Spin-orbital exclusion principle and the periodic system

Introduction

According to modern understanding, electron distribution in the atoms of chemical elements with respect to the values of quantum numbers (the principal n, the orbital ℓ, the magnetic m_l, and the spin m_s) are regulated by three basic rules: the Bohr Aufbau principle [1], the Pauli exclusion principle [2], and the Hund rule of orbitals filling [3]. Based on these rules, it is theoretically assumed that the filling of the (n+1)^th shell can only start after all the nℓ-subshells with the orbital numbers from ℓ=0 to ℓ=n–1 are filled in the previous n-shell. Sommerfeld [4] called such a sequential filling of electronic shells an ideal system, and it has the following form:

At the same time, the results of spectral analysis [5] indicate that the real sequence of appearing new subshells in neutral atoms with the atomic number Z=1–108 looks as follows:

In addition, it is noted that, in the region of large values of Z, three subshells 4f, 5f, 6d appear with a small delay (4f subshell is preceded by the appearance of one 5d electron in lanthanum, 5f subshell appears after two 6d electrons in actinium and thorium, whereas the main filling of subshell 6d begins after the appearance of one 7p electron in lawrencium). But since these delays are short, they are not shown in sequence (2).

The ideal system (1) of shell filling and the real system of appearing new subshells (2) after filling the 3p subshell do not coincide with each other. This is due to the fact that an ideal system corresponds to hydrogen-like atoms in which each electron is considered independently of the others, while the mutual interaction of electrons influences the real system. Accordingly, it seems tempting to supplement the set of the principles of Bohr, Pauli, and Hund with another theoretical principle that takes into account the interrelations of electrons located on different shells and allows explaining the genesis of sequence (2). Obtaining such a rule and interpreting on its basis the sequence of appearance of new subshells in neutral atoms correspond to the solution of the “Löwdin challenge” problem [6], which is connected with the Madelung empirical rule [7], which has no satisfactory explanation yet [8], [9], [10].

The present paper establishes a new exclusion principle, which regulates the distribution of the electron spin values on neighboring subshells with the same value of the orbital quantum number. With the help of this principle, a theoretical explanation is presented for the real system (2) of the new subshell appearance, and its comparative analysis with the ideal system (1) of shell filling is performed. Moreover, in the process of the material presentation, the nature of the Madelung rule is revealed. In addition, on the basis of the established exclusion principle, an eight-period table of chemical elements is compiled, and an explanation is given of such structural specificities of the Mendeleev periodic table as pairing with respect to the number of elements of the second and third, fourth and fifth, sixth and seventh periods, as well as the absence of a pair for the first period with such same number of elements.

Spin-orbital exclusion principle

Let us denote the ordinal number of the appearance of electrons in the nℓ-subshell by the symbol k=1, 2, …, 2(2ℓ+1) and consider electrons with the same values of ℓ and k. Since each subshell with a particular value of the orbital number ℓ is located farther from the nucleus, if its principal quantum number n is larger, it can be said that such electrons form radial groups in the atom, which will be called ℓk-groups. That is, an ℓk-group is a set of electrons with the same set of values of ℓ and k. Since electrons with a particular value of the orbital number ℓ appear in the atom for the value of the principal quantum number n=ℓ+1, the number of electrons in the ℓk-groups equals n^*–ℓ, where n^* is the principal quantum number of the last subshell of the atom with this value of ℓ. A schematic depiction of such radial ℓk-groups, using the example of p-electrons of the krypton atom, is shown in Fig. 1.

Fig. 1:

Diagram of pk-groups of electrons in the krypton atom.

It is obvious that the change in the values of quantum numbers inside the ℓk-groups must take place in a regular fashion and have the same nature for all groups. However, if the orbital ℓ and magnetic m_ℓ quantum numbers inside each ℓk-group have a constant value, while the principal quantum number n increases monotonically from n=ℓ+1 to n=n^*, then the values of the spin quantum number m_s, according to the accepted rules for filling electronic subshells, can have an arbitrary distribution. This arbitrariness is explained by the fact that the Hund rule, requiring that, in filling the subshells, the modulus of the sum of the spins of all electrons assumes the maximum possible value, is indifferent to the value of the spin of the electron with the ordinal number k=1. Since this electron first appears in the subshell and determines the spin value of all other electrons, 2n*−ℓ variants of the distribution of the spin values inside each ℓk-group are allowed. From the standpoint of the regular structure of the atom, such a number of possible variants for the distribution of spin values is not acceptable, and the question arises concerning the optimal variant. Answering this question, one can proceed from the fact that, under the optimal variant of the distribution of spin values, the electrons should experience a stronger attraction to the atom nucleus in comparison with other variants. In order to find such an optimal variant, the process of formation of ℓk-groups for k=1 can be considered.

