  In the earlier unitary-model-operator approach (UMOA), one-body correlations
have been taken into account approximately by the diagonalization of
unitary-transformed Hamiltonians in the $0p0h$ and $1p1h$ space. With this
prescription, the dependence of the harmonic-oscillator energy ($\hbar\omega$)
on calculated observables is not negligible even at larger model spaces. In the
present work, we explicitly introduce the one-body correlation operator so that
it optimizes the single-particle basis states and then reduces the
$\hbar\omega$-dependence. For an actual demonstration, we calculate the energy
and radius for the $^{4}$He ground state with the softened nucleon-nucleon
($NN$) interactions from Argonne v18 (AV18) and chiral effective field theory
($\chi$EFT) up to the next-to-next-to-next leading order (N$^{3}$LO). As a
result, we obtain practically $\hbar\omega$-free results at sufficiently large
model spaces. The present results are reasonably close to those by the other ab
initio calculations with the same $NN$ interactions. This methodological
development enables us more systematic analysis of calculation results in the
UMOA. We also discuss qualitatively the origin of the $\hbar\omega$-dependence
on calculated observables in a somewhat simplified way.
