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\begin{document}

\title{Structural Stability and Thermodynamic Properties of (Y2O3)n(n=1-15)
Clusters Based on Density Functional Theory}



\author[1]{Xin Jiang}%
\author[1]{Zhenming Zhang}%
\author[1]{Diqiang Luo}%
\author[2]{Jinglin You}%
\author[1]{Chaobin Lai}%
\affil[1]{JiangXi University of Science and Technology}%
\affil[2]{Shanghai University}%


\vspace{-1em}



  
  \date{June 13, 2024}


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\selectlanguage{english}
\begin{abstract}
The initial configuration of Yttrium oxide clusters (Y2O3)n(n=1-15) was
creatively constructed by combining artificial bee colony algorithm with
density functional theory. The structures of large and medium-sized
yttrium oxide clusters with molecular number greater than 10 were
established for the first time, and many new structures that are
different from existing research have been obtained. The average binding
energy, second-order difference energy, HOMO-LUMO gap, density of states
and other properties of the clusters were analyzed. The thermodynamic
properties and behavior of nano yttrium oxide clusters at different
temperatures and sizes were discussed. Studies have shown that for
small-sized clusters, the atomic stacking structure is cage-like, while
for medium-sized and large-sized clusters, the composite trapezoidal
structure and ellipsoid-like structure are more stable. The nanoclusters
tend to be stable as a whole, and the relative stability of the cluster
structure is higher when n = 2,4,7,9. The effect of yttrium oxygen
atomic orbital on bonding is analyzed. The heat capacity (Cp), enthalpy
change (H) and entropy (S) of (Y2O3)n (n=1-15) clusters increase with
the increase of temperature (T), and the vibration free energy (Gv)
decreases with the increase of T. The stability of the clusters changes
in the temperature range of 300K-500K.%
\end{abstract}%



\sloppy


\textbf{Structural Stability and Thermodynamic Properties of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) Clusters
Based on Density Functional Theory}

Xin Jiang\textsuperscript{1}, Zhenming Zhang\textsuperscript{1}, Diqiang
Luo\textsuperscript{1},Jinglin You\textsuperscript{2},Chaobin
Lai\textsuperscript{1}

\textsuperscript{1}Faculty of Materials Metallurgy and Chemistry,JiangXi
university of science and technology,GanZhou,China

\textsuperscript{2}ShangHai university,ShangHai,China

\textbf{Correspondence}

Chaobin Lai,Faculty of Materials Metallurgy and Chemistry,JiangXi
university of science and technology,GanZhou,China

Email:lcb5115@126.com

\textbf{Abstract} :The initial configuration of Yttrium oxide clusters
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) was
creatively constructed by combining artificial bee colony algorithm with
density functional theory. The structures of large and medium-sized
yttrium oxide clusters with molecular number greater than 10 were
established for the first time, and many new structures that are
different from existing research have been obtained. The average binding
energy, second-order difference energy, HOMO-LUMO gap, density of states
and other properties of the clusters were analyzed. The thermodynamic
properties and behavior of nano yttrium oxide clusters at different
temperatures and sizes were discussed. Studies have shown that for
small-sized clusters, the atomic stacking structure is cage-like, while
for medium-sized and large-sized clusters, the composite trapezoidal
structure and ellipsoid-like structure are more stable. The nanoclusters
tend to be stable as a whole, and the relative stability of the cluster
structure is higher when n = 2,4,7,9. The effect of yttrium oxygen
atomic orbital on bonding is analyzed. The heat capacity
(C\textsubscript{p}), enthalpy change (H) and entropy (S) of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n} (n=1-15)
clusters increase with the increase of temperature (T), and the
vibration free energy (G\textsubscript{v}) decreases with the increase
of T. The stability of the clusters changes in the temperature range of
300K-500K.

