# Study of Magnesium Diboride Clusters Using Hybrid Density Functional Theory

Authors: D. Rodríguez; D. Soto; E. Ramírez; A. Cruz; A. Santana; G. E. López
Journal: Research Letters in Physics
 (2008)

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## Abstract

Using hybrid density functional theory and a relatively large basis set, the lowest energy equilibrium structure, vibrational spectrum, and natural orbital analysis were obtained for magnesium diboride clusters [(MgB2)x for
x=1,2, and 3]. For comparison, boron clusters
[Bxforx=2,4,and6] were also considered. The
MgB2 and
(MgB2)2 showed equilibrium structures with the boron atoms in arrangements similar to what was obtained for pure boron atoms, whereas, for
(MgB2)3 a different arrangement of boron was obtained. From the population analysis, large electron density in the boron atoms forming the clusters was observed.


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## Body

## 1. Introduction

Materials
composed of diborides have attracted the attention of researchers in recent
years because of its superconductivity properties [1, 2]. In particular, it has
been found that MgB2 behaves as a superconductor at relatively high critical
temperature, Tc=40K. Although Tc for this material is
not as high as other oxides with superconducting properties, its preparation is
less expensive and it has a lower anisotropy, making smoother grain boundaries
and hence increasing current flow. The origin of the superconducting properties
of MgB2 is still a topic of debate [3]. However, the structure of
MgB2 might have a significant importance in this property. This
material shows a simple hexagonal AlB2-type structure typical of
borides. It is formed of layers of boron similar to the basal plane of
graphite, separated by closed-packed layers of magnesium. The magnesium atoms
are located at the center of the hexagons formed by the boron atom and donate
the electron density to the boron layers. It has been postulated [4] that this
electron donation might be related to the superconducting nature of this
material. Also, electronic structure calculations of bulk MgB2 have
shown [5] a metallic character of the material which causes a strong
electron-phonon interaction and hence superconducting properties. Experimental
evidence has suggested [4] that an increment in the Tc temperature
can be obtained if nanosized particles of MgB2 are embedded in inert
materials. This increment in Tc seems to be caused by the change in
the electronic density of states.  The
main objective of this study is to use cluster models to understand the
electronic problem of MgB2. These models have been extensively used
to understand the behavior of bulk materials because of the great theoretical
detail that can be obtained [6, 7].  The
lowest energy equilibrium structure was obtained and corroborated with a
harmonic vibrational analysis. Natural orbital analysis was performed in order
to characterize the electron density of the system. Boron clusters were also
considered for comparison purposes with the results obtained being in agreement
with recent studies [8].


## 2. Methods

The theoretical
methods used in this study were based on ab-initio methods within the density
functional theory approximation. Specific details of the procedure followed can
be found elsewhere [9]. Specifically, the three parameter hybrid functional
B3PW91 was used with the Dunning basis set cc-pvtz augmented with polarization
functions. Hence, the method was termed B3PW91/aug-cc-pvtz. For all systems, a
full geometry optimization was performed and various initial geometries were
used to guarantee the determination of the lowest energy equilibrium structure.
Harmonic vibrational analysis was performed for each system not only to obtain
the vibrational frequencies, but also to characterize the nature of the structure
obtained in the potential energy surface (PES). All calculations were performed
using the Gaussian 03 software package [10] installed on an SGI 200
workstation. In order to analyze the charge distribution in the system, natural
bond orbital (NBO) analysis was performed on the optimized wave function using
the NBO program of the Gaussian 03 package. This analysis provides an
improvement to the Mulliken population analysis usually used in the description
of the charge distribution.


