box (the triplet pair sits over four times farther apart than the singlet), and tie the machinery to the physical system that made it famous — parahelium and orthohelium, the two faces of the same helium atom split by whether the two electrons form a spin singlet or triplet.

Key takeaways

Two electrons have four spin states: one antisymmetric singlet ( $S=0$ ) and three symmetric triplet states ( $S=1$ ), constructed in Singlet and Triplet States and recalled here.
Fermi antisymmetry of the total electron wavefunction locks space to spin: singlet spin (antisymmetric) $\otimes$ symmetric space; triplet spin (symmetric) $\otimes$ antisymmetric space. The other two combinations are forbidden.
The owned result is the two-electron space $\times$ spin selection table: of four space $\times$ spin pairings, only two survive Fermi statistics — proved by multiplying exchange eigenvalues to $-1$ .
The pairing has a spatial signature with no classical analogue: the antisymmetric spatial part carries a "Fermi hole" that keeps triplet electrons apart, while the symmetric part lets singlet electrons approach.
Worked number: for two electrons in a box (levels 1 and 2), the mean-square separation is $0.038\,L^2$ for the singlet (symmetric space) versus $0.168\,L^2$ for the triplet (antisymmetric space) — the triplet pair sits $4.4\times$ farther apart.
This is why the helium spectrum splits into parahelium (singlet, para) and orthohelium (triplet, ortho): the same $1s2s$ configuration lies lower as a triplet because the electrons avoid each other and pay less Coulomb repulsion.
Key idea: for two electrons, Fermi antisymmetry converts a spin choice into a spatial verdict — singlet $\leftrightarrow$ symmetric space (electrons close), triplet $\leftrightarrow$ antisymmetric space (electrons apart) — and the resulting exchange splitting organizes atomic spectra even though the Hamiltonian is spin-blind.

The core insight

For two identical electrons the total wavefunction must be antisymmetric, so spatial and spin symmetry are locked: the spin singlet ( $S=0$ , antisymmetric) pairs with a symmetric spatial state, the spin triplet ( $S=1$ , symmetric) with an antisymmetric one. This purely statistical space $\leftrightarrow$ spin correlation makes triplet electrons avoid each other and produces the exchange splitting (parahelium vs orthohelium) with no spin term in the Hamiltonian.

Where this lives today

The singlet/triplet distinction for electron pairs is one of the busiest ideas in modern physics. In quantum computing, the two-electron singlet and triplet states of a double quantum dot are a leading qubit encoding — the "singlet-triplet qubit" — where the exchange splitting this article describes is the control knob that rotates the qubit, tunable by pushing the two electrons together or apart. In chemistry, whether a molecule's ground state is a singlet or a triplet decides its reactivity: molecular oxygen is famously a ground-state triplet, which is why it is a mild oxidizer at room temperature despite being thermodynamically ravenous, while singlet oxygen is a potent, short-lived reactive species. In magnetism, the Heisenberg exchange Hamiltonian — the map from this singlet/triplet splitting onto an effective spin coupling $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2$ — is the starting point for essentially all theories of magnetic order. And in precision spectroscopy, the intercombination lines that weakly bridge parahelium and orthohelium (forbidden at leading order, allowed only through relativistic mixing) are used as frequency standards and as sensitive tests of quantum electrodynamics in two-electron systems.