\textbf{Keywords:} Nanocluster structure; Density functional theory;
Structural stability;Thermodynamic properties;ABC algorithm

\section*{Introduction}

{\label{introduction}}

Rare earth elements have many unique properties. Adding trace rare earth
elements into metal materials can improve the microstructure, mechanical
properties, oxidation resistance and corrosion resistance of metals to
varying degrees. The results show that the addition of rare earth
elements in the smelting process of metals can play a role in purifying,
modifying inclusions and microalloying\textsuperscript{{[}1-6{]}}. These
effects will change the morphology and distribution of inclusions in
metals, so that more inclusions with very small particle diameter and
dispersed distribution are generated in the solidification process of
metals, and then improve their comprehensive properties. At present, the
mechanism of rare earth's morphology and action form in metals is not
clear, which affects its efficient utilization.As a transition state
between solid state and gas state, the emergence of cluster research
provides a suitable research method for the development and improvement
of the theory of atomic bonding, the existence form and the formation
law of various macromolecules.

Yttrium oxide (Y\textsubscript{2}O\textsubscript{3}) is a deoxidation
product of rare earth treatment of molten steel. It is a typical rare
earth oxide and has the general properties of rare earth oxides. It has
a wide range of technical applications in the fields of electronics,
optics, mechanical engineering, metallurgical engineering and catalyst
support\textsuperscript{{[}7{]}}. At room temperature, yttrium oxide has
a C-type cubic structure of rare earth oxides, belonging to the
body-centered cubic structure of iron manganese ore type (space group
Ia3), similar to fluorite (CaF\textsubscript{2}) structure with a
quarter anion vacancy, and the band gap is 5.8
eV\textsuperscript{{[}8-10{]}}. Yttrium oxide cluster is a kind of
transition state and intermediate product in the nucleation of yttrium
oxide crystal. At present, most of these studies restricted to the small
size with monomer, dioxide, and trioxide
clusters\textsuperscript{{[}11-14{]}}. In the experiment, Wu and
Wang\textsuperscript{{[}15{]}} studied the electronic structure of small
YO\textsubscript{n}\textsuperscript{-} clusters with n=1-5 by
photoelectron spectroscopy (PES), obtained the vibration-resolved
photoelectron spectroscopy of YO\textsubscript{n}\textsuperscript{-},
and measured the electron affinity of YO.
Pramann\textsuperscript{{[}16{]}}et al. studied the electron affinity
and vertical ionization energy of
Y\textsubscript{n}O\textsubscript{m}\textsuperscript{-} clusters with
n=2-10, m=1-3, discussed the evolution process of the electronic
structure of neutral yttrium oxide clusters, and compared the difference
of yttrium oxide dissociation energy.
Knickelbein\textsuperscript{{[}17{]}} calculated the photoelectric
energy spectra of Y\textsubscript{n} and Y\textsubscript{n}O clusters
with n=2-31, and studied the variation law of their vertical ionization
energy. In terms of theoretical calculations, Amol B.
Rahane\textsuperscript{{[}18{]}}et al. studied small-sized yttrium oxide
clusters with molecular number of 1-10, and discussed their stability
and electronic characteristics. Rong Li and Qiyao
Zhang\textsuperscript{{[}19, 20{]}} studied the structure and stability
of (Al\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=7 and
n=15), and obtained many isomers. Xiao
Jianyun\textsuperscript{{[}21{]}}et al. studied
(HgSe)\textsubscript{n}(n=1-6) clusters using density functional theory,
obtained the equilibrium geometry, vertical ionization energy results,
and analyzed the atomic net charge distribution, frontier molecular
orbital characteristics.

Previous studies only studied the structure and photoelectric energy
spectrum of yttrium oxide ions or single element yttrium, and discussed
the effects of different oxygen to metal ratios on their structure and
electronic properties. However, there were few studies on the neutral
Y\textsubscript{2}O\textsubscript{3} clusters structure, especially the
cluster structure with medium and large sizes. In this paper, the
structure and properties of yttrium oxide clusters are studied by
artificial bee colony algorithm combined with quantum chemical
calculation, in order to improve the thermodynamic data of nano yttrium
oxide clusters, and lay the foundation for exploring the nucleation
process of rare earth inclusions in metal materials, and provide
technical guidance for the process of adding rare earth yttrium in metal
and the size control of inclusions.

