  Understanding, optimizing, and controlling the optical absorption process,
exciton gemination, and electron-hole separation and conduction in low
dimensional systems is a fundamental problem in materials science. However,
robust and efficient methods capable of modelling the optical absorbance of low
dimensional macromolecular systems and providing physical insight into the
processes involved have remained elusive. We employ a highly efficient linear
combination of atomic orbitals (LCAOs) representation of the Kohn--Sham (KS)
orbitals within time dependent density functional theory (TDDFT) in the
reciprocal space ($k$) and frequency ($\omega$) domains, as implemented within
our LCAO-TDDFT-$k$-$\omega$ code, and apply the derivative discontinuity
correction of the exchange functional $\Delta_x$ to the KS eigenenergies. In so
doing we are able to provide a semi-quantitative description of the optical
absorption, conductivity, and polarizability spectra for prototypical 0D, 1D,
2D, and 3D systems within the optical limit ($\|\bf{q}\|\to0^+$) as compared to
both available measurements and from solving the Bethe$-$Salpeter equation with
quasiparticle $G_0 W_0$ eigenvalues ($G_0 W_0$-BSE). Specifically, we consider
0D fullerene (C$_{60}$), 1D metallic (10,0) and semiconducting (10,10)
single-walled carbon nanotubes (SWCNTs), 2D graphene (GR) and phosphorene (PN),
and 3D rutile (R-TiO$_2$) and anatase (A-TiO$_2$). For each system, we also
employ the spatially resolved electron-hole density to provide direct physical
insight into the nature of their optical excitations. These results demonstrate
the reliability, applicability, efficiency, and robustness of our
LCAO-TDDFT-$k$-$\omega$ code, and open the pathway to the computational design
of macromolecular systems for optoelectronic, photovoltaic, and photocatalytic
applications $in$ $silico$.
