Zeeman interaction in the Δ13 state of HfF+ to search for the electron electric dipole moment

Abstract

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 $^{3}\mathrm{\ensuremath{\Delta}}_{1}$ state are calculated as functions of the external electric field. The minimal value for the difference between the $g$ factors of $\mathrm{\ensuremath{\Omega}}$-doublet levels, $\mathrm{\ensuremath{\Delta}}\mathrm{g}=3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, is attained at the electric field 7 V/cm. The body-fixed $g$ factor, ${G}_{\ensuremath{\parallel}}$, 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 ${G}_{\ensuremath{\parallel}}=0.0115$ agrees very well with the ${G}_{\ensuremath{\parallel}}=0.0118$ obtained with our fitting procedure. The calculated ab initio value ${D}_{\ensuremath{\parallel}}=\ensuremath{-}1.53$ a.u. for the molecule-frame dipole moment (with the origin in the center of mass) is in agreement with the experimental datum ${D}_{\ensuremath{\parallel}}=\ensuremath{-}1.54(1)$ a.u. [H. Loh, Ph.D. thesis, Massachusetts Institute of Technology, 2006.].