The first electron of the ℓk-group is simultaneously the first electron in the atom with this value of the orbital number ℓ. The direction of the spin of this electron can be arbitrary, which follows, for example, from the fact that, in the formation of a hydrogen molecule, two atoms participate, the electrons of which have opposite spins. Therefore, in order to identify the general nature of changes in the spin values in all ℓk-groups, one can assume that the first electrons with a new value of ℓ have a positive spin. Considering now the second electron of the ℓk-group for k=1, it can be claimed that an exchange interaction arises between it and the first electron [11], which is characterized by the fact that electrons with parallel spins repel each other more strongly than with the antiparallel ones [12]. It follows that the second electron must have a negative spin for a stronger attraction to the nucleus of the atom. Moreover, the spins of both electrons are mutually compensated. Correspondingly, the spin of the third electron must be positive, of the fourth one, again negative, and so on. According to the Hund rule, all the electrons with the ordinal numbers 1≤k≤2ℓ+1 will have the same distribution of spin values, whereas for the ordinal numbers 2ℓ+1<k≤2(2ℓ+1), the spin values will have the opposite sign. That is, to improve the overall stability of the atom, the compensation of electron spins must take place not only on the orbitals (orbital compensation) but also inside each ℓk-group (radial compensation).

Thus, it is possible to draw a conclusion that the optimal variant of the distribution of the spin values inside the ℓk-groups is their sequential alternation. A visual representation of such a spin alternation is provided by Table 1, where, by the example of p-electrons of the xenon atom, the spin direction for the optimal and traditional distribution variants is shown by arrows (the upward arrows correspond to positive spin values and the downward arrows, to negative values).

It should be noted that the same conclusion about the optimal distribution of spin values can be made based on the idea that ℓk-groups with the same value of ℓ and the same number n^*–ℓ of electrons should have the same value of the sum of the total electron moments ∑j, j=ℓ+m_s. The reason is that it is traditionally assumed that all subshells begin with electrons with the same spin value. Moreover, in most cases, it is assumed by tacit agreement that the first electron has a positive spin. Such identical value of the spin of the first electrons of all subshells creates, for the same value of ℓ, an asymmetry of ℓk-groups in terms of the value of the sum ∑j, which contradicts the regular structure of the atom. For example, the pk-groups of the xenon atom, under the traditional distribution of spin values (Table 1), have the sum ∑j=6 for k=1, 2, 3 (p¹, p², p³), while for k=4, 5, 6 (p⁴, p⁵, p⁶) this sum equals ∑j=2. The indicated asymmetry of the sum of total moments is completely eliminated or becomes minimal, if the subshells with the same value of ℓ alternate in the direction of the spin of their first electrons. In this case, the complete elimination of asymmetry takes place when ℓk-groups consist of an even number of electrons, while in the case of an odd number, the values of the sum ∑j for 1≤k≤2ℓ+ become greater by one than for 2ℓ+1<k≤2(2ℓ+1). For example, in the same xenon atom, all pk-groups consist of four electrons and, accordingly, when alternating the directions of the spin, each group has the value ∑j=4, that is, there is no asymmetry. In turn, as an example of the minimum asymmetry of the sum of total moments, one can consider pk-groups of the krypton atom, consisting of three electrons. In this case, for k=1, 2, 3, pk-groups have the sum ∑j=3.5, whereas for k=4, 5, 6 this sum is equal to 2.5 (while under the traditional distribution of spin values, these sums are equal to 4.5 and 1.5, respectively).

The same conclusion, obtained in different ways, about the optimality of alternation of spin values inside radial ℓk-groups allows making the following general conclusion: the neighboring electron subshells with the same value of the orbital quantum number (n, ℓ and n±1, ℓ) cannot start with electrons with the same spin value. Since the conclusion made is prohibitive in nature and regulates the distribution of spin values in subshells with the same orbital quantum number, it can be called the spin-orbital exclusion principle.

