Abstract
The whole sets of critical points of analytical functions corresponding to the long‐range two‐body interaction between two, three and four dipole vectors set at the vertices of several polygons are determined using topology theorems. Betti numbers associated to the configuration spaces are first obtained, then stationary points are located via an analytical gradient method till the Morse inequalities are checked.
Similar content being viewed by others
References
R.G.A. Bone, T.W. Rowlands, N.C. Handy and A.J. Stone, Mol. Phys. 72 (1991) 33.
J.G.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt, Chem. Phys. Lett. 237 (1995) 560.
M. Greenberg, in: Lectures on Algebraic Topology(Benjamin, New York, 1966).
W. Klopper and M. Schütz, Chem. Phys. Lett. 237 (1995) 536.
D. Liotard, Thesis, Université de Pau, France (1979).
D. Liotard and M. Rérat, Theoret. Chim. Acta 86 (1993) 297.
S.A.C. McDowell, J. Chem. Phys. 105 (1996) 4180.
P.G. Mezey, in: Potential Energy Hypersurfaces(Elsevier, New York, 1987).
O. Mó, M. Yañez and J. Elguero, J. Chem. Phys. 97 (1992) 6628.
M. Morse, in: Calculus of Variation in the Large, Amer. Math. Colloq. Publ. 18 (1934).
F. Pacella, Arch. Rat. Mech. Anal. 97 (1987) 59.
M. Rérat and D. Liotard, J. Chim. Phys. 91 (1994) 1401.
M. Rérat, D. Liotard and J.M. Robine, Theoret. Chim. Acta 88 (1994) 285.
M. Schütz, W. Klopper, H.P. Lüthi and S. Leutwyler, J. Chem. Phys. 103 (1995) 6114.
E.G. Spanier, in: Algebraic Topology(McGraw-Hill, New York, 1966).
D.J. Wales and T.R. Walsh, J. Chem. Phys. 106 (1997) 7193.
S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, The Advanced Book Program (Addison-Wesley, Redwood City, CA, 1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rérat, M., Gélize, M. & Liotard, D. Topology of potential hypersurfaces of two, three and four dipoles interacting at long distances. Journal of Mathematical Chemistry 22, 235–247 (1997). https://doi.org/10.1023/A:1019136217083
Issue Date:
DOI: https://doi.org/10.1023/A:1019136217083