The physical picture

Set two electrons loose in an atom and forget, for a moment, about the force between them. They still are not free to arrange themselves however they like. Being identical fermions, their combined wavefunction has to flip sign when you swap them. That wavefunction is a product of a spatial part and a spin part, and a product flips sign only if exactly one of its factors flips. So the two symmetries are not independent — pick the spin symmetry and you have fixed the spatial symmetry, and vice versa. For spin, the choices are already catalogued. Two spin-½ electrons make four states, sorted by total spin: a lone singlet with $S=0$ , which is antisymmetric under exchange, and a triplet of three states with $S=1$ , all symmetric. That construction — the raising-and-lowering ladder that produces $|1,1\rangle,|1,0\rangle,|1,-1\rangle$ and the orthogonal $|0,0\rangle$ — belongs to Singlet and Triplet States, and we take it as given. What this article adds is the electron-specific consequence: the spin symmetry, through Fermi antisymmetry, dictates the spatial symmetry. The dictation runs one way in each case. A singlet is antisymmetric in spin, so to make the total antisymmetric the spatial part must be symmetric — the electrons are allowed to sit on top of each other. A triplet is symmetric in spin, so the spatial part must be antisymmetric — the electrons are forbidden from coinciding, and the antisymmetric spatial wavefunction carves out a hole around $x_1 = x_2$ . Nothing in the Hamiltonian did this; the electrostatic energy does not know about spin. It is the antisymmetry postulate alone that turns a spin label into a rule about where the electrons are allowed to be.

Figure

The space–spin ledger for two electrons: only the anti-diagonal balances to −1 (Fermi-allowed)

The figure is the whole article in one grid. Four ways to combine a spatial symmetry with a spin symmetry; two of them multiply to $-1$ and are allowed for electrons; two multiply to $+1$ and are thrown out. The surviving pair is the anti-diagonal: symmetric-space-with-singlet and antisymmetric-space-with-triplet. Everything downstream — the helium spectrum, Hund's first rule, the exchange interaction of magnetism — is a consequence of which cells survive.

The pairing rule and its exchange eigenvalues

The four spin states of two electrons come pre-sorted by their behaviour under exchange, which is exactly what we need. Recall them from Singlet and Triplet States:

|0,0\rangle=\tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle\big),\qquad |1,1\rangle=|\!\uparrow\uparrow\rangle,\ \ |1,0\rangle=\tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle+|\!\downarrow\uparrow\rangle\big),\ \ |1,-1\rangle=|\!\downarrow\downarrow\rangle \qquad (1)

where: the singlet $|0,0\rangle$ carries total spin $S=0$ ; the triplet $|1,m\rangle$ carries $S=1$ . The physical read: apply the exchange operator that swaps the two spins and the singlet returns minus itself while every triplet state returns plus itself — the singlet is antisymmetric, the triplet symmetric. (The eigenvalue credentials $\hat S^2\to0$ or $2\hbar^2$ and the ladder that builds these are owned by Singlet and Triplet States; here we only use their exchange signs.) Now impose the electron law. The total state is spatial $\times$ spin, and its exchange eigenvalue is the product of the two factors' eigenvalues. For electrons that product must be $-1$ :

\hat P_{12}\,\Psi_{\text{tot}}=(\varepsilon_{\text{space}})(\varepsilon_{\text{spin}})\,\Psi_{\text{tot}}\overset{!}{=}-\Psi_{\text{tot}} \qquad (2)

where: $\varepsilon_{\text{space}},\varepsilon_{\text{spin}}\in\{+1,-1\}$ are the exchange eigenvalues of the two factors. The physical read: since the product is pinned to $-1$ , the two factors must have opposite signs. Antisymmetric spin ( $-1$ , the singlet) demands symmetric space ( $+1$ ); symmetric spin ( $+1$ , the triplet) demands antisymmetric space ( $-1$ ). This is the pairing rule, and it is a theorem, not a tendency.

The Fermi hole and the exchange splitting

The pairing rule would be a bookkeeping curiosity if it had no physical bite. Its bite is spatial. The spatial parts of the two allowed states are built from two single-electron orbitals $\psi_a$ and $\psi_b$ :

\psi_\pm(x_1,x_2)=\tfrac{1}{\sqrt2}\big[\psi_a(x_1)\psi_b(x_2)\pm\psi_b(x_1)\psi_a(x_2)\big] \qquad (3)

where: $\psi_+$ is the symmetric (para/singlet) spatial state and $\psi_-$ the antisymmetric (ortho/triplet) one. The physical read: at $x_1=x_2$ the antisymmetric combination $\psi_-$ vanishes identically — two triplet electrons can never be found at the same point — while $\psi_+$ has no such node and lets the electrons overlap. The antisymmetric state carries a "Fermi hole" around coincidence; the symmetric one carries an anti-hole (a Fermi heap). The statistical separation this induces is captured by the exchange-correlation theorem — the mean-square distance between the two electrons — which is derived and owned by Symmetric and Antisymmetric States. That article proves that for a pair built from $\psi_a,\psi_b$ ,