\section*{Computational details}

{\label{computational-details}}

The molecular orbital theory program based on density functional theory
(DFT) in the Gaussian 09\textsuperscript{{[}22{]}} software package was
used to optimize the structure and calculate the thermodynamic
properties. First, use the ABC
algorithm\textsuperscript{{[}23-25{]}}(artificial bee colony algorithm)
to construct the initial configuration of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15), that
is, set up different boxes for different system sizes, and add a certain
number of oxygen atoms and yttrium atoms inside,automatically generates
a large number of initial configurations according to the internal
algorithm of the program. Then, preliminary optimization is carried out
under the semi-empirical method, and the more stable partial structures
are screened out according to the energy level of dozens to thousands of
calculated structures for the next step of precise calculation and
optimization. In order to ensure the accuracy of the ground state
structure, on the basis of fully considering the spin multiplicity, for
each cluster size, as many initial structures as possible are selected
for geometric optimization. Finally, the global optimal structure is
obtained and the frequency calculation is performed to obtain various
thermodynamic properties of the cluster. In the precise calculation, the
BP86 functional and mixed basis set are used, that is, the
all-electronic basis set 6-311G* is used for oxygen elements. For the
heavier rare earth yttrium element, the Stuttgart pseudopotential and
the corresponding pseudopotential are used considering the relativistic
effect\textsuperscript{{[}26-28{]}}. Finally, the optimal structure of
different isomers with n=1-15 and the structure with lower energy are
obtained.

\section*{Results and discussion}

{\label{results-and-discussion}}

\subsection*{Geometry structures}

{\label{geometry-structures}}

Using the method of combining ABC and DFT, a series of structures of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) were
obtained. When n=1-5, most of the global optimal structures obtained are
consistent with existing reports, but when n=6-15, many new stable
structures have been searched out. These structures will be discussed in
detail below.

\subsubsection*{n=1-5}

{\label{n1-5}}

Figure 1 shows the global optimum and low energy structure of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-5) clusters,
where red is oxygen atom and blue is yttrium atom. As shown in Figure 1,
when n=1, the global optimal structure of
Y\textsubscript{2}O\textsubscript{3} is symmetrical tetrahedron,
D\textsubscript{3h} symmetry, kite-like 1B structure is 1.56ev higher
than the optimal structure, and its spatial configuration is the global
Al\textsubscript{2}O\textsubscript{3} optimal
structure\textsuperscript{{[}20{]}}. The 1C linear structure is 4.1ev
higher in energy than 1A, and the Y-O bond is shorter.For
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{2}, the global
optimal structure is a cage-like structure (2A) with
T\textsubscript{d}symmetry, this configuration is also the optimal
structure of
(Al\textsubscript{2}O\textsubscript{3})\textsubscript{2}\textsuperscript{{[}19{]}}.
The 2B structure is 1.56ev higher in energy than the 2A structure. 4
oxygen atoms and 2 yttrium atoms form a plane, and are connected with 2
YO units. This configuration is a relatively stable structure of
(Fe\textsubscript{2}O\textsubscript{3})\textsubscript{2}\textsuperscript{{[}29{]}}.
When n=3, five different structures are given, and the cap structure
shown in 3A is considered to be the global optimal structure with
C\textsubscript{1} symmetry. The structure of 3B is similar to that of
3A, only 0.07ev higher in energy than the latter, with
C\textsubscript{S} symmetry. The structures of 3C, 3D and 3E have been
reported in previous studies of alumina clusters. 4A is a newly
discovered structure with C\textsubscript{1} symmetry and the lowest
energy among all configurations. The 4B structure has also been studied
by others. They believe that the optimal structure is 0.41ev lower in
energy than the 4B structure, while 4A is 0.64ev lower than 4B.
Therefore, the newly discovered 4A structure can be considered as the
global optimal structure. Compared with others, the 5A structure has
lower electron energy and C\textsubscript{2v} symmetry, which is
consistent with the results obtained by Amol B.
Rahane\textsuperscript{{[}18{]}} et al. 5B, 5C, and 5E are similar to
the alumina cluster structure reported by
RongLi\textsuperscript{{[}20{]}} et al.

