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 al42.2. The 1D nature of the original ABC algorithmNow 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 pis given by:uni03C4=+-+kk1pp pp p,1,1(6)And similarly, the evolution time from ito fcan 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 ). A pre-constructed 2D PES with a rough landscape, for representing the system evolution from ito fwith multiple competing processes. (b) A variation of the evolution time with temperature sampled by full-catalog kMC (with two different rc), ABC, and ABC-E.uni239Buni239Duni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Euni23A0uni239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Buni239Duni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Euni23A0uni239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Buni239Duni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Cuni239Euni23A0uni239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi239Funi03C4uni03C4uni03C4-uni22EF+-+uni22EF+-+uni22EFuni22EEuni22F1uni22EEuni22EF+-uni2192uni2192uni22EEuni2192=-uni22EE---kkkkkkkkkkkkkkkttt110001000100001•.iiin nn nn fiffnfin1,1,1,21,21,1,22,12,12,32,32,12,3,1,1,11(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 uni2192tif. 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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