Abstract. To reveal the origin of the (ZrO2)n (n = 1 ÷ 11) cluster stability and to study the
structural trends in the zirconium oxide neutral cluster distribution, (ZrO2)n clusters have
been constructed and calculated employing Density Functional Theory (DFT). In this work,
we calculate the formation energy, electronic structures, stabilities and Raman spectra for
varying isomers of (ZrO2)n clusters. The total binding energy of the cluster (ZrO2)n indicates
that energies of stabilization does not decrease monotonically with increasing size and the
energies change very slow from n = 5. The results of the calculated Raman spectra of the
clusters (ZrO2)n were compared with the experimental data. We also consider anionic clusters
and analyze the both the neutral and anionic clusters (ZrO2)n. This calculation results are
useful for studies on nanometer-sized photocatalysts.

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92
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0033
Natural Sciences 2018, Volume 63, Issue 6, pp. 92-99
This paper is available online at
A STUDY OF THE (ZrO2)n (n = 1 ÷ 11) CLUSTERS
BY DENSITY FUNCTIONAL THEORY
Nguyen Minh Thuy and Nguyen Phuong Lien
Faculty of Physics, Hanoi National University of Education
Abstract. To reveal the origin of the (ZrO2)n (n = 1 ÷ 11) cluster stability and to study the
structural trends in the zirconium oxide neutral cluster distribution, (ZrO2)n clusters have
been constructed and calculated employing Density Functional Theory (DFT). In this work,
we calculate the formation energy, electronic structures, stabilities and Raman spectra for
varying isomers of (ZrO2)n clusters. The total binding energy of the cluster (ZrO2)n indicates
that energies of stabilization does not decrease monotonically with increasing size and the
energies change very slow from n = 5. The results of the calculated Raman spectra of the
clusters (ZrO2)n were compared with the experimental data. We also consider anionic clusters
and analyze the both the neutral and anionic clusters (ZrO2)n. This calculation results are
useful for studies on nanometer-sized photocatalysts.
Keywords: Zirconium, clusters, absorption, Raman frequency, Density functional theory.
1. Introduction
Zirconium ZrO2 is an important ceramic materials with high melting point, low thermal
conduction and high ionic conductivity, so that ZrO2 possesses excellent thermal, dielectric,
mechanical, chemical and biocompatibility properties [1, 2]. Zirconium is also a useful catalysis
or an important support material for catalysis, having acidic and alkaline properties. The
zirconium nanoparticles will be widely applied in high-performance structural engineering
ceramics and catalyst industry. The lack of understanding on the electronic structure and catalytic
activity of metal oxides particles and surfaces is still under debate. Many studies have been carried
out on zirconium oxide clusters to correlate their properties with those of the bulk.
For efficient and cost effective applications of nanomaterials based on ZrO2, their structural
and physical properties are required at molecular and atomic scale. Due to complexity and very
high cost of equipment, only limited experiments have been conducted to characterize required
properties at nanoscale. This led to computational modeling and simulation of various structures
and properties of ZrO2 based on atomistic and continuum approaches [3].
A study of small metal oxide cluster reactivity as a function of cluster size can help us to find
which role played by each of these properties in cluster activity. In all applications, surface
properties are of major importance [3].
Received July 25, 2018. Revised August 6, 2018. Accepted August 14, 2017.
Contact Nguyen Minh Thuy, e-mail address: thuynm@hnue.edu.vn
A study of the (ZrO2)n (n = 1 ÷ 11) clusters by density functional theory
93
In this work the structures, stabilities and properties of small zirconium oxide clusters (ZrO2)n
(n = 1÷11) are studied by density functional theory calculations. We calculate the formation
energy, electronic structures and stabilities for varying configuration of (ZrO2)n (n = 1÷11)
clusters.
By comparing with experimental investigation, a theoretical analysis by computer simulation
is expected to clarify varies effects in detail [4, 5]. With this aim, DFT semi-core pseudopotentials
(DSPP) method within the framework of density functional theory (DFT) [6, 7] has been adopted
in this work. We calculate the Raman modes of the zirconium oxide clusters (ZrO2)n (n = 1÷11)
and their anions. The calculation results have been compared with the experimental and other
calculated results.
