  The Hardy-Rellich inequality in the whole space with the best constant was
firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher
dimensions $N\geq 5$. Then it was extended to lower dimensions $N\in \{3, 4\}$
by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011)
by applying totally different techniques.
  In this note we refine the method implemented by Tertikas and Zographopoulos,
based on spherical harmonics decomposition, to give an easy and compact proof
of the optimal Hardy-Rellich inequality in any dimension $N\geq 3$. In
addition, we provide minimizing sequences which were not explicitly mentioned
in the quoted papers, emphasizing their symmetry breaking in lower dimensions
$N\in \{3,4\}$. We also show that the best constant is not attained in the
proper functional space.
