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Shielding of Electrons in Atoms and Ions (Part II) 

This is a continuation of a study on the ionization energy for electrons in different positions within atoms and ions. Ionization energy IE, or as it is usually called the ionization potential, for an electron is the amount of energy required to dislodge it. The previous study was limited to only five cases for each electron position. The more extensive database is displayed in the appendix.
The Bohr model of a hydrogenlike atom or ion indicates that the energy required to remove an electron, called the ionization potential, should follow the formula
where R is the Rydberg constant (approximately 13.6 electron Volts (eV), Z is the net charge experienced by the electron and n is the principal quantum number, effectively the shell number.
Here are the graphs of the ionization potential of the three innermost electrons.
Clearly the relationships are very regular and quadratic. The graph for the first electron exhibits an unexplained jump in ionization energy after ten protons. The data for none of the other electron positions exhibit such a jump.
Let p denote the number of protons in the nucleus of the atom or ion. The value of Z in the above Bohr formula is the number of protons in the nucleus p less the shielding ε by the electrons in inner shells or in the same shell. Thus the ionization energy would be
There may be other phenomena that affect the ionization energy besides the charge of the nucleus and the shielding. For example there may be energy associated with the spin pairing of electrons. Thus the formula should be
where the ζ term includes the ε² term and any other factors affecting ionization energy.
The Bohr model is strictly for a hydrogenlike atom or ion; i.e., one in which there is a single electron in the outermost shell. However the regression equation also fits very well the cases of multiple electrons in the outer shells if charge shielding is taken into account. That shielding is by the electrons in the inner shells and may also be by electrons in the same shell. But the shielding by electrons in the same shell is likely only for a fraction of their charge. As it turns out, shielding even for electrons in the inner shells the shielding is less than the full value of their charges.
Here is the rationale for the shieldings. If inner shell electrons execute trajectories that take them over a spherical shell it is as though their charges are smeared over a spherical shell and their effect on outer shell electrons is the same as though the charges of the inner shell electrons are concentrated at the center of the atom or ion and thus cancel out an equal number of positive charges.
The electrons in a lower subshell may be entirely interior to an outer subshell. Thus the analysis should be in terms of shielding by electrons in inner shells and subshells versus shielding by electrons in the same subshell.
The shielding by electrons in the same subshell is a bit more complicated. The effect of a charge distributed over a spherical shell on an electron entirely within that spherical shell is zero. If the electron is entirely outside of the spherical shell the effect is the same as if the charge were concentrated at the center of the spherical shell. But if the center of the electron is located exactly on the shell then roughly half of the electron is inside of the spherical shell and is unaffected by its charge. Thus an electron is shielded by an amount approximately equal to one half of the charges in the same shell. That is the rough theory. It needs to be tested empirically.
This partial shielding by electrons in the same shell explains how there can be negative ions. In a negative ion such as O^{=} there are outer electrons clinging to a structure with overall negative charge. That appears to be a puzzle. The oxygen nucleus has eight protons. There are two electrons in the first shell and six in the second shell of the oxygen atom. Overall that is electrostatically neutral. But for a seventh electron in the second shell the two electrons in the inner shell and other six electrons in the second shell shield only a portion of the positive charge of the nucleus. Thus there is a positive attraction for that seventh electron. And since the seventh electron shields only a fraction of a unit charge for the eighth there is a positive attraction for the eighth electron. However another electron would go into a third shell and the ten (2+8) other electrons would be inner shell electrons and could more than shield the eight positive charges of the nucleus. Thus there would be a repulsion of an eleventh electron.
The extended Bohr formula for ionization energy suggests a regression equation of the form
Such a form does give a very good fit to the data. The value of ε can be found from the regression coefficients as
For the second electron the quadratic regression that fits the data is
Note that the Rydberg constant is 13.60569 eV.
The coefficient of determination for this equation is 0.999999673 and the standard error of the estimate is 0.9945 eV.
The figures in square brackets are the tratios for the regression coefficients 1above them; i.e., the ratios of the regression coefficients to their standard deviations. `A tratio of less than 2 would indicate that at the 95 percent degree of confidence the perceived relationship between the independent variable for the coefficient and the dependent variable is not real; i.e., is just due to chance. The tratios for the above regression equation are astronomically high.
The estimate of ε which comes from this equation
Thus the shielding by another electron in the same shell is real and is approximately 0.5 electron charges.
The ssubshell of the second shell can contain at most two electrons. Therefore the third and fourth electrons are in that subshell. The "expected" shieldings for the third and fourth electrons are 2.0 and 2.5, respectively. This expectation is in the nature of "naively expected."
For the third electron (the first in the second shell) the regression equation is
The coefficient of determination for this equation is 0.999999872 and the standard error of the estimate is 0.14129 eV.
The estimate of ε which comes from this equation
Full shielding by the two electrons in the first shell would give ε=2.0.
The coefficient of p² is R/n² where n for the second shell is 2. Thus the R value for this case is 2²(3.436413)=13.7456517, close to the Rydberg constant of 13.60569 eV.
The fifth through tenth electrons are in the psubshell of the second shell. Therefore their "expected" shieldings are 4.0, 4.5, 5.0, 5.5, 6.0 and 6.5, respectively. The shielding values found for the fifth through eighth electrons are 3.16858, 3.82658, 4.52719 and 5.31449. Thus there is a close match for the eighth electron. The measured shielding for the ninth electron is 5.99813, reasonably close to 6.0.
The measured shielding for the tenth electron is 6.81818, surprisingly exceeding the "expected" value 6.5.
Now it is worthwhile to apply the above methodology to the case of the first electron. There is no shielding in this case so ε should be zero. But it should not be applied to interval that includes the jump.
The regression equation for the first electron for p=1 through p=11 is
The coefficient of determination for this equation is 0.999999965 and the standard error of the estimate is 0.11637 eV.
The estimate of ε which comes from this equation is 0.01006 charges. This is essentially zero, thus confirming the methodology.
The eleventh electron is in the third shell. There are ten electrons in the first two shells so the shielding would be expected to be 10.0. Its measured value is 8.17159, about 0.8 of its "expected" value.
The measured value of the shielding for the twelfth electron is 8.17159, about 0.8 of its "expected" value. 8.78703 charges, 0.61544 charges higher than the value for the eleventh electron.
The table of the above values and the values for subsequent electron positions is:
Shielding of Electrons in Different Positions in Atoms and Ions  

