Dispersion-corrected density functional theory (DFT) calculations examined energetically preferable hybrid structures between [13]cycloparaphenylene ([13]CPP) and an armchair (m,m) tube (5 ≤ m ≤ 9). Tube-in-ring and ring-on-tube structures, denoted by (m,m)@[13]CPP and [13]CPP–(m,m), respectively, were considered. DFT-based energy-decomposition analyses revealed that the key factors determining the stability of the hybrid structures are attractive long-range interactions between [13]CPP and the (m,m) tube and the energy penalty required for the deformation of [13]CPP. The long-range interactions in (m,m)@[13]CPP become more significant with an increase in the tube diameter. The [13]CPP deformation in (m,m)@[13]CPP varies with the degree of deviation between the radius difference between [13]CPP and a tube (Δr) and the van der Waal contact distance between two carbon atoms (3.35 Å). Accordingly, the stability of (m,m)@[13]CPP depends on the tube diameter. Particularly, (8,8)@[13]CPP, whose Δr is close to 3.35 Å, is highly stabilized among (m,m)@[13]CPP because of a negligible energy penalty for the [13]CPP deformation. In contrast, the stability of [13]CPP–(m,m) is less sensitive to the tube-diameter, and their attractive long-range interactions are weaker than those in (m,m)@[13]CPP. Consequently, a tube-in-ring structure whose radius difference is close to 3.35 Å is the most preferable among hybrid structures between [13]CPP and an (m,m) tube.
Dispersion-corrected density functional theory (DFT) calculations were employed to investigate stable structures of triiodobenzenes (BzI3) inside an armchair (m,m) carbon nanotube (m = 7–9), denoted by n × BzI3@(m,m), where n is the number of guests (n = 1–4). Three BzI3 isomers were considered: 1,3,5-, 1,2,4-, and 1,2,3-BzI3. DFT calculations found that nanotube confinement has an impact on n × BzI3@(m,m) structures, featuring guest orientation, positioning, and alignment. Then, we estimated a threshold value of the diameter of a tube containing BzI3 monomer in a certain orientation without repulsive host–guest interactions (D threshold). To represent the degree of nanotube confinement in 1 × BzI3@(m,m), we obtained ΔD small values by subtracting the smallest limit of D threshold of the guest from the host–tube diameter (D). Negative ΔD small values indicate strong confinement, and positive values less (much larger) than 1.0 Å indicate medium (weak) confinement. The strong and medium confinement in n × BzI3@(m,m) form a linear alignment of guests in parallel orientation with the smallest D threshold value. Reflecting from the different degrees of nanotube confinement, various types of intermolecular interactions operate. DFT calculations found that I–I intermolecular interactions are key in enhancing the second-order nonlinear optical properties of n × BzI3@(m,m). As a result, n × 1,2,4-BzI3@(8,8) with medium nanotube confinement has the most significant hyperpolarizability.
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