QUANTUM CHEMICAL MODELING OF THE STRUCTURE AND PROPERTIES OF SnO 2 NANOCLUSTERS

By the method of density functional theory with exchange-correlation functional B3LYP and basis set 3-21G (d), the structural and energy characteristics have been considered of the molecular models of SnO 2 nanoclusters of different size and composition with the number of Sn atoms from 1 to 10. Incompletely coordinated surface tin atoms were terminated by hydroxyl groups. It has been shown that the Sn–O bond length in nanoclusters does not depend on the cluster size and on the coordination number of Sn atoms, but is determined by the coordination type of neighboring oxygen atoms. Namely, the bond length Sn–O (3) (  2.10 Å) is greater than that of Sn–O (2) (  1.98 Å). The calculated values of Sn–O (3) bond lengths agree well with the experimental ones for crystalline SnO 2 samples (2.05 Å). The theoretically calculated width of the energy gap decreases naturally with increasing cluster size (from 6.14 to 3.46 eV) and approaches the experimental value of the band gap of the SnO 2 crystal (3.6 eV). The principle of additivity was used to analyze the energy characteristics of the considered models and to estimate the corresponding values for a cassiterite crystal. According to this principle, a molecular model can be represented as a set of atoms or atomic groups of several types that differ in the coordination environment and, therefore, make different contributions to the total energy of the system. The calculated value of the atomization energy for SnO 2 is 1661 kJ/mol and corresponds satisfactorily to the experimentally measured specific atomization energy of crystalline SnO 2 (1381 kJ/mol). It has been shown that a satisfactory reproduction of the experimental characteristics of crystalline tin dioxide is possible when using clusters containing at least 10 state atoms, for example, (SnO 2 ) 10  14H 2 O.


INTRODUCTION
Tin dioxide is the most common and most thermodynamically stable among tin oxides (enthalpy of formation ΔH 0 298 = -581 kJ/mol) [1,2]. Under normal conditions, tin(IV) oxide crystallizes in the tetragonal structure of rutile with lattice parameters a = 4.737 Å, c = 3.186 Å (spatial group P4 2 /mnm). For SnO 2 , this is only one stable phase. In the nature it occurs as a mineral cassiterite [2].
The SnO 2 unit cell consists of two formula units. Each tin atom is six-coordinated, its coordination environment is almost a regular octahedron (Fig. 1). The four oxygen atoms (equatorial) lie in the same plane and have shorter Sn-O bonds (1.98 Å), the other two bonds are slightly longer and are axial (2.1 Å). Oxygen atoms are tricoordinated. SnO 6 octahedra combine into chains elongated along the crystallographic direction c. Each SnO 6 octahedron has two common edges with ajacent octahedra [2,3].
The crystalline tin(IV) oxide is a wide-band n-type semiconductor with a band gap of 3.6 eV [1,3]. It is characterized by low electrical resistance (3.410 3 Ohm/cm) [4], high optical transparency in the visible region of the spectrum [5] and chemical stability at high temperatures [6]. Although SnO 2 is transparent in the visible range, it has a high reflectivity in the infrared range [3].
Real SnO 2 crystals contain various bulk and surface defects, mostly due to oxygen deficiency, which leads to the formation of regions corresponding to the lower oxides of Sn 3 O 4 and SnO [3]. In most cases, a tin atom exhibits two possible oxidation states: Sn 2+ and Sn 4+ (the latter is more stable), and oxides of SnO and SnO 2 are formed, respectively. The capability of tin atoms to be in two oxidation states determines the redox properties of the SnO 2 surface. Reduction of Sn(+4) to Sn(+2) can occur quite easily either due to chemisorption of donor molecules, or due to the capture of electrons released by the lattice anions during the formation of an oxygen vacancy [7]. A change in the composition of the oxide causes a change in the conductivity that can be measured. This is a basis for the use of tin dioxide in the production of gas sensors, in which it is used in the form of polycrystalline powder [8][9][10]. Tin dioxide is one of the classic materials, the research of which has remained promising for more than a decade. The wide range of applications of SnO 2 -based materials is due to a combination of a number of unique optical and electrophysical properties.
