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We investigate roles of electron correlation effects in the determination of$g_j$ factors of the $ns~^2S_{1/2}$ ($n$=5,6,7), $np~^2P_{1/2,3/2}$ ($n$=5,6),$5d~^2D_{3/2,5/2}$, and $4f~^2F_{5/2,7/2}$ states of the singly ionized cadmium(Cd$^+$) ion. Single and double excited configurations along with importantvalence triple excited configurations through relativistic coupled-cluster(RCC) theory are taken into account for incorporating electron correlationeffects in our calculations. We find significant contributions from the triplesto the lower $S$ and $P$ states for attaining high accuracy results. Thecontributions of Breit interaction and lower-order quantum electrodynamicseffects, such as vacuum polarization and self-energy corrections, are alsoestimated using the RCC theory and are quoted explicitly. In addition, wepresent energies of the aforementioned states from our calculations and comparethem with the experimental results to validate $g_j$ values. Using the $g_j$factor of the ground state, systematical shift due to the Zeeman effect in themicrowave clock frequency of the $|5s~^2S_{1/2}, F=0,m_F=0 \rangle\leftrightarrow |5s~^2S_{1/2}, F=1,m_F=0 \rangle$ transition in $^{113}$Cd$^+$ion has been estimated.

physical review. a/physical review, a

Roles of electron correlation effects for accurate determination of  $g_j$ factors of low-lying states of $^{113}$Cd$^+$ and their applications to  atomic clock