An exact formulation of hyperdynamics simulations

Abstract

We introduce a new formula for the acceleration weight factor in the hyperdynamics simulation method, the use of which correctly provides an exact simulation of the true dynamics of a system. This new form of hyperdynamics is valid and applicable where the transition state theory (TST) is applicable and also where the TST is not applicable. To illustrate this new formulation, we perform hyperdynamics simulations for four systems ranging from one degree of freedom to 591 degrees of freedom: (1) We first analyze free diffusion having one degree of freedom. This system does not have a transition state. The TST and the original form of hyperdynamics are not applicable. Using the new form of hyperdynamics, we compute mean square displacement for a range of time. The results obtained agree perfectly with the analytical formula. (2) Then we examine the classical Kramers escape rate problem. The rate computed is in perfect agreement with the Kramers formula over a broad range of temperature. (3) We also study another classical problem: Computing the rate of effusion out of a cubic box through a tiny hole. This problem does not involve an energy barrier. Thus, the original form of hyperdynamics excludes the possibility of using a nonzero bias and is inappropriate. However, with the new weight factor formula, our new form of hyperdynamics can be easily implemented and it produces the exact results. (4) To illustrate applicability to systems of many degrees of freedom, we analyze diffusion of an atom adsorbed on the (001) surface of an fcc crystal. The system is modeled by an atom on top of a slab of six atomic layers. Each layer has 49 atoms. With the bottom two layers of atoms fixed, this system has 591 degrees of freedom. With very modest computing effort, we are able to characterize its diffusion pathways in the exchange-with-the-substrate and hop-over-the-bridge mechanisms.

Fig. 1

Mean square displacement (unit: ${\epsilon }_0{\tau }_0^2/m$) vs time (unit: τ₀) for free diffusion. The solid line is the analytical solution. The dashed line that completely overlaps the solid line is the result of the hyperdynamics simulation. The dotted line is the result of the Langevin dynamics simulation. The inverse temperature is β=10/ε₀ and the damping constant is γ = 2.5/τ₀.

Fig. 2

The Kramers potential V(x) (solid) and the boosted potential V_b(x) (dashed). The unit of potential is ε₀ and the unit of x is $d_0={\tau }_0\sqrt{{\epsilon }_0/m}$.

Fig. 3

Kramers rate × 10³ (unit: 1/τ₀) vs inverse temperature (unit: 1/ε₀). The line is the Kramers formula and the points are the results of the present hyperdynamics study. The damping constant is taken as γ = 2.5/τ₀.

Fig. 4

Probability (× 10⁵) for a particle to escape the box as a function of time (unit: τ₀). The solid curve is the result of our hyperdynamics study and the dashed one is from the unbiased Langevin dynamics simulation for 10⁵ paths sampled. γ=1/τ₀ and β=10/ε₀.

Fig. 5

(Color) Ni on Ni(001) surface before (left), at transition state (center), and after (right) an exchange-with-the-substrate event.

FIG. 6

Potential energy V(t) (left) and action functional βI_ξ(t) (right) along the diffusion pathway for an exchange-with-the-substrate event. The hyperdynamics simulation is executed during the time interval shown. Outside this time interval, regular Langevin dynamics prevail.

Fig. 7

(Color) Ni on Ni(001) surface before (left), at transition state (center), and after (right) a hop-over-the-bridge event.

Fig. 8

Potential energy V(t) (left) and action functional βI_ξ(t) (right) along the diffusion pathway for a hop-over-the-bridge event. The hyperdynamics simulation is executed during the time interval shown. Outside this time interval, regular Langevin dynamics prevail.