J. Phys. Chem. A 2010, 114, 12075–12082
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Theoretical Studies of CH4 Inside an Open-Cage Fullerene: Translation-Rotation Coupling and Thermodynamic Effects Keith E. Whitener, Jr.* Department of Chemistry, Yale UniVersity, P.O. Box 208107, New HaVen, Connecticut 06520-8107, United States ReceiVed: May 19, 2010; ReVised Manuscript ReceiVed: October 2, 2010
Molecules trapped inside fullerenes exhibit interesting quantum behavior, including quantization of their translational degrees of freedom. In this study, a theoretical framework for predicting quantum properties of nonlinear small molecules in nonsymmetric open-cage fullerenes (OCFs) has been described along the lines of similar theories which treat small molecules inside C60 and clathrate cages. As an example, the coupled translational-rotational energy structure has been calculated for the case of CH4 inside a known OCF. The calculated energy levels have been used to calculate the equilibrium fraction of incorporated CH4 as well as the translational heat capacity for the encapsulated molecule. The heat capacity shows an anomalous maximum at 239 K for CH4 and 215 K for CD4 which are not present in free methane. 1. Introduction In their paper announcing the discovery of C60, Kroto et al. noted the possibility of inserting atoms and small molecules into the interior of the cage, forming an endohedral complex.1,2 Since then, a number of theoretical and experimental studies have elucidated the properties of several different families of endohedral fullerenes, including metallofullerenes,3-8 trimetallic nitride-containing fullerenes,9-12 beam-implanted atoms inside C60,13-19 and fullerenes which incorporate small molecules and noble gases into their cages.20-33 Significant advances in the synthesis of endohedral fullerenes have been made recently by employing the concept of “molecular surgery”.28,34-40 This idea, outlined in Scheme 1, consists of three main steps. First, one or more of the C-C bonds on the C60 are scissioned to create an orifice on the fullerene cage. This species is referred to as an open-cage fullerene (OCF). An endohedral dopant is then inserted through the newly formed orifice, often under conditions of elevated pressure and temperature. Finally, the broken bonds on the C60 cage are repaired by chemical means. The molecular surgery approach to endohedral fullerenes is exemplified in the work of Komatsu et al. in the synthesis of
[email protected] In their work, a 13-member orifice is created synthetically on the C60 cage. Molecular hydrogen is then inserted through the orifice at high pressures and temperatures, and the orifice is closed using a series of 4 synthetic steps to form H2@C60 in high yield.28 To date, this method is the only one to have achieved full cage closure while keeping the endohedral molecule inside the fullerene. One drawback to the method of Murata et al. is that the orifice formed on the C60 cage is large enough to accommodate only molecular hydrogen and helium. However, many other methods exist for opening larger holes on C60. In particular, Iwamatsu et al. synthesized an OCF (1) with a 20-member orifice which spontaneously incorporates a single H2O molecule (Figure 1).37 Our group, in collaboration with Iwamatsu’s group, has shown that molecules as large as CH4 and NH3 can fit through the * To whom correspondence should be addressed. Current address: Department of Chemistry and Biochemistry, University of Colorado at Boulder, UCB 215, Boulder, CO, 80309-0215. E-mail: keith.whitener@ colorado.edu.
orifice of 1 to form stable compounds such as
[email protected] However, there does not yet exist a method for closing the cage of 1 while simultaneously retaining the endohedral guest. Theoretical studies on endohedral fullerenes have focused almost exclusively on those species where the fullerene cage is intact.30-33,45-49 The icosahedral symmetry of C60 substantially simplifies electronic structure calculations, and the corrugation of the potential energy surface inside the fullerene is small enough that the cage can be roughly approximated as spherically symmetric. Mamone et al. and Xu et al. performed quantum calculations on H2@C60,30-32 and Cross29 carried out similar studies on CO@C60 and N2@C60 which took advantage of the high symmetry of the fullerene to expedite calculation of the infrared and Raman spectra of the guest molecules inside the cage. Examples of this type of calculation being performed in an anisotropic cage include work by Horsewill et al.,49 who performed energy level calculations on H2 inside an open-cage fullerene using an ellipsoidal rotational potential. Also, Xu et al. 50 performed similar calculations on H2 inside the anisotropic SCHEME 1: Outline of the Molecular Surgery Method for Synthesizing Endohedral Fullerenesa
a In the first step (A), bonds on the pristine C60 cage are broken chemically. In the second step (B), a dopant molecule is inserted through the orifice. In the third step (C), the C60 cage is reformed to give the final endohedral product.
Figure 1. Structure of Iwamatsu’s large-orifice OCF 1.44
10.1021/jp104601g 2010 American Chemical Society Published on Web 10/25/2010
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TABLE 1: Single-Point Energy Calculations for Three Orientations of Methane inside 1a orientation vertex edge face
energy 0.0 +7.577 (633.4) +9.130 (763.2)
a The CH4 carbon is located 3.51 Å from the nearest carbons in the radial direction and 3.80 and 3.74 Å from the methylene groups on the orifice, respectively. All energies are given in kJ/mol (cm-1).
