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Chapter 4
Multiple Time Steps Algorithms For
Large Size Flexible Systems with
Strong Electrostatic Interactions
In the previous sections we have described how to obtain multiple time step integrators given a certain
potential subdivision and have provided simple examples of potential subdivision based on the inter/intra
molecular separation. Here, we focus on the time scale separation of model potentials of complex molecular
systems. Additionally, we provide a general potential subdivision applying to biological systems, as well
as to many other interesting chemical systems including liquid crystals. This type of systems are typically
characterized by high flexibility and strong Coulombic intermolecular interactions. Schematically, we can
then write the potential V as due to two contributions:
V
=
Vbnd + Vnbn .
(4.1)
Here, the “bonded” or intramolecular part Vbnd is fast and is responsible for the flexibility of the system.
The “non bonded” or intermolecular (or intergroup) term Vnbn is dominated by Coulombic interactions.
The aim of the following sections is to describe a general protocol for the subdivision of such forms of the
interaction potential and to show how to obtain reasonably efficient and transferable multiple time step
integrators valid for any complex molecular system.
4.1
Subdivision of the “Bonded” Potential
As we have seen in Sec. 2.3 the idea behind the multiple time step scheme is that of the reference
system which propagates for a certain amount of time under the influence of some unperturbed reference
Hamiltonian, and then undergoes an impulsive correction brought by the remainder of the potential. The
exact trajectory spanned by the complete Hamiltonian is recovered by applying this impulsive correction
onto the “reference” trajectory. We have also seen in the same section that, by subdividing the interaction
potential, we can determine as many “nested” reference systems as we wish. The first step in defining
a general protocol for the subdivision of the bonded potential for complex molecular systems consists in
identifying the various time scales and their connection to the potential. The interaction bonded potential
in almost all popular force fields is given as a function of the stretching, bending and torsion internal
coordinates and has the general form
Vbnd
= Vstretch + Vbend + Vtors ,
(4.2)