24806
2006, 110, 24806-24811 Published on Web 11/15/2006
Hydrogen Molecule in the Small Dodecahedral Cage of a Clathrate Hydrate: Quantum Five-Dimensional Calculations of the Coupled Translation-Rotation Eigenstates Minzhong Xu, Yael S. Elmatad, Francesco Sebastianelli, Jules W. Moskowitz, and Zlatko Bacˇ ic´ * Department of Chemistry, New York UniVersity, New York, New York 10003 ReceiVed: September 30, 2006; In Final Form: October 31, 2006
We report quantum five-dimensional (5D) calculations of the energy levels and wave functions of the hydrogen molecule, para-H2 and ortho-H2, confined inside the small dodecahedral (H2O)20 cage of the sII clathrate hydrate. All three translational and the two rotational degrees of freedom of H2 are included explicitly, as fully coupled, while the cage is treated as rigid. The 5D potential energy surface (PES) of the H2-cage system is pairwise additive, based on the high-quality ab initio 5D (rigid monomer) PES for the H2-H2O complex. The bound state calculations involve no dynamical approximations and provide an accurate picture of the quantum 5D translation-rotation dynamics of H2 inside the cage. The energy levels are assigned with translational (Cartesian) and rotational quantum numbers, based on calculated root-mean-square displacements and probability density plots. The translational modes exhibit negative anharmonicity. It is found that j is a good rotational quantum number, while the threefold degeneracy of the j ) 1 level is lifted completely. There is considerable translation-rotation coupling, particularly for excited translational states.
I. Introduction Clathrate hydrates are a large group of inclusion compounds which incorporate guest molecules inside the polyhedral cages of the host framework made up of hydrogen-bonded water molecules.1 Recently, clathrate hydrates have been synthesized with hydrogen molecules as guests, at high pressures of ∼200 MPa.2 They have the classical structure II (sII), whose unit cell with 136 H2O molecules is composed of 16 pentagonal dodecahedron (512) small cages and 8 hexakaidodecahedron 51264 large cages.2 The small and the large cage consist of 20 and 28 H2O molecules, respectively. This work has generated much interest and stimulated numerous further studies of both pure H23 and binary clathrate hydrates,4-7 largely because of the potential application to hydrogen storage.8,9 In addition, clathrate hydrates of hydrogen provide an excellent opportunity for investigating the intriguing quantum dynamics of H2 molecules and small clusters in confined geometries. Initially, it was reported that two H2 molecules occupy the small 512 cage, while four H2 molecules occupy the large 51264 cage.2 A subsequent neutron diffraction study of the pure sII hydrogen hydrate3 found only one D2 molecule in the small cage and up to four D2 molecules in the large cage. Single D2 occupancy of the small cage was established by neutron diffraction also for the binary sII clathrate hydrate with tetrahydrofuran (THF) as the second guest.7 Several theoretical treatments of the pure H210-12 and the binary H2-THF hydrate13 have been reported. Their main focus was on the thermodynamic stability of the clathrates with different H2 occupancies, employing a variety of approaches: ab initio electronic structure calculations, alone10 or combined with a statistical mechanical model,11 classical molecular* Corresponding author. E-mail:
[email protected] 10.1021/jp066437w CCC: $33.50
dynamics simulations,12,13 and lattice dynamics calculations14 using empirical force fields. The dynamics of the coupled translational and rotational motions of a hydrogen molecule inside the clathrate cage is highly quantum mechanical. Therefore, its quantitative description can be achieved only by calculating accurately the multidimensional translation-rotation eigenstates of the encapsulated H2. This has not been done in the theoretical studies of the hydrogen clathrates to date. Quantum dynamics of a hydrogen molecule in confined geometries has so far been investigated for para- and ortho-H2 on amorphous ice surfaces using quantum Monte Carlo simulations,15 and for H2 within (rigid) carbon nanotubes by means of quantum 4D calculations.16-18 Calculation of the rotational Raman spectrum of H2 in ice has also been reported,19 with a quantal treatment of the H2 translational motion. This situation has motivated us to initiate a comprehensive theoretical investigation of the quantum dynamics of one or more hydrogen molecules inside the clathrate hydrate cages. In this paper, we present the results of the first stage of the program, the quantum 5D bound state calculations of the energy levels of a single hydrogen molecule confined within the small dodecahedral (512) cage (subsequently referred to as “small”). The three translational and the two rotational degrees of freedom of (rigid) H2 are treated explicitly, while the small cage is treated as rigid. The quantum mechanical energy levels of this 5D system are calculated exactly for the H2-cage potential energy surface employed, taking fully into account all the mode couplings and anharmonicities, by diagonalizing the 5D translation-rotation Hamiltonian. The eigenstates are assigned with the Cartesian (or “translational”) and rotational quantum numbers. Our results elucidate for the first time the quantum dynamics of the coupled translational and © 2006 American Chemical Society
