1. Introduction
TiO2 is one of the most widely used materials in science [1] and technology [2]. Recent
studies show that TiO2 nanostructures give high results in photocatalytic activity [3-7]. TiO2
nanoparticles have been synthesized and applied in various regions, especial in photochemical
applications [8]. Photocatalytic activities of TiO2 nanoparticles have been improved by different
methods, the two most common are doping [9-11] and compositing with other materials [12, 13].
To predict the photocatalytic activities of TiO2-based materials, different models of TiO2 clusters
have been created, optimized and used in modified TiO2 models [14].
Xiaohui et al. [15] show that the magnetism of TiO and TiO2 clusters depend on their
shapes but the stable structures had not been discussed. Chiodo et al. [16] studied the four most
common types of TiO2 clusters: linear chain, ring, rutile-like and anatase-like structures but the
most stable clusters have not been clearly shown. Zhang et al. [17] created the most common
structures and studied the geometry structures, the coordinates and binding energies of (TiO2)n
clusters with n =1÷9. The compact structures are more energetically favorable than other
structures like quasi-linear or circular ones while Blagojevic [18] showed the contrary results.
The previous results do not show a clearly view on the most stable structures of TiO2
clusters and lead to the difficult in the application of calculations on modified TiO2 materials. In
this research, all of possible structures of (TiO2)n clusters in the range of n = 1÷10 have been built,
optimized, calculated to find the most stable structures for each value of n. These stable structures
can be used to create models of composite materials of TiO2.

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52
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0069
Natural Sciences 2018, Volume 63, Issue 11, pp. 52-58
This paper is available online at
A TDDFT STUDY ON TiO2 CLUSTERS
Duong Quoc Van and Nguyen Minh Thuy
Faculty of Physics, Hanoi National University of Education
Abstract. In this research, first principle calculations based on Time-Dependent Density
Functional Theory (TDDFT) have been used to study the geometry structures, the binding
energies and electronic properties of (TiO2)n clusters with n = 1÷10. All possible structures of
TiO2 clusters have been built, optimized and studied. The geometry optimization calculations
indicate that almost stable structures with lowest total energies also have high symmetric
shapes than the others. The calculated binding energies of stable models show that TiO2
clusters tend to group together to form larger and more stable structures.
Keywords: TiO2 clusters, TDDFT, stable structures, binding energy, electronic structures.
1. Introduction
TiO2 is one of the most widely used materials in science [1] and technology [2]. Recent
studies show that TiO2 nanostructures give high results in photocatalytic activity [3-7]. TiO2
nanoparticles have been synthesized and applied in various regions, especial in photochemical
applications [8]. Photocatalytic activities of TiO2 nanoparticles have been improved by different
methods, the two most common are doping [9-11] and compositing with other materials [12, 13].
To predict the photocatalytic activities of TiO2-based materials, different models of TiO2 clusters
have been created, optimized and used in modified TiO2 models [14].
Xiaohui et al. [15] show that the magnetism of TiO and TiO2 clusters depend on their
shapes but the stable structures had not been discussed. Chiodo et al. [16] studied the four most
common types of TiO2 clusters: linear chain, ring, rutile-like and anatase-like structures but the
most stable clusters have not been clearly shown. Zhang et al. [17] created the most common
structures and studied the geometry structures, the coordinates and binding energies of (TiO2)n
clusters with n =1÷9. The compact structures are more energetically favorable than other
structures like quasi-linear or circular ones while Blagojevic [18] showed the contrary results.
The previous results do not show a clearly view on the most stable structures of TiO2
clusters and lead to the difficult in the application of calculations on modified TiO2 materials. In
this research, all of possible structures of (TiO2)n clusters in the range of n = 1÷10 have been built,
optimized, calculated to find the most stable structures for each value of n. These stable structures
can be used to create models of composite materials of TiO2.
Received August 30, 2018. Revised October 12, 2018. Accepted October 19, 2018.
Contact Duong Quoc Van, e-mail address: vandq@hnue.edu.vn
A TDDFT study on TiO2 clusters
53
2. Content
2.1. Computational methods
Initial TiO2 clusters have been manually built using visualization package from Materials
Studio [19] - a commercial software widely used in modelling materials. TDDFT-based
calculations were performed using DMol3 module [20, 21] in Materials Studio software.
Generalized Gradient Approximation (GGA) were used for the exchange-correlation functional
with the parametrization presented by Perdew, Burke and Ernzerhof (PBE) [22]. The electron-ion
interactions were modeled using norm-conserving pseudopotentials within The Density
Functional Semi-core PseudoPotentials (DSPP) method; the valance configurations of the atoms
were 3s
2
3p
6
3d
2
4s
2
for Ti and 2s
2
2p
4
for O. The convergence threshold for self-consistent iterations
was set at 10
-6
eV. The energy change, maximum force and maximum displacement tolerances in
the geometry optimization processes were set at 10
−5
Ha/atom, 0.002 Ha/Å and 0.005 Å,
respectively.
