Quantum Effects on H2 Diffusion in Zeolite RHO: Inverse Kinetic

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Quantum Effects on H2 Diffusion in Zeolite RHO: Inverse Kinetic Isotope Effect for Sieving Lu Gem Gao, Rui Ming Zhang, Xuefei Xu, and Donald G. Truhlar J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.9b06506 • Publication Date (Web): 31 Jul 2019 Downloaded from pubs.acs.org on July 31, 2019

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Journal of the American Chemical Society

revised, 7/27/2019

Quantum Effects on H2 Diffusion in Zeolite RHO: Inverse Kinetic Isotope Effect for Sieving Lu Gem Gao,1,2 Rui Ming Zhang,1 Xuefei Xu1,*, and Donald G. Truhlar2,* 1

Center for Combustion Energy, Department of Energy and Power Engineering, and Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China 2

Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, USA

Abstract We use canonical variational theory (CVT) with small-curvature tunneling (SCT) contributions to investigate quantum effects on the H2 diffusion process in the pure-silica zeolite RHO. At low temperature we find an inverse kinetic isotopic sieving effect in that the heavier isotopic species diffuses faster than the lighter one. Three quantum effects contribute to this kinetic isotope effect (KIE). The first one is quantum mechanical tunneling; this – on its own – would lead to a normal kinetic isotopic sieving effect, in which lighter diprotium diffuses faster than dideuterium. The second factor, which we find to dominate in the present case, is zero-point energy (ZPE); deuterium has a lower ZPE, which leads to a smaller barrier in the effective barrier for tunneling because the transition state has a larger ZPE than the precursor stable state; this results in an inverse KIE. The third factor, the thermal vibrational energy (computed from the quantized vibrational partition function), also favors a normal KIE, but it is outweighed by the ZPE effect. The vibrations of the zeolite host framework are found to play an important role at low temperatures, and our calculations consider up to 7296 degrees of freedom at the equilibrium structure and the saddle point and up to 221 degrees of freedom along the reaction path. The importance of quantum considerations on the dynamics is elucidated by comparison to a purely classical treatment. 1. Introduction Isotopic species are sometimes separated by expensive and complicated processes such as centrifugation, laser isotope separation, and thermal diffusion. Microporous and nanoporous materials are widely used for separating gaseous mixtures, and it would be very advantageous if they could be used for the separation of isotopes by sieving, as an alternative to the conventional methods.1 However in many cases the sieving is based on the size or shape of pores and molecules or on electronic effects like polarity.2 But these effects are negligible for isotopologs, so the separation must be based on mass. The mass effects are dominated by quantum effects like zero point energy and tunneling, so classical mechanical simulations would not be adequate for understanding such separations.

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Many molecular simulations and experimental studies focusing on hydrogen isotope separation in microporous materials (zeolites, metal-organic frameworks, nanotubes, and porous carbon) have been published.3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 In the present work we consider hydrogen isotope separation in all-silica zeolites. (Molecular sieve action may be different in aluminosilicates with charge-balancing extraframework cations in the pores, where, for example, a trapdoor mechansim may apply.24,25) Quantum sieving has been first proposed to study the hydrogen isotopes in microporous materials, and it was found that under equilibrium conditions heavier isotopes are adsorbed more strongly than lighter ones.3,26,27 This is the so called equilibrium quantum sieving (EQS) effect. By using metal-organic frameworks (MOFs), FitzGerald et al. reported chemical affinity quantum sieving (CAQS) effect by which heavier isotopes are preferentially adsorbed due to attractive interaction sites of MOFs;28 this may be considered to be special case of EQS since it is still based on equilibrium thermodynamic properties. Besides equilibrium factors, one must also consider dynamics, and the present study is devoted to dynamics. The importance of quantum effects on the dynamics was studied theoretically by Kumar et al. using molecular dynamics simulations.29,30 By analogy to EQS and CAQS, this is called kinetic quantum sieving (KQS), and several studies were carried out focusing on hydrogen isotope separation in microporous materials by this effect.11212,13,22,31,32,33,34,35 In the study of hydrogen isotope separation in AlPO4-25, Kumar et al.31 used a simplified transition state theory expression36 with a transmission coefficient evaluated from a trajectory simulation in which quantum effects are included by using an effective potential obtained via the Feynman-Hibbs path integral formalism.37 In another treatment, Hankel et al. reported TST simulations including quantum effects in the partition functions for a model with four degrees of freedom.33 Although there have been many studies incorporating the lattice vibrations of host atoms in the diffusion of adsorbates, they are inconclusive about the importance of these motions,38,39,40,41 thus studies on the quantum effects on diffusion dynamics have always treated the host microporous materials as essentially rigid. We show here that this treatment overestimates the diffusion rates, especially at low temperatures. Zeolite RHO has been studied several times to show the reverse kinetic selectivity for H2/D2 mixtures.1,30,31 In the present paper, we study the quantum effects on dynamics for the case of H2 diffusion in zeolite RHO including the vibrations of host microporous materials and including quantum effects both by using quantized partition functions (which includes both the zero point energy effects and the steric effects due to the isotopic differences in quantal vibrational amplitudes42) and by treating the tunneling by a multidimensional semiclassical method. We include quantum effects in partition functions with up to 7296 degrees of freedom at the equilibrium structure and the saddle point and up to 221 degrees of freedom along the reaction path. This allows for the flexibility of the framework, which is expected to be important because RHO is known43,44 to be a particularly flexible zeolite.


