Fast Calculation of Cohesive Energy for Large Diamond Nanoparticles: The Orbital-Free Simulation

A full-electron orbital-free modeling method, developed in the framework of the density functional theory (DFT), was used for calculation of the cohesive energy of cubic diamond nanoparticles. Numbers of included atoms were changed from 64 up to 1000. The kinetic energy functional was constructed using Kohn-Sham (KS) calculations for a single carbon atom and a C8 cubic particle. Interatomic distances were changed to find the total energy minima. The calculation speed was found much higher than that of the KS method; the calculation time grows linearly with increasing of the system.

In the last 20 years many research groups see for example [1-6]. Made great efforts to develop on the basis of the density functional theory. A method for modeling large polyatomic systems, namely, an orbital-free (OF) method, dispensing with wave functions, working only with electron density, however they did not achieve serious success yet. In our opinion the reason of difficulties is that those works rely on the authority of Hohenberg and Kohn [7]. Who declared that there is a universal functional of kinetic energy and the task consists only in that to find it. However, as shown recently [8,9]. The idea of Hohenberg and Kohn about existence of a universal density functional leading to the energy minimum is not strictly proved. Thus, there are bases to believe that attempts to construct an OF method based on use 

at the condition where V(r) a potential of atomic nuclei,  is the Hartree potential,  and  are of any universal functional are obviously doomed to failure. Our alternative OF approach is based on the kinetic energy functionals specified for the each type of atoms. In previous works [10-14]. We demonstrated ability of our method to describe in the pseodopotential approximtion the structure and energy of atomic systems consisted of two, three, and four atoms of various types. Recently we have developed a full-electron version of the orbital free modeling method [15]. Which allows us to study large systems. Now we demonstrate how this full-electron method can simulate large nanoparticles of diamond. 

In short, our approach is as follows. The total energy of the polyatomic system E is found by minimizing the functionalexchange-correlation and kinetic energy functionals, N is the total number of electrons in the systems.Minimization of (1) with theleads to an

Where is the electron chemical potential,  are

exchange-correlation and kinetic potentials. In this work we used the both Local Density Approximation (LDA) [16]. Generalized Gradient Approximation (GGA) [17]. For calculation of the exchange-correlation potential. For the kinetic potential we used a technique described in our recent work this technique allows us to find the kinetic potential for a multi-atomic system if we know the kinetic potentials of single atoms. The single atom kinetic potential may be calculated easily using the Kohn-Sham (KS) approach [18]. An analysis of the C₈ cubic system in the frame of the KS method (using the full-electron ELK code [19]. Had allowed us to find for the carbon multiatomic systems the following dependence of on the average distance d between atoms: 

the values of corresponded cohesive energies are shown in the (Figure 1). 

Figure 1: Dependence of the cohesive energy of cubic diamond particles on numbers of atoms.

One can see in the Table 1 that the average interatomic distance increases with increasing of nanoparticles and tends to the value corresponded to bulk diamond [20,21]. The demonstrates that the calculated cohesive energy also grows with increasing of particles and its value tends to the cohesive energy of bulk diamond (7.5 eV) (Table 1). It should be noted that the calculated values of cohesive energy and interatomic distances in the GGA are larger than in the LDA, as is usually the case in DFT calculations. The main advantage of the OF approach is its high speed resulting from the rejection of wave functions. This is what makes it attractive for modeling of large nano systems (Figure 2).

Thus, we have shown that our full-electron orbital-free method allows us to describe the energetics of large carbon diamond nanoparticle. Namely, we have found that the cohesive energy grows with increasing of particles and its value tends to the the cohesive energy of bulk diamond. Interatomic distances also tends to the bulk diamond ones with increasing of particles. The calculation speed is much more than that of the Kohn-Sham method; the calculation time grows linearly with increasing of the system. 

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