  Dolgachev surfaces are simply connected minimal elliptic surfaces with
$p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by
logarithmic transformations of rational elliptic surfaces. In this paper, we
explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein
smoothing of singular rational surfaces with two cyclic quotient singularities.
This construction is based on the paper by Lee-Park. Also, some exceptional
bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing
are constructed based on the idea of Hacking. In the case if Dolgachev surfaces
were of type $(2,3)$, we describe the Picard group and present an exceptional
collection of maximal length. Finally, we prove that the presented exceptional
collection is not full, hence there exist a nontrivial phantom category in the
derived category.
