Classical and quantum transition state theory for the diffusion of helium in silica sodalite

Journal of Physical Chemistry B

Volumn 101, Issue 4, 1997, Pages 491-503

  Murphy, Michael J ^a   Voth, Gregory A ^b,c   Bug, Amy L R ^a  

^a SWARTHMORE COLLEGE   (United States)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

^c UNIVERSITY OF UTAH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACTIVATION ENERGY; APPROXIMATION THEORY; COMPUTER SIMULATION; ELECTRON DIFFRACTION; ELECTRON TUNNELING; ENTROPY; EQUATIONS OF STATE OF GASES; HELIUM; MONTE CARLO METHODS; QUANTUM THEORY; SILICA; ZEOLITES;

QUANTUM TRANSITION STATE THEORY; SILICA SODALITES;

DIFFUSION IN GASES;

EID: 0030734213     PISSN: 15206106     EISSN: None     Source Type: Journal    
DOI: 10.1021/jp9620678     Document Type: Article

References (130)

1. Zeolites: Science and Technology; Ribeiro Ramoa, F., Rodrigues, A. E., Rollmann, L. D., Naccache, C., Eds.; Nijhoff: Boston, 1984. Bhatia, S. Zeolites Catalysis: Principles and Applications; CRC Press: Boca Raton, FL 1990. Suib, S. L. Chem. Rev. 1993, 93, 803.
2. Zeolites: Science and Technology; Ribeiro Ramoa, F., Rodrigues, A. E., Rollmann, L. D., Naccache, C., Eds.; Nijhoff: Boston, 1984. Bhatia, S. Zeolites Catalysis: Principles and Applications; CRC Press: Boca Raton, FL 1990. Suib, S. L. Chem. Rev. 1993, 93, 803.
3. Zeolites: Science and Technology; Ribeiro Ramoa, F., Rodrigues, A. E., Rollmann, L. D., Naccache, C., Eds.; Nijhoff: Boston, 1984. Bhatia, S. Zeolites Catalysis: Principles and Applications; CRC Press: Boca Raton, FL 1990. Suib, S. L. Chem. Rev. 1993, 93, 803.
4. For example: Karger, J.; Ruthven, D. M. Diffusion in Zeolites and other Microporous Solids; John Wiley and Sons: New York, 1992. Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, 1123, 1143. Chen, N. Y.; Degnan Jr., T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; VCH Publishers: New York, 1994.
5. For example: Karger, J.; Ruthven, D. M. Diffusion in Zeolites and other Microporous Solids; John Wiley and Sons: New York, 1992. Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, 1123, 1143. Chen, N. Y.; Degnan Jr., T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; VCH Publishers: New York, 1994.
6. For example: Karger, J.; Ruthven, D. M. Diffusion in Zeolites and other Microporous Solids; John Wiley and Sons: New York, 1992. Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, 1123, 1143. Chen, N. Y.; Degnan Jr., T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; VCH Publishers: New York, 1994.
7. Ozin, G. A.; Kuperman, A.; Stein, A. Angew. Chem., Int. Ed. Engl. 1989, 28, 359. Stucky, G. D.; Srdanov, V. I.; Harrison, W. T. A.; Gier, T. E.; Keder, N. L.; Moran, K. L.; Haug, K.; Metiu, H. I. In Supramolecule Architecture: Synthetic Control in Thin Films and Solids; Bein, T., Ed.; American Chemical Society: Washington, DC, 1992.
8. Ozin, G. A.; Kuperman, A.; Stein, A. Angew. Chem., Int. Ed. Engl. 1989, 28, 359. Stucky, G. D.; Srdanov, V. I.; Harrison, W. T. A.; Gier, T. E.; Keder, N. L.; Moran, K. L.; Haug, K.; Metiu, H. I. In Supramolecule Architecture: Synthetic Control in Thin Films and Solids; Bein, T., Ed.; American Chemical Society: Washington, DC, 1992.
9. Weitkamp, J.; Fritz, M.; Ernst, S. In Proceedings from the Ninth International Zeolite Conference: II; von Ballmoos, R., Higgins, J. B., Treacy, M. M. J., Eds.; Butterworth Heinemann: Boston, 1992.
