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J. Phys. Chem. C 2008, 112, 2654-2659
Chemical Reactivity within Carbon Nanotubes: A Quantum Mechanical Study of the D + H2 f HD + H Reaction Tun Lu and Evelyn M. Goldfield* Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202
Stephen K. Gray Chemical Sciences and Engineering DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: October 1, 2007; In Final Form: NoVember 12, 2007
Chemical reactivity may be significantly altered when reagents are confined to move within a nanoscale environment. Chemical reactions inside carbon nanotubes (CNTs), in particular, have been the focus of some attention. To help lay theoretical foundations for understanding such nanoscale-confined chemistry, we study the quantum dynamics of the D + H2 f HD + H exchange reaction, one of the most fundamental reactions in gas-phase chemistry, within a CNT. A five-dimensional Hamiltonian model for the system is developed, and numerous wavepacket calculations are carried out. Quantum reaction probabilities are compared with gas-phase reaction probabilities. Several different sized CNTs are considered. The smaller CNT diameter reaction probabilities are considerably higher than the gas-phase ones.
1. Introduction There is a growing body of research on chemical reactions within carbon nanotubes (CNTs). In pioneering early experimental work, silver “nanobeads” were produced in CNTs via reduction of AgNO3.1 More recent experimental examples include polymerization of C60O fullerene derivatives inside single-walled CNTs2 and enhanced ethanol production from CO and H2 in CNTs containing Rh-nanoparticle catalysts.3 Although more limited, there has also been some interesting theoretical work on chemical reactivity inside CNTs. An electronic structure theory study, for example, predicted that the Menshutkin SN2 reaction H3N + H3CCl f H3NCH3+ + Cl- should be catalyzed in CNTs owing to a lowering of the activation energy due to polarization effects.4 Although such static calculations are very instructive, there is a also need for fundamental theoretical dynamics studies on the effects of confinement on chemical reactions. In particular, in order to develop a theoretical dynamical underpinning, dynamical studies of very simple chemical reactions, such as the reaction of an atom with a diatomic molecule, are necessary. Such reactions have been well-studied in the gas phase, and it is relatively straightforward to assess the effects of the confining media on their reactivity. When a chemical system is confined in a nanoporous material such as a CNT, the effects of confinement on reactivity can be both chemical and physical in nature. Chemical effects include intermolecular interactions between the atoms of the confined system and the atoms of the confining environment. These interactions may differentially stabilize reactants, products, intermediates, or transition structure and thus have a very significant effect on the equilibria, the rates, and the products of the reactions. Chemical confinement effects are determined primarily by the chemical composition of the confining media. The Menshutkin example noted in the above paragraph is one such example.4 Physical effects pertain to the ways the size and * Corresponding author. E-mail:
[email protected].
the geometry of the confined space influences and restricts the orientations and conformations accessible to the confined systems. Restriction of accessible structures could possibly render a high-energy gas phase conformation the most stable conformation in the confined space. Physical or steric restriction may also reduce the number of available reaction paths and alter the rates or the products of a reaction. Collisions with the walls of the confining media could also be an important factor in determining the course of a reaction. The size and shape of the confining media will play a large role in determining physical confinement effects. The interplay between physical and chemical confinement effects is both fascinating and exploitable for practical applications Here we discuss the D + H2 f HD + H reaction confined to occur within a CNT. The H + H2 reaction (and its isotopic variants) is one of the simplest of all chemical reactions. It has played a fundamental role in chemical reaction dynamics both in terms of establishing concepts and in terms of detailed comparisons between theory and experiment5 and is still the subject of intense study.6-11 The reaction has a collinear transition state, so one might expect that confinement within a cylindrical environment will enhance reactivity. We refer to the CNT-confined system as D + H2@CNT. Our previous quantum studies on CNT-confined H2 and its isotopomers12,13 showed that for smaller radii CNTs both translation and rotation were highly hindered and the molecule preferred to be aligned along the long nanotube axis. Quantum dynamics results for a fivedegree-of-freedom model of D + H2@CNT are discussed here. We calculate the energy-resolved reaction probabilities of the D + H2 reaction inside four carbon nanotubes of various sizes using a time-dependent wavepacket method. 