Chun-Yaung (Albert) Lu
ResearchOne application of finding MEP is to estimate the reaction rate. According to the transition state theory (TST), the rate constant kTST is given by
where is the average velocity normal to the dividing surface in the direction of product, Q++ / QR is the ratio of transition state to reactant partition fuction. The reversible work based TST introduces the configuration integrals Qz of the system confined to a hyperplane:
The last term includes calculating free energy difference dA between a given hyperplane and the dividing surface. For a reaction with well defined reaction coordinate (denoted by ), its dA can be estimated from the reversible work done by moving the hyperplane uphill against the average force from the energy minimum (x=0) to the saddle point (x=s).
The total work done by the force acts upon the hyperplane at different position R in the hyperplane of normal nx is expressed in the above equation. From the mechanical analogy, the total work can be separated in to two components, i.e. translational (Atrans) and rotational work (Arot). In practice, we insert hyperplanes alone the MEP (Fig. 1) obtained from NEB, and then sample the force in the plane "i" to calculated the work based on the following recipe:
In Fig. 1, we demonstrate an example of a simple 2-D LEPS system. The rotational and translational works are shown in Fig. 2.
Figure 1 Contour plot of LEPS potential with MEP(red), and hyperplane(blue, not all hyperplanes used in the calculation are shown in the plot.)
Figure 2 Works calculated from the hyperplane sampling and the resulting total work (red) shows great agreement with direct integration result of Eq. (1) (black)
Once the free energy is obtained from hyperplane sampling, we use Eq. (2) to calculate the rate constant. The rest part of the equation can be determined by using the following equations:
The partition function ratio in Eq. (6) is obtained by running MD in the reactant region, which includes summing up all segment lengths ri - ri + 1 of point pairs cross the minimum hyperplane of normal nz.
The LEPS rate constants calculated at various temperatures are summarized in Fig. 3. The values estimated from hTST, and direct MD are also shown in the plot for comparison.
Figure 3 LEPS rate constants estimated by different methods over a range of temperature.
The hyperplane method can also be applied to the systems with higher dimension. In Figure 4, we calculated the rate of Al adatom hopping on Al(100) surface at different temperatures. The system is composed of 101 movable and 200 frozen atoms, and thus the total degrees of freedom is 303.
Figure 4 rate constants of Al hopping mechanism estimated by hyperplane sampling at various temperature.
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