Quantum States of Hydrogen and Its Isotopes Confined in Single

María Pilar de Lara-Castells , Andreas W. Hauser , Alexander O. .... Herwig Peterlik , Hans Kuzmany , Francesco Zerbetto , Manuel Melle-Franco , Luke...
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1742

J. Phys. Chem. B 2006, 110, 1742-1751

Quantum States of Hydrogen and Its Isotopes Confined in Single-Walled Carbon Nanotubes: Dependence on Interaction Potential and Extreme Two-Dimensional Confinement Tun Lu,†,§ Evelyn M. Goldfield,*,† and Stephen K. Gray‡,| Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202, and Chemistry DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: August 11, 2005; In Final Form: October 20, 2005

Quantum mechanical energy levels are computed for the hydrogen molecule and its homonuclear isotopes confined within carbon nanotubes of various sizes and structures using three different interaction potentials. Two translational and two rotational degrees of freedom are treated explicitly. We study the dependence on the interaction potential and the size of the nanotube of several features, including zero-pressure quantum sieving selectivities, ortho-para energy splittings, and wave function characteristics. We show that large quantum sieving selectivities, as well as large deviations from gas phase ortho-para splittings, occur only under the condition of extreme two-dimensional confinement, when the characteristic length of the hydrogencarbon interaction potential is nearly equal to the radius of the nanotube.

I. Introduction The subject of this paper is quantum effects that arise because of the confinement of H2 in carbon nanotubes (CNTs). The quantum mechanics of hydrogen molecules confined in CNTs has generated much recent interest,1-13 in part because of potential applications of nanotubes as storage media, quantum sieves, and catalysts. The structure and dynamics of small molecules confined in structures with very regular geometries are also of fundamental theoretical interest. In this work we study isotope effects, including quantum sieving and the rotational structure of confined H2 molecules. The focus of this work is the dependence of these quantities on parameters of the interaction potential. Our work builds on the theoretical studies of such systems of Johnson and co-workers,2,5,7 Hathorn, Sumpter, and Noid6 and Yildirim and Harris.13 Interesting studies of spin and isotope effects of hydrogen adsorbed in carbon nanotubes have also been carried out by Gordillo et al.8 and by Trasca et al.12 Garberoglio et al.14 have recently examined the effect of the potential on quantum sieving selectivities, and we compare our results to theirs. For hydrogen molecules confined to the inside of CNTs of various sizes and chiralities, we explore in depth the dependence of both quantum sieving effects and rotational structure on the interaction potential between the hydrogen atoms and the nanotube carbon atoms. In a previous paper10 (paper I), we treated the full quantum mechanics of confined rigid molecular hydrogen and its isotopomers using a modified Brenner potential that included a Lennard-Jones (LJ) potential for the long-range C-H interactions. This paper was the first, and to our knowledge the only, paper to treat coupled translational and rotational motion. The potential we used, however, gave an incorrect representation of the C-H interactions. As detailed * Corresponding author. E-mail: [email protected]. † Wayne State University. ‡ Argonne National Laboratory. § E-mail: [email protected]. | E-mail: [email protected].

below, we used a Lennard-Jones length parameter (σCH) that was unrealistically small, and the Lennard-Jones well depth was too large by a factor of 2. With these potential parameters, we did not see the large quantum sieving effects found by previous authors.2,5,12 There is a clear need to explore the sensitivity of quantum sieving and the rotational structure to the nature of the underlying potential, and this was the motivation for the present work. In this work, we apply our method to three more realistic H-C potentials: (1) the Novaco and Wroblewski (NW) potential15 used to study rotational states of H2 and its isotopes adsorbed on graphite, (2) the Frankland and Brenner (FB) potential16 used in a study of hydrogen Raman shifts in CNTs, and (3) the Williams and Starr (WS77) potential17,18 used by Yildirim and Harris13 to study the spectrum of hydrogen in CNTs and other environments. The first two potentials have the LJ form while the WS77 potential has an exponential repulsive wall and an R-6 attractive term. As discussed below, their differences and similarities elucidate the most important factors determining the spectral properties of confined hydrogen. Classical molecular sieves separate molecules based on size, shape, or chemical affinity. Thus, since isotopes differ only in their masses, one does not expect molecular sieves to work for separating isotopes. Quantum sieves, however, separate isotopes based upon preferential adsorption of heavier isotopes because of the difference in quantum mechanical energy levels of the atoms or molecules confined in the CNT or other porous material.2,19 We show that the characteristics of the potential that lead to high isotope selectivities also lead to other interesting phenomena, including dramatic effects upon the rotational spectra of the confined molecules resulting in selectivities of nuclear spin states. To date, quantum sieves have not been built from CNTs. Recent experiments20 on H2 and its isotopomers adsorbed on carbon “nanohorns”, however, show that even at 77 K there is significant differential adsorption of H2 and D2 in the narrow cone portion of the nanohorn.

10.1021/jp0545142 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/07/2006

Quantum States of Hydrogen and Its Isotopes

J. Phys. Chem. B, Vol. 110, No. 4, 2006 1743

TABLE 1: Nanotube Diameter and the Potential Energy Minimum for Confined H2 for the Three Interaction Potentialsa potential minimum cm-1

nanotube

radius Å

translational unit cell Å

number of unit cells

NW re ) 3.12 Å w ) 18 cm-1

FB re ) 3.46 Å w ) 19.2 cm-1

WS77 re ) 3.60 Å w ) 12 cm-1

(3, 6) (8, 0) (2, 8) (6, 6) (10, 10)

3.11 3.13 3.59 4.07 6.78

11.42 4.32 6.60 2.49 2.49

7 7 7 13 15

-1309.6 -1296.7 -996.2 -806.5 -547.1

-1459.6 -1501.3 -1624.2 -1243.4 -745.0

-723.2 -776.1 -1153.9 -896.1 -504.16

a

Also given for each interaction potential is the equilibrium H-C distance re and the H-C well depth, w. For the two LJ potentials, the H-C well depth is equal to the potential parameter, .

Figure 1. Coordinates used in our 4-D calculations.

In comparing the effects of the three different H-C interaction potentials on the energy levels, we have identified three different types of confinement: one-dimensional (1-D), twodimensional (2-D), and what we call extreme two-dimensional (X2-D) confinement. The first two types of confinement have been identified previously by us and others,2,5,10,13 but the concept of X2-D confinement is new to this work. We show that very large quantum sieving effects result when and only when the H-C interaction potential leads to X2-D confinement. There is no reason to believe that this phenomenon is restricted to hydrogen, although the high zero-point energy of H2 leads to a dramatic illustration of the phenomenon. Section II outlines our model and computational methods, and section III presents our results. Section IV presents an analysis of the resulting wave functions to explain the dramatic effects of X2-D confinement on rotational structure of the energy levels. Our conclusions are in section V. II. Computational Methods II.A. Computation of Energy Levels. The methods that we use to compute the quantum mechanical energy levels have been given in some detail in paper I,10 and we only give a brief summary here. The coordinate system for our four-degree-offreedom model is displayed in Figure 1. In Figure 2, we illustrate our model by showing a hydrogen molecule within a (6, 6) nanotube. In our model, x and y are Cartesian coordinates representing translational motion of the system orthogonal to the nanotube axis; θ is the angle between the molecular bond and the nanotube axis, z; and the azimuthal angle, φ, represents rotation about the z-axis. The molecular center-of-mass is fixed at a particular value of z in the center of the nanotube. The results are not greatly influenced by what particular value of z is chosen. The size of the translational unit cell is given in Table 1. To test the effects of corrugation along z, for each nanotube, we varied the center of mass of the H2 along the unit cell and minimized the potential energy with respect to our four coordinates. For all nanotubes except the (8, 0) the variation was negligible. For the (8, 0) nanotube with the FB potential,

