  The ground state energy of a many-electron system can be approximated by an
variational approach in which the total energy of the system is minimized with
respect to one and two-body reduced density matrices (RDM) instead of
many-electron wavefunctions. This problem can be formulated as a semidefinite
programming problem. Due the large size of the problem, the well-known interior
point method can only be used to tackle problems with a few atoms. First-order
methods such as the the alternating direction method of multipliers (ADMM) have
much lower computational cost per iteration. However, their convergence can be
slow, especially for obtaining highly accurate approximations. In this paper,
we present a practical and efficient second-order semi-smooth Newton type
method for solving the SDP formulation of the energy minimization problem. We
discuss a number of techniques that can be used to improve the computational
efficiency of the method and achieve global convergence. Extensive numerical
experiments show that our approach is competitive to the state-of-the-art
methods in terms of both accuracy and speed.
