Energy barrier of pairing change

In our simulation, the movement of one Au atom in the nanoparticle is attempted for each kMC step. The movement involves the breaking and formation of various Au-Au pairs. For large nanoparticles, it is hard to compute the energy barrier of the movement using the first-principles density functional theory. An reliable approach holding for the microscopic reversibility principle is highly desired to calculate the barriers E_a.

Generally, the energy barrier E_a should be between ΔE and E_b for endothermic reactions or between 0 and E_b for exothermic reactions, where ΔE is the total energy change and E_b is the total energy to break Au-Au pairs in the movement, respectively. The total energy change ΔE = E_b − E_f, where E_f is the absolute value of the total energy to form Au-Au pairs. Thus a possible expression satisfying the microscopic reversibility principle is

where the co-efficient α (0 ≤ α ≤ 1) characterizes the location of the transition state along the reaction coordinate. As a coarse approach, we take unitary α in Eq. (9) and obtain Clearly represents the energy needed to break Au-Au pairs in the attempting movement.

The energy barrier in our previous study¹⁹ was calculated as

Compared with , the fractional α makes the transition state move towards the final state. The energy barrier with even small α that fits the principle of microscopic reversibility are also made available by

Obviously, these energy barriers with small α correspond to the late transition states.

The choice of Eqs (10) to (12), , depends only on the position of the transition state along the reaction coordinate. Once E_b and E_f are known, the calculation of the energy barriers does not require any empirical parameter, which is usually present when applying the Brönsted-Evans-Polanyi principles22^,23. Furthermore, the movement of Au atoms in different coordination environments, such as the corner, edge and surface of the nanoparticle, involves different Au-Au pairs; thus, the calculated E_a and rate constants of the Au movements are distinguished as a consequence in our model.