  Attempts to go beyond the local density approximation (LDA) of Density
Functional Theory (DFT) have been increasingly based on the incorporation of
more realistic Coulomb interactions. In their earliest implementations, methods
like LDA+$U$, LDA + DMFT (Dynamical Mean Field Theory), and LDA+Gutzwiller used
a simple model interaction $U$. In this article we generalize the solution of
the full Coulomb matrix involving $F^{(0)}$ to $F^{(6)}$ parameters, which is
usually presented in terms of an $\ell m_\ell$ basis, into a $jm_{j}$ basis of
the total angular momentum, where we also include spin-orbit coupling; this
type of theory is needed for a reliable description of $f$-state elements like
plutonium, which we use as an example of our theory. Close attention will be
paid to spin-flip terms, which are important in multiplet theory but that have
been usually neglected in these kinds of studies. We find that, in a
density-density approximation, the $jm_j$ basis results provide a very good
approximation to the full Coulomb matrix result, in contrast to the much less
accurate results for the more conventional $\ell m_\ell$ basis.