Spin-orbital specificities of real and ideal systems

The distribution of electrons, taking into account the spin-orbital exclusion principle, leads to the fact that in many-electron atoms, for each value of the orbital quantum number, two types of electron subshells are formed, which are mirror-symmetrical to each other in the direction of spin of their electrons. These subshell types can be classified as positive (n, ℓ)⁺ and negative (n, ℓ)⁻, keeping in mind that, in the first case, subshells begin with electrons with a positive spin, and in the second case, with a negative one. That is, to each value of the orbital quantum number ℓ=n–1, n=1, 2, …, there corresponds in a many-electron atom a sequence of positive and negative subshells:

Moreover, in the process of filling each subshell, the number of electrons in the corresponding ℓk-groups increases, and the spins of these electrons are periodically radially compensated.

Let us now subdivide the subshells of the real system (2) into positive and negative ones according to (3) and analyze the resulting configuration. Moreover, for the convenience of analysis, in order to observe the same number of positive and negative subshells for ℓ=3, a subshells 7p, 8s can be added to the real system. That is, a real system for Z=120 is considered in the following form:

The configuration (4) shows that, from the perspective of the spin-orbital exclusion principle, the process of filling electronic subshells in a real system has a clear periodic character and develops in cycles consisting of two periods with the same number of subshells. Moreover, the first period of the cycle consists of positive subshells, whereas the second period consists of negative ones. Altogether, for Z=120 four cycles T_N, N=1, 2, … and eight periods t_r, r=1, 2, …, are distinguished, which form a spin-orbital periodicity in the electron configuration of the atom. The detailed structure of this periodicity with the indication of the corresponding chemical elements is presented in Table 2.

It can be seen from Table 2 that each spin-orbital cycle begins with a subshell with a new value ℓ^new of the orbital quantum number. Moreover, in each period the subshells follow each other in the decreasing order of the values of this number from ℓ=ℓ^new to ℓ=0. This suggests that, at the beginning of the first period of each cycle, new ℓk-groups appear in the atom in the form of one-element sets of electrons with the value of the orbital number ℓ=ℓ^new, after which the number of electrons in all previously formed groups is successively increased by one. In the second period, this process of the successive increase in the number of electrons in all ℓk-groups repeats and differs from the process in the first period only in that the new electrons have the opposite spin. Moreover, the principal quantum number determines the energy level of the electrons inside each ℓk-group and successively increases in passing from one subshell to another in the first period from n=ℓ^new+1 to n=2ℓ^new+1, and in the second period, from n=ℓ^new+2 to n=2(ℓ^new+1).

It follows from the above that, within each cycle, the distribution of spin values with respect to electronic states occurs in such a way that, in the first period of the cycle, there takes place accumulation of electrons with radially uncompensated spins, which are then successively mutually compensated by the spins of the electrons of the second period. Correspondingly, at the end of each cycle, all electrons in the atom have a radially compensated spin, after which a new cycle begins and everything repeats. A vivid illustration of this is Fig. 2, which shows the dynamics of the change in a real system of the number of electrons with radially uncompensated spins with the increase in the atomic number.

Fig. 2:

Dependence of the number of electrons with radially uncompensated spins on the atomic number in a real system.

Consider now the ideal system (1) from the viewpoint of the spin-orbital exclusion principle; to this end, according to (3), one can identify in it positive and negative subshells and present the configuration of seven known shells K, L, M, N, O, P, Q in Table 3.

Analysis of Table 3 shows that the equality of the number of positive and negative subshells in the ideal system is observed only once when a subshell (2s)⁻ appears, whereas already in the filling of the (2p)⁺ subshell, an irreversible excess of the number of positive subshells over the negative ones occurs. That is, the electronic structure of the atom, “drawn” by the ideal system, is always asymmetric with respect to the relation between positive and negative subshells for Z>4. Accordingly, the simultaneous radial compensation of the spin for all electrons never occurs, which is clearly shown in Fig. 3 for the successive increase in the number of electrons to the value Z=120.

Fig. 3:

Dependence of the number of electrons with radially uncompensated spins on the atomic number in the ideal system.

Comparing the dynamics of changes in the number of electrons with radially uncompensated spins in the real (Fig. 2) and ideal (Fig. 3) systems leads to the conclusion that, from the positions of the spin-orbital exclusion principle, the evolution of the electronic structure of the atom in the process of increasing the charge of the nucleus goes not along the path of successive formation of shells consisting of electrons with the same value of the principal quantum number, but along the path of a periodic increase in the number of electrons in the radial ℓk-groups. In this case, ℓk-groups with a new value of the orbital quantum number are formed in the atom immediately after the electron system reaches a state in which all electrons have a radially compensated spin. It is in this evolutionary development of electronic systems that the reason for the discrepancy between the ideal and the actual systems lies. Let us demonstrate this using particular examples and, first of all, explain why the 4s-electrons, not the 3d ones (as should be according to the ideal system) appear in the potassium and calcium atoms in the real system.