Figure 2. View of H2 in the center of a (6, 6) nanotube.

it was ∼60 cm-1, which is only 4% of the binding energy. Our nanotubes are long enough to contain all relevant potential interactions. The number of unit cells that we included is given in Table 1. We fix the bond length of the diatomic in its equilibrium position, and we do not treat motions of the atoms in the carbon CNT. The Hamiltonian for this system is given by

[

]

p 2 ∂2 ∂2 + 2 2 2M ∂x ∂y 1 ∂ ∂ 1 ∂2 (sin θ) + 2 + V(x, y, θ, φ) (1) B sin θ ∂θ ∂θ sin θ ∂φ2

H ˆ )-

{(

)

}

where M is the total mass of the diatomic and B, the rotational constant, is given by B ) p2/2µRe2 where µ is the reduced mass of the molecule and Re is the equilibrium bond length of the diatom. The potential includes all interactions between atoms of the diatomic and the carbon atoms of the CNT. The wave function is expanded in a set of real eigenfunctions of the rotational part of the Hamiltonian, jmax j

ψ(x, y, θ, φ) )

P h jm(cos θ)(Cj,m(x, y) cos mφ + ∑ ∑ j)0 m)0 Dj,m(x, y) sin mφ) (2)

where P h jm is a normalized associated Legendre function and j,m C and Dj,m are expansion coefficients. The Cartesian coordinates, x and y, are represented on evenly spaced grids and a sinc-DVR is used to compute the action of the kinetic energy operator.21 The homonuclear symmetry of the diatomic molecule results in a decoupling of the Hamiltonian matrix into even (j ) 0, 2, ...) and odd (j ) 1, 3, ...) blocks. Using each of the three interaction potentials, we computed the energy levels of H2, D2, and T2 for five CNTs of various structures and sizes determined by their chiral indices (n, m). We considered two armchair structures with (n, m) ) (6, 6) and (10, 10), one zigzag (8, 0), and two chiral CNTs (3, 6) and

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Lu et al.

(2, 8). The diameters of these nanotubes, ranging from 6.0 to 13.6 Å, are given in Table 1. Since the even and odd symmetry j-blocks are treated independently, there are a total of 90 separate calculations. For each CNT, we computed the relevant energy levels and wave functions for H2, D2, and T2 using the ARPACK (Arnoldi package) software library,22,23 which is a collection of subroutines designed to solve large-scale eigenvalue problems iteratively. ARPACK is especially useful if one has degenerate and/or nearly degenerate eigenvalues, as we do in this problem, and if one wishes to inspect the actual eigenfunctions. Except for the potentials used, the calculations are carried out as in paper I. The number of grid points in x and y is determined by the diameter of the nanotube and the grid spacing. For H2, we use a grid spacing of 0.165 Å for the (n, m) ) (3, 6) and (8, 0) CNTs and a larger grid spacing of 0.33 Å for the others. The spacings for the heavier D2 and T2 isotopes were reduced by factors of 2-1/2 and 3-1/2, respectively. The value of jmax was taken to be 10 for the (8, 0) and (3, 6) nanotubes and 5 for the others. The number of grid points for H2 ranged from 17 in x and in y for the (2, 8) and (6, 6) nanotubes, 21 for the (3, 6) and (8, 0) nanotubes, and 35 for the (10, 10) nanotube. II.B. Interaction Potentials. The H2-CNT interactions are dispersion interactions, resulting entirely from instantaneous electron correlation. Density functional theory is not capable of computing such interactions reliably, and higher levels of ab initio theory, such as coupled cluster theory, are (currently) not feasible for these large systems. Therefore, empirical longrange H-C potentials are used in studying these H2-CNT interactions. We emphasize that for each point in our fourdimensional grid we determine the potential by computing the interaction of each hydrogen atom with each carbon in the nanotube. In this work, we compare three H-C interaction potentials which we designate as NW, FB, and WS77. The first two are LJ 6-12 interaction potentials in which the H-C interactions are given by

V(r) ) 4

(

)

σ12 σ6 r12 r6

(3)

For the NW potential,15  ) 18 cm-1 and σ ) 2.78 Å; for the FB potential,16  ) 19.2 cm-1 and σ ) 3.08 Å. The WS77 potential13 is given by a Buckingham form,

V ) Be-Cr -

A r6

(4)

with A ) 5.94 eV Å6, B ) 678.2 eV, and C ) 3.67 Å-1. The two important characteristics of the H-C interaction potentials are the well depth, w, and the value of r at the minimum in the two-body H-C interaction, re. For the LJ potentials, re ) 21/6σ and w ) . The values of w and re for each interaction potential are given in Table 1. The well depths of the two LJ interaction potentials are quite similar whereas w is significantly smaller for the WS77 potential. However, as can be seen in Table 1, re is smaller for the NW potential than for the other two potentials. The radii for the two smallest CNTs, given in Table 1, are comparable to re for the NW potential and smaller than re for the other two potentials. It is worth pointing out that the NW value of σ ) 2.78 Å is small compared to other commonly used potentials.14 The H-C LJ interaction potential originally used by us10,11 (not explicitly stated in the papers) had σ ) 2.54 Å, corre-

