This section gives a basic overview of the Configurational-bias Monte Carlo (CBMC) algorithm that
is implemented into Towhee. CBMC algorithm development remains a major research activity, and a particular
favorite of at least one of the Towhee developers. For more information about the algorithms implemented
into Towhee the best two places to look are the references for each of the moves, and the code itself.
The Martin and Siepmann 1999 paper is a good place
to start learning about CBMC as it is implemented into Towhee.
Early CBMC algorithms
CBMC was developed on lattice by J. Ilja Siepmann 1990
as a method for sampling chain molecules in a simple model of a monolayer. A variety of researchers
(Siepmann and Frenkel 1992,
Frenkel et al. 1992,
Laso et al. 1992, and
Siepmann and McDonald 1992) brought the method to
continuous space in 1992. This version of the CBMC algorithm worked well for the linear chain molecules
studied in the mid-nineties and was combined with the Gibbs ensemble to compute some of the first vapor-liquid
coexistence curves for chain molecules (such as united-atom n-alkanes).
The basic concept is that molecules are grown atom by atom into a dense fluid in such a way that the local
space for each new atom is sampled and the lower energy positions are more likely to be chosen to continue the
growth of the molecule. This accumulates a bias that is then removed in the acceptance rule. The net effect
is a large increase in the acceptance rate for insertions of polyatomic molecules into liquids. At this point
intramolecular interactions (vibrations, bending angles, dihedrals etc.) were very simple and handled with a
Boltzmann rejection scheme.
In 1998 Vlugt et al. 1998 developed a cost-saving version
of the CBMC algorithm. Instead of using the full nonbonded cutoff during the CBMC move, a shorter range cutoff is
used and this makes the computation less expensive in dense systems. The full potential is then computed for the
final structure and the energy difference between the true potential, and the one used to generate the growth trial,
is incorporated into the acceptance rule to remove this bias. In Towhee this algorithm is implemented using the
rcutin variable for the inner cutoff used during the growth and rcut for the cutoff used to computed
the "true" energy of the system. Proper setting of rcutin can decrease the simulation time by a factor of two.
Note that the chemical potential computed using dual-cutoff CBMC has not been proven to be correct, and empirical
evidence suggests that it is not correct in certain cases.
Coupled-Decoupled CBMC 'Martin and Siepmann 1999' formulation
Work on branched alkane adsorption in silicalite by
Vlugt et al. 1999 revealed a flaw in the Boltzmann
rejection technique if a molecule contains any atom that is bonded to three or more other atoms. In addition, this
method became very slow for molecules with atoms bonded to 4 or more other atoms.
One of the methods developed to resolve this problem was the coupled-decoupled CBMC algorithm of
Martin and Siepmann 1999.
The intramolecular terms were now also generated using a biasing procedure with appropriate corrections
in the acceptance rule. Bond lengths were still rigid in that paper, although the algorithm was later generalized
to include decoupled flexible bond lengths and class 2 force field term by
Martin and Thompson 2004. The flexible bond angles were decoupled
from the torsions, which were coupled to the nonbonded terms. What this means, is the bond angles are selected based
solely on the bond angle energies and the phase space terms and then those angles are used in all subsequent selections
(torsion and nonbond). Thus, the bond angle selection is decoupled from the other selections. In contrast,
for each nonbond trial a full selection is done to generate torsional angles so these two selections are coupled.
Fixed Endpoint CBMC
Cyclic molecules are substantially more difficult to grow using CBMC because their conformational space is severely
limited by the constraints of having cyclic portions of the molecule. Attempting to grow a cyclic molecule using
standard CBMC methods, and just hoping it closes itself up properly, has an acceptance rate that is nearly zero.
It is generally believed that an additional biasing is required during the growth procedure in order to nudge the
growth into positions that will result in reasonable ring closures. There are a variety of biasing procedures in
the literature. One that is notable, but not currently implemented into Towhee, is the self-adapting fixed-endpoint
(SAFE) CBMC algorithm of Wick and Siepmann 2000.
The problem with SAFE-CBMC is it can use a large amount of memory in order to keep track of all of the adapting
fixed-endpoint biasing functions. The version implemented into Towhee is currently unpublished, but uses some
analytical biasing functions based upon a crude, but consistent, transformation of the distance between growth atoms
and target ring atoms, into a bias function based loosely upon dihedral, bending, and vibrational energies.
Considerable research is still needed in this area to determine optimal biasing strategies. The algorithm implemented
into Towhee was first used, but not satisfactorily described, in
Martin and Thompson 2004.
Arbitrary Trial Distribution CBMC
Traditionally, the trials for things like bond lengths, bending angles, and torsional angles are generated according to
a distribution that would be the true distribution if there were no potential energy terms. The trials are then accepted
or rejected based upon factors related to the potential energy terms. While this split into "entropic" and "energetic" terms
is convenient, it is not necessarily the best way to handle the trial generation.
Martin and Biddy 2005 used a new method where the bond lengths and
bending angles are generated according to a Gaussian distribution and then this is corrected in the acceptance rules.
There is nothing special about using a Gaussian and any arbitrary trial distribution can be used to generate the trials
so long as it is strictly positive throughout the appropriate ideal range. The strictly positive requirement comes from the
acceptance rule which divides out the arbitrary trial distribution in order to remove this bias (division by zero is
not a good idea). Several options for the arbitrary trial distributions are implemented into Towhee and this is a subject
of continuing research (and hopefully a few journal articles describing the method).
Coupled-Decoupled CBMC 'Coupled to pre-nonbond' formulation
This is an experimental new algorithm that is not yet published. With the invention of the arbitrary trial distribution
method it is now possible to get good acceptance rates using only a single trial for things like bond lengths and bending
angles, instead of the normal 100 to 1000 required when using the ideal trial distribution method. This enables a rethinking
of the coupled-decoupled algorithm as the motivation for decoupling terms was to keep the expense down for the terms that
occur early in the growth step, like vibrations and bending angles. Now that these steps are considerably less expensive
it makes sense to explore various strategies to try and improve the acceptance rate for challenging molecular geometries
(such as strongly branched and cyclic molecules). The 'Coupled to pre-nonbond' formulation implemented into Towhee adds a
new selection process in between the dihedral selection and the nonbond selection (a pre-nonbond selection). Bond lengths,
bending angles, and dihedral angles are all decoupled from each other, but coupled to the pre-nonbond selection. Preliminary
work suggests this can improve the acceptance rate for cyclic molecules. Research in this area is extremely active right
now and this implementation is currently in ongoing testing so if you wish to use it be sure to check the website
frequently for updates as the debugging process continues.
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