The electron system of the argon atom (Z=18), which precedes the potassium, taking into account the sign of the subshells, has the configuration of (1s)⁺, (2s)⁻, (2p)⁺, (3s)⁺, (3p)⁻. That is, by the time of appearance of an electron with the ordinal number Z=19 in the electron system of the atom, all p-electrons, as well as s-electrons of the subshells (1s)⁺ and (2s)⁻, have a radially compensated spin; whereas the electron spins of the subshell (3s)⁺ remain uncompensated. This means that, for argon, after filling the (3p)⁻ subshell, the fourth spin-orbital period t₄, with which the second cycle T₂ ends, remains incomplete. Accordingly, until the spin of the electrons of the (3s)⁺ subshell is radially compensated, the third spin-orbital cycle T₃, at the beginning of which the d-electrons should appear for the first time, cannot begin its formation. For this reason, in the potassium and calcium atoms, it is precisely the (4s)⁻ subshell that is being filled, the electron spins in which are mutually compensated by the electron spins of the (3s)⁺ subshell. After that, in the scandium atom (Z=21), the first d-electron expectedly appears, that is, the filling of the (3d)⁺ subshell is initiated, which is the first subshell of the third spin-orbital T₃ cycle. The fact that the (6s)⁻ subshell is filled before the (4f)⁺ one is explained in a similar fashion. This is due to the fact that (6s)⁻ is the last subshell of the T₃ cycle, whereas (4f)⁺ is the first subshell of the next cycle T₄. Accordingly, the first f-electron appears in the atom only after the completion of the T₃ cycle, that is, after filling the (6s)⁻ subshell.

With regard to the discrepancies between the real and ideal systems, such as earlier filling of the (5s)⁺ and (7s)⁺ subshells with respect to the (4d)⁻ and (5f)⁻ subshells, the following can be said. These discrepancies take place inside the spin-orbital cycles T₃ and T₄ and are connected with the replacement of the periods of accumulation of electrons with radially uncompensated spin (t₅ and t₇) by the periods in which radial compensation of the spin occurs (t₆ and t₈). That is, in the T₃ cycle, the (5s)⁺ subshell completes the first period of the cycle, after which the (4d)⁻ subshell, with which the second period starts, is regularly filled. Similarly, in the T₄ cycle, the (7s)⁺ subshell is filled at the end of the first period, after which the (5f)⁻ subshell is formed, which is the first subshell of the second period of the cycle.

Quantitative aspects of the spin-orbital periodicity

Let us now demonstrate the relationship between the ordinal numbers of spin-orbital cycles and periods with the value of the quantum numbers of their subshells, after which determine the number of electrons in each cycle and period, and also find the ordinal numbers of electrons for which new values of the orbital quantum number should appear for the first time.

Each cycle T_N begins with the electrons with a new value of the orbital quantum number ℓ^new, which is increased by one when passing from cycle to cycle. Since the minimum value of this number is zero, the ordinal number N of the cycle is expressed by ℓ^new via a simple relation:

It was previously demonstrated that inside the spin-orbital periods, the values of the principal and orbital quantum numbers successively change in the opposite manner. It follows that the sum n+ℓ of these numbers within each period preserves its constant value. Moreover, the minimum value of n increases when passing from the first period of the cycle to the second, whereas the maximum value ℓ increases with the transition from cycle to cycle. Since in the first period t₁ the principal and orbital quantum numbers have the values n=1 and ℓ=0, the value of the sum n+ℓ in each period t_r is equal to its ordinal number r, that is:

In connection with obtaining equation (6), it is necessary to say a few words about Madelung’s rule of the “n+ℓ” sum [7], which is also known as Klechkovsky’s rule [13]. This rule says that as the atomic number Z increases, the subshells are filled in the order of increasing the sum of the principal n and orbital ℓ quantum numbers, and when these sums are equal, the subshells follow each other in the order of increasing values of n and decreasing values of ℓ. Such a change in the values of quantum numbers n and ℓ is observed in each period t_r and, correspondingly, from the standpoint of equality (6), it can be said that the sum “n+ℓ” rule simply reflects the sequence of formation of the spin-orbital periods and the order of filling the subshells within them. This, in turn, allows us to say that equation (6) together with Table 2 can be considered as a version of solution to the “Löwdin challenge” problem [6], in which it is required to theoretically explain the equivalent of the Madelung rule, postulated by Bohr.

