Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem

The auxiliary field Monte Carlo (AFMC) technique has advantages over other ab initio quantum Monte Carlo methods for fermions, as it does not seem to require approximations for alleviating the sign problem and is directly applicable to excited states. Yet, the method is severely limited by a numerical instability, a numerical sign problem, prohibiting application to realistic electronic structure systems. Recently, the shifted contour auxiliary field method (SC-AFMC) was proposed for overcoming this instability. Here we develop a theory for the AFMC stabilization, explaining the success of SC-AFMC. We show that the auxiliary fields can be shifted into the complex plane in a manner that considerably stabilizes the Monte Carlo integration using the exact one-electron density. Practical stabilization can be achieved when an approximate Hartree–Fock density is used, proposing that an overwhelming part of the sign problem is removed by taking proper account of the Fermion mean-field contribution. The theory is demonstrated by application to ${\mathrm{H}}_2.$