Variational Principles in Conformation Dynamics - FU Berlin, FB MI

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14 II. The variational principleFor a symmetric operator like T (τ), theRayleighcoefficientisalsocalledmatrix element orsimply expectation value. Thenexttworesultswillrevealtousthesignificanceofautocorrelationfunctions when it comes to approximating eigenvalues and -functions, [Noé, 2011,sec. 2.3]:Lemma II.5: The transfer operator’s eigenfunctions ψ i satisfy the relation:acf(ψ i ,ψ i ; τ) =λ i .(II.34)Proof. All we have to do is recognize the action of T (τ) on its eigenfunctions and use theorthogonality relation between those:acf(ψ i ,ψ i ; τ) =T (τ)ψ i | ψ i µ= λ i ψ i | ψ i µ= λ i .(II.35)Theorem II.6 (Variational principle): For any function ψ ∈ L 2 µ(Ω) which is orthogonalto ψ 1 ,wehave:acf(ψ, ψ; τ) ≤ λ 2 .(II.36)Proof. First, expand ψ in terms of the orthonormal basis {ψ i }.Notethatthereisnooverlapwith ψ 1 :acf(ψ, ψ; τ) =T (τ)ψ | ψ µ=∞c i c j T (τ)ψ i | ψ j µ,i,j=2(II.37)where c i = ψ | ψ i µ.Likeinthepreviousproof,weusetheactionofthetransferoperatoron its eigenfunctions:acf(ψ, ψ; τ) =∞c i c j λ i ψ i | ψ j µ=i,j=2∞c 2 i λ i .i=2(II.38)Finally, we recall that the eigenvalues are sorted in decreasing order, and that in an orthonormalbasis expansion, the squares of the coefficients sum up to unity:acf(ψ, ψ; τ) ≤ λ 2∞i=2c 2 i = λ 2 .(II.39)
II.4. The Ritz method and the Roothan-Hall method 15Remark II.7: Likewise, we can prove that for any function ψ which is orthogonal to thefirst k eigenfunctions, the expression acf(ψ, ψ; τ) is bounded from above by λ k+1 .Among all candidate functions, namely those which are orthogonal to the first eigenfunctionψ 1 ,thetruesecondeigenfunctionmaximizestheRayleighcoefficient.Forsomefunctionψ,T (τ)ψ | ψ µcan be viewed as a measure of how well it approximates the second eigenfunction,and that is what we will be trying to do: Maximize the Rayleigh coefficient amonganumberofansatzfunctions,andcomparedifferentchoicesofcandidatefunctions. Variationalmethods of that sort are used in a number of different fields as well, e.g. for theapproximation of electronic wavefunctions in quantum mechanics. In the next section, weshow how the Rayleigh coefficient can be maximized within the linear span of some giventest functions.II.4. The Ritz method and the Roothan-Hall methodFor all of the upcoming results, we follow [Szabo, Ostlund, 1989, ch.1.3].Theorem II.8 (Ritz method): Let χ 1 ,...,χ m ∈ L 2 µ(Ω) be mutually orthonormal. TheRayleigh coefficient is maximized among these function by the eigenvector b 1 correspondingto the greatest eigenvalue ξ 1 of the matrix eigenvalue problemH b 1 = ξ 1 b 1 ,(II.40)where H is the density matrix with entries h ij =acf(χ i ,χ j ; τ) =ΩΩdx dy χ i (x)p(x, y; τ)µ(x)χ j (y).(II.41)More precisely, the maximal Rayleigh coefficient is found by taking the linear combination ofthe functions χ i with coefficients taken from b 1 .Proof. Let us first, for a function ˆψ = mi=1 b iχ i ,expresstheRayleighcoefficientintermsof the b i :acf( ˆψ, ˆψ;τ) =ˆψ |T(τ) | ˆψ =mb i b j χ i |T(τ) | χ j =i,j=1mb i b j h ij .i,j=1(II.42)
Page 1: Freie Universität BerlinInstitut f
Page 5: Table of contentsI. Introduction 3I
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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
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Page 33 and 34: III.5. Two-well potential 29molecul
Page 35 and 36: IV.Application to moleculesIn this
Page 37 and 38: IV.1. The example system 33221.81.8
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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