The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of N-representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a stan
dard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N-representability conditions perform very well for this problem.
The half-life of the $alpha$ decaying nucleus $^{221}$Fr was determined in different environments, i.e. embedded in Si at 4 K, and embedded in Au at 4 K and about 20 mK. No differences in half-life for these different conditions were observed within
0.1%. Furthermore, we quote a new value for the absolute half-life of $^{221}$Fr of t$_{1/2}$ = 286.1(10) s, which is of comparable precision to the most precise value available in literature.
