The Hydrogen Molecule Ion H2+ as the Reference System

H₂⁺ is far from a museum piece. It is the dominant molecular cation in the interstellar medium and in diffuse molecular clouds, formed by cosmic-ray ionization of H₂, and its reaction with H₂ to form H₃⁺ is one of the most important ion-molecule reactions in astrochemistry, seeding the entire network of interstellar chemistry that eventually builds complex organic molecules. Its exact spectroscopic energy levels, computed to extraordinary precision using the Born–Oppenheimer machinery set up here (plus small non-adiabatic and relativistic corrections), serve as benchmarks for testing quantum electrodynamics in molecules and for extracting fundamental constants like the proton-to-electron mass ratio from precision rovibrational spectroscopy. In mass spectrometry, H₂⁺ and its fragmentation thresholds (as in the worked appearance-energy example above) are routine calibration and diagnostic tools. In plasma physics and fusion research, H₂⁺ (along with H₃⁺) is a significant species in hydrogen plasmas and neutral beam injection systems. Pedagogically and computationally, H₂⁺ remains the standard first test case for every new electronic structure method — Hartree-Fock theory, density functional approximations, explicitly correlated methods — because its Born–Oppenheimer potential curve is known to extremely high numerical accuracy and any new method's performance on it is immediately diagnostic. Even the most advanced quantum chemistry codes, aimed ultimately at drug-sized molecules or solid-state materials, are still validated in part by their ability to reproduce the exact H₂⁺ curve that Burrau first computed by hand nearly a century ago.

Ask the tutor: How do the LCAO bonding and antibonding trial wavefunctions built from two 1s orbitals actually reproduce (and fall short of) this exact H₂⁺ energy curve?

Picture two bare protons, labeled A and B, separated by a distance $R$, with a single electron somewhere in the space around them. This is the entire physical system: three charged particles interacting only through Coulomb's law. Unlike the hydrogen atom, there is no spherical symmetry — the electron feels two attracting centers instead of one, and the geometry is only cylindrically symmetric about the line joining the two nuclei (the internuclear axis, conventionally the $z$-axis). The full time-independent Schrödinger equation for this three-particle system, before any approximation, involves the kinetic energy operators of both nuclei and the electron, plus three Coulomb interaction terms: electron–proton A, electron–proton B, and proton–proton repulsion. Solving this exactly, nuclei and electron together, is intractable — but it does not need to be solved that way. The mass of a proton is about 1836 times the electron mass, so the nuclei move sluggishly on the timescale of electronic motion. This is the physical content of the Born-Oppenheimer approximation: freeze the nuclei at a fixed separation $R$, solve the electronic problem exactly at that geometry, and only afterward let $R$ vary to trace out a potential energy curve. Once the nuclei are frozen, the remaining problem is genuinely a one-electron problem: a single particle of charge $-e$ moving in the static electric field of two point charges $+e$ separated by $R$. This is why H₂⁺ occupies such a privileged position — it is structurally identical to the hydrogen atom problem, except that spherical symmetry has been replaced by a lower, cylindrical symmetry. The natural coordinate system for such a two-center problem is not spherical polar coordinates but prolate spheroidal coordinates, built from the two nuclear foci. In these coordinates, remarkably, the Schrödinger equation separates completely into two ordinary differential equations, one depending only on a coordinate that measures "distance from both nuclei combined" and one that measures "difference of distances." No other molecule enjoys this luxury: the moment you add a second electron, electron–electron repulsion couples the two electronic coordinates and the equation no longer separates in any coordinate system. H₂⁺ is thus the last exactly solvable case before approximation becomes mandatory.

The Hamiltonian sketched in the figure has an intuitive reading: kinetic energy of the electron, attraction to nucleus A, attraction to nucleus B, and a constant nuclear repulsion term $1/R$ that (because the nuclei are frozen) simply shifts the electronic energy up or down without affecting the electron's wavefunction at all. This last point trips up many students: the $1/R$ term does not enter the electronic Schrödinger equation as an operator acting on anything — it is a number, added after the electronic eigenvalue problem is solved, to get the total Born-Oppenheimer energy at that geometry. Qualitatively, the exact ground-state electronic wavefunction of H₂⁺ looks like a distorted 1s orbital that has been smeared between the two protons: near each nucleus it looks locally like a hydrogen 1s function (cusped at the nucleus, as required by the Kato cusp condition), while in between it interpolates smoothly, building up amplitude in the internuclear region. This buildup of electron density between the nuclei — screening their mutual repulsion and simultaneously being pulled toward both — is the microscopic origin of the chemical bond, and it is exactly this feature that the LCAO approximation is built to mimic using nothing more than a sum of two atomic 1s orbitals.