\langle(x_1-x_2)^2\rangle_\pm=\langle x^2\rangle_a+\langle x^2\rangle_b-2\langle x\rangle_a\langle x\rangle_b\mp2\,|X_{ab}|^2,\qquad X_{ab}=\int\psi_a\,x\,\psi_b\,\mathrm{d}x \qquad (4)

where: the top sign ( $-2|X_{ab}|^2$ ) is the symmetric (singlet) case and the bottom sign ( $+2|X_{ab}|^2$ ) the antisymmetric (triplet). We recall this from Symmetric and Antisymmetric States and use it below; what this article contributes is the electron-specific reading: the sign of the cross term is now dictated by spin through the pairing rule. A singlet's electrons sit closer by $2|X_{ab}|^2$ ; a triplet's sit farther by the same amount. Because the Coulomb repulsion $e^2/4\pi\varepsilon_0 r_{12}$ grows as the electrons approach, the state that keeps them apart pays less repulsion. The triplet, forced apart, lies lower — the origin of the exchange splitting.

Figure

The Fermi hole in the joint density: triplet electrons avoid coincidence, singlet electrons pile onto it

Derivations

Derivation AThe space–spin selection table for two electrons

Prove that of the four ways to pair a spatial exchange symmetry with a spin exchange symmetry, exactly two are allowed for two electrons: symmetric-space with the spin singlet, and antisymmetric-space with the spin triplet. The goal is to turn "spin fixes space" from a slogan into a two-line theorem grounded in explicit exchange eigenvalues, and to show that the two forbidden combinations are forbidden absolutely — not merely energetically disfavoured but excluded by the antisymmetry postulate. The result is the master table that every downstream fact about two-electron atoms rests on.

Step 1. Write the total two-electron state as a product $\Psi_{\text{tot}}=\Psi_{\text{space}}\otimes\chi_{\text{spin}}$ . Under exchange of the two electrons, each factor is an eigenstate of the swap with eigenvalue $\pm1$ , so the total picks up the product $\varepsilon_{\text{space}}\,\varepsilon_{\text{spin}}$ .

Step 2. Read the spin eigenvalues off the states of Eq. (1). Applying the spin-swap matrix, the singlet returns eigenvalue $\varepsilon_{\text{spin}}=-1$ and each triplet state returns $\varepsilon_{\text{spin}}=+1$ — computed exactly, and cross-checked against $\hat S^2$ (singlet $0$ , triplet $2\hbar^2$ ) and $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2$ ( $-\tfrac34\hbar^2$ singlet, $+\tfrac14\hbar^2$ triplet).

Step 3. Impose the electron law $\varepsilon_{\text{space}}\varepsilon_{\text{spin}}=-1$ . With $\varepsilon_{\text{spin}}=-1$ (singlet) the equation forces $\varepsilon_{\text{space}}=+1$ (symmetric space). With $\varepsilon_{\text{spin}}=+1$ (triplet) it forces $\varepsilon_{\text{space}}=-1$ (antisymmetric space). The remaining two combinations — symmetric-space/triplet and antisymmetric-space/singlet — give product $+1$ and are forbidden.

Step 4. Confirm the count. Of $2\times2=4$ space $\times$ spin pairings, exactly the two anti-diagonal entries survive; the two diagonal entries are excluded. The table is complete and there is no third option, because $\hat P_{12}^2=\mathbb 1$ leaves only the eigenvalues $\pm1$ for each factor. Verified: `verify_aq-qm-singlet-triplet-two-electron.py` — the singlet/triplet spin-swap eigenvalues ( $-1$ / $+1$ ), the $\hat S^2$ ( $0$ / $2$ ) and $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2$ ( $-3/4$ / $+1/4$ ) credentials, and the Fermi-allowed selection (only the two anti-diagonal cells) computed exactly.