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J. Phys. Chem. B, Vol. 110, No. 49, 2006 24807
rotational motions of H2 molecule within the small cage of the sII clathrate hydrate. II. Theory In this work, the small dodecahedral clathrate cage is treated as rigid, as was done also by Patchkovskii and Tse11 in their study of the thermodynamics stability of hydrogen clathrates. The quantum translation-rotation eigenstates of the (rigid) hydrogen molecule inside the rigid cage are calculated rigorously, as fully coupled. We believe this 5D model treatment captures the salient features of the quantum translation-rotation dynamics of the encapsulated hydrogen molecule for the following reasons: (i) The H2 molecule is much lighter than the (H2O)20 cage. (ii) The interaction between the hydrogen molecule and the cage is much weaker than the hydrogen bonds between the H2O molecules of the cage. (iii) Every water molecule of the cage is hydrogen-bonded to three neighboring molecules, creating a fairly rigid framework. (iv) In the clathrate hydrate, the cage is a part of an extended lattice, which further enhances its rigidity. The geometry of the small cage used in the quantum 5D calculations is shown in Figure 1. The water molecules forming the cage have the geometry employed in the calculations of 5D (rigid-monomer) PES for the H2O-H2 complex,20 used by us to construct the H2-cage PES described below; the O-H bond length is 1.836 au and the H-O-H angle is 104.69°. The oxygen atoms were arranged following Patchkovskii and Tse: 11 eight oxygen atoms occupy the corners of a perfect cube (O ), h at the distance of 7.24 au from the cage center, while the remaining twelve oxygen atoms, 7.48 au from the center of the cage, are arranged so as to transform according to the Th subgroup of Oh. This configuration of the O atoms corresponds closely to that from the X-ray diffraction experiments for the sII clathrate hydrates.21 However, the H atoms of the water molecules are configurationally disordered. There is one H atom on each edge of the cage, forming a hydrogen bond between two O atoms at the corners connected by the edge. Since the dodecahedron has thirty edges, 10 water molecules must be double donors, with both their H atoms participating in the hydrogen bonds with two neighboring O atoms. The other 10 water molecules are single donors, having only one H atom in a hydrogen bond, while the second O-H bond of the molecule is free, nonbonded, dangling from the surface of the cage. For a dodecahedral (H2O)20 cage, there are over 30 000 possible hydrogen-bonding topologies.22 The one shown in Figure 1 was constructed by aiming to distribute the nonbonded O-H bonds rather evenly over the cage exterior. We believe that the quantum dynamics of the hydrogen molecule does not depend strongly on the particular topology of the hydrogen bonding of the cage. To test this assumption, the bound-state calculations were repeated for another structure, with a pattern of hydrogen bonds different from that in Figure 1; the results obtained for these two structures were very similar. For the description of the translation-rotation (T-R) dynamics of a rigid H2 molecule inside the cage, a set of five coordinates (x, y, z, θ, φ) was chosen; x, y and z are the Cartesian coordinates of the center of mass (c.m.) of the hydrogen molecule, while the two polar angles θ and φ specify its orientation. The origin of the coordinate system is at the c.m. of the cage, and its axes, shown in Figure 1, are aligned with the principal axes of the cage. The (H2O)20 cage can be safely assumed to be infinitely heavy (in comparison to H2) and nonrotating. For this case, the 5D Hamiltonian for the T-R motions of the confined H2 molecule is
Figure 1. Geometry of the small dodecahedral cage (top). The Cartesian X,Y,Z coordinate axes coincide with the three principal axes of the cage. The distances shown are in bohrs. Also shown (bottom) are three 3D isosurfaces, at -800, 0, and 600 cm-1, respectively, for the H2-cage PES, obtained by minimizing the H2-cage interaction with respect to the two angular coordinates of the H2 molecule, at every position of its center of mass.