Electron Position  Inner Electrons 
Same Shell Electrons  Shielding  "Expected" Shielding 
1  0  0  0.010  0.000 
2  0  1  0.678  0.500 
3  2  0  1.647  2.00 
4  2  1  2.221  2.500 
5  4  0  3.169  4.00 
6  4  1  3.827  4.500 
7  4  2  4.527  5.000 
8  4  3  5.314  5.500 
9  4  4  5.998  6.000 
10  4  5  6.818  6.500 
11  10  0  8.17159  10.000 
12  10  1  8.78703  10.500 
13  10  2  10.13256  11.000 
14  10  3  10.61864  11.500 
15  10  4  11.26194  12.000 
16  10  5  12.01621  12.5 
17  10  6  12.68347  13.0 
18  10  7  13.33016  13.500 
19  18  0  16.45756  18.000 
20  18  1  17.37618  18.500 
Before proceeding further it is necessary to note a feature of the electron structure of atoms beyond Argon (p=18). The 19th electron goes into the ssubshell of the fourth electronic shell even though there is additional capacity in the dsubshell of the third shell. It does this because of energy considerations.
The 20th electron also goes into the ssubshell of the fourth shell but subsequent electrons go into the dsubshell of the third shell. This creates a problem of the identification of the data for the 21st through the 30th electron. Apparently this is for the outermost electron. For the cases of 21 through 28 there are two electrons in the ssubshell of the fourth shell. Cu (p=29) is the anomaly it has only one electron in the ssubshell of the fourth shell. Zn (p=30) again has two electrons in the ssubshell of the fourth shell.
Shielding of Electrons in Different Positions in Atoms and Ions  

