# Minimal Energy Configurations of Finite Molecular Arrays

^{*}

## Abstract

**:**

## 1. Introduction

**Notation:**

## 2. Equivariant Bifurcation from a Simple Eigenvalue

**Theorem**

**1**

- 1.
- ${\mathcal{C}}_{\mathcal{H}}$ is unbounded in ${\mathbb{R}}^{n+1}$;
- 2.
- the closure of ${\mathcal{C}}_{\mathcal{H}}$ intersects the boundary $\partial \mathcal{U}$ of $\mathcal{U}$;
- 3.
- ${\mathcal{C}}_{\mathcal{H}}$ intersects $\mathcal{T}$ at a point $({\overrightarrow{\mathbf{x}}}_{*},{A}_{*})$ where ${A}_{*}\ne {A}_{0}$.

## 3. The Three Particle Case

#### 3.1. Existence and Stability of Trivial States

**Lemma**

**1.**

**Theorem**

**2.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 3.2. Existence and Stability of Nontrivial Solutions

**Theorem**

**3.**

**Proof.**

## 4. Four Particles in a Tetrahedron

- R permutes $(a,b,c)$ and $(A,B,C)$ with the same permutation of three elements;
- Q is any permutation of $(a,b,c,A,B,C)$ in which the base of the tetrahedron is changed to another face. For example, $(c,A,B,C,a,b)$ corresponds to reorienting the tetrahedron so that the base is given by $(C,a,b)$.

#### 4.1. Existence and Stability of Trivial States

**Lemma**

**2.**

**Theorem**

**4.**

**Proof.**

**Example**

**4.**

- 1.
- Assume that ${\delta}_{2}\in (2,6]$. Then the second condition in (43) is automatically satisfied and the first condition holds if and only if $V<{V}_{0}$, where ${V}_{0}$ is determined from the condition (cf. (41)):$$\frac{{c}_{1}{\delta}_{1}({\delta}_{1}-2)}{{r}_{0}^{{\delta}_{1}+2}}}-{\displaystyle \frac{{c}_{2}{\delta}_{2}({\delta}_{2}-2)}{{r}_{0}^{{\delta}_{2}+2}}}=0,\phantom{\rule{1.em}{0ex}}{r}_{0}={a}_{{V}_{0}},$$$${V}_{0}={\displaystyle \frac{\sqrt{2}}{12}}{\left[\frac{{c}_{1}{\delta}_{1}({\delta}_{1}-2)}{{c}_{2}{\delta}_{2}({\delta}_{2}-2)}\right]}^{\frac{3}{{\delta}_{1}-{\delta}_{2}}}.$$Thus, in this case the regular tetrahedron ${\overrightarrow{\mathbf{a}}}_{V}$ is a (local) solution of (39) if and only if $V<{V}_{0}$.
- 2.
- If ${\delta}_{2}>6$, then the second condition in (43) holds if and only if $V<{V}_{1}$, where ${V}_{1}$ is determined from the condition:$$\frac{{c}_{1}{\delta}_{1}({\delta}_{1}-6)}{{r}_{1}^{{\delta}_{1}+2}}}-{\displaystyle \frac{{c}_{2}{\delta}_{2}({\delta}_{2}-6)}{{r}_{1}^{{\delta}_{2}+2}}}=0,\phantom{\rule{1.em}{0ex}}{r}_{1}={a}_{{V}_{1}},$$$${V}_{1}={\displaystyle \frac{\sqrt{2}}{12}}{\left[\frac{{c}_{1}{\delta}_{1}({\delta}_{1}-6)}{{c}_{2}{\delta}_{2}({\delta}_{2}-6)}\right]}^{\frac{3}{{\delta}_{1}-{\delta}_{2}}}.$$Since ${\delta}_{1}>{\delta}_{2}>6$, it follows that ${V}_{1}<{V}_{0}$. Thus, in this case the regular tetrahedron ${\overrightarrow{\mathbf{a}}}_{V}$ is a (local) solution of (39) if and only if $V<{V}_{1}$.

**Example**

**5.**

**Example**

**6.**

#### 4.2. Existence and Stability of Nontrivial States

- ${\mu}_{1}(V)={\varphi}^{\u2033}({a}_{V})+\frac{3}{{a}_{V}}{\varphi}^{\prime}({a}_{V})$ with algebraic and geometric multiplicity three, and corresponding eigenvectors:$${(0,-1,0,0,1,0,0)}^{t},\phantom{\rule{1.em}{0ex}}{(0,0,-1,0,0,1,0)}^{t},\phantom{\rule{1.em}{0ex}}{(0,0,0,-1,0,0,1)}^{t}.$$
- ${\mu}_{2}(V)={\varphi}^{\u2033}({a}_{V})+\frac{7}{{a}_{V}}{\varphi}^{\prime}({a}_{V})$ with algebraic and geometric multiplicity two, and corresponding eigenvectors:$${(0,-1,1,0,-1,1,0)}^{t},\phantom{\rule{1.em}{0ex}}{(0,-1,0,1,-1,0,1)}^{t}.$$

**Remark**

**1.**

#### 4.2.1. The Eigenvalue ${\mu}_{1}(V)$

**Theorem**

**5.**

**Remark**

**2.**

**Theorem**

**6.**

**Remark**

**3.**

#### 4.2.2. The Eigenvalue ${\mu}_{2}(V)$

**Theorem**

**7.**

**Remark**

**4.**

## 5. Numerical Examples

## 6. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Generic graphs of the functions F and G appearing in the stability condition (26) for a Buckingham potential.

**Figure 3.**Bifurcation diagram for the system (13) in the case of a Lennard–Jones potential for a larger interval of values of A. There are secondary bifurcations into stable scalene triangles.

**Figure 4.**Solution set for the system (13) in the case of a Lennard–Jones potential without the A dependence.

**Figure 5.**Solution set for the system (13) in the case of a Lennard–Jones potential without the A dependence with the branch of trivial solutions coming out of the page.

**Figure 6.**Dependence of the a and c components on the parameter A for the system (13) in the case $a=b$ and for a Buckingham potential.

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Negrón-Marrero, P.V.; López-Serrano, M.
Minimal Energy Configurations of Finite Molecular Arrays. *Symmetry* **2019**, *11*, 158.
https://doi.org/10.3390/sym11020158

**AMA Style**

Negrón-Marrero PV, López-Serrano M.
Minimal Energy Configurations of Finite Molecular Arrays. *Symmetry*. 2019; 11(2):158.
https://doi.org/10.3390/sym11020158

**Chicago/Turabian Style**

Negrón-Marrero, Pablo V., and Melissa López-Serrano.
2019. "Minimal Energy Configurations of Finite Molecular Arrays" *Symmetry* 11, no. 2: 158.
https://doi.org/10.3390/sym11020158