The H2 Molecule: The Two-Electron Bond

Every electronic structure calculation performed today — density functional theory on a drug-binding pocket, coupled-cluster calculations on a transition-metal catalyst, or a simple Hartree–Fock geometry optimization in an undergraduate computational chemistry lab — is, at its core, still solving a version of the two-electron problem set up here, just with vastly more electrons and vastly more sophisticated machinery for approximating $1/r_{12}$. The central lesson of H₂ — that electron correlation is real, physically significant, and omitted by any naive single-product wavefunction — is exactly why post-Hartree–Fock methods (configuration interaction, coupled cluster, Møller–Plesset perturbation theory) exist at all, and why density functional theory needs an explicit "exchange-correlation functional" bolted onto an otherwise one-electron-looking Kohn–Sham framework. In materials science, the singlet/triplet symmetry argument developed for H₂ reappears directly in the theory of magnetic exchange coupling in transition-metal complexes and molecular magnets, where the sign and magnitude of the exchange interaction between two spin centers (ferromagnetic vs antiferromagnetic coupling) is governed by exactly the same spin–space symmetry logic worked out here for the simplest possible two-electron system. In photochemistry and photobiology, singlet and triplet excited states of organic chromophores — governing fluorescence versus phosphorescence, and the operation of photodynamic therapy agents that rely on triplet oxygen sensitization — trace their behavior to the same antisymmetry-forced spin–space link established for H₂. In quantum computing and quantum information, two-electron spin qubits in coupled quantum dots are engineered and controlled using precisely the singlet-triplet energy splitting and exchange coupling that Heitler and London first quantified for the hydrogen molecule. The H₂ molecule also remains the standard benchmark for testing new electronic structure methods: because its Born–Oppenheimer potential energy curve has been computed (by Kołos and Wolniewicz and successors) to accuracy far beyond spectroscopic resolution, any new quantum chemistry method — a new density functional, a new coupled-cluster variant, a new quantum-computing-based eigensolver — is routinely validated first against the tiny, two-electron, exactly-benchmarked H₂ molecule before being trusted on anything larger, precisely because H₂ is the smallest system where the electron-electron repulsion problem introduced in this article shows up at all.

Ask the tutor: How does the Heitler–London wavefunction turn this spin-and-symmetry framework into an actual quantitative bond energy for H₂?

Picture the H₂⁺ problem first, purely to see what changes. There, one electron moves in the field of two fixed nuclear charges separated by distance $R$; the Hamiltonian is a sum of a kinetic energy operator and two attractive Coulomb terms, and although the equation is not solvable in simple closed form in Cartesian coordinates, it separates exactly in prolate spheroidal coordinates and can be solved to essentially unlimited numerical precision. Bonding and antibonding molecular orbitals, $\sigma_g$ and $\sigma_u^*$, emerge naturally as the two lowest solutions. Now add a second electron and a second proton's worth of nuclear charge to make H₂. Label the electrons 1 and 2, and the nuclei A and B, held apart by distance $R$ (the Born–Oppenheimer clamped-nuclei approximation, exactly as used for H₂⁺, is retained: nuclei are fixed and only electronic coordinates are dynamical). The full electronic Hamiltonian is

Every term up through the four nuclear-attraction terms is exactly the H₂⁺ Hamiltonian, once for electron 1 and once for electron 2 — that part is separable, and if the $1/r_{12}$ term were switched off, the two-electron problem would collapse back into two independent, exactly solvable H₂⁺-type problems. It is the single term $e^2/(4\pi\varepsilon_0 r_{12})$, the mutual repulsion between electron 1 and electron 2, that ruins this. Because $r_{12} = |\vec r_1 - \vec r_2|$ depends simultaneously on both electrons' positions, $\hat H$ cannot be written as $\hat h(1) + \hat h(2)$ for any single-particle operator $\hat h$. The variables do not separate, and the exact eigenfunctions of $\hat H$ are not products of one-electron functions. This is the fundamental complication that appears in H₂ and persists — in more elaborate form — in every atom beyond hydrogen and every molecule beyond H₂⁺. The second new ingredient is indistinguishability. Electrons 1 and 2 are identical fermions, so no physical prediction can depend on which electron is arbitrarily labeled "1" and which "2." The probability density $|\Psi(1,2)|^2$ must be invariant under exchanging the full coordinates (space and spin) of the two electrons, which forces the wavefunction itself to be antisymmetric:

This is not a small refinement — it is the reason electron pairing looks the way it does in chemistry. A wavefunction is built from a spatial part and a spin part. If the spatial part is symmetric under $1\leftrightarrow 2$ exchange, the spin part must be antisymmetric (the singlet), and vice versa. This spin–space entanglement is invisible in H₂⁺ (one electron, no exchange symmetry to worry about) but is the central organizing fact of H₂.

The picture above makes the structural point visually: four dashed lines are exactly the H₂⁺ attractions, duplicated for each electron. The single solid red line, $r_{12}$, is what makes the whole equation non-separable, since it ties electron 1's position to electron 2's position no matter how the nuclei are held fixed. Now consider what a naive trial wavefunction might look like, and why it instantly runs into the antisymmetry requirement. Suppose, in the crudest picture, electron 1 sits on atom A in a 1s orbital $1s_A$, and electron 2 sits on atom B in $1s_B$. The "obvious" guess is the simple Hartree product

This function is wrong in a very specific way: it says electron 1 is always near A and electron 2 is always near B, but electrons are identical, so this assignment is physically meaningless — nature does not label them. Swapping the labels, $1s_A(2)\,1s_B(1)$, is an equally valid description of the same physical arrangement, and the true wavefunction must be built from both possibilities combined into a properly symmetric or antisymmetric combination, not either one alone. This is the direct motivation for the covalent (Heitler–London) combination

which is exactly where the next construction in this line of ideas begins. The point to absorb here is only that the naive single product is illegitimate, and some symmetrized/antisymmetrized combination — paired with the correct spin function — is mandatory.

The exact non-relativistic, Born–Oppenheimer electronic Hamiltonian for H₂ at fixed internuclear distance $R$ is

where:

• $\hat h(i) = -\dfrac{\hbar^2}{2m_e}\nabla_i^2 - \dfrac{e^2}{4\pi\varepsilon_0}\left(\dfrac{1}{r_{iA}}+\dfrac{1}{r_{iB}}\right)$ is the one-electron H₂⁺-like Hamiltonian for electron $i$ moving in the field of both fixed nuclei,
• $e^2/(4\pi\varepsilon_0 r_{12})$ is the instantaneous Coulomb repulsion between the two electrons,
• $e^2/(4\pi\varepsilon_0 R)$ is the (constant, for fixed $R$) nuclear-nuclear repulsion, which shifts the total energy but does not affect the electronic wavefunction's shape,
• $R$ is a fixed parameter, not a dynamical variable — the whole calculation is repeated at a series of $R$ values to build a potential energy curve.

Reading: if the $1/r_{12}$ term were deleted, $\hat H$ would be a sum of two commuting H₂⁺-type operators, $\hat h(1)+\hat h(2)$, and the exact wavefunction would be a simple product of one-electron molecular orbitals, $\psi(1)\psi(2)$, with energy $2\varepsilon$. The presence of $1/r_{12}$ is the entire reason H₂ is harder than H₂⁺, and it is the same term (in more electrons) that makes every polyelectronic atom and molecule require approximation. The exact electronic Schrödinger equation reads

where:

• $\Psi(1,2)$ is the exact two-electron spatial-and-spin wavefunction (a function of six spatial coordinates plus two spin coordinates),
• $E(R)$ is the exact electronic energy at that $R$, which combined with $R$ dependence traces out the potential energy curve $U(R) = E(R)$.

Reading: because $\hat H$ does not separate, $\Psi(1,2)$ cannot in general be written as a product $\psi(1)\phi(2)$ of one-electron functions — the true motions of the two electrons are correlated, each electron's instantaneous position depends on where the other one is, an effect called electron correlation that any single-product wavefunction fundamentally omits. The requirement imposed by particle indistinguishability plus the Pauli exclusion principle is the antisymmetry postulate:

where:

• $\hat P_{12}$ is the permutation operator that exchanges all coordinates (spatial and spin) of electrons 1 and 2,
• the minus sign is specific to fermions (electrons); for bosons the sign would be $+1$.