2. D + H2@CNT Model and Computational Methods 2.1. Five-Dimensional Model. To fully describe a three-atom system within a rigid CNT environment, nine degrees of freedom are required. This is because the DH2 center-of-mass
10.1021/jp077737w CCC: $40.75 © 2008 American Chemical Society Published on Web 01/25/2008
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J. Phys. Chem. C, Vol. 112, No. 7, 2008 2655
(CM) velocity vector and angular momentum vector are no longer constants of motion. Six space-fixed (SF) coordinates and three body-fixed (BF) Jacobi coordinates may be used for this purpose. (For a discussion of SF and BF coordinates, see refs 14 and 15.) The BF coordinates are the Jacobi coordinates R, r, and R, where R is the distance from the D atom to the CM of the H2 molecule, r is the distance between the two H atoms, and R is the angle between R and r. We define a SF Cartesian axis system with the Z axis lying along the long axis of the nanotube. The DH2 CM position vector is denoted by (DX, DY , DZ). Rotational motion is described by the three Euler angles θ, φ, and η.14,15 If we choose r to be our BF z-axis, then θ is defined as the angle between r and Z. Note that this choice is not the usual choice in gas-phase three-atom problems, although it has been used in some studies and is also termedr-embedding.14-17 The two azimuthal angles φ and η denote rotation about the z and Z axes, respectively. In these coordinates, the Hamiltonian for the nine-degree-of-freedom system is
Η ˆ )-
(
)
p 2 ∂2 ∂2 ∂2 p 2 ∂2 p 2 ∂2 + + + 2M ∂D 2 ∂D 2 ∂D 2 2µR ∂R2 2µr ∂r 2 X Y Z (Jˆ - Lˆ ) 2µr r
2
2
+
Lˆ + V(DX,DY,DZ,θ,φ,η,R,r,R) (1) 2µRR2 2
where M ) MD + 2MH is the total mass of the three-atom system and the reduced masses µr and µR are
Figure 1. Coordinates used in the five-degree-of-freedom calculations.
atoms approaching H2 in a direction parallel to the long axis of the nanotube, Z. The model, however, is not strictly planar in that we continue to use a Legendre basis (see below). Otherwise it would be difficult to compare with the J ) 0 three-degreeof-freedom calculations on the isolated system. Rather, we simply set K and M to zero, thereby eliminating the Coriolis and centrifugal terms that arise from the Jˆ‚Lˆ operator. Combined with our restriction of the CM motion to be along the X axis, the five-degree-of-freedom Hamiltonian is
H ˆ )-
p2 ∂2 p2 ∂2 p2 ∂2 Jˆ 2 Lˆ 2 + + + 2M ∂D 2 2µR ∂R2 2µr ∂r 2 2µ r 2 2µ r 2 X
r
Thus, our wavepackets are expanded as follows
µr ) MH/2 µR ) 2MDMH/(MD + 2MH)
r
Lˆ 2 + V(DX,θ,R,r,R) (4) 2µRR2
Jmax Lmax
(2)
The angular momentum operators Jˆ and Lˆ correspond to DH2 angular momentum and orbital angular momentum, respectively. Note that, unlike the isolated system, the potential is dependent on all nine degrees of freedom. We expand the wavepacket in rotational channels
ψL,J,K,M(DX,DY,DZ,θ,φ,η,R,r,R,t) ) J (θ,φ,η) (3) CL,J,K,M(DX,DY,DZ,R,r,t) PKL (R) D*K,M
where t refers to time and PKL (R) is an associated Legendre *J (θ,φ,η) is a normalized Wigner D-matrix, and M function, DK,M is the projection of DH2 angular momentum J on the SF Z axis, which is parallel to the long axis of the CNT. Additionally, K is the projection of DH2 angular momentum and of orbital angular momentum on the body fixed z-axis, that is, the H-H bond distance, r. In the isolated system M is a good quantum number, but in the confined environment it is not. Unfortunately, the nine-degree-of-freedom problem of eq 1 is currently too large to be exactly solved. We therefore develop a reduced dimension model that eliminates four degrees of freedom (DZ, DY, and the two azimuthal Euler angles). The first dynamical approximation is to ignore the motion of the system CM along the nanotube axis, thereby eliminating DZ. Because the interaction potential does not change much along the nanotube axis, this is a reasonable approximation. The second, more drastic approximation, which eliminates the other three variables, is to confine the D + H2 system to lie in the XZ plane thereby ignoring the out-of-plane motion. There are two aspects to this approximation: (1) it assumes that the reaction is largely planar, which is known to be true at low energies, and (2) the important configurations must involve a plane containing the nanotube axis. Thus, our model system is consistent with D
