1. Selection and description of used approach
Main problem at calculation of multielectron atoms, which elementary case are heliumlike atoms, eventually is the necessity to take into account of interaction between electrons. The review of approximate methods of the solution of this problem (as of 2000 year) it is possible to find in [1].
In the present work we use the oneconfiguration first approximation of a variational method (further VM_{1}), in which one parameter  charge of the nucleus varies only, and all interaction of electrons is with each other reduced only to mutual shielding by them of this charge. Thus VM_{1} is most simple from all possible methods, which use approximation of independent partiсles. The description VM_{1} can be found in many sources (for example  in [2]) and there is no necessity to consider it here in full details.
We note, that we not take into account in this approach exchange effects. The wave function of twoelectron atom in this approximation is look as
ψ = ψ_{1}· ψ_{2} , (1)
where ψ_{1} and ψ_{2}  wave functions of separate electrons, which differ from hydrogenlike functions only by replacement in them of a real charge of the nucleus Z on effective charges Z_{e1} and Z_{e2}, which are variational parameters.
Eventually formula of energy (hereinafter everywhere in the text as units of energy is used Rydberg Ry) twoelectron atom calculated by the method VM_{1} is noted as
Е_{He}_{1} = ε_{1} ·(Z_{e}_{1}/n_{1} )^{2} + ε_{2}·^{ }(Z_{ }_{e}_{2}/n_{2} )^{2} , (2)^{ }
where n_{1} and n_{2}  general quantum numbers of appropriate electrons, and ε_{1 }and ε_{2 } factors, which take into account relativistic effects and influence of the nucleus. In the present work we have taken their value from experiment under the formula
ε = Е_{H exp} · ( n/Z)^{2} , (3)
where Е_{H exp}  experimental value of energy of hydrogenlike atom for given Z and given n.
2. Correlation energy
The residual between precision energy of atom and energy calculated by one from methods of approximation of independent partiсles is named as correlation energy.
We can estimate accuracy of approach VM_{1} with the help of correlation energy. In the present work we have analyzed of correlation energy of a method VM_{1} for nl_{ 1}nl _{2 }states of heliumlike atoms, where n  main quantum number, common for both electrons, and l_{ 1 }and l _{2}  orbital quantum numbers of appropriate electrons. The similar analysis for a ground state of twoelectron atoms can be found in various sources (for example in [2]), but we could not find any publication of similar investigation for doubly excited states (DES) and we claim for superiority in this problem. We have analyzed of correlation energy for all nl_{ 1}nl _{2 }states of Helike atoms, for which it was possible to find the experimental data now, but because of limitation of the size of the article we shall present here as an example only data obtained for nsns(1S) states. The values Е_{He1}, calculated by a method VM_{1}, values of energy E_{Не}_{ exp}, taken from experiment, and also value of correlation energy Е_{cor}= E_{Не}_{ exp}  Е_{He1}, for nsns (1S) states are given in a table 1.
We have analyzed relation of correlation energy from Z and n, and have detected, that the nl_{ 1}nl _{2 }states are divided into two groups. The first group involves lowest states for the given configuration, i.e. states with least possible n. It is states, in which at least one electron is on the such orbit, for which n = l + 1. In the old quantum theory this condition corresponds to the special case of circular orbits. Second group involves all remaining states of configurations, i.e. states with n > l + 1 for everyone from two electrons. We obtained the formulas
Е_{cor} = (1/2)·C·(1/n)·Z·[ k_{1}  k_{2} ·(Z1)/Z ] (4)
and
Е_{cor} = (1/2)· C·(1/n)· Z· k_{1} (5)
accordingly for the first and second group of states, where k_{1} and k_{2}  integer factors, which values are given in a table 2, and C  numerical factor. The values C, at which formula (4) and (5) give precision coincidence with experiment, are given in a table 1 for nsns (1S) states. It is easy to see, that C with a large degree of probability is a constant and is in range 0,1100,116, i.e. is close to 1/9.
Table 1. The data of experiment and calculations for nsns (1S) states of heliumlike atoms for n from 1 up to 3
State 
Z 
E_{Не }_{exp} (Ry) 
Е_{He}_{1} (Ry) 
Е_{cor} (Ry) 
C 
Е_{He}_{ }(Ry) 
Ref. E_{Не exp} 
1s1s (1S) 
1 
1,0550 
0,9448 
0,1101 
0,110 
1,0559 
[3] 

2 
5,8068 
5,6948 
0,1120 
0,112 
5,8059 
[4] 

3 
14,5597 
14,4457 
0,1140 
0,114 
14,5568 
[4] 

4 
27,3131 
27,1987 
0,1143 
0,114 
27,3099 
[4] 

5 
44,0699 
43,9561 
0,1137 
0,114 
44,0672 
[4] 

6 
64,8318 
64,7202 
0,1116 
0,112 
64,8313 
[4] 

7 
89,6035 
89,4946 
0,1089 
0,109 
89,6057 
[4] 

