Stepsize threshold

More generally, even without the presence of bonds, the stepsize threshold in the simulation of molecular models by an explicit method like Verlet arises because of the shape of the energy surface. The understanding of this for a nonlinear system like the Lennard-Jones cluster requires examining the linearization of the problem near equilibrium points and even then the discussion is always quite technical... 

Among explicit symplectic Partitioned Runge-Kutta methods this is the maximum stability threshold [74]. In a similar way one can analyze the stability of the Verlet and other methods and one thus obtains conditions on the stepsize that must hold for the equilibrium points to be stable in the linearization. Analyzing the stability of both continuous and discrete iteration is much more compUcated for... 

This indieates that the Implicit Midpoint method has no stability threshold for the stepsize when solving a linear system This remarkable property may inspire some optimism that Implicit Midpoint would allow much larger stepsize in molecular dynamics simulation than does Verlet. 

---