\subsubsection*{n=6-10}

{\label{n6-10}}

For medium-sized clusters, the spatial structure is more complex. Figure
2 shows the global optimum and lower energy structure diagram of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=6-10)
clusters. When n=6, the global optimal structure of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{6} is a ladder
structure with C\textsubscript{S} symmetry, which is different from the
research results of Amol B. Rahane\textsuperscript{{[}18{]}} et al. The
structures of 6B and 6C are relatively similar, and the energy
difference is only 0.22 energy from 6A. 6D and 6E are newly discovered
structures. For(Y\textsubscript{2}O\textsubscript{3})\textsubscript{7},
we consider that 7A shows the optimal structure,
C\textsubscript{S}symmetry, which is different from the results reported
by Amol B. Rahane\textsuperscript{{[}18{]}}. 7B is a newly searched
configuration with cage structure and C\textsubscript{1} symmetry, which
is 0.06 ev higher than 7A in energy. 7C, 7D and 7E are similar to 7B in
atomic stack, but the energy is higher than that of the latter. When
n=8, the global optimal structure is a pyramid-like structure, which has
C\textsubscript{S} symmetry. The structures of 8B, 8D and 8E are
similar, which is consistent with the partial structure of
(Al\textsubscript{2}O\textsubscript{3})\textsubscript{8}\textsuperscript{{[}20{]}}.
8C is a new structure, C\textsubscript{S} symmetry, energy is 0.2ev
higher than the optimal structure. The global optimal structure shown in
9A is similar to the alumina cluster studied by Qiyao
Zhang\textsuperscript{{[}19{]}}et al., which has C\textsubscript{1}
symmetry and the lowest electron energy. The structures such as 9B and
9C are similar to 9A, and the energy difference is small. When n=10, the
10A with C\textsubscript{60}-like structure is the global optimal
structure, which consists of 19 yttrium atoms and 30 oxygen atoms
forming a space sphere-like surface with one yttrium atom wrapped
inside. This is a newly discovered configuration. The first five
structures with the lowest energy in global search are all spherical
structures.

\subsubsection*{n=11-15}

{\label{n11-15}}

For larger clusters, the system gradually evolves into ellipsoidal
structure. Fig. 3 shows the schematic diagram of the global optimum and
low energy structure of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=11-15)
clusters, which is the first calculated large
Y\textsubscript{2}O\textsubscript{3} cluster structure. For
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{11}, the
tower-shaped spatial three-dimensional structure is the global optimal
structure, which evolves from 10A and has C\textsubscript{S} symmetry,
which has the same spatial structure as
(Al\textsubscript{2}O\textsubscript{3})\textsubscript{11} reported by
Qiyao Zhang\textsuperscript{{[}19{]}} et al. The other isomers are
similar in structure but slightly higher in energy. When n=12, the
global optimal structure of the system is ellipsoidal (12A), which has
C\textsubscript{S} symmetry. The hollow ellipsoidal surface is formed by
22 yttrium atoms and 31 oxygen atoms. The interior contains two yttrium
atoms and five oxygen atoms, and the distance between five oxygen atoms
and the central yttrium atom is equal. The other structures (12B, 12C,
12D, etc.) also have similar spatial configurations, but the energy is
higher than that of 12A. When n=13, the optimal structure (13A) evolves
from 12A, which also has ellipsoidal structure and
C\textsubscript{S}symmetry. The extra two yttrium atoms and three oxygen
atoms are distributed inside the ellipsoidal surface and the surface,
respectively.
For(Y\textsubscript{2}O\textsubscript{3})\textsubscript{14}, we believe
that 14A is the global optimal structure, which is a spherical structure
with slightly flat ends and slightly bulging middle, and has
C\textsubscript{1} symmetry. The 14A configuration has atomic
stratification, and there are 11 layers of atoms from top to bottom and
they are symmetrical. Through the comparison of energy, it is found that
this configuration has high stability. 14B has similar spatial
configurations.Finally, when n=15, the global optimal structure is shown
in 15A, and a Y atom connects three oxygen atoms from the ellipsoid
body, which is different from the above structure. 15B, 15C have similar
configurations.

\subsection*{Structural stability}

{\label{structural-stability}}

The average binding energy E\textsubscript{b} of the cluster is usually
used to characterize the stability of the cluster structure. The average
binding energy E\textsubscript{b} of the cluster reflects the energy
released by each atom in the cluster. The larger the average binding
energy is, the more stable the system is.The calculation method of
E\textsubscript{b} is as follows :

E (Y), E (O) and E
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n} represent the
total energies of single yttrium, oxygen atoms and clusters,
respectively.

