He$_2^{3+}$ and HeH$^{2+}$ molecular ions in a strong magnetic field: the Lagrange mesh approach

Accurate calculations for the ground state of the molecular ions He$_2^{3+}$ and HeH$^{2+}$ placed in a strong magnetic field $B\gtrsim 10^{2}$ a.u. ($\approx 2.35 \times 10^{11}$G) using the Lagrange-mesh method are presented. The Born-Oppenheimer approximation of zero order (infinitely massive centers) and the parallel configuration (molecular axis parallel to the magnetic field) are considered. Total energies are found with 9-10 s.d. The obtained results show that the molecular ions He$_2^{3+}$ and HeH$^{2+}$ exist at $B > 100$,a.u. and $B > 1000$,a.u., respectively, as predicted in \cite{Tu:2007} while a saddle point in the potential curve appears for the first time at $B \sim 80$ a.u. and $B \sim 740$ a.u., respectively.

💡 Research Summary

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This paper presents highly accurate non‑relativistic calculations of the ground‑state properties of two one‑electron molecular ions, He₂³⁺ (a helium‑helium system with total charge +3) and HeH²⁺ (helium‑hydrogen with total charge +2), immersed in extremely strong magnetic fields ranging from 10² to 4.4 × 10¹³ G (≈ 10²–10⁴ a.u.). The authors adopt the Born‑Oppenheimer approximation of zero order, treating the two nuclei as infinitely massive point charges fixed along the magnetic‑field direction (parallel configuration), which is known to be energetically optimal for sufficiently strong fields.

The theoretical framework starts from the non‑relativistic Hamiltonian in the symmetric gauge A = (B/2)(‑y, x, 0). The two nuclei have charges Z₁ = 2 (helium) and Z₂ = 1 (hydrogen) and are separated by a distance R along the z‑axis. The electron’s orbital angular momentum component L_z commutes with the Hamiltonian, allowing the authors to focus on the m = 0 (σ_g) state.

To solve the Schrödinger equation, the authors employ the Lagrange‑mesh method, which combines a basis of Lagrange‑interpolating functions with Gaussian quadrature at the mesh points. They use prolate spheroidal coordinates (ξ, η, φ) defined by ξ = (r₁ + r₂)/R – 1 and η = (r₁ – r₂)/R, where r₁ and r₂ are the electron‑nucleus distances. The wavefunction ψ₀(ξ, η) is expanded as a double sum over N_ξ × N_η basis functions F_{ij}(ξ, η) with coefficients c_{ij}. The kinetic‑energy matrix elements are taken from the literature, while the potential is evaluated at the zeros of the Laguerre polynomial L_{N_ξ}(x) (scaled by a parameter h) and the Legendre polynomial P_{N_η}(η). This discretisation reduces the problem to a generalized eigenvalue equation