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Quantum Theory and the Interactions
of Electrons in Multi-Electron Atoms

Quantum Mechanics works wonderfully well analytically and empirically for the hydrogen atom, but for multiple electron atoms the analysis is intractable. For an indication of the intractibility of the problem even for the simplest multiple electron atom, that of helium, see the Appendix. However, motions of the bodies of the solar system are also potentially incredibly complex but the reality is that they move in relatively simple patterns that prevail over time. Thus the potentially complex motion of the electrons in helium and other multi-electron atoms may be only potential rather than actual. The standard mathematical analysis of a helium atom maintains the potential complexity. However note that the wave function for an electron in the lowest energy state in a hydrogen atom is a spherically distributed shell. The center for such a distribution is at the center of the nucleus. A second electron might have the same distribution. This is quantum mechanically possible if the second electron has the opposite spin.

For electrons having spherical probability density functions the interaction is considerably simplified. For the outer electron (blue) the effect of the inner electron (red) is the same as if it were concentrated at the center of its distribution, which in this case is the center of the nucleus. Thus the effect of the inner electron on the outer electron is the same as the cancellation of one unit of the two units of positive charge in the nucleus.

On the other hand, the outer electron has no effect on the inner electron because the overall effect cancels out.

There is one more case to consider; i.e., the two electrons are in the same shell and their probability density functions coincide. In this case approximately one half of the probability density is closer to the nucleus than the mean distance of the other electron. Thus one half of a unit of negative charge would cancel the effect of a half unit of positive charge in the nucleus. The one half ratios are only approximations; later some empirical estimates of the shielding ratios will be given.

The Single Electron Atom

The quantum mechanical analysis, both old and new, indicate that an electron in a central force field of charge Z can have energies only of the following form

En = RZ²/n²

where R is the Rydberg constant (13.6 electron volts) and n is an integer. The orbit radius is determined by a similar quantization condition; i.e.,

rn = Bn²/Z

where B is a constant.

The Helium Atom

Consider the first electron added to a bare helium nucleus. The system satisfies the conditions for the Bohr model with Z=2. Its quantum number would be n=1. It would form a spherical shell of finite thickness. The value of Z is 2. A definite radius and energy would be determined for it.

Now consider the second electron. If the second electron has the opposite spin from the first one they would form a pair located in the same shell. Each would shield about a half unit of charge from the other. This means that each would experience a net charge from the nucleus of 1.5. This would constitute the extent of their interaction.

This suggests that the effective charge Z experienced by an electron is the positive charge of the nucleus #p less the charge shielded by the inner electron to a shell and the electrons in the same shell. There are measurements of the energy required to dislodge the electrons in the various shells. This energy is called the ionization potential.

Empirical Ionization Potential Functions

According to the theory sketched above the ionization potential IE for an electron in a particular place in a shell should be given by

IE = (R/n²)(#p−ε)²
where #p is the number of protons in the nucleus.

This equation can be put in the form

IE = (R/n²)((#p)² − 2(#p)ε + ε²)

A regression equation of the form

IE = c0 + c1(#p) + c2(#p)²

gives a very good fit to the data. The coefficient of determination goes as high as 0.9999998+.

Here is a plot of the data for one case.

The value of ε is found as

ε = −½c1/c2

However it also should be that c0/c2 should be ε² and thus equal to the square of the value found from c1 and c2. The regression coefficients are not constrained to achieve that equality. Thus effectively the form assumed for the relationship for ionization potential is

IE = (R/n²)[(#p−ε)² + ζ]

where R is an empirical value, rather than necessarily being the Rydberg constant, and ζ is a constant. The values of R are however notably close to the Rydberg constant. The values for some cases are given below are for the first electrons in several shells.

R (eV)

The value for the fourth shell is not close to the Rydberg constant.

The value of ε, as was indicated above, is found as −½c1/c2. The values for the first electrons in the first few shells are given below.


There can be only two electrons in the first shell, eight in the second and third shells, and eighteen in the fourth and fifth shells. The standard model presumes that ε is zero for the first shell, two for the second shell, ten for the third shell and eighteen for the fourth shell. The results indicate that there is only partial shielding by the electrons in inner shells. This could be due to deviations from spherical symmetry.

Now for the values of ζ. For the fourth shell the value of epsilon; is 17.57065433 and this squared is 308.7278935. On the other hand the ratio of c0 to c2 is 309.3893215, only 0.661427979 more than the value of ε². This is remarkably close. The value of ζ for case of the first electron in the fourth shell is thus 0.661427979. The values of ζ for the other shells and this case are given in the table below.


The Parameters Values for the Empirical Relationships

The previous results indicated that there is only partial shielding by the electrons in the inner shells.

The values given below are for the case of the two electrons of the first shell.

Parameter Values and Statistical Characteristics of the Empirical
Ionization Function for the Electrons in the First Shell
ζCoefficient of
10.17976 13.892541.237150.999998221
20.77739 13.85999 0.868520.999998302

The fact that ε is not zero for the first electron is an enigma. The amount that the first electron shields the second electron adds to the shielding is the difference, (0.77739−0.17976)=0.59763. Thus the shielding ratio for the first electron is about 0.6.

