MINIMUM ENERGY AND STEEPEST DESCENT PATH ALGORITHMS FOR QM/MM APPLICATONS
Abstract
A number of new methods are presented to determine the reaction path both for chemical
systems where the transition state(TS) is known and for the more complicated case
when only the endpoints are available. To determine the minimum energy path(MEP) two
algorithms were developed.The first MEP method is a quadratic string method (QSM)
which is based on a multiobjective optimization framework.
In the method, each point on the MEP is integrated in the descent direction perpendicular
to path. Each local integration is done on an approximate quadratic surface with an
updated Hessian allowing the algorithm to take many steps between energy and gradient
calls. The integration is performed with an adaptive step size solver, which is
restricted in length to the trust radius of the approximate Hessian. The full algorithm
is shown to be capable of practical superlinear convergence, in contrast to the linear
convergence of other methods. The method also eliminates the need for predetermining
such parameters as step size and spring constants, and is applicable
to reactions with multiple barriers. The method is demonstrated for the Muller Brown
potential, a 7-atom Lennard-Jones cluster and the enolation of acetaldehyde to vinyl
alcohol.The second MEP method is referred to as the Sequential Quadratic Programming
Method (SQPM). This method is based on minimizing the points representing the path
in the subspace perpendicular to the tangent of the path while using a penalty term
to prevent kinks from forming. Rather than taking one full step, the minimization
is divided into
a number of sequential steps on an approximate quadratic surface. The resulting method
is shown to be capable of super-linear convergence. However,
the emphasis of the algorithm is on its robustness and its ability to determine the
reaction mechanism efficiently, from which transition state can be easily
identified and refined with other methods. To improve the resolution
of the path close to the transition state, points are clustered close to
this region with a reparametrization scheme. The usefulness of the algorithm is demonstrated
for the Mu$ller Brown potential, amide hydrolysis and an 89 atom cluster taken from
the active site
of 4-Oxalocrotonate tautomerase (4-OT) for the reaction which catalyzes 2-oxo-4-hexenedioate
to the intermediate 2-hydroxy-2,4-hexadienedioate.When the TS is known we present
two methods for integrating the steepest descent path (SDP). Also the concepts of
stability and stiffness are elaborated upon. The first SDP method is an optimally
combined explicit-implicit method for following the reaction path
to high accuracy. Although the SDP is generally considered to be a stiff ODE,
it is shown that the reaction path is not uniformly stiff and instead is only stiff
near stationary points. The optimal algorithm is developed by combining the explicit
and implicit methods with a simple criterion, based on the stiffness, to switch between
the two. Using two different methods an algorithm is developed to efficiently integrate
the SDP. This method is tested on a number of small molecules.The final method given
is based on the diagonally implicit Runge-Kutta framework, which is shown to be
a general form for constructing stable, efficient steepest descent reaction path integrators,
of any order. With this framework tolerance driven, adaptive step-size methods can
be constructed
by embedding methods to obtain error estimates of each step without additional computational
cost. There are many embedded and non-embedded, diagonally implicit Runge-Kutta methods
available from the numerical analysis literature and these are reviewed for orders
2,3 and 4. New embedded methods are also developed which are tailored to the application
of reaction path following. All integrators are summarized and compared
for three systems.
Type
DissertationDepartment
ChemistrySubject
Chemistry, PhysicalPermalink
https://hdl.handle.net/10161/377Citation
Burger, Steven Knox (2007). MINIMUM ENERGY AND STEEPEST DESCENT PATH ALGORITHMS FOR QM/MM APPLICATONS. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/377.Collections
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