The Skewed Momenta Method

Methods such as skewed momenta are expected to have an additional advantage in high-dimensional systems. Puddle jumping is efficient in such systems only if the puddle can be selectively applied across a few pertinent degrees of freedom. In contrast, the skewed momenta method can be applied without modification to trajectories involving concerted changes to many degrees of freedom. 

We will now turn our attention to the reconstruction of free energy profiles using the Jarzynski identity. This identity can be cast in terms of an equilibrium average, (8.49), as explained in Chap. 5. We can then bias the dynamics to follow the motion of the pulling potential, enhancing sampling of the low-work tail of the work distribution and thereby increasing the accuracy of the calculation. 

As alluded to above, the method relies upon the identification of a 3A-dimensional vector in configuration space, es, which points along a favored direction for the motion of the system. We then choose the initial momenta for the trajectory ensemble from a Gaussian distribution artificially extended in the direction of es, as illustrated in the right panel of Fig. 8.2. In the case of free energy reconstructions from (8.49), we wish to induce motion along a predefined pulling direction, and so es can be found by inspection. 

In the case of (8.49), the situation is once again simplified by prior knowledge of the pulling direction es can be taken without loss of generality to lie along one of the natural Cartesian axes of the system, so that the p and p systems are equivalent, and no axis rotations need be performed. 

At equilibrium, the components of the momentum vector p are drawn from the distribution 

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We start with some background on existing methods that alter the initial distributions in the reactant basin, focusing in particular on the puddle jumping method of Tully and coworkers [69, 110], which is the inspiration for the skewed momenta method developed in the following section. We continue with a description of the skewed momenta method, as applied to (8.48), with numerical examples for each case. We end with a concluding discussion. 

Unlike the methods mentioned above, the skewed momenta method involves accentuating the dynamics only along particularly relevant directions. Because the... 

In this section we explore the use of the skewed momenta method for estimating the equilibrium free energy from fast pulling trajectories via the Jarzynski identity [104]. The end result will be that generating trajectories with skewed momenta improves the accuracy of the calculated free energy. As described in Chap. 5, Jarzynski s identity states that... 

With Jarzynski s identity in the form of (8.46), we can apply to it the skewed momenta method simply by setting A[r(f)] = exp(— /3Wt). However, we anticipate that the method will be most useful in the particular case of calculating free energy profiles from pulling experiments, for which Hummer and Szabo have provided a modified form of Jarzynski s expression [106]. 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

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