Open Phys. 2016; 14:617–620
Research Article
Open Access
Levan Chkhartishvili*
Quasi-planar elemental clusters in pair
interactions approximation
DOI 10.1515/phys-2016-0070
Received September 13, 2016; accepted December 20, 2016
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 quasiplanar 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
*Corresponding Author: Levan Chkhartishvili: Department of
Engineering Physics, Georgian Technical University, Tbilisi, Georgia, E-mail: [email protected]
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 abovementioned issue can be correctly addressed within the
pair interactions approach. The application of the old, socalled 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
© 2016 Levan Chkhartishvili, published by De Gruyter Open.
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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.
(where e is the elementary charge). As for the total shared
electron charge, it would be
Q = −enν.
(4)
Within the pair interactions approximation, it is obvious
to assume that this charge between the atoms is divided
proportionally to their coordination numbers:
qi = −
enνC i
.
j=n
∑︀
Cj
(5)
j=1
2 Binding energy and bonds
polarity
In the first approximation within the pair-interactions approach, a real cluster Xn 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 ∑︁i=n ∑︁k i =C i
E=
E ik i .
(1)
i=1
k i =1
2n
Here, the factor 1/2 is introduced to correct the double
sum which includes every pair twice. Denote ⃗r ik i 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
E≈
i=n
E0 ∑︁
Ci .
2n
And the corresponding changes in atomic charges are:
⎛
⎞
nC
i
⎜
⎟
⎜
⎟
q i − q = eν ⎜1 −
(6)
⎟.
j=n
⎝
⎠
∑︀
Cj
j=1
This implies that the atoms develop non-zero, effective
static atomic charges with charge numbers
⎞
⎛
nC
i
⎟
⎜
⎟
⎜
(7)
Z i = ν ⎜1 −
⎟,
j=n
⎠
⎝
∑︀
Cj
j=1
respectively.
Consequently, the binding energy correction per atom
related to the polarity of interatomic bonds is:
E1 = −
q = −eν
(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:
(2)
(3)
(8)
i=1 k i =1
⃗r ik i + ⃗r k i l i + ⃗r l i i = 0.
i=1
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
i=n k i =C i
e2 ∑︁ ∑︁ Z i Z k i
8πε0 n
r ik i
(9)
3 Equilibrium molar binding energy
Thus, we have obtained the cluster equilibrium molar
binding energy in the pair interactions approach:
E(r ik i ) ≈ E0 + E1 =
=
i=n
i=n k i =C i
E0 ∑︁
e2 ∑︁ ∑︁ Z i Z k i
Ci −
.
2n
8πε0 n
r ik i
i=1
i=1 k i =1
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(10)
Quasi-planar elemental clusters in pair interactions approximation |
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
References
[1]
U(r ik i ) =
i=n
i=n k i =C i
1 ∑︁ ∑︁
E ∑︁
Ci +
=− 0
2n
8n
i=1
(︃
2
Mω (r ik i
i=1 k i =1
e2 Z i Z ki
− d0 ) +
πε0 r ik i
2
)︃
,
[2]
(11)
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:
(︃
)︃
i=n k∑︁
i =C i
∑︁
⃗r ik i
e2 Z i Z ki
r ik i − d0 −
+ ⃗λ ik i = 0, (12)
r ik i
2πε0 Mω2 r2ik i
i=1 k i =1
[3]
[4]
[5]
[6]
where ⃗λ
ik i stands for the Lagrange multipliers. These equations can be linearized taking into account that deviations
of bond radius-vectors ⃗r ik i in a polarized cluster, from their
counterparts ⃗d ik i in 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
)︃
)︃
(︃ (︃
i=n k∑︁
i =C i
∑︁
e2 Z i Z ki
⃗r ik i 1 −
− ⃗d ik i + ⃗λ ik i = 0 (13)
2πε0 Mω2 r3ik i
[7]
[8]
i=1 k i =1
⃗r ik i + ⃗r k i l i + ⃗r l i i = 0
[9]
k i = 1, . . . , C i
i = 1, . . . , n,
from which we find Lagrange multipliers ⃗λ ik i and then the
radius-vectors ⃗r ik i of 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.
[10]
[11]
[12]
[13]
[14]
4 Conclusions
[15]
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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