∑ ∑ ψJ,L(DX,θ,R,r,R,t)
ψ(DX,θ,R,r,R,t) )
(5)
J)0 L)0
ψJ,L(DX,θ,R,r,R,t) ) CJ,L(DX,R,r,t) PL(cos R) PJ(cos θ) (6) where PJ and PL are normalized Legendre polynomials. Insertion of eqs 4-6 into the time-dependent Schro¨dinger equation leads to a set of coupled equations
∂CJ,L(DX,R,r,t) ip
) dt (Tˆ Rad + TRot)CJ,L(DX,R,r,t) +
∑VJL,J′L′CJ′,L′(DX,R,r,t)
(7)
J′,L′
where
Tˆ Rad ) -
p2 ∂ 2 p2 ∂ 2 p 2 ∂2 2M ∂D 2 2µR ∂R2 2µr ∂r 2
(8)
X
TRot )
J(J + 1)p2 2µr r 2
+
L(L + 1)p2 2µrr 2
+
L(L + 1)p2 2µRR2
(9)
and
VJL,J′L′ )
∫∫PL(cos R) PJ(cos θ) V(Dx,R,r,R,θ)
PL′(cos R) PJ′(cos θ) d cos R d cos θ (10)
As discussed in 2.4 below, we do not directly compute the potential matrix, VJL,J′L′. The coordinate system used to describe the system is shown in Figure 1. 2.2. Potential Interactions. The potential of the system is divided into two portions
V ) VDHH + WDHH-C
(11)
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Lu et al.
where VDHH represents the interaction of the D atom with the two hydrogen atoms and WDHH-C represents the interaction of the H atoms and the D atom with each of the carbon atoms in the nanotube. Unlike the Menshutkin reaction, where the polarizability of the nanotube was central,4 the important interactions of our system with the nanotube are dispersion interactions, which cannot be reliably calculated using density functional theory methods, for example. The higher levels of electronic structure theory that would be required are much too computationally intensive for such a large system. Thus, we use an empirical potential to describe the interactions with the nanotube carbons. We take VDHH to be the well-known LSTH potential,18,19 and WDHH-C is a sum of pairwise Lennard-Jones (L-J) potentials with the form
(
)
σ12 σ6 r12 r6
TABLE 1: Computational Parameters (All Units Are Atomic Units) Rmin, Rmax, ∆R rmin, rrmax, ∆r ∆DX R0, δ, k0 Jmax Lmax ∆t r† Rabs,Cabsa rabs,cabsa a
0.15, 9.0, 0.15 0.15, 5.3, 0.15 0.15 4.4, 0.4, 10.98 12 12 4.5 3.3 7.5, 0.1 4.5, 0.1
The absorption is given by exp[-Cabs(R - Rabs)2], R g Rabs.
TABLE 2: Features of Potential Energy Surface and the Nanotubes
(12)
nanotube type
potential minimum (eV)
transition state (eV)
reaction barrier (eV)
nanotube radius (Å)
where r is the H or D atom to C atom distance. The L-J parameters are taken to be ) 19.2 cm-1, σ ) 3.08 Å as given by Frankland and Brenner.20 Nanotube structures were generated using the tubegen algorithm.21 Our potential is further simplified in two respects. First, we ignore all potential variation along the long (Z) axis of the nanotube. The variation of the potential along the Z axis is much smaller than that along the X axis. Thus, in our simplified model, the interaction potential of each atom with the nanotube is determined by its X coordinate. Second, we ignore short-range interactions between the isolated D (or H) atoms and the nanotube, precluding the possibility of an isolated D (or H) atom interacting with the nanotube. It is not possible to treat such interactions properly with a rigid nanotube model. We have studied collisions of H and D atoms with the walls of the nanotube using classical dynamics, using a potential with short-range interactions and a flexible nanotube. Our results show reactions between the D atoms and the nanotube walls do not occur for the range of the collision energy we used in this study. 2.3. Initial Wavepacket. The initial wavepacket is given by
(8,0) (2,8) (6,6) (10,10) gas phase
-0.28 -0.30 -0.23 -0.14 0
0.16 0.14 0.21 0.31 0.43
0.44 0.44 0.44 0.45 0.43
3.13 3.60 4.07 6.78
V(r) ) 4
ψ(t ) 0) ) G(R) φ(DX,θ,R) χ(r)
(13)
where
(
G(R) ∝ exp -
)
(R - R0)2 2δ2
exp(ik0R)
(14)
is a Gaussian wavepacket centered at R0 with initial momentum centered at pk0, χ(r) is the ground vibrational state wavefunction of H2 and φ(DX,θ,R) is the lowest eigenstate of the threedimensional Schro¨dinger equation obtained by setting R ) R0 and r equal to the equilibrium value of H2, re:
(
)
-p2 ∂2 Jˆ 2 Lˆ 2 Lˆ 2 + + + + V φ ) Eφ (15) 2M ∂D 2 2µ r 2 2µ r 2 2µ R2 X
r
r
R
The parameters used in our calculations are given in Table 1. 2.4. Propagation and Analysis. We computed the energyresolved reaction probabilities of the D + H2 reaction in four nanotubes (8,0), (2,8), (6,6), and (10,10) with radii ranging from 3.13 to 6.78 Å as listed in Table 2. (Here we use the standard two integer component chiral vector to denote the different CNTs.22) When studying H2 and its isotopomers in CNT’s,12,13 we looked at two small nanotubes of nearly equal diameter but very different structure: the (8,0) and (3,6) nanotubes whose
