10ħωshell-model calculation of the ground state and the first excited state ofHe4

The ground state and the first excited state of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are investigated in harmonic oscillator shell-model bases including all the configurations up to 10ħω excitation energy. The Gogny et al. potential and versions I/III and V of the Malfliet-Tjon potential are used in the calculation. The difficulties in describing these states in a consistent manner are not overcome by substantially enlarging the shell-model bases. The lowest eigenvalue with a 1ħω center-of-mass wave function is also extracted from the energy matrices, providing the absolute energy of the first $J^{\Pi }$ ${=1\mathrm{}}^{\mathrm{-}}$, T=0 level. It is found that the agreement with experiment for this level is significantly improved when the model space is enlarged.