Enhanced sensitivity to variations of fundamental constants in highly charged molecules from analytic perturbation theory

Abstract

Quasi-forbidden electronic transitions in atoms and quasi-degenerate vibronic transitions in molecules serve as powerful probes of hypothetical temporal variations of fundamental constants. Computation of the sensitivity of a transition to a variation of the fine-structure constant is conventionally performed by numerical variation of the speed of light in sophisticated electronic structure calculations, and therewith several individual calculations have to be performed. An approach is presented herein that obtains sensitivity coefficients as first order perturbation to the Dirac-Coulomb Hamiltonian and allows their computation as expectation values of the relativistic kinetic energy and rest-mass operators. These are available in essentially all \emph{ab initio} relativistic electronic structure codes. Additionally, the corresponding operators for two-component Hamiltonians are derived, explicitly for the zeroth order regular approximation Hamiltonian. The approach is applied to demonstrate great sensitivity of highly charged polar molecules that were recently proposed for high-precision spectroscopy in [Zülch \emph{et al.}, arXiv:2203.10333[physics.chem-ph]]. In particular, a high sensitivity of a wealth of quasi-degenerate vibronic transitions in PaF$^{3+}$ and CeF$^{2+}$ to temporal variations of the fine-structure constant and the electron-proton mass ratio is shown.

Figures (5)

• Figure 1: Potential energy curves for the seven energetically lowest electronic states relative to the energetic minimum of the electronic ground state of CeF^2+ computed on the level of DC-FSCCSD.
• Figure 2: Adiabatic excitation energy of CeF^2+ as a function of the variation in $\alpha$ computed on the ZORA-cGHF level. The bond lengths for the different electronic states are listed in Tab. \ref{['tab:props']} and were not changed when varying $\alpha$. Dots correspond to the values obtained when changing $\alpha$ numerically with the corresponding 4th order polynomial fit shown as dashed lines. Solid lines correspond to the analytically obtained second order expansion with $\Delta q_1$ and $\Delta q_2$ values from Tab. \ref{['tab:cghf_dhf']}.
• Figure 3: Adiabatic excitation energy of CeF^2+ as a function of the variation in $\alpha$ computed on the AOC-DHF level. The bond lengths for the different electronic states are listed in Tab. \ref{['tab:props']} and were not changed when varying $\alpha$. Dots correspond to the values obtained when changing $\alpha$ numerically with the corresponding 4th order polynomial fit shown as dashed lines. Solid lines correspond to the analytically obtained second order expansion with $\Delta q_1$ and $\Delta q_2$ values from Tab. \ref{['tab:cghf_dhf']}.
• Figure 4: Adiabatic excitation energy of CeF^2+ as a function of the variation in $\alpha$ computed on the FSCCSD level. The bond lengths for the different electronic states are listed in Tab. \ref{['tab:props']} and were not changed when varying $\alpha$. Dots correspond to the values obtained when changing $\alpha$ numerically with the corresponding 4th order polynomial fit shown as dashed lines. For comparison, curves with slopes and curvatures obtained analytically obtained on the ZORA-cGHF level are shown with their y-intercepts set to the corresponding FSCCSD values.
• Figure 5: Born-Oppenheimer potential energy curves for CeF^2+ (top) and PaF^3+ (bottom, zulch:2022) computed on the level of ZORA-cGHF shown with the respective vibrational energy levels obtained herein with a DVR approach. For CeF^2+, we emphasise the similarity to the potential energy curves reported in Ref. simpson:2025 due to a similar basis sets employed (see computational details). Exemplary states with pronounced predicted wavenumber shifts due to a variation of the fine-structure constant and the electron-proton mass-ratio are depicted. The surface plot on the bottom shows the frequency shift between transitions $(1)7/2(11) \leftarrow (\mathrm{X})5/2(0)$ and $(\mathrm{X})5/2(18) \leftarrow (\mathrm{X})5/2(0)$ in PaF^3+ as function of $\delta\alpha/\alpha$ and $\delta\mu/\mu$, where the $\alpha$-variation was computed using the respective equilibrium structures.