36 IV. Application to moleculesexpensive. It turns out to be quite difficult to f<strong>in</strong>d sett<strong>in</strong>gs match<strong>in</strong>g these requirements. Inparticular, we have to turn away from values used <strong>in</strong> real world examples a number of times.Acompletelistoftheparametersweusedisshown<strong>in</strong>Table 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 dom<strong>in</strong>ant slow process to show up.We start with a small basis set, consist<strong>in</strong>g of no more than two Gaussian functions centredat the two m<strong>in</strong>imum positions, <strong>in</strong> 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 m<strong>in</strong>ima, with variance equal to 0.1. Detailsaboutthebasis sets used <strong>in</strong> this chapter are given <strong>in</strong> 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 <strong>in</strong>to two sets along the dihedral coord<strong>in</strong>ate. The other one is a f<strong>in</strong>ediscretization <strong>in</strong>volv<strong>in</strong>g 100 sets along the same coord<strong>in</strong>ate. Results show<strong>in</strong>g the estimatedimplicit time scales, the dom<strong>in</strong>ant eigenvalues, as well as the first two eigenfunctions, aredepicted <strong>in</strong> Figure IV.4. Weseethat<strong>in</strong>bothcases,theslowprocesscanberesolved,andtheimplicit time scales converge as the lag time τ <strong>in</strong>creases. As expected, the f<strong>in</strong>e discretizationsconverge much faster than the coarse ones. Most importantly, we see that the large basisset Roothan-Hall approximation is as good as the f<strong>in</strong>e MSM for most lag times. Also, bothmethods yield a very good approximation of the stationary density, compared to the statisticalestimate obta<strong>in</strong>ed from sampl<strong>in</strong>g, and a conv<strong>in</strong>c<strong>in</strong>g estimate of the second eigenfunction,display<strong>in</strong>g the characteristic sign change between the m<strong>in</strong>ima. This is a good result, it showsus that the variational method yields an approximation which is comparable to a f<strong>in</strong>e MSM.We also learn from our attempts that the choice of basis functions is very <strong>in</strong>fluential 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 imag<strong>in</strong>ary parts. But as wesaid <strong>in</strong> the beg<strong>in</strong>n<strong>in</strong>g of the chapter, the shape of the energy function can be used to choosethe right functions and move <strong>in</strong> the right direction.For system B, the process will now display four metastable regions, but they will be populatedwith different frequencies. Whereas the m<strong>in</strong>imum 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 dom<strong>in</strong>ant slow processes to appear, describ<strong>in</strong>gthe transitions between different m<strong>in</strong>ima. Aga<strong>in</strong>, we apply the Roothan-Hall method withtwo different basis sets, a small one consist<strong>in</strong>g of four functions, and a large one consist<strong>in</strong>gof 35 functions, which are distributed closely around the four m<strong>in</strong>ima. We f<strong>in</strong>d that threedom<strong>in</strong>ant slow processes are present, and compare the estimates for implicit time scales and
IV.3. Application of the Roothan-Hall method 37eigenvalues with a four set and a 100 set MSM discretization. The results are shown <strong>in</strong>Figure IV.5 and Figure IV.6. Theyaresimilartothepreviousexample,botheigenvaluesandtime scales as well as the eigenfunctions are approximated comparably well by both the largeset Roothan-Hall method and the f<strong>in</strong>e MSM.These examples had <strong>in</strong> common that the metastable behaviour was only dependent on ones<strong>in</strong>gle coord<strong>in</strong>ate. As a last step, we take a look at system C, where more than one coord<strong>in</strong>ateis relevant for the description of slow processes. For the Roothan-Hall method, wecomb<strong>in</strong>e the two basis sets used <strong>in</strong> the previous examples, that means we have 35 basisfunctions which depend on the first dihedral coord<strong>in</strong>ate and 27 which cover the second. Wecompare the results to those stemm<strong>in</strong>g from a Markov state model discretization, whichuses all possible comb<strong>in</strong>ations of 15 sets along each coord<strong>in</strong>ate, i.e. 225 sets altogether.The second, third and fourth eigenvalue and time scale, estimated with both methods, areshown <strong>in</strong> Figure IV.7. WeseethatthetimescalesestimatedwiththeRoothan-Hallmethodconverge earlier. This is a good sign, because we are able to estimate the eigenvalues andtime scales of the larger system by simply add<strong>in</strong>g up the basis functions used for the smallersystem, without hav<strong>in</strong>g to use all possible comb<strong>in</strong>ations of these functions.
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