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3

We also compare to a completely classical treatment (in which vibrations are treating classically and tunneling is neglected) to show the difference of the present treatment from a classical molecular dynamics simulation. 2. Methodology and model Our calculations are for the completely siliceous RHO, also called all-silica RHO. We start with calculations on the periodic solid, and then the dynamics are done using a cluster model carved from the periodic structure. 2.1 The force field The potential energy surface is a combination of Lennard-Jones, valence, and electrostatic terms. The Lennard-Jones parameters are taken from the MM2 force field, which has been shown to perform well for silicalite structures.45,46 The valence and electrostatic terms are obtained from the CVFF47 force field with one changed parameter. The modification is the force constant for the SiO-Si angle, which is increased by 12% because the standard CVFF parameters do not yield a reasonable symmetric unit cell structure, since the Si-O-Si bond angle bend is highly anharmonic.48 A full description of the force field terms is summarized in Supporting Information. 2.2 Structure of the zeolite and cluster model The unit cell of all-silica RHO has the formula Si48O96. First we optimized the unit cell structure with periodic boundary conditions. Previous work has shown that this can result in one of two cubic space groups, a centric Im3m form and an acentric I 43m form,49 as shown in Figure 1. Since a pure silica version has not been synthesized yet, we calculated both of these geometries, and we found the acentric I 43m form space group geometry with lattice constants a = b = c = 14.4146 Å has a lower energy (by 94.075 kcal/mol unit cell) than the Im3m structure (with lattice constants a = b = c = 14.7643 Å), so the cluster model used for dynamics calculation is based on the acentric I 43m unit cell. The cluster model used to study the diffusion corresponds to a (2.5 × 2.5 × 2.5) unit cell structure of zeolite RHO. Each dangling bond created by carving out the cluster is capped with a hydrogen atom, which yielded a capped cluster with 2702 atoms. All the outermost Si atoms (a total of 270 atoms) in the cluster model are frozen during optimizations and dynamics in order to simulate the periodic boundary conditions; this means that 2432 atoms are optimized and are allowed to move for the dynamics. 2.3 Quantum consideration in CVT/SCT method The rate constant for site-to-site hopping is calculated by canonical variational theory50 (CVT) with small-curvature tunneling51 (SCT), in which the rate constant is52 𝑘𝐶𝑉𝑇/𝑆𝐶𝑇 = 𝜅(𝑇)𝑘𝐶𝑉𝑇

(

k T Q‡ exp -V ‡ RT 𝑘𝐶𝑉𝑇 = 𝛤(𝑇) B R h Q

(1)

)


(2)


