Reaction coordinates

It is frequently the case that the progress of some chemical,
mechanical, or thermodynamics process can be followed by
following the evolution of a small subset of generalized
coordinates in a system. When generalized coordinates
are used in this manner, they are typically referred to
as *reaction coordinates*, *collective variables*,
or *order parameters*, often depending on the context
and type of system. Whenever referring to these coordinates,
we will refer to them as *reaction coordinates*, although the
reader should be aware that the other two designations are also used
in the literature.

As an example of a useful reaction coordinate, consider
a simple gas-phase diatomic dissociation process AB
A+B.
If
and
denote the Cartesian
coordinates of atom A and B, then a useful generalized coordinate
for following the progress of the dissociation is simply the
distance
. A complete set of generalized coordinates
that contains as one of the coordinates is the
set that contains the center of mass
,
the magnitude of the relative coordinate
,
and the two angles
and
, where
, , and are the components of the relative
coordinate
. Of course, in the
gas-phase, where the potential between A and B likely only
depends on the distance between A and B, is really the
*only* interesting coordinate. However, if the reaction
were to take place in solution, then other coordinate
such as and become more relevant as
specific orientations might change the mechanism or
thermodynamic picture of the process, depending on the
complexity of the solvent, and averaging over
these degrees of freedom to produce a free energy
profile in alone will wash out some
of this information.

As another example, consider a gas-phase proton transfer
reaction A-HB
AH-B.
Here, although the distance
can be used to monitor the progress of the proton away from A
and the distance
can be used to
monitor the progress of the proton toward B, neither distance
alone is sufficient for following the progress of the
reaction. However, the difference
can be used to follow the progress of the proton transfer
from A to B and, therefore, is a potentially useful
reaction coordinate. A complete set of generalized
coordinates involving can be constructed
as follows. If
,
and
denote the Cartesian coordinates of the three atoms,
then first introduce the center-of-mass
, the relative coordinate
between A and B,
, and a third
relative coordinate between H and the center-of-mass
of A and B,
. Finally, is transformed into
spherical polar coordinates,
, and
from and , three more coordinates are formed:

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While reaction coordinates or collective variables are potentially very useful constructs, they must be used with care, particularly when enhanced sampling methods are applied to them. Enhanced sampling of a poorly chosen reaction coordinate can bias the system in unnatural ways, leading to erroneous predictions of free energy barriers and associated mechanisms. A dramatic example of this is the autodissociation of liquid water following the classic reaction 2HO HO + OH, which ostensibly only requires transferring a proton from one water molecule to another. If this notion of the reaction is pursued, then a seemingly sensible reaction coordinate would simply be the distance between the oxygen and the transferring proton or the number of hydrogens covalently bonded to the oxygen. These reaction coordinates, as it turns out, are inadequate for describing the true nature of the reaction and, therefore, fail to yield reasonable free energies (and hence, values of the autoionization constant ). Chandler and coworkers showed that the dissociation reaction can only be considered to have occurred when the HO and OH ions are sufficiently far apart that no contiguous or direct path of hydrogen-bonding in the liquid can allow the proton to transfer back to the water or its origin. In order to describe such a process correctly, a very different type of reaction coordinate would clearly be needed.

Keeping in mind such caveats about the use of reaction coordinates,
we now proceed to describe a number of popular methods designed
to enhance sampling along pre-selected reaction coordinates.
All of these methods are designed to generate, either directly
or indirectly, the probability distribution function
of a subset of reaction coordinates
of interest in a system. If these reaction coordinates
are obtained from a transformation of the Cartesian
coordinates
,
, then the probability density
that these coordinates will have
values
in the canonical ensemble is