Ab Initio Finite Temperature Auxiliary Field Quantum Monte Carlo

We present an \textit{ab initio} auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the \textit{ab initio} phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, \textit{ab initio} finite temperature method. We demonstrate that the method produces internal energies within chemical accuracy of exact diagonalization results across a wide range of temperatures for H$_{2}$O (STO-3G), C$_{2}$ (STO-6G), the one-dimensional hydrogen chain (STO-6G), and the multi-orbital Hubbard model. Our method effectively controls the phase problem through importance sampling, \textit{often even without invoking the phaseless approximation}, down to temperatures at which the systems studied approach their ground states and may therefore be viewed as exact over wide temperature ranges. This technique embodies a versatile tool for studying the finite temperature phase diagrams of a plethora of systems whose properties cannot be captured by a Hubbard U term alone. Our results moreover illustrate that the severity of the phase problem for model Hamiltonians far exceeds that for many molecules at all of the temperatures studied.