By prof. L. Kaliambos (Λευτέρης Καλιαμπός) Τ. Ε. Institute of Larissa Greece ( March 2014)

**PREFACE**

Despite the enormous success of the Bohr model (1913) and the Schrodinger equation in three dimensions (1926) based on the well-established electromagnetic laws in explaining the principal features of the spectrum of one-electron atomic systems, neither was able to provide a satisfactory explanation of the two-electron atoms with two-electron orbitals of opposite spin. Note that the discovery of the electron spin (1925) gives a peripheral velocity greater than the speed of light. Therefore under the influence of the invalid relativity physicists developed theories of the so-called qualitative symmetric properties including not the real magnetic attractions between the two electrons of opposite spin . In fact, in my paper “Nuclear structure is governed by the fundamental laws of electromagnetism (2002) I discovered that the velocity (u>>c) of the spinning electrons of opposite spin gives a magnetic attraction of short range which overcomes the electric repulsion of long range at short inter-electron separations.

It is well-known that in 1925 Goudsmit and Uhlenbeck discovered the electron spin

S = [s(s+1)]^{0.5} (h/2π) where s = 1/2

which gives a peripheral velocity u greater than the speed of light (u >> c ) invalidating Einstein’s relativity. Taking into account my discovery of the PHOTON-MATTER INTERACTION I discovered also that the velocity u >>c cannot be related with the absorption of photons in the PHOTOELECTRIC EFFECT OF LAWS . Such an enormous velocity (u >> c ) for two electrons of opposite spin (S=0) gives stronger magnetic attraction than the electric repulsion at a very short inter-electron separation r < 578.8/10^{15} m.

Thus, in the absence of such a detailed knowledge great theoretical physicists under the strong influence of the invalid relativity abandoned the natural laws of electromagnetism and developed theories with qualitative approaches. Following the work, of Pauli (1925) who suggested the qualitative exclusion principle for two electrons of opposite spin, physicists studied the properties of many-electron atoms. Though their efforts shed much light on the subject, the fundamental nature of the forces that contributes to the structure of two-electron atoms remained mysterious. For example Heisenberg tried to explain the ground state energy of the simplest Helium atom but without success.

So Einstein’s CONTRADICTING RELATIVITY THEORIES did much to retard not only the progress of nuclear physics but also the progress of atomic physics. Under this physics crisis I published my papers “ Nuclear structure is governed by the fundamental laws of electromagnetism” (2003) and “Spin-spin interaction of electrons and also of nucleons create atomic molecular and nuclear structures”.(2008). In that papers I showed the DISCOVERY OF NUCLEAR FORCE AND STRUCTURE and the DISCOVERY OF TWO-ELECTRON ATOMS .

**CALCULATION OF THE BOUND STATE ENERIES OF ONE-ELECTRON AND TWO-ELECTRON ORBITALS FOR n =1**

The ionization energies of the Beryllium atom is 153.8966 eV( third), and 217.71865 eV (fourth). So for n = 1 the bound state energy of the one-electron orbital 1s^{1} is given by the Bohr model as

E(1s^{1}) = (-13.6) Z^{2}/n^{2} = -217.71865 eV

Since Z = 4 one gets n^{2 }= (-217.6)/(-217.71865) = 0.999455 or n = 0.999

That is, this value differs slightly from the n =1 because of the penetrating orbital 1s^{1} .

Then, to calculate the bound state energy E(1s^{2}) for the two electron orbital 1s^{2 }we may write

E(1s^{2}) = -153.8966 – 217.71865 = - 371.61525 eV

However no one was able to find the solution of this problem , because in the absence of a detailed knowledge about the magnetic interaction of spinning electrons pysicists abandoned the natural laws in favor of wrong theories. Under thisphysics crisis taking into account the successful application of the basic Coulomb law in the Schrodinger equation in my paper “Nuclear structure is governed by the fundamental laws of electromagnetism”(2002) I applied the fundamental laws of Coulomb and Biot-Savart on the charges of the spinning electrons. Because of the antiparallel spin along the radial direction the interaction of the electron charges gives an electromagnetic force

F_{em} = F_{e} -F_{m}

Therefore in my research the integration for calculating the mutual electromagnetic force F_{em} between the two electrons of mass m and charge (-e) at an inter-electron separation r led to the following expression

F_{em} = Ke^{2}/r^{2 }– (Ke^{2}/r^{4}) (9h^{2}/16π^{2}m^{2}c^{2})

Here the electric repulsion F_{e} is of long range while the magnetic attraction F_{m} is of short range. Of course for F_{e }= F_{m} one gets the equilibrium separation

r_{o} = 3h/4πmc = 578.8/10^{15} m

That is, for r < r_{o } the electros exert an attractive electromagnetic force. As a consequence this situation provides the physical basis for understanding the pairing of two electrons with S=0 . Note that in the presence of an external magnetic field the two electrons operate with parallel spin (S=1) giving F_{em} = F_{e }+ F_{m} . Whereas for the two electrons with opposite spin I discovered that for r < r_{o} a motional EMF produces in eV a vibration energy.

E_{v} = 16.95Z - 4.1

For understanding this situation one may see my DISCOVERY OF TWO-ELECTRON ATOMS . Therefore the two electrons of 1s^{2} orbital behave like one particle circulating the nucleus under the rules of the quantum mechanics with a bound state energy

E(1s^{2}) = ( -27.2Z^{2 }+16.95Z - 4.1)/n^{2} = -153.8966 -217.71865 = -371.61525 eV

Since Z=4 one gets n^{2} = (-435.2 + 67.8 -4.1) / (- 371.61525) and n = 0.999

which differs slightly from the n=1 because of the penetrating orbital 1s^{2 }.

**CALCULATION OF THE BOUND STATE ENERGIES OF ONE-ELECTRON AND TWO-ELEXTRON ORBITALS FOR n=2 **

The ionization energies is 9.3227 eV (first) and 18.211 (second). So for the one-electron orbital 2s^{1} since the two-electron orbital of -2e screens the charge Ze = 4e of nucleus the bound state energy E(2s^{1}) can be calculated by using the effective charge ζe > (Ze-2e) as

E(2s^{1}) = (-13.6ζ^{2}/n^{2} = -18.211 eV

Whereas for the two electron orbital 2s^{2} the bound state energy E(2s^{2}) is calculated as

E(2s^{2}) = (-27.2ζ^{2} +16.95ζ -4.1)/n^{2} = -9.3227 – 18.211 = - 27.5337 eV

Thus to find ζ the first equation can be written as

13.6ζ^{2}/n^{2} = 18.211 or 0.7468ζ^{2} = n^{2}

Whereas the second equation can be written as

(27.2ζ^{2} -16.95ζ + 4.1)//27.5337 = n^{2 } or 0.2411ζ^{2} -0.6156ζ + 0.1489 = n^{2}

Then equating one writes

0.2411ζ^{2} – 0.6156ζ + 0.1489 = 0

and gets ζ =2.282745 and n = 1.9727

Here one sees that ζ > (Z-2) and n < 2

because both repulsions as 1s^{2}-2s^{1} and 1s^{2}-2s^{2} contribute not only to the deformation of the spherical symmetries bur also to the reduction of n because of the penetrating orbitals.