10. Monson, P. R. Energy Res. Ahstr. 1982, 7, No. 42509.
11. Fraissard, J.; Ito, T. Zeolites 1988, 8, 350. Tsiao, C.; Kauffman, J. S.; Corbin, D. R.; Abrams, L.; Carroll Jr., E. E.; Dybowski, C. J. Phys. Chem. 1991, 95, 5586. Jameson, C. J. Chem. Phys. 1996, 104, 1709.
12. Fraissard, J.; Ito, T. Zeolites 1988, 8, 350. Tsiao, C.; Kauffman, J. S.; Corbin, D. R.; Abrams, L.; Carroll Jr., E. E.; Dybowski, C. J. Phys. Chem. 1991, 95, 5586. Jameson, C. J. Chem. Phys. 1996, 104, 1709.
13. Fraissard, J.; Ito, T. Zeolites 1988, 8, 350. Tsiao, C.; Kauffman, J. S.; Corbin, D. R.; Abrams, L.; Carroll Jr., E. E.; Dybowski, C. J. Phys. Chem. 1991, 95, 5586. Jameson, C. J. Chem. Phys. 1996, 104, 1709.
14. Barrer, R. M. Zeolites and Clay Minerals as Adsorbers and Molecular Sieves; Academic Press: London, 1978; Chapter 11.
15. Walker, P. L., Jr.; Austin, L. G.; Nandi, S. P. Chem. Phys. Carbon 1966, 2, 257.
16. Rutheven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley and Sons: New York, 1984; Chapter 5.
17. Pollard, W. G.; Present, R. D. Phys. Rev. 1948, 73, 762.
18. Hill, T. L. Introduction to Statistical Thermodynamics; Addision Wesley: Reading, MA, 1960; p 172. Ruthven, D. M.; Doetsch, I. H. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1043.
19. Hill, T. L. Introduction to Statistical Thermodynamics; Addision Wesley: Reading, MA, 1960; p 172. Ruthven, D. M.; Doetsch, I. H. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1043.
20. 4 summarized in: Jobic, H. Spectrochim. Acta 1992, 48A, 293, and molecular dynamics studies like: Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016; Cohen De Lara, E.; Goulay, A. M. J. Chem. Phys. 1989, 90, 7482; or Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1 provide some examples.
21. 4 summarized in: Jobic, H. Spectrochim. Acta 1992, 48A, 293, and molecular dynamics studies like: Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016; Cohen De Lara, E.; Goulay, A. M. J. Chem. Phys. 1989, 90, 7482; or Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1 provide some examples.
22. 4 summarized in: Jobic, H. Spectrochim. Acta 1992, 48A, 293, and molecular dynamics studies like: Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016; Cohen De Lara, E.; Goulay, A. M. J. Chem. Phys. 1989, 90, 7482; or Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1 provide some examples.
23. 4 summarized in: Jobic, H. Spectrochim. Acta 1992, 48A, 293, and molecular dynamics studies like: Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016; Cohen De Lara, E.; Goulay, A. M. J. Chem. Phys. 1989, 90, 7482; or Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1 provide some examples.
24. 4 summarized in: Jobic, H. Spectrochim. Acta 1992, 48A, 293, and molecular dynamics studies like: Demontis, P.; Yashonath, S.; Klein, M. L. J. Phys. Chem. 1989, 93, 5016; Cohen De Lara, E.; Goulay, A. M. J. Chem. Phys. 1989, 90, 7482; or Yashonath, S.; Santikary, P. Mol. Phys. 1993, 78, 1 provide some examples.
25. Some representative studies: Hufton, J. R. J. Phys. Chem. 1991, 95, 8836. Catlow, C. R. A.; Freeman, C. M.; Vessal, B.; Tomlinson, S. M.; Leslie, M. J. Chem. Soc., Faraday Trans. 1991, 87, 1947. Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. Ibid. 1991, 87, 1951. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1992, 96, 1051. Nicholas, J. B.; Trouw, F. R.; Mertz, J. E.; Iton, L. E.; Hopfinger, A. J. J. Phys. Chem. 1993, 97, 4149.
