Quantum Behavior of Substituted-Chrysene Molecular Assemblies: Bipartite Schur-Complement Transfer Anisotropy, Quadratic-to-Linear Stark Switching Across (D, I, T), and Registry-Gated Ballistic Transport

Abstract

A uniform-hopping model on the 18-vertex chrysene graph connects directional transfer, electric response and registry-dependent transport. The C6--C12 and C3--C9 terminal pairs have first nonzero Schur-complement moments at orders five and seven, respectively, proving inequivalence of their scalar transfer functions. At the gap center their isolated Green-function magnitudes are $1/(2t)$ and $1/(8t)$, giving an exact squared-amplitude ratio of 16; its conductance interpretation requires matched weak contacts and appropriate level alignment. Chiral sublattice symmetry forces two named same-sublattice resolvent entries to vanish at this energy. For the gapped closed-shell reference Hamiltonian, corrected response calculations establish unequal diagonal polarizabilities, with $\alpha_{xx}^{(0)}/\alpha_{yy}^{(0)}\simeq0.6882$. Inversion-symmetric site potentials on $D=\{6,12\}$ and $T=\{3,6,9,12\}$ force zero permanent model dipole and an even Stark energy branch. The asymmetric $I=\{3,6,12\}$ and mixed-arm $Dm$ cases permit a linear term when their actual electronic dipole is nonzero; geometric locant moments are treated separately. Finally, a coherent $2N$-site chain with $2N-1$ registry bits and matched leads has exact zero-energy transmission $T(0;\eta)=\operatorname{sech}^2(\Delta m(\eta)\ln\tau)$. Reversal and global orientation complement preserve this transmission. At six sites, the 32 fields split into counts $(10,15,6,1)$ by $|\Delta m|$, while the uniform-ensemble mean scales as $\Theta(N^{-1/2})$ for fixed $0<\tau<1$. The results supply explicit operator realizations and identify the additional molecular and experimental conditions needed to apply them.