Variational Principles in Conformation Dynamics - FU Berlin, FB MI

36 IV. Application to moleculesexpensive. It turns out to be quite difficult to find settings matching these requirements. Inparticular, we have to turn away from values used in real world examples a number of times.AcompletelistoftheparametersweusedisshowninTable IV.1.IV.3. Application of the Roothan-Hall methodWith a sufficiently long simulation trajectory at hand, we can turn to the approximation ofeigenvalues and time scales. For system A, we expect one dominant slow process to show up.We start with a small basis set, consisting of no more than two Gaussian functions centredat the two minimum positions, in order to get a rough approximation of the eigenvalues.Furthermore, we apply the method to a large basis set of twenty-seven functions, most ofthem centred closely around the two minima, with variance equal to 0.1. Detailsaboutthebasis sets used in this chapter are given in Table IV.2. As a reference, we also computethe eigenvalues and -vectors of two Markov state model discretizations. One of them justsplits up the state space into two sets along the dihedral coordinate. The other one is a finediscretization involving 100 sets along the same coordinate. Results showing the estimatedimplicit time scales, the dominant eigenvalues, as well as the first two eigenfunctions, aredepicted in Figure IV.4. Weseethatinbothcases,theslowprocesscanberesolved,andtheimplicit time scales converge as the lag time τ increases. As expected, the fine discretizationsconverge much faster than the coarse ones. Most importantly, we see that the large basisset Roothan-Hall approximation is as good as the fine MSM for most lag times. Also, bothmethods yield a very good approximation of the stationary density, compared to the statisticalestimate obtained from sampling, and a convincing estimate of the second eigenfunction,displaying the characteristic sign change between the minima. This is a good result, it showsus that the variational method yields an approximation which is comparable to a fine MSM.We also learn from our attempts that the choice of basis functions is very influential to theapproximation quality. If the functions are too strongly peaked, they are not sufficiently correlatedand the eigenvalues go down. However, if they are too widely spread, they correlatetoo much and the H-matrix is just poorly estimated. As a consequence, the computationreturns eigenvalues which are greater than one or have non-zero imaginary parts. But as wesaid in the beginning of the chapter, the shape of the energy function can be used to choosethe right functions and move in the right direction.For system B, the process will now display four metastable regions, but they will be populatedwith different frequencies. Whereas the minimum which is closest to atom number one willbe visited much more often because of the electrostatic attraction, the others will be lesspopulated. We therefore expect a number of dominant slow processes to appear, describingthe transitions between different minima. Again, we apply the Roothan-Hall method withtwo different basis sets, a small one consisting of four functions, and a large one consistingof 35 functions, which are distributed closely around the four minima. We find that threedominant slow processes are present, and compare the estimates for implicit time scales and