### On Filling Energy Levels and Sub-Levels

As it is known, the filling of energy levels and sub-levels is produced in accordance with V. M. Klechkowski I and II rules [1-3] which are based on the two solutions of the E. Schroedinger equation [4,5] the principal n and the orbital l quantum numbers, but nothing is said about the third solution – the magnetic ml quantum number. It is true that the order obtained by increasing the energy corresponds to the experimental data, but in our opinion it is better to use a single unified rule in which to the sum of the quantum numbers (n and l) will be added the third term. The third solution of the Schroedinger equation ml determines the number of orbitals on the sub-levels ${k}_{{m}_{l}}\text{\hspace{0.33em}}\left[{k}_{{m}_{l}}\text{\hspace{0.33em}}(s)\text{\hspace{0.33em}}=\text{\hspace{0.33em}}1,\text{\hspace{0.33em}}{k}_{{m}_{l}}\text{\hspace{0.33em}}(p)\text{\hspace{0.33em}}=\text{\hspace{0.33em}}3,\text{\hspace{0.33em}}{k}_{{m}_{l}}\text{\hspace{0.33em}}(d)\text{\hspace{0.33em}}=\text{\hspace{0.33em}}5,\text{\hspace{0.33em}}{k}_{{m}_{l}}\text{\hspace{0.33em}}(f)\text{\hspace{0.33em}}=\text{\hspace{0.33em}}7\right]$. Our input is to add the inverse values of these numbers (1, 3, 5 and 7) $\frac{1}{{k}_{{m}_{l}}}\text{\hspace{0.33em}}-\text{\hspace{0.33em}}1,\text{\hspace{0.33em}}0.33,\text{\hspace{0.33em}}0.2\text{\hspace{0.33em}}\text{and}\text{\hspace{0.33em}}0.14$ to this sum. With this in mind, the sum of three numbers $\left(\sum \text{\hspace{0.33em}}=\text{\hspace{0.33em}}n\text{\hspace{0.33em}}+\text{\hspace{0.33em}}1\text{\hspace{0.33em}}+\text{\hspace{0.33em}}\frac{1}{{k}_{{m}_{l}}}\right)$ for all sub-levels of the 7 periods of the periodic table of the elements is given in Table 1.

The layout of the sub-levels according to the sum increase is given in Table 2.

The new filling rule will be formulated as follows:

**The electron sub-levels are filled in accordance with the sum of the principal n, the orbital l quantum numbers and the inverse values of the number of orbitals on the sub-level - $\sum \text{\hspace{0.33em}}=\text{\hspace{0.33em}}n\text{\hspace{0.33em}}+\text{\hspace{0.33em}}1\text{\hspace{0.33em}}+\text{\hspace{0.33em}}\frac{1}{{k}_{{m}_{l}}}$.**

Based on this, the principle of least energy is formed:

**The electronic sub-levels are arranged in the following order according to the energy increase:**

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p < 6f < 7d < 7f.

## References

- Klechkowski VM, Doklady (1951) 80: 603.
- Klechkowski VM, Zh. Exsperim. i Teor. Fiz (1952) 23: 115.
- Klechkowski VM, Zh. Exsperim. i Teor. Fiz 1962 41: 465 (Transl. Soviet Physics JETP (1962) 14: 334).
- Schrödinger, E. (1926) An undulatory theory of the mechanics of atoms and molecules. Physical Review 28: 1049-1070.
- Gray HB. (1965) Electrons and chemical bonding. WA Benjamin, Inc.

## Corresponding Author

Z. V. Pachulia, Faculty of Natural Sciences, Mathematics, Technology and Pharmacy, Sokhumi State University, 0186 Tbilisi, Georgia.

## Copyright

© 2023 Pachulia ZV. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.