Thermostatted molecular dynamics: How to avoid the Toda demon hidden in Nosé-Hoover dynamics

The Nos\'e-Hoover thermostat, which is often used in the hope of modifying molecular dynamics trajectories in order to achieve canonical-ensemble averages, has hidden in it a Toda ``demon,'' which can give rise to unwanted, noncanonical undulations in the instantaneous kinetic temperature. We show how these long-lived oscillations arise from insufficient coupling of the thermostat to the atoms, and give straightforward, practical procedures for avoiding this weak-coupling pathology in isothermal molecular dynamics simulations.