Schematic of spillover mechanism for hydrogen on (5,5) SWNT. Free molecules are catalytically dissociated to form H atoms at the catalyst, indicated by the large oval attached to the top of the SWNT. The H atoms then chemisorb to the SWNT and migrate along the SWNT exterior.
Five unit cells of a (5,5) SWNT, showing the four sites of highest symmetry. The gray lines represent carbon-carbon bonds. The center of a hexagon is designated as the “-site.” The carbon atoms themselves constitute the “-sites.” The bridge site of an equatorial carbon-carbon bond is call an “-site,” whereas that of a zigzag bond along the SWNT axis is the “-site.”
Cylindrical sites within a single reduced unit cell, corresponding to and coordinate values of grid points used for explicit DFT calculations (and PES fitting) for the (5,5) SWNT-H system. The solid black circles represent the sites explicitly computed using DFT. The solid gray circles are obtained from these via inversion about one of four -sites (the one represented by the lowest solid black circle in the rightmost column of solid black circles). The black open circles are then obtained by applying —a horizontal reflection about —and the dashed open circles from a subsequent vertical reflection. The lower-rightmost gray filled circle is located at the origin of the reduced unit cell coordinates, and —also known as an “-site.” The four corners of the reduced unit cell also correspond to -sites, with the midpoints of the four rectangular boundaries corresponding to the -sites.
PES slices along high-symmetry cylindrical sites, as a function of radial distance , for the (5,5) SWNT-H system (subtract 3.45 Å from to obtain distance from SWNT wall). The dashed black, solid black, dashed gray, and solid gray curves correspond to -, -, -, and -sites, respectively, as obtained from the global analytically fit PES of this paper. The graph indicates a physisorptive region from about 5.75 to 7.5 Å, and a chemisorptive region between 4.45 and 5.0 Å or so (with a binding energy of around −0.75 eV). A barrier of lies between the physi- and chemisorption wells. The circles along the bottom of the graph indicate the radial locations of the ab initio grid points, with corresponding DFT values also plotted as circles along the corresponding curves. The inset shows a closeup of the physisorption well, with local minimum along the -site.
Effect of SWNT symmetry operations on cylindrical coordinates, and . The horizontal and vertical dashed lines represent the and reflection planes, respectively. The filled black circles represent centers of inversion . The open circles are symmetrically equivalent sites on the SWNT, with and − representing the relative sign values for a wave function belonging to the irrep.
Contour plots for the global analytic PES developed in this paper, as computed for three different “slices” along the cylindrical coordinates with fixed values. (a) , the local minimum for the physisorption well, located at the -site (center of plot). (b) , the chemi-physisorption isomerization transition state, located just off the -site, in the direction of the -site. (c) , the global minimum for the chemisorption well, located almost directly at the -site. Thick black lines indicate the SWNT carbon-carbon bonds.
Zigzag reaction pathway, connecting two adjacent (and equivalent) chemisorption minima, for the (5,5) SWNT-H system. Panel (a) shows how the radial coordinate changes along the reaction path, as a function of the reaction coordinate . Panel (b) shows how the angular coordinate changes along the reaction path, as a function of . 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 .
Radial probability densities (left) and cylindrical probability densities (right) for four bound rovibrational states of the (5,5) SWNT-H system. The former is obtained by integrating the eigenstate density over and ; the latter is obtained by integrating over . Note that darker color correlates with higher density in these plots. (a) ground state, , , with energy of −0.4966 eV. (b) excited state, , with energy of −0.1594 eV, corresponding to a single excitation in and a double excitation in . This state shows a large migration enhancement, despite higher excitation in than . (c) excited state, , with energy of −0.1284 eV, corresponding to a single excitation in both and . This state shows the greatest migration enhancement. (d) highly excited state, with energy of −0.003 85 eV, corresponding to a single excitation in in the physisorption region; density is now centered around the -site (center of right plot). A state label cannot be easily assigned to (d).
Subtable of the character table, showing all singly degenerate irreps used to label rovibrational states of the (5,5) SWNT-H system.
Fitting parameter values for the radial functions used in the PES functional form as given by Eq. (4). Those values explicitly set equal to zero are denoted by the entry “0” (see footnotes).
Correlations between cylindrical Fourier basis functions and the irreps of .
Computed rovibrational bound-state energies of the (5,5) SWNT-H system for all eight singly degenerate irreps. The number of digits listed corresponds to the convergence accuracy of the calculation, or to the number of digits needed to distinguish migration (i.e., tunneling) splittings, whichever is greater. All energies are given in eV, relative to the desorption threshold. Although the listed energies are all less than zero, the minus sign has been removed for visual clarity.
Geometries and energies (relative to desorption threshold) for all stationary points (local minima and transition states) of the (5,5) SWNT-H system, as determined using the global analytical PES of Sec. II C. The “type” column distinguishes between minima and transition states, and also specifies where they occur (physisorption well, chemisorption well, or in between). The two distinct -site and -site transition states that lie between adjacent chemisorption wells are referred to as “chemi--chemi” and “chemi--chemi,” respectively. All stationary points were found using the NMINIMIZE function within MATHEMATICA.
Irrep pairings associated with energy level splittings for each type of chemisorptive migration pathway, and . The first irrep listed in each pair has character under the defining symmetry operation for the corresponding migration pathway (i.e., for migration and for migration), whereas the second has −1 character. The first two pairs for each migration pathway exhibit character under , and thus are grouped together; the second two pairs for each pathway exhibit −1 character under , and are also grouped together.
Computed migration rates for each energy grouping of four bound rovibrational eigenstates, in . Column 1: state assignments, vis-a-vis number of excitations in . Column 2: average energy eigenvalue, in eV. Columns 3 and 4: the two migration rates, with the first corresponding to the irrep pair with character under . Column 5: average of columns 3 and 4. Columns 6 and 7: the two migration rates, with the first corresponding to the irrep pair with character under . Column 8: average of columns 6 and 7.
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