2. Content
2.1. Computational methodology
The structures and stabilities of (ZrO2)n (n = 1÷11) clusters are calculated using DFT.
Geometry optimizations with full relaxation of all coordinates are done for these clusters in
attempt to find the global minimum on the cluster potential hypersurface of each cluster size.
Calculations of total energy and electronic structure were carried out using the DMOL3 package
within the framework of DFT. The Perdew – Burke – Ernzerhof (PBE) parameterization of the
generalized gradient approximation (GGA) [5] was adopted for the exchange-correlation potential.
The atomic orbital was specified by double numerical plus polarization (DNP) and the core
electrons were described by DFT semi-core pseudopotentials (DSPP). The Zirconium 4s, 4p, 4d,
5s and 5p; the Oxygen 2s, 2p and 3d electrons were treated as part of the valence states. Cut off
element values for the Zr and O were chosen for fine calculation and were assumed equal for both,
is 10.2 Å.
The self- consistent ﬁeld (SCF) convergence criterion is set to 1 × 10-6 Ha for the total energy.
A 0.005 Ha/Å smearing was applied to the system to facilitate convergence of the electronic
structure. The point symmetry group of the resulting clusters is determined with the tolerance of
0.001 Å. Total density of state (DOS) and projected DOS (PDOS), adsorption energy, charge
transfer, Raman modes calculations was performed.
2.2. Results and discussion
2.2.1. Stability of the (ZrO2)n clusters (n = 1÷11)
There are many stable isomers corresponding to different local minima (LM) on the cluster
potential hypersurface for each cluster size [8]. The most stable isomers (the global minima (GM)
on the cluster potential surface) are found from a number of different optimized configurations.
We have considered many different structures for each cluster size. We built 2, 4, 12 different
configurations for n = 1, 2, 3 respectively. For the n > 3, the DFT energy calculation predicted that
the lowest energy configuration is the boomerang arrangement or close - rigid ion model [9].
Many of the local minima found for the larger size clusters (n > 4) can be constructed from
smaller local minima clusters and they are in a good agreement with report [9].
The most stable configurations of the neutral (ZrO2)n clusters are shown in Figure 1.
In order to quantify the relative stabilities of these clusters, we have calculated the average
total binding energies Eb(n) for the most stable (ZrO2)n configurations for cluster size n = 1÷11.
Figure 2 shows the binding energy per ZrO2 unit as a function of n. Eb(n) is a measure of average
energy per ZrO2 unit and indicates the stability of a particular cluster in terms of the (ZrO 2)n.
Nguyen Minh Thuy and Nguyen Phuong Lien
94
All clusters (n > 1) have positive Eb and are stable with respect to a single ZrO2 unit. Eb(n)
approaches that of the bulk phase structure while the cluster size increases. The Eb(n) is defined by
22 n n
b
nE Zr nE O E Zr O
E n
n
where E (Zr) = -1938,145 eV, E(O) = -2040.678 eV are the energies of a Zr and O atom,
respectively (when they were as separate atoms and not bonding to other atoms).
The energies of the GM clusters are shown from figure 2 as a function of cluster size n.
Figure 1. The most stable configurations of the neutral (ZrO2)n clusters for n = 1÷11. Light
blue and red indicate zirconium (Zr) and oxygen (O) atoms, respectively
Table 1. The binding energy per ZrO2 unit of the neutral (ZrO2)n clusters
Cluster size n Eb(n)(eV)
1 17.78
2 19.80
3 20.64
4 21.10
5 21.60
6 21.84
7 22.03
8 22.30
9 22.48
10 22.53
11 22.53
1 2
3
4
5
6 7
8
9 10
11
A study of the (ZrO2)n (n = 1 ÷ 11) clusters by density functional theory
95
Figure 2.The binding energy per ZrO2 unit of the neutral (ZrO2)n clusters
The average total binding energies Eb(n) for the most stable form in each cluster size does not
rise monotonically as the cluster increases in size. There is a slight increase in these energies from
n = 5. As can be seen from this graph, the binding energy diﬀerences between clusters of a larger
size tend to be smaller.