sponding to re ) 2.851 Å, which is about 9% smaller than the NW value and about 20% smaller than the FB and WS77 values. The smaller re value accounts for the relatively smaller quantum sieving ratios we observed with this potential. We also employed  ) 40.2 cm-1, a well depth that is more than twice the corresponding NW, FB, and WS77 values. The version of the Brennermd code24 that we used to compute the potential contained the unrealistic σ value, but a programming error on our part was responsible for  being twice as large. As we discuss in section III.A below, the error in  turns out to have a smaller effect on the results than the small value of σ. The minimum energy for each potential for hydrogen confined in each of the CNTs is given in Table 1. For the NW interaction, the largest well depth occurs for the smallest (3, 6) nanotube and the well depth decreases monotonically with increasing nanotube radius. The situation is different for the other two potentials, where the (2, 8) nanotube has the largest well depth. Differences between the three H-C interaction models are further illustrated in the one-dimensional cuts of the potentials shown in Figure 3. Because the potentials for the (3, 6) and (8, 0) CNTs are so similar, they are nearly coincident in Figure 3. For all three interaction potentials, the larger (10, 10) and (6, 6) CNTs exhibit double wells with the minima away from the center of the nanotube. While motion along the nanotube axis is almost free, the molecule may experience confinement in the directions transverse to the nanotube axis. If, for example, the molecule is confined to the center of the nanotube (2-D confinement), it will be a quasi one-dimensional system. The confinement of the larger nanotubes is primarily in the radial direction and is thus denoted as one-dimensional (1-D) confinement. A qualitative description of 1-D confinement in the larger nanotubes could be given by a particle-in-a-ring model. As the diameter of the nanotube decreases, the wells move closer together until they finally merge into one well for the smallest CNTs. The molecule is then confined to the center of the nanotube. A qualitative description of this two-dimensional (2-D) confinement is a 2-D harmonic oscillator. The merging together of the two wells with decreasing nanotube radius is illustrated most clearly in the (2, 8) CNT for the NW potential, which lies very close to the transition between 1-D and 2-D confinement but nevertheless exhibits two distinct minima. For the FB and WS77 potentials however, the (2, 8) potential minima is clearly in the center of the nanotube showing 2-D confinement. Although for these interaction potentials, the (2, 8) CNT has the greatest well depth, its potential is much broader than for the smaller (3, 6) and (8, 0) nanotubes where the potential is very steep and the H2 (or D2 or T2) molecule is strongly confined to the center of the nanotube. II.C. Quantum Sieving. The theory of quantum sieving is based upon the idea that because of the difference in their energy levels, heavier isotopes will preferentially bind to the nanotube. To a first approximation, this preference is a result of the zeropoint energy differences that can be attributed to the difference in masses.2,5-7 We can define a low-pressure selectivity factor2, S0(2/1), for isotope 2 over isotope 1, which is the ratio of equilibrium constants for adsorption:

S0(2/1) )

K2 Qads,2 Qfree,1 ) K1 Qfree,2 Qads,1

(5)

where Kn is the equilibrium constant for adsorption for molecule n and Qads and Qfree are the corresponding adsorption and free

Quantum States of Hydrogen and Its Isotopes

J. Phys. Chem. B, Vol. 110, No. 4, 2006 1745 the nanotube axis, z, as free motion so that the translational partition function, Qtrans,z, is the same in the adsorbed and free species and will cancel out. While this is an approximation, we expect that it is a very good one. The potential varies only a small amount along the nanotube axis. It is highest at the end of the nanotube and lowest at the center, where we do our calculation. The typical variation from end to center is only 25-30 cm-1. The translational partition function, Qtrans,xy, results in a ratio of masses in S0, and in fact, this ratio inhibits the selectivity of the heavier isotope. Because we are interested in hydrogen at low temperature where the separation between rotational energy levels with respect to kbT will be large, we treat the rotational partition function quantum mechanically. Following Hathorn, Sumpter, and Noid,6 for the purposes of computing isotopic selectivities, we assume that the adsorbed molecules maintain a statistical distribution of ortho and para states. This assumption vastly simplifies the computation of S0. Activated carbon has been shown to interconvert the two species.25 It is not clear, however, whether nanotubes are catalytically active for this conversion. The ideal gas rotational partition function is given by6

Qrot ) ge

∑

(2j + 1) exp[-B(j(j + 1))/(kbT)] +

j)even

go

∑

(2j + 1) exp[-B(j(j + 1)/(kbT)] (8)

j)odd

where the ge and go are spin-statistical weighting factors. For hydrogen and tritium, with a half-integral nuclear spin, they are 1 and 3, respectively. For deuterium, with a nuclear spin of 1, they are 6 and 3, respectively. We compute Qads using the discrete energy levels, Ej, obtained from our four-degree-offreedom quantum mechanical calculation,

Qads ) ge

∑n exp[-Ene/(kbT)] + go∑n exp[-Eno/(kbT)]

(9)

In eq 9, the first and second terms represent computed energies that include only the even rotational and odd rotational basis states, respectively. If, rather than assume a statistical distribution of ortho and para states, we assume that they either do not interconvert or do so very slowly, then we can treat ortho and para hydrogen as separate species. In this case, we can compute an ortho/para selectivity for hydrogen adsorbed inside the nanotube as follows:

Figure 3. One-dimensional cuts of the potential energy surfaces as a function of x, with y ) 0, minimized with respect to θ and φ: (a) NW, (b) FB, and (c) WS77.

partition functions, respectively. The free partition function is a product of translational and rotational partition functions,

Qfree ) Qtrans,xyQrot

(6)

accounting for two free translational degrees of freedom as well as free molecular rotation. We use a simple classical translational partition function,

Qtrans,xy )

(

)

2πmkbT h2

V2/3

(7)

where V is the volume of the system. We treat the motion along

Sop )

go Qoads Qerot ge Qo Qe rot

(10)

ads

where the superscripts o and e denote partition functions computed using the only odd and even levels, respectively (and no nuclear degeneracy factors). III. Results III.A. Quantum Sieving Isotope Selectivities. Table 2 gives the quantum sieving selectivities, S0(D2/H2) and S0(T2/H2) for all three potentials in all six nanotubes at 20 K. As shown in Figure 4, quantum sieving selectivities are only significant at low temperatures. For a given H-C potential, the magnitude of the quantum sieving effect is a strong function of the nanotube radius. Table 2 shows that very large quantum sieving effects occur only for the smallest radius (3, 6) and (8, 0) CNTs and then only for the FB and WS77 H-C potentials.

1746 J. Phys. Chem. B, Vol. 110, No. 4, 2006

Lu et al.

TABLE 2: Quantum Sieving Selectivities at 20 Ka S0(D2/H2) nanotube NW (3, 6) (8, 0) (2, 8) (6, 6) (10, 10) a

283 236 3.0 3.2 5.11

S0(T2/H2)

FB

WS77

NW

FB

WS77

6.6 × 106 4.1 × 106 28.9 2.18 5.91

3.7 × 105 2.0 × 105 31.3 2.2 3.21

5547 4162 4.4 6.0 12.2

1.7 × 1010 8.2 × 109 129 3.13 15.56

2.9 × 108 1.7 × 108 166 2.93 5.80

Values indicating X2-D confinement are indicated in bold.

Figure 4. Quantum sieving selectivity, S0, as a function of temperature for the (3, 6) nanotube using the FB potential.

