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

Authors: Victor G. Zavodinsky; Olga A. Gorkusha
Journal: ISRN Nanomaterials
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

It is well known [1] that the electron ground state energy Eel of a quantum system may be found by minimization of the energy functional which depends on the total electron density ρ as follows:
(1)Eel[ρ]=Et[ρ]+Eex[ρ]+Ecor[ρ]+EH[ρ]-∫V(r⃑)ρ(r⃑)d3r→,
where V, Et, Eex, Ecor, and EH are an external potential, the kinetic, exchange, correlation, and Hartree energies, correspondingly, and EH is given as follows:
(2)EH[ρ]=12∫ρ(r→)ρ(r′→)|r→-r→′|d3r→d3r→′.The first attempt to study the quantum system without wave functions was made by Thomas and Fermi [2, 3]. They considered the interrelation of the electron density, the electronic kinetic energy, and the electrostatic potential of an isolated atom and have found a simple one-dimensional equation for the potential. Then their idea has been developed by a number of authors [4–6], however still nowadays this theory is used only for single-atomic and other radial symmetric systems, or for jellium-approached ones [7–9]. Our work demonstrates a possibility to calculate the total energy and the electron density for neutral many-atomic systems directly from the minimization of the energy functional equation (1).


## 2. Results and Discussion

The variation principle together with the constant number of electronsN gives
(3)δE[ρ]δρ=μwith∫ρ(r→)d3r→=N.Thus we have the equation for finding the equilibriumρ(r→)(4)F[ρ]≡δE[ρ]δρ=-V(r→)+φ(r→)+μt(ρ)+μex(ρ)+μcor(ρ)-μ=0,
where φ(r→)=∫(ρ(r′→)/|r→-r′→|)d3r′→, μt(ρ)=δT[ρ]/δρ, μex(ρ)=δEex[ρ]/δρ, and μcor(ρ)=δEcor[ρ]/δρ.Equation (4) may be solved using any iteration method if you can calculate all its components. For simplicity, we have limited our current consideration by the local density approximation (LDA), in which
(5)μt=15(3π2ρ)2/3,εt=310(3π2)2/3ρ5/3,μex=-(3ρπ)1/3,εex=-34π(3π2)1/3ρ4/3.The correlation potentialμcor and the correlation energy εcor were taken as it was proposed in works [10, 11]. Namely,
(6)εcor=γ(1+β1rs+β2rs),μcor=(1-rs3ddrs)εcor=εcor1+(7/6)β1rs+(4/3)β2rs1+β1rs+β2rs,
for rs=(4/3πρ)-1/3≥1.And(7)εcor=Aln⁡⁡rs+B+Crsln⁡⁡rs+Drs,μcor=Aln⁡⁡rs+(B-13A)+23Crsln⁡⁡rs+13(2D-C)rs,
for rs=(4/3πρ)-1/3<1.HereA=0.0311, B=-0.048, C=0.002, D=-0.0116, γ=-0.1423, β1=1.0529, and β2=0.3334 (in a.e.).The well-known Newton’s iteration method was used to findρ(r→) as a self-consisted solution for (4):
(8)ρi+1=ρi-F[ρi]dF[ρi]/dρi.The main problem is to calculate the Coulomb potentialφ(r→) for a many-atomic system, and it is seems that this is a main reason why there was a little progress on this way. However, at the last years, the efficient methods for calculation of Coulomb integrals were developed. Namely, we used the supercell-cut-off technique [12, 13] and the fast Fourier transformation code [14].The second problem is to operate with all-electrons atomic potentials which have strong singularities leading to very sharp 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 FHI98PP code [15] and taking into account only valence electrons. For simplicity we have considered only diatomic systems (Si2, Al2, N2), thus the total external potential V(r→) was calculated as a sum of pseudopotentials Vp(r→-R→1) and Vp(r→-R→2) centered on the positions of two atoms, R→1 and R→2. The sum of pseudoatomic densities, ρp(r→-R→1)+ρp(r→-R→2), was used as a start density for the Newton’s iteration procedure. To find the equilibrium interatomic distance (d0) we considered the total energy Etot as a sum of the electron energy, Eel, and the energy of the “ion-ion” repulsion, Erep, that is, the interaction energy of positive-charged point nucleus
(9)Erep=Zv2|R→1-R→2|,
where Zv is the number of valence electrons: Zv(Si)=4, Zv(Al)=3, and Zv(N)=5.With the self-consistedρ(r→), obtained from (4), we calculated the total energy Etot as a function of the interatomic distance d=|R→1-R→2|, Etot=Eel+Erep.Values of the equilibrium distancesd0 and binding energies Eb for studied species are listed in Table 1 comparing with published data.Table 1
The equilibrium distances and binding energies for Si2, Al2, and N2. The published data is shown in brackets.
Si2
Al2
N2
d
0, Å
2.2 (2.27a)
2.6 (2.51b)
1.1 (1.098c)
E
b, eV
3.3 (3.07a)
1.4 (1.55b)
7.3 (9.42d)
aCalculated [16]; bcalculated [17]; cexperimental [18]; dexperimental [19].


## 3. Summary

Our results show that the calculated equilibrium binding energies and distances are close to published data; 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; however, 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 from the number of electrons; it depends linearly on the volume of the system, or in other words 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.



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*Source: 531965-2012-08-29.xml*