Magnetic Hückel theory: framework for the Hermitian adjacency matrix of graphs

We introduce a magnetic extension of the Hückel molecular orbital (HMO) method in which the effect of an external magnetic field is incorporated through the Peierls substitution. In this formulation, the π-electron Hamiltonian of a conjugated molecule becomes a complex-weighted hopping operator defined on the hydrogen-depleted molecular graph. Under the standard Hiickel assumptions, the corresponding one-electron Hamiltonian matrix is proportional to the Hermitian adjacency matrix of the graph, providing a physical realization of Hermitian adjacency operators within molecular electronic structure theory. The magnetic-HMO framework enables the calculation of global flux-dependent observables such as orbital magnetization and orbital susceptibility from the magnetic adjacency spectrum, while local observables such as bond and ring currents are determined by the corresponding eigenvectors or one-particle density matrix. Applications to polycyclic aromatic hydrocarbons show that the magnetic response of π-electron systems is strongly controlled by molecular topology. In particular, linearly fused systems exhibit regular flux-dependent oscillations in the orbital magnetization, whereas nonlinear and extended molecules display more complex spectral rearrangements associated with multiple conjugation pathways.