Variational Principles in Conformation Dynamics - FU Berlin, FB MI

IV.Application to moleculesIn this chapter, we are going to apply the variational method to diffusion processes in ahigher dimensional state space. For systems with many degrees of freedom, the potentialenergy function often possesses multiple minima, and contains at times highly complicatedinteractions between several of the coordinates. Evaluation of the partition function becomesvery difficult, and requires specialized algorithms like Markov chain Monte Carlo methods.Computing eigenvalues and characteristic time scales can be done using Markov state models,but the choice of adequate sets requires suitable clustering methods. This also leads to ahuge computational effort if the state space is very high-dimensional. We have the hope thatthe use of variational methods can help to reduce this effort.IV.1. The example systemLet us consider a system which is like a very much simplified small molecule. Let it consistof N atoms, with N being small, either equal to 4 or to 5 in what follows. Denote theirposition vectors by r i ∈ R 3 , i ∈{1,...,N}. Neighbouringatomsareconnectedbyabond.Denote the distance vectors between those atoms by r ij := r j − r i and the distances byr ij := r ij .Forthreeneighbouringatoms,wedefinethebondangleθ ijk byθ ijk := cos −1 − r ij | r jk r ij r jk, (IV.1)which is the angle between r ij and r jk . Additionally, for four atoms, we can define thedihedral angle ψ ijkl as the angle between the plane spanned by r ij , r jk and the one spannedby r jk , r kl .Usingthenormalvectorsn ijk and n jkl ,givenbyn ijk := r ij × r jk ,(IV.2)which are perpendicular to the respective planes, we find that the dihedral is the anglebetween the two normal vectors: ψ ijkl =cos −1 nijk | n jkl . (IV.3)n ijk n jkl