  We present an improved version of the second order Gaussian Poincar\'e
inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and
Reinert (2009). These novel estimates are used in order to bound distributional
distances between functionals of Gaussian fields and normal random variables.
Several applications are developed, including quantitative CLTs for non-linear
functionals of stationary Gaussian fields related to the Breuer-Major theorem,
improving previous findings in the literature and obtaining presumably optimal
rates of convergence.