C70 cage. On a related note, in a recent study by Matanovic et al.,33 the translational-rotational behavior of CH4 trapped inside icosahedral ice clathrate structures was examined. This paper presents a similar methodology to the Matanovic work, specializing that paper’s results to cylindrical coordinates and providing a novel derivation of the Hamiltonian matrix elements. In the case of H2@C60 and H2 in an open-cage fullerene, the compounds are readily synthesized so that theoretical predictions can be compared against experimental results. Xu and coworkers were able to get qualitative agreement between theory and experiment using a simple two-site Lennard-Jones 12-6 potential, 31,32 which they then refined to a three-site Lennard-Jones potential which yielded quantitative agreement with experiment.50 Mamone et al.30 and Horsewill et al.49 also achieved quantitative agreement using various global three-parameter potentials fitted to the experimental data. The fact that such a simple potential model can yield insight into the effect of the fullerene cage on the endohedral dopant is encouraging. However, as mentioned before, there are a number of cases in which endohedrally doped OCFs are simple to prepare but for which there does not yet exist a method to convert them to endohedrally doped C60. In light of this fact, we extended the methods employed in the studies mentioned above to analyze the dynamics of CH4 in Iwamatsu’s OCF 1. In particular, we analyzed a remarkable feature common to the dynamics of all molecules trapped in small containerssnamely, the quantization of the translational motion of the dopant and its interaction with the dopant’s rotationsand used the results from an energy level analysis to predict observable thermodynamic quantities. 2. Theory 2.1. Preliminary Electronic Structure Calculations. In order to establish preliminary information about the nature of the potential that CH4 experiences inside the cage, several electronic structure calculations were carried out using the Gaussian 03 ab initio quantum chemistry package.51 Unless otherwise indicated, all electronic structure calculations were performed using the ONIOM multilayer method52 where the fullerene is treated at the B3LYP/6-31G(d) level of theory and the endohedral methane is treated at the MP2/6-311++G(3df,3pd) level of theory. This method does not explicitly include dispersion effects, which are the main stabilizing forces for CH4 inside the fullerene. However, the restoring force for the rattling motion of the methane molecule inside the cage is dominated by electrostatic repulsion, which is accounted for in this model. This approximation was necessary as the size of the system does not permit extensive ab initio treatment. For a molecule inside a closed C60 cage, the variation from free rotation is very small (on the order of 1 cm-1) and the cage potential can be treated as spherically symmetric to a good approximation.47,53 However, single-point energy calculations of CH4@1 indicate that this approximation does not hold in general in OCFs. Table 1 shows the results of calculations of methane oriented with a vertex,
Figure 2. Plots of energy vs position for translation of CH4 along the z axis (parallel to the orifice) and the x and y axes (perpendicular to the orifice) of 1. The origin is located 3.51 Å from the nearest carbons in the radial direction and 3.80 and 3.74 Å from the methylene groups on the orifice, respectively. The z axis potential is fitted to a sixthorder polynomial, while the x- and y-axis potentials are fitted to fourthorder polynomials. The activation barrier along the z axis for insertion of methane was calculated as 13 046 cm-1 at the B3LYP/6-31G(d,p)// B3LYP/3-21G level of theory in an earlier work.43
edge, and face of its tetrahedral geometry toward the orifice of the OCF. These calculations indicate a barrier to free rotation of several hundred wavenumbers. Further single-point energy calculations were carried out where the CH4 was translated along the z axis through the center of the cage orifice as well as perpendicular to it along the x and y axes. The origin of the coordinate system was taken to be the position of the CH4 carbon at the energy minimum of the geometry-optimized structure obtained using the level of theory indicated above. The results of these rigid potential energy scans are shown in Figure 2. A polynomial fit was obtained to sixth order for translation in the axial direction and fourth order for translations in the radial directions. The fitted polynomial coefficients and their R2 values are given in the Supporting Information Table S1. The results of these polynomial fits indicate that the axial potential is asymmetric and perturbed by higher order even and odd terms while the radial potential is roughly symmetric about the origin and perturbed by a quartic term, with small linear and cubic corrections. It is also apparent that translation in the two orthogonal radial directions gives similar potentials, suggesting that it might be possible to treat the translational motion of CH4 inside the cage as being approximately cylindrically symmetric. Below, we assume that the cage potential is cylindrical and construct a Hamiltonian matrix which we diagonalize to obtain energy levels for translational and rotational motion. We then use the energy spectrum obtained from this procedure to predict thermodynamic properties of methane trapped in a fullerene. 2.2. Translational Basis Set. With the above results in mind, the translational motion of the methane inside the fullerene was modeled using cylindrical harmonic oscillator eigenfunctions as basis functions. In all of the following sections, the assumption is made that the endohedral dopant collides elastically with the walls of the fullerene; no energy is transferred from the methane to the cage. Performing a separation of variables on the Schro¨dinger equation yields a Hamiltonian of the form
