The surroundings of an atom are assumed as a dielectric medium, where the electrostatic laws hold. The nucleus is assumed motionless.

In order to find the equilibrium positions of electrons in an atom, the electrons are initially located on the outer surface (assumed spherical) of the nucleus, as shown in Figure 1(a), by using as guide a rhombic dodecahedron [20] , which will be described in Section 2.5. The electrons are initially uniformly and sparsely arranged, so that to avoid very small distances between neighboring electrons, which create numerical problems.

Then, by taking into account nucleus-electron interactions (attractive and repulsive), described by a Lennard-Jones curve [13] , which will be described in Section 2.4, and mutual electron-electron repulsions, as shown in Figure 1(b), and by using a simple and short computer program [14] , described by the flow-chart of Figure 2, we find that all the electrons reach in equilibrium on the outer surface of the same sphere, concentric and larger than nucleus, as shown in Figure 1(c).

According to the flow-chart of Figure 2, the proposed computer program of step-by-step nonlinear static analysis for finding the equilibrium positions of electrons, in an atom, is as follows:

First, the general input data are read, which are: the atomic number Z of the Element, equal to the number of its protons [1] , the nuclear radius, given by the formula r n = r H Z 3 , where r H = 37   pm nuclear radius of hydrogen [1] , which is reasonable for assumption of homogeneous spherical nuclei. The number ν of electrons and the steplength Δu of the algorithm, e.g. Δu = 0.001 pm, for which information will be given in Section 2.4.

Then, the initial locations r i ¯ = { x i y i z i } , i = 1 , ⋯ , ν of the electrons are read. In every step of the algorithm, as shown in Figure 2, for every electron i, first its distance r, from nucleus center O is determined and, by calling subroutine NUCLEL, the force F i ¯ (attractive or repulsive), by which the nucleus acts on the electron, is found.

Then, the distance r i k of electron i from every other electron k is determined and, by calling the subroutine ELEL, the repulsive force F i k ¯ , by which electron k acts on electron i is found.

The forces F i k ¯ , from all the other electrons k, on the electron i are summed to its force F i ¯ from nucleus and the resultant force F i ¯ on any electron i is determined.

If the maximum absolute value of the out-of-balance forces of all the electrons is greater than the desired accuracy, e.g. if max | F i | > 0.01   mdyn for i = 1 , ⋯ , ν , then every electron i is displaced by r i ¯ ← r i ¯ + Δ u F i ¯ / max | F | , as in the program of [14] , and we go to the next step of the algorithm.

Whereas, if max | F i | < 0.01   mdyn for i = 1 , ⋯ , ν , the equilibrium positions of

all the electrons r i ¯ = { x i y i z i } , i = 1 , ⋯ , ν are obtained, they are written as output, and the algorithm is interrupted.

The program is simple and short. The main program has about 100 Fortran instructions and the subroutines, NUCLEL and ELEL, about 15 and 10 instructions, respectively. So, the program has totally about 125 instructions.

As the steplength Δ u of the algorithm is very small, e.g. Δ u = 0.001   pm , millions of steps of the algorithm are performed; however, this happens in a few seconds of computing time.

A second computer program of step-by-step nonlinear static analysis is proposed, for estimation of Ionization Energies, which can be connected to the first program of previous Section 2.1, as a continuation of it.

This second program starts from the electrons being located on the outer surface of an equilibrium sphere, as shown in Figure 3(a), and one of these electrons is subjected to gradual forced displacement, radially and outwards with respect to center O of nucleus. In reality, the displacement increment Δ u , of this single electron, is very small; however, in Figure 3(a), it is presented with expanded scale, so that to become visible. After every forced displacement increment of this electron, we wait for the remaining electrons to equilibrate on the outer surface of a slightly smaller concentric sphere, by performing, again for them, the first program described in Section 2.1.

