Origin of the linear relationship between CH₂/NH/O-SWNT reaction energies and sidewall curvature: Armchair nanotubes

The origin of the linear relationship between the reaction energy of the CH₂/NH/O exo and endo additions to armchair (n, n) single-walled carbon nanotubes (SWNTs) and the inverse tube diameter (1/d) measuring sidewall curvature was elucidated using density functional theory and density functional tight binding methods for finite-size SWNT models with n = 3, 4, ..., 13. A nearly perfect linear relationship between ΔE and 1/d all through exohedral (positive curvature) and endohedral (negative curvature) additions is due to cancellation between the quadratic contributions of the SWNT deformation energy and the interaction energy (INT) between the deformed SWNT and CH ₂/NH/O adducts. Energy decomposition analysis shows that the quadratic contributions in electrostatic, exchange, and orbital terms mostly cancel each other, making INT weakly quadratic, and that the linear 1/d dependence of INT, and therefore of ΔE, is a reflection of the 1/d dependence of the back-donative orbital interaction of b₁ symmetry from the occupied CH₂/NH/O pπ orbital to the vacant C=C π* LUMO of the SWNT. We also discuss the origin of the two isomers (open and three-membered ring) of the exohedral addition product and explain the behavior of their associated minima on the C-C potential energy surfaces with changing d.