  Molecular dynamics simulations are used to examine the relationship between
water-like anomalies and the liquid-liquid critical point in a family of model
fluids with multi-Gaussian, core-softened pair interactions. The core-softened
pair interactions have two length scales, such that the longer length scale
associated with a shallow, attractive well is kept constant while the shorter
length scale associated with the repulsive shoulder is varied from an inflexion
point to a minimum of progressively increasing depth.
  As the shoulder well depth increases, the pressure required to form the high
density liquid decreases and the temperature up to which the high-density
liquid is stable increases, resulting in the shift of the liquid-liquid
critical point to much lower pressures and higher temperatures. The pair
correlation entropy is computed to show that the excess entropy anomaly
diminishes when the shoulder well depth increases. Excess entropy scaling of
diffusivity in this class of fluids is demonstrated, showing that decreasing
strength of the excess entropy anomaly with increasing shoulder depth results
in the progressive loss of water-like thermodynamic, structural and transport
anomalies. Instantaneous normal mode analysis was used to index the overall
curvature distribution of the fluid and the fraction of imaginary frequency
modes was shown to correlate well with the anomalous behaviour of the
diffusivity and the pair correlation entropy. The results suggest in the case
of core-softened potentials, in addition to the presence of two length scales,
energetic and entropic effects associated with local minima and curvatures of
the pair interaction play an important role in determining the presence of
water-like anomalies and the liquid-liquid phase transition.