8 
118,3845 
118,2829 
0,1016 
0,102 
118,3940 
[4] 

9 
151,1885 
151,0900 
0,0985 
0,099 
151,2011 
[4] 

10 
188,0111 
187,9201 
0,0910 
0,091 
188,0312 
[4] 
2s2s (1S) 
1 
0,2972 
0,2444 
0,0528 
0,106 
0,2999 
[5] 

2 
1,5571 
1,4436 
0,1135 
0,114 
1,5547 
[6] 

3 
3,8077 
3,6431 
0,1646 
0,110 
3,8098 
[7] 

4 
7,0788 
6,8434 
0,2354 
0,118 
7,0656 
[7] 

5 
11,3354 
11,0452 
0,2902 
0,116 
11,3229 
[8] 
3s3s (1S) 
2 
0,7206 
0,6431 
0,0776 
0,116 
0,7171 
[9] 

6 
7,4304 
7,2260 
0,2044 
0,102 
7,4482 
[10] 

7 
10,2379 
9,9849 
0,2530 
0,108 
10,2442 
[11] 

8 
13,4581 
13,1900 
0,2681 
0,101 
13,4863 
[10] 
Table 2. Factors k1 and k2
State 
k_{1} 
k_{2} 
nsns(1S) 
2 
2 
npnp(1D) 
2 
2 
nsnp(3P) 
2 
0 
npnp(3P) 
1 
0 
nsnp(1P) 
1 
3 
npnp(1S) 
0 
4 
Finally we have following semiempirical formula for calculation of full energy of heliumlike atom for nl_{ 1}nl _{2 }states
Е_{He} = Е_{He1 }+ Е_{cor}, (6)
where Е_{He1} is calculated by a method VM_{1} and follows from the formula (2), and Е_{cor} follows from the formulas (4) and (5) and is entered because of analysis of the experimental data.
In that specific case of states nsns (1S), npnp (1D), ndnd (1G) etc., when both electrons are on same orbit, or, as speak, occupy the same quantum cell, the formulas (4) and (5) receive especially simple kind
Е_{cor} = C·(1/n) (7)
for n = l + 1 and
Е_{cor} = C·(1/n)· Z (8)
for n > l + 1.
We note, that our refusal to take into account of exchange degeneration is justified for these states from any point of view.
The particular case of the formula (7) for n = 1 corresponds to a ground state of heliumlike atoms (1s1s) and results to Е_{cor} = C, i.e. the correlation energy in this case does not depend from Z and it was marked by Bethe as the curious fact in [2]. The HartreeFock method also results in independence of correlation energy of a nuclear charge in case of a ground state 1s1s , but the constant obtained at it, has average value 0,085 unlike 0,111 for case of a method VM_{1}. The correlation energy of a HartreeFock method was a theme of many works at the end of 1950s and at the beginning of 1960s, however after appearance of the first experimental data on DES of atoms in a middle of 1960s the point of view has prevailed, that the HartreeFock method and generally approximation of independent partiсles is inapplicable for the description of a similar class of states because of large increase for them of correlation energy. The obtained above formulas show, that at least in case of a method VM_{1} the approximation of independent partiсles can be applied successful for DES if to consider correlation energy not as an annoying error, but as the simply taken into account correction with interesting physical properties which will be discussed below. Moreover, simplicity of obtained expressions results that the correlation energy is transformed from a problem into a proof of effectiveness of approximation of independent partiсles. Already now it is possible to use algorithm of calculation , offered us, as the simple and at the same time rather precision semiempirical approximation to describe known lines of spectra of heliumlike atoms and to predict or to help to identify until now unknowns lines.
The data of calculation of energy Е_{He} under the formula (6), in the supposition, that C = 1/9, are given in table1 for those of nsns (1S) states, for which the experimental data are known . Goodness of fit of calculations with experiment quite satisfactory, taking into account errors of measurements and approximations at calculations. The data of the same calculation for a unknown for today nsns (1S) states at n from 4 up to 10 and Z from 1 up to 10 are given as an example in a table 3.
Table 3. The data of calculation for nsns (1S) states of heliumlike atoms for n from 4 up to 10
Z 
E_{He}(Ry) 
E_{He}(Ry) 
E_{He}(Ry) 
E_{He}(Ry) 
E_{He}(Ry) 
E_{He}(Ry) 
E_{He}(Ry) 