Fig. 4 shows the change trend of binding energy E\textsubscript{b} of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) clusters
and Y\textsubscript{n}\textsuperscript{{[}30{]}} clusters with the
increase of cluster size.It can be seen from Figure 4 that the average
binding energy of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) clusters
increases with the increase of cluster size, and gradually tends to be
saturated. From the perspective of growth trend, the growth rate of
yttrium oxide clusters is close to that of Y\textsubscript{n} clusters
in the growth process, and the trends of the two are similar. Compared
with Y\textsubscript{n} clusters, the average binding energy of yttrium
oxide clusters increased more obviously at the initial stage of the
cluster system. This is mainly due to the fact that the valence bond
orbital of oxygen atom is less than half full (2P\textsuperscript{4}),
while the valence bond orbital of yttrium atom is full
(5S\textsuperscript{2}), and the probability of electron filling p
orbital is much larger than that of filling s orbital. Therefore, it
shows that the average binding energy growth trend of yttrium clusters
is slower than that of mixed clusters. With the increase of the system,
the average binding energy of yttrium oxide clusters increases slowly
and gradually tends to saturation, which is similar to that of pure
yttrium clusters. This is because the coordination number of all atoms
is almost saturated, the contribution to the binding energy decreases,
and the average binding energy becomes flat. In addition, the average
binding energy of mixed clusters is larger than that of pure yttrium
clusters, which indicates that the stability of doped clusters is higher
than that of pure yttrium clusters.

Second-order energy difference \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}E is a sensitive
physical quantity to judge the relative stability of clusters in cluster
physics.In the experiment, the fine structure of the cluster mass
spectrum distribution mainly depends on the difference between the
binding energy of the two adjacent clusters, and the relatively stable
cluster corresponds to the peak position of the mass
spectrum\textsuperscript{{[}31, 32{]}}. Therefore, the relative
stability of the cluster structure corresponds to the second-order
energy difference, and the larger the difference score is, the higher
the stability is compared with the adjacent clusters. The calculation
formula is as follows :

where E\textsubscript{total}(n+1) indicates that the system increases
the total energy of an atom, E\textsubscript{total}(n-1) indicates that
the system reduces the total energy of an atom, and
E\textsubscript{total} (n) indicates the total energy of the system. The
energy levels of the highest occupied orbital (HOMO) and the lowest
space orbital (LUMO) can reflect the strength of the electron gain and
loss ability of the cluster molecules, HOMO-LUMO energy gap is an
important parameter to characterize the electronic structure and
stability of clusters. The size of HOMO-LUMO energy gap reflects the
ability of electrons to transition from occupied orbit to empty orbit,
which represents the ability of molecules to participate in chemical
reactions to a certain extent. Higher energy gap means that higher
energy is needed to change the electronic structure of clusters, so
clusters have higher stability and lower chemical properties.

Figure 5 shows the trend of second-order energy
difference\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}E and H-L energy gap with the increase of
clusters size. It can be seen that with the increase of the number of
atoms, the second-order difference energy changes alternately. When the
number of atoms in the system is 10, 20, 35, 45, 60, it is at the peak,
indicating that these structures have strong structural stability
relative to the nearby size. When the number of atoms is 10, the curve
reaches the maximum, indicating that
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{2} has the highest
stability in the whole cluster evolution process. It can be seen from
the H-L energy gap curve that when the cluster size is in the range of
5-55 atoms, the energy gap decreases with the increase of the total
atomic number of the cluster. When the atomic number is greater than or
equal to 55, the H-L energy gap fluctuates. It can be seen from the
above analysis that the change trend of the second-order energy
difference \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}E and the H-L energy gap is generally
consistent.