diameters differ by ∼0.04 Å. We obtained nearly identical results for every quantity we calculated. We thus concluded that the diameter of the nanotube would be far more important than its specific structure in determining the effects of confinement on this chemical reaction. The length of each nanotube was set to a large value so that further increasing the length of the nanotube did not change the potential in the interaction region. The actual lengths of these nanotubes were set to 24 Å in our calculation. To propagate the wavepackets in time, we used a symplectic propagator (“m ) 6, n ) 4”) developed by Gray and Manolopoulos.23 This is a very straightforward method for propagating the full wavepacket. Because of the exploratory nature of this project, we chose to use the symplectic propagator as opposed to the real wavepacket or Chebyshev-based propagator that we have used more extensively.24 The radial coordinates are represented on an evenly spaced grid, and the DFFD method25 is used to compute the action of the relevant kinetic energy operators. In order to compute the action of the potential energy operator, we use a method employed in previous calculations.12,13 We transform the wavepacket from the Legendre basis (J, L) to a grid representation in θ and R where the potential energy operator is diagonal. After multiplication by the potential, the result is transformed back to the Legendre basis representation. As noted in Table 1, the grid spacing in the three radial coordinates is 0.15 a0 and Lmax ) Jmax ) 12. For the (8, 0) nanotube, we also computed reaction probabilities using a smaller grid spacing of 0.1 a0 and more rotational states with Lmax ) Jmax ) 24. The reaction probabilities calculated between 0 and 1.4 eV with this set of values are about 1-3% higher than those with the cruder basis. Thus, we consider that our basis is adequate. Wavepackets reaching the end of the grid are absorbed by a damping function of Gaussian form. The reaction probability is calculated by computing the flux at r† ) 3.3 a0 at which distance the H2 bond is considered to be broken. Specifically, this time-dependent flux is Fourier transformed to energy space and normalized by the energy content of the initial wavepacket, as in ref 26. We propagated for 0.2 ps (2000 time steps). Depending on the size of the nanotube, the calculations took from 24 to 72 h. Our parallel MPI code25,27 ran on the eight Itanium 2 processors of an SGI Altix 3300 multiprocessor distributed shared memory
Chemical Reactivity within Carbon Nanotubes
J. Phys. Chem. C, Vol. 112, No. 7, 2008 2657
Figure 3. One-dimensional cuts of the potential energy surfaces in the various nanotubes as a function of DX, minimized with respect to all other coordinates.
Figure 2. Reduced density plots of the initial wavepacket in the (8, 0), (2,8), (6,6), (10,10) nanotubes: (a) density shown as a function of distance of the center of mass of the system from the nanotube axis, DX; (b) density shown as a function of the angles R and θ.
computer with 16 GB of RAM. Memory requirements also depend on the size of the nanotube; calculations for the largest nanotube essentially used all of the available memory. 3. Results and Discussion 3.1. Initial Wavepacket. Densities associated with the initial wavepackets are plotted in Figure 2a and b. Figure 2a shows the density function of the wavefunction as function of DX averaged over other degrees of freedom. From Figure 2a, we see that the DH2 CM is centered on the X axis for nanotube all but the (10, 10) nanotube, where it is ∼3 Å from the center. Figure 2b shows the wavepacket density as a function of the angle θ and R averaged over other degrees of freedom. For the two smaller nanotubes (8, 0) and (2, 8), the maximum density is near θ ) R ) 0 (π) corresponding to collinear D-H-H, parallel to the Z axis. For the larger nanotubes, the initial density is much broader with a maximum density near θ ) R ) π/2. This corresponds to a configuration in which the H2 is perpendicular to the Z axis with the D atom approaching parallel to the Z axis. In general, the system prefers configurations in which the D atom approaches parallel to the Z axis, which is indicated by the higher density when θ + R ) π. Thus, for all nanotubes the preferred direction of approach is the same. The broader distributions in the larger nanotubes indicate that, as expected, the confinement effects are weaker in these nanotubes.
