  This paper studies the rational homotopy groups of the group
$\mathrm{Diff}(S^4)$ of self-diffeomorphisms of $S^4$ with the
$C^\infty$-topology. We present a method to prove that there are many `exotic'
non-trivial elements in $\pi_*\mathrm{Diff}(S^4)\otimes \mathbb{Q}$
parametrized by trivalent graphs. As a corollary of the main result, the
4-dimensional Smale conjecture is disproved. The proof utilizes Kontsevich's
characteristic classes for smooth disk bundles and a version of clasper surgery
for families. In fact, these are analogues of Chern--Simons perturbation theory
in 3-dimension and clasper theory due to Goussarov and Habiro.
