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) are satisfied. Thus, we can find and for this single atom as functions of space coordinates:

In order to calculate the kinetic energy we have to make integration firstly on ρ and then on r:

Therefore, we have to pass from the coordinate determination of and to their determination through ρ_s and ρ_p and then coming back to the coordinate determination to make integration over the space.

Therefore, we can write the partial kinetic functionals and:

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 and (Figure 2) to their determination through ρ_s and ρ_p. Let us compare dependencies and 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.

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, , , and.

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 signifi-

cantly exceed the maximum value, for which the single-atom function is determined. How to construct and in this case? How to extend dependences and to higher den-

sities? Probably, the exact answer to these questions will be found in some future, but now we offer other, an 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 d₀ 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 Z₁ and Z₂ are positive charges of ions with numbers 1 and 2 situated at R₁ and R₂ points. The binding energy per atom E_b was determined as follows:

where E_tot is the energy of two interacted atoms, E₁ and E₂ 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 Si₂ 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 d₀ 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, , , and.

For B:

For C:

For N:

For O:

Thus, we can conclude that using of special expressions for the kinetic energy functional (unique for each kind of atoms) allows us to describe interactions of atoms with compact densities. Note that these elements (B, C, N, and O) are very important components of many useful materials, molecules and compounds.