The H₂⁺ molecular ion — exact separation in spheroidal coordinates

H₂⁺ ion

The only system of the collection with two centres: one electron in the field of two protons. Exact separation in prolate spheroidal coordinates, Coulomb spheroidal functions (confluent Heun), binding curve \(E(R)\) and vibrational levels. The eigensolve (matching of the separation constant) is done by the Python backend (via gw2py).

Hydrogen, Morse and Pöschl–Teller live on hypergeometric functions; H₂⁺ has two centres and climbs one step up: it is the “fully Heun” representative of the collection.

The H₂⁺ molecular ion in the Born–Oppenheimer approximation (nuclei fixed at distance \(R\)); atomic units.

The two-centre problem

One electron in the field of two protons:

\[ \hat H_{\rm el}=-\tfrac12\nabla^2-\frac{1}{r_a}-\frac{1}{r_b},\qquad E_{\rm tot}=E_{\rm el}+\frac1R. \]

There is no spherical symmetry: spherical-harmonic expansions do not separate. The natural geometry is the prolate spheroidal one, with the two nuclei at the foci.

Prolate spheroidal coordinates

\[ \xi=\frac{r_a+r_b}{R}\in[1,\infty),\quad \eta=\frac{r_a-r_b}{R}\in[-1,1],\quad \varphi\in[0,2\pi). \]

In these coordinates the term \(1/r_a+1/r_b=\dfrac{4}{R}\dfrac{\xi}{\xi^2-\eta^2}\) and the Laplacian separate: \(\psi=\Lambda(\xi)\,M(\eta)\,e^{im\varphi}\).

The two separated equations

With \(p^2=-E_{\rm el}R^2/2\) (bound state: \(p^2>0\)) and separation constant \(A\):

\[ \frac{d}{d\eta}\!\Big[(1-\eta^2)\frac{dM}{d\eta}\Big]+\Big[A+p^2\eta^2-\frac{m^2}{1-\eta^2}\Big]M=0, \] \[ \frac{d}{d\xi}\!\Big[(\xi^2-1)\frac{d\Lambda}{d\xi}\Big]+\Big[-p^2\xi^2+2R\xi-A-\frac{m^2}{\xi^2-1}\Big]\Lambda=0. \]

These are two Coulomb spheroidal equations — forms of the confluent Heun equation. Quantization arises from the condition that the same \(A\) and the same \(p^2\) satisfy both: \(A_{\rm ang}(p^2)=A_{\rm rad}(p^2)\).

States and nuclear motion

The \(\sigma\) states (\(m=0\)) that are even in \(\eta\) are bonding (\(1s\sigma_g\)); the odd ones antibonding (\(2p\sigma_u\)). Repeating the solution for every \(R\) yields the binding curve \(E_{\rm tot}(R)\), on which the nuclei vibrate and rotate: solving the nuclear equation \(-\frac{1}{2\mu}\chi''+E_{\rm tot}(R)\chi=E_v\chi\) (\(\mu=m_p/2\)) gives the vibrational levels.

Geometry and state

Internuclear distance R = 2.00 a₀ Electronic state 3D isovalue = 0.18 · max

Method

Exact separation in the backend: angular equation in a Legendre basis (spectral), radial equation by finite differences, matching of the separation constant \(A\) in \(p^2\). The nuclear motion solves a second 1D problem on the σg curve \(E_{\rm tot}(R)\).

The button computes the \(E(R)\) curves of σg and σu, the orbital at the chosen \(R\) and the vibrational levels; after the first run, moving \(R\) or changing the state refreshes the orbital from the backend.

3D isosurface |ψ| = iso

red: ψ>0 · blue: ψ<0 · drag to rotate · wheel to zoom

Section in the plane of the nuclei

ψ |ψ|²

vertical axis = internuclear axis z · black dots = nuclei

Binding curve E(R) and vibrational levels

run the computation

State

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Total energy

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Parameter p²

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Equilibrium R

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D_e (binding)

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Vibrational ω_e

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Why H₂⁺ “breaks the pattern” of the suite — and how it is tamed anyway.

Outside the ₁F₁ / ₂F₁ dichotomy

Hydrogen, Morse, Pöschl–Teller live on hypergeometric functions (\(_1F_1\), \(_2F_1\)) because they are one-dimensional or spherically symmetric. H₂⁺ has two centres: its separated equations are confluent Heun (the Coulomb spheroidal functions), one step above the hypergeometric. It is the “fully Heun” representative of the collection.

The visualization cannot be the “corkscrew”

The complex one-dimensional \(\Psi\) of the other pages makes no sense here: the orbital is intrinsically 3D (with cylindrical symmetry about the internuclear axis). This is why the page shows the 2D section in the plane of the nuclei, the 3D isosurface (a surface of revolution, exploiting \(m=0\)), the density \(|\psi|^2\) and the binding curve with the vibrational levels.

Numerical method

The angular equation is solved in a Legendre basis (spectral: at \(p^2=0\) it reproduces exactly \(A=q(q+1)=0,2,6,12\)); the radial one by finite differences on \(\xi\in[1,\xi_{\max}]\); the separation constant \(A\) is matched by a root search in \(p^2\) (hence in \(E\)). The nuclear motion is a second 1D problem on the \(E_{\rm tot}(R)\) curve. All in the Python backend (numpy/scipy), as for the FD pages.

Computational evidence

The solver reproduces the H₂⁺ reference values:

\(1s\sigma_g\): \(R_e\simeq 2.00\,a_0\), \(E_{\rm tot,min}\simeq -0.6026\) Ha, \(D_e\simeq 2.79\) eV;
\(2p\sigma_u\) at \(R=2\): \(-0.168\) Ha (antibonding, above dissociation);
fundamental vibrational level \(\omega_e\simeq 2190\) cm⁻¹.

Angular part validated against the Legendre eigenvalues; cross-checked with numpy/scipy.

References

D. R. Bates, K. Ledsham, A. L. Stewart, «Wave functions of the hydrogen molecular ion», Phil. Trans. R. Soc. A 246, 215 (1953). doi.
L. I. Ponomarev, L. N. Somov, J. Comput. Phys. 20, 183 (1976) — two-centre spheroidal functions.
M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, ch. 21 (spheroidal functions).
NIST DLMF, ch. 30 (spheroidal functions) and 31 (Heun). dlmf.nist.gov/30, /31.
J. M. Peek, «Eigenparameters for the 1sσg and 2pσu Orbitals of H₂⁺», J. Chem. Phys. 43, 3004 (1965). doi.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: molecular ion, H2+, prolate spheroidal coordinates, confluent Heun, Coulomb spheroidal functions, binding curve, bonding orbital, Born-Oppenheimer