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where 𝜅 is the SCT tunneling transmission coefficient, 𝑘𝐶𝑉𝑇 is the quasiclassical CVT rate constant, 𝛤 is the recrossing transmission coefficient, 𝑘𝐵 is Boltzmann’s constant, T is the temperature, h is Planck’s constant, 𝑉‡ is the classical energy height, Q ‡ is the quantized partition function at the transition state, Q R is the quantized partition function of reactants, and we note that these partition functions are calculated relative to the energy minimum for R or relative to the saddle point energy for ‡. The quasiclassical CVT rate constant is the quasiclassical dynamical flux through a hypersurface separating the product site from the reactant site; it is called quasiclassical because all degrees of freedom except the reaction coordinate are quantized; the reaction coordinate motion is classical. The transition state hypersurface is variationally optimized.50,53 Quantum effects on reaction coordinate motion, including the multidimensional coupling of the reaction coordinate to another degrees of freedom, are included by the tunneling transmission coefficient in the SCT approximation, which includes both tunneling and nonclassical reflection from the effective barrier. Thus quantum effects are included both by the tunneling transmission coefficient 𝜅 and by using quantum mechanical partition functions (including zero point energy) in 𝑘CVT . The CVT/SCT dynamical method has been well validated for gas-phase reactions,54,55 diffusion at metal surfaces,56 liquid-solution reactions,57,58 and enzyme-catalyzed reactions.59,60,61 We will report the results as unimolecular rate constants for a single hopping direction, but if desired, the reader can convert them to diffusion coefficients by the following argument. In our case, diffusion is anisotropic and occurs along a one-dimensional channel, so we are calculating the diffusion coefficient in the direction parallel to that channel; call this D∣∣. Because the system can hop to the left or the right, the frequency of hopping is f = 2k CVT/SCT

(2a)

Furthermore, for one-dimensional diffusion, the Einstein diffusion law gives

D|| = (d 2 2) f

(2b)

where δ is the hopping distance. The hopping distance is the distance between the initial and final equilibrium structures. From the equilibrium structures optimized in section 3.1, we calculate δ = 4.355 Å. Only the nondegenerate ground electronic state is considered, so the electronic partition function cancels out and is not needed. Since some atoms are frozen in the cluster model, there is no translational or rotational partition function.62 So the total partition function equals the vibrational partition function. The reaction path for calculating the CVT recrossing coefficient and the tunneling transmission coefficient is taken as the minimum energy path (MEP) through mass-scaled coordinates. The calculation of the recrossing coefficient requires evaluating the vibrational partition function along the reaction path. This is given quantum mechanically by


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5

3𝑁−1

𝑄vib (𝑠) = ∏ 𝑚=1

exp⁡(−𝛽ℏ 𝜔𝑚 (𝑠)⁄2) 1 − exp(−𝛽ℏ𝜔𝑚 (𝑠)) 𝐹

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ = exp(−𝛽𝐸𝑍𝑃 (𝑠)) ∏ 𝑚=1

(3)

1 1 − exp(−𝛽ℏ𝜔𝑚 (𝑠))

= exp(−𝛽𝐸𝑍𝑃 (𝑠)) 𝑄̃vib (𝑠)

where s denotes the position along the reaction path, N is the number of movable atoms,⁡𝛽 is 1⁄𝑘𝐵 𝑇, 𝜔𝑚 (s) is the generalized-normal-mode vibrational frequency of mode m at location s, ̃ (𝑠) is the vibrational partition 𝐸𝑍𝑃 (𝑠) is the local vibrational zero-point energy at s, and 𝑄 vib function relative to the zero-point level. The transition-structure enthalpy of activation at 0 K is given by (4) The effective potential for tunneling is isotope-dependent and is given by49,52 (5) GT (s) is the frequency of where VMEP (s) is the potential energy along the MEP, and w m generalized normal mode m on the reaction path at a signed distance s from the saddle point. Note ‡

that VMEP (s = 0) is equal to V .

2.4 Reaction zone To evaluate the recrossing transmission coefficient and to evaluate the effective potential and effective mass for tunneling requires calculating the 3N – 1 generalized transition state vibrational frequencies not only at the transition structure (i.e., saddle point) but also at each step along the reaction path. As explained in Section 2.2, there are 2432 nonfixed atoms in our cluster model, so 3N – 1 is 7295, but there is no need to use all of them (which would be very expensive) for dynamics. Therefore we developed smaller models by allowing only atoms in a reaction zone to move during the vibrational-mode calculations, where the reaction zone is defined as a sphere centered on the center of H2 at the saddle point geometry. Then we increased the radius of the reaction zone until DH 0‡ is converged. Table 1 shows the convergence test and also compares the results to the case where all vibrations of the zeolite host framework are omitted. This test was performed for s = 0 (i.e., at the saddle point).