Go deeper · Derivation B · 1 minThe Fermi hole as a computed separation

Derivation BThe Fermi hole as a computed separation

Quantify the spatial consequence of the pairing rule by computing the mean-square separation of two electrons in a concrete system — a one-dimensional box with the electrons in levels $1$ and $2$ — for the singlet (symmetric space) and triplet (antisymmetric space). The goal is to replace "electrons avoid each other" with an actual number and its ratio, showing that the triplet pair sits several times farther apart than the singlet pair, driven entirely by the sign of the exchange cross term. This is the exchange separation that will control the helium ortho/para splitting.

Step 1. Take the two orbitals to be the box eigenstates $\psi_1,\psi_2$ of a well of width $L$ . The single-orbital moments are standard: the cross moment is $X_{12}=\int_0^L\psi_1\,x\,\psi_2\,\mathrm{d}x=-\dfrac{16L}{9\pi^2}$ , the same integral that Symmetric and Antisymmetric States evaluates for its box pair.

Step 2. Assemble the mean-square separation from Eq. (4). The distinguishable (no exchange) value is $\langle(x_1-x_2)^2\rangle_{\text{dist}}=0.10334\,L^2$ . The exchange cross term is $2|X_{12}|^2=2(16/9\pi^2)^2L^2=0.06489\,L^2$ .

Step 3. Apply the sign dictated by the pairing rule. The singlet rides the symmetric spatial state, which subtracts the cross term: $\langle(x_1-x_2)^2\rangle_{\text{singlet}}=0.10334-0.06489=0.03845\,L^2$ . The triplet rides the antisymmetric spatial state, which adds it: $\langle(x_1-x_2)^2\rangle_{\text{triplet}}=0.10334+0.06489=0.16823\,L^2$ .

Step 4. Read the verdict. The triplet mean-square separation is $0.16823/0.03845=4.4$ times the singlet's — the triplet electrons sit over four times farther apart in mean-square distance, purely because their spin is symmetric and Fermi statistics then forces the spatial part to have a node at coincidence. No force did this; the antisymmetry postulate did. Verified: `verify_aq-qm-singlet-triplet-two-electron.py` — $X_{12}=-16L/9\pi^2$ , the distinguishable value $0.10334\,L^2$ , the cross term $0.06489\,L^2$ , and the singlet $0.03845\,L^2$ / triplet $0.16823\,L^2$ separations with ratio $4.4$ , all exact.

Limiting cases

Case	Result	Meaning
Same spatial orbital, $\psi_a=\psi_b$	antisymmetric space vanishes; only singlet survives	two electrons in one orbital must be spin-paired — this is the Pauli Exclusion Principle seen from the two-electron side
Spin-independent Hamiltonian, $[\hat H,\hat S^2_{\text{tot}}]=0$	singlet and triplet are exact eigenstates	total spin is conserved, so ortho/para never mix under Coulomb dynamics alone
Vanishing orbital overlap, $X_{ab}\to0$	exchange cross term $\to0$ ; singlet and triplet separation degenerate	distant electrons feel no exchange — the splitting is an overlap effect
Bosonic pair (integer spin) instead	pairing flips: symmetric total ⇒ triplet $\leftrightarrow$ symmetric space	the electron rule is fermion-specific; bosons pair the same symmetries (owned by Bosons and Fermions)
Add spin–orbit coupling	singlet/triplet mix; $S$ no longer exactly good	the ledger is exact only when $\hat H$ is spin-blind; relativistic corrections blur it

Common trapThe pairing is total-antisymmetry, not an energy preference — and the singlet is the antisymmetric spin state