Reading: this is not an approximation — it is an exact symmetry requirement on the true wavefunction of any two-electron system, and it must also be imposed, as best as possible, on every approximate trial wavefunction, or the variational method will simply produce nonsense (or the wrong state entirely). Writing $\Psi(1,2) = \chi(\vec r_1,\vec r_2)\,\Theta(\sigma_1,\sigma_2)$ as a product of a spatial function $\chi$ and a two-electron spin function $\Theta$ (valid whenever spin-independent Hamiltonians are used, which is the case here since $\hat H$ contains no spin operator), the antisymmetry condition factors into

where:

• there are exactly two ways to satisfy this: $\chi$ symmetric and $\Theta$ antisymmetric, or $\chi$ antisymmetric and $\Theta$ symmetric,
• the two-electron spin functions split exactly into one antisymmetric singlet $\Theta_{S=0} = \frac{1}{\sqrt2}\big(\alpha(1)\beta(2)-\beta(1)\alpha(2)\big)$ and three symmetric triplet functions $\Theta_{S=1}$: $\alpha(1)\alpha(2)$, $\frac{1}{\sqrt2}(\alpha(1)\beta(2)+\beta(1)\alpha(2))$, $\beta(1)\beta(2)$.

Reading: this factorization is the single most important structural fact for understanding covalent bonding. It says spatial symmetry and spin multiplicity are not independent choices — they are rigidly linked. A spin-paired (singlet) state is forced to have a spatially symmetric wavefunction, and a spin-parallel (triplet) state is forced to have a spatially antisymmetric wavefunction. Since chemists routinely describe bonding as "pairing of electrons with opposite spin," this equation is the rigorous statement of why that pairing language is physically necessary rather than a convenient fiction.

Why this predicts bonding versus repulsion before any calculation is done

Consider the two symmetry-adapted spatial combinations built from two atomic-like functions localized near each nucleus, $\phi_A$ and $\phi_B$ (not yet specified in detail — that specification is the job of the Heitler–London construction):

where:

• $\chi_+$ is symmetric under $1\leftrightarrow2$ exchange and must be paired with the singlet spin function,
• $\chi_-$ is antisymmetric under $1\leftrightarrow2$ exchange and must be paired with a triplet spin function.

Reading: in $\chi_-$, setting $\vec r_1 = \vec r_2$ makes the whole expression vanish identically — the two electrons can never occupy the same point in space in the triplet spatial function, a rigorous consequence of the Pauli principle known as the Fermi hole. This keeps the electrons apart, which lowers electron-electron repulsion but also empties out the region between the nuclei where bonding charge needs to accumulate. In $\chi_+$, no such node is forced, and probability density can build up constructively in the internuclear region. This single symmetry argument — with no numbers computed yet — already predicts that the singlet state should be bonding and the triplet state should be repulsive (or at best very weakly bound), a prediction borne out by every accurate calculation and by experiment: the ground state of H₂ is a singlet, and the lowest triplet state of H₂ (denoted $^3\Sigma_u^+$) is dissociative, with no minimum in its potential energy curve.

Example 1 — Foundational: counting the terms in the H₂ Hamiltonian and checking the "turn off repulsion" limit. Write out every term of $\hat H(1,2)$ explicitly for H₂ at $R = 0.741$ Å (the experimental equilibrium bond length) and confirm that switching off $1/r_{12}$ recovers two independent H₂⁺ problems, each with electronic energy $\varepsilon_{\text{H}_2^+}(R)$. The full Hamiltonian, in atomic units ($\hbar = m_e = e = 4\pi\varepsilon_0 = 1$, energies in hartree, distances in bohr) is

At $R = 0.741\ \text{Å} = 1.400\ a_0$ (using $1\ a_0 = 0.529177\ \text{Å}$, so $0.741/0.529177 = 1.4003$), the nuclear repulsion term is

Now imagine (unphysically, but as a diagnostic) deleting the $1/r_{12}$ term. The remaining operator is