Studies of CH4 Inside an Open-Cage Fullerene
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]
-p 1 ∂2R 1 ∂R 1 ∂2Z m2 + + + 2mCH4 R ∂r2 rR ∂r Z ∂z2 r2 1 1 m ω2r2 + mCH4ωz2z2 ) Er + Ez (1) 2 CH4 r 2
(
)
The Schro¨dinger equation separates into a radial part and an axial part. The axial part of the equation is a simple onedimensional harmonic oscillator, and the Hamiltonian is diagonal in the azimuthal quantum number m. The required matrix elements 〈V′|zk|V〉 for the axial part of the Hamiltonian are derived and tabulated in the Supporting Information Table S2. The energy levels and eigenfunctions for the radial part of the cylindrical harmonic oscillator are
Er ) pωr(2n + m + 1)
Rmn (r)
)
(
) ( 1/
2mCH4ωrn! pΓ(|m| + n + 1)
2
exp(-mCH4ωrr
mCH4ωr
2
p
(2)
) (
|m|
×
r
/2p)L|m| n
mCH4ωrr2 p
)
[
(
[
)
]
1 d |m| β2r2 Aˆ† ) r + +n+1 2 dr 2 2
(
)
]
(8)
where β ) (mCH4ω r/p). Since the radial potential is assumed to be cylindrically symmetric, it is an even function and therefore its Taylor expansion includes only even-order terms, so eq 8 gives a convenient method to find the matrix elements for all the nonzero terms of the Taylor expansion. 2.3. Rotational Basis Set. The rotation of methane inside 1 involves three degrees of freedom given by the Euler angles. The topology of the rotational potential has the form of a threedimensional surface of a four-dimensional sphere. The Wigner D matrices provide a complete, orthogonal set which covers this topology, and therefore, the potential can be expanded in terms of these functions.55 However, since the potential is purely real valued and the Wigner D-matrices are complex in general, it is convenient to express the rotational potential in terms of real-valued functions
V(R, Z, R, β, γ, q) )
j j (R, Z, q)∆µ′,µ (R, β, γ) ∑ Vµ′,µ
j,µ′,µ
(9)
(4)
]
[
(|m| + 2n + 1) - (Aˆ + Aˆ†) k , k ) 0, 1, 2, ... β2
(3)
where mCH4 is the mass of the oscillator, ωr is the harmonic frequency of the oscillator, and m and n are azimuthal and radial quantum numbers, respectively. Ln|m| denotes a Laguerre polynomial indexed by the aforementioned quantum numbers. Deviations from harmonicity in the radial portion of the Hamiltonian require matrix elements of the form 〈n′m′|rk|nm〉. These matrix elements can be obtained either through direct integration or by constructing ladder operators in a technique that has been modified from the work of Dong et al.54 Both methods are presented in full detail in the Supporting Information. The result of the ladder operator technique is given here. We construct the following operators
|m| 1 d β2r2 Aˆ ) - r + +n 2 dr 2 2
r2k )
where R, β, and γ are the Euler angles, R and Z are the translational degrees of freedom, and q denotes internal j is related to the (vibrational) degrees of freedom, and ∆µ′,µ Wigner D-matrices by a unitary transformation j j ∆µ′,µ ) U-1Dµ′,µ U
)
j j Dµ′,µ ∑ cµ′,µ
(10)
j,µ′,µ
The full mathematical details for this transformation are given in the Supporting Information. It is assumed that the vibrational motion of the methane molecule can be decoupled from its translational and rotational motion inside the cage. For small displacements, we assume that the axial and radial parts of the potential are separable and the coefficients νjµ′µ can be expressed as
(5) j j j V µ′,µ (R, Z) ) V µ′,µ (R) + V µ′,µ (Z)
(11)
which, upon application to the radial eigenfunctions, yield
AˆR|m| n
)
|m| aRn-1 ,
a ) √n(|m| + n)
(6)
and † |m| Aˆ†R|m| n ) a Rn+1,
a† ) √(n + 1)(|m| + n + 1)
We note that
Aˆ + Aˆ† ) (|m| + 2n + 1) - β2r2 (|m| + 2n + 1) - (Aˆ + Aˆ†) r2 ) β2 which gives our desired result
(7)
For higher energies, some coupling between axial and radial modes is expected to occur. This coupling was not evaluated explicitly in the present work; however, these terms can be obtained by direct integration of the matrix elements using the formulas supplied in the Supporting Information. Setting j (0,0) ) 0 and assuming that the results from the aforemenVµ′µ tioned electronic structure calculations allow the expansion of the coefficients a power series j j j Vµ′,µ (R) ) a2µ′,µ R2 + a4µ′,µ R4 + · · · j j j Vµ′,µ (Z) ) b1µ′,µ Z + b2µ′,µ Z2 + · · ·
(12)
The coupled translational-rotational Schro¨dinger equation now reads
ˆ Ψ(R, Z, Φ, R, β, γ) ) EΨ(R, Z, Φ, R, β, γ) H
(13)
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where
(
)
2 2 2 2 ˆ ) -p 1 ∂ R ∂ + 1 ∂ + ∂ + p L2 H 2 2 2 2mCH4 R ∂R ∂R 2I R ∂Φ ∂Z 1 j + Vj (R, Z)∆µ′,µ 2 j,µ′,µ µ′,µ (14)