In each cycle T^N, the orbital quantum number of subshells assumes twice the same series of values ℓ=ℓ^new, …, 0. Accordingly, taking into account that, according to the Pauli principle, a filled subshell contains 2(2ℓ+1) electrons, the total number of electrons in the cycle |T_N| equals:

Substituting in equation (7) the sum ℓ^new+1 by N according to (5), one can obtain a simple dependence of the number of electrons in the cycle on its ordinal number:

Since each cycle T_N consists of two equal-sized periods, it follows from (8) that the number of electrons in any period |t_r| depends on the number N of the cycle, in which this period lies and is determined from the condition:

Condition (9) indicates that the number of electrons in an arbitrarily taken period t_r does not have a one-valued relation with its ordinal number r. However, this relation can be expressed separately for the odd (r=2N–1) and even (r=2N) periods. That is, in accordance with condition (9), one can get:

Carrying out calculations by formulas (8), (10), (11), one can arrive at the following values of the number of electrons in the spin-orbital cycles (N=1÷4) and periods (r=1÷8):

As was already indicated, spin-orbital cycles start with electrons with a new value of the orbital quantum number ℓ^new. Therefore, the ordinal number Zlnew of these electrons is equal to the increased by one sum of the number of electrons forming the previous cycles. Accordingly, on the basis of (8) taking into account (5), one can obtain:

Calculations by formula (14) for known s, p, d, f – electrons yield the following numerical series:

The obtained values (15) with respect to s, p, d-electrons completely coincide with the ordinal numbers of the chemical elements in whose atoms they first appear (₁H, ₅B, ₂₁Sc), and with respect to the f-electron, there is a deviation of one element from its actual appearance in the ₅₈Ce atom. The explanation of this single deviation is beyond the scope of the present paper.

Spin-orbital periodic table

The structure of the spin-orbital periodicity, expressed by the numerical series (12) and (13), as well as the obtained understanding of the alternation of spin values in the radial ℓk-groups of electrons allow giving a graphical expression of the periodic system in the form of an eight-period table, which is presented in Fig. 4.

Fig. 4:

Spin-orbital periodic table of elements.

The presented spin-orbital periodic table is subdivided in the horizontal direction into four cycles, each of which includes two periods with the same number of electrons. Moreover, the elements in the periods are characterized by the fact that in odd periods, when passing from one element to another, the number of electrons with radially uncompensated spin increases in the electron system of the atom, whereas in even-numbered periods successive radial compensation of spin takes place. That is, besides the traditional quantum-mechanical division of elements into s, p, d, f types, in the spin-orbital periodic table all elements are divided into two sets depending on whether the last electron of the atom has a radial compensation of the spin.

In the vertical direction of the table, the groups of elements located one below the other correspond to the radial ℓk-groups of electrons. That is, the elements of the traditional s, p, d, f-blocks are subdivided in the spin-orbit table into 2 sk-groups, 6 pk-groups, 10 dk-groups, and 14 fk-groups. Moreover, in each block, the first half of the ℓk-groups start with the elements, the last electron of which has a positive spin, while the second half, with the elements having the last electron with a negative spin. That is, for example, boron, carbon, and nitrogen, which are the first elements of the radial groups p1, p2, p3, have the last electron with a positive spin, whereas for oxygen, fluorine, and neon, as the first elements of the groups p4, p5, p6, the last electron has a negative spin.

An analysis of the interrelation between the physicochemical properties of elements and the spin-orbital periodicity is not among the authors’ present objectives, but it is necessary to consider one interesting point. The fact is that in the vertical groups of the canonical Mendeleev periodic table there is a secondary periodicity [14], which comes down to the fact that the properties of the elements change from top to bottom not monotonously, but periodically. A typical example of such periodicity is given in [15], which states that the strength of the oxygen compounds of halogens increases with the transition from fluorine to chlorine, decreases with the further transition to bromine, and again increases for iodine compounds. In the presented table, it can be seen that fluorine and bromine are elements of odd periods with a radially uncompensated spin of the last electron, while chlorine and iodine are elements of even periods with a radially compensated spin. That is, the strength of the oxygen compounds of halogens in even periods is higher than in odd ones. This example suggests that the secondary chemical periodicity is somehow connected with the spin-orbital periodicity.