26. Some representative studies: Hufton, J. R. J. Phys. Chem. 1991, 95, 8836. Catlow, C. R. A.; Freeman, C. M.; Vessal, B.; Tomlinson, S. M.; Leslie, M. J. Chem. Soc., Faraday Trans. 1991, 87, 1947. Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. Ibid. 1991, 87, 1951. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1992, 96, 1051. Nicholas, J. B.; Trouw, F. R.; Mertz, J. E.; Iton, L. E.; Hopfinger, A. J. J. Phys. Chem. 1993, 97, 4149.
27. Some representative studies: Hufton, J. R. J. Phys. Chem. 1991, 95, 8836. Catlow, C. R. A.; Freeman, C. M.; Vessal, B.; Tomlinson, S. M.; Leslie, M. J. Chem. Soc., Faraday Trans. 1991, 87, 1947. Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. Ibid. 1991, 87, 1951. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1992, 96, 1051. Nicholas, J. B.; Trouw, F. R.; Mertz, J. E.; Iton, L. E.; Hopfinger, A. J. J. Phys. Chem. 1993, 97, 4149.
28. Some representative studies: Hufton, J. R. J. Phys. Chem. 1991, 95, 8836. Catlow, C. R. A.; Freeman, C. M.; Vessal, B.; Tomlinson, S. M.; Leslie, M. J. Chem. Soc., Faraday Trans. 1991, 87, 1947. Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. Ibid. 1991, 87, 1951. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1992, 96, 1051. Nicholas, J. B.; Trouw, F. R.; Mertz, J. E.; Iton, L. E.; Hopfinger, A. J. J. Phys. Chem. 1993, 97, 4149.
29. Some representative studies: Hufton, J. R. J. Phys. Chem. 1991, 95, 8836. Catlow, C. R. A.; Freeman, C. M.; Vessal, B.; Tomlinson, S. M.; Leslie, M. J. Chem. Soc., Faraday Trans. 1991, 87, 1947. Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. Ibid. 1991, 87, 1951. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1992, 96, 1051. Nicholas, J. B.; Trouw, F. R.; Mertz, J. E.; Iton, L. E.; Hopfinger, A. J. J. Phys. Chem. 1993, 97, 4149.
30. June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1991, 95, 8866.
31. El Amrani, S.; Vigne-Maeder, F.; Bigot, B. J. Phys. Chem. 1992, 96, 9418.
32. See the recent review by Hanggi, P.; Talkner, P.; Borkovec, M. Rev. Mod. Phys. 1990, 62, 251 and references therein.
33. Laidler, K. J.; King, M. C. J. Phys. Chem. 1983, 87, 2657.
34. Eyring, H. J. Chem. Phys. 1935, 3, 107.
35. Evans, M. G.; Polyani, M. Trans. Farad. Soc. 1935, 31, 875. Wigner, E. Trans. Faraday Soc. 1938, 34, 29.
36. Evans, M. G.; Polyani, M. Trans. Farad. Soc. 1935, 31, 875. Wigner, E. Trans. Faraday Soc. 1938, 34, 29.
37. Laidler, K. J.; Tweedale, A. Adv. Chem. Phys. 1971, 21, 113.
38. Pollak, E.; Child, M. S.; Pechukas, P. J. Chem. Phys. 1980, 72, 1669.
39. Keck, J. C. Adv. Chem. Phys. 1967, 13, 85. Pechukas, P. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B. Pollak, E. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV. Tucker, S. C. In New Trends in Kramers Reaction Rate Theory; Talkner, P., Hangii, P., Eds.; Kluwer Academic: Boston, 1995; p 5.
40. Keck, J. C. Adv. Chem. Phys. 1967, 13, 85. Pechukas, P. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B. Pollak, E. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV. Tucker, S. C. In New Trends in Kramers Reaction Rate Theory; Talkner, P., Hangii, P., Eds.; Kluwer Academic: Boston, 1995; p 5.
41. Keck, J. C. Adv. Chem. Phys. 1967, 13, 85. Pechukas, P. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B. Pollak, E. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV. Tucker, S. C. In New Trends in Kramers Reaction Rate Theory; Talkner, P., Hangii, P., Eds.; Kluwer Academic: Boston, 1995; p 5.