Within each forced displacement increment, the produced work increment is easily determined according to Figure 3(a), Δ E = F × Δ u and these increments

are gradually summed E ← E + Δ E , to obtain the total work E. As this single electron is gradually removed, the attractive force F diminishes and tends to zero, F → 0 . So, Δ E → 0 , too, thus the total work E = ∑ i E i no more significantly increases and tends to an upper bound E u , which is the Ionization Energy I E = E u and the procedure is interrupted. After the removal of this single electron, the remaining electrons equilibrate on a lower shell, that is, on the outer surface of a sphere smaller than the previous equilibrium sphere, but larger than nucleus, as shown in Figure 3(b).

In Figure 3(c), is shown how the total work E gradually increases and tends asymptotically to an upper bound E u , which is the Ionization Energy, I E = E u . This is usually achieved for x = r r n ≈ 100 , where r is the distance of removed electron from the center O of nucleus and r n is the radius of nucleus sphere.

The mutual repulsive force F, in a couple of electrons, is given by the following formula, with respect to the distance r between the two electrons,

F = C p 1 r 2 , (1)

where the permittivity constant is C p = 2.3072 × 10 − 28 J ⋅ m .

As example, let us consider the Element Argon with atomic number Z = 18 and nuclear radius r n = 97.0   pm . According to following Section 2.5, can be proved that the minimum distance between two electrons, in the proposed model, for 14 < Z ≤ 26 , is min r = 97.0   pm [ 0.577 2 + 2 ( 0.707 − 0.577 ) 2 ] 1 / 2 = 58.77   pm

By substituting this value in Equation (1), is obtained

max F = 2.3072 × 10 − 28   N ⋅ m 2 10 24 58.77 2   m 2 = 6.680   mdyn ,

represented in Figure 4.

The derivative of function F(r), with respect to r, according to Equation (1), which can be called stiffness K, is

K = d F / d r = − 2 C p 1 r 3 ,

and for r = minr = 58.77 pm, gives

max K = − 2 × 2.3072 × 10 − 28 N ⋅ m 2 10 36 58.77 3 m 3 = − 0.2273   mdyn / pm ,

also shown in Figure 4.

The definite integral of function F(r), from r 1 up to ∞, is

∫ r 1 ∞ F d r = − C p [ 1 r ] r 1 ∞ = − C p 1 r 1

and, by referring to 1.0 mole of the Element, for Avogadro constant C A = 6.022 × 10 23 / mole , the work produced for separation of couples of atoms

from an initial mutual distance r 1 = 200   pm up to infinity, is:

E = − C p 1 r 1 C A = − 2.3072 × 10 − 28 J ⋅ m 10 12 200   m 6.022 × 10 23 / mole = − 694.7   kJoules / mole (2)

For the nucleus-electron interaction (attractive and repulsive), a Lennard-Jones function [13] is adopted,

φ ( x ) = − 1 x 2 n + 1 x n , (3)

where x = r / r n , r distance of electron from nucleus center and r n nuclear radius. The first above term is repulsive and the second one attractive.

In Figure 5, curves φ ( x ) are shown, corresponding to the above function of Equation (3), for three different values of the parameter n: 1.05, 1.5, 2.0. It is observed, in this Figure 5, that, as the value of parameter n increases from 1.05 up to 2.0, the attractive potential of the L-J function φ ( x ) is gradually weakened.

The complete equation, giving the force F with respect to the distance r, is

F = Z C p 1 r n 2 ( − 1 x 2 n + 1 x n ) ,

where Z atomic number of the Element, C p permittivity constant, r n nuclear radius and x = r / r n . Let us try and example.

For Argon, Z = 18, r n = 97.0   pm , and the parameter of L-J curve, for the last four Ionization Energies (15^th up to 18^th IE), is n = 1.057, as we will see in Section 3 of applications. So,

F = 18 × 2.3072 × 10 − 28 N ⋅ m 2 10 24 97.0 2 m 2 φ ( x ) = 44.138   mdyn ( − 1 x 2.114 + 1 x 1.057 ) ,

represented by the curve of Figure 6, where the maximum value of φ ( x ) is 0.25, so maxF = 11.035 mdyn.

