A New Formulation of a Many-Level Shell Model: A Method of Constructing Orthonormal Basis States Analytically

In the jj-coupling shell model 1), 2) with multishell (or multilevel) states, one of the most important problems is to efficiently construct antisymmetric orthonormalized basis states and to calculate the Hamiltonian matrix. In the traditional method, we first construct orthonormalized antisymmetric states in each subshell (or level), usually by using the coefficients of fractional parentage (CFPs) for each level, and then we couple the states to a completely antisymmetric state. In this algorithm, the method to calculate the CFPs at each level has been established, 2) but the orthonormalization of all multilevel states is, in general, a rather troublesome procedure. 3) We need to carry out many steps of angular-momentum coupling and sometimes numerical orthogonalization of overcomplete sets of states. If we can apply this procedure in an analytic form, then we could analytically represent the Hamiltonian matrix. This would be convenient for numerical calculations in the shell model. The present paper is devoted to this purpose. In §2 we introduce a new type of CFP to couple the orthonormalized basis states in each subshell (or level) to a completely antisymmetric state. In §3 we show how to analytically represent the matrix elements of the Hamiltonian using the new type of CFPs. Some concluding remarks are given in §4.