Zeeman interaction in the ${}^3\mathrm{\Delta }_1$ state of ${\mathbf{HfF}}^{+}$ to search for the electron electric dipole moment

A theoretical study devoted to suppression of magnetic systematic effects in ${\mathrm{HfF}}^{+}$ cation for an experiment to search for the electron electric dipole moment is reported. The $g$ factors for $J=1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the ${}^3\mathrm{\Delta }_1$ state are calculated as functions of the external electric field. The minimal value for the difference between the $g$ factors of $\mathrm{\Omega }$-doublet levels, $\mathrm{\Delta }\mathrm{g}=3\times {10}^{-6}$, is attained at the electric field 7 V/cm. The body-fixed $g$ factor, $G_{\parallel }$, was obtained both within the ab initio electronic structure calculations and with our fit of the experimental data [H. Loh, K. C. Cossel, M. C. Grau, K.-K. Ni, E. R. Meyer, J. L. Bohn, J. Ye, and E. A. Cornell, Science 342, 1220 (2013)]. For the electronic structure calculations we used a combined scheme to perform correlation calculations of ${\mathrm{HfF}}^{+}$, which includes both the direct four-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 with our fitting procedure. The calculated ab initio value $D_{\parallel }=-1.53$ a.u. for the molecule-frame dipole moment (with the origin in the center of mass) is in agreement with the experimental datum $D_{\parallel }=-1.54\left(1\right)$ a.u. [H. Loh, Ph.D. thesis, Massachusetts Institute of Technology, 2006.].

Figure 1

Calculated $g$ factors for upper (${\mathrm{g}}^u$) and lower (${\mathrm{g}}^l$) levels of $\mathrm{\Omega }$ doublets for the $J=1,F=3/2,M_F=3/2$ levels of the ${}^3\mathrm{\Delta }_1$ state of ${\mathrm{HfF}}^{+}$ as functions of the electric field.