Minimal Energy Configurations of Finite Molecular Arrays
In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a tria...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/2/158 |
| _version_ | 1811262720215875584 |
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| author | Pablo V. Negrón-Marrero Melissa López-Serrano |
| author_facet | Pablo V. Negrón-Marrero Melissa López-Serrano |
| author_sort | Pablo V. Negrón-Marrero |
| collection | DOAJ |
| description | In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to a fixed area in the case of the triangular array, and to a fixed volume in the tetrahedral case. For a general class of intermolecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first-order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non-homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard⁻Jones and Buckingham potentials. The numerics show that there exist non-homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well. |
| first_indexed | 2024-04-12T19:32:00Z |
| format | Article |
| id | doaj.art-5a9dd40a82ac4465911c318fe4d92afc |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-12T19:32:00Z |
| publishDate | 2019-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-5a9dd40a82ac4465911c318fe4d92afc2022-12-22T03:19:19ZengMDPI AGSymmetry2073-89942019-01-0111215810.3390/sym11020158sym11020158Minimal Energy Configurations of Finite Molecular ArraysPablo V. Negrón-Marrero0Melissa López-Serrano1Department of Mathematics, University of Puerto Rico, Humacao, PR 00791-4300, USADepartment of Mathematics, University of Puerto Rico, Humacao, PR 00791-4300, USAIn this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to a fixed area in the case of the triangular array, and to a fixed volume in the tetrahedral case. For a general class of intermolecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first-order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non-homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard⁻Jones and Buckingham potentials. The numerics show that there exist non-homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well.https://www.mdpi.com/2073-8994/11/2/158molecular arraysconstrained optimizationequivariant bifurcation theory |
| spellingShingle | Pablo V. Negrón-Marrero Melissa López-Serrano Minimal Energy Configurations of Finite Molecular Arrays Symmetry molecular arrays constrained optimization equivariant bifurcation theory |
| title | Minimal Energy Configurations of Finite Molecular Arrays |
| title_full | Minimal Energy Configurations of Finite Molecular Arrays |
| title_fullStr | Minimal Energy Configurations of Finite Molecular Arrays |
| title_full_unstemmed | Minimal Energy Configurations of Finite Molecular Arrays |
| title_short | Minimal Energy Configurations of Finite Molecular Arrays |
| title_sort | minimal energy configurations of finite molecular arrays |
| topic | molecular arrays constrained optimization equivariant bifurcation theory |
| url | https://www.mdpi.com/2073-8994/11/2/158 |
| work_keys_str_mv | AT pablovnegronmarrero minimalenergyconfigurationsoffinitemoleculararrays AT melissalopezserrano minimalenergyconfigurationsoffinitemoleculararrays |