( ) ∑
The variable I denotes the moment of inertia for methane, and L is the principal rotational quantum number, while K denotes projection onto the molecule-fixed z axis, and M is the projection onto the space-fixed z axis. With some algebraic manipulation, the full potential matrix elements can be expanded as
2〈L′K′M′n′m′V′|Vˆ |LKMnmV〉 ) 〈n′m′|R2 |nm〉
∑ j j b1µ′µ + 〈V′|Z|V〉 ∑ Xµ′µ j,µ′,µ j j + 〈V′|Z2 |V〉 ∑ Xµ′µ b2µ′µ + ···
+ 〈n′m′|R4 |nm〉
j j a2µ′µ ∑ Xµ′µ
j,µ′,µ
j j Xµ′µ a4µ′µ + ···
j,µ′,µ
j,µ′,µ
(15) where j j Xµ′µ ) 〈L′K′M′|∆µ′,µ |LKM〉
(16)
Because methane is a spherical top, its rotational eigenfunctions are given by Wigner D-matrices. The fact that the ∆jµ′,µ functions can be expanded in terms of D-matrices as well gives the j following result for Xµ′µ
(
)(
)
∫
2L + 1 1/2 2L′ + 1 1/2 L′ j L × dΩD-K′,-M′ *∆µ′,µ D-K,-M 8π2 8π2 2L + 1 1/2 2L′ + 1 1/2 L′ j j L ) dΩD-K′,-M′ *×( cµ′,µ Dµ′,µ )D-K,-M 8π2 8π2 j,µ′,µ 2L + 1 1/2 ) 2L′ + 1 j Xµ′µ )
(
×
∑
)(
)
∫
(
∑
)
j cµ′,µ δµ′-K,-K′δµ-M,-M′〈L, -K, j, µ′|L′, -K′〉〈L, -M, j, µ|L′, -M′〉
j,µ′,µ
(17) where the angled brackets in the last equality denote ClebschGordan coefficients. 2.4. Full Quantum Simulation. Following the method of Cross29 and Xu et al.,31,32 we use a four-parameter two-site Lennard-Jones 12-6 potential of the following form for the interaction between the endohedral CH4 molecule and the fullerene cage 60
V)
∑ i)1
{ [( ) ( ) ] 4εCC
σCC riC
12
4
σCC riC
∑ 4εCH
R)1
6
+
[( ) ( ) ]} σCH riR
12
-
σCH riR
6
(18)
where the Lennard-Jones parameters εCC, εCH, σCC, and σCH were obtained by modifying standard parameters from the interaction of methane with graphite.56 The modification pro-
parameter
simulation value
εCC/kB (K) εCH/kB (K) σCC (Å) σCH (Å)
47.68 17.00 3.188 2.598
cedure involved fitting the parameters to a model calculation of CH4@1 using the ONIOM method MP2/6-311++G(3df,3pd)// B3LYP/6-31G(d) described previously, such that the parameters provided an energy minimum for this geometry. This minimum energy point was used as the origin of the coordinate system in the simulation. The modified Lennard-Jones parameters are given in Table 2. The full simulation program operates by first calculating the Lennard-Jones potential for a particular orientation of methane at the center of the cage, then rotating around an Euler angle and recalculating the potential, and so forth until the entire rotation space has been covered. The potential map is decomposed into its D-matrix components using the sampling theorem of Kostelec and Rockmore,57 and the coefficient of each D-matrix is stored in memory. After this, the methane is translated by a small amount in the radial direction and the process of mapping and decomposition is repeated. Once the methane has been stepped in the radial direction a preset number of times, the coefficients for each D-matrix are fitted to a fourthorder polynomial. The coefficients for this polynomial are then j coefficients in eq 15. The same procedure identified as the anµ′µ is repeated, but the methane is stepped along the z axis and the D-matrix coefficients are fitted to a sixth-order polynomial to j coefficients in eq 15. Once these coefficients are give the bnµ′µ obtained, the full Hamiltonian matrix can be constructed and diagonalized to give eigenvalues and eigenvectors for the system. The Hamiltonian obtained is an N × N matrix, where the size of the matrix is determined by the maximum quantum numbers for rotation, Lmax, radial translation, nmax, and axial translation, Vmax N)
(2Lmax + 3)(2Lmax + 2)(2Lmax + 1) (nmax + 1)2(Vmax + 1) 6
The routine for calculating little d-matrices is based on the algorithm of Blanco et al.,58 and the routine for calculating Clebsch-Gordan coefficients is based on the work of Stevenson.59 3. Simulation Results The translational energy levels calculated for CH4@1 by the simulation can be compared with simple calculations for the energy levels of a cylindrical harmonic oscillator with the same harmonic frequencies. Likewise, the rotational energy levels calculated by the simulation can be compared with calculations for the energy levels of a simple spherical top. These comparisons are shown in Figure 3. The plots reveal several features about the motion of the guest inside the cage. First, Figure 3a shows that the axial motion of the methane is closely approximated by an anharmonic oscillator, and the lowest 8 energy levels follow a pattern that can be estimated by the function
EV 1 1 + ωexe V + ) ωe V + hc 2 2
(
)
(
2
)
(19)
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Figure 3. Simulation results for CH4@1 using a D-matrix sampling bandwidth of 8 with (a) translational levels V ) 0-7 for the axial mode of motion, (b) translational levels, including azimuthal (m) sublevels, n ) 0-4 for the radial mode of motion, (c) rotational levels L ) 0-4, including all sublevels (K, M quantum numbers), and (d) coupled translational-rotational energy levels, with V ) 0-3, n ) 0-3, and L ) 0-2, including all sublevels. The graph in d is cut off above the 1900th energy level because higher levels show anomalous energies due to basis set truncation error.