Two traps. First, the space $\leftrightarrow$ spin correlation is not an energetic tendency the electrons could violate at a cost — it is forced by the antisymmetry postulate and holds exactly for any two-electron state, occupied or virtual. The exchange energy splitting is a separate, downstream consequence (owned by Exchange Forces and Exchange Energy); do not conflate the kinematic pairing rule with the energetic splitting it enables. Second, watch the sign: the singlet $|0,0\rangle=\tfrac1{\sqrt2}(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle)$ is the antisymmetric spin state (swap eigenvalue $-1$ ), and therefore pairs with symmetric space; the triplet is symmetric (+1) and pairs with antisymmetric space. It is easy to flip these — a common error is to call the singlet "symmetric" because it has $S=0$ . Spin magnitude and exchange symmetry are different labels; the $S=0$ singlet is antisymmetric under exchange.

Worked examples

Example 1 — the box-pair separation. Two electrons occupy levels $1$ and $2$ of a box of width $L$ . Compute the mean-square separation $\langle(x_1-x_2)^2\rangle$ for the parahelium-like (singlet) and orthohelium-like (triplet) arrangements, and report the ratio.

Step 1. Assemble the ingredients. The distinguishable mean-square separation is $\langle(x_1-x_2)^2\rangle_{\text{dist}}=0.10334\,L^2$ , and the exchange cross moment is $X_{12}=-16L/9\pi^2$ , giving $2|X_{12}|^2=0.06489\,L^2$ .

Step 2. Apply the pairing rule. The singlet is symmetric in space, so it subtracts: $0.10334-0.06489=\mathbf{0.03845\,L^2}$ . The triplet is antisymmetric in space, so it adds: $0.10334+0.06489=\mathbf{0.16823\,L^2}$ .

Step 3. Boundary reading — the ratio. The triplet-to-singlet ratio is $0.16823/0.03845=\mathbf{4.4}$ . The triplet electrons sit over four times farther apart in mean-square distance than the singlet electrons — and they pay correspondingly less Coulomb repulsion, which is why for excited configurations the triplet lies lower. Nothing in the Hamiltonian selected this; the antisymmetry postulate did.

Example 2 — the helium $1s2s$ ordering. In helium's first excited configuration $1s2s$ , one electron stays in $1s$ and one moves to $2s$ . Explain, using the pairing rule, why orthohelium ( $2\,^3S$ , triplet) lies below parahelium ( $2\,^1S$ , singlet), and place the splitting.

Step 1. Identify the spatial states. With two distinct orbitals $1s$ and $2s$ available, both symmetric ( $\psi_+$ ) and antisymmetric ( $\psi_-$ ) spatial combinations exist, so both singlet and triplet are physically realized — unlike the ground state $1s^2$ , where only the symmetric orbital product exists and only the singlet survives.

Step 2. Apply the rule. The triplet ( $S=1$ , symmetric spin) rides the antisymmetric spatial state $\psi_-$ , which has a Fermi hole keeping the two electrons apart; the singlet ( $S=0$ ) rides the symmetric $\psi_+$ , letting them approach. Because Coulomb repulsion is smaller when the electrons avoid each other, the triplet configuration has lower repulsion energy.

Step 3. Boundary — place the number. Measured spectroscopically, the $1s2s$ triplet ( $2\,^3S$ ) lies about $0.80$ eV below the singlet ( $2\,^1S$ ) — a splitting set entirely by the exchange integral, with no spin-dependent force involved. The famous metastability of orthohelium ( $2\,^3S_1$ lives roughly $7900$ s against decay to the ground state) is the same physics: the triplet cannot easily reach the singlet ground state because that would require flipping a spin. The full exchange-energy accounting is owned by The Helium Atom and Exchange Forces and Exchange Energy; here the pairing rule supplies the reason the ordering is triplet-below-singlet.