This does separate: with $\hat h(1)\psi(\vec r_1) = \varepsilon\,\psi(\vec r_1)$ solved exactly as the H₂⁺ bonding-orbital problem at $R=1.4003\ a_0$ (numerically, $\varepsilon_{\sigma_g} \approx -0.5865$ hartree for the bonding MO energy including electron kinetic and both nuclear attractions, a standard tabulated H₂⁺ result at this geometry), the two-electron product wavefunction $\Psi = \psi(1)\psi(2)$ solves $\hat H_{\text{no-rep}}$ exactly with total electronic energy

Comparing to twice the isolated hydrogen atom energy, $2\times(-0.5\ \text{hartree}) = -1.0\ \text{hartree} = -27.21\ \text{eV}$, this crude repulsion-free estimate predicts a huge apparent bond energy of

which is roughly three times the true experimental bond energy of 4.75 eV. This dramatic overestimate is the clearest possible demonstration that electron-electron repulsion cannot be neglected: putting two electrons into the same bonding orbital without their mutual repulsion wildly overbinds the molecule, because nothing in the "no-repulsion" Hamiltonian penalizes both electrons piling into the same spatial region between the nuclei. Restoring the true $1/r_{12}$ term (which raises the energy, since it is a positive/repulsive contribution) is essential, and doing so accurately is exactly the hard part of the two-electron problem — as a limiting-case check, note that as $R\to\infty$ both formulations correctly reduce to $E\to -1.0$ hartree (two isolated ground-state H atoms), since all internuclear and interelectronic terms that involve $1/R$-type separations vanish, confirming the dissociation limit is handled consistently either way even though the bonding region is not.

Example 2 — Applied: verifying the singlet/triplet symmetry prediction against the measured H₂ potential curves. Confirm numerically, using tabulated spectroscopic data, that the ground (singlet) state of H₂ is strongly bound while the lowest triplet state is unbound, consistent with the spin–space symmetry argument derived above. The ground electronic state of H₂ is $X\,^1\Sigma_g^+$, a singlet, with experimentally measured

This is the spatially symmetric combination $\chi_+$ paired with the antisymmetric singlet spin function, exactly as required by the antisymmetry postulate for a symmetric spatial part. The lowest triplet state of H₂ is $b\,^3\Sigma_u^+$, built from the spatially antisymmetric combination $\chi_-$ paired with a symmetric triplet spin function. Spectroscopically, this state is measured to have no potential energy minimum at all — its potential energy curve decreases monotonically as $R$ increases from short range, meaning the triplet state of H₂ is purely repulsive (dissociative) at all internuclear distances; it does not support a single bound vibrational level. This is a dramatic, experimentally unambiguous confirmation of the prediction made purely from the symmetry argument: forcing the two electrons into a spatially antisymmetric arrangement (a "Fermi hole" between them) removes the constructive buildup of charge density between the nuclei that is required for bonding, leaving only the residual nuclear-nuclear and electron-electron repulsions to dominate. As a check on relative magnitudes: converting $D_e = 4.7477$ eV to a per-mole quantity,

which matches the commonly quoted H–H bond dissociation energy to within rounding, and the equilibrium bond length $R_e = 0.74144$ Å in bohr is

consistent with the value used in Example 1. The stark qualitative difference between a deeply bound singlet ($D_e \approx 4.75$ eV) and a wholly repulsive triplet, with no adjustable parameters beyond the exchange symmetry of the spin state, is one of the most convincing pieces of evidence in all of quantum chemistry that the Pauli antisymmetry principle — not some separate "bonding force" — is doing real chemical work.

Example 3 — Exam/competition level: estimating the magnitude of electron correlation energy from the variational principle. A student proposes the crude "no-correlation" trial wavefunction from Example 1 — the simple molecular-orbital product $\Psi_{\text{MO}} = \psi_{\sigma_g}(1)\psi_{\sigma_g}(2)\,\Theta_{S=0}$, using the exact H₂⁺ bonding orbital $\psi_{\sigma_g}$ at $R = 1.4011\ a_0$, then computes the true expectation value $\langle\Psi_{\text{MO}}|\hat H|\Psi_{\text{MO}}\rangle$ including the full $1/r_{12}$ operator (not omitting it as in Example 1). Standard published results for this simple MO treatment of H₂ (using the exact H₂⁺ orbital rather than a variationally reoptimized effective-nuclear-charge orbital) give a total electronic energy of approximately $-1.0996$ hartree at $R_e$, corresponding to a computed bond energy of about

Compare this to the experimental $D_e = 4.7477$ eV, and estimate what fraction of the true bond energy this simple wavefunction recovers, then explain physically the source of the remaining discrepancy.