The derivative dF/dr of the above function F(r), which can be called stiffness K, is

K = d F d r = Z C p 1 r n 3 ( + 2 n x 2 n + 1 − n x n + 1 ) ,

and, by substituting the values of variables, for x = r / r n = 1.0 , is obtained

max K = 18 × 2.3072 × 10 − 28 N ⋅ m 2 10 36 97.0 3 m 3 × 1.057 = 0.4811   mdyn / pm ,

also represented in Figure 6.

By observing the maximum values of F, in Figure 4 and Figure 6, maxF = 6.680 and 11.035 mdyn, respectively, it is obtained that an accuracy of 0.01 mdyn is sufficient in calculations. Also, by observing the maximum stiffnesses maxK = 0.2273 and 0.4811 mdyn/pm, in the same Figure 4 & Figure 6, we can choose a steplength Δu, for the step-by-step nonlinear static analysis algorithm of Section 2.1, according to the inequality

0.4811   mdyn / pm × Δ u < 0.01   mdyn ⇒ Δ u < 0.02078   pm .

So, a steplength Δu = 0.02 pm is sufficiently small. However, an even smaller steplength Δu can be used, up to Δu = 0.001 pm, for more accuracy, because millions of steps of the proposed algorithm are performed in a few seconds of computing time.

The definite integral of the L-J function φ ( x ) of Equation (3) is

∫ 1 x φ d x = ( 1 n − 1 − 1 2 n − 1 ) − [ 1 ( n − 1 ) x n − 1 − 1 ( 2 n − 1 ) x 2 n − 1 ] (4)

which, for n > 1.0, exhibits an upper bound

φ u = ∫ 1 ∞ φ d x = 1 n − 1 − 1 2 n − 1 . (5)

So, in the specific example under consideration, in the last 18^th IE (Ionization Energy) of Argon, with only one electron, for the removal of this one electron in every atom of Argon, from r n up to infinity, for1.0 mole of Element Argon, with the accurate value of parameter n = 1.05727, we have

φ u = 1 1.05727 − 1 − 1 2 × 1.05727 − 1 = 17.461 − 0.897 = 16.564 .

And the upper bound of produced work E u , that is the Ionization Energy, is

18 thIE = E u = Z × C p × 1 r n × φ u × C A = 18 × 2.3072 × 10 − 28 J ⋅ m 10 12 97.0   m × 16.564 × 6.022 × 10 23 / mole = 427063 ≈ 427066   kJoules / mole (6)

that is, it agrees accurately with the value of database [19] .

As mentioned in Section 2.1, in the proposed step-by-step program of nonlinear static analysis, a rhombic dodecahedron [20] is used as a guide, in order to determine the initial locations of the electrons on the outer spherical surface of nucleus, so that the electrons be uniformly and sparsely arranged and avoid very small distances between neighboring electrons, which create numerical problems.

In Figure 7, is shown how a rhombic dodecahedron is formed from a regular octahedron and a cube. In Figure 8, the plan and elevation of rhombic dodecahedron are shown and the order by which, one-by-one, the electrons are added, first to the 6 vertices of regular octahedron, then to the 8 vertices of cube, projected on the circumscribed sphere and the centers of 12 faces of rhombic dodecahedron, coinciding with the middles of edges of regular octahedron, also projected on the outer surface of circumscribed sphere.

Attention is paid to the location of the fourth electron, so that not to lie on the same plane with the first three electrons.

In order to extend the use of rhombic dodecahedron to Elements with higher atomic numbers, Z > 26, we can also use the 12 ´ 4/2 = 24 middles of edges, and the 12 ´ 4 = 48 middles of distances of every face center from the four vertices of corresponding rhomb. All, projected on the outer surface of circumscribed sphere, can be used as guide for initial electrons locations.

So, according to the above, the total number of electrons, which can be accommodated by the rhombic dodecahedron, is

ν = 6 + 8 + 12 + 24 + 48 = 98 ,

20 less than the 118 Electrons of the last existing Element.

In the present work, the applications will not exceed the 18 electrons. So, the rhombic dodecahedron, which is very simple, is enough here and will be used as guide for the initial locations of electrons.