42. Keck, J. C. Adv. Chem. Phys. 1967, 13, 85. Pechukas, P. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B. Pollak, E. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV. Tucker, S. C. In New Trends in Kramers Reaction Rate Theory; Talkner, P., Hangii, P., Eds.; Kluwer Academic: Boston, 1995; p 5.
43. Chandler, D. J. Chem. Phys. 1978, 68, 2959.
44. Chandler, D.; Pratt, L. R. J. Chem. Phys. 1976, 65, 2925. Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV.
45. Chandler, D.; Pratt, L. R. J. Chem. Phys. 1976, 65, 2925. Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV.
46. Kramers, H. A. Physica 1940, 4, 284.
47. Berne, B. J.; Borkovec, M.; Straub, J. E. J. Phys. Chem. 1988, 92, 3711.
48. A related quantity is the "conversion coefficient" of Anderson (Anderson, J. B. J. Chem. Phys. 1973, 58, 4684), which is the ratio of successful, reactive crossings to all crossings of the TS.
49. Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University: New York, 1987; Chapter 8.
50. Berne, B. J. In Multiple Time Scales; Brackbill, J. V., Cohen, B. I., Eds.; Academic Press: New York, 1985. Montegomery, J. A.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70, 4056. Straub, J. E.; Hsu, D. A.; Berne, B. J. J. Phys. Chem. 1985, 89, 5188.
51. Berne, B. J. In Multiple Time Scales; Brackbill, J. V., Cohen, B. I., Eds.; Academic Press: New York, 1985. Montegomery, J. A.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70, 4056. Straub, J. E.; Hsu, D. A.; Berne, B. J. J. Phys. Chem. 1985, 89, 5188.
52. Berne, B. J. In Multiple Time Scales; Brackbill, J. V., Cohen, B. I., Eds.; Academic Press: New York, 1985. Montegomery, J. A.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70, 4056. Straub, J. E.; Hsu, D. A.; Berne, B. J. J. Phys. Chem. 1985, 89, 5188.
53. Vinyard, G. J. Phys. Chem. Solids 1957, 3, 121. Franklin, W. M. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975; p 1. Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. In Zeolite Science 1994: Recent Progress and Discussions; Karge, H. G., Weitkamp, J., Eds.; Elsevier Science: New York, 1995.
54. Vinyard, G. J. Phys. Chem. Solids 1957, 3, 121. Franklin, W. M. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975; p 1. Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. In Zeolite Science 1994: Recent Progress and Discussions; Karge, H. G., Weitkamp, J., Eds.; Elsevier Science: New York, 1995.
55. Vinyard, G. J. Phys. Chem. Solids 1957, 3, 121. Franklin, W. M. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975; p 1. Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. In Zeolite Science 1994: Recent Progress and Discussions; Karge, H. G., Weitkamp, J., Eds.; Elsevier Science: New York, 1995.
56. Bennett, C. H. p 73.
57. Wert, C.; Zener, C. Phys. Rev. 1949, 76, 1169.
58. Wert, C. A. Phys. Rev. 1950, 79, 601.
59. Rice, S. A. Phys. Rev. 1958, 112, 804.
60. Ruthven, D. M.; Derrah, R. I. Trans. Faraday Soc. 1972, 68, 2332.
61. Barrer, R. M. Trans. Faraday Soc. 1941, 37, 590. Ruthven, D. M. Can. J. Chem. 1974, 52, 3523. Theodorou, D.; Wei, J. J. Catal. 1983, 83, 205.
62. Barrer, R. M. Trans. Faraday Soc. 1941, 37, 590. Ruthven, D. M. Can. J. Chem. 1974, 52, 3523. Theodorou, D.; Wei, J. J. Catal. 1983, 83, 205.
63. Barrer, R. M. Trans. Faraday Soc. 1941, 37, 590. Ruthven, D. M. Can. J. Chem. 1974, 52, 3523. Theodorou, D.; Wei, J. J. Catal. 1983, 83, 205.
64. Sargent, R. W. H.; Whitford, C. J. In Molecular Sieve Zeolites II; Gould, R. I., Ed.; American Chemical Society: Washington, DC, 1971; p 144. Betemps, M.; Jutard, A. J. Phys. D: Appl. Phys. 1980, 13, 423.
