Real-World Predictions from Ab Initio Molecular Dynamics Simulations - Multiscale Molecular Methods in Applied Chemistry - page 131

A very excellent introduction to string methods is found in [ 62 ] in which the
authors introduced a combined approach of the string method with a sampling
technique to determine minimum free energy paths (MFEP). The string method was
originally designed to determine reaction pathways without making any a priori
assumption about the mechanism [ 63 ]. With the string method, an evolution
equation for a parameterized curve is constructed, coined a string, which is such
that any initial guess for the string converges to a solution as time evolves. The
solution has the condition that the force must be everywhere tangent to the
minimum energy path (MEP). Maragliano et al. stated [ 62 ] that the string method
is very robust and efficient at determining minimum MFEP in a given landscape,
requiring as input only the calculation of the mean force. The cost of the string
method calculations performed locally scaled linearly with the number of points
along the discretized string. However, it was independent of the dimension which
was in marked contrast with free energy mapping techniques, for which the cost
increases exponentially with the number of reaction coordinates used to describe
the reaction. In order to eliminate potential difficulties for the string method when
being applied in the original state space of the system, the authors removed
fast degrees of freedom. They found that working in free energy space was easier
as well, because the free energy landscape is in general much smoother than
the original potential energy landscape of the system [ 62 ], The applicability of
the new technique was demonstrated with the example of the dipeptide alanine
isomerization transition. Most importantly, the example showed that the transition
mechanism can be described by using the four dihedral angles; however, it was not
captured using only two of them [ 62 ].
Michaelides and coworkers explained the nudged elastic band (NEB) method
[ 64 ] as taking a set of system images between the initial and the final states
and optimizing them simultaneously in a subspace perpendicular to the imaginary
line connecting the images [ 61 ]. Additional forces keep the images evenly
distributed along the pathway between the initial and final states. According to
Michaelides the location of the transition state with the NEB method can be
achieved using the climbing image procedure, where the highest energy image
is moved with the aid of the modified real force. The modified real force is obtained
by flipping the component of the force parallel to some direction [ 61 ]. In their
study the authors evaluated the performance of a number of methods for locating
transition states with DFT. They also introduced three new algorithms. A major
conclusion from their work was that the NEB method is relatively fast, especially
when just a single (climbing) image is used [ 61 ].
3.5.2 Metadynamics
Metadynamics is a method based on MD simulations that allows thorough sampling
of a predefined multidimensional configurational space and provides, at the same
time, the direct reconstruction of the explored free energy surface (FES) [ 65 - 71 ].