Origins · 1 minspin triplet and how this was discovered

spin triplet

two-electron spectroscopy · para/orthohelium

The spin triplet is the three-fold $S=1$ multiplet formed when two spin-½ electrons combine so that their spins are symmetric under exchange — the states $|1,1\rangle,|1,0\rangle,|1,-1\rangle$ , distinguished from the lone $S=0$ singlet by both their total spin and their exchange sign. The physical importance of the singlet/triplet distinction was recognized in the spectra of two-electron atoms long before its origin was understood: helium's lines fall into two families that never combine, and these were named parahelium (singlet system, para) and orthohelium (triplet system, ortho) — for decades a puzzle, since the two "kinds" of helium behaved almost like different elements. Werner Heisenberg's 1926 treatment of the helium atom showed that the split arises from exchange symmetry, not from any new force: the triplet's antisymmetric spatial wavefunction keeps its electrons apart and lowers their mutual repulsion. The triplet's role runs far beyond helium — Hund's first rule (maximize total spin), the Heisenberg exchange interaction of magnetism, and the spin structure of the chemical bond all rest on the same symmetric-spin/antisymmetric-space correlation. The singlet-versus-triplet choice is one of the most consequential binary distinctions in atomic and condensed-matter physics.

History

1926

— Heisenberg treats the helium atom, tracing the para/ortho split to exchange symmetry rather than a new force

1926

— the singlet/triplet spin construction for two spin-½ particles is formalized (Heisenberg, Dirac)

1927

— Hund organizes atomic ground-state terms, with maximal spin (triplet-favouring) as the first rule

1929

— Heisenberg's exchange interaction is applied to ferromagnetism, mapping spin symmetry to magnetic order

1959

— orthohelium

2\,^3S_1

metastability (lifetime

\sim7900

s) established as a benchmark of forbidden transitions

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §5.1–5.2 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 3 (spin addition) and Ch. 7 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 10 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. XIV · Bransden & Joachain, Physics of Atoms and Molecules, Ch. 6.

Provenance. The four-state spin construction (singlet $|0,0\rangle$ and triplet $|1,m\rangle$ ), the ladder derivation, and the eigenvalue credentials $\hat S^2\in\{0,2\hbar^2\}$ , $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2\in\{-\tfrac34\hbar^2,+\tfrac14\hbar^2\}$ are owned by Singlet and Triplet States and recalled here. The exchange-correlation theorem $\langle(x_1-x_2)^2\rangle_\pm$ with $X_{12}=-16L/9\pi^2$ is owned by Symmetric and Antisymmetric States; this article applies it under the electron pairing rule. The exchange energy and its $E_\pm=E_0+J\pm K$ structure are owned by Exchange Forces and Exchange Energy, and the full helium accounting by The Helium Atom — both recalled here by name, not re-derived. Every number added by this article — the singlet $0.038\,L^2$ / triplet $0.168\,L^2$ separations, their ratio $4.4$ , the exchange cross term $0.06489\,L^2$ , and the spin eigenvalues — is recomputed in the committed verification script. The helium $1s2s$ singlet-triplet gap ( $\sim0.80$ eV) and the orthohelium $2\,^3S_1$ metastable lifetime ( $\sim7900$ s) are standard atomic-data values (Griffiths §5.2, Bransden & Joachain Ch. 6).

Builds on: Indistinguishability of Identical Particles · Permutation Symmetry and the Exchange Operator · Bosons and Fermions · Symmetric and Antisymmetric States Leads to: Pauli Exclusion Principle · Exchange Forces and Exchange Energy · The Helium Atom · Hund's Rules · Slater Determinants

Why does spin determine the spatial symmetry if the Hamiltonian has no spin term?

It is not the Hamiltonian that does it — it is the antisymmetry postulate. The total electron wavefunction must flip sign under exchange, and it is a product of a spatial and a spin part. A product flips sign only if exactly one factor flips, so once you fix the spin symmetry the spatial symmetry is forced to be the opposite. The Coulomb Hamiltonian never appears in this argument; the pairing is pure kinematics of identical fermions. What the Hamiltonian does is assign different energies to the two allowed pairings, because the spatial correlation changes the electron–electron repulsion.

Is the singlet symmetric or antisymmetric? I keep getting confused.

