A Simple Quantum Mechanics Way to Simulate Nanoparticles and Nanosystems without Calculation of Wave Functions

It is shown that the variation principle can be used as a practical way to find the electron density and the total energy in the frame of the density functional theory (DFT) without solving of the Kohn-Sham equation. On examples of diatomic systems Si2, Al2, and N2, the equilibrium interatomic distances and binding energies have been calculated in good comparison with published data. The method can be improved to simulate nanoparticles containing thousands and millions atoms.

It is shown that the variation principle can be usecl as a practical wav to find the electron density and the total energy in the frame of the density functional theory (DFT) without solving of the Kohn-Sham equation. On examples of diatomic systems Si2, A13, and N2, the equilibrium interatonric distances and binding energies irave been calculated in good comparison with publisired data. The method can be improved to simulate nanoparticles containing thousands and rnillions atoms.

l.Introduction
It is rvcll known I I ] that the electron ground state energy Ecl of a qr-rantum system may be found by rninimization of the energ)r tirnctional rvhich depends on the dotal electron density p as lbllows: where V, E , E.*, Etot, and EH are an external potential, the kinetic, cxchange, correlation, and Hartree energies, correspondingly, and EH is given as fbllolvs: EHinr= LleolP.q-) d3;d3f .
(z) " Lrr-2) li_i'l 'llhe lirst attempt to study the quantum system without wave {irnctions was made by'i'homas and Fermi [2, 31. I'hey con,siclcrcd the interrelation of the electron density, the electr:onic kinetic energy, and the electrostatic potential of an isolated atom and have found a sirnple one-dimensional equation for the potential. Then their idea has been developed by a number of authors [4][5], however still non'adays (1) this theory is used only for single-atomic and other radial symmetric systems, or fbr jellium-approached ones [7][8][9]. Our work demonstrates a possibilitv to calculate the total energy and the electron density for neutral many-atomic systems directly from the minirnization of the energy functional equation (1).

Results and Discussion
The variation principle together with the constant number of electrons N gives   ,,= Lr(3ntp)t", "Calculatetl [16];t'calculated [17];cexperimental [18];dexperimental ll9l. distance (do) we considered the total energy tsot as a sum of the electron energy' ,E'1, and the energy of the "ion-ion" repulsion, E"P, that is, the interaction energy of positivecharged point nucleus Fo= -(+)"" It,or = Av r,* (ulo) . The well-known Newton's iteration method ll'as used to find p(l') as a self-consisted solution fbr (4): t:lpt] Pi+r: Pi -arAtn' (8) lfhe main problem is to calculate the Coulomb Potential g(i) fbr a many-atomic system, and it is seems that this is a main reason why there was a little progress on this way' Horvever, at the last years, the efficient methods for calculation <lf Coulomb integrals were developed'. Namely, we used the supcrcell-cut-off technique [12, L3l and the fast Fourier trans{blrnation code I l4].
l'he second problem is to operate rvith all-electrons atomic potentials which have strong singularities leading to ver1, shirp density peaks and, as a result, to impossibility to calculate the Coulomb potential correctly through the Fourier expansion with limited number of plane waves' Therefore, we worked within the frame of a pseudopotential version of the DFT theory, constructing pseudopotentials using FHIgSPP code [15] and taking into account only valence electrons. For simplicity lve have considered only diatomic systems (Siz, Alz' Nz), thurs the total external potcntiai lt(r-) was calculated as a sunt of pseudo;rotentials With the self-consisted p(f), obtained from (4), we calculated the total energy Etot as a function of the interatomic distance d : lR-t -R-, I, E'ut : ;cl 16rep.
Values of the equilibrium distances do and binding energies E6 for studied species are listed in Table 1 comparing with published data.

Summary
Our results show that the calculated equilibrium binding energies and distances are close to published daia; thus we can be sure that this approach may be used for modeling of huge particles, probably up to million atoms' -Our consideration is limited by the spin-restricted case; hor.r'ever, we believe that the spin polarization can be included as well as the general gradient approximation.
The main advantage of the developed method consists in the independence of the calculation time frorn the number of electrons; it depends linearly on the volume of the system, or in other r.r'ords on the number of atoms' Certainly, it does not give us electron states, but they can be easily calculated through the usual Kohn-Sham technique if we know the electron density. We also believe that the proposed approach can be successfully improved in future for all-electrons atoms.