Electron Position  Inner Electrons 
Same Shell Electrons  Shielding  "Expected" Shielding 
21  19  1  18.33305  19.500 
22  20  1  19.32662  20.500 
23  21  1  20.23488  21.500 
24  22  1  21.99433  22.500 
25  23  1  23.01113  23.500 
26  24  1  24.57486  24.500 
27  25  1  25.67367  25.500 
28  26  1  23.49360  26.500 
29  28  0  24.05092  28.000 
30  28  1  25.17227  28.500 
Shielding of Electrons in Different Positions in Atoms and Ions  

Electron Position  Inner Electrons  Same Shell Electrons  Shielding  "Expected" Shielding 
31  30  0  26.5035  30.000 
32  30  1  27.24866  30.500 
33  30  2  28.46114  31.000 
34  30  3  29.01879  31.500 
35  30  4  29.98610  32.000 
36  30  5  29.93740  32.500 
37  36  0  34.97742  36.000 
38  36  1  37.42557  36.500 
39  37  1  38.69343  37.500 
40  39  0  39.10448  39.000 
The "expected" shieldings of an electron based on the number of electrons in inner subshells and shells and the number of electrons in the same subshell are not precise estimates of the measured shieldings but are rough approximations. It might be the ratios used of 1.0 for the inner electrons and 0.5 for the same subshell electrons are not the best ratios for predicting the shieldings.
There are forty observations in the above empirical analysis which can be used to obtain the best ratios. Let Inner and Same be the number of the shielding electrons. Then the regression equation presumed is
There is no intercept constant for the equation because if there are no shielding electrons there is no shielding.
The regression results are:
Lo and behold! The ratio values of 1.0 and 0.5 are very close to the best values for predicting the shielding of an electron. The coefficient of determination for the regression equation is 0.99685 and the standard error of the estimate is 1,20 eV. Thus about 99.7 percent of the variation in the shielding ε is explained by the variation in the numbers of shielding electrons.
If a constant is included in the regression the tratio for its estimate is −1.2, indicating the constant is not significantly different from zero at the 95 percent level of confidence.
(To be continued.)
The energy required to dislodge an electron from a position in an atom is determined by the net positive charge it experiences from the nucleus. The net experienced charge is the charge of the nucleus less the amount that it is shielded from by the electrons in the inner shells and subshells and also by the electrons in the same subshell. The shielding by the electrons in the same subshell is a fraction of their charge, roughly one half. The shielding by electrons in inner shells or shells is generally less than one for one, but roughly 0.95.
Fractional shielding accounts for the existence of negative ions.
This data was compiled from a database published by the Physics Department of Ohio State University.



Electron Number  
Elem.  Number of Protons  1  2  3  4  5  6  7  8  9  10 
H  1  13.598  
He  2  54.416  24.587  
Li  3  122.451  76.638  5.392  
Be  4  217.713  153.893  18.211  9  
B  5  340.217  259.368  37.93  25.154  8.298  
C  6  489.981  392.077  64.492  47.887  24.383  11.26  
N  7  667.029  552.057  97.888  77.472  47.448  29.601  14.534  
O  8  871.387  739.315  138.116  113.896  77.412  54.934  35.116  13.618  
F  9  1103.089  953.886  185.182  157.161  114.24  87.138  62.707  34.97  17.422  
Ne  10  1362.164  1195.797  239.09  207.27  157.93  126.21  97.11  63.45  40.962  21.564 
Na  11  1648.659  1465.091  299.87  264.18  208.47  172.15  128.39  98.91  71.64  47.286 
Mg  12  2304.08  1761.802  367.53  327.95  265.9  224.94  186.5  141.26  109.24  80.143 
Al  13  2673.108  2085.983  442.07  398.57  330.21  284.59  241.43  190.47  153.71  119.99 
Si  14  3069.762  2437.676  523.5  476.06  401.43  351.1  303.17  246.52  205.05  166.77 
P  15  3494.099  2816.943  611.85  560.41  479.57  424.5  371.73  309.41  263.22  230.43 
S  16  3946.193  3223.836  707.14  651.63  564.65  504.78  447.09  379.1  328.23  280.93 
Cl  17  4426.114  3658.425  809.39  749.74  656.69  591.97  529.26  455.62  400.05  348.28 
Ar  18  4933.931  4120.778  918  854.75  755.73  686.09  618.24  538.95  478.68  422.44 
K  19  5469.738  4610.955  1034  968  861.77  787.13  714.02  629.09  564.13  503.44 
Ca  20  5129.045  1157  1087  974  895.12  816.61  726.03  656.39  591.25  
Sc  21  926  829.79  755.47  685.89  
Ti  22  940.36  861.33  787.33  
V  23  974.02  895.58  
Cr  24  1010.64  1010.64  
Mn  25  1136.2  1136.2  
Fe  26  1266.1  1266.1  
Co  27  1403  1403  
Ni  28  1547  
Cu  29  1698  
Zn  30  1856 



