# Zeeman interaction in $^3\Delta_1$ state of HfF$^+$ to search for the electron electric-dipole-moment

## TL;DR

This paper presents a theoretical study on the Zeeman interaction in HfF$^+$ to optimize conditions for electron EDM measurements, focusing on g-factor suppression and electronic structure calculations.

## Contribution

It provides detailed calculations of g-factors and dipole moments in HfF$^+$, combining electronic structure methods with experimental data fitting to improve electron EDM search accuracy.

## Key findings

- Minimum g-factor difference at 7 V/cm electric field
- Calculated g-factor $G_{\parallel}$ matches experimental fit
- Molecular dipole moment agrees with experimental value

## Abstract

We report the theoretical investigation of the suppression of magnetic systematic effects in HfF$^+$ cation for the experiment to search for the electron electric dipole moment. The g-factors for $J = 1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the $^3\Delta_1$ state are calculated as functions of the external electric field. The lowest value for the difference between the g-factors of $\Omega$-doublet levels, $\Delta g = 3 \times 10^{-6}$, is attained at the electric field 7 V/cm. The body-fixed g-factor, $G_{\parallel}$, was obtained both within the electronic structure calculations and with our fit of the experimental data from [H. Loh, K. C. Cossel, M. C. Grau, K.-K. Ni, E. R. Meyer, J. L. Bohn, J. Ye, and E. A. Cornell, Science {\bf 342}, 1220 (2013)]. For the electronic structure calculations we used a combined scheme to perform correlation calculations of HfF$^+$ which includes both the direct 4-component all-electron and generalized relativistic effective core potential approaches. The electron correlation effects were treated using the coupled cluster methods. The calculated value $G_{\parallel}=0.0115$ agrees very well with the $G_{\parallel}=0.0118$ obtained in the our fitting procedure. The calculated value $D_{\parallel}=-1.53$ a.u. of the molecule frame dipole moment (with the origin in the center of mass) is in agreement with the experimental value $D_{\parallel}=-1.54(1)$ a.u. [H. Loh, Ph.D. thesis, Massachusetts Institute of Technology (2006)].