  The structural, electronic, and magnetic properties of bulk GdCu (CsCl-type)
are investigated using spin density functional theory, where highly localized
$4f$ orbitals are treated within LDA+$U$ and GGA+$U$ methods. The calculated
magnetic ground state of GdCu using collinear as well as spin spiral
calculations exhibits a C-type antiferromagnetic configuration representing a
spin spiral propagation vector
$\mathbf{Q}=\frac{2\pi}{a}(\frac{1}{2},\frac{1}{2},0)$. The parameters of the
effective Heisenberg Hamiltonian are evaluated from a self-consistent
electronic structure and are used to determine the magnetic transition
temperature. The estimated N\'{e}el temperature of the cubic GdCu using GGA+$U$
and LDA+$U$ density functionals within the mean field and random phase
approximations are in good agreement with the experimentally measured values.
In particular, the theoretical understanding of the experimentally observed
core Gd $4f$ levels shifting in photoemission spectroscopy experiments is
investigated in detail. By employing the self-consistent constrained
random-phase approximation we determined the strength of the effective Coulomb
interaction (Hubbard $U$) between localized $4f$ electrons. We find that, the
shift of Gd-$4f$ states in GdCu with respect to bulk Gd within DFT+$U$ is
sensitive to choice of lattice parameter. The calculations for $4f$-level
shifts using DFT+$U$ methods as well as Hubbard-1 approximation are not
consistent with the experimental findings.
