Extended Data Fig. 2: The detailed electronic structure of [Gd₂C]²⁺·2e⁻ obtained by DFT calculations.

a, Relaxed crystal structure of the nine-slab model for the slab calculations. a × b × 3c supercell with a vacuum of 20 Å along the c-axis was used. For the relaxation of the nine-slab model, the optimised bulk structure was used for the central three-layer, while the three layers at both ends of the slab were relaxed. b,c, Calculated band structure and the Fermi surface of [Gd₂C]²⁺·2e⁻ using the nine-slab model, respectively. Blue and red colours indicate spin up and spin down components, respectively. Band thickness represents the contribution of the surface electrons. The thicker line reflects a contribution mainly from the surface character. d,e, Band structure and the Fermi surface (top view along k_z direction) of [Gd₂C]²⁺·2e⁻ obtained by bulk calculation, respectively. In contrast to the result from the nine-slab model, the bulk calculation does not reproduce the parabolic band dispersion with cylindrical 2D Fermi surface observed by ARPES (Fig. 2a and Extended Data Fig. 3a–c).