H)-
(
)
p 2 ∂2 ∂2 ∂2 + + + Bj2 + V(x, y, z, θ, φ) 2m ∂x2 ∂y2 ∂z2 j2
(1)
In eq 1, m is the mass of the H2 molecule, while B and are the rotational constant and the angular momentum operator of the diatomic, respectively. V(x, y, z, θ, φ) in eq 1 is the 5D PES described in the following paragraph. The energy levels and wave functions of the 5D Hamiltonian in eq 1 were calculated exactly for the PES employed using the computational methodology which we developed for the 5D intermolecular vibrational eigenstates of Arn HF clusters,23-26 where the Arn subunit was treated as rigid. Detailed description of the method can be found in ref 24. In brief, this methodology relies on the 3D direct-product discrete variable representation (DVR)27,28
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TABLE 1: Lowest 10 Excited Translation-Rotation Energy Levels of a p-H2 Molecule in the Small Dodecahedral Cavity of the Clathrate Hydrate, from the Quantum 5D Bound State Calculationsa
TABLE 2: Lowest 13 Excited Translation-Rotation Energy Levels of an o-H2 Molecule in the Small Dodecahedral Cavity of the Clathrate Hydrate, from the Quantum 5D Bound State Calculationsa
n
E
∆E
∆x
∆y
∆z
(Vx, Vy, Vz)
n
E
∆E
0 1 2 3 4 5 6 7 8 9 10
-707.600
0.00 52.35 66.81 77.74 124.51 130.98 141.11 149.97 159.06 177.21 205.70
0.75 1.15 0.70 0.69 1.24 1.14 1.07 0.66 0.68 0.68 1.30
0.66 0.63 1.04 0.64 0.79 0.87 0.63 1.15 0.97 0.71 0.95
0.62 0.58 0.61 0.99 0.58 0.61 0.93 0.77 0.92 1.08 0.56
(0,0,0) (1,0,0) (0,1,0) (0,0,1) (2,0,0) (1,1,0) (1,0,1) (0,2,0) (0,1,1) (0,0,2) (2,1,0)
0 1 2 3 4 5 6 7 8 9 10 11 12 13
-616.306
0.00 26.91 51.49 60.57 65.72 77.63 82.23 96.51 105.98 112.87 123.28 127.91 132.00 138.94
They all correspond to the j ) 0 state of p-H2. The excitation energies ∆E are relative to the lowest even-j level designated n ) 0, whose energy E is listed. All energies are in cm-1. Also shown are the root-mean-square (rms) amplitudes ∆x, ∆y, and ∆z (in au), and the Cartesian quantum number assignments (νx,νy,νz). a
for the x, y, and z coordinates and the spherical harmonics for the angular, θ and φ coordinates. The size of the final Hamiltonian matrix is reduced drastically by means of the sequential diagonalization and truncation procedure developed by Bacˇic´ and Light,27,29,30 without any loss of accuracy. Diagonalization of this truncated Hamiltonian matrix yields the desired 5D T-R energy levels and wave functions. The 5D PES for the H2-cage system was constructed by summing over the interactions between the H2 molecule and each of the 20 water molecules of the cage. For the pair interaction between H2 and H2O we use the ab initio 5D (rigid monomer) PES for the H2-H2O complex by Hodges and coworkers,20 whose global minimum is at -240.8 cm-1. In the quantum 5D bound-state calculations, the rotational constant for H2 in eq 1 has the value B ) 59.322 cm-1.31,32 The dimension of the sine-DVR basis was 50 for each of the three Cartesian coordinates x, y, and z. The sine-DVR grid spanned the range -2.65 au e λ e 2.65 au (λ ) x, y, z). The angular basis included functions up to jmax ) 5. The energy cutoff parameter for the intermediate 3D eigenvector basis24 was set to 580 cm-1, resulting in the final full 5D Hamiltonian matrix of dimension 2800. III. Results and Discussion Molecular hydrogen exists in two quite distinct species, parahydrogen (p-H2) with the total nuclear spin I ) 0 and only even rotational angular momentum j allowed (j ) 0, 2, 4, ...), and ortho-hydrogen (o-H2), which has the nuclear spin I ) 1 and odd rotational angular momentum only (j ) 1, 3, ...). The lowerlying quantum 5D translation-rotation (T-R) energy levels of p-H2 and o-H2 in the dodecahedral cage are given in Tables 1 and 2, respectively, together with their Cartesian (translational) quantum number assignments (Vx, Vy, Vz). Table 1 also displays for each state the root-mean-square (rms) amplitudes ∆x, ∆y, and ∆z; the rms amplitudes are not shown in Table 2, since they are very similar for the T-R states of p-H2 and o-H2 which have the same Cartesian quantum numbers (Vx, Vy, Vz). For all the T-R eigenstates, the expectation values of the Cartesian coordinates 〈x〉, 〈y〉, and 〈z〉 are equal to 0, corresponding to the c.m. of the cage. The Cartesian quantum numbers could in most cases be assigned by inspecting the rms amplitudes. Table 1 shows clearly that the fundamental excitation of any one of the translational modes along x, y, or z markedly increases its rms amplitude,
[0.00] (0.00) [50.72] [69.61] [79.07] (52.30) (67.34) (78.38)
(Vx, Vy, Vz)
(j,|m|)
(0,0,0) (0,0,0) (1,0,0) (0,0,0) (0,1,0) (1,0,0) (0,0,1) (0,1,0) (0,0,1) (1,0,0) (2,0,0) (0,1,0) (1,1,0) (0,0,1)
(1,0) (1,1)l (1,0) (1,1)u (1,0) (1,1)l (1,0) (1,1)l (1,1)l (1,1)u (1,0) (1,1)u (1,0) (1,1)u
a
The excitation energies ∆E are relative to the lowest odd-j level designated n ) 0, whose energy E is listed. Also shown are the Cartesian (νx,νy,νz) and the rotational (j,|m|) quantum number assignments. The energies in [ ] brackets are relative to the (1,1)l state (0,0,0) [n ) 1], while the energies in ( ) brackets are relative to the (1,1)u state (0,0,0) [n ) 3]. They are the translational fundamentals for (1,1)l and (1,1)u, respectively. All energies are in cm-1.
by ∼0.4 bohr, relative to that in the ground state (0,0,0). Interestingly, adding one more quantum of excitation to the x, y, or z mode increases its rms amplitude by only ∼0.1 bohr. This undoubtedly reflects the restricted space available for the translational motions of the H2. Figure 1 (bottom) shows that at the energies of interest, below -500 cm-1, the cage walls are only ∼2 au away from the center of the cage. In order to confirm the (Vx, Vy, Vz) assignments, and to help visualize the 5D T-R wave functions, we calculate the 3D reduced probability density
Fn(x,y,z) )
∫ψ/n (x, y, z, θ, φ)ψn(x, y, z, θ, φ) sinθ dθ dφ
(2)
where ψn(x, y, z, θ, φ) is the nth T-R eigenstate of the encapsulated p-H2 or o-H2. Figures 2 and 3 display Fn(x, y, z) for a few selected states discussed below. By projecting an eigenstate on the rotational basis, we can determine the contribution of each rotational basis function. In this way, we established that j is a good quantum number for all of the T-R eigenstates considered, to a very high degree. The T-R states of p-H2 in Table 1 are almost pure (98-99%) j ) 0 states. Likewise, the o-H2 states shown in Table 2 are pure (> 99%) j ) 1 states. As discussed below, the threefold degeneracy of the j ) 1 states is completely lifted in the cage. Therefore, the states in Table 2 are additionally assigned with the rotational quantum numbers (j, |m|). These assignments are made with the help of another reduced probability density, now in the angular coordinates:
Fn(θ, φ) )
∫ψ/n (x, y, z, θ, φ)ψn(x, y, z, θ, φ) dx dy dz
(3)
To distinguish the two states having |m| ) 1 and the same Cartesian quantum numbers, the one with the lower energy is labeled in Table 2 as (1,1)l and the higher-energy state as (1,1)u. Figure 4 shows Fn(θ, φ) for (0,0), (1,0), (1,1)l and (1,1)u states; clearly the (j, |m|) assignment is readily made by inspecting these contour plots. The global minimum of the H2-cage PES is at -884.8 cm-1. Since the lowest energy level of p-H2 lies at -707.60 cm-1,
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J. Phys. Chem. B, Vol. 110, No. 49, 2006 24809
Figure 2. 3D isosurfaces of the reduced probability density in the Cartesian coordinates, defined by eq 2, of the p-H2 states (a) n ) 1, (1,0,0); (b) n ) 2, (0,1,0); (c) n ) 3, (0,0,1). They are listed in Table 1. The isosurfaces are drawn at 20% of the maximum value of the density.
