6 A note on finding the activation energy barrier values in principle the

6 a note on finding the activation energy barrier

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6. A note on finding the activation energy barrier values: in principle, the minimized energy just before the system escapes into a new minimum energy state represents the energy of the saddle point. The energy difference between the saddle point and the initial minimum state is the activation energy for that particular pathway. However, because of the penalty functions, the activation energy can be overestimated in the ABC or ABC-E methods. We take the pathways (i.e. mechanisms) identified by steps 1–5 above. To enable better accuracy in the com- putation of kinetics based on those pathways, we employ the nudged elastic band method to calculate the precise values of activation energies connecting these pathways determined by ABC-E. By doing so, a series of states that neighbor the original state are found. The order of finding these new states is with J. Phys.: Condens. Matter 26 ( 2014 ) 365402
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Y Fan et al 6 increasing activation barriers. The more the identified tran- sitions, the more accurate the results. The criteria for stop- ping the search for more states can be defined in two ways. An easy criterion is to set a maximum number of total pen- alty functions to be added to the entire simulation in steps 1–5 described above, N steps MAX , and consider all the identified transition pathways within this upper bound of search steps. Specifically, let us denote n 1 as the number of penalty func- tions added before detecting the first connected state to the original minimum. Once the first connected state is found, the system is set back to the original state, following the proce- dures explained above. Let us assume it then takes another n 2 steps to find the second connected state to the original min- imum-energy state. The search is terminated after observing the i th state, such that uni2211 uni2211 uni2264 < = = = = + n N n j j i j steps MAX j j i j 1 1 1 , where n j represents the number of ABC steps in the search for the j th state. This criterion is suitable for situations in which there are not too many different transition states, because under such a circumstance, a reasonable choice of N steps MAX can be enough to search the complete transition states. For example, our experi- ences on a hypothetical 2D PES and on the anisotropic dif- fusion of point defects in Zr (discussed in the next section) suggest that assuming N steps MAX to be in the order of a few hun- dred is sufficient to get converged results. A second criterion can be defined by the following. We assume the number of the already-observed states is N states obs , with the associated barriers, from low to high, as E obs 1 , E E , , obs N obs 2 states obs . Once a new state is found with the barrier E new , we compare the relative probability of this transition with respect to the previously observed states at a given tem- perature by calculating uni239C uni239F uni239B uni239D uni239E uni23A0 uni2211 uni03B1 = - - ( ) T E k T ( ) exp exp .
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  • Summer '19
  • Transition state, KMC

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