The cyclic and linear paraphenylenes (*N* = 5−18) analyzed in this work are shown in Figure . All quantum chemical calculations were carried out with the parameter-free PBE0 hybrid density functional that incorporates a fixed combination of 25% Hartree−Fock exchange and Perdew’s GGA corrections in the correlation contribution.^{26} Previous investigations by Tretiak et al. have shown that the use of pure local and gradient-corrected functionals (i.e., LDA and PBE) results in unphysical unbound exciton states, whereas hybrid functionals partially overcome this problem by mixing in a fraction of nonlocal Hartree−Fock exchange.^{24,25} On the basis of their studies, I have chosen the PBE0 functional for this work because the PBE0 kernel provides a balanced description of neutral excitons (both singlets and triplets) in conjugated polymers and carbon nanotubes.^{24,25,27−30}

Ground-state geometries of all cyclic and linear paraphenylenes were optimized at the PBE0/6-31G(d,p) level of theory. Geometry optimizations were calculated without symmetry constraints, and root-mean-squared forces converged to within 0.00003 au. At the optimized ground-state geometries, TDDFT calculations were performed with a larger, diffuse 6-31+G(d,p) basis set to obtain the lowest four singlet vertical excitations. For both the ground-state and the TDDFT calculations, I used a high-accuracy Lebedev grid consisting of 96 radial and 302 angular points. All ab initio calculations were performed with a locally modified version of GAMESS.^{31}

Within the TDDFT formalism, one obtains the excited-state electron density, ρ(

**r**)

|ψ(

**r**)|

^{2} = ∑

_{i=1}^{N}|ϕ

_{i}(

**r**)|

^{2} composed of

*N*-occupied molecular orbitals ϕ

_{i}(

**r**) as solutions from the time-dependent Kohn−Sham equations. In the same way that one can calculate transition density matrices from time-dependent Hartree−Fock orbitals, one can use orbitals from the noninteracting Kohn−Sham system to form transition densities that include many-body correlation effects from the TDDFT formalism. Following the two-dimensional real-space analysis approach of Tretiak et al.,

^{21−25} one can construct coordinate

**Q**_{v} and momentum

**P**_{v} matrices with elements given by

where ψ

_{g} and ψ

_{v} are ground and excited states, respectively. The Fermi operators

*c*_{i}^{†} and

*c*_{i} represent the creation and annihilation of an electron in the

*i*th basis set orbital in ψ. For the cyclic and linear paraphenylenes analyzed in this work, the

**Q**_{v} and

**P**_{v} matrices each form a two-dimensional

*xy* grid running over all the carbon sites along the

*x* and

*y* axes. The specific ordering of the carbon sites used in this work is shown in Figure . The (

*P*_{v})

_{mn} momentum matrix represents the probability amplitude of an electron−hole pair oscillation between carbon sites

*m* and

*n*, and the (

*Q*_{v})

_{mn} coordinate matrix gives a measure of exciton delocalization between sites

*m* and

*n*. Each of these matrices provides a global view of electron−hole coherence and exciton delocalization for optical transitions within the paraphenylene systems.