For the Bohr model extended to hydrogen-like atoms the energy E of the outermost electron is given by the formula

E = −mZ²K²/(2h²n²)
which may be rewritten as
E = −m(Z/n)²K²/(2h²)

where m is the mass of an electron, Z is the net positive charge experienced by the electron, K is the constant for the electrostatic force between two charges, h is Planck's constant divided by 2π, and n is an integer called the principal quantum number.

The net charge experienced by the electron Z is the positive charge of the nucleus less the shielding by the inner electrons. The charge shielded by the inner electrons may be equal to the number of inner electrons, but it could be different as a result of the deviations from sphericity of the charge distributions of the inner electrons.

The model can be extended beyond the hydrogen-like atoms by allowing some portion of the charge of the electrons in the same shell shield an electron. That proportion or shielding factor should be in the vicinity of one half. For more on this see Ionization.

The potential energy of an electron separated from a net charge of Z by a distance s is

E = −KZ/s
and thus
s = KZ/|E|

In an atom with the nucleus being so much more massive than an electron the separation distance s and the orbit radius r are essentially the same. Thus

−mZ²K²/(2h²n²) = −KZ/r
which reduces to
mZK/(2h²n²) = 1/r
and hence
r = 2h²n²/(mZK)

If the orbit radius for electrons depends only upon the principal quantum number n then any electrons with the same n would be in the same shell with the same orbit radius.

There are however three other quantum numbers for an electron state. There is its magnetic quantum number k which can range from 0 to (n-1). The orbital quantum number l can range from −k to + k. Finally there is the spin quantum number s which can have the values ±½. This means there are 2(2k+1) electron states with the same magnetic quantum number k. The electron states having the same magnetic quantum number k can be characterized as belonging to the k subshell for the shell corresponding to the principal quantum number n.

The Subshells and Their Structure

For each principal quantum number n there is a set of subshells containing {2, 6, 10, …} states. The value of the magnetic quantum number k, in principle could go up to (n-1) but in practice it often does not do so. Based upon the atomic numbers for the noble gases the capacities for the electron shells are {2, 8, 8, 18, 18, 32}. This means that the value of k for the first shell is 0; i.e., n=1 minus 1. For the second shell k equals 1; i.e., n=2 minus 1. But for the third shell k also is equal to 1. This is because the energy for n=3 and k=2 is greater than the energy for n=4 and k=0. Thus the fourth shell starts filling after the third shell has eight electrons. Likewise for the fourth shell the maximum k is 2 rather than 3. And in the fifth shell the maximum k is 3 rather than 4.

For each shell, the subshell corresponding to a value of k has 2(2k+1) states and the same spatial structure in each shell.

In the first shell there is only the subshell corresponding to the magnetic quantum number being zero. These are depicted in the above diagram as the circles colored violet. In the second shell there are two subshells, depicted in the above diagram as violet and green, corresponding to the magnetic quantum numbers of 0 and 1. In the third shell there are three subshells corresponding to the magnetic quantum numbers of 0, 1 and 2 and depicted as violet, green and yellow circles. In the fourth shell there are the subshells for the magnetic quantum number ranging from 0 to 3, with the subshell for 3 depicted as a red circle.

The Sommerfeld Model for the
Electron States in an Atom

Arnold Sommerfeld developed a model of the atom in which electrons may have elliptical orbits. In his model there are two quantum numbers, n_θ and n_r. The quantum number n_θ is called the azimuthal quantum number and represents the number of angular momentum units of h for the state. The other quantum n_r quantizes the degree of eccentricity of the electron orbit. It can take on any nonnegative integral value; i.e.,

n_r = 0, 1, 2, …

The sum of the two quantum numbers, n_θ+n_r, is defined as the principal quantum number n. The azimthal quantum number n_θ can then take on the integral values 1, 2, …, n.

The energy E of an electron is then given by

E = −μZ²K²/(2h²n²)[1 + (α²Z²/n)(1/n_θ − (3/4)(1/n)]
which may be rewritten as
E = −μ(Z/n)²K²/(2h²)[1 + (α²n(Z/n)²)(1/n_θ − (3/4)(1/n)]

where μ is the reduced mass of the system and α is the fine structure constant, which is approximately 1/137. For a system of an electron and a nucleus the reduced mass is essentially the same as the mass of the electron.

Thus for the Sommerfeld model as well as the Bohr model the ionization energy is a function of (Z/n).