where ωe ) 63.55 ( 1.20 cm-1 and ωexe ) 3.65 ( 0.15 cm-1. Higher energy levels begin to show a divergence from this pattern. Second, the rotation of methane inside the cage is very nearly free, with the energy levels being well approximated by the function
EL ) BeL(L + 1) hc
(20)
where Be ) 5.2427 cm-1 is the rotational constant, which is only slightly larger than that of free methane, Be ) 5.24059 cm-1.60 The fact that the rotation of methane inside of an opencage fullerene can be treated as free is somewhat surprising, given that the cage potential is strongly anisotropic, and so it is reasonable to assume that there might be a barrier to rotation of methane induced by this anisotropy. However, small deviations from spherical top behavior do in fact appear in the results as a breaking of the degeneracy of the rotational sublevels by a few wavenumbers at higher rotational energies, as seen in Figure 3c. The splitting for L ) 1 is 0.024 cm-1, and the splitting for L ) 2 is 4.00 cm-1. A comparison of these results with the splittings of the rotational levels of methane inside clathrate hydrates reveals that the fullerene-induced splittings are much smaller for L ) 1 (0.024 cm-1 in the fullerene versus 0.3 and 0.6 cm-1 in the small and large clathrate hydrates, respectively) and L ) 2 (4.00 cm-1 in the fullerene versus 17.0 and 19.9 cm-1 in the clathrate cages). This is likely caused by the weaker Lennard-Jones interaction between CH4 and the fullerene than the interaction between methane and the water molecules making up the clathrate.33 The radial levels show significant deviation from harmonic behavior in Figure 3b. This deviation occurs because of the quartic term in the potential that was found with electronic
structure calculations. The radial energy levels can be estimated by a function
Enm ) ωe(|m| + 2n + 1) + ωexe(|m| + 2n + 1)2 hc where ωe ) 132.29 cm-1 and ωexe ) 11.47 cm-1. One might reasonably be surprised by the result that the radial modes exhibit more significant anharmonicity than the axial mode. It is clear that the axial mode is more anharmonic at excitation energies approaching the activation energy of the complex, since the reaction coordinate for dissociation of the complex is along this direction. A likely cause of this discrepancy is that the equilibrium position of the CH4 molecule in the cage is nearer to the walls of the cage in the radial directions than in the axial direction (3.51 Å in the radial direction versus 3.88 Å in the axial direction). Thus, slight displacements from equilibrium in the axial direction tend to appear more harmonic at lower excitation energies. The total translational-rotational simulation shows that the main deviation from the simple harmonic oscillator/spherical top picture comes from the quartic term in the radial potential, as seen in Figure 3d. The spacing between energy levels gets further apart as the energy of the system increases; thus, the system displays the same negative anharmonicity as was found by Xu et al. in H2@C60 and
[email protected],50 The mixing of translational and rotational modes also softens the abrupt jumps in the energy levels associated with the simple model, presumably by breaking the large degree of degeneracy associated with that model. 4. Statistical Mechanics and Thermodynamics With the energy levels in hand, it is trivial to calculate a partition function for methane inside Iwamatsu’s fullerene 1.
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For the incorporation reaction of CH4 into 1
CH4(g) + 1 h CH4@1
(21)
the equilibrium constant with respect to the behavior of the methane is a simple ratio of partition functions
Keq )
exp(D0 /kT)qCH4@1 qCH4(g)
(22)
where D0 is the incorporation energy minus the zero-point energy of translation of methane inside the fullerene. This energy is dependent on the Lennard-Jones parameters in Table 2. We can ignore the vibrational contribution to the partition function at low temperatures, and from the results presented in section 3, we can treat the rotation of methane as essentially free. Therefore, the equilibrium constant becomes a ratio of translational partition functions. Using the Lennard-Jones parameters from before and assuming experimental conditions of 190 °C and 136 atm CH4,43 we obtain an incorporation fraction of 43.8%, whereas the experimental incorporation fraction is typically between 25% and 35%.43 For perdeuterated methane, the theoretical incorporation fraction climbs to 52.0%. This increase is most likely caused by a lowering of the zero-point energy of the methane inside the fullerene cage. Although at first the theoretical incorporation fraction appears to be in conflict with the experimental data, it should be noted that the theoretical result is dependent on the exponential of D0, as seen in eq 22, and is therefore exquisitely sensitive to changes in the Lennard-Jones parameters. Adjusting the Lennard-Jones parameters can bring the experimental and theoretical values in closer agreement at the cost of losing agreement with the optimized geometry calculated from electronic structure theory, which may or may not be slightly inaccurate itself. Moreover, at the temperature of the experiment, the fullerene begins to decompose and there may be several unforeseen factors that affect the equilibrium incorporation of methane into the fullerene.43 A study of these potential effects was not attempted at present. The constant volume heat capacity CV is another measurable quantity that may be useful in providing information about the translational degrees of freedom in this system, since it is likely that these dynamics are virtually inaccessible to spectroscopy. For methane encapsulated in an open-cage fullerene, the rotational dynamics are essentially unaltered from free methane but the translational degrees of freedom are very different. The vibrational degrees of freedom, being only slightly altered from the free case, probably do not contribute much to the heat capacity. Therefore, the ensuing analyses focus on the translational dynamics of methane in the cage and their effect on the heat capacity. A fundamental result in statistical mechanics shows that the heat capacity is proportional to the variance of the internal energy of a system61
CV )
σE2 kT2
(23)
This formula allows for ready calculation of the heat capacity for the system, given the energy levels. In a first approximation to the potential exerted on methane by the fullerene cage, the
Figure 4. Plot of calculated heat capacity vs temperature. The black line denotes the heat capacity for the CH4 translational levels derived from the simulation, while the green line denotes the same quantity for CD4. The blue line is the heat capacity of a harmonic oscillator having the same fundamental frequency as the CH4 simulation, and the red line is the heat capacity of a particle with the mass of CH4 in a 1.0 Å cubic box.
