Hybrid Monte Carlo 297arises from the use of scaled coordinates in the configurational partition function in
eqn. (7.3.45). Similarly, the acceptance probability of a trial deletionmove is
A(N− 1 |N) = min[
1 ,
λ^3 N
V
e−βμe−β[U(r′)−U(r)]]
, (7.3.47)
wherer′is the configuration of an (N−1)-particle system generated by the deletion
andris, again, the originalN-particle configuration. Particle insertion and deletion
moves require calculation of only the change in potential energy dueto the addi-
tion or removal of one particle, which is computationally no more expensive than the
calculation associated with a particle displacement. Thus, particle insertion and dele-
tion moves can be performed with greater frequency than can volume moves in the
isothermal-isobaric ensemble. In general, one Monte Carlo pass consists ofNparticle
displacements andNi/d insertion/deletion attempts. Note thatNiandNdare new
parameters that need to be optimized for each particular system.
7.4 Hybrid Monte Carlo
In the remaining sections of this chapter, we will discuss several algorithms that build
on the basic ideas developed so far. Although we will focus on samplingthe canon-
ical distribution, the techniques we will introduce can be easily generalized to other
ensembles.
In Section 7.3, we showed how the canonical distribution can be generated using
uniform trial particle displacements. We argued that we can only attempt to move one
or just a few particles at a time in order to maintain a reasonable average acceptance
probability. We also noted that a key difference between Monte Carloand molecular
dynamics calculations is the ability of the latter to generate moves ofthe entire system
(global moves) with acceptance probability 1. In molecular dynamics, however, such
moves are deterministic and fundamentally limited by the time step ∆t, which needs
to be sufficiently small to yield reasonable energy conservation. In this section, we
describe thehybrid Monte Carloapproach (Duaneet al., 1987), which is a synthesis
of M(RT)^2 Monte Carlo and molecular dynamics and, therefore, derives advantages
from each of these methods.
Hybrid Monte Carlo seeks to relax the restriction on the size of ∆tin a molecular
dynamics calculation by introducing an acceptance criterion for molecular dynamics
moves with a large ∆t. Of course, when ∆tis too large, the numerical integration
algorithm for Hamilton’s equations leads to large changes in the HamiltonianH, which
should normally be approximately conserved. In principle, this is not aproblem since
in the canonical ensemble,His not constant but is allowed to fluctuate as the system
exchanges energy with a surrounding thermal bath. By using molecular dynamics as
an engine for generating moves, the system naturally tends to move toward regions
of configuration space that are energetically favored, and hencethe moves are more
“intelligent” than simple uniform displacements. However, when we use a time step
that is too large, we are simply performing a “bad” molecular dynamicscalculation,
and the changes inHcaused by inaccurate integration of the equations of motion will
not be consistent with the canonical distribution. Thus, we need a device for ensuring