In order to further analyze the contribution of different elements in
the cluster system to bond formation, the molecular orbital composition
of (Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15)
clusters was analyzed by using the wave function analysis program
Multiwfn\textsuperscript{{[}32, 33{]}}. Fig. 6 shows the schematic
diagram of the total state density (TDOS), local state density (PDOS,
which reflects the contribution curve of specific fragments to TDOS) and
overlapping population state density (OPDOS) of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{14}. Discrete lines
represent original data, the curves broadened from which have been
scaled by factor of 0.03. Left axis is for total and partial density of
states, right axis is for overlap density of states. Notice that only
relative rather absolute height of curves is meaningful. The graph
clearly exhibits orbital characteristics in different energy ranges. It
is obvious that the major contribution from S basis function of oxygen
(magenta curve) is due to low-lying MOs instead of frontier MOs.The
major compositions of MOs around -7.00eV are P\textsubscript{x},
P\textsubscript{y} and P\textsubscript{z} orbitals of oxygen (blue
curve) and yttrium atom (red curve). Inspection of the green OPDOS
curve, which expresses the bonding between oxygen P\textsubscript{x},
P\textsubscript{y}, P\textsubscript{z} and yttrium atom, suggests that
oxygen P\textsubscript{x}, P\textsubscript{y} and
P\textsubscript{z}orbitals are important for stabilization of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{14}. HOMO is almost
purely contributed by yttrium orbitals. For all virtual MOs, OPDOS curve
is in negative region and show antibonding characteristic. This is due
to the unfavorable overlapping in orbital phase,as can been seen from
LUMO isosurface.

\subsection*{Thermodynamic property}

{\label{thermodynamic-property}}

In order to further analyze the influence of the structure and size of
nanoclusters on their thermodynamic properties, gaussian09 was used to
calculate the frequency of yttrium oxide
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15), and the
thermodynamic data of different sizes at 0-2000K were obtained by
thermodynamic analysis software Shermo\textsuperscript{{[}34{]}}. The
results are shown in Figure 1. The calculated energy
E\textsubscript{0}of yttrium oxide clusters at 0K is shown in table 1.

Fig.7(a)(c) shows the variation of the heat capacity
(C\textsubscript{p}) and entropy (S) of the calculated nanocluster
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) with
temperature (T) and the molecular number n of the clusters. It can be
seen that in the temperature range of 0-2000 K, at the same temperature,
the heat capacity and entropy increase with the increase of n; For
clusters with the same n value, the heat capacity and entropy increase
with increasing temperature. In addition, the change rate of the two
decreases with the increase of temperature, that is, in the low
temperature stage, the change is faster with the increase of
temperature. In the high temperature stage, the increase rate decreases
with the increase of temperature, and finally tends to be constant. It
can be seen that the variation law of heat capacity and entropy of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n} with temperature
is consistent with that of conventional materials, but the two change
with n (molecular number, representing different structures and also
changing the size to a certain extent), which reflects the influence of
structure and size on thermodynamic properties.

Fig.7(b) shows the variation of the enthalpy (H) of nanoclusters
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) with
temperature (T) and clusters size. In the temperature range of 0-2000K,
at the same temperature, H increases with the increase of n; For
clusters with the same n value, H also tends to increase with the
increase of temperature. In addition, the rate of change of H with
temperature gradually increases with the increase of temperature. That
is, in the low temperature stage, the heat capacity changes slowly with
the increase in temperature, and in the higher temperature stage, the
increase in the heat capacity gradually increases with the increase in
temperature.

Fig.7(d) shows the variation of vibrational free energy
(G\textsubscript{v}) of nanoclusters
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) with
temperature (T) and clusters size. The internal amplification diagram is
the variation law in the temperature range of 300-500K. In this
temperature range, the G\textsubscript{v} curves of different clusters
are pairwise intersecting. It can be seen that when the temperature is
below 350 K, at the same temperature, G\textsubscript{v} increases with
the increase of n; When the temperature is higher than 450K,
G\textsubscript{v} decreases with increasing n at the same temperature.
For clusters with the same n value, G\textsubscript{v} decreases with
increasing temperature. In addition, the change rate of
G\textsubscript{v} with temperature of clusters with larger n value is
greater than that of clusters with smaller n value. In the low
temperature stage, the G\textsubscript{v} of large clusters is larger
than that of small clusters, but with the increase of temperature, the
G\textsubscript{v} of large clusters decreases with the increase of
temperature. In the high temperature stage, the G\textsubscript{v} of
large clusters is lower than that of small clusters. In general, the
variation of G\textsubscript{v} with temperature of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) clusters
is consistent with that of macroscopic thermodynamics, but the special
law of vibrational free energy with temperature reflects the
inconsistency of structural stability of large and small clusters with
temperature, which is completely different from macroscopic materials
and is the embodiment of special thermodynamic properties of
nanoclusters.