65. Sargent, R. W. H.; Whitford, C. J. In Molecular Sieve Zeolites II; Gould, R. I., Ed.; American Chemical Society: Washington, DC, 1971; p 144. Betemps, M.; Jutard, A. J. Phys. D: Appl. Phys. 1980, 13, 423.
66. Banavar, J. R.; Cohen, M. H.; Gomer, R. Surf. Sci. 1981, 107, 113.
67. Barrer, R. M.; Jost, W. Trans. Faraday Soc. 1949, 45, 928.
68. Voter, A. F.; Doll, J. D. J. Chem. Phys. 1984, 80, 5814, 5832.
69. Voter, A. F.; Doll, J. D. J. Chem. Phys. 1985, 82, 80.
70. Zhang, Z.; Haug, K.; Metiu, H. J. Chem. Phys. 1990, 93, 3614.
71. Yashonath, S.; Santikary, P. J. Phys. Chem. 1993, 97, 3849.
72. See the reviews of: Miller, W. H. Acc. Chem. Res. 1976, 9, 306. Pechukas, P. Annu. Rev. Phys. Chem. 1981, 32, 159. Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664.
73. See the reviews of: Miller, W. H. Acc. Chem. Res. 1976, 9, 306. Pechukas, P. Annu. Rev. Phys. Chem. 1981, 32, 159. Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664.
74. See the reviews of: Miller, W. H. Acc. Chem. Res. 1976, 9, 306. Pechukas, P. Annu. Rev. Phys. Chem. 1981, 32, 159. Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664.
75. Voth, G. A.; Chandler, D.; Miller, W. H. J. Chem. Phys. 1989, 91, 7749. Voth, G. A. J. Phys. Chem. 1993, 97, 8365.
76. Voth, G. A.; Chandler, D.; Miller, W. H. J. Chem. Phys. 1989, 91, 7749. Voth, G. A. J. Phys. Chem. 1993, 97, 8365.
77. Feynman, R. P.; Hibbs, A. R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, 1965; Chapter 10.
78. Feynman, R. P. Statistical Mechanics; Benjamin/Cummings: Reading, MA, 1972; Chapter 3.
79. Chandler, D.; Wolynes, P. G. J. Chem. Phys. 1981, 74, 4078.
80. Gillan, M. J. J. Phys. C 1987, 20, 3621. Additionally, Valone, Voter, and Doll (ref 77) used what amounts to a variational approximation to PI-QTST; see: Voth, G. A. J. Chem. Phys. 1991, 94, 4095.
81. Gillan, M. J. J. Phys. C 1987, 20, 3621. Additionally, Valone, Voter, and Doll (ref 77) used what amounts to a variational approximation to PI-QTST; see: Voth, G. A. J. Chem. Phys. 1991, 94, 4095.
82. Richardson Jr., J. W.; Pluth, J. J.; Smith, J. V.; Dytrych, W. J.; Bibby, D. M. J. Phys. Chem. 1988, 92, 243.
83. Kiselev, A. V.; Lopatkin, A. A.; Shulga, A. A. Zeolites 1985, 5, 261.
84. Wahnstrom, G.; Haug, K.; Metiu, H.; Chem. Phys. Lett. 1988, 148, 158. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1989, 90, 540.
85. Wahnstrom, G.; Haug, K.; Metiu, H.; Chem. Phys. Lett. 1988, 148, 158. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1989, 90, 540.
86. Sholl, D. S.; Skodje, R. T. Physica D 1994, 71, 168.
87. Machta, J.; Zwanzig, R. Phys. Rev. Lett. 1983, 50, 1959.
88. Jameson, C. J.; Jameson, A. K.; Gerald, R. II; de Dios, A. C. J. Chem. Phys. 1992, 96, 1676. Auerbach, S. M.; Henson, N. J.; Cheetham, A. K.; Metiu, H. I. J. Phys. Chem. 1995, 99, 10600.
89. Jameson, C. J.; Jameson, A. K.; Gerald, R. II; de Dios, A. C. J. Chem. Phys. 1992, 96, 1676. Auerbach, S. M.; Henson, N. J.; Cheetham, A. K.; Metiu, H. I. J. Phys. Chem. 1995, 99, 10600.