Quantum sieving, to a first approximation, can be correlated with the zero-point energy (ZPE). Table 3 displays the corresponding ZPE values, which indeed parallel the S0 trends of Table 2. We can understand the ZPE trends as follows. When the CNT radius is much larger than re for the H-C potential, the optimal position of the H2 is closer to the CNT walls than to the center of the nanotube, and we are in the limit of 1-D confinement. When the CNT radius is reduced so that it is comparable to re, the molecule will optimize its attractive interactions with the nanotube carbons by positioning itself in the center of the nanotube, preferentially aligned along the nanotube axis. This corresponds to 2-D confinement, which tends to have a larger ZPE than the 1-D confinement case. If the CNT radius is smaller than re, the molecule will be even more strongly confined to this position, resulting in Very steep potentials and very high ZPE values. For the (3, 6) and the (8, 0) CNTs, the ZPE values for the FB and WS77 potentials, highlighted in bold, are considerably larger than the others. The molecules, as described by these interaction potentials, exhibit extreme 2-D (X2-D) confinement. We will show that X2-D confinement is qualitatively different from ordinary 2-D confinement and that these differences are seen not only in the high quantum sieving selectivities but in many other quantities, including the energy separations, the number of negative energy states, the rotational spectrum, the ortho-para separations, and the wave functions themselves. A 2-D confined system may

exhibit only slightly hindered rotation. We will show, however, that highly hindered rotation is a feature of X2-D confined systems. Unlike the FB and WS77 potentials, the NW H-C potential case does not exhibit X2-D confinement for the (3, 6) and (8, 0) CNTs. The NW potential has considerably smaller S0 and ZPE values (Tables 2 and 3). This is consistent with the fact that the NW potential has a smaller re than the FB and WS77 potentials, which, following the arguments above, implies weaker 2-D confinement. Of the FB and WS77 potentials, which do exhibit X2-D confinement, the FB confinement effect is larger as indicated by the larger ZPE values and higher selectivities. Finally, we note the quantum sieving and zeropoint energy do not simply decrease as one increases the CNT radius but rather display a minimum. This minimum corresponds to a transition from 2-D to 1-D confinement and corresponds to the smallest CNT radius that exhibits 1-D confinement.3,6,8,11 A comparison of the 20 K quantum sieving results with those in paper I illustrates further the importance of re. For example, we previously had11 S0(D2/H2) ) 20.0, 19.7, 3.1, 5.2, and 6.7 for the (3, 6), (8, 0), (2, 8), (6, 6), and (10, 10) sequence of CNTs. Of the three potentials studied here, these older results agree best with the NW results of Table 2, although the 2-D confined (3, 6) and (8, 0) results are an order of magnitude bigger with the NW potential. Recall from section II.B above that re for the paper I potential is 20% smaller than the FB and WS77 values but just 9% smaller than the NW value. The well depth of the Paper I potential is actually more than twice that of the ones studied here. This may partly compensate for its small re value because the steepness of the interaction does increase somewhat with increasing well depth. However the re effect, that is, smaller re values being associated with less severe confinement, less steep potential interactions, and smaller quantum sieving selectivities, dominates.14 Thus, as with the NW potential, the unrealistically small value of re used in the H-C potential of paper I prevented us from finding very large quantum sieving selectivities or any X2-D confinement. The results shown in Table 2 for the WS77 potential are roughly the same as those previously reported by Johnson and co-workers.5 It must be emphasized, however, that they used a different potential and did not include rotational contributions to S0. Since the quantum sieving selectivities have been shown to be dependent on the potential parameters and since the rotational contributions can be quite large,14 the agreement is somewhat fortuitous. A better comparison is with the recent results of Garberoglio et al. using the FB potential where we agree quite well.14 For example, using a more approximate method for including rotational contributions to S0, they obtain S0(T2/H2) ) 1.51 × 1010 for the (3, 6) CNT at 20 K, which agrees quite well with our value of 1.7 × 1010 at 20 K. III.B. Number of Negative Energy Levels. One distinguishing characteristic of X2-D confinement is the small number of “bound” energy levels, that is, those states that are lower in energy than a hydrogen molecule an infinite distance away from

TABLE 3: Zero-Point Energies in cm-1 a H2

D2

T2

nanotube

NW

FB

WS77

NW

FB

WS77

NW

FB

WS77

(3, 6) (8, 0) (2, 8) (6, 6) (10, 10)

306.06 295.56 74.77 101.04 135.02

723.36 701.68 191.55 71.63 141.83

621.31 605.37 212.92 54.59 98.22

218.89 210.87 50.81 82.62 107.28

487.21 472.71 135.34 54.54 112.46

430.51 419.89 155.8 35.19 77.47

178.6 171.89 41.67 72.99 95.13

381.08 369.82 111.08 47.86 97.70

340.90 332.66 129.91 27.68 65.47

a

Values indicating X2-D confinement are indicated in bold.

Quantum States of Hydrogen and Its Isotopes

J. Phys. Chem. B, Vol. 110, No. 4, 2006 1747

TABLE 4: Ortho/Para Energy Separations in cm-1 a H2

a

D2

T2

nanotube

NW

FFB

WS77

NW

FB

WS77

NW

FB

WS77

(3, 6) (8, 0) (2, 8) (6, 6) (10, 10) free

69.6 70.1 116.2 116.0 111.0 122

8.5 9.1 93.8 118.3 109.4

18.8 19.7 91.7 117.8 114.0

21.2 21.7 57.5 54.4 50.9 61

0.40 0.47 38.4 58.3 49.6

1.6 1.7 35.5 58.2 53.4

8.7 9.1 38.2 34.2 31.5 41

0.05 0.06 21.1 38.2 30.6

0.24 0.27 18.2 38.5 33.6

Values indicating X2-D confinement are indicated in bold.

TABLE 5: Energy Differences in cm-1 between Ground and First Excited States within the Even-j and Odd-j Blocksa nanotube

H2

D2

T2

even levels

NW

FB

WS77

NW

FB

WS77

NW

FB

WS77

(3, 6) (8, 0) (2, 8) (6, 6) (10, 10)

259.1 251.7 76.8 15.9 0.54

431.2 422.5 179.9 50.9 0.68

390.2 384.9 182.9 66.7 0.76

166.7 160.4 43.4 5.9 0.22

288.0 279.9 117.0 25.5 0.31

248.3 243.8 121.7 39.3 0.36

124.2 120.1 4.1 3.3 0.11

234.0 226.4 90.6 16.3 0.18

201.0 196.8 95.9 28.7 0.23

nanotube

H2

D2

T2

odd levels

NW

FB

WS77

NW

FB

WS77

NW

FB

WS77

(3, 6) (8, 0) (2, 8) (6, 6) (10, 10)

91.9 90.6 7.4 5.7 0.43

358.8 349.5 43.1 4.7 0.36

267.3 261.0 47.9 6.3 0.47

80.8 79.1 4.1 3.5 0.22

281.2 272.9 36.5 2.9 0.11

229.7 224.4 43.4 4.0 0.14

75.0 73.1 2.7 1.5 0.11

233.1 225.4 33.6 2.3 0.07

197.4 192.9 41.3 3.0 0.05

a

Values indicating X2-D confinement are indicated in bold.

