Quasi-planar elemental clusters in pair interactions approximation

Levan Chkhartishvili

doi:10.1515/phys-2016-0070

Article Open Access

Quasi-planar elemental clusters in pair interactions approximation

Levan Chkhartishvili

Published/Copyright: December 30, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

The pair-interactions approximation, when applied to describe elemental clusters, only takes into account bonding between neighboring atoms. According to this approach, isomers of wrapped forms of 2D clusters – nanotubular and fullerene-like structures – and truly 3D clusters, are generally expected to be more stable than their quasi-planar counterparts. This is because quasi-planar clusters contain more peripheral atoms with dangling bonds and, correspondingly, fewer atoms with saturated bonds. However, the differences in coordination numbers between central and peripheral atoms lead to the polarization of bonds. The related corrections to the molar binding energy can make small, quasi-planar clusters more stable than their 2D wrapped allotropes and 3D isomers. The present work provides a general theoretical frame for studying the relative stability of small elemental clusters within the pair interactions approximation.

Keywords: atomic clusters; bonds polarity; molar binding energy; relative stability; pair interactions approximation

PACS: 61.46.Bc

1 Introduction

The main goals of research in the field of elemental clusters have been formulated by I. Boustani in [1]. These aim at developing, simulating, modeling and predicting novel nanostructures of boron with specific predefined properties. The structures of boron clusters are divided into four groups: quasi-planar, tubular, convex and spherical. Transitioning quasi-planar structures into tubules or cages may be viewed as rolling up these atomically thin sheets into cylinders or spheres, respectively. The boron case shows that small quasi-planar elemental clusters can be more stable than their structural isomers in the form of nanotubular and fullerene-like atomic surfaces or true 3D structures (see our recent reviews on boron nanostructures for examples [2–5]).

Based on the experimental studies and computational simulations available for all-boron clusters [6–11], one can conclude that small elemental clusters should have a quasi-planar structure, but when n, the number of atoms constituting cluster, exceeds a certain critical value (in the case of boron n ~ 20), such clusters have to transform into a nanotubular structures.

In any clustered form of elemental substance, molar binding energy serves as a key factor determining relative stabilities and consequently, concentrations of clusters with different numbers of atoms. This means that at a fixed number of atoms in clusters, the isomers with symmetrical shapes and without “holes” (vacancies) in their structure, i.e. with the maximal number of interatomic bonds, are expected to be more stable. Therefore, the main problem in the theoretical study of atomic clusters is the calculation of their molar binding energy.

In the case of small elemental clusters, the above-mentioned issue can be correctly addressed within the pair interactions approach. The application of the old, so-called diatomic model [12] to atomic structures, including clusters, is based on the property of interatomic bonding being saturated. In the first approximation, the binding energy of a structure is equal to the sum of the energies of the pair interactions between neighboring atoms. Within the pair interactions approximation, the microscopic theory of expansion and its generalization to periodical structures allows correct estimation of the thermal expansion coefficient for many crystalline substances [13]. Despite its simplicity, the diatomic model is still successfully used to calculate various anharmonic effects in solids [14]. An analogous approach is successfully used to explain various isotopic effects in all-boron lattices [15–17].

Our recent studies [18, 19] were devoted to the calculations of the molar binding energy and dipole moments of planar boron clusters within the pair interactions approximation. As for the theoretical scheme of calculations of the chemical bond length in elemental planar clusters and its numerical realization in the boron case, corresponding results will be published elsewhere [20, 21].

Within the frame of the pair interactions approach, further refinement of the clusters’ binding energy and other ground-state parameters can be achieved by abandoning the requirement for equality of all the bond lengths. In this work, we aim to provide a general theoretical frame for studying the relative stability of small elemental clusters in the pair interactions approximation.

2 Binding energy and bonds polarity

In the first approximation within the pair-interactions approach, a real cluster X_n built of n ≥ 2 identical X-atoms located at certain distances from each other is modeled by the perfect cluster, in which the lengths of all the X– X bonds are equal. Denote this single structural parameter of the model as d₀.

Suppose that the index i, numbers the atoms constituting the cluster, i = 1, . . . , n, and C_i is their respective coordination number. Now let the index k_i, numbers the nearest neighboring atoms to the i-atom, k_i = 1, . . . , C_i. If the energy of binding between the i- and k_i-atoms is E_{ik_i}, the cluster molar binding energy in the initial approximation would be

(1)E=12n∑i=1i=n∑ki=1ki=CiEiki.

