Gradient-dependent upper bound for the exchange-correlation energy and application to density functional theory

We propose a simple gradient-dependent bound for the exchange-correlation energy (sLL), based on the recent non-local bound derived by Lewin and Lieb. We show that sLL is equivalent to the original Lieb-Oxford bound in rapidly-varying density cases but it is tighter for slowly-varying density systems. To show the utility of the sLL bound we apply it to the construction of simple semilocal and non-local exchange and correlation functionals. In both cases improved results, with respect to the use of Lieb-Oxford bound, are obtained showing the power of the sLL bound.