Variational Principles in Conformation Dynamics - FU Berlin, FB MI

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32 IV. Application to molecules‘-168.931.49Figure IV.1.: Schematic drawing of the four atom system, corresponding either to system Aor B. It shows the dihedral angle in blue. Created with [VMD, 1996].AsketchofsuchasystemisshowninFigure IV.1.Let us now define a potential function V ,whichgeneratesaforcefieldactingonthemolecule.Abondbetweentwoatomsistypicallymodelledlikeaspring. Therefore,wedefineharmonicpotentials V ij ,dependingonthedistancer ij ,foreachpairofconnectedatoms:V ij := 1 2 k ij (d ij − r ij ) 2 .(IV.4)Here, k ij is a constant defining the strength of the potential and d ij is the distance of minimalenergy. The system will drive the bond lengths towards the minimum distances d ij .Similarly,we set up harmonic potentials V ijk for each bond angle:V ijk := 1 2 k ijk (d ijk − θ ijk ) 2 .(IV.5)The dihedral angles usually have a number of favoured positions. In order to model this, wedefine the dihedral potential asV ijkl := k ijkl (1 − cos(nψ ijkl )) ,(IV.6)for ψ ijkl ∈ [−π, π). The position ψ ijkl = π is identical to ψ ijkl = −π, consequently,itisenough to have V ijkl defined on the above interval. This potential has n minimum positions,
IV.1. The example system 33221.81.81.61.61.41.41.21.2V(ψ)1V(ψ)10.80.80.60.60.40.40.20.20−4 −3 −2 −1 0 1 2 3 4ψ0−4 −3 −2 −1 0 1 2 3 4ψFigure IV.2.: The potential energy function V (ψ ijkl )=k ijkl (1 − cos(nψ ijkl )) for the dihedralangle, with k ijkl =1and n =2, n =4,respectively.as displayed in Figure IV.2. Ifn =2,thesetwominimaaresituatedatψ ijkl =0;−π. Lastly,we would also like to include Coulomb potentials V C defined by:V C (r ij )= 14π 0e 1 e 2r ij, (IV.7)which describe the electrostatic repulsion or attraction between two atoms. Here, 0 is theelectric constant, and e 1 , e 2 are the charges of the two atoms. In our simulation, the chargeswill be equal to one elementary charge e =1.602 · 10 −19 C in absolute value, but can haveopposite signs. The total potential energy function V is then the sum of all individualenergies.For our computations, we will consider three small systems, which we call system A, systemB and system C.Thefirstisafouratomsystemwhichonlyincludesthethreeharmonicbondinteractions, two bond angle interactions, and a dihedral potential with n =2.Thesecondone also consists of four atoms, but we set n =4for the dihedral and add an attractiveCoulomb interaction between atoms one and four. In the last case, we have N =5,fourbonds, three bond angles and two dihedrals, one with n =4and the other with n =2.If we now add random perturbations and Smoluchowski dynamics, we obtain a diffusionprocess with invariant distributionµ(x) = 1 Zexp(−βV (x)),(IV.8)where Z is the partition function, as before, and x ∈ R 3N is the full position vector containingall coordinates of the atoms.It is important to point out once more that the stationary distribution and all the othereigenfunctions are defined on the 3N-dimensional Euclidean space. As such, they probablytake a highly complicated shape. At least for our examples however, the state of the system
Page 1: Freie Universität BerlinInstitut f
Page 5: Table of contentsI. Introduction 3I
Page 8 and 9: 4 I. IntroductionFigure I.1.: An in
Page 10 and 11: 6 II. The variational principlerega
Page 12 and 13: 8 II. The variational principleIn E
Page 14 and 15: 10 II. The variational principleThe
Page 16 and 17: 12 II. The variational principlesim
Page 18 and 19: 14 II. The variational principleFor
Page 20 and 21: 16 II. The variational principleSin
Page 22 and 23: 18 II. The variational principleThe
Page 25 and 26: III.Diffusion processes andapplicat
Page 27 and 28: III.2. Diffusion process 23time ∆
Page 29 and 30: III.3. Numerical considerations 25W
Page 31 and 32: III.4. Diffusion in a quadratic pot
Page 33 and 34: III.5. Two-well potential 29molecul
Page 35: IV.Application to moleculesIn this
Page 39 and 40: IV.2. Simulation 35Quantity Symbol
Page 41 and 42: IV.3. Application of the Roothan-Ha
Page 47 and 48: V. Summary and outlookWe have prese
Page 49 and 50: A. AppendixA.1. Diffusion in a quad
Page 51 and 52: A.1. Diffusion in a quadratic poten
Page 53: A.1. Diffusion in a quadratic poten
Page 57 and 58: Bibliography[Behrends, 2011] Ehrhar

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