Electron Number  
Elem.  Number of Protons  11  12  13  14  15  16  17  18 
Na  11  5.139  
Mg  12  15.035  7.646  
Al  13  28.447  18.828  5.986  
Si  14  45.141  33.492  16.345  8.151  
P  15  65.023  51.37  30.18  19.725  10.486  
S  16  88.049  72.68  47.3  34.83  23.33  10.36  
Cl  17  114.193  98.03  67.8  53.46  39.61  23.81  12.967  
Ar  18  143.456  124.319  91.007  75.02  59.81  40.74  27.629  15.759 
K  19  175.814  154.86  117.56  100  82.66  60.91  45.72  31.625 
Ca  20  211.27  188.54  147.24  127.7  108.78  84.41  67.1  50.908 
Sc  21  249.832  225.32  180.02  158.7  138  111.1  91.66  73.47 
Ti  22  291.497  265.23  215.91  193.2  168.5  140.8  119.36  99.22 
V  23  336.267  308.25  255.04  230.5  205.8  173.7  150.17  128.12 
Cr  24  384.3  355  298  270.8  244.4  209.3  184.7  161.1 
Mn  25  435.3  404  343.6  314.4  286  248.3  221.8  196.46 
Fe  26  489.5  457  392.2  361  330.8  290.4  262.1  235.04 
Co  27  546.8  512  444  411  379  336  305  276 
Ni  28  607.2  571  499  464  430  384  352  321.2 
Cu  29  671  633  557  520  484  435  401  368.8 
Zn  30  738  698  619  579  542  490  454  419.7 



Electron Number  
Elem.  Number of Protons  19  20  21  22  23  24  25  26  27  28  29  30 
K  19  4.341  
Ca  20  11.871  6.113  
Sc  21  24.76  12.8  6.54  
Ti  22  43.266  27.491  13.58  6.82  
V  23  65.23  46.707  29.31  14.65  6.74  
Cr  24  90.56  69.3  49.1  30.96  16.5  6.766  
Mn  25  119.27  95  72.4  51.2  33.667  15.64  7.435  
Fe  26  151.06  125  99  75  54.8  30.651  16.18  7.87  
Co  27  186.13  157  129  102  79.5  51.3  33.5  17.06  7.86  
Ni  28  224.5  193  162  133  108  75.5  54.9  35.17  18.168  7.635  
Cu  29  266  232  199  166  139  103  79.9  55.2  36.83  20.292  7.726  
Zn  30  310.8  274  238  203  174  134  108  82.6  59.4  39.722  17.964  9.394 
Ga  31  64  30.71  20.51  
Ge  32  93.5  45.71  34.22  
As  33  127.6  62.63  50.13  
Se  34  155.4  81.7  68.3  
Br  35  192.8  103  88.6  
Kr  36  230.39  126  111  
Rb  37  277.1  150  136  
Sr  38  324.1  177  162  
Y  39  374  206  191 
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