Variational Principles in Conformation Dynamics - FU Berlin, FB MI

18 II. The variational principleThe only problem left is the requirement that the basis functions were so far assumed toorthonormal. However, this can easily be circumvented. Again, let χ 1 ,...,χ m ∈ L 2 µ(Ω) benormalized basis functions as before, except that they are no longer assumed to be orthogonal.We can still derive a similar result:Theorem II.10 (Roothan-Hall method): For basis functions as above, the linear combinationthat maximizes the Rayleigh coefficient is given by the solution b 1 corresponding tothe greatest eigenvalue ξ 1 of the generalized eigenvalue problemH b 1 = ξ 1 S b 1 ,where H is as in Theorem II.8 and S is the overlap matrix with entriess ij = χ i | χ j µ.(II.55)(II.56)Proof. Starting out the same way as in Theorem II.8, wefirsthavetoreformulatethenormalization constraint. Equation II.43 then becomes:1= ˆψmm2 = b i b j χ i | χ j µ= b i b j s ij .(II.57)i,j=1i,j=1i,j=1This results in an effective functional of the form:mmF (b 1 ,...,b m )= b i b j h ij − ξ( b i b j s ij − 1).Maximization requires the solution of:⇒ ξ0=2mb j s ij =j=1i,j=1mb j h ij − 2ξj=1mb j h ij ,j=1mb j s ij .j=1(II.58)(II.59)(II.60)for every i ∈{1,...,m}. Thisisthegeneralizedeigenvalueequationstatedabove.SinceH issymmetric and S is positive definite, this problem has m real eigenvalues ξ i and correspondingeigenvectors b i which are orthonormal with respect to the S-weighted scalar product, i.e. wehave b T i S b j = δ ij . Therefore, similar to Equation II.47 - Equation II.48, wefindforthecorresponding functions ˆψ i := mj=1 b i,jχ j :ˆψi |T(τ) | ˆψ mmi = b i,j b i,k χ j |T(τ) | χ k = b i,j b i,k h jk (II.61)j,k=1j,k=1 m= ξ i b i,j b i,k s j,k = ξ i b T i S b i = ξ i .j,k=1(II.62)