the nanotube, which is taken as the zero of energy. Of course all of the computed energy levels, that is, negative and positive energy levels, correspond to wave functions that are bound with respect to motion transverse to the nanotube axis. When motion along the nanotube axis is considered, however, the positive energy states could exit the nanotube. The number of negative energy levels is a function of both the energy level spacing and the ratio of the ZPE to the well depth. Energy levels are more closely spaced for 1-D than for 2-D confinement and for the heavier isotopomers. For example, there may be well over 1000 negative even or odd energy levels for T2 in the (10, 10) CNT. By contrast, for H2 in the (3, 6) CNT, the calculation using the NW interaction potential results in 34 even and 34 odd negative levels. The same calculation using the FB potential results in only eight even and five odd negative levels. For the WS77 potential, there is only one even and one odd negative level, which may not be too surprising when one considers that the ZPE is ∼85% of the well depth. III.C. Hindered Rotation and Ortho/Para Energy Separations and Selectivities. One issue that will dominate much of the rest of this paper is the degree to which rotation is hindered within a nanotube. One way to measure the overall effects of hindered rotation is to look at the ortho/para separation, which in our case is the difference in energy between the lowest even and lowest odd energy levels. For free H2, the ortho/para separation is the difference between the 3-fold degenerate ortho j ) 1 level and the para j ) 0 level, which is 2B ) 121 cm-1. Brown et al.4 measured the j ) 0 f 1 transition of hydrogen physisorbed onto the exterior surface of CNTs and concluded that the CNT-hydrogen interaction provided only a small barrier to rotation. One neutron scattering study of H2 adsorbed in the interstitial tunnels of bundles of carbon nanotubes also observed nearly free rotation.26 Another study, however, also of interstitially adsorbed H2 found considerable shifts in the rotational spectra from bulk H2.9 When placed within a CNT, the ortho/ para energy separation decreases with increasing hindrance to

rotation. The reasons for this will be made clear in section III.D below where we discuss the nature of the wave functions. The ortho/para separations for H2, D2, and T2 adsorbed into the interior of the nanotube are given in Table 4, along with those for the free diatomics. As one can see, regardless of the interaction potential, there is a significant reduction in the separation for all of the nanotubes. As in quantum sieving, the smallest effect is for the smallest 1-D confined nanotube. What is striking, however, is the extremely small splitting between the lowest even and odd energy levels for the (3, 6) and (8, 0) CNTs when the FB or WS77 interaction potentials are employed as shown in bold in Table 4. These small splittings indicate that hindered rotation might be playing a major role in determining the energy spectrum in the X2-D confined systems. By contrast, as seen in Table 5, the energy separations between the lowest two levels within a given j-block are much larger in the case of X2-D confinement. For H2 in its vibrational ground state, the energy difference between the two lowest even states for unconfined H2 would simply be the difference between the j ) 0 and j ) 2 energy levels or 366 cm-1. For the odd levels, the three lowest states of freely rotating H2 are degenerate. The impact of confinement for all of the nanotubes is to lift the j ) 1 degeneracy, an indication that there are preferred orientations. These preferences become more pronounced as the nanotube radius decreases, particularly in the case of X2-D confinement (shown in bold). For the even levels, except for the X2-D confined cases, the splitting between the ground and first excited state is smaller than 366 cm-1, indicating that these states may both be primarily j ) 0 states. (Indeed, inspection of the wave functions shows this to be the case.) For X2-D confinement, the energy difference between the lowest and first excited even level are all larger than the freely rotating diatomic, particularly true for T2 in the (3, 6) CNT where the splitting is 244.9 cm-1, over twice the energy difference of T2(j ) 2) and T2(j ) 0). These large energy differences indicate very strong

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TABLE 6: The Ratio of Adsorbed Ortho to Adsorbed Para Hydrogen Computed Using the FB Potential at 20 and 50 K nanotube

20 K

50 K

(3, 6) (8, 0) (2, 8) (6,6 ) (10, 10)

3460 3308 8.0 3.1 5.0

26 25.6 3.5 3.1 3.3

orientation preferences and rotational hindrance in the case of the X2-D confinement. The strong effect of X2-D confinement on the rotational energy levels results in a very large enhancement of ortho over para hydrogen adsorbed in the nanotube at low temperatures as can be seen in Table 6. For larger nanotubes or higher temperature, the ortho/para, Sop, is nearly statistical (3/1). For the (3, 6) and (8, 0) nanotubes, however, Sop is enhanced by 3 orders of magnitude. Similarly large enhancements of ortho over para hydrogen were predicted by Trasca et al.12 Thus, it is possible that the smaller carbon nanotubes could be used to separate ortho and para hydrogen.

ψ ≈ ψnx,ny(x, y) ψrot(θ, φ)

IV. Analysis of the Wave Functions To gain insight into the cause of these dramatic effects, we examine our wave functions. To reduce the dimensionality of the wave functions for the purposes of analysis, we compute two reduced probability densities:

Fn(x. y) )

The notation (1, 0) ( (0, 1) in Table 7 indicates that the state is either (1, 0) + (0, 1) or (1, 0) - (0, 1). The total number of translational quanta is e2 for the first 20 excited states. The coefficients of the dominant rotational basis functions jm are given by the reduced density Fjm n and defined as Fn g 0.1. The dominant rotational contributions to the wave functions are from j e 2 (even) or j e 3 (odd). In all the cases that we examined there is no mixing of different values of m, where m is associated with the projection of rotation on the nanotube axis. Because of the strong preference for alignment along this axis, lower values of m give rise to lower energy states. When m ) 0, the rotational basis functions combine to form lowenergy/high-energy pairs. For the even j-block, in addition to states 0 and 5, the degenerate states 3 and 4 form this type of pair with states 15 and 16. In the odd j-block, states 3 and 4 are coupled in this manner to states 19 and 20. IV.A. Approximate Separable Wave Functions. If we assume, for the purposes of analysis only, that rotation and translation are separable, we may write the wave function as a product of translational and rotational wave functions,

(14)

We identify the probability densities of these wave functions with the reduced probabilities densities as

|ψnx,ny(x, y)|2 ) F(x, y)

∫ψn*(x, y, θ, φ) ψn(x, y, θ, φ) d cos θ dφ

(11)

|ψrot(θ, φ)|2 ) F(θ, φ)

∫ψn*(x, y, θ, φ) ψn(x, y, θ, φ) dx dy

(12)

From Table 7 we see that the dominant contributions to the wave function of the lowest even state, the ground state, comes from P h 00 and P h 20 rotational states. The ground state and the fifth excited state form a low-energy/high-energy pair of states. Both of these states are in the translational ground state. From the information in Table 7, we can deduce approximate wave functions for the ground and fifth excited even state. The assignment of phases to these approximate wave functions can be justified through plots of cuts of the actual wave function.

Fn(θ, φ) )

where ψn is the nth eigenstate in either the even or odd j-blocks. To estimate the contribution to an eigenstate from an individual rotational state, we project the wave function onto the rotational basis, take the square modules, and integrate over x and y to obtain

Fjm n )

∫(|Cjmn |2 + |Djmn |2) dx dy

(13)

Yildirim and Harris13 describe a model for the energy levels of H2 confined in a cylindrical potential. Using the WS77 potential, they apply their model to both 1-D and 2-D confinement. Their model assumes that j is a good quantum number and that coupling of rotation with translation only involves the projection quantum number, m. Inspection of Fjm n for our computed eigenstates reveals that for the (2, 8), (6, 6), and (10, 10) CNTs j is, in fact, a good quantum number. For the smaller (3, 6) and (8, 0) nanotubes, however, the assumption that j is a good quantum number does not hold. For the NW potential, the two lowest even states of H2 are primarily j ) 0 states, but for D2 and T2, there is a significant amount of j ) 2 mixed in. For the FB and WS77 potentials, the lowest even state for all three isotopomers is a mixture of j ) 2 and j ) 0. The reduced densities given in eqs 11 and 13 are used to assign the 21 lowest even and odd states for the (3, 6) CNT as described by the FB potential. These assignments are presented in Table 7. The “translational” quantum numbers, nx and ny, can be thought of as quantum numbers for a 2-D harmonic oscillator. They are obtained from inspection of the nodal patterns of contour plots of F(x, y). We emphasize that because (1) the interaction potential is not harmonic, (2) the nanotube is not isotropic, and (3) translation and rotation are coupled quite strongly in the smaller nanotubes, the actual probability densities differ from those of a simple isotropic 2-D harmonic oscillator.

