New Orbital-Free Approach for Density Functional Modeling of Large Molecules and Nanoparticles

Development of the orbital-free (OF) approach of the density functional theory (DFT) may result in a power instrument for modeling of complicated nanosystems with a huge number of atoms. A key problem on this way is calculation of the kinetic energy. We demonstrate how it is possible to create the OF kinetic energy functionals using results of Kohn-Sham calculations for single atoms. Calculations provided with these functionals for dimers of sp-elements of the C, Si, and Ge periodic table rows show a good accordance with the Kohn-Sham DFT results.


Introduction
The modern materials science and macromolecular chemistry combining nano, micro, and macro scales represent special inquiries to modeling of atomic interactions. When the system contains millions or billions of electrons, the task to find its quantum-mechanical state using wave functions (orbitals) becomes almost insoluble. Intensive attempts to develop an orbital-free (OF) approach for modeling of polyatomic systems based on the density functional theory (DFT) [1] were made by a number of groups in last two decades [2]- [10]. Most of them were developed within a pseudopotential approach, however, recently even an all-electron version of the OF method appeared [11]. Unfortunately, still now there is not a good comparison between OF and the KS results. Perhaps, the reason of this unluckiness is that all OF works stand on idea of using some universal functionals for kinetic energy-in approaches of Tomas-Fermi [12] [13], Weizsacker [14], and their modifications and combinations [10] [11]. However, in recent years a number of works appeared [15] [16] in which it was shown that the hypothesis of existence of universal density functional was incorrect, and that first of all it concerned the functional of kinetic energy (FKE). Nevertheless, the problem of development of the efficient approach for the OF-modeling of polyatomic systems remains tempting and actual. In our recent papers [17] [18], we have showed how it is possible, using single-atoms calculations by the Kohn-Sham DFT method (KS-DFT) [19], to find numerically the kinetic energy functionals for atoms, and then to use them for orbital-free modeling of atomic interactions. We calculated equilibrium distances, and binding energies for dimers contained Al, Si, and P atoms. Now we present an expansion of our method for other sp-species.

Methodology
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As it is known the DFT claims that the energy E el of the ground state of any quantum system can be found by minimization of the some functional depending only on the electronic density of this system ρ: where V ext is an external potential, E kin is kinetic energy, E ex is exchange energy, E c is correlation energy, and E H is Hartree energy: The total energy E tot is given by integral: Minimization of (1) with a condition where μ is the Lagrange parameter having a sense of the electron chemical potential.
There are some realistic approximations for exchange Quantum mechanical pseudopotentials are usually constructed for different angular states. Thus, we have to present the total density as a sum of partial densities: For the s-p case, we may write the equations

Single Atoms
If we calculate the ground state of an atom in the KS-DFT pseudopotential approach, we may declare that its partial electron densities minimize their energy functionals or in other words, Equation (6) In order to calculate the kinetic energy we have to make integration firstly on ρ and then on r: Let us consider Si, Al, and P atoms as typical atoms with s, p electrons. For constructing of pseudo potentials and calculation of equilibrium densities and energies we will use the FHI98pp [20] and FHI96md [21] packages widely used for KS-DFT calculations. Exchange and correlation potentials we will consider in the local density approximation [22] [23]. Then the partial densities ρ s (r) and ρ p (r) will look for SI as follows (Figure 1).
As it was said above we can use dependencies ρ s (r) and ρ p (r) for passing from the coordinate determination µ ρ for Al, Si, and P atoms as they are shown in Figure 3. Analyzing these curves we can see that universal dependence doesn't exist. There are essential differences between s and p components; and the curves received for various atoms, considerably differ from each other. However, we see also important common features of these curves.   µ ρ for Al, Si, and P single atoms. Solid curves describe functionals outside pseudopotential spheres, the hole quadrates demonstrate behavior of functionals inside pseudopotential spheres.
All of them have regions with reverse motion (i.e. two-branches), and besides, each curve has the limited range of definition corresponding to the maximum value of density for this atom. Reverse motions of curves correspond to densities situated inside pseudopotential spheres. To avoid ambiguity, we will separately determine densities inside spheres and outside and will speak about separate kinetic functionals

Atomic Dimers
An attempt to consider atomic interactions in dimers leads immediately to the problem: atomic densities are summarized and under certain conditions (a close arrangement of atoms) the total electronic density can significantly exceed the maximum value, for which the single-atom function approximate way for finding of equilibrium density and energy of interacting atoms. It seems to us that one of the possible practical ways is to construct analytical formulas, which can extrapolate data calculated for single atoms. In the best case, such formulas may be the same for some different kinds of atoms, but in the general case, each specie needs its own expressions for kinetic energy functionals. Calculations of equilibrium interatomic distances and binding energies for atomic dimers may be used for testing of constructed formulas. We have tested the following expressions for some s-p dimers: Calculations were performed using a cubic cell of 30 Bohr size (1 Bohr = 0.0529 nm) divided by the 100 × 100 × 100 grid for integration. To find the binding energies E b and equilibrium distances d 0 between atoms in dimers we consider the total energy E tot as a sum of the electron energy E el and the energy of "ion-ion" repulsion where Z 1 and Z 2 are positive charges of ions with numbers 1 and 2 situated at R 1 and R 2 points. The binding energy per atom E b was determined as follows: where E tot is the energy of two interacted atoms, E 1 and E 2 are energies of free atoms. Results of calculations were compared with results obtained in the framework of the KS-DFT approach using the FHI96md package. The preference was given to this package because the FHI96md package applies the same pseudopotentials and exchange-correlation approximations as we used. Figure 4 demonstrates the dependence of the binding energy (per atom) for Si 2 on the distance between atoms; and Figure 5 shows changes of the total electron density during the iteration process for various interatomic distances. We can see that our results correlate with results obtained by the KS-DFT method: the both methods lead to similar changes of density in the process of calculation.
Results (equilibrium distances d 0 and binding energies E b ) for atomic dimers of the Na-Cl and K-Br rows are collected in Table 1 and Table 2, correspondingly, in comparison with results obtained by us using the FHI96md package. One can seen that our OF results for the most of dimers are close to the KS-DFT ones. Exceptions are dimers of bivalent elements (Mg, Ca) and dimers of six-seven-valence atoms (S, Cl and Se, Br); the nature of this peculiarity will be studied later in a special work.
In must be noted that we have met some troubles in attempts to expanse our approach to dimers of small atoms (B, C, N, and O). Namely, to obtain good results for equilibrium distances and binding energies (shown in Table 3) we used for them different expressions for

Conclusion
We have demonstrated a principal possibility to find equilibrium densities, interatomic distances, and binding energies in the orbital-free approach using simple functionals of kinetic energy. However, in general case, FKE cannot be described by any universal formula and must be constructed for the each species. This work does not represent the finished algorithms for modeling of polyatomic systems; we have showed only a basic possibility to create such code. Still, it is necessary to include in the GGA options, spin polarization, and certainly to expand it to d-atoms. Besides, as it is our main future task, it is necessary to add calculations of forces operating on atoms in order to pass from dimers to more complicated objects.