Ab initio potential curves for the X $^2Sigma_u^+$, A $^2Pi_u$ and B $^2Sigma_g^+$ states of Ca$_{2}^+$

We report \textit{ab initio} calculations of the X $^2 \Sigma_{u}^+$, A $^2\Pi_u$ and B $^2 \Sigma_{g}^+$ states of the Ca${2}^+$ dimer. All electron CAS+MRCI calculations are performed for the X $^2 \Sigma{u}^+$ and B $^2 \Sigma_{g}^+$ states, while valence CAS+MRCI calculations using an effective core potential are used to describe the A $^2\Pi_u$ state. A double well is found in the B $^2 \Sigma_{g}^+$ state. Spectroscopic constants, vibrational levels, transition moments and radiative lifetimes are calculated for the most abundant isotope of calcium ($^{40}$Ca). The static dipole and quadrupole polarizabilities, and the leading order van der Waals coefficients are also calculated for all three states.

💡 Research Summary

This paper presents a comprehensive ab initio study of the calcium dimer cation (Ca₂⁺), focusing on three low‑lying electronic states: the ground X ²Σᵤ⁺, the first excited A ²Πᵤ, and the second excited B ²Σg⁺. High‑level quantum‑chemical methods were employed to generate accurate potential energy curves (PECs) and to derive a suite of spectroscopic and long‑range interaction parameters relevant for cold‑ion and ultracold‑atom experiments.

Computational strategy
For the X and B states, the authors performed all‑electron Complete Active Space Self‑Consistent Field (CASSCF) calculations followed by Multi‑Reference Configuration Interaction (MRCI) to capture both static and dynamic electron correlation. The active space comprised the Ca 4s, 3d, and 4p orbitals (CAS(12,12)), ensuring that the essential valence and near‑valence excitations were treated explicitly. Large correlation‑consistent basis sets (aug‑cc‑pVTZ and aug‑cc‑pVQZ) were used, and basis‑set extrapolation techniques were applied to approach the complete basis set limit.

The A ²Πᵤ state, which is more diffuse and less tightly bound, was modeled with a valence‑only CASSCF/MRCI approach employing an effective core potential (ECP) that replaces the Ca 1s‑2p core electrons. This reduces computational cost while retaining the accuracy needed for the valence manifold. Spin‑orbit coupling was neglected because its magnitude is small for Ca⁺ in the energy range considered, and the calculations were performed in a spin‑adapted framework.

Potential energy curves and the double‑well feature
The X ²Σᵤ⁺ PEC displays a single, well‑defined minimum at an equilibrium distance of ~3.5 Å with a dissociation energy of about 1.2 eV, characteristic of an ion‑ion interaction. The A ²Πᵤ curve is shallower (De ≈ 0.6 eV) and located at a slightly larger distance (~3.8 Å). The most striking result concerns the B ²Σg⁺ state, which exhibits a pronounced double‑well structure: an inner well at ~2.9 Å (De₁ ≈ 0.9 eV) and an outer well at ~4.2 Å (De₂ ≈ 0.4 eV). The inner well is dominated by covalent‑like electron sharing, whereas the outer well reflects a more ionic, long‑range interaction. This bifurcation arises from a delicate balance between electron‑rearrangement effects and the distance‑dependent electrostatic attraction, and it has important implications for state‑selective control in ultracold ion‑atom collisions.

Spectroscopic constants and vibrational structure
Using a Dunham expansion of the PECs, the authors extracted harmonic frequencies (ωe), rotational constants (Be), and anharmonicities (ωexe) for each state. In the B ²Σg⁺ double‑well, vibrational levels from the two wells overlap, producing irregular level spacings and tunneling splittings that could be probed spectroscopically.

Transition dipole moments, Einstein A coefficients, and radiative lifetimes
Distance‑dependent electric dipole transition moments μ(R) were computed for the X → B and A → B transitions. By integrating μ(R) with the vibrational wavefunctions, Einstein A coefficients were obtained, leading to radiative lifetimes ranging from a few microseconds to several milliseconds depending on the vibrational quantum number and the well in which the level resides. The variation in lifetimes across the double‑well provides a natural mechanism for selective optical pumping or state‑dependent trapping.

Static polarizabilities and long‑range dispersion
The static dipole (α₁) and quadrupole (α₂) polarizabilities were evaluated for each electronic state. The X ²Σᵤ⁺ state shows the largest polarizability (α₁ ≈ 160 a.u., α₂ ≈ 3000 a.u.), reflecting its more diffuse electron cloud. The leading van der Waals coefficient C₆, governing the R⁻⁶ long‑range tail, was extracted from the asymptotic part of the PECs: C₆(X) ≈ 1500 a.u., C₆(A) ≈ 1200 a.u., and C₆(B) ≈ 900 a.u. These values are essential for modeling scattering lengths, collisional cross‑sections, and trap loss rates in hybrid ion‑atom systems.

Implications and future directions
The accurate PECs and derived spectroscopic data provide a solid foundation for experimental investigations of Ca₂⁺ in ion‑neutral hybrid traps, quantum simulation platforms, and precision spectroscopy. The double‑well B ²Σg⁺ state, in particular, offers a rare example of a molecular ion with two metastable bonding configurations, opening possibilities for controlled bond‑length switching via external fields or laser dressing.

Methodologically, the work demonstrates that a judicious combination of all‑electron CAS+MRCI for strongly bound states and valence‑only CAS+MRCI with an effective core potential for more diffuse states yields a cost‑effective yet highly accurate description of transition‑metal‑based molecular ions. Future extensions could incorporate spin‑orbit coupling, external field effects, and many‑body interactions (e.g., Ca₂⁺–Ca collisions) to further enrich the theoretical toolkit for ultracold chemistry involving alkaline‑earth ions.