In this computational study, we investigate the electronic properties of zigzag graphene nano-parallelograms (GNPs), which are parallelogram-shaped graphene nanoribbons of various widths and lengths, using thermally-assisted-occupation density functional theory (TAO-DFT). Our calculations reveal a monotonic decrease in the singlet-triplet energy gap as the GNP length increases. The GNPs possess singlet ground states for all the cases examined. With the increase of GNP length, the vertical ionization potential and fundamental gap decrease monotonically, while the vertical electron affinity increases monotonically. Besides, as the GNP length increases, the symmetrized von Neumann entropy increases monotonically, denoting an increase in the degree of multi-reference character associated with ground-state GNPs. The occupation numbers and real-space representation of active orbitals indicate that there is a transition from the nonradical nature of the shorter GNPs to the increasing polyradical nature of the longer GNPs. In addition, the edge/corner localization of active orbitals is found for the wider and longer GNPs.

For the ground-state properties of gas-phase nanomolecules with multi-reference character, thermally-assisted-occupation density functional theory (TAO-DFT) has been recently found to outperform the widely used Kohn-Sham density functional theory, when the conventional exchange-correlation energy functionals are employed. Aiming to explore solvation effects on the ground-state properties of nanomolecules with multi-reference character at a minimal computational cost, we combine TAO-DFT with the polarizable continuum model (PCM). To show its usefulness, TAO-DFT-based PCM (TAO-PCM) is employed to study the electronic properties of linear acenes in three different solvents (toluene, chlorobenzene, and water). According to TAO-PCM, in the presence of these solvents, the smaller acenes should have nonradical nature, and the larger acenes should have increasing polyradical nature, showing remarkable similarities to the past findings in the gas phase.

Thermally-assisted-occupation density functional theory (TAO-DFT) has been an efficient electronic structure method for studying the ground-state properties of large electronic systems with multi-reference character over the past few years. To explore the time-dependent (TD) properties of electronic systems (e.g., subject to an intense laser pulse), in this work, we propose a real-time (RT) extension of TAO-DFT, denoted as RT-TAO-DFT. Besides, we employ RT-TAO-DFT to study the high-order harmonic generation (HHG) spectra and related TD properties of molecular hydrogen H2 at the equilibrium and stretched geometries, aligned along the polarization of an intense linearly polarized laser pulse. The TD properties obtained with RT-TAO-DFT are compared with those obtained with the widely used time-dependent Kohn-Sham (TDKS) method. In addition, issues related to the possible spin-symmetry breaking effects in the TD properties are discussed.