4s4s 
5s5s 
6s6s 
7s7s 
8s8s 
9s9s 
10s10s 
1 
0,0893 
0,0616 
0,0459 
0,0360 
0,0293 
0,0245 
0,0210 
2 
0,4175 
0,2762 
0,1980 
0,1500 
0,1183 
0,0962 
0,0802 
3 
0,9958 
0,6508 
0,4612 
0,3457 
0,2699 
0,2174 
0,1794 
4 
1,8242 
1,1854 
0,8356 
0,6230 
0,4839 
0,3879 
0,3186 
5 
2,9029 
1,8802 
1,3212 
0,9820 
0,7605 
0,6078 
0,4979 
6 
4,2319 
2,7352 
1,9180 
1,4227 
1,0997 
0,8771 
0,7170 
7 
5,8114 
3,7504 
2,6260 
1,9452 
1,5014 
1,1959 
0,9763 
8 
7,6417 
4,9261 
3,4455 
2,5494 
1,9657 
1,5641 
1,2758 
9 
9,7230 
6,2623 
4,3763 
3,2355 
2,4927 
1,9818 
1,6152 
10 
12,0556 
7,7592 
5,4183 
4,0035 
3,0823 
2,4490 
1,9943 
3. Results
3.1. From a point of view of physics of the formula (4) and (5) result to exotic, i.e. in nonclassical relation of energy of interaction of charged partiсles to a distance between them.
If in the formulas, obtained by us, to present n as a radius of atom r (remembering, that in hydrogenlike atoms r ~ n^{2}), we shall receive for different parts of full energy of atom of relation
Е ~ 1/r^{k} ,
where for terms, calculated by the method VM_{1}, k = 1, that completely corresponds to the classical law of the Coulomb, and for correlation energy, k = 1/2. Thus here there is an additional interaction decreasing on a distance slower, than Coulomb force.
The additional calculations, carried out by us, have shown, that the relation Е ~ 1/r^{1/2}, or accordingly Е ~ 1/n, occurs only in that case, when both electrons have identical general quantum numbers n. We found proofs of existence of similar relation of correlation energy from a general quantum number of external electrons not only in heliumlike atoms, but also in atoms with large number of electrons, and in molecules and crystals.
That fact, that coincidence of general quantum numbers n at both electrons is required for appearance of relation Е ~ 1/n, can indicate resonant character of additional interaction. Moreover it results in electronelectron attraction, instead of repulsion and very strongly depends on a configuration of spin and orbital moment, that makes it even less similar to electrostatic interaction, but similar to interaction, which exist between protons in the nucleus.
3.2. From a mathematical point of view it is obvious, that if in twoelectron atom there were no nonclassical interactions, the method VM_{1} would give precisiouly analytical solution of a threebody problem. The simplicity of the formulas (48) allows to hope, that the analytical solution is possible and with taking into account of nonclassical interactions. It would become possible after an evaluation of correlation energy Е_{cor} and constant C from certain general principles.
3.3. From a practical point of view it is interesting, that since some value n usual Coulomb repulsion electrons (decreasing as 1/n^{2}) will become less additional not  Coulomb attractions (decreasing as 1/n). It can result in macroscopic case to join of electrons in certain stable or metastable structures  quasineutral (if number of electrons equally to number of positive ions) or charged, just as the protons are integrated in the charged nucleus. The similar processes could spontaneously happen in strong rarefied plasma of intersidereal space, in high layers of an ionosphere of the Sun, Earth and other cases, in which the large distances between charged partiсles ( appropriate to the large values n ) are realized. It is possible, that the similar effects could explain at least some from anomalous plasmalike effects observed in atmosphere and an ionosphere, such as a ball lightning etc.
To receive a similar new states of substance in experiment, it is necessary, that the electrons of substance were excited synchronously, i.e. had identical energy and identical values n in each instant . To the present moment not much of similar (doubly exited) states is obtained even for twoelectron atoms . For molecules them it is known even less. Moreover both in case of atoms, and in case of molecules the values n are not reached yet value, at which the attraction between electrons exceeds a repulsion between them. In case of macroscopic skew fields the problem of synchronous excitation of electrons up to maximum large n till now not to pose, though technically it is not unattainable, since the similar problems are decided at creation of quantum generators of coherent electromagnetic radiation.
Let´s remind also, that the explanation of a superconductivity involves appearance of additional electronelectron attraction , which exceeds Coulomb repulsion under certain conditions.
Moreover there are the direct analogies between additional correlation energy of electrons in superconductors and additional correlation energy of electrons in separate atoms in the literature (see for example [3]), and the Cooper pair sometimes is represented as two electrons moving round an induced positive charge, and is compared to atom of a helium.
All of this makes probablis boseeinstein condensation of synchronously excited electrons both in atoms, and in macroscopic skew fields from that moment, when not  Coulomb attraction of electrons will begin to exceed Coulomb repulsion. The similar superconductivity already could be named superhightemperature.
Conclusion
The approach based on separation of full energy of multielectron systems on classical Coulomb and nonclassical not  Coulomb parts, allows on the one hand to simplify calculations, and with another  to see interesting regularities, which were not visible at use of more complex methods. Most interesting is the discovery of that fact, that under certain conditions electronelectron attraction exceeds electronelectron repulsion. The most important practical consequence it is the capability of existence of ordered structures of a new type in the special way exited substance.
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The work was submitted to international scientific conference «Prospects for the development of university science», Dagomys (Sochi), 2023 September 2008, came to the editorial office on 01.09.2008.