^{1}

^{2}

The Paulie’s principle is used for development of the orbital-free (OF) version of the density functional theory. On the example of the three-atomic clusters, Al
_{3}, Si
_{3}, and C
_{3}, it is shown that the OF approach may lead to equilibrium configurations of atomic systems with both the metallic and covalent bonding. The equilibrium interatomic distances, interbonding angles and binding energies are found in good accordance with the known data. Results will be useful for developing of theoretical study of huge molecules and nanoparticles.

A number of works (for instance [

There is a significant progress in modeling of two-atomic systems [

An essential trouble is the fact that the electronic density of a single (isolated) atom is spherical, i.e. the “orbital-free” atom has a shape of a ball, but balls form the close packed structures. For example, three identical atoms are obliged to form an equilateral triangle with corners of 60 degrees. At the same time, it is known that three atoms of silicon form an isosceles triangle with the main corner of about 80 degrees [

In our recent papers [

As in our recent works we used here the pseudopotential approach with the FHI98pp package [

Transfer from dimers to trimers is a principal point because trimers characterized by some angles between atomic bonds.

To describe the angular dependence of interatomic bonding in the framework of the OF approach we must analyze the reasons of this dependence in the standard quantum-mechanical approach, which uses wave functions and electronic states. For example, it is specified in the work [_{3} are defined by the Yang-Teller effect caused by existence of the energy gap between occupied and empty states. In other words, the difference of structures of semiconductor and metal small clusters is connected with the difference of their electronic states and bond wave functions. Namely, covalent atoms have full occupied bonding states corresponded to localized functions orientated between nearest atoms; these states are separated by the energy gap from above lied non-occupied states. Metallic atoms have dispersed functions without orientation in the space and without energy gaps between them.

In our case wave functions are absent, electronic states are absent too, and, therefore, we cannot speak about any energy gap. In the OF approach we deal only with the electronic density which defines all energy and structure of the polyatomic system. However the main quantum-mechanical rules remain fair and in this case. We have not to forget that quantum mechanics is not only a set of wave functions and electron states. We have not to forget the Paulie’s principle specifying that in one quantum state there can be only two electrons (without taking into account a spin). In our case this principle may be paraphrased by a following way: a covalent bond is formed by two electrons, the common wave function of which is localized in the space between two nearest atoms. It is obvious that the quantity of the electrons which are responsible for this bond doesn’t change as the distance between atoms changes (if, of course, the bond isn’t broken at all and the electronic structure isn’t reconstructed completely). In case of the metals the states are close each other and electrons can easily “flow” from one state to another during the changing of the atomic geometry.

The above mentions may be reformulated in the language of the electronic density: the density integral (n_{int}) between atoms with covalent bonding, has to remain its value at change of distance between atoms; in case of metal bonding the value of the integral n_{int} can be any possible. Certainly, there is a question: on what space we have to provide integration? And what have we to do with intermediate cases, with atoms of different types? We will leave these questions for the future, and now we will try to explain the difference in structure of covalent and metal systems on the example of homogeneous clusters Al_{3}, Si_{3} and C_{3}.

It is obvious that the space of integration has to be rather local and at the same time it has to give us the information on quantity of the electrons included in a covalent bond. In the present work we used the space having a shape of the slab situated between two nearest atoms (_{int} was normalized to one point.

As it was said above the integral of electron density between the nearest atoms is the critical parameter for covalent bonding. It cannot be larger than some value that is a characteristic of a so called saturated bond. Let us

take the interatomic bond in a dimer as a standard saturated bond. Let us call the value

of n_{int} of a trimer to n_{int} of a dimer) as “the bonding strength” and accept that for covalent bonds the value of P shouldn’t exceed 1.0 at change of the distance between atoms. For metallic bonds the value of P can be any possible.

Calculated values P for clusters Al_{3}, Si_{3}, and C_{3} are presented in _{3} and Si_{3} trimers are approximately equal.

In order to check reliability of our results we repeated the calculations using thicker slabs (4ΔL and 8ΔL) for integration of values n_{int} (dimer) and n_{int} (trimer) between atoms in dimers and trimers and we found that results have changed no more than by 2 per cent.

In _{3}, Si_{3} and C_{3} trimers without restriction of the “bonding strengths”. From these curves it is clear that in all three cases the minimums of the binding energy correspond to triangular clusters, “bonding strengths “in which significantly exceed the corresponding values, characteristic for linear chains. This result looks natural for aluminum as its states have the metallic, not localized character; but for the clusters of Si_{3} and C_{3} having covalent bonds it is necessary to introduce the restrictions on values of P stipulated above. We have taken into account this condition (P = 1) and found dependences of the binding energy on the angle between bonds in the Si_{3} and C_{3} clusters (

Equilibrium values of interatomic distances d, angles α, and binding energies E_{b} (absolute values) for the trimers Al_{3}, Si_{3}, and C_{3} are collected in

It is interesting to compare our results for interatomic densities with results of standard KS-DFT calculations. For this purpose we calculated “bonding strengths” P for Si_{3} and Al_{3} using the popular KS-DFT code FHI96md [_{3}, but it increased up to 1.3 for the Al_{3} equilateral triangle. These results are in excellent agreement with our OF ones.

We showed that the use of the restriction principle for the interatomic density (following from the Paulie’s

Trimer | Source of data | α, deg | d, Å | E_{b}, eV |
---|---|---|---|---|

Si_{3} | Our method | ≈80 | 2.1 | 3.7 |

Other calculations | 77.8^{a} 78.10^{c} 79.6^{e} | 2.26^{b} 2.177^{c} | 2.51^{b} 2.93^{d} | |

Al_{3} | Our method | 60 | 2.3 | 2.4 |

Other calculations | 60^{f} 60^{h} | 2.50^{f} 2.55^{h} | 1.74^{g} 1.96^{f} | |

С_{3} | Our method | 180 | 1.2 | 7.0 |

Other calculations | 180^{i} 180^{j} | 1.29^{c} 1.3^{j} 1.316^{k} | 6.8^{i} 5.0^{k} |

Notations: ^{a} [^{b} [^{c} [^{d} [^{e} [^{f} [^{g} [^{h} [^{i} [^{j} [^{k} [

principle) allowed us to describe correctly angular dependences of the interatomic bonding in polyatomic clusters in the framework of the orbital-free version of the density functional theory. In particular, it is possible to show that for the Al_{3} cluster, the equilateral triangle is favorable; the Si_{3} trimer is characterized by the isosceles triangle with angles of 80 and 50 degrees, and three atoms of carbon built the linear chain. Calculated equilibrium interatomic distances and the values of binding energy are well compared with the known data.

As the problem of the correct description of angles between interatomic bonds is a key point in modeling of polyatomic systems, it is possible to consider that our work opens a direct way to design an effective method of modeling of big nanosystems and supermolecules.

We have to note that the consideration which is carried out above is directly applicable only to the systems consisting of identical atoms and requires a special development for application to more complicated systems.

Victor G. Zavodinsky,Olga A. Gorkusha, (2016) Development of the Orbital-Free Density Functional Approach: The Problem of Angles between Covalent Bonds. Modeling and Numerical Simulation of Material Science,06,11-16. doi: 10.4236/mnsms.2016.62002