Figure 1 Illustration of the system evolution from one minimum energy state to

Figure 1 illustration of the system evolution from

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Figure 1. Illustration of the system evolution from one minimum energy state to another on the PES in the form of a nodal network, with the initial state ‘ i ’ and the final state ‘ f ’. J. Phys.: Condens. Matter 26 ( 2014 ) 365402
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Y Fan et al 4 2.2. The 1D nature of the original ABC algorithm Now let’s revisit the transition network discussed above with the original ABC method. The ABC algorithm evolves the system mainly towards the pathway with the lowest barrier, while neglecting other, higher-barrier pathways. Therefore, for the same system, ABC will provide an evolution chain that connects the transition paths in 1D instead of a network (seen in figure  2 ). Since for this 1D chain, each state is only connected with one state before it and one state after on the nodal network of states, the residence time for state p is given by: uni03C4 = + - + k k 1 p p p p p , 1 , 1 (6) And similarly, the evolution time from i to f can be calcu- lated by solving the following linear equations: Figure 2. Illustration of the system evolution from one minimum energy state to another on the PES in the form of a nodal network, with the initial state ‘ i ’ and the final state ‘ f ’. The collection of states and connectivities are the same as in figure  1 . An evolution chain is provided by the ABC algorithm as the red path on the left, corresponding to the PES of a 1D chain of transitions illustrated on the right. Figure 3. ( a ) (Adapted with permission from [ 32 ]). A pre-constructed 2D PES with a rough landscape, for representing the system evolution from i to f with multiple competing processes. ( b ) A variation of the evolution time with temperature sampled by full-catalog kMC (with two different r c ), ABC, and ABC-E. uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni03C4 uni03C4 uni03C4 - uni22EF + - + uni22EF + - + uni22EF uni22EE uni22F1 uni22EE uni22EF + - uni2192 uni2192 uni22EE uni2192 = - uni22EE - - - k k k k k k k k k k k k k k k t t t 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 . i i i n n n n n f i f f n f i n 1, 1, 1,2 1,2 1, 1,2 2,1 2,1 2,3 2,3 2,1 2,3 , 1 , 1 , 1 1 (7) The dimension for the above matrix is the length of the 1D chain, which is significantly smaller than the total number of nodes in the system. In addition, the matrix has the banded tridiagonal structure, which is much simpler than the matrix in the last section for a full catalog of events and transitions. kMC can also be employed based on this reduced matrix to calculate the evolution time uni2192 t i f . Because of the reduced dimension and simpler underlying mathematical structure, the ABC-based kMC method saves substantial computational load compared to the full-catalog kMC calculations.
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  • Summer '19
  • Transition state, KMC

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