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Table 1 shows that ignoring the vibrations of the zeolite host framework underestimates the energetic requirement for diffusion by 0.243 kcal/mol, but the enthalpy of activation at 0 K is converged when the radius of reaction zone reaches 7.1 Å. As a result of this test we chose to calculate the reaction rates with two values of the reaction zone radius, namely 7.1 and 7.6 Å. A comparison of the rate constants calculated with these two choices will be presented in section 3.2, and it complements Table 1 by providing a finite-temperature convergence check on the size of the reaction zone.

Table 1. Enthalpy of activation as a function of reaction zone size Reaction zone radius (Å)

Atoms in reaction zone

3N – 1

DH 0‡ (kcal/mol)

Deviationa (kcal/mol)

N/Ab

2

5

14.03

0.243

3.6

14

41

14.17

0.102

5.0

26

77

14.17

0.105

6.2

44

131

14.22

0.060

7.1

68

203

14.27

0.008

7.6

74

221

14.27

0.006

∞

2432

7295

14.28

0.000

Deviation of DH 0‡ from last row b All vibrations of the zeolite host framework are omitted for this row of the table. a

2.5 Software and computational details Unit cell optimizations were carried out by the CP2K package.63 All the cluster model structure calculations were carried out by the Gaussian 09 package64 by using the option to specify parameters using external parameter files so that the force files are as specified in section 2.1 . This was accomplished with the keyword uff=(softfirst,vrange=12,crange=12,switch=charmmsq)

where uff=softfirst means that external parameter files were used for the molecular mechanics calculations, vrange and crange control the cutoffs of Lennard-Jones interactions and Coulomb interactions, respectively, and switch=charmmsq must be specified to ensure a correct Hessian calculation.


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For transition state optimization, the nomicro option was used in the opt keyword. The CVT/SCT rate constants were determined by direct dynamics calculations using the Polyrate65 and Gaussrate66 programs. 3. Results and discussion 3.1 Structure optimization We first found the stationary points along the diffusion path. The saddle point structure is shown in Figure 3, which shows that the H2 molecule is located at the center of a six-member rings shared by two zeolite RHO cages and is oriented perpendicular to this ring. Following the minimum energy path that begins with the imaginary-frequency vibrational mode of this structure, we found that it connects the minimum energy structures in two different cages, and this confirms that this is a transition structure for diffusion. 3.2 Rate constants In the first step we compared canonical variational theory to conventional transition state theory. We found, for both isotopes and all temperatures, that the maximum free energy of activation occurs for a dividing hypersurface at the saddle point, so 𝛤 = 1. Table 2 shows the CVT/SCT rate constants for two values of the radius of the reaction zone. We can see they agree well with each other, so calculations of the kinetic isotope effect (given in section 3.3) are carried out only for the radius equal to 7.1 Å, which corresponds to 68 atoms in the reaction zone.

Table 2. CVT/SCT hopping rate constants (s-1) for H2 T(K)

k(N = 2)

k(N = 68)

k(N = 74)

k(2)/k(68)

k(68)/k(74)