90. Zhou, H-X.; Zwanzig, R. J. Chem. Phys. 1991, 94, 6147.
91. Voter, A. F. J. Chem. Phys. 1985, 82, 1890.
92. Barrer, R. M.; Vaughan, D. E. W. J. Phys. Chem. Solids 1971, 32, 731.
93. Nicholas, J. B.; Hopfinger, A. J.; Trouw, F. R.; Iton, L. E. J. Am. Chem. Soc. 1991, 113, 4792.
94. Evans, D. J.; Morriss, G. P. Comput. Phys. Rep. 1984, 1, 297.
95. Voter, A. F. Phys. Rev. Lett. 1989, 63, 167. Voter, A. F.; Doll, J. D.; Cohen, J. M. J. Chem. Phys. 1989, 90, 2045.
96. Voter, A. F. Phys. Rev. Lett. 1989, 63, 167. Voter, A. F.; Doll, J. D.; Cohen, J. M. J. Chem. Phys. 1989, 90, 2045.
97. Karger, J.; Ruthven, D. M. Diffusion in Zeolites and other Microporous Solids; John Wiley and Sons: New York, 1992; Chapter 2.
98. Demontis, P.; Fois, E. S.; Suffritti, G. B.; Quartieri, S. J. Phys. Chem. 1990, 94, 4329.
99. Demontis, P.; Suffritti, G. B.; Fois, E. S.; Quartieri, S. J. Phys. Chem. 1992, 96, 1482.
100. Gillan, M. J. Phys. Rev. Lett. 1987, 58, 563.
101. See for example Volkl, J.; Alefeld, G. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975. Electronic Structure and Properties of Hydrogen in Metals; Jena, P., Satterthwaite, C. B., Eds.; Plenum: New York, 1983. Metal-Hydrogen Systems: Fundamentals and Applications; Marchester, F. D., Ed.; Elsevier Sequoia: Lausanne, France, 1990; Vol. I.
102. See for example Volkl, J.; Alefeld, G. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975. Electronic Structure and Properties of Hydrogen in Metals; Jena, P., Satterthwaite, C. B., Eds.; Plenum: New York, 1983. Metal-Hydrogen Systems: Fundamentals and Applications; Marchester, F. D., Ed.; Elsevier Sequoia: Lausanne, France, 1990; Vol. I.
103. See for example Volkl, J.; Alefeld, G. In Diffusion in Solids: Recent Developments; Nowick, A. S., Burton, J. J., Eds.; Academic Press: New York, 1975. Electronic Structure and Properties of Hydrogen in Metals; Jena, P., Satterthwaite, C. B., Eds.; Plenum: New York, 1983. Metal-Hydrogen Systems: Fundamentals and Applications; Marchester, F. D., Ed.; Elsevier Sequoia: Lausanne, France, 1990; Vol. I.
104. Bouchard, J. P.; Cohen De Lara, E.; Kahn, R. Europhys. Lett. 1992, 17, 583.
105. Landau, L.; Lifschitz, E. Statistical Physics, Vol. 1; Pergamon Press: New York, 1980; Chapter 33.
106. Some reviews on path integral techniques are: Berne, B. J.; Thirumalai, D. Annu. Rev. Phys. Chem. 1987, 37, 401. Doll, J. D.; Freeman, D. L.; Beck, T. L. Adv. Chem. Phys. 1990, 78, 61. Ceperley, D. Rev. Mod. Phys. 1995, 67, 279.
107. Some reviews on path integral techniques are: Berne, B. J.; Thirumalai, D. Annu. Rev. Phys. Chem. 1987, 37, 401. Doll, J. D.; Freeman, D. L.; Beck, T. L. Adv. Chem. Phys. 1990, 78, 61. Ceperley, D. Rev. Mod. Phys. 1995, 67, 279.
108. Some reviews on path integral techniques are: Berne, B. J.; Thirumalai, D. Annu. Rev. Phys. Chem. 1987, 37, 401. Doll, J. D.; Freeman, D. L.; Beck, T. L. Adv. Chem. Phys. 1990, 78, 61. Ceperley, D. Rev. Mod. Phys. 1995, 67, 279.
109. Sprik, M.; Klein, M.; Chandler, D. Phys. Rev. B. 1985, 31, 4234.
110. Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987; Chapter 10.
