标题： 有限分子阵列的最低能量构型 显示英文标题

标题： Minimal energy configurations of finite molecular arrays

摘要： 本文研究了由分子间势能引起的有限粒子系统最小能量构型的特征问题。我们局限于三种粒子以三角形排列和四种粒子以四面体排列的情况。对于三角形排列，最小化被约束在固定面积下；而对于四面体排列，则被约束在固定体积下。 对于一类广义的分子间势能，我们给出了阵列均匀配置（即等边三角形或正四面体）稳定的条件，即系统势能的极小值条件。为了确定是否存在其他稳定状态，我们将一阶必要条件的最小化问题作为分歧问题来处理，其中面积或体积变量作为分歧参数。 由于问题中存在的对称性，我们可以应用等变分歧理论的技术，证明在面积或体积参数的特定值处，从均匀解的平凡分支分岔出非均匀解的分支，此时均匀配置的稳定性发生变化。 对于三角形阵列，我们数值构造了Lennard-Jones势和Buckingham势的分歧图。数值结果显示存在非均匀的稳定状态，面积参数的某些区间内存在多重稳定状态，以及次级分岔现象。 显示英文摘要

摘要： In this paper we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attracting or repulsive forces given by a certain inter molecular 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 fixed area in the case of the triangular array, and to fixed volume in the tetrahedral case. For a general class of inter molecular 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.