  While recent work towards the development of tight-binding and ab-initio
algorithms has focused on molecular dynamics, Monte Carlo methods can often
lead to better results with relatively little effort. We present here a
multi-step Monte Carlo algorithm that makes use of the possibility of quickly
evaluating local energies. For the thermalization of a 1000-atom configuration
of {\it a}-Si, this algorithm gains about an order of magnitude in speed over
standard molecular dynamics. The algorithm can easily be ported to a wide range
of materials and can be dynamically optimized for a maximum efficiency.
