Continuous Symmetry Measures. 5. The Classical Polyhedra

Inorganic Chemistry

Volumn 37, Issue 21, 1998, Pages 5575-5582

  Pinsky, Mark ^a   Avnir, David ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0542373319     PISSN: 00201669     EISSN: None     Source Type: Journal    
DOI: 10.1021/ic9804925     Document Type: Article

References (54)

1. Reviews: (a) Avnir, D.; Katzenelson, O.; Keinan, S.; Pinsky. M.; Pinto, Y.; Salomon, Y.; Zabrodsky Hel-Or, H. In Concepts in Chemistry; Rouvray, D. H., Ed.; Research Studies Press: Somerset, 1997; Chapter 9,
2. Reviews: (a) Avnir, D.; Katzenelson, O.; Keinan, S.; Pinsky. M.; Pinto, Y.; Salomon, Y.; Zabrodsky Hel-Or, H. In Concepts in Chemistry; Rouvray, D. H., Ed.; Research Studies Press: Somerset, 1997; Chapter 9, (b) Avnir, D.; Zabrodsky Hel-Or, H.; Mezey, P. G. In Encyclopedia of Computational Chemistry; von Rauge Schleyer, P., Ed.; Wiley: Chichester, in press. (b) Avnir, D.; Zabrodsky Hel-Or, H.; Mezey, P. G. In Encyclopedia of Computational Chemistry; von Rauge Schleyer, P., Ed.; Wiley: Chichester, in press. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
3. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
4. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
5. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
6. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
7. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
8. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
9. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
10. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
11. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
12. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
13. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
14. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466.
15. Some examples for symmetry measurements (see also citations in ref 1): (a) Maaskant, W. J. A. J. Phys.; Condens. Matter 1997, 9, 9759. (b) Zimpel, Z. J. Math. Chem. 1993, 14, 451. (c) Auf der Heyde. T. P. E.; Bürgi, H.-B. Inorg. Chem. 1989, 28, 3960. (d) Murray-Rust, P.; Bürgi, H. B.; Dunitz, J. D. Acta Crystallogr. 1978, B34, 1787. The problematics of selecting a specific reference structure was discussed in another paper by these authors: Murray-Rust, P.; Burgi, H.-B.; Dunitz, J. D. Acta Crystallogr. 1979, A35, 703. (e) Cammi, B.; Cavalli, E. Acta Crystallogr. 1992, B48, 245. Cavalli, E.; Cammi, R. Comput. Chem. 1994, 18, 405. (f) Mexey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D. H., Ed.; Academic Press: San Diego, 1997; pp 139, (g) Korobko, V. I. Symmetry; Culture and Sci. 1995, 6, 308. (h) Klein, D. J. J. Math. Chem. 1995, 18, 321. (i) Kuz'min, V. E.; Stel'mach. I. B.; Bekker, M. B.; Pozigun, D. V. J. Phys. Org. Chem. 1992, 5, 295. (j) Toporova, A. P.; Toporov, A. A.; Ishankhodzhaeva, M. M.; Parptrev, N. A. Russ. J. Org. Chem. 1996, 41, 466. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233. (k) Grunbaum, B. Proc. Symp. Pure Math.; Am. Math. Soc. 1963, 7, 233.
16. (a) Zabrodsky Hel-Or, H.; Peleg, S.; Avnir, D. J. Am. Chem. Soc. 1992, 114, 7843.
17. (b) Zabrodsky Hel-Or, H.; Peleg, S.; Avnir, D. J. Am. Chem. Soc. 1993, 115, 8278.
18. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below.
19. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below.
20. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below.
21. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below.
22. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744.
