Transition Path Sampling and the Calculation of Free Energies

Abstract

Many interesting physical, chemical and biological processes occur on time scales that exceed those accessible by molecular dynamics simulation by orders of magnitude. Often, these long time scales are due to high free energy barriers (large compared to kBT ) that the system must cross when moving between long-lived stable states. Examples for rare barrier crossing events include chemical reactions, the nucleation of first order phase transitions and the folding of proteins. One approach to treat such rare events in computer simulations is based on selecting a putative reaction coordinates and then determining the free energy as a function of that coordinate using the methods described in Chapter 4. By combining the free energy calculation with information obtained from dynamical trajectories initiated on a dividing surface separating the long-lived stable states, also reaction rates can be calculated [1–3]. To define a good reaction coordinate it is necessary to identify those degrees of freedom that capture the essential physics of the process. In complex systems, however, such information is often unavailable and this approach fails. In order to illustrate this problem in greater detail let us consider, as an example, a volume of pure liquid water carefully cooled below the freezing temperature. Although under such conditions the solid (ice) is the more stable phase, the system can remain liquid for hours or days even for strong undercooling. The reason for this behavior is that the phase transition from the liquid to the solid proceeds through the formation of a small crystalline nucleus somewhere in the liquid. The nucleus can then grow and the crystalline region eventually encompasses the whole sample. While overall the free energy of the system decreases during the transition, the initial stages of the freezing process are free energetically uphill. This free energetic cost is associated with the formation of an interface between the solid and the liquid. Whithin classical nucleation theory the free energy of the system as a function of the radius r of a spherical crystallite can be expressed as