The full molecular Hamiltonian, before any approximation, includes the kinetic energy of both nuclei ($T_A$, $T_B$) and the electron ($T_e$), plus all Coulomb interactions:

where: $M_A, M_B$ are the nuclear masses (equal, since both nuclei are protons), $m_e$ is the electron mass, $r_A$ and $r_B$ are the instantaneous electron–nucleus distances, and $R$ is the instantaneous internuclear distance. Reading: this is simply Coulomb's law applied to three point charges plus their kinetic energies — nothing here is unique to quantum mechanics except the operator form of the kinetic energies.

The Born–Oppenheimer separation exploits $M_A, M_B \gg m_e$ to argue that nuclear kinetic energy can be dropped from the electronic problem, and $R$ treated as a fixed parameter:

where: $\hat{H}_{\text{el}}(R)$ is the clamped-nucleus electronic Hamiltonian, parametrically dependent on $R$, and the last term is a constant (not an operator) at fixed $R$. Reading: solving $\hat{H}_{\text{el}}(R)\,\psi(\mathbf{r}; R) = E_{\text{el}}(R)\,\psi(\mathbf{r}; R)$ at many values of $R$ generates the potential energy curve $E_{\text{el}}(R)$ that governs nuclear motion (vibration, dissociation) — the entire logic of thinking about a "bond" as existing on a potential surface.

In atomic units ($\hbar = m_e = e = 4\pi\varepsilon_0 = 1$, lengths in bohr $a_0$, energies in hartree $E_h$), the electronic Hamiltonian takes the compact and universally quoted form:

where: all lengths are in bohr radii and all energies in hartree ($1\ E_h = 27.211$ eV). Reading: this is the operator every subsequent LCAO treatment of H₂⁺ is built to approximate; its exact eigenvalues are the target that variational trial functions try to reach from above.

The total Born–Oppenheimer energy at a given geometry is the electronic eigenvalue plus the (already-included, but sometimes separated for clarity) nuclear repulsion:

where: because $1/R$ is already inside $\hat{H}_{\text{el}}$ as written above, $E_{\text{el}}(R)$ is the total energy directly; some texts instead solve for a "purely electronic" eigenvalue $\varepsilon(R)$ that excludes $1/R$, in which case $E_{\text{total}}(R) = \varepsilon(R) + 1/R$ — the two conventions must never be mixed. Reading: whichever convention is used, plotting $E_{\text{total}}$ against $R$ produces the molecular potential energy curve, with a minimum at the equilibrium bond length and $E_{\text{total}} \to E_{1s}\ (=-0.5\ E_h)$ as $R \to \infty$.

Example 1 — Foundational: writing and evaluating the exact electronic Hamiltonian at a specific geometry. Write the electronic Hamiltonian for H₂⁺ at $R = 2.00\ a_0$ (close to its equilibrium separation) and evaluate the potential energy term at a specific electron position. In atomic units,

With $R = 2.00\ a_0$, the constant nuclear repulsion contribution is

Now place the electron at a specific asymmetric point: $1.20\ a_0$ from nucleus A and $1.60\ a_0$ from nucleus B (consistent with these two distances and $R = 2.00\ a_0$ forming a valid triangle, since $1.20 + 1.60 > 2.00$). The potential energy at this point is

Evaluating term by term:

Summing,

Converting to electronvolts using $1\ E_h = 27.211$ eV gives $V = -26.08$ eV. This single-point potential energy evaluation is the raw ingredient that goes into any variational or numerical solution of $\hat{H}_{\text{el}}\psi = E\psi$; note that the value is asymmetric precisely because the chosen electron position was not equidistant from the two nuclei, so both Coulomb terms had to be evaluated separately rather than combined by symmetry.

Limiting check: as $r_A, r_B \to \infty$ with $R$ fixed, both attraction terms vanish and $V \to +1/R$, the bare nuclear repulsion — as expected for an electron removed to infinity.

Example 2 — Applied: estimating H₂⁺ formation energetics relevant to mass spectrometry and astrochemistry. H₂⁺ is generated whenever H₂ is ionized, for instance by electron impact in a mass spectrometer or by cosmic-ray/UV ionization in interstellar clouds. The relevant thermochemical cycle uses the ionization energy of H₂ and the bond dissociation energies of H₂ and H₂⁺. Given: the ionization energy of H₂ is $IE(\text{H}_2) = 15.43$ eV, the bond dissociation energy of neutral H₂ is $D_0(\text{H}_2) = 4.52$ eV, and the bond dissociation energy of H₂⁺ (from the exact Born-Oppenheimer solution, referenced later) is $D_0(\text{H}_2^+) = 2.65$ eV (after correcting the electronic well depth of 2.79 eV for zero-point vibrational energy). Question: what is the appearance energy for producing H⁺ + H(1s) starting from H₂, via the pathway H₂ → H₂⁺ + e⁻ followed by H₂⁺ → H⁺ + H? Construct the thermochemical cycle:

The total energy required is the sum of the two steps:

This appearance energy — the threshold electron-impact energy at which H⁺ fragment ions first appear in a mass spectrum of H₂ — is a directly measurable quantity, and its accepted experimental value is close to $18.1$ eV, in good agreement with this cycle. The calculation shows explicitly how the "purely academic" exact H₂⁺ binding energy feeds directly into a laboratory-measurable quantity in mass spectrometry.