Figure 3. 3D isosurfaces of the reduced probability density in the Cartesian coordinates, defined by eq 2, of the p-H2 states (a) n ) 4, (2,0,0); (b) n ) 7, (0,2,0); (c) n ) 9, (0,0,2). They are listed in Table 1. The isosurface are drawn at 10% of the maximum value of the density.
the zero-point energy is 177.2 cm-1, or 20% of the well depth. We now discuss the nature of the translational motions of the confined H2. For this, we focus on the j ) 0 states of p-H2 in Table 1. The frequencies of the x-mode fundamental (1,0,0), the y-mode fundamental (0,1,0), and the z-mode fundamental (0,0,1) are 52.4, 66.8, and 77.7 cm-1, respectively; the frequency spread of the three fundamentals is due primarily to the non-symmetric arrangement of the H atoms of the cage. The translational modes display negatiVe anharmonicity, since the frequencies of the first-overtone states (2,0,0), (0,2,0), and (0,0,2) are more than twice the frequencies of the corresponding fundamentals. The nodal patterns of the reduced probability densities Fn(x, y, z) of the three translational fundamentals shown
in Figure 2 are regular. For the states (2,0,0), (0,2,0), and (0,0,2), Fn(x, y, z) displayed in Figure 3 are more distorted and tilted with respect to the Cartesian axes, evidence of the stronger mode coupling at higher excitation energies; still, two nodal planes are clearly visible in each isosurface plot. For free o-H2 molecule, the j ) 1 rotational level is threefold degenerate. However, Table 2 shows that the confinement to the small cage lifts this degeneracy completely, due the anisotropy of the environment. For the ground translational state (0,0,0), the states (1,1)l (n ) 1) and (1,1)u (n ) 3) are 26.91 and 60.57 cm-1, respectively, above the (1,0) state (n ) 0). As a result of this lifting of the degeneracy, the energy difference between the lowest levels of p-H2 and o-H2, -707.60 and
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Figure 4. Contour plots of the reduced probability density in the angular coordinates, defined by eq 3, of four states with the following rotational (j,|m|) quantum numbers: (a) (0,0), n ) 0 state in Table 1; (b) (1,0), n ) 0 state in Table 2; (c) (1,1)l, n ) 1 state in Table 2; (d) (1,1)u, n ) 3 state in Table 2. The contour levels are relative to the maximum value of the density.
-616.31 cm-1, respectively, is decreased to 91.29 cm-1, from 2B ) 118.64 cm-1 for the freely rotating H2. Breaking of the degeneracy of the j ) 1 levels was observed also for H2 inside single-walled carbon nanotubes.16,17 For (0,0,0), the calculated splitting of the j ) 1 triplet of states is reproduced very well by the following simple model, which considers only the angular (θ, φ) degrees of freedom of the problem. The three rotational wave functions are φ(1,0) ) 0.5(3/π)1/2 cos θ, φ(1,1)l ) 0.5(3/π)1/2 sin θ sin φ, and φ(1,1)u ) 0.5(3/π)1/2 sin θ cos φ. The H2 molecule is placed at the center of the cage, and the angular part of the H2-cage PES is averaged over each of the three rotational wave functions:
〈V〉j,|m| ) 〈φj,|m||V(0, 0, 0, θ, φ)|φj,|m|〉
(4)