2.2. Results and discussion
2.2.1. Structures of TiO2 clusters
1.1
2.1
2.2
2.3
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Figure 1. Initial structures of (TiO2)n clusters with n = 1÷5
In general, there are many possible structures of (TiO2)n corresponds to a certain value of n
(especially for large n) but not all of them are able to exist. All initial structures of (TiO2)n with
n = 1÷5 are shown in Figure 1, total number of clusters is 21 (1 structure for n = 1; 3 structures for
Ti O
Duong Quoc Van and Nguyen Minh Thuy
54
n = 2; 4 structures for n = 3; 6 structures for n = 4 and 7 structures for n = 5). For n from 6 to 10,
there are 32 possible structures (12 structures for n = 6; 7 structures for n = 7; 4 structures for n = 8; 6
structures for n = 9 and 3 structures for n = 10). These structures are shown in Figure 2.
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
7.1
7.2
7.3
7.4
7.5
7.6
7.7
8.1
8.2
8.3
8.4
9.1
9.2
9.3
9.4
9.5
9.6
10.1
10.2
10.3
Figure 2. Initial structures of (TiO2)n clusters with n = 6÷10
Ti O
A TDDFT study on TiO2 clusters
55
To find the most stable structure, geometry optimization were performed for all initial clusters
(TiO2)n with n from 1 to 10. The optimization processes show that a few local minimums might
appear in optimization process or not but the final structures always have the lowest energy. The
total energies for all clusters were shown in Figure 3.
Figure 3. Calculated total energies of (TiO2)n clusters
For n = 1, there is only one possible structure so it is not necessary to find the lowest energy
model. For n = 1÷5, the lowest total energy structures have linear shapes while the corresponding
structures for larger n are more compact ones, is inconsistent with the results of Qu [23] and
Blagojevic [18]. For n = 6÷10, compact structures are more favorable, in a good agreement with
previous studies [18, 23].
The most stable structures for each value of n were shown in Figure 4. Figure 4 also show
that all stable structures are more symmetry than the others, this can be used to guest the stable
structures for larger n values.
Duong Quoc Van and Nguyen Minh Thuy
56
n =1 n = 2 n = 3 n = 4 n = 5
n = 6 n = 7 n = 8 n = 9 n = 10
Figure 4. The most stables structures of (TiO2)n clusters with n = 1÷10
2.2.2. Binding energies of TiO2 clusters
For each value of n, the model which has smallest total energy is the most stable structure.
To investigate the formation processes of TiO2 clusters, the average binding energies for the most
stable structures of (TiO2)n with n = 1÷10 were calculated using following formula [17]
02( ) Ti n
nE nE E
BE n
n
(1)
where ETi is the energy of a Ti atom in the bulk form, E0 is the energy of a O atom in a free O2
molecule and En is the total energy of the corresponding cluster. In this paper, the values of ETi
and E0 are 1943.5259 eV and -2048.4620 eV, respectively.
The dependence of average binding energy of different TiO2 clusters on the value of n
(from 1 to 10) was shown in Figure 5. Figure 5 also shows that the increase rate of average
binding energy of TiO2 clusters is larger for n = 1÷3 and smaller for n = 4÷10 so it can be guest
that the binding energy can converge at larger value of n. This prediction has not been checked in
this research.
Figure 5. The change of binding energy of (TiO2)n on n value
A TDDFT study on TiO2 clusters
57
2.2.3. Electronic structure of TiO2 clusters
Figure 6 shows the total density of states of (TiO2)n clusters with n = 1÷10 and the projected
corresponding to s, p and d orbitals (Figure 6b, 6c and 6d). The total density of state of TiO2
clusters on Figure 6a show the differences between two groups – group of small clusters with
n = 1÷10 and group of larger clusters with n = 6÷10. It can be seen that the p electrons play an
important role in the formation of TiO2 clusters while s and d electrons do not. For small cluster,
the p electrons are more mobile but for larger cluster, they are localized in the states around 0 eV.
The differences in mobility of p electrons could be the effects of cluster shapes – ring shapes of
small clusters and compact shapes of larger clusters.
Figure 6. (a) Total density of states and (b,c,d) projected density of states of (TiO2)n clusters
3. Conclusions
The most stable structures with lowest total energies, the average binding energies and
electronic properties of TiO2 clusters were obtained from TDDFT-based calculations using norm
conserving pseudopotentials, double numerical plus polarization (DNP) basis and GGA-PBE
formulation of correlation and exchange energy. Almost symmetry structures are more stable than
the remains, this can be used to predict the shapes of larger clusters. For n = 1÷5, the favourable
structures have the ring-shape while for n = 6÷10, the compact shapes are easier to form. The
calculated average binding energies show that TiO2 clusters prefer to form larger clusters than stay
as single small clusters. Stable structures of TiO2 will be used to model for hybrid materials such
as TiO2/CNTs, TiO2/Graphene or other TiO2-modified materials.
a b
c d
Duong Quoc Van and Nguyen Minh Thuy
58
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