In order to accommodate Elements with very high atomic number Z > 98, another regular polyhedron [21] is revealed, which is obtained by the projections of the 8 cube vertices on the circumscribed sphere of rhombic dodecahedron. The plan view of this regular polyhedron is shown in Figure 9(b), in comparison with the very simple plan view of corresponding rhombic dodecahedron, shown in Figure 9(a). In both Figure 9(a) & Figure 9(b), only the visible faces are shown.

The above revealed new regular polyhedron has the 6 vertices of regular octahedron and the 8 vertices of cube, totally 6 + 8 = 14 vertices, 6 ´ 4 = 24 triangular faces and 24 ´ 3/2 = 36 edges. Vertices, gravity centers of triangular faces and middles of edges, all projected on the circumscribed sphere, can be used as guide for initial electrons locations.

Also, the 24 ´ 3 = 72 middles of distances of gravity centers of triangular faces from corresponding triangle vertices, projected on the circumscribed sphere, can be used.

According to the above, the maximum number of electrons, which can be accommodated by the new regular polyhedron, is

ν = 6 + 8 + 24 + 36 + 72 = 146 ,

that is 28 more than the 118 electrons of last existing Element. These 146 electrons are demonstrated in Figure 10(b), in comparison with the 98 electrons that can be accommodated by the rhombic dodecahedron, shown in Figure 10(a).

The two last Ionization Energies of Argon, the 18th and 17th IE, can be investigated by a hand calculator having the option of fractional exponent, on the basis of Figure 11(a) & Figure 11(b).

For the last 18th IE (Figure 11(a)), the following formula is used, according to

previous Equation (6)

18 thIE = E u = ∫ r n ∞ F d r = Z C p 1 r n φ u C A , (7)

where Z = 18 atomic number of Argon, C p = 2.3072 × 10 − 28 J ⋅ m permittivity constant, r n = 97.0   pm nuclear radius, C A = 6.022 × 10 23 / mole Avogadro constant and the upper bound φ u of L-J function integral, for parameter n > 1.0 , is,

φ u = ∫ 1 ∞ φ ( x ) d x = 1 n − 1 − 1 2 n − 1 ,

according to previous Equation (5).

The database [19] gives, for the last 18th IE of Argon, the value 427,066 kJoules/mole.

By substituting the above values in Equation (7), is obtained

427066   kJoules / mole = 18 × 2.3072 × 10 − 28 J ⋅ m 10 12 97.0   m φ u × 6.022 × 10 23 / mole ⇒ φ u = 1 n − 1 − 1 2 n − 1 = 16.654 ( dimensionless ) .

By using the diagram φ n − n of Figure 12, the value of parameter n, of L-J curve, is obtained graphically, for the last 18th IE of Argon, which is n = 1.05727. In the diagram φ n − n of Figure 12(a), are found the values of parameter n of L-J curve, for the last IE of 12 Elements of 3 first Periods, from Argon (Z = 18) up to Nitrogen (Z = 7), all in the same way as for Argon, with help of database [19] . The same diagram φ n − n is continued to Figure 12(b), with contracted n scale, where the values of parameter n of L-J curve, for the last IEs of the remaining 6 Elements of 3 first Periods, from Carbon (Z = 6) up to Hydrogen (Z = 1), are found in the same way.

Based on Figure 11(b), we can estimate the 17th IE of Argon, too, by subtracting, from 18th IE, the negative work produced by the separation of the two electrons, which is very small, as follows, according to Equation (2)

E = ∫ r 0 ∞ F d r = − C p 1 r 0 C A ,

where r 0 = 2 × 98   pm = 196   pm , according to Figure 11(b); permittivity C p = 2.3072 × 10 − 28 J ⋅ m and Avogadro constant C A = 6.022 × 10 23 / mole .

By substituting these values on the above equation, is obtained

E = − 2.3072 × 10 − 28 J ⋅ m 10 12 196   m 6.022 × 10 23 / mole = − 708.9   kJoules / mole

Thus, for the 17th IE of Argon, the value 427,066 − 708.9 = 426,357 kJoules/mole results. The database [19] gives a somewhat smaller value, 17th IE = 397,605 kJoules/mole for Argon, which is, however close to the last 18th IE = 427,066 kJoules/mole, too.