For the electrons in the second shell the parameter values are:

Parameter Values and Statistical Characteristics of the Empirical
Ionization Function for the Electrons in the Second Shell
ε R
ζCoefficient of
1 1.78258 13.917341.23715 0.999967446
2 2.30587 13.8918440.86852 0.999995889
3 3.26152 13.93007-0.26319 0.999998748
4 3.80646 13.72633-1.57611 0.999999107
5 4.47257 13.68296-2.10579 0.999999898
6 5.31574 13.69648-3.41429 0.999999584
7 6.01266 13.71371-3.95581 0.999999288
8 6.75131 13.83482-4.34685 0.999999695

The fact that the shielding for the first electron is 1.782576199 rather than 2.0 is notable. Instead it is about 89 percent of that value.

Taking the successive differences in the values of ε for the electrons gives their shielding ratios; i.e.,


For the third shell the parameter values are:

Parameter Values and Statistical Characteristics of the Empirical
Ionization Function for the Electrons in the Third Shell
ε R
ζCoefficient of
1 8.2336414.12815 -3.899270.999999245
2 8.8500614.20187 -4.904170.999999245
3 9.56661413.63683 -10.214650.999698201
4 11.0195514.62630 -3.078460.999573295
5 11.1925514.10977 -8.181900.999999218
6 11.9601313.99266 -10.339580.999998189
7 12.605314.03815 -11.519210.999997968
8 13.5291814.47872 -10.190810.999993592

The shielding ratios for the electrons of the third shell are then

4 0.17300

The value for the third electron is anomalously high and the one for the fourth electron is anomalously low; their average is 0.81297.

The data only allows for the computation for the first five electrons in the fourth shell. Those values are:

Parameter Values and Statistical Characteristics of the Empirical
Ionization Function for the First Five Electrons in the Fourth Shell
εζCoefficient of
117.57065 0.66143 0.997488104
218.53151 0.99805 0.997193112
319.54117 1.06776 0.99726095
420.61611 1.31067 0.9969434
521.40504 0.54656 0.997849703

The value of the parameter R is not given for this case. The R values are found by multiplying the regression coefficient for (#p)² by n², where n is the principal quantum number for the shell. For shells one through three the value of n which seems appropriate is equal to the shell number. The principal quantum number could be based upon the maximum occupancy of the shell but this does not seem to fit for third shell where the occupancy is 8 and thus n would be 2. Instead a value of n=3 seems to fit. Thus for shells one through three the value of n is equal to the shell number. This is not the case for the fourth shell. For a value of n=4 the value R for the fourth shell is not close to the Rydberg constant. To get a value R equal to the Rydberg constant the regression coefficient for (#p)² would have to be multiplied by about 6.5 instead of 3²=9 or 4²=16. In other words, n would have to be approximately 2.55. A value of 2½ seems to be appropriate.

The shielding ratios for the electrons in the fourth shell are:

1 0.96086
2 1.00966
3 1.07494
4 0.78893

For the fourth shell the shielding seems to be nearly 100 percent.

Patterns to the Parameter Values

The shielding by electrons in the same shell should be a function of the number of electrons in the shell, #e, less one; i.e., #e−1. The plot of the shielding versus #e−1 is shown below.

The regression results are

IE = 0.991596406 + 0.715998632(#e-1)
[14.2] [51.7]
R² = 0.997755065

This indicates that the shielding due to the two electrons in the inner shell of electrons is 0.992 units instead of 2.0. The additional shielding from each additional electron in the second shell is about 0.716 of a unit charge. The regression coefficients for the first through fourth shells are as follows:


where εI stands for the shielding by the inner shell electrons and γ stands for the additional shielding per additional electron in the shell.

(To be continued.)



The Analytics of the Electrons in a Helium Atom

Consider two electrons captured in the field of the helium nucleus. Assume for a while that the electrons are point particles of mass m revolving about the nucleus. Let r1 and r2 be the position vectors of the two electrons from the nucleus and r1 and r2 be the corresponding magnitudes the two vectors. Likewise let p1 and p2 be the electrons' linear momenta.

The Hamiltonian function for the version of the helium atom is then

H = p1²/(2m) + p2²/(2m) − 2K/r1² − 2K/r2² + K/|r2r1|

The last term on the right in the above equation is called the interaction term. This term congers up images of electrons moving in intricate, chaotic trajectories. However because the electrons repel each other they might end up opposite each other behaving like a rotor.

The Hamiltonian operator H is derived from the above equation by replacing pj by ihj and replacing its exponent with the order of the differentiation. The symbol i denotes the imaginary unit √−1 and h is Planck's constant divided by 2π. Thus,

H = −h²∇1²/(2m) −h²∇2²/(2m) − 2K/r1 − 2K/r2 + K/|r2r1|

Let Φ(r1, r2) be the wave function for the helium atom. The Schrödinger equation for the two electrons of a helium atom would then be

[−h²∇1²/(2m) −h²∇2²/(2m) − 2K/r1 − 2K/r2 + K/|r2r1|]Φ = EΦ

If the electrons are not point particles then the position vectors might be for the centers of the wave functions. This seems a reasonable modification of the model.

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