On bound state spectra of the one-electron diatomic ions

Abstract: The total energies of a large number of diatomic (or two-center) one-electron $A^{+} B^{+} e<sup>{-}$ ions with unit electrical charges are determined numerically to high accuracy. Based on these results we derive some accurate mass-interpolation formulas for the total energies of such three-body systems (ions). These formulas can be applied to the both symmetric $A^{+} A^{+} e<sup>{-}$ and non-symmetric $A^{+} B^{+} e<sup>{-}$ diatomic ions. Based on the results obtained in this study we also consider a few actual and currently unsolved problems, which are known for the two-center (or diatomic) one-electron ions.

• The paper presents a novel variational method using complex exponential basis functions that significantly outperforms traditional adiabatic approximations.
• It derives precise mass-interpolation formulas to accurately predict total energies as functions of nuclear masses in diatomic ions.
• Extensive numerical benchmarks confirm the method’s high precision and its practical implications for spectroscopy, astrophysics, and theoretical modeling.

Highly Accurate Bound State Calculations of One-Electron Diatomic Ions

Introduction and Motivation

This study presents a rigorous numerical analysis of bound state spectra in one-electron diatomic ions of the form $A^{+}B^{+}e^{-}$, addressing both symmetric and asymmetric cases with unit nuclear charges. The focus is on an accurate computation of ground ($1s\,\sigma$) state energies, complemented by the derivation and validation of precise mass-interpolation formulas for predicting total energies as functions of nuclear masses. The relevance of these ions extends to atomic, molecular, plasma, and astrophysical physics, where they serve as benchmarks for theoretical and computational advances.

Limitations of the Adiabatic Approximation

A central assertion is the inadequacy of the adiabatic (Born-Oppenheimer) approximation for highly accurate bound state calculations in diatomic one-electron ions with finite-mass nuclei. The traditional parameter $r = m_e / M$ is substantially smaller than the true expansion parameter $\tau = \sqrt[4]{m_e/M}$ that governs convergence in adiabatic-based variational methods. For typical ions, $\tau$ is not sufficiently small, leading to slow convergence and limiting achievable numerical precision. The failure is especially pronounced for relatively light nuclei, as evidenced by benchmarks for ions such as H$_2^+$, HD$^+$, and HT$^+$. The study demonstrates that variational expansions based on complex exponentials—free from the adiabatic assumption—yield significantly improved convergence and accuracy.

Three-Body Hamiltonian Formulation and Variational Approach

The non-relativistic three-body Hamiltonian is analyzed in both laboratory and relative coordinate systems. The rotationally and translationally invariant nature of the exponential basis sets in either relative ($r_{ij}$) or perimetric ($u_i$) coordinates allows extensive variational calculations. The basis functions incorporate complex non-linear parameters to capture subtle correlation effects, including nuclear vibrations and inter-nuclear localization that are inaccessible to traditional adiabatic-based expansions.

For ground states ($1s\,\sigma$0), the variational ansatz is: $1s\,\sigma$1 subject to the positivity constraint on $1s\,\sigma$2 for convergence. Numerical optimization of these parameters enables attainment of precision up to 116 decimal digits.

Analysis of Adiabatic Divergence

The manuscript provides a detailed mathematical account of the adiabatic divergence in three-body variational expansions. It is shown that, as nuclear mass increases, the Hamiltonian’s dependence on derivatives with respect to inter-nuclear separation vanishes, leading to wavefunctions with $1s\,\sigma$3-type localization—a scenario poorly modeled by real-parameter exponential expansions. Complex-parameter expansions resolve this deficit, accurately capturing the "adiabatic" vibrational and localization regime.

The convergence parameter $1s\,\sigma$4 in the expansion's asymptotics drops drastically for heavier nuclei, indicating the divergence. For example, $1s\,\sigma$5 decreases from $1s\,\sigma$6 in $1s\,\sigma$7 to below $1s\,\sigma$8 in H$1s\,\sigma$9, correlating with slower convergence and reduced accuracy.