Concluding the brief description of the spin-orbital periodic table, it is also necessary to indicate that, in its form, number, and length of periods, this table completely coincides with Janet’s left-step table [16], which is currently being actively discussed [17], [18], [19], [20] and considered as a serious alternative to the traditional Mendeleev periodic table. In this regard, it should be noted that in compiling his table, Janet was guided by empirical ideas that the elements whose last electrons have the same sum of the principal and orbital quantum numbers should be in the same period. This circumstance, taking into account equality (6), allows considering Janet’s table as a table of spin-orbital periods, in which there is no association of periods into cycles and the division of electronic subshells into positive and negative types is not taken into account. That is, Janet’s table can be considered to be an empirical prototype of the spin-orbital periodic table presented in Fig. 4.

The Mendeleev table and the spin-orbital periodicity

The empirically established chemical periods (t*) of the Mendeleev table start with alkali metals, in the atoms of which the first s-electrons appear of the new n-shells, and end with inert gases with completely filled np-subshells. The number of elements in the seven known periods thus forms a numerical series:

At the same time, the spin-orbital periodicity is characterized by the fact that each cycle T_N begins with electrons with a new value of the orbital quantum number, whereas all periods t_r end with the filling of the ns-subshell. Moreover, if the first period with the number of elements |t₁|=2 is removed from the structure of the spin-orbital periodicity (13), an exact copy of the structure of the Mendeleev periodic table is obtained (16). This suggests that the number of elements in the chemical period with the ordinal number n equals the number of elements in the spin-orbital period with the number r=n+1, that is,

Equality (17) indicates that in a set of chemical elements arranged in the order of increasing Z, two types of periodicity occur: the spin-orbital periodicity, which is due to the alternation of spin values inside the radial ℓk-groups of electrons and the chemical periodicity due to the regular repetition of the configuration of the outer subshells ns and np, formed by valence electrons. Since the outer subshells reflect the configuration of the last electrons in the radial ℓk-groups, the spin-orbital periodicity is the primary periodicity in genetic terms, while the chemical periodicity lags behind it in its development by one period of two elements and repeats its structure taken without the first period t₁.

Turning now to such structural features of the Mendeleev periodic table, as the first period t1∗ having no pair with the same number of elements and the paired relationship in terms of the number of elements for all other periods (|t2*|=|t3*|, |t4*|=|t5*|, |t6*|=|t7*|), the following can be said. The reason for the absence of a pair for the first Mendeleev period is the genetic lag by one period with two elements of the chemical periodicity from the spin-orbital periodicity, whereas the paired relationship for all other periods is a consequence of the equality of the number of electrons in the pairs of spin-orbital periods forming the cycles.

This leads to the conclusion that the periodic system of elements cannot have any single optimal variant of its graphic expression. Each of the considered types of periodicity should have its own periodic table. In other words, it is necessary to recognize two independent versions of the periodic tables, one of which is chemical and the other quantum-mechanical.

Conclusion

The paper discusses the distribution of the values of quantum numbers in the radial groups of electrons with the same value of the orbital quantum number and the same order of appearance in subshells. A new (spin-orbital) exclusion principle is established, according to which neighboring subshells with the same orbital quantum number cannot start with electrons with the same spin value. Using this principle, an analysis of the real sequence of the appearance of new subshells in neutral atoms is carried out and the structure of Mendeleev’s periodic table is considered. In brief, the main results are reduced to the following:

1. The spin-orbital periodicity is established of the electronic structure of the atom, which, in the process of increasing the total number of electrons, develops in cycles, having an increasing number of electrons and consisting of two equal-sized periods.

2. An explanation is given of the real sequence of appearing of new subshells in neutral atoms and the nature of the Madelung rule is revealed, which corresponds to the solution of the “Löwdin challenge” problem.

3. It is concluded that the evolution of the electronic structure of the atom proceeds not along the path of successive filling of the shells, but along the path of the periodic increase in the number of electrons in the radial groups.

4. It is shown that the Mendeleev chemical periodicity lags behind the spin-orbital periodicity by one period of two elements and repeats its structure taken without the first period. On this basis, the explanation is given of such structural specificities of the Mendeleev table as the first period having no pair with the same number of elements and the paired relationship for all other periods.

5. An eight-period spin-orbital table of elements is compiled, the prototype of which is Janet’s left-step table.

The listed results of the conducted research allow considering the general content of the paper as a contribution to the theory of the electronic structure of the atom and the theory of the periodic system of elements.