Electronic Properties of Ladder-Type Phenylenes Studied by Thermally-Assisted-Occupation Density Functional Theory

Accurate prediction of the electronic properties of extended $\pi$-conjugated systems with radical character remains a major theoretical challenge. To overcome this challenge, thermally-assisted-occupation density functional theory (TAO-DFT) has been developed in recent years. In this study, we employ TAO-DFT to systematically explore the electronic properties of ladder-type phenylenes ($n$-LPs), containing $n$ fused four-membered carbon rings. Our calculations reveal that all investigated $n$-LPs (with $n$ = 1 ~ 40) possess singlet ground states. With increasing $n$-LP size, we observe a transition from the nonradical character of smaller $n$-LPs to the polyradical character of larger $n$-LPs. Notably, we identify a distinct damped periodic oscillation in the radical character of $n$-LPs, featuring highly correlated hotspots at specific intervals ($n=3k-1$, where $k$ are positive integers). The real-space representation of active orbitals confirms this global electronic behavior is governed by the structural localization of active orbitals along the central ladder framework. These findings provide fundamental insights into the size-dependent electronic properties of $n$-LPs, guiding their rational design for future nanomaterial applications.