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Graphical abstract The band structures of Ag(1 1 1) slabs of various thicknesses. Highlights • Splitting of surface bands of clean films depends on the thickness of slabs. • The splitting stems from the interaction between the surfaces. • For clean Cu(1 1 1), Ag(1 1 1), and Au(1 1 1) films, spin-splitting is absent. • Existence of a surface itself does not destroy inversion symmetry of the system. • Hydrogen monolayer on one face of the films leads to Rashba splitting. Abstract The splitting of the surface bands of Cu(1 1 1), Ag(1 1 1), and Au(1 1 1) films is studied by means of relativistic density-functional theory (DFT) calculations. It is found that the splitting of the bands of clean films obtained in semirelativistic approximation, in agreement with previous studies, stems from the interaction between the surfaces and hence depends on the thickness of the slabs. Since the Kramers' degeneracy is not broken by the presence of the surface, the spin-orbit coupling does not lead to the spin-splitting of the bands. An H monolayer, deposited on one face of the film, breaks the inversion symmetry and results in the appearance of the Rashba-type splitting of the bands. [ABSTRACT FROM AUTHOR]