Limiting check: if H₂⁺ were unbound ($D_0(\text{H}_2^+) \to 0$), the appearance energy would reduce to just the ionization energy of H₂ itself, $15.43$ eV — correctly reflecting that no additional energy would be needed to fall apart into H⁺ + H if there were no bond left to break.

Example 3 — Exam/competition level: virial-theorem consistency check on the exact H₂⁺ potential curve. The molecular virial theorem states that at the equilibrium geometry of any Born–Oppenheimer potential curve (where $dE/dR = 0$), the exact electronic kinetic and potential energies must satisfy

where $E_{\text{el}}$ is the total electronic energy (including nuclear repulsion) at that geometry, provided $E_{\text{el}}$ is measured relative to the separated-atom limit and the system sits exactly at a stationary point of $E(R)$. Given data for H₂⁺ near its equilibrium bond length: the total electronic energy relative to a proton plus a ground-state hydrogen atom is $E_{\text{el}} = -2.79$ eV at $R_e = 2.00\ a_0$ (the accepted exact Born–Oppenheimer well depth). Question: what are the expected kinetic and potential energy contributions (relative to the separated-atom reference) at equilibrium, and does a naive "electron simply sees more positive charge" argument correctly predict the sign of $\langle T \rangle$? Apply the virial theorem directly:

Check: $\langle T \rangle + \langle V \rangle = 2.79 + (-5.58) = -2.79$ eV $= E_{\text{el}}$ ✓, confirming internal consistency of the two virial relations. The result $\langle T \rangle > 0$ (kinetic energy increases relative to the isolated atom) is the well-known and initially counterintuitive finding first emphasized in rigorous virial analyses of chemical bonding (Ruedenberg and others): naive intuition says the electron, now attracted by two nuclei instead of one, should simply "relax" and lower its kinetic energy. The exact virial-consistent answer is the opposite — binding in H₂⁺ is accompanied by a net increase in kinetic energy, more than compensated by a larger decrease in potential energy, so that the total energy still drops. Physically, this reflects that binding requires contraction of the wavefunction toward the increased effective nuclear charge felt in the bonding region (via the potential energy drop from being pulled by both nuclei at once), and by the uncertainty principle, spatial contraction of a wavefunction raises its kinetic energy — even though the dominant energetic story is the potential energy stabilization from sitting between two attractive centers.

Limiting check: as $R \to \infty$, $E_{\text{el}} \to 0$ (by construction, relative to the separated-atom reference), so both $\langle T \rangle \to 0$ and $\langle V \rangle \to 0$ — correctly recovering the reference atomic kinetic and potential energies with no residual "molecular" contribution, exactly as required at infinite separation.

The exact solution of H₂⁺ is inseparable from the birth of quantum chemistry itself. In 1927, just a year after Schrödinger published his wave equation, Øyvind Burrau, a Danish physicist working at the University of Copenhagen under Niels Bohr, carried out the first numerically exact solution of the H₂⁺ electronic Schrödinger equation using confocal elliptic (prolate spheroidal) coordinates. Burrau's calculation was a landmark: it showed, for the first time from first-principles quantum mechanics (rather than the old Bohr-Sommerfeld quantized-orbit models that had failed for this exact system), that a stable chemical bond could emerge purely from Schrödinger's wave mechanics applied to electrostatic attraction and repulsion. His computed equilibrium bond length and dissociation energy for H₂⁺ agreed well with the (then still developing) spectroscopic picture, and gave quantum chemists their first quantitatively trustworthy target for approximate theories to reproduce. That same year, Walter Heitler and Fritz London published their treatment of H₂ using a valence-bond approach, explaining covalent bonding via exchange energy between two electrons — a landmark for the two-electron problem, running in parallel to Burrau's one-electron exact result. Shortly afterward, Friedrich Hund and Robert Mulliken developed the molecular orbital viewpoint more broadly, using H₂⁺-like one-electron orbitals delocalized over both nuclei as the building blocks for describing many-electron molecules — directly building on the conceptual foundation that Burrau's exact solution had validated.