van der Waals corrected density functionals for cylindrical surfaces: Ammonia and nitrogen dioxide adsorbed on a single-walled carbon nanotube

We extend the damped Zaremba-Kohn model (dZK) for long-range dispersion interaction between a molecule and a planar surface [J. Tao, H. Tang, A. Patra, P. Bhattarai, and J. P. Perdew, Phys. Rev. B 97, 165403 (2018)] to molecules adsorbed on a curved cylindrical surface, and employ this extended model as an additive correction to the semilocal density functionals PBE (Perdew-Burke-Ernzerhof) and SCAN (strongly constrained and appropriately normed). The resulting PBE+vdW (van der Waals)-dZK and SCAN+vdW-dZK are applied to two systems, $\mathrm{N}{\mathrm{H}}_3$ and $\mathrm{N}{\mathrm{O}}_2$ molecules adsorbed on a single-wall carbon nanotube (CNT), for calculations of binding energies and equilibrium distances. For comparison, the results from vdW nonlocal functionals, such as SCAN+rVV10 and PBE+rVV10, are also presented. The binding energies from PBE+rVV10 (Vydrov and Van Voorhis), SCAN+rVV10, PBE+vdW-dZK, and SCAN+vdW-dZK are about 70–115 meV for the system of $\mathrm{CNT}+\mathrm{N}{\mathrm{H}}_3$ and 300–500 meV for the system of $\mathrm{CNT}+\mathrm{N}{\mathrm{O}}_2$. The results from PBE+vdW-dZK and SCAN+vdW-dZK are closer to each other than those from PBE+rVV10 and SCAN+rVV10 are. The relatively closer results from PBE+vdW-dZK and SCAN+vdW-dZK indicate the consistency of our developed vdW−dZK model for cylindrical surfaces. All methods, including PBE, SCAN, PBE+rVV10, SCAN+rVV10, PBE+vdW-dZK, and SCAN+vdW-dZK, give approximately the same binding energy differences between two adsorption configurations (types I and II) for the two systems. This implies that the two adsorption sites have approximately the same adsorption stability. The exponent of the vdW interaction power law from our vdW-dZK model for the two systems is about 0 at short distance, largely due to the damping factor, and tends slowly to −4 to −4.5 at distances $D$ about 20–50 Å. At even larger distances, the vdW power-law exponent approaches −5. This feature is very similar to the one calculated with random-phase approximation and renormalization group approaches, supporting the applicability of our methods. Our developed vdW-dZK method provides a highly efficient and reliable method for large systems with cylindrical surfaces, such as vdW interactions with nanotubes.

Figure 1

Schematic diagram of an interacting system consisting of a dipole and a dielectric cylinder. The origin of the coordinate system is at the center of the dipole. The cylinder has a relative dielectric constant $\varepsilon$ and a radius $a$. The relative dielectric constant of the surrounding vacuum is ${\varepsilon }_v=1$.

Figure 2

(10,0) CNTs with adsorbed $\mathrm{N}{\mathrm{H}}_3$ or $\mathrm{N}{\mathrm{O}}_2$ molecules. (a) Top view of the $\mathrm{N}{\mathrm{H}}_3$ molecule attached to the CNT in type I geometry, (b) in type II geometry. (c) Side view of the $\mathrm{N}{\mathrm{O}}_2$ molecule attached to the CNT when the plane of the $\mathrm{N}{\mathrm{O}}_2$ is parallel to the axis of the CNT (type I). (d) Side view of the $\mathrm{N}{\mathrm{O}}_2$ molecule attached to the CNT when the plane of the $\mathrm{N}{\mathrm{O}}_2$ is perpendicular to the axis of the CNT (type II).

Figure 3

Flow chart of the computational process for the vdW interaction energies, DFT energies, and total energies for the developed PBE+vdW-dZK and SCAN+vdW-dZK methods.

Figure 4

vdW interaction energy power-law exponents from the vdW-dZK model for (a) $\mathrm{N}{\mathrm{H}}_3$ and (b) $\mathrm{N}{\mathrm{O}}_2$ molecules adsorbed on CNT as a function of adsorption distance $D$.