The study of Silver Nanoparticles in basis of Slater functions

The electronic structure of the silver nanoparticles were investigated by semi-empirical Wolfsberg – Helmholz method. It is avariant of the molecular orbitals method. Molecular orbitals are represented as a linear combination of valence atomic orbitals of the atoms of the nanoparticle. The atomic orbitals used 5s-, 5py-, 5pzand 5pxSlateratomic orbitals of silver atoms. The exponential parameters of Slater functions were calculated and defined the analytic expression of the basis functions. The numerical values of the unknown coefficients of the linear combination are found by solution of equations of molecular orbitals method. Calculations were carried out with computer program. The orbital energies, potential ionization, total electronic energy and the effective charge of atoms of silver nanoparticles were also calculated. The results indicate that the silver nanoparticles are tough, electrophile and stable dielectric material.


Introduction
The silver nanoparticles have a wide range of applications due to their novel properties. These nanoparticles are used in the preparation of different transmitters in electronics, in medicine the diagnostics of various diseases,and in the chemical processes as a catalysts, and its application fields is expanding . For this reason, the study of electronic structure of the silver nanoparticles ( Fig.1) by quantum mechanics methods has a great importance [1,2,3]. It is obviously known that thestructure and properties of nanoparticles is determined by the sizes and number of atoms in nanoparticles. The size of nanoparticles which consists ofN atoms is given in the following formula [2,18,19,24,25].  (1) Here, N -number of atoms, M-molar mass,  -material density and NA-Avaqadro number . The calculated size of silver nanoparticles consist of N = 12 atoms by the formula (1) was D = 0.813 nm. Univ. Aden J. Nat. and Appl. Sc. Vol. 24 No.1 -April 2020 286

2.Methodology
In this work, the electronic structure of the Ag12silver nanoparticles were investigated by semiempirical Wolfsberg -Helmholz (WH) method. It is known that the (WH) method is a simple semi-empirical variant ofthe molecularorbital (MO) method [3,4,5,6,7,8,18,24,25]. In MO the state of the electron is described with one electron wave function so-called molecular orbital. Molecular orbitals representedas as linear combinationof valence atomicorbital(MCLO) of the atomsof the nanoparticles. Molecular orbitals ( i U ) are multicenter functions.Thus, the distances of electron from avariety nucleus of atoms was included into their expression.There are various ways to construct molecular orbitals. One of them isMolecular orbitals as aliner combination of atomic orbitals (MO LCAO) approximation, In this approximation, the molecular orbitals are written as a linear combinations of valence atomic orbitals of atoms: Where, qi C -the unknown coefficients, q  -atom orbitals given as basis functions. In this work,as basic functions, the real slater type atomic orbitals were used(STO's). It is well-known that the calculation of multicenter matrix elements over exponential type orbitals (ETO's) is of great importance for accurate evaluation of problems in quantum chemistry and physics. Among the ETO'scommonly used are the Gaussian type orbitals (GTO's) and STO's. The STO's represent the real situation for the electron density in valence region, but are not so good nearer to the nucleus. Many calculations over the years have been carried out with STO's [9][10][11][12][13][14]. The real STO's are determined as ) , are the principial, orbital and magnetic quantum numbers,  is exponential parameter which is determined by the following [16].
where Z is the atom number,and  ; Shielding constant.Usually, in quantum mechanics thecalculations of electronic structure molecules are satisfied considering only the atomic orbitals of valence electrons.For the creation of molecular orbitals of silvernanoparticles,4 valence atomic orbitals(5s, 5py, 5pz, 5px) were taken from each silver atomic using 48Slater'satomic orbitals. The analytic expressions of atomic orbitals are considered as follow : e r Ag s 4 1 0,5269031 ) The study of Silver Nanoparticles in basis of Slater …………Tawfik Mahmood Mohammed Ali Univ. Aden J. Nat. and Appl. Sc. Vol. 24 (10) In the expressions of (7) -(10)  , , r are spherical coordinates of electron.Based on the formula (2) 48 molecular orbitals have been used. The nanoparticle which was created from 12 silver atoms has 12 1 * 12 = valence electrons.They fill 6 low energetic levels. The basic functions of other silver atoms are determined in a similar manner.The unknowncoefficients qi C are found by solving the following system of equations [4,20]: where the following definitions are introduced: The quantity Hpqis matrix elements of effective Hamiltonian (14), for one electron moving in a molecule in some effective field independent from other.Thus, for solving of the system of equations(11), i.e.for the determinations of the orbitals energies i  and the corresponding sets of coefficients Cqi , one must know numerical Hpq and Spq values. However, Hpq values can not be calculated exactly because the explicit expressions for the operator is unknown. So, we need to estimate them by various ways, one of which based quantum chemical semi-empirical method (WH). Accoding to the method (WH),each diagonal matrix elements Hpqis guest equal to potential of ionization accoding to the valence state of the given atoms.The non-diagonal elements are defined byaratio [4,5,9,10,11,19,24].
where the coeficient(K=1.33)is establishedt heoretically from the condition of minimum of energy or from comparision with experimental data. As seen from (11) and (15)the expression for the implementation quantum mechanicalcal culating by VH method is important to know the value of overlapintegrals in molecular coordination system. In this work, for the calculation of overlap integrals in basis of STO's,we used the expressions from [17][18][19][20].
On the basis of these expressions for calculating,overlap integrals, m n , , quantum number,exponential parametersof atomic orbitalsand the cartesian coordinates of atomsshould be included.

The study of Silver Nanoparticles in basis of Slater …………Tawfik Mahmood Mohammed Ali
Univ. Aden J. Nat. and Appl. Sc. Vol. 24

The computer calculations for silver Ag12 nanoparticles by the Wolfsberg-Helmholz method
Total electronic energy E = -61797.168985a.u. Potential ionizationIp= 2.464504eV  The accuracy of the result, came from the paper, has been checked bythe test calculations of the Ag2 and Ag16 molecules, using the other methods, comparing the result of different methods for Ag2 and Ag16 are given in (table 3).

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As seen from Table 3 ,there are differences in the results. These difference occur because of the type and number of the basic function , and the variety of the electrons which were used in the calculations.

Conclusion
The electronic structure of thesilver nanoparticles is investigated bysemi-empirical Wolfsberg -Helmholz (WH) method in basis of Slater functions.STO's are used as atomic orbitals. The results of the calculations indicate that STO's are useful in the investigation of properties of nanoparticle in valence electronic approximation .The computer calculations were carried out by software operating system Delphi Studio systemunder the operating system Windows. The orbital energies, ionization potential, the total electronic energyand effective charge of atoms ofsilver nanoparticles were calculated. The results of calculations show that silver nanoparticleis tough, electrophile and stable dielectric material, and the using of Slater functions in study and application of nanosystems are appropriate.