  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.