3.2. Potential Energy Surface. The global minimum occurs at the reactant (product) asymptote, as one would expect because there is no minimum on the LSTH surface. Figure 3 shows cuts of the potential surface as a function of DX, minimized with respect to all other variables. For the (8, 0) and (2, 8) nanotubes, the minimum is at the center of the tube. The two larger nanotubes, however, display double-well potentials. The (6, 6) potential exhibits very shallow minima at DX ) (0.44 Å, and the (10, 10) nanotube exhibits more pronounced minima at DX ) (3.7 Å. For the smaller (8, 0) and (2, 8) nanotubes, the global minimum corresponds to collinear D + H2 aligned parallel to the nanotube Z axis. For the large nanotube, the situation is more complex because, due to the less restrictive confinement effects, there are several equivalent minima at large R, corresponding to the D atom and the H2 molecule both being on the same or on opposite sides of the long axis of the nanotube. For each nanotube, the transition states have the same geometry as the isolated molecule on the LSTH surface, with an HH (HD) bond distance of 0.926 Å and R equal to 0 or π. Additionally, θ is also 0 or π, which tells us that the collinear transition state is lying parallel to the nanotube Z axis. The CM position for the transition state, however, differs in different nanotubes. In the smaller (8, 0) and (2, 8) nanotubes, the CM position transition-state position is at the center but for the larger nanotubes it lies off-center with DX ) (0.47 Å (6, 6) and DX ) (3.65 Å (10, 10). The energies of the global minima as well as that of the transition state are given in Table 2. It can be seen that within the nanotubes there is a lowering of energies. However, the transition state is not stabilized more than the reactants. Thus, this model does not predict any significant changes in the bare reaction barrier, defined as the difference in energy between the transition state and reactants, as a result of confinement. We do not believe that the small increase in the reaction barrier for the confined system seen in Table 2 is numerically significant. Because there is no variation along the Z axis in our potential, the energy for a collinear minimum energy path aligned parallel to the Z axis will vary from that of the isolated system by a constant. Significant differences from the potential for the isolated system occur, however, when the system strays from that minimum energy path. 3.3. Reaction Probabilities. Quantum reaction probabilities as a function of collision energy for the D + H2 system confined within the four nanotubes are shown in Figure 4a. As expected, the reaction probabilities from the smaller (8, 0) and (2, 8)
2658 J. Phys. Chem. C, Vol. 112, No. 7, 2008
Lu et al. that there is no enhancement of the reaction probability in the larger nanotubes. The 2D calculation, of course, takes place entirely on the minimum energy path and thus one might expect high reaction probabilities at lower collision energies, although at higher collision energy a recoil effect comes into play, lowering the reaction probability quite dramatically. For a full 3D calculation, however, the system is no longer constrained to follow the minimum energy path. In this case, there are two requirements for being close to the minimum energy path: (1) low impact parameter collisions and (2) the H2 molecule being aligned parallel to the incoming D atom. The 3D J ) 0 calculation from ground-state reactants meets the first of these requirements. The H2 molecule in its rotational ground state, however, is not aligned at all. If we instead consider a calculation with J ) 0, but now jHH ) 1, then we will meet both of these requirements. As shown in Figure 4c, the reaction probability for this case is comparable to that of confinement in the (8, 0) nanotube. Thus, by constraining the conformational space available to the reactants the smaller nanotube essentially “selects out” the conditions that favor reaction from the large number of initial conditions available in the gas phase. Besides the increasing of probability, another interesting phenomenon occurs: The reaction threshold in all nanotubes is ∼0.13 eV lower than 3D and comparable to 2D. Thus, although as stated above, the bare potential barrier is not lowered in the nanotube, the actual reaction barrier is. To explain this phenomenon, we consider the zero-point energy (ZPE) of the reactants and at the transition state. For the 2D case, there is the ZPE of one stretching vibration, the H2 stretch of the reactants and the symmetric stretch at the transition state. In the 3D case, however, there is an additional degree of freedom at the transition state; the free rotation of the reactants becomes the D-H-H bend, adding to the ZPE at the transition state. Within the nanotube, all of the rotational and translational motions of the reactants are hindered. So rather than free rotation turning into the D-H-H bend, it is highly hindered rotational and translational motion of the reactants that contributes to the bend at the transition state. The confinement effects that increase the reaction probability of D + H2 inside the nanotube are different from those that lower the reaction threshold. As noted, there is no enhancement of the reaction probability in the larger nanotubes; nevertheless the reaction threshold is still reduced from the 3D case. In contrast, the 3D ( jHH ) 1) calculation gives a reaction probability similar to that in the (8, 0) nanotube, but with the reaction threshold as from ground-state reactants. Thus, an optimal initial alignment of the reactants favors high reaction probability. The reaction threshold lowering, however, results from the fact that the translational and rotational motions of the reactants are restricted.