By substituting the value of parameter n = 1.05727 of L-J function, found for the last 18th IE of Argon, in the Equation (4) of definite integral of L-J function

∫ 1 x φ ( x ) d x = ( 1 n − 1 − 1 2 n − 1 ) − [ 1 ( n − 1 ) x n − 1 − 1 ( 2 n − 1 ) x 2 n − 1 ] ,

we find how the work produced by the removal of a single electron in every atom, referred to 1.0 mole of the Element, varies, as the electrons are gradually removed from the atoms.

E ( x ) = 18 × 2.3072 × 10 − 28 J ⋅ m 10 12 97.0   m ∫ 1 x φ ( x ) d x = 25783   kJoules / mole [ 16.564 − 1 ( n − 1 ) x n − 1 − 1 ( 2 n − 1 ) x 2 n − 1 ]

for n = 1.05727.

The graphical representation of the above function E(x) is shown in Figure 13, for x = r / r n up to 10⁸, that is for r up to about 1.0 cm. The produced work reaches up to about 280,000 kJoules/mole and tends asymptotically to the upper bound 427,066 kJoules/mole.

The produced work E(x), corresponding to the 17th IE, is slightly lower than that of last 18th IE, according to what mentioned above, and almost coincides with it.

All the Ionization Energies of Argon, from last 18th IE, with 1 electron, up to first IE, with 18 electrons, have run by the proposed two step-by-step nonlinear static analysis programs of Sections 2.1, 2.2, and the results are represented in Figure 14 up to 16, and in Table 1.

In Figure 14, according to the results of first program, the radius of equilibrium sphere varies from r 1 = 97.0   pm , which is the nuclear radius (18th IE with 1 electron), up to 122.8 pm, which corresponds to 1st IE, with 18 electrons ( ν = 18 = Z ) ; so, this is the atomic radius predicted by the first program. Also, an intermediate radius is shown, with value 110.4 pm, which corresponds to 10th IE with 9 electrons. More details for radii of equilibrium spheres of electrons can be found in Table 1.

The atomic radius 122.8 pm of Figure 14, predicted by the first program, is compared with three corresponding values of published databases [15] [18] , 71, 106 and 188 pm, and found close to their average, ( 71 + 106 + 188 ) / 3 = 121.7 ≈ 122.8   pm . The proposed model is static. So, the three values of databases can be considered as due to dynamic oscillations around a mean static value.

In Figure 15, obtained by the second program, is shown a group of curves representing the variation of work E(x) produced by removal of one electron, referred to 1.0 mole of the Element, for x = r / r n ranging up to 100, that is for r up to about 10   nm = 10 − 8   m . Every curve corresponds to an IE, from 18th IE, with 1 electron, up to 1st IE with 18 electrons.

It is observed, in Figure 15, that, for number of electrons 10 ≥ ν ≥ 7 , these curves E(x) tend to an upper bound E u = I E for x ≈ 100 . Whereas, for ν ≥ 11 , the curves E(x) tend to an upper bound E u = I E earlier, for x ≈ 50 .

Particularly, it is observed, in Figure 15, that between 9th and 8th IE, that is for number of electrons ν between 10 and 11, an abrupt fall occurs, in these curves, with the value of I E = E u falling from about 40,000 to about 14,000 kJoules/mole.

In Table 1, for all the Ionization Energies, from last 18th IE with only 1 electron, up to 1st IE with 18 electrons, the corresponding values of the following quantities are given, obtained by the proposed two step-by-step programs of nonlinear static analysis and by the database [19] .

1) Radius r ν , for ν = 1 , ⋯ , 18 , of the equilibrium sphere of electrons, obtained by the first program, increasing gradually from 97.0 up to 122.8 pm.

2) Ratio r ν / r n , for ν = 1 , ⋯ , 18 , of radius r ν of equilibrium sphere of electrons to nuclear radius r n , increasing from 1.0 up to 1.266 (dimensionless).

