Mathematical modeling of the electronic structure of Titanium dioxide ( TiO 2 ) 6 nanoparticles

The calculation of the number of atoms of the given dimensional nanoparticle, composed of different type atoms has been researched in this work. The calculations have been carried out for nanoparticles of titanium dioxide. Theoretical visual models have been configured, and quantum – mechanical calculations have been carried out for (TiO2)6 nanoparticle. The calculations for titanium dioxide nanoparticle have been carried out on the basis of Gaussian atomic orbitals. Besides, Gaussian functions have been used as atomic orbitals. The numerical values of unknown coefficients of the linear combination of atomic orbitals of the atoms of the titanium nanoparticle have been found from the solution of Hartree–Fock–Roothaan (HFR) equations.The values of orbital energies, ionization potential, and the total electronic energy of titanium dioxide nanoparticles have been determined . The calculations show that, titanium dioxide nanoparticle is tough, electrophile, and stable dielectric, material. The effective charge of atoms have been calculated, and the theoretical visual mode of titanium dioxide nanopartical have been constructed.


1.Introduction
Titanium dioxide nanoparticle material is a wide class of material used in electronics, optoelectronics,and piezo technic. For this possibility of application , it has a great importance to investigate electronic structure of titanium dioxide by quantum mechanical methods [1,9,10,12,14]. In this work, electronic structure and properties of Titanium dioxide nanoparticle have been studied by Hartree-Fock-Roothaan (HFR) method. As HFR method, the state of electron in the molecule is represented with one-electronic wave functions called Ui-molecular orbitals. Ui is represented as linear combinations of q atomic orbitals of atoms of molecules [4,5,8,12,14], with : (1) q orbitals are considered as known. Unknown coefficients Cqi are found from the solution of HFR equations. We can express these equations in matrix form as following: FC=εSC (2) Here, εorbital energies of electrons, S-overlap matrix elements between P and q atomic orbitals, C-matrix of unknown coefficients. F is matrix elements of Fock operator, it depends on the C unknown coefficients. By unitary conversion method (2), it is possible to converse generalized eigenvalues equations to ordinary eigenvalues equations. As a result of calculations εiorbital energies, and values of coefficients Cqi have been found. On the basis of values of coefficients Cqi ,the analytical expression of molecular orbitals can be obtained. This allows to calculate some parameters of nanostructures, such as effective charge of atoms. On the basis of the values of εi it is possible to calculate the total energy, the values of ionization potential, electric conductivity, and strength and other properties of Titanium dioxide nanoparticle .During calculations as q atomic orbitals, 1s-, 2s-, 2px-, 2py-, 2pz-, 3s-, 3px-, 3py-, 3pz-, 3dx2-, 3dy2-, 3dz2-, 3dxy-, 3dxy-, 3dxz-, 3dyz-, 4s-, 4px-, 4py-, 4pz-atomic orbitals of titanium atoms, 1s-, 2s-, 2px-, 2py-, 2pz-atomic orbitals of oxygen atoms have been used .Gaussian functions have been used as atomic orbitals.

2.Theoretical methodology
The properties of nanoparticles depend on their dimensions, and the number of atoms of nanoparticles. The following formula is used for defining the number of atoms of composed same type atoms, and given dimensional nanoparticle [1,3,6,12,15]: Here, nanoparticle is considered as 'dense packing' sphere. N -The number of atoms, ρdensity of material, Ddiameter of sphere formed nanoparticle, NA = Avogadro's number, molar mass. It is not possible to use (3) for nanostructures, composed with different atoms. In literature, various ways are suggested to define dimensions, and number of atoms in such nanostructures. These ways are complicated, and are hard to make calculations with them. In this work, when the dimensions of such nanostructures are known, some ways are suggested to find out the number of atoms of nanostructure. Nanoparticles is considered as sphere form. the dimension of spherical formed TiO2 ( fig.1) compound, the calculation of (TiO2)6 nanoparticle can be found by :  2). Then, the formula for, n, nanoparticles with radius R can be calculated as: Here, = − 2 ℎ . The calculations are carried out for nanoparticle with radius R= 0.46 nm, and n ≈ 6. The number of atoms in nanoparticle are N=18. The theoretical visual model of (TiO)2 nanoparticle ( fig. 3) is configured [2,3,4,12] and quantum mechanical calculations are carried out. fig. 3. The theoretical visual model of (TiO2)6 nanoparticle

3.Result and Discussion:
Calculations of (TiO2)6 nanoparticles have been carried out by Hartree-Fock-Roothaan (HFR) method. As basis atomic orbitals,1s-, 2s-, 2px-, 2py-, 2pz-, 3s-, 3px-, 3py-, 3pz-, 3dx2-, 3dy2-, 3dz2-, 3dxy-, 3dxy-, 3dxz-, 3dyz-, 4s-, 4px-, 4py-, 4pz-atomic orbitals of titanium atoms, 1s-, 2s-, 2px-, 2py-, 2pz-atomic orbitals of oxygen atoms have been used. Gaussian functions have been used as atomic orbitals. Molecular orbitals have been represented as linear combination of mentioned atomic orbitals [2] . While calculating, computer programs have been used . As a result of this calculations the values of orbital energies, ionization potential, and the total electronic energy of titanium dioxide nanoparticle have been determined(TableS1and 2) . The result of calculations for (TiO2)6 nanoparticle have been given as follows : Total energy = -5926.024383 (a.u.) Ionization potential IP = 2.19959 In Table(3),we compared the accuracy of result of the titanium dioxide nanoparticle on the basis of HFR(GTO , s) theory with the results which we previously had when we studied for (Au16),(Ag12),and(CdS)9 on the basis of WH(STO , s) method [3,10,11,12,14].  The result of Table(3) indicates a variety on the calculations accuracy. these varieties of the calculations accuracy because of the type of the basic functions, and the variety of number of electrons which were used in the calculations in each compound.

Interpretation of the results for (TiO2)6 nanoparticle:
Starting from the lowest energy level 228 electrons of (TiO2)6 nanoparticles are placed in level two by two levels. The value of band gap can be calculated as = − . Here, is the energy of the lowest empty molecular orbital, and is the highest energy of molecular orbital occupied by electrons.

5.Conclusion:
Titanium dioxide nanoparticle have been investigated by HFR method. Orbital energies, ionization potential, values of total electron energies, and effective charges of atoms of titanium dioxide nanoparticle have been calculated. The results of calculations show that titanium dioxide nanoparticle is tough, electrophile and stable dielectric material.