Variational Principles in Conformation Dynamics - FU Berlin, FB MI

III.5. Two-well potential 29molecular simulations, a harmonic potential is often used to model bonds between atoms,as well as bond angles. The harmonic potential will therefore frequently show up in applications.III.5. Two-well potentialAnother important example is a two-well potential. In general, this is some potential functiondisplaying two minimum positions which are separated by a significant energy barrier, andwhich rises to infinity both to the left and to the right of the minima. For our calculations,we choose V (x) =k(x 4 − 2x 2 +1) for some k>0, whichhastwominimaatx = −1and x =1.TheenergyfunctionandtheinvariantdistributionaredisplayedinFigure III.4aand Figure III.4b. Thetwo-wellpotentialisagoodmodelforcertainmolecularinteractionswhere a certain degree of freedom favours two characteristic positions. The regions aroundthe potential minima correspond to metastable states, we therefore expect one dominantslow process apart from the stationary process. We apply the Roothan-Hall method usingthirteen Gaussian functions centred around the two minima, with uniform variance equalto 0.5. In order to evaluate the results, we also computed the eigenvalues of an MSMtransition matrix. Here, we used a fine discretization of the state space into 100 sets, mostof them situated close to the potential minima. A comparison of the results can be seenin Figure III.4. Most importantly, we see that the implied time scales are estimated verywell. Both methods converge quickly to about the same value. The stationary distributionis very well approximated, and we obtain a convincing result for the second eigenfunction,clearly displaying a characteristic sign change between the two metastable regions. Thecomputational effort required by the Roothan-Hall method is much smaller, since we onlyneed to compute an eleven by eleven matrix instead of a 100 × 100 matrix.