2.2.2. Properties of the neutral and anionic (ZrO2)n clusters for n = 8
Considering the different geometry configurations of the (ZrO2)n clusters can clarify the
structures and growth modes of the large size clusters, what related to the final bulk phase
structure ZrO2. As shown in Figure 2, the larger size (ZrO2)n clusters (n > 8) have only a slight
increase of the average binding energies, so we choose (ZrO2)8 clusters for the more detail
configuration research.
Figure 3. The relaxed configurations of the neutral (ZrO2)8 clusters and labeled 8x X,
where X indicate its symmetry point group and x are the isomers of neutral (ZrO2)8 clusters
8a C
1
8b C
1
8c C
i
8d C
1
8e C
1
8f C
1
8g C
1
8h D2d
Nguyen Minh Thuy and Nguyen Phuong Lien
96
We have built 12 different (ZrO2)8 typical cluster structures and charged them with -1 (e).
After geometry optimizing, these neutral (ZrO2)8 clusters configurations are divided into typical
two groups. Group I consists of fourconfigurations (8a, 8b, 8c, 8d) in which there are one or two
terminal O atoms. The 8b and 8d structure each have one terminal O atom. The 8a and 8c
structure each have two terminal O atoms binding to inverse Zr atoms in opposite corners. In these
structures, there are some three – fold coordinated O atoms and almost all of Zr atoms have four -
foldcoordination. Group II includesfourconfigurations (8e, 8f, 8g, 8h) in which all structures have
no terminal O atom. All optimized configurations of (ZrO2)8 cluster are shown in Figure 3. Anion
configurations are structurally similar to neutral configurations and differences are not much.
The point symmetry group of the relaxed clusters is determined with the tolerance of 0.001Å.
The symmetry group of neutral (ZrO2)8 clusters was predicted to be C1 for the 8a, 8b, 8d, 8e, 8f,
8g structures, Ci for the 8c structure, and D2d for the 8h structure.
* Density-of-state (DOS) calculation
Figures 4 and 5 show the DOS for the neutral and their anionic (ZrO2)8 clusters. In the
anionic (ZrO2)8, the extra electron enters the predominantly Zr – 4d state based on LUMO of
neutral (ZrO2)8, which should evolve into the conduction band of the bulk. The HOMO of the
neutral (ZrO2)8 clusters derives primarily from O – 2p state and should evolve into the bulk
valence band. At higher energy from 0 eV there is a relatively board band of O - 2p state that are
hybridized with Zr - 4d state. There is a clear peak in the O - 2p state ranging from -0.5 to 5eV is
seen for the group I clusters, reflecting the local environment of terminal O atoms in configurations
of the group I. These terminal O atoms can be related to some states in the gap, which are likely to
affect the photocatalytic properties of ZrO2 clusters.
Figure 4. Total density-of-states (DOS) for neutral (ZrO2)8 clusters corresponding group I
configurations (8a, 8b) and group II configurations (8e, 8f)
A study of the (ZrO2)n (n = 1 ÷ 11) clusters by density functional theory
97
Figure 5 shows that the DOS of anionic (ZrO2)8 clusters tend to shift to lower energy,
however the DOS structures for the neutral and anionic (ZrO2)8 clusters do not change
significantly.
Figure 5. Total density-of-states (DOS) for the structure 8a corresponding to the neutral (left)
and anionic (right) (ZrO2)8 clusters
* HOMO-LUMO energy
Energy difference between HOMO and LUMO orbital is called as energy gap (Eg) which is
an important stability for structures. Thus, we have calculated the HOMO (EHUMO), LUMO (ELUMO)
energies, and the energy gap Eg (=EHUMO - ELUMO), of all clusters in order to elucidate the
electronic properties. In general, there is a direct relation between the stability and Eg of the
clusters, and a larger Eg implies a higher stability of a particular system. Table 2 shows the
calculated HOMO and LUMO energies of the neutral and the anionic (ZrO2)n cluster, in eV unit.
