The energy spectrum of ${\mathrm{O}}^{16}$ has been investigated theoretically using the basis functions of the nuclear (harmonic-oscillator) shell model. Random-phase-approximation (RPA) techniques are used to decouple the core state from the excited configurations, thereby overcoming the problem of the exaggerated ground-state depression found in standard shell-model calculations. The general equations of the higher RPA are reduced to a more restricted but tractable form which contains both the Tamm-Dancoff approximation and the standard (first) RPA as special cases. For the even-parity states, it is found that the resulting secular equation reduces in good approximation to that of a shell-model calculation with the core state removed. Interaction matrices between all one-hole one-particle and two-hole two-particle states at $1\hslash \omega$ and $2\hslash \omega$ excitation were diagonalized to obtain the level structure of ${\mathrm{O}}^{16}$ in the absence of spurious states of center-of-mass motion. A Gaussian central force with Rosenfeld exchange was employed, the strength being determined by a rough fit to the lowest $J=0+ \mathrm{and} 2+$, $T=0\mathrm{}$ levels. Despite the restricted nature of this fit, a remarkably good agreement between the calculated and observed energy spectrum is obtained. The $T=0\mathrm{}$ spectrum is well represented apart from two states (with $J=4+ \mathrm{and} 6+$) which are calculated below the lowest corresponding observed levels. Moreover, the $T=1\mathrm{}$ states of both parities fall correctly in relation to each other and to the fitted states, and the $T=2\mathrm{}$ states are well reproduced.

• Received 1 August 1966

DOI:https://doi.org/10.1103/PhysRev.153.1039