To date, tin dioxide, as a bulk (3D) material, is quite well studied. However, different nanostructured types of SnO 2 show better properties compared to bulk types for both gas analysis and a wide range of other applications. As shown in [20][21][22], the chemical and electrophysical properties of tin dioxide in the nanocrystalline state significantly depend on the particle size. Thus, in [23] the values of the band gap for nanoscale and filamentous nanostructures are determined, which are in the range from 3.85 to 4.2 eV and from 2.8 to 3.4 eV, respectively. Nanosized powders are characterized by slightly inflated values of the band gap, and for filamentous structures, the value of the band gap is slightly lower compared to the value characteristic of crystalline samples of cassiterite (3.6 eV).
A large number of incompletely coordinated atoms on the surface significantly changes the physical and chemical properties of low-size oxide materials compared to their bulk counterparts. During the transition to nanoscale, with the increase of the specific surface on which Sn(4+) and Sn(2+) ions coexist, the unique redox properties of the SnO 2 surface intensify. As a result, nanosized oxide materials, such as nanoparticles, nanospheres, nanotubes, nanowires, nanobands are of great interest [3].
Various applications of tin dioxide have led to the interest of researchers in the construction and systematic analysis of the properties of theoretical models for crystals and nanoparticles of this substance to study the dependence of structure -properties, in particular with the use of quantum chemistry [24]. Periodic [25][26][27][28][29] and cluster [30][31][32][33][34] approaches to modeling the structure and properties of SnO 2 are known in the literature. The latter are more convenient for reproducing properties of the nanoparticles of tin dioxide, because they do not have a long-range order. Cluster models are based on the capability to model a crystal or its surface with a small number of atoms that form it. The use of cluster models allows one the use of powerful computer programs designed to calculate the properties of molecules, but require careful selection of a molecular cluster itself. The results obtained using a cluster model depend on the size of the cluster, its stoichiometry and shape.
Based on the methods of computer modeling of the surface of the tin dioxide [1,3,28,29], a row of the catalytic activity of its faces has been proposed, namely: (110) < (001) < (100) < (101). It is known [3] that the face (110) is the most stable in oxides with the structure of rutile, and makes up most of the surface area of crystalline samples.
Using computer simulation methods that use periodic boundary conditions, the types are examined of structural defects that can be formed on the surface of tin dioxide. Thus, in [26], within the method of density functional theory, it has been found that oxygen vacancies are the main cause of n-type conductivity due to the formation of donor levels on the bottom of the conduction band due to the transition of electrons from Sn(II) to Sn(IV) sites. This applies to oxygen vacancies, which can be found both in the bulk of the crystal and on its surface. The energy of formation of oxygen vacancies in SnO 2 is very small, so such defects are formed relatively easily [29].
In [25], the bulk properties of the cassiterite crystal and the properties of (100), (110), (001), (101) SnO 2 surfaces, both pure and with water molecules adsorbed on them, were studied by the density functional theory method with imposition of periodic boundary conditions. It is shown that dissociative adsorption of water molecules is the most favorable for all considered faces of SnO 2 .
The process of doping of the tin dioxide (110) surface with Co, Cu, and Zn atoms was studied in [30] within the framework of a cluster approach. When modeling the stoichiometric surface, the Sn 15 O 30 cluster was chosen, whereas the Sn 15 O 29 and Sn 15 O 28 clusters were selected for the reconstructed surface. It has been shown that in the transition from an oxidized surface to a reduced one, the absence of bridged oxygen atoms generates electron levels in the band gap due to 5s/5p orbitals of four-and fivecoordinated Sn atoms and 2p orbitals of O atoms placed on the surface. The process of addition of Co atoms is exothermic, while addition of Cu and Zn atoms is endothermic.
Using the Sn 10 O 16 cluster as a surface tin dioxide model, its interaction with a formaldehyde molecule and the process of its partial oxidation at different temperatures were studied. Analysis of the results of calculations of SnO 2 clusters with different functionals has shown that the most acceptable functional is B3LYP [34].
The paper considers molecular models of SnO 2 of different size and composition with the number of Sn atoms from 1 to 10. The results obtained for these models are compared with the available in the literature experimental data.