Setting up the comparison. The simple single-product (single-configuration) treatment recovers

that is, only about 57% of the true bond energy — the remaining 43%, roughly 2.0 eV, is missing. Since 2.0 eV is comparable in size to typical chemical reaction barriers and far larger than $k_BT$ at room temperature ($0.025$ eV), this is not a minor correction; a wavefunction missing this much energy would badly mispredict reaction thermochemistry, equilibrium constants, and vibrational frequencies.

Diagnosing the source. The variational principle guarantees $\langle\Psi_{\text{trial}}|\hat H|\Psi_{\text{trial}}\rangle \ge E_{\text{exact}}$ for any normalized trial function, so the single-product treatment necessarily gives energy too high (bond too weak) relative to the true ground state — consistent with $D_{e,\text{MO}} < D_{e,\text{exact}}$ above. The physical deficiency is that $\Psi_{\text{MO}} = \psi_{\sigma_g}(1)\psi_{\sigma_g}(2)$ places both electrons in the same spatial orbital, so it assigns equal weight to configurations where both electrons sit simultaneously near nucleus A (an "ionic" $\text{H}^-\text{H}^+$-like configuration) as to configurations where they sit on opposite nuclei (a "covalent" $\text{H}\cdot\cdot\text{H}$-like configuration) — because $\psi_{\sigma_g} \propto (1s_A+1s_B)$ spreads over both nuclei, expanding the product $\psi_{\sigma_g}(1)\psi_{\sigma_g}(2)$ generates ionic terms like $1s_A(1)1s_A(2)$ with the same coefficient as covalent terms like $1s_A(1)1s_B(2)$. Physically, however, having both electrons on the same atom simultaneously is energetically costly (it is exactly the repulsive configuration that $1/r_{12}$ penalizes most), so the true wavefunction should suppress the ionic terms relative to the covalent ones — an effect called left–right (or ionic–covalent) correlation, missing entirely from a single molecular-orbital product. Recovering this piece of correlation — by mixing in configuration interaction, or by building a wavefunction from the outset that favors the covalent arrangement — is precisely the next step in constructing a quantitatively better two-electron wavefunction for H₂, and is the physical motivation for moving beyond the single MO product toward valence-bond-flavored trial functions.

The two-electron treatment of H₂ is inseparable from the names Walter Heitler and Fritz London, whose 1927 paper marked the birth of quantum chemistry as a discipline distinct from atomic physics. Heitler, a German theoretical physicist trained under Arnold Sommerfeld and later working with Max Born in Göttingen, and London, working alongside him at Zürich under Erwin Schrödinger's influence, set out to answer a question that had defeated Bohr-era physics entirely: why do two neutral hydrogen atoms, each electrically neutral and each already "satisfied" in its own 1s orbital, attract each other strongly enough to form a molecule at all? Classical electrostatics between two neutral, spherically symmetric charge distributions gives essentially no net attraction; some genuinely quantum mechanism was needed. Heitler and London's insight — built on exactly the antisymmetrization and spin-pairing logic developed here — was that treating the two electrons with a properly antisymmetrized, spin-paired wavefunction (rather than as classical charge clouds, or as independent particles in separate orbitals) produces a large exchange stabilization with no classical electrostatic analogue. Their 1927 calculation, using the simplest possible covalent trial function, recovered roughly two-thirds of the true H₂ bond energy — a landmark result at the time, since it was the first demonstration from first-principles quantum mechanics that the chemical bond is fundamentally a quantum phenomenon rooted in electron exchange and the Pauli principle, not a classical electrostatic effect. This calculation is the direct ancestor of the modern valence-bond method and is treated in full in its own right as the next natural step from the framework built here.