Nuclear binding energies from moment methods: Harmonic oscillator Hamiltonian

Total binding energies for systems of $N$ neutrons and $Z$ protons near $N=Z=8\mathrm{}$ are obtained for the harmonic oscillator Hamiltonian via moment methods and are compared with the exact results. We examine the accuracy of employing only the few lowest moments of the eigenvalue density to predict the ground state (g.s.) energy as a function of several ingredients in the method. We find systematic errors are strongly dependent on the scheme chosen for truncating the single particle space and rather weakly dependent on the size of the model space in many cases of interest. Two functional forms approximating the exact eigenvalue distribution, the Gram-Charlier series and the Weibull distribution, give results of comparable accuracy in cases where three moments are employed. Best overall accuracy is obtained using the traditional truncation scheme where all single particle states below a fixed energy are retained. For this truncation scheme, the distribution of energy eigenvalues is significantly skewed. However, the Gaussian approximation, which requires only the two lowest moments, yields one of the better estimates of the g.s. energy. This is true for both the total distribution of eigenvalues and for the distribution restricted to states of specific total angular momentum. The absolute error in the g.s. energy estimate is generally found to grow with increasing model space size; however, most of the error is systematic, and the relative binding energies of adjacent nuclei are found to be predicted with considerably greater accuracy. Isolation of these systematic errors plus demonstration of small errors in relative energies enhances the prospects for obtaining reliable total nuclear binding energies from realistic (no-core) Hamiltonians using moment methods.