  We study $^5$He variationally as the first $p$-shell nucleus in the
tensor-optimized antisymmetrized molecular dynamics (TOAMD) using the bare
nucleon--nucleon interaction without any renormalization. In TOAMD, the central
and tensor correlation operators promote the AMD's Gaussian wave function to a
sophisticated many-body state including the short-range and tensor correlations
with high-momentum nucleon pairs. We develop a successive approach by applying
these operators successively with up to double correlation operators to get
converging results. We obtain satisfactory results for $^5$He, not only for the
ground state but also for the excited state, and discuss explicitly the
correlated Hamiltonian components in each state. We also show the importance of
the independent optimization of the correlation functions in the variation of
the total energy beyond the condition assuming common correlation forms used in
the Jastrow approach.
