Results and discussion

Geometry structures

Using the method of combining ABC and DFT, a series of structures of (Y2O3)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.

n=1-5

Figure 1 shows the global optimum and low energy structure of (Y2O3)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 Y2O3 is symmetrical tetrahedron, D3h symmetry, kite-like 1B structure is 1.56ev higher than the optimal structure, and its spatial configuration is the global Al2O3 optimal structure[20]. The 1C linear structure is 4.1ev higher in energy than 1A, and the Y-O bond is shorter.For (Y2O3)2, the global optimal structure is a cage-like structure (2A) with Tdsymmetry, this configuration is also the optimal structure of (Al2O3)2[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 (Fe2O3)2[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 C1 symmetry. The structure of 3B is similar to that of 3A, only 0.07ev higher in energy than the latter, with CS symmetry. The structures of 3C, 3D and 3E have been reported in previous studies of alumina clusters. 4A is a newly discovered structure with C1 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 C2v symmetry, which is consistent with the results obtained by Amol B. Rahane[18] et al. 5B, 5C, and 5E are similar to the alumina cluster structure reported by RongLi[20] et al.

n=6-10

For medium-sized clusters, the spatial structure is more complex. Figure 2 shows the global optimum and lower energy structure diagram of (Y2O3)n(n=6-10) clusters. When n=6, the global optimal structure of (Y2O3)6 is a ladder structure with CS symmetry, which is different from the research results of Amol B. Rahane[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(Y2O3)7, we consider that 7A shows the optimal structure, CSsymmetry, which is different from the results reported by Amol B. Rahane[18]. 7B is a newly searched configuration with cage structure and C1 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 CS symmetry. The structures of 8B, 8D and 8E are similar, which is consistent with the partial structure of (Al2O3)8[20]. 8C is a new structure, CS 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[19]et al., which has C1 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 C60-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.

n=11-15

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 (Y2O3)n(n=11-15) clusters, which is the first calculated large Y2O3 cluster structure. For (Y2O3)11, the tower-shaped spatial three-dimensional structure is the global optimal structure, which evolves from 10A and has CS symmetry, which has the same spatial structure as (Al2O3)11 reported by Qiyao Zhang[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 CS 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 CSsymmetry. The extra two yttrium atoms and three oxygen atoms are distributed inside the ellipsoidal surface and the surface, respectively. For(Y2O3)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 C1 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.

Structural stability

The average binding energy Eb of the cluster is usually used to characterize the stability of the cluster structure. The average binding energy Eb 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 Eb is as follows :
E (Y), E (O) and E (Y2O3)n represent the total energies of single yttrium, oxygen atoms and clusters, respectively.
Fig. 4 shows the change trend of binding energy Eb of (Y2O3)n(n=1-15) clusters and Yn[30] clusters with the increase of cluster size.It can be seen from Figure 4 that the average binding energy of (Y2O3)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 Yn clusters in the growth process, and the trends of the two are similar. Compared with Yn 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 (2P4), while the valence bond orbital of yttrium atom is full (5S2), 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 Δ2E 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[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 Etotal(n+1) indicates that the system increases the total energy of an atom, Etotal(n-1) indicates that the system reduces the total energy of an atom, and Etotal (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 differenceΔ2E 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 (Y2O3)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 Δ2E 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 (Y2O3)n(n=1-15) clusters was analyzed by using the wave function analysis program Multiwfn[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 (Y2O3)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 Px, Py and Pz orbitals of oxygen (blue curve) and yttrium atom (red curve). Inspection of the green OPDOS curve, which expresses the bonding between oxygen Px, Py, Pz and yttrium atom, suggests that oxygen Px, Py and Pzorbitals are important for stabilization of (Y2O3)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.

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 (Y2O3)n(n=1-15), and the thermodynamic data of different sizes at 0-2000K were obtained by thermodynamic analysis software Shermo[34]. The results are shown in Figure 1. The calculated energy E0of yttrium oxide clusters at 0K is shown in table 1.
Fig.7(a)(c) shows the variation of the heat capacity (Cp) and entropy (S) of the calculated nanocluster (Y2O3)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 (Y2O3)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 (Y2O3)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 (Gv) of nanoclusters (Y2O3)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 Gv curves of different clusters are pairwise intersecting. It can be seen that when the temperature is below 350 K, at the same temperature, Gv increases with the increase of n; When the temperature is higher than 450K, Gv decreases with increasing n at the same temperature. For clusters with the same n value, Gv decreases with increasing temperature. In addition, the change rate of Gv with temperature of clusters with larger n value is greater than that of clusters with smaller n value. In the low temperature stage, the Gv of large clusters is larger than that of small clusters, but with the increase of temperature, the Gv of large clusters decreases with the increase of temperature. In the high temperature stage, the Gv of large clusters is lower than that of small clusters. In general, the variation of Gv with temperature of (Y2O3)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 (Y2O3)n(n=1-15) with temperature (T) and molecular number n, G=Gv+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 Gv value, so the size of G is mainly determined by the size of E (0K), and Gv 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 Gv 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[35, 36]et al.
The regression equations of thermodynamic properties of (Y2O3)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.