Reaction Path Optimization with Holonomic Constraints and Kinetic Energy Potentials

Abstract

Two methods are developed to enhance the stability, efficiency, and robustness of reaction path optimization using a chain of replicas. First, distances between replicas are kept equal during path optimization via holonomic constraints. Finding a reaction path is, thus, transformed into a constrained optimization problem. This approach avoids force projections for finding minimum energy paths (MEPs), and fast-converging schemes such as quasi-Newton methods can be readily applied. Second, we define a new objective function - the total Hamiltonian - for reaction path optimization, by combining the kinetic energy potential of each replica with its potential energy function. Minimizing the total Hamiltonian of a chain determines a minimum Hamiltonian path (MHP). If the distances between replicas are kept equal and a consistent force constant is used, then the kinetic energy potentials of all replicas have the same value. The MHP in this case is the most probable isokinetic path. Our results indicate that low-temperature kinetic energy potentials (<5 K) can be used to prevent the development of kinks during path optimization and can significantly reduce the required steps of minimization by 2-3 times without causing noticeable differences between a MHP and MEP. These methods are applied to three test cases, the C7eq-to-Cax isomerization of an alanine dipeptide, the (4)C1-to-(1)C4 transition of an α-d-glucopyranose, and the helix-to-sheet transition of a GNNQQNY heptapeptide. By applying the methods developed in this work, convergence of reaction path optimization can be achieved for these complex transitions, involving full atomic details and a large number of replicas (>100). For the case of helix-to-sheet transition, we identify pathways whose energy barriers are consistent with experimental measurements. Further, we develop a method based on the work energy theorem to quantify the accuracy of reaction paths and to determine whether the atoms used to define a path are enough to provide quantitative estimation of energy barriers.

Reaction Path Optimization and Sampling Methods and Their Applications for Rare Events

“…Recently, Chu and coworkers (Brokaw et al 2009) applied an equal distance holonomic constraint in the RPATH framework. Given two states of a molecular system with N atoms, r 0 and r k , a chain of K+1 replicas can be constructed to connect these two states.…”

“…The RPATH/constraint was applied to study the helix-to-sheet transition of a GNNQQNY heptapeptide. (Brokaw et al 2009) An initial reaction path with 100 replicas was generated. Then a stable intermediate was chosen to divide the pathway into two segments in which the number of replicas was doubled.…”

Improved constrained optimization method for reaction-path determination in the generalized hybrid orbital quantum mechanical/molecular mechanical calculations

“…The two bond lengths change smoothly in a concerted manner as reported in previous works. 31 The number of energy/gradient calculations is 11 000 and 1200 for HF and MP2, respectively. We did not need to calculate MP2 a lot because we started from the structure obtained from HF.…”

“…[15][16][17][18][19][20][21][22][23][24][25] NEB and string methods work equally well if a suitable optimizer is used. 26,27 There are also several alternatives for calculating RPs, namely, conjugate peak refinement, 28 combined approach, 29 geometric minimum action, 30 holonomic constraints approach, 31 swarm walk approximation, 32 and constrained broyden dimer method. 33 Some of these methods have been combined with the quantum mechanical/molecular mechanical (QM/MM) technique for chemical reactions.…”

Distance and angular holonomic constraints in molecular simulations

“…In minimization and reaction-path dynamics, 7,12,13 it is useful to use such distance and angle constraints, for example, to study the approach of a molecule toward a catalytic site. Hard constraints in minimization can be solved by ͑1͒ a Lagrangian approach or ͑2͒ projection methods.…”