Introduction of the one-body correlation operator in the unitary-model-operator approach

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.