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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, the4-dimensional Smale conjecture is disproved. The proof utilizes Kontsevich'scharacteristic classes for smooth disk bundles and a version of clasper surgeryfor families. In fact, these are analogues of Chern--Simons perturbation theoryin 3-dimension and clasper theory due to Goussarov and Habiro.

Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$