translational motions of the guest were treated as a 3D harmonic oscillator. From the energy equipartition theorem, the hightemperature limit of the translational heat capacity is CV ) 3R. At ambient temperatures, it is clear from the calculated harmonic frequencies that the high-temperature regime has not been achieved and, therefore, the heat capacity must be calculated from the variance directly. Furthermore, analysis of the energies shows considerable deviation from harmonic behavior, so the heat capacity should not be expected to even approach the limit of 3R even at high temperatures. Calculation of the heat capacity over the temperature range from 1 to 500 K using the first 1000 translational (radial and axial) energy levels yields an interesting trend, as shown in Figure 4. As is evident on the graph in Figure 4, the heat capacity for CH4 increases from zero to a maximum at 239 K and then begins slowly decreasing. The heat capacity for perdeuterated methane, CD4, is plotted for comparison, giving a maximum at 215 K. This phenomenon is unusual in that the heat capacities of almost all gases rise quickly from zero and asymptotically approach a maximum value monotonically from below. The behavior calculated above for encapsulated methane can be explained by looking at a similar plot of heat capacity for methane in a 1 Å cubic box. As is shown in the figure, similar anomalous heat capacity behavior is present for the case of a particle in a box potential, where the heat capacity rises quickly to a maximum and begins decreasing asymptotically to the classical hightemperature limit given by the energy equipartition theorem (3R/2 in this case).62 The standard calculation for the translational part of the partition function involves assuming a particle in a box with macroscopic dimensions and rewriting the sum as an integral. However, as is clear from above, if the dimensions of the box are small enough, this approximation fails. For the case of encapsulated methane, the radial modes in particular are dominated by an r4 term, which at higher energies begins to resemble a particle in a box. The cause of the nonmonotonicity in heat capacity as the system approaches the behavior of a particle in a box is that the energy levels are not evenly spaced, as they are in a harmonic oscillator. As the quantum number increases, the energy levels get further apart, which leads
Studies of CH4 Inside an Open-Cage Fullerene to the calculated anomaly in the heat capacity. This phenomenon is similar to the anomalous heat capacity observed in some glasses, which is generated by low-frequency anharmonic modes arising from the motion of weakly bound noncrystalline clusters in the glass.63 5. Conclusion The translational and rotational motions of CH4 inside an open-cage fullerene were simulated using a simple fourparameter potential in a generalization of a method used previously to calculate the energy structure of diatomic molecules encapsulated in C60. It was found that even though the open-cage fullerene provides a highly anisotropic environment, the rotational motion of the endohedral methane is approximately free. The translational motion of methane inside the cage is strongly anharmonic, especially in the case of motion which is perpendicular to the cage orifice. The partition function for a methane molecule inside the fullerene can be calculated easily from the energy levels found in the simulation, allowing theoretical determination of thermodynamic properties. The equilibrium constant for incorporation of methane into the fullerene was found in this manner, using the Lennard-Jones parameters which yielded a geometry consistent with that found in high-level electronic structure calculations. These parameters gave a calculated equilibrium incorporation fraction of 43.8% CH4 in 1, which compares favorably with the observed value of 25-35% incorporation. The energy levels obtained by the simulation were further used to find the constant volume heat capacity of the encapsulated methane over a range of temperatures. The heat capacity curve showed an anomalous maximum at 239 K for CH4 and 215 K for CD4, a feature likely attributable to the significant anharmonicity of the translational motions of the methane inside the fullerene. It should be noted that similar heat capacity curves are obtained experimentally for various glasses,63 and theoretically for such systems as a particle in a box,62 free rotor,64 and two-level system.65 This effect should be observable in a carefully controlled calorimetric experiment as the difference in heat capacity between empty and filled fullerenes. Acknowledgment. The author would like to acknowledge Prof. R. James Cross for his valuable insight into the theory of molecules inside fullerenes. Supporting Information Available: Full derivations for all major results presented in section 2; table with polynomial fit of data shown in Figure 2; table with matrix elements for axial degree of freedom. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Heath, J. R.; O’Brien, S. C.; Zhang, Q.; Liu, Y.; Curl, R. F.; Tittel, F. K.; Smalley, R. E. J. Am. Chem. Soc. 1985, 107, 7779–7780. (2) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162–163. (3) Johnson, R. D.; de Vries, M. S.; Salem, J.; Bethune, D. S.; Yannoni, C. S. Nature 1992, 355, 239–240. (4) McElvany, S. W. J. Phys. Chem. 1992, 96, 4935–4937. (5) Yannoni, C. S.; Hoinkis, M.; de Vries, M. S.; Bethune, D. S.; Salem, J. R.; Crowder, M. S.; Johnson, R. D. Science 1992, 256, 1191–1192. (6) Guo, T.; Odom, G. K.; Scuseria, G. E. J. Phys. Chem. 1994, 98, 7745–7747. (7) Cagle, D. W.; Thrash, T. P.; Alford, M.; Chibante, L. P. F.; Ehrhardt, G. J.; Wilson, L. J. J. Am. Chem. Soc. 1996, 118, 8043–8047. (8) Ohtsuki, T.; Masumoto, K.; Ohno, K.; Maruyma, Y.; Kawazoe, Y.; Sueki, K.; Kikuchi, K. Phys. ReV. Lett. 1996, 77, 3522. (9) Ge, Z.; Duchamp, J. C.; Cai, T.; Gibson, H. W.; Dorn, H. C. J. Am. Chem. Soc. 2005, 127, 16292–16298.
