Theor Chem Acc (2015) 134:134
1 3
gradually replaced by molecular structures of fewer and
fewer equivalences among their nuclear charges. To follow
these changes, it is advantageous to use the 3 N -dimensional
confi guration space (instead of the reduced nuclear confi guration space M of (3 N − 6) dimensions), for the following
reasons:
The case of (3 N )D PES allows for such Euclidean representation; however, in the case of PES defi ned over M ,
no such representation is possible generally, and in fact
one deals with not an Euclidean space but with a manifold
with boundary, involving many, only locally valid coordinate systems. Therefore, in this context, it is advantageous
to use for reference cluster studies the full, 3 N -dimensional
confi gurational space.
A contrast exists between two effects of increasing symmetry. If the symmetry of nuclear charges is maximal in Z
space, as it happens for the reference cluster, then this typically implies the highest energy among all isoprotonic species within the same nuclear charge space Z for these clusters. Yet, if an increase in the actual 3D symmetry of the
nuclear framework renders two, previously non-equivalent
clusters equivalent, then the energy inequalities, such as
those for the clusters produced by the metals atoms Re(75),
Os(76), Ir(77), Pt(78), Au(79) of the examples above,
become simpler, often implying tighter energy bounds.
An interesting problem arises concerning the manifestation of the gradual reductions in the 3 N -dimensional symmetries of the reference cluster to actual molecule transformations as these are represented in the reduced internal
confi guration space of (3 N − 6) dimensions. In this lowerdimensional space, the recognition of such symmetries is
less straightforward; however, the information is fully preserved in some different form. These relations will be the
subject of a forthcoming study.
Acknowledgments This study has been supported by the Canada
Research Chair (CRC) Program of Canada, the Scientifi c Modeling
and Simulation Laboratory (SMSL), and the Memorial University of
Newfoundland.
References
1. Thirring W (1975) Acta Phys Austriaca (Suppl) 14:631–635
2. Narnhofer H, Thirring W (1975) Acta Phys Austriaca
41:281–287
3. Lieb EH, Simon B (1978) J Phys B 11:L537–L542
4. Mezey PG (1981) Theor Chim Acta 59:321–332
5. Mezey PG (1981) Int J Quant Chem Symp 15:279–285
6. Mezey PG (1982) Mol Phys 47:121–126
7. Mezey PG (1982) Int J Quant Chem 22:101–114
8. Mezey PG (1982) Chem Phys Lett 87:277–279
9. Mezey PG (1983) Int J Quant Chem 24:523–526
10. Mezey PG (1984) Can J Chem 62:1356–1357
11. Mezey PG (1984) J Chem Phys 80:5055–5057
12. Mezey PG (1985) J Am Chem Soc 107:3100–3105
13. Mezey PG (1985) Surf Sci 156:597–604
14. Mezey PG (1986) Int J Quant Chem 29:85–99
15. Mezey PG (1986) Int J Quant Chem 29:333–343
16. Otto P, Ladik J, Mezey PG (1987) J Math Chem 1:85–96
17. Cassam-Chenaï P, Chiaramello J-M, Mezey PG (2008) J Math
Chem 44:981–987
18. Mezey PG (2007) AIP Conf Proc 963:513–516
19. Mezey PG (2012) AIP Conf Proc 1504:725–728
20. Mezey PG (2015) J Phys Chem A 119:5305–5312
21. Mezey PG (2015) Topological tools for the study of families of
reaction mechanisms: the fundamental groups of potential surfaces in the universal molecule context. In: Alikhani E, Chauvin
R, Lepetit C, Silvi B (eds) Applications of topological methods
in molecular chemistry. Springer, New York (in press)
22. Mezey PG (1987) Potential energy hypersurfaces. Elsevier,
Amsterdam
23. Mezey PG (1989) Topology of molecular shape and chirality. In: Bertran J, Csizmadia IG (eds) New theoretical concepts
for understanding organic reactions. Kluwer Academic, The
Netherlands
30
Reprinted from the journal
1 3
gradually replaced by molecular structures of fewer and
fewer equivalences among their nuclear charges. To follow
these changes, it is advantageous to use the 3 N -dimensional
confi guration space (instead of the reduced nuclear confi guration space M of (3 N − 6) dimensions), for the following
reasons:
The case of (3 N )D PES allows for such Euclidean representation; however, in the case of PES defi ned over M ,
no such representation is possible generally, and in fact
one deals with not an Euclidean space but with a manifold
with boundary, involving many, only locally valid coordinate systems. Therefore, in this context, it is advantageous
to use for reference cluster studies the full, 3 N -dimensional
confi gurational space.
