Six-body hyperspherical solution toLi6bound states

We apply the hyperspherical method to the $N$-body Schrödinger equation and solve for the bound states of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ with the center of mass motion correctly treated. It is shown how to construct antisymmetric hyperharmonic polynomials of definite $J$, $J_z$, $T$, and $T_z$ from Slater determinants with shell model coordinates. A set of coupled one dimensional differential equations are obtained and solved in $K_{min}$ approximation. A super-soft- core potential that provides a good fit to the two-nucleon scattering phase shifts is given a slight state dependence and is used for the two-nucleon potential. The proper handling of the potential energy matrix element with the center of mass motion excluded is detailed. A prescription for obtaining effective interactions for shell model calculations is presented. The difference in the potential energy matrix element of four $S_{\frac{1}{2}}$ nucleons in a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ nucleus or in a ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ nucleus is shown not to be zero. A $J$, $T=1, 1$ state of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ is calculated to have a binding energy of 15.245 MeV. This state may resemble a nuclear molecule of ${}_{}{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.