Derivation of Mass-Interpolation Formulas

The total energies of one-electron diatomic ions are shown to obey mass-interpolation formulas, parameterized by nuclear masses: $r = m_e / M$0 where $r = m_e / M$1 are dimensionless functions of the masses, and $r = m_e / M$2 is a universal function determined by benchmarks. For symmetric ions ($r = m_e / M$3), the interpolation is a regular power series in $r = m_e / M$4. However, inclusion of the adiabatic ($r = m_e / M$5) limit requires use of Puiseux series containing rational exponents ($r = m_e / M$6), reflecting changes in dynamical symmetry.

For non-symmetric ($r = m_e / M$7) ions, the interpolation incorporates an antisymmetry parameter, and optimal coefficients are derived via least-squares fitting to numerically computed energies. The methodology enables predictions of energies for arbitrary mass combinations with high accuracy, circumventing direct bound-state calculations.

Partial Adiabatic Limits

The research identifies "partial adiabatic limits," a phenomenon where, in a sequence of ions with one fixed nucleus and the other tending toward infinite mass, the ground state energy approaches a finite value determined uniquely by the fixed mass. These limits are derived and tabulated for $r = m_e / M$8 values of $r = m_e / M$9, $\tau = \sqrt[4]{m_e/M}$0, and $\tau = \sqrt[4]{m_e/M}$1, illustrating their convergence properties and practical implications.

Bound State Spectra: Multiplicities and Operator Classification

The spectral structure of bound states is systematically analyzed. For increasing nuclear masses, the number of bound states grows rapidly. The discrete spectra are classified via the convergence properties of Hilbert-Schmidt sums. For finite nuclei, the discrete spectrum is finite and the Hamiltonian is Hilbert-Schmidt. In contrast, for the purely adiabatic $\tau = \sqrt[4]{m_e/M}$2H$\tau = \sqrt[4]{m_e/M}$3 ion, the Hamiltonian of discrete states is a nuclear (kernel) completely continuous operator, exhibiting unique spectral properties and infinite bound state multiplicity at fixed internuclear separation.

Completeness of Exponential Basis Sets

A rigorous mathematical appendix establishes the completeness of the exponential basis in perimetric coordinates for three-body wavefunctions, utilizing the Stone-Weierstrass theorem. This ensures that, for any compact subset in the coordinate space, the exponential expansion can approximate any continuous function arbitrarily well, validating its application to both ground and rotationally excited states.

Numerical Results

Extensive tabulations furnish benchmark energies and expectation values for ground-state properties across a wide range of symmetric and asymmetric diatomic ions, including lighter (e.g., H$\tau = \sqrt[4]{m_e/M}$4, D$\tau = \sqrt[4]{m_e/M}$5, T$\tau = \sqrt[4]{m_e/M}$6), intermediate, and heavy model nuclei. Achieved numerical precision exceeds $\tau = \sqrt[4]{m_e/M}$7 to $\tau = \sqrt[4]{m_e/M}$8 for energies. Detailed convergence studies confirm robustness and efficiency of the complex exponential variational approach.

Implications and Future Directions

The findings decisively establish the limitations of adiabatic approximations in achieving high-precision results for one-electron diatomics and present the complex exponential variational method as the reference standard for such calculations. The mass-interpolation framework is critical for spectroscopic predictions, isotope effect analysis, and modeling of exotic systems with arbitrary masses (including muonic and hypothetical particles).

Theoretical implications include enhanced understanding of the transition from general three-body dynamics to the adiabatic regime and the nature of spectral operator classes in few-body quantum systems. Practically, the results and interpolation tools offer immediate application to high-precision spectroscopy and astrophysical modeling.

Future developments may involve extension to multi-electron diatomic ions, non-unit charges, highly excited states, and exploration of mass singular points using analytic tools grounded in perturbation theory and operator analysis.

Conclusion

This work provides a comprehensive and highly technical account of bound state spectra in one-electron diatomic ions, numerically and analytically establishing the superiority of three-body variational approaches over adiabatic approximations. Precise mass-interpolation formulas, operator classifications of spectra, and extensive benchmarking advance both practical and theoretical understanding of these quintessential quantum systems. The study opens the door for further methodological refinement and wider application in atomic and molecular physics.