I. INTRODUCTION The potential energy surface (PES) , U ( r ), is the configurational energy of a system of N particles as a function of the 3 N -dimensional position vector, r . Energy landscape approaches focus on the connections between crucial topographical features of the PES and the thermodynamic and kinetic properties of liquids. 1–3 Such studies have focused largely on simple liquids where interactions are dominated by strong, short-range, repulsions with weak, long-range attractions. 4 Since the hard-sphere fluid with a single length scale is a very good zeroth-order model for such systems, the relationship between the energy landscape, thermodynamics and mobility is expected to be relatively simple compared to anomalous fluids. Recent work, however, demonstrates that softening the core-repulsions in liquids with isotropic, pair-additive interactions allows one to generate a range of anomalous behaviour that mimics the behaviour of structurally more complex fluids. 5–19 In this study, we explore the energy landscape of such a core-softened fluid to under- stand the microscopic origins of water-like liquid state anomalies. Our analysis of the po- tential energy surface focuses on understanding the entropy scaling relationships that are very useful for connecting structure, mobility and entropy for a wide range of simple 20–26 and anomalous 27–30 liquids, confined fluids and polymeric melts. 31 The excess entropy ( S$_{ex}$ ) measures the reduction in the entropy ( S ) of a liquid relative to an ideal gas ( S$_{id}$ ) at the same temperature and density due to structural correlations. The effect of fluid structure on the entropy can be formally expressed as: where S$_{n}$ is the entropy contribution due to n -particle spatial correlations. 32–36 The pair correlation contribution to the excess entropy per particle of a one-component fluid of struc- tureless particles is given by where g ( r ) is the radial distribution function and s ∗ 2 = S$_{2}$/Nk$_{B}$ . The structural correlations which lower the entropy may intuitively be expected to reduce mobility by enhancing cage effects due to formation of shells of neighbouring particles. This correlated decrease in entropy and mobility can be semi-quantitatively captured through excess entropy scaling 2 s ∗ 2 = − 2 πρ ∫ ∞ 0 { g ( r ) ln g ( r ) − [ g ( r ) − 1] } r $^{2}$dr (2) S$_{ex}$ = S − S$_{id}$ = S$_{2}$ + S$_{3}$ + . . . (1)