Essentials of Computational Chemistry

76 3 SIMULATIONS OF MOLECULAR ENSEMBLES
p
Figure 3.2 An actual phase-space trajectory (bold curve) and an approximate trajectory generated
by repeated application of Eq. (3.17) (series of arrows representing individual time steps). Note that
each propagation step has an identical t, but individual p values can be quite different. In the
illustration, the approximate trajectory hews relatively closely to the actual one, but this will not be
the case if too large a time step is used
atoms moving seemingly randomly. The very high energies of the various steps will preclude
their contributing in a meaningful way to any property average.)
Of course, we know that in the limit of an infinitesimally small time step, we will recover
Eqs. (3.10) and (3.12). But, since each time step requires a computation of all of the molecular
forces (and, presumably, of the property we are interested in), which is computationally
intensive, we do not want to take too small a time step, or we will not be able to propagate
our trajectory for any chemically interesting length of time. What then is the optimal length
for a time step that balances numerical stability with chemical utility? The general answer is
that it should be at least one and preferably two orders of magnitude smaller than the fastest
periodic motion within the system. To illustrate this, reconsider the 1-D harmonic oscillator
example of Figure 3.1: if we estimate the first position of the mass after its release, given
that the acceleration will be computed to be towards the wall, we will estimate the new
position to be displaced in the negative direction. But, if we take too large a time step, i.e.,
we keep moving the mass towards the wall without ever accounting for the change in the
acceleration of the spring with position, we might end up with the mass at a position more
negative than −b. Indeed, we could end up with the mass behind the wall!
In a typical (classical) molecular system, the fastest motion is bond vibration which, for
a heavy-atom–hydrogen bond has a period of about 10 −14 s. Thus, for a system containing
such bonds, an integration time step t should not much exceed 0.1 fs. This rather short time
t