Variational Principles in Conformation Dynamics - FU Berlin, FB MI

IV.3. Application of the Roothan-Hall method 3912100.70.6Estimated φ1, Roothan−HallEstimated φ1, sampling80.5V(ψ)64φ1(ψ)0.40.30.220.10−4 −3 −2 −1 0 1 2 3 4ψ(a)0−4 −3 −2 −1 0 1 2 3 4ψ(b)Second eigenvalue λ21.00510.9950.990.9850.98Estimated λ2, large Roothan−HallEstimated λ2, large MSMEstimated λ2, small MSMEstimated λ2, small Roothan−HallImplied time scale t23530252015100.9750.970 50 100 150Time lag τ(c)Estimated t2, large Roothan−Hall5Estimated t2, large MSMEstimated t2, small Roothan−HallEstimated t2, small MSM00 50 100 150Time lag τ(d)0.60.40.2φ2(ψ)0−0.2−0.4−0.6Estimated φ2, Roothan−HallEstimated φ2, MSM−0.8−4 −3 −2 −1 0 1 2 3 4ψ(e)Figure IV.4.: Approximation results for system A. We used every third step of a 30 milliontrajectory corresponding to a sampling time step ∆t =10 −3 ps. We comparethe results obtained with the small and the large basis set with a 2 set anda 100 set MSM. The functions displayed were computed from the large basissets. (a) Potential energy for the dihedral coordinate. (b) Projection of firsteigenfunction φ 1 ,comparedtodirectestimatefromthesample. (c)Secondeigenvalue λ 2 . (d) Second implied time scale t 2 . (e) Projection of secondeigenfunction φ 2 .