Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming

Determining the energy and properties of an $N$-electron molecule through a two-electron variational optimization has been a dream for more than half a century. While optimizations, using two-electron reduced density matrices constrained to represent $N$ electrons, have recently been achieved, the computational costs are prohibitive. In this report an efficient algorithm with an order-of-magnitude reduction in floating-point operations and memory usage is presented. Because the optimization occurs on the space of two electrons, this method automatically treats strong, multireference correlation. Application is made to ${\mathrm{N}}_2$ and ${\mathrm{H}}_6$ where the method yields consistent accuracy at all geometries.