67

8.00E-32

2.00E-32

1.99E-32

3.99

1.01

71

4.67E-30

1.00E-30

9.88E-31

4.67

1.01

77

2.25E-27

4.60E-28

4.56E-28

4.89

1.01

83

7.22E-25

1.60E-25

1.59E-25

4.50

1.01

91

6.47E-22

1.65E-22

1.63E-22

3.93

1.01

100

4.27E-19

1.23E-19

1.22E-19

3.48

1.00

150

1.83E-09

7.82E-10

7.77E-10

2.34

1.01

200

1.09E-04

5.54E-05

5.52E-05

1.96

1.00

250

6.93E-02

3.91E-02

3.90E-02

1.77

1.00

298.15

4.11E+00

2.46E+00

2.45E+00

1.67

1.00

300

4.67E+00

2.80E+00

2.80E+00

1.67

1.00



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400

7.74E+02

4.97E+02

4.96E+02

1.56

1.00

500

1.49E+04

9.92E+03

9.90E+03

1.50

1.00

600

1.01E+05

6.88E+04

6.87E+04

1.47

1.00

800

1.04E+06

7.19E+05

7.18E+05

1.44

1.00

1000

4.02E+06

2.82E+06

2.81E+06

1.43

1.00

1200

9.75E+06

6.87E+06

6.86E+06

1.42

1.00

1400

1.82E+07

1.29E+07

1.28E+07

1.41

1.00

1600

2.89E+07

2.05E+07

2.04E+07

1.41

1.00

1800

4.12E+07

2.93E+07

2.92E+07

1.41

1.00

2000

5.47E+07

3.89E+07

3.88E+07

1.41

1.00

We also calculated the CVT/SCT rate constants by treating the zeolite host framework as rigid, that is, ignoring the vibrations of the zeolite support. This result is shown in Table 2 in the column for N = 2. This treatment always overestimates the rates, with the degree of overestimate decreasing with temperature. The physical reason for the overestimate is that the vibrations of the framework tighten up due to steric effects as one proceeds from the equilbirium structure to the transition state for diffusion. The vibrational effect is more significant at low temperature because it is dominated by changes in zero point energy; these have a larger Boltzmann factor at low temperature. The overestimate is a factor of four at 67 K, two at 200 K, and 1.7 at room temperature. This shows the importance of treating the framework as flexible. 3.3 Quantum effects on diffusion In the isotope effect literature, when the lighter isotopic rate constant is larger than that of the heavier isotopic case, the kinetic isotope effect (KIE) is called normal, and when the rate constant for the heavier case is large it is called an inverse KIE.67,68 Tunneling plays an important role in the diffusion process at low temperatures. To examine the contributions of the different kinds of quantum effect, we calculated the hopping rate constants for both H2 and D2 diffusing in zeolite RHO, and we analyzed the KIE by factorizing it as follows:

𝐾𝐼𝐸 =

𝑘𝐻2 𝜅𝐻2 = 𝑘𝐷2 𝜅𝐷2

æ Q‡ ç H2 ç R ç QH è 2

Q‡ ö÷ 𝜅 exp⁡(−𝛽Δ𝐸 ≠ ) D2 𝐻 𝑍𝑃,𝐻2 = 2 ÷ ≠ QDR ÷ 𝜅𝐷2 exp⁡(−𝛽Δ𝐸𝑍𝑃,𝐷2 ) 2ø

where


(6) = 𝜂1 𝜂2 𝜂3

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9

𝜂1 =

𝜅𝐻2 𝜅𝐷2

≠ 𝜂2 = exp⁡[−𝛽(Δ𝐸≠ 𝑍𝑃,𝐻2 − Δ𝐸𝑍𝑃,𝐷2 )]

(7)

(8)

(9)

where Δ𝐸≠ 𝑍𝑃 is the difference of ZPE between the saddle point and the equilibrium structure, and

denote the vibrational partition functions relative to the respective zero point levels,

subscripts H2 and D2 correspond to the H2 and D2 diffusion process, 𝜂1 is the contribution to KIE from tunneling, 𝜂2 is the contribution to KIE from ZPE, 𝜂3 is the contribution to KIE from the thermal component of the quantized vibrational partition function. The calculated KIEs and their factorizations are shown in Figure 4. Figure 4 shows that all three factors must be considered to understand the KIE quantitatively. We can see a significantly inverse KIE at low temperatures, and Figure 4 shows that the major contribution to the inverse KIE is the ZPE factor. The ZPE at the saddle point is 4.72 kcal/mol larger than at the equilibrium structure for H2 diffusion, whereas the difference is only 3.41 kcal/mol for D2. This explains the large deviation of 𝜂2 from unity in Figure 4. Figure 5 shows the calculated small-curvature tunneling transmission coefficients SCT. Quantum tunneling gives a normal (not inverse) contribution to the KIE, as would be expected from a onedimensional tunneling model, and here we verify that this is still the case from a more accurate multidimensional tunneling calculation. Figure 6 shows the effective barrier for tunneling in the multidimensional tunneling calculation. The higher barrier for H2 reflects the larger change in ZPE for coordinates normal to reaction coordinate for H2, and – because the barrier is plotted in isoinertial coordinates scaled to the same reduced mass for both cases – the narrower barrier for H2 reflects the smaller mass for motion along the reaction coordinate. We see that SCT for H2 becomes very large when the temperature is below 83 K, and this leads to a very large difference from D2, as shown by 𝜂1 in Figure 4. The large tunneling effect at low temperature leads to a very large KIE. This may be compared to the work of Kumar et al. in their reduced-dimensional simulation of H2/D2 separation in a rigid model of zeolite RHO, where they observed high quantum selectivity when the temperature is low, with a peak value of 22 at about 65 K.29 The lower value below 65 K was attributed to a stronger decrease of the transport coefficient for D2 due to quantum effects. The kinetic isotope effect we calculated here with a different geometric model of zeolite RHO, with a nonrigid model of the zeolite, and wih a more reliable treatmnt of tuneling is 13 at 65 K. Kumar et al found differences of a factor of ~1.5 at 65 K by using different geometric models of zeolite RHO.1,30