Here, the factor 1/2 is introduced to correct the double sum which includes every pair twice. Denote r→iki as the radius-vector of k_i-atom, drawn from the i-atom.

In the standard diatomic model, bond lengths and binding energies between each pair of adjacent atoms are equal, r_{ik_i} ≡ d₀ and E_{ik_i} ≡ E₀, and it turns out that

(2)E≈E02n∑i=1i=nCi.

However, differences in the coordination numbers of atomic sites of a cluster – a finite structure of atoms – lead to the redistribution of the outer valence shell electrons’ charge and, as a result, to different binding energies E_{ik_i} of atomic pairs.

If the outer shell of an isolated X-atom contains ν electrons, the outer shell valence charge of the X-atom equals

(3)q=−eν

(where e is the elementary charge). As for the total shared electron charge, it would be

(4)Q=−enν.

Within the pair interactions approximation, it is obvious to assume that this charge between the atoms is divided proportionally to their coordination numbers:

(5)qi=−enνCi∑j=1j=nCj.

And the corresponding changes in atomic charges are:

(6)qi−q=eν1−nCi∑j=1j=nCj.

This implies that the atoms develop non-zero, effective static atomic charges with charge numbers

(7)Zi=ν1−nCi∑j=1j=nCj,

respectively.

Consequently, the binding energy correction per atom related to the polarity of interatomic bonds is:

(8)E1=−e28πε0n∑i=1i=n∑ki=1ki=CiZiZkiriki

(ε₀ is the electrical constant).

In the polarized structure, interatomic distances become non-equal: r_{ik_i} ≠ d₀. It means that the initially perfect structure is converted into a quasi-perfect one. Anyway, relative radius-vectors of atomic sites should satisfy following relations:

(9)r→iki+r→kili+r→lii=0.

3 Equilibrium molar binding energy

Thus, we have obtained the cluster equilibrium molar binding energy in the pair interactions approach:

(10)E(riki)≈E0+E1==E02n∑i=1i=nCi−e28πε0n∑i=1i=n∑ki=1ki=CiZiZkiriki.

The equilibrium bond lengths r_{ik_i}, accounting for the polarity of bonds under the constraints given by Eqs. (9), should be determined by minimizing the potential energy

(11)U(riki)==−E02n∑i=1i=nCi+18n∑i=1i=n∑ki=1ki=CiMω2(riki−d0)2+e2ZiZkiπε0riki,

where M denotes the mass of the atom (consequently, M/2 is the reduced mass of the diatomic system) and ω is the cyclic frequency of relative vibrations of bonded pairs of atoms near the bond length of d₀.

Minimization yields the following equation:

(12)∑i=1i=n∑ki=1ki=Cir→ikirikiriki−d0−e2ZiZki2πε0Mω2riki2+λ→iki=0,

where λ→iki stands for the Lagrange multipliers. These equations can be linearized taking into account that deviations of bond radius-vectors r→ikiin a polarized cluster, from their counterparts d→ikiin a non-polarized perfect cluster with equal lengths d_{ik_i} ≡ d₀, are expected to be too small. Finally, we get a set of linear equations

(13)∑i=1i=n∑ki=1ki=Cir→iki1−e2ZiZki2πε0Mω2riki3−d→iki+λ→iki=0r→iki+r→kili+r→lii=0ki=1,…,Cii=1,…,n,

from which we find Lagrange multipliers λ→ikiand then the radius-vectors r→ikiof the bonds.

For small clusters, i.e. with small numbers of equations in the set,solutions can be found easily. Inserting the obtained values of bond lengths r_{ik_i} into the expression of the molar binding energy E(r_{ik_i} ), one can find its value and estimate relative stability of atomic clusters.

4 Conclusions

In summary, on the basis of the pair interactions approximation and taking into account the possibility of polarization of bonding, even in elemental clusters if they are critically small, a set of linear equations is proposed which determines chemical bond lengths and directions in elemental clusters.

Solving this system allows the calculation of molar binding energies of elemental clusters and evaluation their relative stability.

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Received: 2016-9-13

Accepted: 2016-12-20

Published Online: 2016-12-30

Published in Print: 2016-1-1

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