(15)

h 00(cos θ) + 0.61P h 20(cos θ)) ψ0 ≈ ψ0,0(x, y)(0.79 P h 00(cos θ) - 0.79P h 20(cos θ)) (16) ψ5 ≈ ψ0,0(x, y)(0.61P We plot the reduced probability density, |ψrot(θ)|2, for these two states in Figure 5. As can be seen, the ground state minimizes density about θ ) π/2 and maximizes it about θ ) 0 and θ ) π. The excited-state density, however, has a maximum at π/2. Because of the smaller values of the rotational constants for D2 and T2, the mixing in of excited rotational states is even more pronounced. For example, the coefficient on |P h 20|2 in the reduced probability density for the ground state is 0.37 (H2), 0.52 (D2), and 057 (T2). The mixing of the j ) 0 and j ) 2 states for the case of X2-D confinement is necessitated by the behavior of the potential as a function of the angle, θ. When θ ) 0 or π, the H2 axis is parallel to the z (nanotube) axis; when θ ) π/2, the H2 axis is perpendicular to the Z axis. In all cases that we have examined

V ⊥ - V| > 0

(17)

where V⊥ and V| are defined as the value of the H2-nanotube interaction potential for θ ) π/2 and θ ) 0 (or π), respectively, averaged over the remaining coordinates. For the larger nano-

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J. Phys. Chem. B, Vol. 110, No. 4, 2006 1749

TABLE 7: Wave Function Assignments for the Lowest 21 Even and Odd States for H2 in the (3, 6) CNT Calculated Using the FB Interaction Potential state

energy (even)

(nx, ny)

0

-736.3

(0, 0)

1

-303.1

(0, 0)

2

-303.1

(0, 0)

3

-253.0

(0, 1)

4

-253.0

(1, 0)

5

-172.6

(0, 0)

dominant rotational functions

energy (odd)

(nx, ny)

dominant rotational functions

-727.8

(0, 0)

0.90 (1, 0) c

-369.0

(0, 0)

0.30 (2, 1) s 0.64 (2, 1) c

-369.0

(0, 0)

-247.4

(0, 1)

-247.4

(1, 0)

74.4

(0, 0)

0.62 (0, 0) c 0.37 (2, 0) c 0.64 (2, 1) s 0.30 (2, 1) c

117.4

(1, 0) ( (0, 1)

0.93 (2, 2) s

143.9

(0, 0)

(1, 0) ( (0, 1)

0.46 (2, 1) s 0.46 (2, 1) c

143.9

(0, 0)

206.9

(1, 0) ( (0, 1)

0.46 (2, 1) s 0.46 (2, 1) c

159.7

(1, 0) ( (0, 1)

10

206.9

(1, 0) ( (0, 1)

0.46 (2, 1) s 0.46 (2, 1) c

159.8

(1, 0) ( (0, 1)

11

243.7

(1, 0) ( (0, 1)

170.1

(0, 0)

12

247.9

(2, 0) - (0, 2)

170.1

(0, 0)

0.93 (3, 2) s

13

248.6

(1, 1)

0.46 (2, 1) s 0.46 (2, 1) c 0.52 (0, 0) c 0.45 (2, 0) c 0.52 (0, 0) c 0.45 (2, 0) c

0.67 (1, 1) s 0.13 (1, 1) c 0.17 (3, 1) s 0.13 (1,1) s 0.67 (1, 1) c 0.17 (3, 1) c 0.88 (1, 0) c 0.12 (3, 0) c 0.88 (1, 0) c 0.12 (3, 0) c 0.12 (1, 0) c 0.85 (3, 0) c 0.38 (1, 1) s 0.38 (1, 1) c 0.12 (3, 1) s 0.12 (3, 1) c 0.10 (1, 1) s 0.14 (1, 1) c 0.28 (3, 1) s 0.43 (3, 1) c 0.14 (1, 1) s 0.10 (1, 1) c 0.43 (3, 1) s 0.28 (3, 1) c 0.34 (1, 1) s 0.35 (1, 1) c 0.14 (3, 1) s 0.15 (3, 1) c 0.35 (1, 1) s 0.34 (1, 1) c 0.15 (3, 1) s 0.14 (3, 1) c 0.93 (3, 2) c

2.4

(1, 0) ( (0, 1)

14

268.0

(2, 0) + (0, 2)

254.4

(2, 0) - (0, 2)

15

367.7

(0, 1)

255.1

(1, 1)

16

367.7

(1, 0)

276.1

(2, 0) + (0, 2)

17

546.2

(1, 0) ( (0, 1)

411.3

(0, 0)

18

548.2

(1, 0) ( (0, 1)

411.6

(0, 0)

19

569.2

(1, 0) ( (0, 1)

586.6

(1, 0)

20

569.2

(1, 0) ( (0, 1)

586.6

(0, 1)

0.34 (1, 1) s 0.34 (1, 1) c 0.15 (3, 1) s 0.15 (3, 1) c 0.82 (1, 0) c 0.14 (3, 0) c 0.82 (1, 0) c 0.14 (3, 0) c 0.83 (1, 0) c 0.16 (3, 0) c 0.45 (3, 3) s 0.51 (3, 3) c 0.51 (3, 3) s 0.45 (3, 3) c 0.13 (1, 0) c 0.74 (3, 0) c 0.13 (1, 0) c 0.74 (3, 0) c

6

-8.62

(0, 0)

0.56 (0, 0) c 0.42 (2, 0) c 0.56 (0, 0) c 0.42 (2, 0) c 0.38 (0, 0) c 0.55 (2, 0) c 0.93 (2, 2) c

7

-8.59

(0, 0)

8

172.6

9

0.52 (0, 0) c 0.45 (2, 0) c 0.38 (0, 0) c 0.45 (2, 0) c 0.38 (0, 0) c 0.45 (2, 0) c 0.45 (2, 2) s 0.44 (2, 2) c 0.45 (2, 2) s 0.44 (2, 2) c 0.40 (2, 2) s 0.42 (2, 2) c 0.42 (2, 2) s 0.40 (2, 2) c

a Energies are in cm-1. Approximate harmonic-like translational quantum numbers, nx and ny, are computed by inspecting the nodal patterns of the reduced probability density Fn(x, y). The coefficients of the dominant rotational basis functions are obtained from Fjm n , eq 13. An “s” or “c” following the (j, m) quantum numbers indicates sin(mφ) or cos(mφ) functions, respectively.

tubes (1-D confinement), this energy difference is very small, on the order of a few wavenumbers. For 2-D confinement, the energy difference is larger but still less than the H2 rotational spacing. For X2-D confinement, however, V⊥ - V| is many times larger than the rotational spacing. The isotropic j ) 0 state must mix with the j ) 2 state in order to build amplitude in the energetically favorable regions of the potential energy surface.

