How to couple the $^3p_2$ with $d_{5/2}$?

The first two electrons’ LS term is $^3p_2$,how can this term coupling with another electron $d_{5/2}$?(this electron in another shell)

for $^3p_2$,$L_{12}=1$,$S_{12}=1$,and only the orientation of $L_{12}$ and $S_{12}$are the same,the $J_{12}=2$
for $d_{5/2}$,$L_3=2$,$S_{3}=1/2$,and only the orientation of $L_{3}$ and $S_{3}$are the same,the $J_{3}=5/2$

if the orientation of $L_{12}$is opposite to $L_{3}$,then the total $L=1$,total $S=1/2$,total $J=1/2$,then the LS term is $^2P_{1/2}$

However,I found this coupling has only $^2P_{3/2}$ from this paper?

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.101.032508
(TABLE 1,Though the electron numbers of LS term have been changed)