J. Phys. Chem. A, Vol. 114, No. 45, 2010 12081 (10) Stevenson, S.; Rice, G.; Glass, T.; Harich, K.; Cromer, F.; Jordan, M. R.; Craft, J.; Hadju, E.; Bible, R.; Olmstead, M. M.; Maitra, K.; Fisher, A. J.; Balch, A. L.; Dorn, H. C. Nature 1999, 401, 55–57. (11) Stevenson, S.; Fowler, P. W.; Heine, T.; Duchamp, J. C.; Rice, G.; Glass, T.; Harich, K.; Hajdu, E.; Bible, R.; Dorn, H. C. Nature 2000, 408, 427–428. (12) Olmstead, M. M.; Bettencourt-Dias, A. d.; Duchamp, J. C.; Stevenson, S.; Marciu, D.; Dorn, H. C.; Balch, A. L. Angew. Chem., Int. Ed. 2001, 40, 1223–1225. (13) Jime´nez-Va´zquez, H. A.; Cross, R. J.; Saunders, M.; Poreda, R. J. Chem. Phys. Lett. 1994, 229, 111–114. (14) Knapp, C.; Dinse, K. -.; Pietzak, B.; Waiblinger, M.; Weidinger, A. Chem. Phys. Lett. 1997, 272, 433–437. (15) Pietzak, B.; Waiblinger, M.; Murphy, T. A.; Weidinger, A.; Ho¨hne, M.; Dietel, E.; Hirsch, A. Carbon 1998, 36, 613–615. (16) Weidinger, A.; Pietzak, B.; Waiblinger, M.; Lips, K.; Nuber, B.; Hirsch, A. The 12th International Winterschool on Electronic Properties of Novel Materials: Progress in Molecular Nanostructures. AIP Conference Proceedings, Volume 442, pp 363-367, 1998. (17) Weidinger, A.; Waiblinger, M.; Pietzak, B.; Almeida Murphy, T. Appl. Phys. A: Mater. Sci. Process. 1998, 66, 287–292. (18) Dietel, E.; Hirsch, A.; Pietzak, B.; Waiblinger, M.; Lips, K.; Weidinger, A.; Gruss, A.; Dinse, K. J. Am. Chem. Soc. 1999, 121, 2432– 2437. (19) Lips, K.; Waiblinger, M.; Pietzak, B.; Weidinger, A. Phys. Stat. Solidi (a) 2000, 177, 81–91. (20) Weiske, T.; Bo¨hme, D. K.; Hruscarona´k, J.; Kra¨tschmer, W.; Schwarz, H. Angew. Chem., Int. Ed. Engl. 1991, 30, 884–886. (21) Saunders, M.; Jimenez-Vazquez, H. A.; Cross, R. J.; Poreda, R. J. Science 1993, 259, 1428–1430. (22) Buehl, M.; Thiel, W.; Jiao, H.; Schleyer, P. v. R.; Saunders, M.; Anet, F. A. L. J. Am. Chem. Soc. 1994, 116, 6005–6006. (23) Becker, L.; Poreda, R. J.; Bada, J. L. Science 1996, 272, 249–252. (24) DiCamillo, B. A.; Hettich, R. L.; Guiochon, G.; Compton, R. N.; Saunders, M.; Jimenez-Vazquez, H.; Khong, A.; Cross, R. J. J. Phys. Chem. 1996, 100, 9197–9201. (25) Saunders, M.; Cross, R. J.; Jimenez-Vazquez, H. A.; Shimshi, R.; Khong, A. Science 1996, 271, 1693–1697. (26) Yamamoto, K.; Saunders, M.; Khong, A.; Cross, R. James; Grayson, M.; Gross, M. L.; Benedetto, A. F.; Weisman, R. B. J. Am. Chem. Soc. 1999, 121, 1591–1596. (27) Syamala, M. S.; Cross, R. J.; Saunders, M. J. Am. Chem. Soc. 2002, 124, 6216–6219. (28) Murata, M.; Murata, Y.; Komatsu, K. J. Am. Chem. Soc. 2006, 128, 8024–8033. (29) Cross, R. J. J. Phys. Chem. A 2008, 112, 7152–7156. (30) Mamone, S.; Ge, M.; Huvonen, D.; Nagel, U.; Danquigny, A.; Cuda, F.; Grossel, M. C.; Murata, Y.; Komatsu, K.; Levitt, M. H.; Room, T.; Carravetta, M. J. Chem. Phys. 2009, 130, 081103. (31) Xu, M.; Sebastianelli, F.; Bacˇic´, Z.; Lawler, R.; Turro, N. J. J. Chem. Phys. 2008, 129, 064313. (32) Xu, M.; Sebastianelli, F.; Bacˇic´, Z.; Lawler, R.; Turro, N. J. J. Chem. Phys. 2008, 128, 011101. (33) Matanovic, I.; Xu, M.; Moskowitz, J. W.; Eckert, J.; Bacˇic´, Z. J. Chem. Phys. 2009, 131, 224308. (34) Hummelen, J. C.; Prato, M.; Wudl, F. J. Am. Chem. Soc. 