IV.B. Energy Separations. The large separation in energy between the first and second states within a given j-block evident in Table 7 is a reflection of the strong orientation preference in these nanotubes. In both the even and the odd j-blocks, the first excited states are m ) 1 states with significant probability density perpendicular to the nanotube axis. However, in section III.C we noted that the energy difference between the first even and first odd j-block states, or ortho/para separation, actually

1750 J. Phys. Chem. B, Vol. 110, No. 4, 2006

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Figure 5. Approximate reduced probability densities for the ground and fifth excited even states, respectively, of H2 confined in the (3, 6) nanotube using the FB potential.

decreases with increasing confinement. We can also understand this phenomenon using the separable approximation. We can write the Hamiltonian operator of eq 1 as

H ˆ ≈ Tˆ trans(x, y) + Tˆ rot(θ, φ) + Vtrans(x, y) + Vrot(θ, φ) ˆ rot(θ, φ) ≈H ˆ trans(x, y) + H

(18)

In eq 18, Tˆ trans + Tˆ rot denote the kinetic energy operators given by the first and second bracketed terms in eq 1. Thus, we can write the energy as

E ≈ Etrans + Erot

(19)

In particular, the rotational contribution to the energy will be

ˆ rot|ψrot〉 ) 〈ψrot|Tˆ rot|ψrot〉 + 〈ψrot|Vrot|ψrot〉 Erot ) 〈ψrot|H (20) Suppose also that the lowest even and lowest odd energy wave functions share the same translational wave function, differing only by their rotational wave functions. The energy difference between the odd and the even states is then approximated by

Eo - Ee ≈ (〈ψroto|Tˆ rot|ψroto〉 - 〈ψrote|Tˆ rot|ψrote〉) + (〈ψroto|Vrot|ψroto〉 - 〈ψrote|Vrot|ψrote〉) ) ∆Trot + ∆Vrot (21) (Rather than the separable wave function ansatz of eq 14, a separable density matrix ansatz can be used to arrive at eq 21 without assuming a separable Hamiltonian.27) Equation 21 shows that there are kinetic and potential energy components to the ortho/para energy splitting. As previously noted, the potential energy is generally lowered by aligning the H2 along the nanotube axis, which means that m ) 0 states will have somewhat lower energy than m ) 1 states. This holds for all the cases we have examined. Let us consider two limiting cases. In the first case rotation is only modestly hindered, as is the case for 1-D and ordinary 2-D confinement. There is little mixing of the rotational states so that the even ground state will be primarily the spherically symmetric j ) m ) 0 state. The degeneracy of the j ) 1 states will be lifted, however, since the j ) 1, m ) 0 state will be aligned along the z axis (P h j)1m)0 ∝ cos(θ)), while the j ) 1, m ) 1 states will be aligned perpendicular to it. Thus the lowest odd state will be mostly j

) 1, m ) 0. The splitting of the lowest even and odd states is given by 2B + ∆Vrot. We expect that ∆Vrot will be negative because the odd state is more optimally aligned along the z axis. This explains the decrease in ortho/para splitting with hindrance noted in section III.C above. This analysis is appropriate for (3, 6) and (8, 0) CNTs as described by the NW potential since there is negligible rotational mixing in the lowest states. In the second case, appropriate for X2-D confined systems, the ground state even wave function contains significant components of j ) 2, m ) 0. In this case, ∆Trot is reduced, which leads to a further reduction in the ortho/para splitting. Indeed, with a mixing of 0.37 j ) 2, m ) 0, as in the (3, 6) even ground state of Table 6 for the FB potential, it is easy to see that ∆Trot ≈ 2B - 0.37 × 6B ) -0.22B < 0. Since the lowest odd state is still higher in energy than the lowest even state, it must be the case that ∆Vrot > 0. Here the situation is the opposite of that for slightly hindered rotation described above. Unlike H2, in the case of D2 and T2 the lowest odd state is not completely dominated by P h 10 but rather has a considerable contribution from P h 30. Thus, it is not difficult to see how the energy difference between the lowest even and lowest odd states will be very much altered from its free space value, and in fact, as seen in Table 4, this difference will practically disappear. IV.C. Rotational-Translational Coupling and Anharmonicities. When we consider the coupling of rotation to translation, we must consider both the effects of radial motion of the center of mass (for example, motion away from the center of the nanotube toward the walls) and angular motion of the center of mass in the x-y plane. In 2-D and particularly in X2-D confinement, the coupling of radial motion to translational motion is Very strong. As the center of mass of the molecule gets closer to the walls, the forces requiring it to align along the nanotube axis will grow, that is, the potential as a function of polar angle, θ, will be very steep. We discussed this in paper I in a case of much weaker 2-D confinement than we are discussing here.10 Garberoglio et al.14 were able to estimate the effects of rotational-translational coupling on quantum sieving probabilities and found them to be extremely large. Thus we must emphasize that our separable model is intended only to give qualitative insight and that our quantum mechanical calculations contain the full rotational-translational coupling. We expect that the coupling of angular translational motion to rotational motion will be rather weak. This coupling occurs in the x-y plane and involves the azimuthal angle φ. As discussed in detail by Yildirim and Harris,13 even in an isotropic cylinder, rotation of the molecule in the x-y plane will be correlated to its translational motion in that plane. For example, if the molecule lying in the x-y plane prefers to be parallel to the wall, then as its center of mass moves in a circular direction in the x-y plane, the molecule will adjust its orientation so that it remains parallel to the wall. To estimate this weaker translational-rotational coupling, we consider several groups of states with translational wave functions that are described as linear combinations of states with nx ) 0, ny ) 1 and nx ) 1, ny ) 0. They are coupled to rotational states that are also combinations of states with m ) 1 or m ) 2. In the even block these are states 8-11 and 17-20, and in the odd j-block they are states 6, 9, 10, and 13. These combinations give rise to four possible wave functions such as

h 21(cos θ)(cos φ ( sin φ) ψ ∝ (ψ0,1(x, y) ( ψ1,0(x, y))P

(22)

that approximately describe states 8-11 in the even block. In an isotropic cylinder, without coupling of rotation to translation, the states within each group would be degenerate.13 In fact, the

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J. Phys. Chem. B, Vol. 110, No. 4, 2006 1751

four states are not all degenerate and the pattern of splitting is similar to that predicted for the isotropic nanotubes.13 Therefore, we define a coupling constant, R, such that

E8 ) E 0 - R E9 ) E10 ) E0 E11 ) E0 + R

(23)

