The geometric and electronic structures of the title compounds are calculated with scalar relativistic, gradient-corrected density functional theory. The most stable geometry of ThCp₄ (Cp = η⁵-C₅H₅) and UCp₄ is found to be pseudo-tetrahedral (S₄), in agreement with experiment, and all the other AnCp₄ compounds have been studied in this point group. The metal–Cp centroid distances shorten by 0.06 Å from ThCp₄ to NpCp₄, in accord with the actinide contraction, but lengthen again from PuCp₄ to CmCp₄. Examination of the valence molecular orbital structures reveals that the highest-lying Cp π_2,3-based orbitals split into three groups of pseudo-e, t₂ and t₁ symmetry. Above these levels come the predominantly metal-based 5f orbitals, which stabilise across the actinide series, such that in CmCp₄, the 5f manifold is at more negative energy than the Cp π_2,3-based levels. The stability of the Cm 5f orbitals leads to an intramolecular ligand→metal charge transfer, generating a Cm(III) f⁷ centre and increased Cm–Cp centroid distance. Mulliken population analysis shows metal d orbital participation in the e and t₂ Cp π_2,3-based orbitals, which gradually decreases across the actinide series. By contrast, metal 5f character is found in the t₁ levels, and this contribution increases four-fold from ThCp₄ to AmCp₄. Examination of the t₁ orbitals suggests that this f orbital involvement arises from a coincidental energy match of metal and ligand orbitals, and does not reflect genuinely increased covalency (in the sense of appreciable overlap between metal and ligand levels). Atoms-in-molecules analysis of the electron densities of the title compounds (together with a series of reference compounds: C₂H₆, C₂H₄, Cp⁻, M(CO)₆ (M = Cr, Mo, W), AnF₃CO (An = U, Am), FeCp₂, LaCp₃, LaCl₃ and AnCl₄ (An = Th, Cm)) indicates that the An–Cp bonding is very ionic, increasingly so as the actinide becomes heavier. Caution is urged when using early actinide/lanthanide comparisons as models for minor actinides/middle lanthanides.