Per-adsorbate potential energies (in eV, relative to the second desorption threshold) for the SWNT–H–H system. The black circle represents the location of the first H atom, placed at the (single-adsorbate) equilibrium value of 1.15 Å from the SWNT wall. Various high-symmetry locations for the second H-atom adsorbate (placed at the same radial distance) are indicated by the numbers, which also denote corresponding per-adsorbate potential energies. Green lines represent C–C bonds. Red and blue backgrounds indicate positive- and negative-energy regions, respectively. This system exhibits very pronounced secondary binding enhancement. The largest enhancement occurs at the azimuthally neighboring site, for which the per-adsorbate binding energy (1.2 eV) is 60% greater than for a single adsorbate (0.755 eV).
Schematic of the H-atom motion in the concerted migration model, showing a portion of the single unit cell used. Shown are four hydrogen atoms, represented by the black and gray, filled and open circles, each at an A-site (left panel) or a Z-site (right panel). The left and right panels represent two different stages of the concerted migration process, i.e., the mimimum and transition state configurations, respectively. More general configurations must adhere to the constraints and symmetry restrictions imposed by the model. Specifically, the short solid arrows indicate concerted motion in the z (vertical) direction, whereas the dashed arrows exhibit opposing motion characterizing displacements in the ϕ (azimuthal) direction.
Radial slices for the 3D concerted migration model PES, for A site (solid curve) and Z site (dashed curve). Vertical axis represents per-adsorbate binding energies in eV; horizontal axis represents radial coordinate values in Å. Left edge of plot represents location of SWNT wall at (3.95 Å). The black, filled circles denote the radial positions used to compute the potential interactions. The two curves have different r → ∞ values, owing to different asymptotic arrangements of the desorbed H atoms. Also, the Z-site curve decreases monotonically throughout, which has important ramifications for the migration dynamics (see text).
Radial slices for the 3D hole migration model PES at various high symmetry sites; H, A, Z, and E, represented respectively by dotted, solid, dashed, and dotted-dashed black curves. Gray curve represents A-site slice for the single-adsorbate PES of Paper I. Black and gray circles denote the radial positions (relative to the SWNT wall) used to compute the potential interactions for the corresponding curves. The asymptotic values of the black curves are determined by removing a second H atom (i.e., by creating a second hole), zig-zag-adjacent to the first. A-site and Z-site curve minima are −2.11 and −0.77 eV, respectively, located at 1.15 and 1.05 Å from the SWNT wall.
Contour plot for the 3D hole migration model PES developed in this paper, as computed for a “slice” along the cylindrical coordinates, (z, ϕ), with radial coordinate fixed to the global minimum value, r = 5.079 Å. The global minimum is located very near to the A site, slightly displaced in the direction of the E site. The bolded contours correspond to the desorption threshold, i.e., V = 0. Thick black lines indicate the SWNT carbon–carbon bonds.
Zig-zag reaction pathway, connecting two adjacent (and equivalent) chemisorption minima, for the 3D hole migration model PES. Panel (a) shows how the radial coordinate, r, changes along the reaction path, as a function of the reaction coordinate, z′. Panel (b) shows how the angular coordinate, ϕ′, changes along the reaction path, as a function of z′. The thick solid line across panel (b) shows the carbon–carbon bond. The inset of panel (b) is a zoomed-out view, in which the entire reduced unit cell is visible. Panel (c) is the reaction profile, showing energy as a function of z′.
Radial r probability densities (left) and cylindrical (z, ϕ) probability densities (right) for four bound rovibrational states of the 3D hole migration model system. The former is obtained by “slicing” the eigenstate density, |Ψ(r, z, ϕ)|2, along the (z, ϕ) site of maximum probability, whereas the latter is the maximum-probability slice along fixed r. Darker color corresponds to higher density. (a) ground state, A 1g , |0 0 0〉, E = −2.034 eV. (b) excited state, A 2u , |2 6 1〉, E = −0.793 eV, corresponding to double excitation in r, hextuple excitation in z, and single excitation in ϕ. This state shows a relatively large Z migration rate. (c) excited B 1u state, |0 9 0〉, E = −0.761 eV, corresponding to nonuple excitation in z. Despite high excitation in z, this state shows a lower Z migration rate than (b). (d) excited A 1g state, |0 1 6〉, E = −0.735 eV, corresponding to single excitation in z and hextuple excitation in ϕ. This state, while being a lower excitation in z and higher excitation in ϕ than (b) shows a larger Z migration rate.
Fitting parameter values for the radial functions, a ij (r), used in the PES functional form as given by Eq. (2). See footnotes.
Correlations between cylindrical Fourier basis functions, and the irreps of D 10h .
Per-adsorbate binding energies (eV) for various SWNT–H n systems, and for H2. The last three lines refer to single-unit-cell calculations, for which n = 20 implies full coverage, and n = 19 implies one hole per unit cell. See footnotes.
Lowest 20 computed rovibrational bound state energies of the SWNT–H19 3D hole migration model system, for each of the eight singly degenerate D 10h irreps. All energies are in eV, relative to the reference-adsorbate desorption threshold. The minus sign has been removed for visual clarity, as have all trailing zeros. Eight digits past the decimal point are presented, in order to distinguish the tunneling splittings of the low-lying states, which are converged to a higher level of accuracy than the energies themselves.
Computed Z migration rates for several bound rovibrational eigenstate quartets, in s−1. For brevity, entries in this table satisfy at least one of three conditions: they are low-lying states; they show a relatively large migration rate; they lie near the Z-site transition state. Column 1: state assignments, vis-à-vis number of excitations in (r, z, ϕ). Column 2: average energy eigenvalue, in eV. Column 3 and 4: the two k Z migration rates, with the first corresponding to the irrep pair with +1 character under . Column 5: average of Columns 3 and 4.
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