23. Examples include the following, (a) Application of centrosymmetry measure as an order parameter in the study of the melting point of icosahedral clusters: Buch, V.; Greshgoren, E.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Phys. Lett. 1995, 247, 149. (b) Quantitative investigation of the chirality properties of the cyclic trimer of water and of its enamiomerization pathways; Pinto, Y.; Zabrodsky Hel-Or, H.; Avnir, D. J. Chem. Soc., Faraday Trans. 1996, 92, 2523. (c) Analysis of the correlation between the degree of cemrosymmetry and hyperpolarizability: Kanis, K. D.; Wong, J. C.; Marks, T. S.; Ratner, M. A.; Zabrodsky, H.; Keinan, S.; Avnir, D. J. Phys. Chem. 1995, 99, 11061. (d) Quantitative analysis of the chirality of large random objects; Katzenelson, O.; Zabrodsky Hel-Or, H.; Avnir, D. Chem. Eur. J. 1996, 2, 174. Avnir, D.; Katzenelson, O.; Zabrodsky Hel-Or, H. Chem. Eur. J. 1996, 2, 744. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below. (e) Analysis of the macroscopic chirality of Pasteur's tartrate crystals: Keinan, S.; Zabrodsky Hel-Or, H.; Avnir, D. Enantiomer 1996, 1, 351 and refs 21 and 22 below.
24. Hargittai, I.; Hargittai, M. Symmetry through the eyes of a chemist, 2nd ed.; Plenum Press: N.Y., 1995.
25. Muller, U. Inorganic Structural Chemistry, 2nd ed.; Wiley & Sons: Chichester, 1992.
26. Rouvray, D. H. Top. Curr. Chem. 1995, 173, 1.
27. Rouvray, D. H. Top. Curr. Chem. 1995, 173, 1. Petitjean, M. J. Chem. Inf. Comput. Sci. 1996, 36, 1038. Petitjean, M. J. Chem. Inf. Comput. Sci. 1996, 36, 1038. Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
28. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
29. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
30. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
31. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
32. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
33. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
34. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69.
35. The term "continuous" is used here in contradistinction to "either/ or". It is a general term that has been used with various connotations. For instance, in physics, continuous symmetry means that by performing a continuous transformation of a set of parameters on a physical system it acts the same way everywhere and at all times (Gross, D. J. Proc. Natl. Acad. Sci. U.S.A. 1993, 93, 14256). Lie groups used for gauge symmetries are another place where this term is used, especially in the context of invariance to continuous changes in the coordinates of observation (Rosen, J. Found. Phys. 1990, 20, 283. Pons, J. M. Mod. Phys. Lett. 1994, 9, 2903). Of relevance are also several studies in spectroscopy. Thus, continuity was suggested for asymmetric rotors (King, G. W.; Heiner, R. M.; Cross, P. C. J. Phys. Chem. 1942, 11, 27), although the parameter developed there does not measure the distance from specific symmetries. Near-symmetry has been also treated in spectroscopy in terms of perturbation theory: Bunker. P. R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979; Chapter 11. Symmetry of nonrigid molecules was treated in the following: Longuet-Higgins, H. C. Mol. Phys. 1967, 6, 445. Louk, J. D.; Galbreith, H. W. Rev. Mod. Phys. 1976, 48, 69. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e. The latter reference was through the use of Eckart vectors. Yet another worth noting approach to the expression of structural deviation is the use of matrix elements, which are a power-series expansion in normal modes displacements: Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1988, 88, 1775, See also refs 2d and e.
36. (a) Zabrodsky Hel-Or, H.; Peleg, S.; Avnir, D. Adv. Mol. Struct. Res. 1995, 1, 1.
37. (b) Zabrodsky Hel-Or, H.; Peleg, S.; Avnir, D. J. Am. Chem. Soc. 1995, 117, 462.
38. This is also a direct outcome from the Caushi-Bunykovsky inequality (Korn, G. A.; Korn, T. M. Mathematical Handbook for Scientists and Engineers; McGraw-Hill; New York, 1968; p 831).
39. 3b (the only perfect polyhedral program we developed by that method) with the results of the algorithm described here, and as should be the case, the two resulting S values are indeed the same.
40. (a) Comba, P.; Zimmer, P. Inorg. Chem. 1994, 33, 5368.
41. (b) Chapter 6.6 in ref 5 and references therein.
42. (a) Stillinger, F. H.; Weber, T. A. J. Chem. Phys. 1984, 81, 5095.
43. (b) Wales, D. J.; Berry, R. S. Phys. Rev. Lett. 1994, 73, 2857.
44. Sokolov, V. I. Introduction to Theoretical Stereochemistry; Gordon and Breach, New York, 1991; Chapter 4.
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