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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 calculatedmagnetic ground state of GdCu using collinear as well as spin spiralcalculations exhibits a C-type antiferromagnetic configuration representing aspin spiral propagation vector$\mathbf{Q}=\frac{2\pi}{a}(\frac{1}{2},\frac{1}{2},0)$. The parameters of theeffective Heisenberg Hamiltonian are evaluated from a self-consistentelectronic structure and are used to determine the magnetic transitiontemperature. The estimated N\'{e}el temperature of the cubic GdCu using GGA+$U$and LDA+$U$ density functionals within the mean field and random phaseapproximations are in good agreement with the experimentally measured values.In particular, the theoretical understanding of the experimentally observedcore Gd $4f$ levels shifting in photoemission spectroscopy experiments isinvestigated in detail. By employing the self-consistent constrainedrandom-phase approximation we determined the strength of the effective Coulombinteraction (Hubbard $U$) between localized $4f$ electrons. We find that, theshift of Gd-$4f$ states in GdCu with respect to bulk Gd within DFT+$U$ issensitive to choice of lattice parameter. The calculations for $4f$-levelshifts using DFT+$U$ methods as well as Hubbard-1 approximation are notconsistent with the experimental findings.

First-principles study of the electronic and magnetic properties of cubic GdCu compound