We obtained 〈V〉(1,0) ) -797.29 cm-1, 〈V〉(1,1)l ) -771.82 cm-1, and 〈V〉(1,1)u ) -737.35 cm-1. The differences 〈V〉(1,1)l 〈V〉(1,0) ) 25.47 cm-1, and 〈V〉(1,1)u - 〈V〉(1,0) ) 59.94 cm-1, match closely the energies of the states (1,1)l and (1,1)u relative to the (1,0) state, 26.91 and 60.57 cm-1, respectively. This implies that for (0,0,0), the translation-rotation coupling is weak, and the angular part of the PES largely determines the splitting of the j ) 1 states. However, Table 2 contains clear evidence for stronger coupling of translation to rotation in excited translational states. The energy separation of the j ) 1 states varies significantly with the Cartesian quantum numbers (Vx, Vy, Vz). Let ∆1l,0 and ∆1u,0 denote the energy differences E(1,1)l - E(1,0) and E(1,1)u - E(1,0), respectively, for given (Vx, Vy, Vz), from Table 2. Then, for (1,0,0), ∆1l,0 ) 26.14 cm-1, ∆1u,0 ) 61.38 cm-1; for (0,1,0), ∆1l,0 ) 30.79 cm-1, ∆1u,0 ) 62.19 cm-1; and for (0,0,1), ∆1l,0
) 23.75 cm-1, ∆1u,0 ) 56.71 cm-1. Evidently, the j ) 1 state splitting changes with the excitation of the translational modes. Another manifestation of the translation-rotation coupling is the dependence of the x, y, and z mode fundamentals on the rotational quantum numbers (j,|m|), evident in Table 2. For example, for (1,0), the x, y, and z mode fundamentals are 51.49, 65.72, and 82.23 cm-1, respectively, while for (1,1)u, they are 52.30, 67.34, and 78.38 cm-1, respectively. IV. Conclusions We have performed rigorous quantum 5D calculations of the translation-rotation energy levels and wave functions of H2 molecule (p-H2 and o-H2) confined inside the small dodecahedral cage of the sII clathrate hydrate. Our results provide an accurate description of the quantum dynamics of the coupled translational and rotational motions of the encapsulated hydrogen molecule. The translational modes exhibit negative anharmonicity. We also found that j is a good rotational quantum number. The threefold degeneracy of the j ) 1 rotational level is completely lifted as a result of the confinement. The results show evidence of considerable coupling of translation to rotation, especially for excited translational states. We are currently investigating the quantum dynamics of one or more H2/D2 molecules in the large cage of the sII clathrate hydrate. Acknowledgment. Z.B. is grateful to the National Science Foundation for partial support of this research, through Grant CHE-0315508. The computational resources used in this work were funded in part by the NSF MRI grant CHE-0420870.
Letters References and Notes (1) Sloan, E. D. Clathrate hydrates of natural gases. Marcel Dekker: New York, 1998. (2) Mao, W. L.; Mao, H. K.; Goncharov, A. F.; Struzhkin, V. V.; Guo, Q.; Hu, J.; Shu, J.; Hemley, R. J.; Somayazulu, M.; Zhao, Y. Science 2002, 297, 2247. (3) Lokshin, K. A.; Zhao, Y.; He, D.; Mao, W. L.; Mao, H. K.; Hemley, R. J.; Lobanov, M. V.; Greenblatt, M. Phys. ReV. Lett. 2004, 93, 125503. (4) Florusse, L. J.; Peters, C. J.; Schoonman, J.; Hester, K. C.; Koh, C. A.; Dec, S. F.; Marsh, K. N.; Sloan, E. D. Science 2004, 306, 469. (5) Lee, H.; Lee, J.-W.; Kim, D. Y.; Park, J.; Seo, Y.-T.; Zeng, H.; Moudrakovski, I. L.; Ratcliffe, C. J.; Ripmeester, J. A. Nature 2005, 434, 743. (6) Kim, D. Y.; Lee, H. J. Am. Chem. Soc. 2005, 127, 9996. (7) Hester, K. C.; Strobel, T. A.; Sloan, E. D.; Koh, C. A.; Huq, A.; Schultz, A. J. J. Phys. Chem. B 2006, 110, 14024. (8) Mao, W. L.; Mao, H. K. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 708. (9) Fichtner, M. AdV. Eng. Mater 2005, 7, 443. (10) Sluiter, M. H. F.; Belosludov, R. V.; Jain, A.; Belosludov, V. R.; Adachi, H.; Kawazoe, Y.; Higuchi, K.; Otani, T. Lect. Notes Comput. Sci. 2003, 2858, 330. (11) Patchkovskii, S.; Tse, J. S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 14645. (12) Alavi, S.; Ripmeester, J. A.; Klug, D. D. J. Chem. Phys. 2005, 123, 024507. (13) Alavi, S.; Ripmeester, J. A.; Klug, D. D. J. Chem. Phys. 2006, 124, 014704.
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