Fig.7(e) shows the variation of absolute Gibbs free energy(G) of
nanoclusters
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) with
temperature (T) and molecular number n, G=G\textsubscript{v}+E(0K). It
can be seen from the graph that G decreases with the increase of n at
the same temperature from 0 to 2000 K. For clusters with the same n
value, the change trend of G is not obvious with the increase of
temperature, which is mainly because for a cluster with a n value, its E
(0K) is far lower than its G\textsubscript{v} value, so the size of G is
mainly determined by the size of E (0K), and G\textsubscript{v} only has
a certain influence on G. Therefore, the G curves of different clusters
are almost parallel, but actually not parallel. The slope is caused by
the G\textsubscript{v} value of different clusters. It can also be seen
from the absolute Gibbs free energy that the structure of the cluster
plays a decisive role in its thermodynamic properties. Thermodynamic
properties of nano yttrium oxide clusters are similar to those of
alumina clusters studied by Guo Cheng Wang\textsuperscript{{[}35,
36{]}}et al.

The regression equations of thermodynamic properties of
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) clusters
with temperature and molecular number are shown in tables 2 and 3. These
thermodynamic properties with temperature and their characteristics can
be used to explore the thermodynamic problems of the evolution of
deoxidation products in the process of yttrium deoxidation chemical
reaction.

\section*{Conclusions}

{\label{conclusions}}

The initial configuration of Yttrium oxide clusters
(Y\textsubscript{2}O\textsubscript{3})\textsubscript{n}(n=1-15) was
creatively constructed by combining artificial bee colony algorithm with
density functional theory. The structures of large and medium-sized
yttrium oxide clusters with molecular number greater than 10 were
established for the first time, and their ground state structures were
determined by structural optimization and frequency analysis in Gaussian
09 software package. The average binding energy, second-order difference
energy, H-L energy gap,density of states and thermodynamic properties of
each structure were calculated and analyzed in detail.the result shows:

\begin{enumerate}
\tightlist
\item
  For small clusters with n\textless{}5, oxygen atoms and yttrium atoms
  tend to form cage-like clusters. With the increase of size (n=6-15),
  the clusters structure change from cage-like to space stair-like, and
  gradually evolves into stable ellipsoid-like structure.
\item
  The stability and molecular orbital of the clusters were analyzed in
  detail, and it was found that the cluster structure was generally
  stable. Because the valence bond orbital of oxygen atom is less than
  half full (2P\textsuperscript{4}), and the valence bond orbital of
  yttrium atom is full (5S\textsuperscript{2}), the probability of
  electron filling p orbital is much larger than that of filling s
  orbital, so the binary mixed yttrium oxide clusters have more stable
  structures than the single yttrium clusters. When the number of
  cluster molecules n=2,4,7,9, the relative stability is higher, and the
  second-order energy difference score is basically consistent with the
  results of H-L energy gap in determining the stability of cluster
  structure. The major contribution from S basis function of oxygen
  (magenta curve) is due to low-lying MOs instead of frontier MOs. HOMO
  is almost purely contributed by yttrium orbitals.
\item
  C\textsubscript{p}, H and S of
  (Y\textsubscript{2}O\textsubscript{3})\textsubscript{n} increased with
  the increase of T, and increased with the increase of cluster size.
  G\textsubscript{v} decreases with the increase of temperature, and the
  change rule with cluster size is greatly affected by T. In the
  temperature range of 300K-500K, the G\textsubscript{v} lines intersect
  each other, indicating that the stability of nanoclusters has changed
  in the temperature range of 300K-500K. The thermodynamic properties of
  (Y\textsubscript{2}O\textsubscript{3})\textsubscript{n} vary greatly
  with the number of clusters, which is similar to the size effect of
  the thermodynamic properties of nanoscale materials.
\end{enumerate}

The structure and related thermodynamic data of nanoclusters calculated
in this study have certain guiding significance for studying the
nucleation of yttrium oxide crystals from the perspective of
nanothermodynamics.

\section*{Acknowledgement}

{\label{acknowledgement}}

The authors gratefully acknowledge the financial support from the
National Natural Science Foundation of China(No.51974139) and Jiangxi
Province Major Scientific and Technological Research and Development
Special Funding Project(No.20194ABC28011).

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