3) Parameter n of L-J (Lennard-Jones) curve (exponent of denominator of attractive term) increasing from 1.057 up to 1.93 (dimensionless). Between 9th and 8th IE, so number of electrons between 10 and 11, a jump of n value, from 1.28 to 1.68 is observed. The increase, in the value of L-J parameter n, means weakening of the attractive potential of L-J curve, as mentioned in Section 2.4.

4) Value of IE (Ionization Energy) in kJoules/mole, obtained from database [19] , varying from 427,066 up to 1521 kJoules/mole. The values of two last IEs, 18th and 17th IE are very high; as mentioned previously and shown in Figure 13, they correspond to very large distances r of electron from nucleus and seem not realistic. Between 9th and 8th IE, that is for number ν of electrons between 10 and 11, an abrupt fall of IE value of database is observed from 40,760 to 13,842 kJoules/mole.

5) Value of IE (Ionization Energy) in kJoules/mole, obtained by the second proposed program, varying from 81,450 up to 1542 kJoules/mole. In the last two IEs, 18th and 17th IE, the differences between the values of database [19] and the corresponding ones of proposed model are very large. As mentioned previously, the very large values of database, for two last IEs, correspond to very large values of removed electron distances from nucleus and seem unrealistic. For the next two IEs, 16th and 15th IE, differences, between values of database and corresponding ones of proposed model, are small. For all the other IEs, from 14th up to1st IE, that is, for number ν of electrons from 5 up to 18, the Ionization Energies of proposed model have been calibrated to those of database, by controlling the parameter n of L-J curve, so a close agreement between them is observed.

Based on the data of Table 1, the three diagrams of Figures 16(a)-(c) have been drawn:

1) Increase of radius r of equilibrium sphere of electrons, from 97.0 up to 122.8 pm, for number of electrons ν = 1…18, obtained by the first proposed step-by-step nonlinear static analysis program.

2) Increase of L-J parameter n from 1.057 up to 1.93, for number of electrons ν = 1 , ⋯ , 18 . Between ν = 10 and 11, a jump of n value from 1.28 to 1.68 is observed. The increase of L-J (Lennard-Jones) parameter n means weakening of L-J curve attractive potential, as mentioned in Section 2.4.

3) Decrease of IE (Ionization Energy) of database [19] , from 427,066 kJoules/mole (corresponding to very large electron removal distance r up to 1.0 cm, see Figure 13), up to 1521 kJoules/mole, as well as decrease of IE obtained by the second proposed step-by-step static analysis program, from 81,450 up to 1542 kJoules/mole (corresponding to reasonably small electron removal distances, up to x = r / r n = 100 , that is, for r up to about 10   nm = 10 − 8   m , as shown in Figure 15). For number ν of electrons between 10 and 11, an abrupt fall of IE is observed, from 40,760 to 13,842 kJoules/mole in database [19] values, and from 39,720 to 14,020 kJoules/mole in the results of second proposed step-by-step nonlinear static analysis program.

On the basis of three diagrams of Figure 16, the following can be observed:

As the number ν of electrons increases, these, by their mutual repulsive forces, are pushed to larger equilibrium spheres, which has as consequence the weakening of attractive potential of the nucleus-electron interaction, manifested by an increase of parameter n of L-J curve, as mentioned in Section 2.4, see Figure 5. So, the IEs (Ionization Energies) values are reduced, the electrons are more loosely connected to the nucleus and can be more easily removed.

Particularly, when the number ν of electrons and the radius r of equilibrium sphere exceed some critical values, here ν = 10 and r = 110.0 pm (average of two extreme values (97.0 + 122.8)/2 = 109.9 pm), then an abrupt fall of attractive nucleus-electron potential occurs, manifested by a jump of L-J parameter n from 1.28 to 1.68, consequently an abrupt fall of IE (Ionization Energy) value, both that of database [19] , as well as that of second proposed step-by-step nonlinear static analysis program, from about 40,000 to about 14,000 kJoules/mole.