Chain of states methods

The procedure and methods for the MEP determination by the NEB and parallel path optimizer methods have been explained in detail elsewhere [25, 27], Briefly, these methods are types of chain of states methods [20, 21, 25, 26, 30, 31]. In these methods the path is represented by a discrete number of images which are optimized to the MEP simultaneously. This parallel optimization is possible since any point on the MEP is a minimum in all directions except for the reaction coordinate, and thus the energy gradient for any point is parallel to the local tangent of the reaction path. 

The path optimizations are carried out by an iterative optimization procedure [25]. In the case of enzyme systems, because of the large number of degrees of freedom, we partition them into a core set and an environmental set. The core set is small and contains all the degrees of freedom that are involved with the chemical steps of the reaction, while all the remaining degrees of freedom are included in the environmental set. In all the QM/MM calculations presented below, the core set is defined by the QM subsystem and the environmental set by the MM subsystem. 

To ensure a smooth and continuous change of the environment set during the optimization, the initial coordinates for the environment atoms are set to be the same for all the replicas along the path. Thus, the initially guessed replicas differ only in the coordinates of the core atoms. In other words, the initial environment for all the points on the path is the same and is taken to be the environment from one of the end points. 

Once the path has been optimized with NEB, it is used as the initial guess for the parallel path optimizer method. In this second step the path is again iteratively optimized with the parallel path optimizer method for the core set, followed by the optimization of the environment set. In this part of the calculation no restraints are imposed on the environment set during the optimization. The iterations are continued until all the convergence criteria are met and the final optimized MEP is obtained. 

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We have compared the potential and free energies from four different calculated paths for the first step of the isomerization of 2o4hex catalyzed by 40T. Two of these paths were determined with a combined chain-of-states method. The remaining two paths were obtained with the coordinate driving method. 

For a description of the NEB method and a comparison of the NEB method with other chain of states methods for determining transition states without the use of the Hessian matrix, see D. Sheppard, R. Terrell, and G. Henkelman, J. Chem. Phys. 128 (2008), 134106. The climbing image NEB method is described in G. Henkelman, B. P. Ubemaga, and H. Jonsson, J. Chem. Phys. 113 (2000), 9901 9904. Improvements to the NEB method including a better estimation of the tangent to the MEP are described in G. Henkelman and H. Jonsson,... 

The NEB method is an example of what has been called a chain-of-states method. The common feature is that several images of the system are connected together to trace out a path of some sort. The simple object fimction for a chain... 

Sevick, Bell and Theodorou" proposed a chain of states method for finding the MEP, but their optimization method, which includes explicit constraints for rigidly fixing the distance between images, requires evaluation of the matrix of second derivatives of the potential and is, therefore, not as applicable to large systems and complex interactions. 

Chain-of-states methods have also been used for finding classical dynamical paths.4 A2 Gillilan and Wilson" suggested using an object function similar to equation 1 for finding saddle points, but this suffers from the comer-cutting and down-sliding problems discussed above. 

Chain-of-States Methods. - The chain-of-states methods (see 132) are methods that are devised for calculating a set of structures between two local total-energy minima. Referring to Figure 3, the idea is to start out from two well-known structures, e.g., A and D in the figure. Subsequently, a set of intermediate structures shall be determined that shall fulfill the following criteria ... 

For the first situation, the last decade or so has seen the development of accurate and reliable methods based on chains-of-states methods or extensions hereof (most notably, the nudged elastic-band method). On the other hand, the second situation remains to be a challenge, and also here it shall be remembered... 

The NEB method [143-145] is part of a family of path minimization techniques known as chain-of-states methods. In these methods, two positions on a PES are connected by a series of images that trace a path on the surface. After the initial path is generated, various optimization techniques can be used to minimize the path to an MEP on the PES. The features of this process are outlined schematically in Fig. 13. In one chain-of-states approach, called the elastic band method, the images are connected by springs and the path is optimized by minimizing the force experienced by each image as the sum of the true force and the spring force. The true force... 

Bell, T.A. Sevick, E.M. Theodorou, D.N. A chain of states method for investigating infrequent event processes in multistate, multidimensional systems. J. Chem. Phys. 1993, 98, 3196-3212. 

A more exact approach is to use sampling techniques to calculate the free energy difference between reactant and transition states, with the rate constant calculated by Eq. (1.117). One possibility is to calculate free energy differences between hyperplanes orthogonal to the IRC [125,126] (see also Chapters 9 and 10 in this book). Another is to use chain-of-states methods to sample fluctuations among different states along the IRC simultaneously [111,118,127,128]. A difficulty with the latter for explicit inclusion of polymer chain fluctuations is the high dimensionality of each state point. 

If subsequent states in the chain of states are related by the rules of the natural underlying dynanucs one should speak about trajectories rather than chain of states. Methods based on trajectories, such as TPS, the string method, and action-based methods are discussed later in this... 

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