  We describe a variational method to solve the Holstein model for an electron
coupled to dynamical, quantum phonons on an infinite lattice. The variational
space can be systematically expanded to achieve high accuracy with modest
computational resources (12-digit accuracy for the 1d polaron energy at
intermediate coupling). We compute ground and low-lying excited state
properties of the model at continuous values of the wavevector $k$ in
essentially all parameter regimes. Our results for the polaron energy band,
effective mass and correlation functions compare favorably with those of other
numerical techniques including DMRG, Global Local and exact diagonalization. We
find a phase transition for the first excited state between a bound and unbound
system of a polaron and an additional phonon excitation. The phase transition
is also treated in strong coupling perturbation theory.
