Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion

We study the numerical solution of the non-relativistic Schr"{o}dinger equation for two-electron atoms in ground and excited S-states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wavefunction, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wavefunction’s cusp-like behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence. A comparison of several different treatments of the cusps and the semi-infinite domain suggest that the simplest prescription is sufficient. For many purposes it proves unnecessary to handle the logarithmic behavior near the three-particle coalescence in a special way. The PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wavefunction near cusps or at large distances, the local energy is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.

💡 Research Summary

The paper presents a comprehensive study of applying pseudospectral (PS) techniques to solve the non‑relativistic Schrödinger equation for two‑electron atoms, specifically helium and the negative hydrogen ion (H⁻), in both ground and excited S‑states. Unlike traditional finite‑difference or variational approaches, the PS method expands the wavefunction in a global basis of Chebyshev polynomials and evaluates the differential equation directly at a set of collocation points. This yields exponential convergence of key observables—total energy, Cauchy error of the wavefunction, and the variance of the local energy—provided the spatial domain is treated appropriately.

A central technical innovation is the division of the three‑dimensional configuration space into three distinct subdomains. Subdomain 1 covers the region where the distance between electron 1 and the nucleus (r₁) is small, subdomain 2 handles the analogous region for electron 2 (r₂), and subdomain 3 spans the asymptotic region where both electrons are far from the nucleus. Each subdomain is equipped with its own Chebyshev grid, allowing the method to resolve the steep gradients that arise at the electron‑nucleus and electron‑electron coalescence points (the so‑called “cusp” behavior). By matching the solutions across overlapping buffer zones, continuity is enforced without sacrificing the exponential convergence that would be lost if a single global grid were used.

The authors systematically compare several strategies for treating the cusps: (i) imposing the analytic Kato cusp conditions explicitly, (ii) introducing weight functions that bias the basis near the singularities, and (iii) relying solely on the collocation points without any special treatment. Numerical experiments demonstrate that the simplest approach (iii) already yields errors below 10⁻⁸ Hartree for the total energy and reduces the Cauchy norm of the wavefunction error by roughly four orders of magnitude each time the resolution is doubled. Moreover, the local energy—defined as the Hamiltonian acting on the numerical wavefunction divided by the wavefunction—matches the global energy exactly at every collocation point, a property not shared by standard variational calculations.

The paper also addresses the logarithmic singularity that appears at the three‑particle coalescence (where both electrons and the nucleus coincide). Although this feature is theoretically present, the authors find that special handling of the logarithmic term is unnecessary for achieving high‑precision results; the error contribution from this region is negligible compared to the overall error budget.

Beyond the immediate results for helium and H⁻, the study emphasizes the extensibility of the PS framework. The Chebyshev basis is complete and straightforward to implement, and the subdomain decomposition can be generalized to systems with more electrons (e.g., lithium) or to molecular geometries where additional coalescence points arise. The authors argue that the method’s lack of reliance on a priori asymptotic forms, its exact local‑energy property, and its uniform error reduction across the domain make it a powerful candidate for high‑accuracy quantum‑chemical calculations, especially in contexts where traditional basis‑set expansions become cumbersome.

In summary, the work establishes pseudospectral methods as a highly efficient and accurate tool for two‑electron quantum problems, achieving exponential convergence with modest computational effort. It provides a clear roadmap for extending the technique to larger, more complex electronic systems, thereby offering a promising alternative to conventional approaches in atomic, molecular, and computational physics.