The maximum deviation from the Bohr model occurs for n=1, n_θ=1. For Z=1 the proportional deviation is then

[1 + (1/137)²(1/4) = 1 + 1.33×10^-5

For Z=10 this becomes

[1 + (1/137)²(10²)(1/4) = 1 + 1.33×10^-3

An Empirical Function for the Ionization Energies of Electrons

The Bohr model and the Sommerfeld model suggest that the ionization energy of an electron may be a quadratic function of Z/n, the ratio of the net charge experienced by the electron to its principal quantum number (shell number).

Rather than approach this question through the use of theory, an empirical approach will be used. Theories are only worthwhile if they are empirically validated. Later theory will be used to refine the analysis.

The Data Set

• E = Ionization Energies (also called potentials). The data comes from the Handbook of Chemistry and Physics, 57th Edition (1976-1977) page E-68. There are 379 data points.
• P = Charge of the nucleus, the number of protons in the nucleus. This is also called the atomic number.
• n = Electron shell number. This is also called the principal quantum number for the electron.
• IE = number of electrons in inner shells.
• SS = number of electrons in the same shell.
• k = magnetic quantum number. This is equal to the subshell number less one.

From the variables P, IE and SS a net charge of Z is computed as

Z = P − σ_IEIE − σ_SSSS

where σ_IE and σ_SS are the shielding ratios for the inner shell electrons and the same shell electrons, respectively. It is expected that σ_IE is in the vicinity of 1.0. If the energy of a state does not depend upon the subshell number then all of the electrons in a shell will have the same orbital radius and σ_SSwill be in the vicinity of 0.5. On the other hand if the energy of an electron does depend upon the subshell then the subshells in a shell will be stacked. If the lower subshells have a smaller orbit radius than the higher ones then the shielding ratio will be in the vicinity if 1.0. If the reverse would be true then the shielding ratio would be in the vicinity of 0.0.

The regression equation includes not only (Z/n)² but also (Z/n) because if the value of (Z/n) is in error by some amount γ then [(Z/n)+γ]² is equal to (Z/n)² + 2γ(Z/n)+γ².

The regression coefficient for (Z/n)² should be positive, but the one for (Z/n) can be of either sign.

The regression of the ionization energies on (Z/n) and (Z/n)² with σ_IE=1.0 and σ_SS=0.5 yields a coefficient of determination of 0.917, but the regression coefficient for (Z/n)² is negative.

The coefficient of determination (R²) was determined for various combinations of shielding ratios. The results are as follows.

When the shielding ratios are set at σ_IE=0.95 and σ_SS=0.9 the coefficient of determination (R²) is 0.969124. This appears to be the maximum for R². The regression equation found is

E = 10.42924(Z/n)² + 28.61054(Z/n) − 14.88728
which can be rearranged to
E = 10.42924[(Z/n) + 1.371650145]² − 16.76869944

The plot of the ionization energies versus the regression estimates is shown below.

The Dependence of the Ionization Energy of
an Electron State Upon its Magnetic Quantum (Subshell) Number

The fact that the best estimate of the shielding ratio for the electrons in the same shell was close to 1.0 (0.9) indicated that the subshells were ordered by distance to the nucleus and hence by energy. The dependence of the ionization energy of an electron state upon its magnetic quantum (subshell) number can also be established by including the magnetic quantum number in the regression equation. The result is

E = −11.0053 + 10.31048(Z/n)² + 28.83251(Z/n) −2.88439k

All of the regression coefficients are significantly different from zero at the 95 percent level of confidence. The coefficient of determination (R²) is 0.96968.

The plot of the ionization energies versus the regression estimates is shown below.

Conclusions

The ionization energy of the outermost electron in atoms and ions is a quadratic function of the ratio of the net charge experienced by the electron to the shell number (principal quantum number). This function explains about 97 percent of the variation in ionization energy of the 379 measured cases. The energy also depends to a much lesser extent on the subshell number (the magnetic quantum number).

The effect of the magnetic quantum number on energy is sufficient to order the subshells by distance such that the orbits of electrons in a shell are stacked so that a lower order subshell within a shell is inside that of any higher order subshell. This results in the shielding ratio for electrons in the same shell being close to 1.0. The best estimate for the shielding ratio for electrons in the same shell is 0.9. The best estimate of the shielding ratio for inner shell electrons is 0.95. The deviations of these ratios from 1.0 can be accounted for by the deviations from sphericity of the electron shells.

(To be continued.)