Figure 4. Reaction probabilities as a function of collision energy: (a) (8, 0), (2, 8), (6, 6), and (10, 10) nanotubes; (b) (8, 0) and (2, 8) nanotubes, gas phase (J ) 0, VHH ) 0, jHH ) 0) and collinear (c) (8, 0) nanotube, gas phase (J ) 0, VHH ) 0, jHH ) 0) and (J ) 0, VHH ) 0, jHH ) 1)
nanotubes are larger than those from the larger (6, 6) and (10, 10) nanotubes. In Figure 4b, we compare the (8, 0) and (2, 8) reaction probabilities with reaction probabilities resulting from two calculations on the isolated system: collinear (2D) and a three-degree-of-freedom J ) 0 calculation from ground-state reactants (3D). It is clear that the reaction probability arising from confinement in the two smaller nanotubes is larger than the 3D results for all energies and for the 2D results at higher collision energy. By comparing Figure 4a and b, one can see
4. Concluding Remarks We presented the first quantum mechanical study of the fundamental D + H2 hydrogen exchange reaction in a carbon nanotube. Several approximations were introduced to make the problem tractable from a quantum mechanical perspective. Nonetheless, we believe the resulting five-degree-of-freedom system can describe the essential features of confinement. We found that chemical reactivity can be enhanced considerably compared to the gas phase for the smaller nanotubes studied. The origin of this enhancement is mostly steric in nature, having to do with alignment of the H2 along the long, Z axis of the tube. We also identified a quantum mechanical zero-point energy effect that effectively lowers the reaction thresholds compared to the gas-phase limit.
Chemical Reactivity within Carbon Nanotubes We have recently completed extensive quasiclassical trajectory (QCT) simulations,28 which of course overcome many of the limitations of our model, albeit at the loss of purely quantum effects such as the pronounced threshold effect noted above. In these studies, we see enhancements of reactivities similar to those reported here. We are also able to explore other types of confinement effects that were not possible in this study such as the role of collisions with the walls of the nanotube. Much recent work on the hydrogen exchange reaction and its isotopic variants has focused on the role of the conical intersection29,30 and the geometric phase,7,8,11,31 and one might wonder how confinement in a nanotube would influence the role of the geometric phase on observables in the reaction. A necessary (but not sufficient) requirement for geometric phase effects to occur is that some significant portion of the wavepacket circles the conical intersection, thus involving geometries that are far from collinear. Because confinement in the smaller nanotubes favors paths that are near collinear, it is likely that geometric phase effects will be even less important in the confined system than they are in the gas phase. Finally, one might ask: Could the D + H2 exchange reaction, or some related simple and well-known gas-phase chemical reaction, ever be studied experimentally in a CNT? In view of some of the remarkable recent experimental progress noted in the Introduction, for example, the formation of ethanol in CNTs containing catalysts via injection of H2 and CO,3 we are inclined to think that it should be possible. We hope that the present theoretical work serves to stimulate such studies. Acknowledgment. E.M.G. acknowledges support from the U.S. Department of Energy, Basic Energy Sciences, grant no. DE-FG02-01ER15212. S.K.G. was supported by the U.S. Department of Energy, Basic Energy Sciences, under contract no. DE-AC02-06CH11357. References and Notes (1) Ugarte, D.; Chatelain, A.; de Heer, W. A. Science 1996, 274, 1897.
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