Essentials of Computational Chemistry

3.3 MOLECULAR DYNAMICS 77
step means that modern, large-scale MD simulations (e.g., on biopolymers in a surrounding
solvent) are rarely run for more than some 10 ns of simulation time (i.e., 10 7 computations of
energies, forces, etc.) That many interesting phenomena occur on the microsecond timescale
or longer (e.g., protein folding) represents a severe limitation to the application of MD to
these phenomena. Methods to efficiently integrate the equations of motion over longer times
are the subject of substantial modern research (see, for instance, Olender and Elber 1996;
Grubmüller and Tavan 1998; Feenstra, Hess and Berendsen 1999).
3.3.3 Practical Issues in Propagation
Using Euler’s approximation and taking integration steps in the direction of the tangent is a
particularly simple integration approach, and as such is not particularly stable. Considerably
more sophisticated integration schemes have been developed for propagating trajectories.
If we restrict ourselves to consideration of the position coordinate, most of these schemes
derive from approximate Taylor expansions in r, i.e., making use of
q(t + t) = q(t) + v(t)t + 1
2! a(t)(t)2 + 1
3!
d 3 q(τ)
dt 3
τ=t
(t) 3 +··· (3.19)
where we have used the abbreviations v and a for the first (velocity) and second (acceleration)
time derivatives of the position vector q.
One such method, first used by Verlet (1967), considers the sum of the Taylor expansions
corresponding to forward and reverse time steps t. In that sum, all odd-order derivatives
disappear since the odd powers of t have opposite sign in the two Taylor expansions.
Rearranging terms and truncating at second order (which is equivalent to truncating at thirdorder,
since the third-order term has a coefficient of zero) yields
q(t + t) = 2q(t) − q(t − t) + a(t)(t) 2
(3.20)
Thus, for any particle, each subsequent position is determined by the current position, the
previous position, and the particle’s acceleration (determined from the forces on the particle
and Eq. (3.13)). For the very first step (for which no position q(t − t) is available) one
might use Eqs. (3.16) and (3.17).
The Verlet scheme propagates the position vector with no reference to the particle velocities.
Thus, it is particularly advantageous when the position coordinates of phase space are
of more interest than the momentum coordinates, e.g., when one is interested in some property
that is independent of momentum. However, often one wants to control the simulation
temperature. This can be accomplished by scaling the particle velocities so that the temperature,
as defined by Eq. (3.18), remains constant (or changes in some defined manner), as
described in more detail in Section 3.6.3. To propagate the position and velocity vectors in a
coupled fashion, a modification of Verlet’s approach called the leapfrog algorithm has been
proposed. In this case, Taylor expansions of the position vector truncated at second order