1995, 117, 7003–7004. (35) Rubin, Y.; Jarrosson, T.; Wang, G.; Bartberger, M. D.; Houk, K. N.; Schick, G.; Saunders, M.; Cross, R. J. Angew. Chem., Int. Ed. 2001, 40, 1543–1546. (36) Murata, Y.; Murata, M.; Komatsu, K. J. Am. Chem. Soc. 2003, 125, 7152–7153. (37) Iwamatsu, S.; Uozaki, T.; Kobayashi, K.; Re, S.; Nagase, S.; Murata, S. J. Am. Chem. Soc. 2004, 126, 2668–2669. (38) Iwamatsu, S.; Murata, S.; Andoh, Y.; Minoura, M.; Kobayashi, K.; Mizorogi, N.; Nagase, S. J. Org. Chem. 2005, 70, 4820–4825. (39) Komatsu, K.; Murata, M.; Murata, Y. Science 2005, 307, 238– 240. (40) Stanisky, C. M.; Cross, R. J.; Saunders, M.; Murata, M.; Murata, Y.; Komatsu, K. J. Am. Chem. Soc. 2005, 127, 299–302. (41) Whitener, K. E.; Frunzi, M.; Iwamatsu, S.; Murata, S.; Cross, R. J.; Saunders, M. J. Am. Chem. Soc. 2008, 130, 13996–13999. (42) Stanisky, C. M.; Cross, R. J.; Saunders, M. J. Am. Chem. Soc. 2009, 131, 3392–3395. (43) Whitener, K. E.; Cross, R. J.; Saunders, M.; Iwamatsu, S.; Murata, S.; Mizorogi, N.; Nagase, S. J. Am. Chem. Soc. 2009, 131, 6338–6339. (44) Iwamatsu, S.; Murata, S. Tetrahedron Lett. 2004, 45, 6391–6394. (45) Bu¨hl, M.; Thiel, W. Chem. Phys. Lett. 1995, 233, 585–589. (46) Son, M.; Kiel Sung, Y. Chem. Phys. Lett. 1995, 245, 113–118. (47) Cross, R. J. J. Phys. Chem. A 2001, 105, 6943–6944. (48) Sears, D. N.; Jameson, C. J. J. Chem. Phys. 2003, 118, 9987–9989.
12082
J. Phys. Chem. A, Vol. 114, No. 45, 2010
(49) Horsewill, A. J.; Panesar, K. S.; Rols, S.; Johnson, M. R.; Murata, Y.; Komatsu, K.; Mamone, S.; Danquigny, A.; Cuda, F.; Maltsev, S.; Grossel, M. C.; Carravetta, M.; Levitt, M. H. Phys. ReV. Lett. 2009, 102, 013001. (50) Xu, M.; Sebastianelli, F.; Gibbons, B. R.; Bacˇic´, Z.; Lawler, R.; Turro, N. J. J. Chem. Phys. 2009, 130, 224306. (51) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al.Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; Gaussian, Inc.: Wallingford, CT, 2004. (52) Svensson, M.; Humbel, S.; Froese, R. D. J.; Matsubara, T.; Sieber, S.; Morokuma, K. J. Phys. Chem. 1996, 100, 19357–19363.
Whitener (53) Stanisky, C. M. Ph.D. Thesis, Yale University, New Haven, CT, 2008. (54) Dong, S.; Sun, G.; Lozada-Cassou, M. Phys. Lett. A 2004, 328, 299–305. (55) Biedenharn, L. C.; Louck, J. D. Angular Momentum in Quantum Physics: Theory and Application; Addison-Wesley: Reading, MA, 1981; Vol. 8. (56) Phillips, J. M.; Hammerbacher, M. D. Phys. ReV. B 1984, 29, 5859. (57) Kostelec, P.; Rockmore, D. J. Fourier Anal. Appl. 2008, 14, 145– 179. (58) Blanco, M. A.; Flo´rez, M.; Bermejo, M. J. Mol. Struct.: THEOCHEM 1997, 419, 19–27. (59) Stevenson, P. D. Comput. Phys. Commun. 2002, 147, 853–858. (60) Susskind, J. J. Mol. Spectrosc. 1973, 45, 457–466. (61) McQuarrie, D. A. Statistical Mechanics; University Science Books: Sausalito, CA, 2000. (62) Rosenstock, H. B. Am. J. Phys. 1962, 30, 38–40. (63) Rosenstock, H. B. J. Non Cryst. Solids 1972, 7, 123–126. (64) Mayer, J. E.; Mayer, M. G. Statistical Mechanics; John Wiley & Sons, Inc.: New York, 1940. (65) Mendelssohn, K. Cryophysics; Interscience Publishers, Inc.: New York, 1960.
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