The singlet $|0,0\rangle=\tfrac1{\sqrt2}(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle)$ is antisymmetric under exchange — swap the two spins and the minus sign flips the state to $-|0,0\rangle$ . The confusion comes from mixing two different labels: the singlet has the smallest total spin ( $S=0$ ), but "small spin" is not "symmetric." Total-spin magnitude and exchange symmetry are independent bookkeeping. Check it by hand: the triplet $|1,0\rangle$ has the same two building blocks with a $+$ sign, and swapping leaves it unchanged (symmetric, $+1$ ), while the singlet's $-$ sign makes it antisymmetric.

Does the pairing rule say anything about the ground state of helium?

Yes — it forces it to be a singlet. In the ground state both electrons occupy the same $1s$ orbital, so the antisymmetric spatial combination $\psi_-$ is built from two identical orbitals and vanishes identically. Only the symmetric spatial state survives, and by the pairing rule that must ride the singlet spin. So helium's ground state is $1\,^1S$ (parahelium), with the two electrons spin-paired. There is no orthohelium ground state — the lowest triplet is the excited $1s2s\ ^3S$ . This is exactly the Pauli exclusion principle expressed through the two-electron singlet/triplet framework.

How is this different from the general singlet–triplet article?

Singlet and Triplet States works in the abstract spin Hilbert space: it builds the four states of any two spin-½ particles, proves their $\hat S^2$ and $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2$ eigenvalues, and maps the splitting onto a spin Hamiltonian. That construction is species-agnostic. This article specializes to two electrons, where the extra ingredient is Fermi antisymmetry of the total state, which turns the spin symmetry into a spatial verdict and produces the Fermi hole and the exchange splitting. In short: the earlier article owns the spin algebra; this one owns the electron-specific space $\leftrightarrow$ spin correlation and its atomic consequences.

Why is orthohelium metastable, living thousands of seconds?

Because relaxing from the lowest triplet ( $2\,^3S_1$ ) to the singlet ground state ( $1\,^1S_0$ ) would require flipping one electron's spin, changing the total spin from $S=1$ to $S=0$ . Electric-dipole radiation cannot do that — it does not act on spin — so the transition is strongly forbidden and proceeds only through feeble relativistic (spin-dependent) couplings. The result is a lifetime of order $7900$ s, astronomically long by atomic standards. Orthohelium behaves almost like a separate, long-lived species, which is exactly why the para/ortho split looked like two "kinds" of helium before its exchange origin was understood.

Do the two forbidden combinations ever appear physically?

Not for two electrons in a nonrelativistic, spin-conserving world. Symmetric-space-with-triplet and antisymmetric-space-with-singlet both make the total wavefunction symmetric, which is forbidden for fermions absolutely — they are not high-energy states, they simply do not exist in the physical Hilbert space. If you added a mechanism that broke the clean space $\otimes$ spin product (strong spin–orbit coupling, say), the labels "singlet" and "triplet" would stop being exactly good and the physical eigenstates would be mixtures — but the total state would still be antisymmetric overall. The antisymmetry of the total wavefunction is never violated; only the convenience of separately labelling space and spin can break down.

How does the exchange splitting become a magnetic interaction?

By a rewriting. The energy difference between the symmetric-space (singlet) and antisymmetric-space (triplet) configurations can be written as $E_{\text{singlet}}-E_{\text{triplet}}=2K$ , where $K$ is the exchange integral. Since the singlet has $\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2=-\tfrac34\hbar^2$ and the triplet $+\tfrac14\hbar^2$ , one can reproduce the same two energies with an effective spin Hamiltonian $\hat H_{\text{eff}}=-2K\,\hat{\mathbf S}_1\!\cdot\!\hat{\mathbf S}_2/\hbar^2+\text{const}$ . This is the Heisenberg exchange interaction: a spin–spin coupling with no actual magnetic force behind it, generated entirely by the Coulomb energy plus Fermi antisymmetry. It is the microscopic origin of ferromagnetism and antiferromagnetism. The mapping itself is owned by Singlet and Triplet States.