A Few Minutes With Michelle R. Tinkham

Conformational Statistics of Ribbon-like Chains

“…The orientation of a “ribbon” segment, { t 1 , t 2 , t 3 }, can be parametrized using the Euler angles Ω = {ϕ, θ, ψ} defined in reference to the laboratory frame. In this work, we adopt the ZYZ convention for the Euler angle. As shown in Figure c, the azimuthal angle ϕ and polar angle θ fix the orientation of t 3 , around which the twisting angle ψ is identified, which subsequently fixes the orientations of t 1 and t 2 .…”

“…Because the Hamiltonian scriptA explicitly depends on all three Euler angles, it is convenient to solve eq by expanding the free-chain Green function using the Wigner scriptD functions as the basis set. The Wigner functions D mj false( l false) have three indices that fall into the ranges: l ∈ [0, ∞), m ∈ [− l , l ], and j ∈ [− l , l ]. , The functions D mj false( l false) have several desirable properties: they explicitly depend on the Euler angles, form a complete basis set, and are orthogonal to each other. Moreover, D mj false( l false) are eigenfunctions of L 2 and L 3 , and the action of L 1 and L 2 only raises or lowers the index j , as shown in Section S.1.…”

“…Moreover, D mj false( l false) are eigenfunctions of L 2 and L 3 , and the action of L 1 and L 2 only raises or lowers the index j , as shown in Section S.1. Then, the free-chain Green function can be formally expanded as G ( Ω | normalΩ 0 ; N ) = prefix∑ l = 0 ∞ ∑ m = − l l ∑ j = − l l ∑ j ′ = − l l g j j ′ false( l false) ( N ) D mj false( l false) ( Ω ) D italicmj false′ ( l ) * ( normalΩ 0 ) Above, only one l index is needed because the functions D mj false( l false) with the same l form an irreducible representation of three-dimensional (3D) rotation . The two scriptD functions in the expansion share a common index m because the chain conformation is invariant with respect to whole chain rotation.…”