  In the first part of the paper we use a new Fourier technique to obtain a
Stein characterizations for random variables in the second Wiener chaos. We
provide the connection between this result and similar conclusions that can be
derived using Malliavin calculus. We also introduce a new form of discrepancy
which we use, in the second part of the paper, to provide bounds on the
2-Wasserstein distance between linear combinations of independent centered
random variables. Our method of proof is entirely original. In particular it
does not rely on estimation of bounds on solutions of the so-called Stein
equations at the heart of Stein's method. We provide several applications, and
discuss comparison with recent similar results on the same topic.