COMPUTATIONAL METHOD
The calculations were performed by the method of density functional theory with the exchange-correlation functional B3LYP [35][36][37] and the basis set 3-21G(d) within the PC GAMESS software package (FireFly 8.2.0). In [34] it was shown that the use of this method with this functional gives results that are in good agreement with the experimental ones in the calculation of the atomic and electronic structure of a number of crystalline oxides, in particular tin dioxide.
The equilibrium spatial structures of the original clusters and adsorption complexes were found by minimizing the norm of the total energy gradient.

RESULTS AND DISCUSSION
The equilibrium structures of the considered SnO 2 cluster models are shown in Fig. 2. The simplest model is molecule of tin(IV) hydroxide -Sn(OH) 4 (Fig. 2 a). The cluster has a tetrahedral configuration. The Sn atom is fourcoordinated, and the four oxygen atoms associated with it are two-coordinated. The length of the Sn (4) -O (2) bonds is 1.96 Å. This cluster cannot describe all the structural properties of a cassiterite crystal, because the coordination number of a tin atom in it is only 4, and not 6, as in a crystal. When two Sn(OH) 4 molecules combine into a coordination dimer (Fig. 2 b), a cluster of (SnO 2 ) 2 4H 2 O is formed, in which two Sn atoms become five-coordinated and oxygen atoms of two hydroxyl groups become tricoordinated. This leads to the elongation of the Sn (5) -O (3) bonds, which are 2.08 and 2.14 Å. The presence of two values of bond lengths in the formed cycle indicates the molecular properties of the model. Molecular models are structures in which individual molecules can be distinguished in a cluster based on the difference between intramolecular and intermolecular bonds, which are usually longer. Unlike molecular models, polyhedral models cannot separate individual molecules because the bond lengths between the same types of atoms are identical.
A further increase in the size of the model leads to the formation of the structure (SnO 2 ) 4 6H 2 O in which two tin atoms acquire an octahedral configuration (Fig. 2 c), i.e. the coordination number becomes equal to 6, which corresponds to the structure of the SnO 2 crystal. The lengths of the Sn (6) -O (3) bonds average 2.13 Å, and the average length of the Sn (5) -O (3) bonds is slightly smaller -2.07 Å. Two types of clusters with 6 tin atoms are considered. Both clusters are characterized by the same quantitative composition, they correspond to the gross formula (SnO 2 ) 6 12H 2 O, but their structures are different. The cluster (Fig. 2 d) has a chain structure in which four tin atoms are sixcoordinated, two edge tin atoms are fivecoordinated. The lengths of the Sn (6) -O (3) bonds average 2.12 Å, and the average length of the Sn (5) -O (3) bonds is slightly smaller -2.10 Å. The second type corresponds to a cluster with a ringshaped structure, in which all tin atoms are sixcoordinated (Fig. 2 e). The average length of Sn (6) -O (3) bonds is 2.15 Å. The cluster with a ring-like structure is thermodynamically more advantageous than that with a chain structure. The largest cluster (SnO 2 ) 10 14H 2 O we considered contains 10 tin atoms, eight of which are six-coordinated (Fig. 2 f). The average length of Sn (6) -O (3) bonds is 2.14 Å, and the Sn (5) -O (3) bonds are 2.04 Å.
For all the clusters considered in the work, the average bond length Sn (5,6) An analysis of the obtained structural parameters of clusters shows that the Sn-O bond length does not depend on the cluster size and the coordination number of Sn atoms, but depends on the type of neighboring oxygen atoms (two-or tricoordinated) the tin atom is connected to. Namely, the bond length Sn-O (3) ( 2.10 Å) > the bond length Sn-O (2) ( 1.98 Å). The obtained Sn-O (3) bond lengths agree well with the experimental values for crystalline SnO 2 samples (2.05 Å). For nanoparticles, the experimentally measured bond lengths range from 1.91 to 2.16 Å.  i.e. in which the tin atoms are bound to the surface OH groups.
The principle of additivity, developed first for organic substances [38] and then adapted for crystals of inorganic substances taking into account the different coordination environment of atoms inside and on the surface of cluster models [39], was used to analyze the energy characteristics of the considered models and to estimate the corresponding values for the cassiterite crystal. According to this principle, a molecular model can be represented as a set of atoms or atomic groups of several types, which differ in the coordination environment and, therefore, make different contributions to the system energy [40]. Since in a molecular model, in contrast to the simulated crystal, there are always surface (terminal) atoms with reduced coordination numbers, to determine the energy characteristics of the formula unit of the crystal it is necessary to use at least two (or more) molecular models that differ in their coordination composition for a quantity, a multiple of the formula unit of the crystal.
The cluster (SnO 2 ) 10 14H 2 O  8Sn VI 2Sn V 18O III 16O II 28H includes eight hexaand two five-coordinated tin atoms, eighteen tricoordinated and sixteen double-coordinated oxygen atoms. According to the principle of additivity, to calculate the energy of the formula unit SnO 2 , one has to subtract the energy of the cluster (SnO 2 ) 2 4H 2 O  2Sn V 2O III 6O II 8H, consisting of two five-coordinated tin atoms and two tricoordinated and six two-coordinated oxygen atoms to obtain (8Sn VI 16O III 10O II 20H  (SnO 2 ) 8 10H 2 O), inherent in the crystal (see formula 1). 8Sn (6) 2Sn (5) The energy of the formula unit (E tot(SnO2) ) was found according to the formula (2): where E((SnO 2 ) 10 14H 2 O), E((SnO 2 ) 2 4H 2 O) and E(H 2 O) -total energies of optimized structures at 0 K. The value of atomization energy (E at ) for SnO 2 was calculated by formula (3): where E tot(Sn) , E tot(O) are the total energies of the tin and oxygen atoms in the triplet state, respectively, and the total energy of the formula unit of tin dioxide. The calculated value is 1661 kJ/mol and satisfactorily corresponds to the experimentally measured specific atomization energy of crystalline SnO 2 (1381 kJ/mol).
For tin dioxide clusters, the energy gap is theoretically calculated as the energy difference between the lower vacant molecular orbital and the higher occupied orbital. As the cluster size increases, the energy gap naturally decreases from 6.48 eV for Sn(OH) 4 to 3.46 eV for (SnO 2 ) 10 14H 2 O (see Table) and approaches the experimental value of the band gap for the SnO 2 crystal (3.6 eV).

CONCLUSION
A set of molecular models for tin dioxide crystals has been designed due to association and condensation of several (from 1 to 10) Sn(OH) 4 molecules. Their geometry and energy characteristics have been simulated within density functional theory. From the results of calculations it follows that satisfactory reproduction of the experimental characteristics of crystalline tin dioxide is possible when using clusters containing at least 10 tin atoms.