Table 2. Calculated HOMO and LUMO energies of the neutral and the anionic (ZrO2)n cluster
(EHUMO, ELUMO, (EHUMO - ELUMO) = Eg, eV)
Structure Neutral clusters Anionic clusters
EHOMO ELUMO Eg EHOMO ELUMO Eg
8a -5.927 -3.291 2.636 -1.312 -0.624 0.688
8b -5.884 -3.309 2.575 -2.786 -0.852 1.934
8c -5.927 -3.291 2.636 -3.222 -0.618 2.604
8d -5.603 -3.109 2.494 -2.623 -0.493 2.130
8e -6.384 -3.046 3.338 -3.09 -0.272 2.818
8f -6.373 -3.033 3.340 -3.089 -0.298 2.791
8g -6.449 -3.845 2.604 -3.096 -0.931 2.165
8h -6.359 -2.985 3.374 -2.914 -0.474 2.440
The result from Table 2 shows that the calculated Eg values range from the visible (2.494 eV) to
ultraviolet region (3.374 eV). The calculated Eg of the n = 8 cluster is smaller than the Eg of the
calculated bulk monoclinic, cubic and tetragonal ZrO2 phases (3.5 to 4.7 eV) [10]. This trend
could lead to design of suitable materials for photocatalytic applications of ZrO2 clusters.
Nguyen Minh Thuy and Nguyen Phuong Lien
98
* Vibrational properties
We have initially calculated Raman spectra for neutral (ZrO2)n clusters. All calculations use
the same condition with room temperature (300 K) and incident light wavelength 488nm. As can
be seen from Figure 6, the prominent Raman (or infrared) peaks are observed mainly in two
regions. Region I has wavenumber values ranging from 0 to 250 cm
-1
, which has peaks assigned
to O – O bondstretching vibrations. Region II has a wavenumber values range of 400 to 900 cm-1,
which has peaks assigned to Zr - O bond stretching vibrations [11]. For the group I configurations,
the sharp peaks ranging from 600 to 900 cm
-1
observed may correspond to terminal O (the Zr-O
vibration [12]).
Figure 6. The calculated Raman spectra for the neutral (ZrO2)8 clusters
At present, there is no direct method for measuring the structure and properties of small
clusters by experimental study. The author of [12] investigated infrared spectra of zirconia
nanoparticles prepared by the precipitation method and compared the experiment result with
theoretical IR spectra. Cubic zirconia has only one IR active peak at 480 cm
-1
, the IF spectra at a
low region between cubic and tetragonal zirconia are very close. The peaks of 500 - 600 cm
-1
for
the cubic-like clusters (8g, 8h configurations in Figure 2) could be ascribed to the characteristic
vibration mode of cubic zirconia. Vibration frequencies between 600 - 700 cm
-1
were assigned to
Zr-O vibrations in the Zr-O-Zr-O ring [12].
3. Conclusion
We have studied the stability, structural, electronic properties of the neutral and anionic
(ZrO2)n clusters, where n = 1÷11 by using DFT. We have observed that the structure and the size
of the clusters have main effects on the stability, electronic and other properties of these clusters
structures. The stability of the clusters increases as the cluster size grows.
The initial calculations of Raman spectra show the influence of terminal O atoms to the peaks
of the Raman spectra.
The calculated HOMO - LUMO energy gap Eg values range from the visible to ultraviolet
regions. This property could lead to design of suitable materials for photocatalytic applications of
ZrO2 clusters.
A study of the (ZrO2)n (n = 1 ÷ 11) clusters by density functional theory
99
Acknowledgements. This work was supported by the Vietnamese National Foundation for
Science and Technology Development (NAFOSTED) under Grant N 103.02-2017.328. A part of
this work was performed at the Environment Physics Project’s Laboratory at the General Physics
Division of the Department of Physics, Hanoi National University of Education.
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