A Recipe for the Computation of the Free Energy Barrier and the Lowest Free Energy Path of Concerted Reactions

Abstract

The recently introduced hills method (Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12562) is a powerful tool to compute the multidimensional free energy surface of intrinsically concerted reactions. We have extended this method by focusing our attention on localizing the lowest free energy path that connects the stable reactant and product states. This path represents the most probable reaction mechanism, similar to the zero temperature intrinsic reaction coordinate, but also includes finite temperature effects. The transformation of the multidimensional problem to a one-dimensional reaction coordinate allows for accurate convergence of the free energy profile along the lowest free energy path using standard free energy methods. Here we apply the hills method, our lowest free energy path search algorithm, and umbrella sampling to the prototype S(N)2 reaction. The hills method replaces the in many cases difficult problem of finding a good reaction coordinate with choosing relatively simple collective variables, such as the bond lengths of the broken and formed chemical bonds. The second part of the paper presents a guide to using the hills method, in which we test and fine-tune the method for optimal accuracy and efficiency using the umbrella sampling results as a reference.

Polymorphic phase transition mechanism of compressed coesite

“…The number of MD steps was determined in such a way that the CV displacement is comparable with the width of the bias energy, avoiding the so-called 'hill surfing problems' 43 . Therefore, the metadynamics timestep was chosen adaptively such that a Gaussian is placed at t i once the following condition is fulfilled:…”

The free energy landscape of small peptides as obtained from metadynamics with umbrella sampling corrections

“…It has been claimed in Ref. 35 that increasing the width of the Gaussians requires an increase of the stride to avoid "hill surfing" (where the collective variable continuously rides the tail of the most recently placed hill) and increasing the amplitude requires increasing the width (and therefore increasing the stride) to avoid steep forces on the collective variable. However, very large strides and small hills place serious constraints on the efficiency of phase space exploration.…”

“…In fact, a "recipe" for computing free energy paths was given in Ref. 35, where the authors developed a method to localize the lowest free energy path that connects two minima and expressed it in the form of a one-dimensional reaction coordinate. The potential along this one-dimensional coordinate was then used to perform umbrella sampling to correct the metadynamics results.…”

Comment on “Communication: Benzene dimer—The free energy landscape” [J. Chem. Phys. 139, 201102 (2013)]

“…According to the standards in these papers, even a relatively high bias potential only amounts to a small fraction of the barrier height, however, and it is also recommended to choose Gaussians not higher than k B T. Another important aspect is that the system should be given time to equilibrate after each energy jump and, in view of the large amount of energy added and the velocities of the molecules at T = 2 K, this will take many time steps. 9 In the supplementary material, 8 we show that the statistical error of the computed free energy surface increases linearly with the Gaussian height and with the update frequency. Unfortunately, the authors did not mention how many time steps were actually taken before the next bias potential was added.…”

“…This is much larger than the energy difference of about 0.2 kcal/mol between the TT and PD structures and the height of the energy barrier that separates these structures, 1 and even larger than the total binding energy of 2.7 kcal/mol. In the metadynamics literature, [9][10][11] it is recommended to use relatively high Gaussian bias potentials in the first stage of the simulation, but subsequently to lower the Gaussian heights in order to reach convergence. According to the standards in these papers, even a relatively high bias potential only amounts to a small fraction of the barrier height, however, and it is also recommended to choose Gaussians not higher than k B T. Another important aspect is that the system should be given time to equilibrate after each energy jump and, in view of the large amount of energy added and the velocities of the molecules at T = 2 K, this will take many time steps.…”