A contrast exists between two effects of increasing symmetry. If the symmetry of nuclear charges is maximal in Z
space, as it happens for the reference cluster, then this typically implies the highest energy among all isoprotonic species within the same nuclear charge space Z for these clusters. Yet, if an increase in the actual 3D symmetry of the
nuclear framework renders two, previously non-equivalent
clusters equivalent, then the energy inequalities, such as
those for the clusters produced by the metals atoms Re(75),
Os(76), Ir(77), Pt(78), Au(79) of the examples above,
become simpler, often implying tighter energy bounds.
An interesting problem arises concerning the manifestation of the gradual reductions in the 3 N -dimensional symmetries of the reference cluster to actual molecule transformations as these are represented in the reduced internal
confi guration space of (3 N − 6) dimensions. In this lowerdimensional space, the recognition of such symmetries is
less straightforward; however, the information is fully preserved in some different form. These relations will be the
subject of a forthcoming study.
Acknowledgments This study has been supported by the Canada
Research Chair (CRC) Program of Canada, the Scientifi c Modeling
and Simulation Laboratory (SMSL), and the Memorial University of
Newfoundland.
References
1. Thirring W (1975) Acta Phys Austriaca (Suppl) 14:631–635
2. Narnhofer H, Thirring W (1975) Acta Phys Austriaca
41:281–287
3. Lieb EH, Simon B (1978) J Phys B 11:L537–L542
4. Mezey PG (1981) Theor Chim Acta 59:321–332
5. Mezey PG (1981) Int J Quant Chem Symp 15:279–285
6. Mezey PG (1982) Mol Phys 47:121–126
7. Mezey PG (1982) Int J Quant Chem 22:101–114
8. Mezey PG (1982) Chem Phys Lett 87:277–279
9. Mezey PG (1983) Int J Quant Chem 24:523–526
10. Mezey PG (1984) Can J Chem 62:1356–1357
11. Mezey PG (1984) J Chem Phys 80:5055–5057
12. Mezey PG (1985) J Am Chem Soc 107:3100–3105
13. Mezey PG (1985) Surf Sci 156:597–604
14. Mezey PG (1986) Int J Quant Chem 29:85–99
15. Mezey PG (1986) Int J Quant Chem 29:333–343
16. Otto P, Ladik J, Mezey PG (1987) J Math Chem 1:85–96
17. Cassam-Chenaï P, Chiaramello J-M, Mezey PG (2008) J Math
Chem 44:981–987
18. Mezey PG (2007) AIP Conf Proc 963:513–516
19. Mezey PG (2012) AIP Conf Proc 1504:725–728
20. Mezey PG (2015) J Phys Chem A 119:5305–5312
21. Mezey PG (2015) Topological tools for the study of families of
reaction mechanisms: the fundamental groups of potential surfaces in the universal molecule context. In: Alikhani E, Chauvin
R, Lepetit C, Silvi B (eds) Applications of topological methods
in molecular chemistry. Springer, New York (in press)
22. Mezey PG (1987) Potential energy hypersurfaces. Elsevier,
Amsterdam
23. Mezey PG (1989) Topology of molecular shape and chirality. In: Bertran J, Csizmadia IG (eds) New theoretical concepts
for understanding organic reactions. Kluwer Academic, The
Netherlands
30
Reprinted from the journal