As can be seen in Table 6, state 8 is ∼34 cm-1 lower in energy than the degenerate pair, states 9 and 10, while state 11 is ∼37 cm-1 higher in energy. The odd states 6, 9, 10, and 13 show the same splitting pattern. However, here R is somewhat higher, ∼43 to 44 cm-1. Consistent with the predictions of Yildirim and Harris,13 states with two quanta of excitation in translation coupled to rotational wave functions with m ) 0 separate into two nearly degenerate states and one slightly higher state. These are even states 1214 and odd states 14-16. In this case, the energy separations result from anharmonicities. The splittings are ∼20 cm-1. The effects of rotational-translational coupling in the x-y plane and of anharmonicities on the energetics of the system, ∼20 to 50 cm-1, are much less important than the effects of optimal alignment along the nanotube axis. We take as a measure of this latter quantity the energy difference between the two lowest states having the same number of quanta in translation and different values of m, that is, even states 1 and 0, even states 8 and 3, odd states 1 and 0, and odd states 6 and 3. These energy differences are 433, 345, 359, and 365 cm-1, respectively. This dramatic preference for alignment parallel to the nanotube axis is the hallmark of X2-D confinement. V. Conclusions We studied molecular hydrogen confined to the interior of various CNTs with three different C-H interaction potentials. These potentials, particularly the FB and WS77 potentials, are more realistic than the potential previously employed by us, and they all show significantly more quantum sieving and higher zero-point energies for small CNTs. These results are in better agreement with the previous work of Johnson and co-workers5,7 and in good agreement with the new calculations of Garberoglio et al.14 We found great sensitivity of the results to the value of the equilibrium distance of the C-H interaction, re, with the largest quantum sieving factors being associated with the use of relatively larger re values. For two of the potentials studied, re was sufficiently large to lead to a new type of confinement in the smallest nanotubes, which we term extreme 2-D or X2-D confinement. The X2-D confined states are characterized by steep potentials, very large zero-point energies, large quantum sieving selectivities, and highly hindered rotation. Analysis of the corresponding wave functions reveals that for the X2-D states there is strong mixing of rotational states, and the assumption, which holds in other cases, that j is a good quantum number no longer can be made. Neutron scattering experiments,28 probing the j ) 0 f j ) 1 transition for molecular hydrogen trapped inside solid C60, show that the hydrogen rotation is nearly free. Induced infrared absorption studies29 arrive at the same conclusion. Neutron scattering experiments of H2 adsorbed on the outer surface4 and interstitial regions26 of carbon nanotubes have also led to the conclusion that in these cases the hydrogen rotation is only slightly hindered. The neutron scattering study of Narehood et al.9 of H2 adsorbed on single walled nanotubes, however, shows rotational transitions that deviate significantly from those of bulk H2. It would be extremely interesting if the rotational transitions

of hydrogen adsorbed within carbon nanotubes could be probed to determine the validity of these and other similar theoretical studies. These studies will be most informative when the technology to create bundles of nanotubes of uniform size and structure is more advanced. Because of their cylindrical shape, carbon nanotubes present a particularly interesting geometry for confinement of molecules. Provided the interactions with the nanotube carbons are not too strong, larger molecules may experience X2-D confinement in nanotubes with larger radii. For more complex molecules or groups of molecules, such confinement may change the potential energy landscape of the system, favoring geometries or reaction paths that are not otherwise accessible. Thus, in addition to their potential use as storage media and quantum sieves, carbon nanotubes may have an important role in catalysis. Acknowledgment. The authors thank Giovanni Garberoglio and Karl Johnson for a careful reading of the manuscript and useful discussions. E.M.G. acknowledges support from DOE Grant No. DE-FG02-01ER15212. S.K.G. acknowledges support from the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, U.S. Department of Energy, under Contract No. W-31-109-ENG-38. References and Notes (1) Wang, Q. Y.; Johnson, J. K. J. Phys. Chem. B 1999, 103, 4809. (2) Wang, Q. Y.; Challa, S. R.; Sholl, D. S.; Johnson, J. K. Phys. ReV. Lett. 1999, 82, 956. (3) Williams, K. A.; Eklund, P. C. Chem. Phys. Lett. 2000, 320, 352. (4) Brown, C. M.; Yildirim, T.; Neumann, D. A.; Heben, M. J.; Gennett, T.; Dillon, A. C.; Alleman, J. L.; Fischer, J. E. Chem. Phys. Lett. 2000, 329, 311. (5) Challa, S. R.; Sholl, D. S.; Johnson, J. K. Phys. ReV. B 2001, 63, 245419. (6) Hathorn, B. C.; Sumpter, B. G.; Noid, D. W. Phys. ReV. A 2001, 64, 22903. (7) Challa, S. R.; Sholl, D. S.; Johnson, J. K. J. Chem. Phys. 2002, 116, 814. (8) Gordillo, M. C.; Boronat, J.; Casulleras, J. Phys. ReV. B 2002, 65, 014503. (9) Narehood, D. G.; Kostov, M. K.; Eklund, P. C.; Cole, M. W.; Sokol, P. E. Phys. ReV. B 2002, 65, 233401. (10) Lu, T.; Goldfield, E. M.; Gray, S. K. J. Phys. Chem. B 2003, 107, 12989. (11) Lu, T.; Goldfield, E. M.; Gray, S. K. J. Theor.Comput. Chem. 2003, 2, 621. (12) Trasca, R. A.; Kostov, M. K.; Cole, M. W. Phys. ReV. B 2003, 67, 035410. (13) Yildirim, T.; Harris, A. B. Phys. ReV. B 2003, 67, 245413. (14) Garberoglio, G.; DeKlavon, M. M.; Johnson, J. K. J. Phys. Chem. B, 2006, 110, 1733-1741. (15) Novaco, A. D.; Wroblewski, J. P. Phys. ReV. B 1989, 39, 11364. (16) Frankland, S. J. V.; Brenner, D. W. Chem. Phys. Lett. 2001, 334, 18. (17) Williams, D. E.; Starr, T. L. Comput. Chem. 1977, 1, 173. (18) The Atom-Atom Potential Method; Perstin, A. J., Kitaigorodsky, A. I., Eds.; Springer-Verlag: Berlin, Germany, 1989; p 89. (19) Beenakker, J. J. M.; Borman, V. D.; Krylov, S. Y. Chem. Phys. Lett. 1995, 232, 379. (20) Tanaka, H.; Kanoh, H.; Yudasaka, M.; Lijima, S.; Kaneko, K. J. Am. Chem. Soc. 2005, 127, 7511. (21) Colbert, D. T.; Miller, W. H. J. Chem. Phys. 1992, 96, 1982. (22) Lehoucq, R. B.; Sorensen, D. C.; Yang, C. http://www.caam.rice.edu/software/ARPACK. (23) Lehoucq, R. B.; Sorensen, D. C.; Yang, C. ARPACK Users’ Guide: Solution of Large-Scale EigenValue Problems with Implicitly Restarted Arnoldi Methods; Society for Industrial and Applied Mathematics: Philadelphia, PA, 1998. (24) Freeman, T. http://www.fungible.com/fungimol. (25) Davidson, N. Statistical Mechanics; McGraw-Hill: New York, 1962. (26) Ren, Y.; Price, D. L. Appl. Phys. Lett. 2001, 79, 3684. (27) Garberoglio, G. Private communication. (28) FitzGerald, S. A.; Yildirim, T.; Santodonato, L. J.; Neumann, D. A.; Copley, J. R. D.; Rush, J. J.; Trouw, F. Phys. ReV. B 1999, 60, 6439. (29) FitzGerald, S. A.; Forth, S.; Rinkoski, M. Phys. ReV. B 2002, 65, 140302.