Multiple time-scale integration 115immensely powerful technique that we term thedirect translationmethod (Martyna
et al., 1996). Because direct translation is possible, we can simply let a factorization
of the classical propagator denote a particular integration algorithm; we will employ
the direct translation technique in many of our subsequent discussions of numerical
solvers.
3.11 Multiple time-scale integration
One of the most ubiquitous aspects of complex systems in classical mechanics is the
presence of forces that generate motion with different time scales. Examples include
long biological macromolecules such as proteins and other types of polymers. In fact,
virtually any chemical system will span a wide range of time scales fromvery fast
bond and bend vibrations to global conformational changes in macromolecules or
slow diffusion/transport molecular liquids, to illustrate just a few cases. To make the
discussion more concrete, consider a simple potential energy model commonly used
for biological macromolecules:
U(r 1 ,...,rN) =∑
bonds1
2
Kbond(r−r 0 )^2 +∑
bends1
2
Kbend(θ−θ 0 )^2+
∑
tors∑^6
n=0An[1 + cos(Cnφ+δn)]+
∑
i,j∈nb{
4 ǫij[(
σij
rij) 12
−
(
σij
rij) 6 ]
+
qiqj
rij}
. (3.11.1)
The first term is the energy for all covalently bonded pairs, which are treated as har-
monic oscillators in the bond lengthr, each with their own force constantKbond.
The second term is the bend energy of all neighboring covalent bonds, and again
the bending motion is treated as harmonic on the bend angleθ, each bend having a
force constantKbend. The third term is the conformational energy of dihedral angles
φ, which generally involves multiple minima separated by energy barriersof various
heights. The first three terms constitute the intramolecular energy due to bonding and
connectivity. The last term describes the so-callednonbonded(nb) interactions, which
include van der Waals forces between spheres of radiusσiandσj(σij= (σi+σj)/2)
separated by a distancerijwith well-depthǫij, expressed as a Lennard–Jones poten-
tial (Lennard-Jones, 1924), and Coulomb forces between particles with chargesqiand
qjseparated by a distancerij. If the molecule is in a solvent such as water, then eqn.
(3.11.1) also describes the solvent–solute and solvent–solvent interactions as well. The
forcesFi=−∂U/∂riderived from this potential will have large and rapidly varying
components due to the intramolecular terms and smaller, slowly varying components
due to long-range contributions to the nonbonded interactions. Moreover, the simple
functional forms of the intramolecular terms renders the fast forces computationally
inexpensive to evaluate while the slower forces, which involve sums over many pairs
of particles, will be much more time-consuming to compute. On the time scale over