## 3. Results and Discussion

The equilibrium
bond lengths of
all the optimized structures are presented in Table1 and the structures are
shown in Figures 1 and 2. Figure 1(a) shows the lowest energy structure of MgB2.
As in previous studies [11, 12] a nonlinear structure was obtained as the lowest energy equilibrium
structure with the boron
atoms at an equilibrium distance 0.09 Å shorter than in the B2  molecule (see
Figure 1(b)). From the NBO analysis shown in Figure 1(b), it can be observed
that the electronic density is concentrated at the boron atoms, that is, the
magnesium is donating electron density to the diborine. Table 2 shows the
harmonic vibrational spectrum of MgB2 and B2 and it can
be observed that all frequencies obtained have a positive sign which means that
the global minimum was obtained. In the case of MgB2, the frequency
that corresponds to the diboride vibration, frequency 1, is 16% higher than in
the B2 molecule because the bond shortening. Figures 1(c) and 1(d) show
the lowest energy for equilibrium structure of (MgB2)2 and B4, respectively. Both structures show a planar arrangement of
atoms, with the boron atoms in a two-dimensional unsymmetrical rhombic
arrangement for (MgB2)2 and a perfect square for B4.
Specifically, two of the boron-boron bonds in the molecular species do not
change and the other two were stretched by 9%. The magnesium atoms were
accommodated at a relatively long distance when compared to the separation of
the boron atoms. The vibrational frequency analysis shown in Table 2
demonstrates that the structure is a minimum in the PES. A direct comparison
between the modes in the molecular species and the pure boron clusters was not possible. Figure 1(c)
shows the NBO density analysis for the dimer and it can be observed that all
the boron atoms forming the 2D rhombic arrangement have a negative charge. The
boron atoms, which are closer to the Mg atoms, showed
larger value of electron density than the other atoms which has the magnesium
at a larger distance. Hence, the electron density is being
concentrated on the boron atoms in a nonsymmetrical fashion and is being
donated by the Mg atoms. Figure 2 shows the lowest energy equilibrium structure
of (MgB2)3 and B6. Boron atoms form a nonsymmetrical
nonplanar arrangement. Although the B6 do not show a planar
structure, the arrangement of atoms is different to the one obtained in the
molecular species. The vibrational analysis shows all positive frequencies and
similar to the previous system, it is not possible to correlate the frequencies
between the B6 and the molecular system. The NBO analysis for the
molecular systems shows a similar electronic distribution to the one observed for the
dimer, namely, the electronic distribution is located on the boron atoms. Two
of the boron atoms show a larger electronic distribution (−0.853) because of their
proximity to the magnesium atoms.Table 1
Optimized bond length in angstroms.
MgB2(MgB2)2(MgB2)3B2B4B6r1 = 1.551r1 = 1.726r1 = 1.545r1 = 1.641r1 = 1.522r1 = 1.797r2 = 2.197r2 = 1.581r2 = 1.585——r2 = 1.586—r3 = 2.157r3 = 1.730——r3 = 1.573—r4 = 2.254r4 = 1.700——r4 = 1.541——r5 = 2.247—————r6 = 2.323—————r7 = 3.352—————r8 = 2.800———Table 2
Harmonic vibrational frequencies in cm−1. Number in parenthesis is used to label the frequency.
MgB2(MgB2)2(MgB2)3B2B4B6(1) 452(1)  64(1) 70(1) 962(1) 313(1) 184(2) 496(2) 191(2) 138—(2) 352(2) 305(3) 1132(3) 248(3) 146—(3) 1063(3) 346—(4) 253(4) 175—(4) 1182(4) 381—(5) 324(5) 254—(5) 1276(5) 560—(6) 428(6) 293—(6) 1277(6) 563—(7) 448(7) 306——(7) 713—(8) 578(8) 336——(8) 838—(9) 606(9) 380——(9) 1074—(10) 889(10) 431——(10) 1120—(11) 1080(11) 460——(11) 1261—(12) 1104(12) 489——(12) 1320——(13) 573—————(14) 581—————(15) 696—————(16) 725—————(17) 819—————(18) 973—————(19) 1078—————(20) 1193—————(21) 1221———Lowest energy
equilibrium structure of (a) MgB2, (b) B2,
(c) (MgB2)2, and (d) B4. For the
magnesium diboride clusters, the natural bond orbital analysis is
included.
(a)(b)(c)(d)Lowest energy
equilibrium structure of (MgB2)3 and
B6. Panel 2(a) and 2(b) show two different perspectives
of (MgB2)3; panel 2(c) shows the NBO for the
lowest energy equilibrium structure of
(MgB2)3; and panel 2(d) shows the lowest
energy equilibrium structure for B6.
(a)(b)(c)(d)


## 4. Conclusions

In all cases
considered, the lowest energy equilibrium structure for MgB2 clusters
showed an arrangement of boron atoms that are two dimensional or quasi-two
dimensional. The electron density is concentrated on the boron atoms and hence
the magnesium atoms are acting as electronic donating species. These results
are in accordance to what is observed for MgB2 in the bulk [13]. If
the superconductivity behavior of this material is increased by the larger
electronic density in a specific plane, then it can be expected that this
clusters should exhibit a similar superconducting behavior.



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*Source: 879017-2007-12-09.xml*