Variational Principles in Conformation Dynamics - FU Berlin, FB MI

II.4. The Ritz method and the Roothan-Hall method 17Moreover, as ˆψ 1 and ˆψ 2 diagonalize the H-matrix:ˆψ1 |T(τ) | ˆψ mm2 = b 1,i b 2,j χ i |T(τ) | χ j = b 1,i b 2,j h ij=i,j=1mξ 2 b 1,i b 2,i =0.i=1i,j=1(II.50)(II.51)Therefore, computing the Rayleigh coefficient for ˆψ yields: ˆψ |T(τ) | ˆψ= x 2 ˆψ1 |T(τ) | ˆψ 1 + y 2 ˆψ2 |T(τ) | ˆψ 2 +2xy ˆψ1 |T(τ) | ˆψ2(II.52)= x 2 ξ 1 + y 2 ξ 2 =(1− y 2 )ξ 1 + y 2 ξ 2 = ξ 1 − y 2 (ξ 1 − ξ 2 ). (II.53)Depending on the choice of y, thisvalueisbetweenξ 2 and ξ 1 .Ifwenowhadλ 2