111. Reference 71, Chapter 4. Valleau, J. P.; Whittington, S. G. In Statistical Mechanics; Berne, B. J., Ed.; Plenum: New York, 1977: Part A.
112. Hall, R. W.; Berne, B. J. J. Chem. Phys. 1984, 81, 3641.
113. Reference 28, Chapter 6.
114. Were we working at extremely low temperatures, a large number of umbrella bins would be necessary. At some point, rather than perform numerous MC samplings in intermediate bins, it might be more efficient to implement, e.g., the MC method of ref 57 involving occasional large displacements specifically designed to take one from a well into the TS.
115. Calhoun, A.; Doren, D. J. Phys. Chem. 1993, 97, 2251.
116. Valone, S. M.; Voter, A. F.; Doll, J. D. Surf. Sci. 1985, 155, 687; J. Chem. Phys. 1986, 85, 7480.
117. Valone, S. M.; Voter, A. F.; Doll, J. D. Surf. Sci. 1985, 155, 687; J. Chem. Phys. 1986, 85, 7480.
118. Lauderdale, J. G.; Truhlar, D. G. J. Am. Chem. Soc. 1985, 107, 4590; Surf. Sci. 1985, 164, 558. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1990, 92, 2083. Haug, K.; Metiu, H. J. Chem. Phys. 1991, 94, 3251. Sun, Y-C.; Voth, G. A. J. Chem. Phvs. 1993, 98, 7451.
119. Lauderdale, J. G.; Truhlar, D. G. J. Am. Chem. Soc. 1985, 107, 4590; Surf. Sci. 1985, 164, 558. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1990, 92, 2083. Haug, K.; Metiu, H. J. Chem. Phys. 1991, 94, 3251. Sun, Y-C.; Voth, G. A. J. Chem. Phvs. 1993, 98, 7451.
120. Lauderdale, J. G.; Truhlar, D. G. J. Am. Chem. Soc. 1985, 107, 4590; Surf. Sci. 1985, 164, 558. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1990, 92, 2083. Haug, K.; Metiu, H. J. Chem. Phys. 1991, 94, 3251. Sun, Y-C.; Voth, G. A. J. Chem. Phvs. 1993, 98, 7451.
121. Lauderdale, J. G.; Truhlar, D. G. J. Am. Chem. Soc. 1985, 107, 4590; Surf. Sci. 1985, 164, 558. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1990, 92, 2083. Haug, K.; Metiu, H. J. Chem. Phys. 1991, 94, 3251. Sun, Y-C.; Voth, G. A. J. Chem. Phvs. 1993, 98, 7451.
122. Lauderdale, J. G.; Truhlar, D. G. J. Am. Chem. Soc. 1985, 107, 4590; Surf. Sci. 1985, 164, 558. Haug, K.; Wahnstrom, G.; Metiu, H. J. Chem. Phys. 1990, 92, 2083. Haug, K.; Metiu, H. J. Chem. Phys. 1991, 94, 3251. Sun, Y-C.; Voth, G. A. J. Chem. Phvs. 1993, 98, 7451.
123. Rick, S. W.; Lynch, D. L.; Doll, J. D. J. Chem. Phys. 1993, 99, 8183.
124. Cao, J.; Voth, G. A. J. Chem. Phys. 1994, 100, 5093, 5106. Cao, J.; Voth, G. A. J. Chem. Phys. 1994, 101, 6157.
125. Cao, J.; Voth, G. A. J. Chem. Phys. 1994, 100, 5093, 5106. Cao, J.; Voth, G. A. J. Chem. Phys. 1994, 101, 6157.
126. Cao, J.; Voth, G. A. J. Chem. Phys. 1994, 101, 6168.
127. Pavese, M.; Voth, G A. Chem. Phys. Lett. 1996, 249, 231.
128. Jacucci, G.; Omerti, E. J. Chem. Phys. 1983, 79, 3051.
129. Pollock, E. L.; Ceperley, D. M. Phys. Rev. B 1984, 30, 2555.
130. In fact, the 6-rings are not formed of coplanar atoms in silica sodalite (though the 4-rings are). But one can define a plane from which the deviations of the O positions average to zero; this is the plane of the TS.