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Figure 7 shows the classical KIE and its comparison to the quantum result. The classical result is 1.35, independent of temperature. For H2 diffusion, the ratio of the classical rate constant to the quantum one is 1.15  109 at 65 K, and it decreases to 18.3 at 300 K and to 1.08 at 2000 K. For D2 diffusion, the ratio of the classical rate constant to the quantum one is 6.4  107 at 65 K, and it decreases monotonically to 5.2 at 300 K and to 1.04 at 2000 K. The poor performance of a purely classical simulation for a problem involving hydrogenic motion is not unexpected, and it has also been demonstrated quantitatively in previous work.69 Conclusions We report an accurate and convenient way to calculate the quantum effects on diffusion with the example H2 diffusion in zeolite RHO by using canonical variational theory (CVT) with smallcurvature tunneling (SCT) contributions. The calculations do not assume a rigid framework; the calculation of zero point energies includes 7296 degrees of freedom, and the quantum dynamics calculations include 222 degrees of freedom. We find that if the vibrations of the host framework are ignored, the rate constants will be overestimated, especially at low temperatures. In addition, by comparing the quantum result to the classical result, the importance of quantum effects on dynamics has been elucidated. Although quantum tunneling and thermal vibrational effects contribute to the direction of a normal kinetic isotope effect (H2 faster than D2), the effect of ZPE is much larger and in the opposite direction because the ZPE increases in proceeding to the saddle point, but the increase is smaller for D2. The method we used to include quantum effects on diffusion is not limited to hydrogen isotope separations in zeolites, but is also applicable to any case where it is necessary to consider the quantum effects on the diffusion process of molecules in microporous crystalline materials.

 ASSOCIATED

CONTENT

Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ S

Functional forms and parameters of the force field (PDF file). Unit cell structures for the Coordinates of the optimized cluster structures (three XYZ files). Results for T2.  AUTHOR

INFORMATION

Corresponding Authors *E-mail: [email protected], [email protected] ORCID Lu Gem Gao: 0000-0002-1666-0594 Rui Ming Zhang: 0000-0002-4880-6391 Xuefei Xu: 0000-0002-2009-0483


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Donald G. Truhlar: 0000-0002-7742-7294 Notes

The authors declare no competing financial interest.  ACKNOWLEDGMENT

This work was supported in part by a scholarship from China Scholarship Council (201806210213), the National Natural Science Foundation of China (91841301 and 91641127) and as part of the Nanoporous Materials Genome Center by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under award DE-FG02-17ER16362.



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Figure 1. Zeolite RHO: (a) Im3m space group (b) I 43m space group.

Figure 2. Cluster model of zeolite RHO with oxygen in red, silicon in yellow and hydrogen in pink.


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13

Figure 3. Saddle point structure for H2 diffusing from one cage to another cage (cages are outlined by purple dotted squares) through the six-member ring (outlined by a black circle) shared by two cages. The view is along the a axis in the top portion, and the view direction is indicated by axes in the bottom portion.



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Figure 4. Calculated quantum KIEs; 𝜂1 is the contribution to the KIE from tunneling, (quantum effects on reaction coordinate motion), 𝜂2 is the contribution to the KIE from zero point energy, and 𝜂3 is the contribution to the KIE from the thermal component of the quantized vibrational partition function.

Figure 5. Small-curvature tunneling transmission coefficients SCT of H2 and D2 diffusion process in zeolite RHO.


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Figure 6. Calculated ground-state vibrationally adiabatic potential curves (𝑉𝑎𝐺 ) of H2 and D2 diffusion process in zeolite RHO as functions of diffusion coordinates s, where both diffusion coordinates are scaled to a reduced mass of 1 amu.

Figure 7. Calculated classical KIEs and quantum KIEs  REFERENCES 1

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Quantum Effects on H2 Diffusion in Zeolite RHO: Inverse Kinetic

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