Description of covalent bond in terms of generalized charges: potential and dissociation energy of homonuclear compound

The theory of generalized charges is an asymptotic approximation of quantum mechanics for interatomic forces. It is based on the model of multicomponent electron gas, which extends the Thomas–Fermi model of inhomogeneous electron gas to pair electronic states. The present research takes in consideration the participation of the field of generalized charges in interatomic bonds. Herein, the definition of generalized charges (GC) by means of the overlap integral of the wave functions of valence electrons is given, and properties of GC are found out. Equations are derived for the potential and length of a homopolar bond, and dependence of these parameters on the generalized charges is shown. The dissociation energy of a diatomic molecule is expressed in terms of the nuclear charges, the function and multiplicity of the covalent bond, the angular momentum, and the vibrational energy of the binding electrons. The parametrization rules are substantiated and an a priori calculation of the dissociation energy of diatomic homonuclear molecules is carried out against the electronic configuration of the bond electrons and the nuclear charge. The calculation results for homonuclear compounds of the elements of the first four rows of the periodic table are shown to be of satisfactory accuracy.


Introduction
The theory of chemical bonds is based on quantum mechanical deductions, which are obtained with using Hartree-Fock-Roothaan, Hohenberg-Kohn-Sham, and Thomas-Fermi-Dirac methods [1][2][3][4]. These numerical methods describe some molecular properties and are widely used in quantum chemistry [5][6][7][8][9]. In the absence of analytic expressions, each particular case requires a separate study [10,11]. The complexity and low accuracy of these methods pose the necessity of using empirical expressions for the potentials of atom-atom interactions, which reduces the quality of the theory of chemical bonds [12][13][14].
The model of multicomponent electron gas is proposed [15][16][17][18] to extend the Thomas-Fermi model [1,3] by adding the pair states of electrons, resulting in a more accurate description of atomic properties [17,18]. The amended model has served as the basis for the theory of interatomic and intermolecular interactions, which is called the theory of generalized charges (TGC) [18,19]. Basic equations of TGC are the equations of the model of electron gas for an asymptotic field remote from nuclei. In such a field, the electric charge is replaced by a new charge -"generalized charge" (GC), which is linked to electric charges of the atomic system by asymptotic relations. Importantly, replacing the electric field with a field of GC, having a quantum character, eliminates the well-known paradox about the impossibility of applying the Thomas-Fermi theory to interatomic forces, which follows from the Teller theorem [20].
The TGC quite accurately describes van der Waals interactions and adsorption phenomena, as demonstrated in a series of publications by the author [21][22][23][24][25][26]. The simplicity and versatility of TGC makes it possible to discover new regularities both in the theory of hydrogen bonds [27,28] and in the well-developed area of covalent interactions [29][30][31].
The fundamentals of TGC are briefly formulated [31], with an emphasis on the derivation of expressions for the energy of interatomic interactions in terms of generalized charges. To reveal the quantum nature of the GC, the concept of a covalent bond function as the overlap integral of wave functions of valence electrons is considered; using TGC, supplemented by expressions for the covalent bond function, relationships for the length of homonuclear compounds are derived as a function of atomic number and state of valence electrons.
This article is aimed at further advancement of the concept of interatomic interactions in terms of GC in order to derive theoretical dependences of the covalent bond characteristics on the electronic structure of atoms and to provide an a priori evaluation of the dissociation energy of diatomic homonuclear molecules.

TGC deductions for covalent bond
The most essential aspects of the TGC are given in a recent paper [31] (see also the Appendix 1). In particular, the definition of the GC is given as the charge of the interatomic field, asymptotically related to the field of the electron gas inside the atom. Its power dependence on the electron density is given as Q ∝ n 1∕b e , where Q is the GC, n e is the electronic density; b is exponent equal to 3/2, 4/3, 7/6, and 1 for four components of the model of electron gas (see the Appendix 2). The additivity of the electron density within a rigid molecular segment, composed of a group of smaller fragments, leads to the law of addition of generalized charges: Q b = ∑ Q b j , or its equivalent representation: The additive value V, called "electron volume" (EV), depends linearly on the quantities of electrons in different states creating the GC field.
The boundary condition for the field equation is given by the "hydrogen normalization," i.e., by joining the electrostatic field and the GC field on the "orbit" of the electron r = 1 of the hydrogen atom, for which a unit generalized charge is postulated. The Poisson equation for the GC field Δu = 4 Qa n q contains a multiplier a , necessary for normalizing the GC density outside the atom: 4 a ∫ n q r 2 dr = 1 .
Comparison with electron density normalization shows that the constant a is equal to the limiting number of electrons that fit in one cell of the phase space of the GC. The constant a = 15 √ 2 � 4 = 16.661 corresponds to the hydrogen normalization.
The TGC shows that, outside the atom, the electron gas separates into two pairs of components: one pair of components participates in the covalent interaction, and the second pair takes part in the van der Waals interaction. By solution of the GC field equations, expressions are obtained for the energy of these types of interatomic interaction. In particular, for the potential energy of a simple covalent bond between rigid molecular segments j and j', an approximate nonempirical expression is derived (in atomic units): According to this formula, the maximum kinetic energy of a binding electron (which in atomic units coincides with the bond potential) is equal to

Definition of generalized charges in terms of the overlap integral
The total overlap integral of the wave functions of the electrons of two atoms, the so-called function of covalent bond (FCB), is approximated as the overlap integral of the functions of the slowest (valence) electrons: In the one-dimensional approximation, the coordinate wave function of the electron of an isolated atom, with the index a = 1, 2 , has the form: where ±k a is the projection of the wave vector onto the bond axis; is the longitudinal coordinate. Substitution of the wave functions yields: For an important case 0 < r ≤ follows: (1) where k is the half-sum of modules of projections of wave vectors for binding electrons. From expression (2) for r = r s ≡ 2k , it follows that = 0 , where r s is the electron screening radius, which is analogous to the Wigner-Seitz radius. Its value is defined as the radius of the spherical cell of the phase space of the electron gas with the maximum kinetic energy of the electron E 1 [29]: (1.710 = (9 ) 1∕3 2 −5∕6 ), from where Substitution of Eq. (4) into (2) allows for expressing the FCB in terms of the bond potential and the internuclear distance: From the definition of the FCB as the overlap integral, we obtain the following equation: where Z a ( a = 1, 2) are atomic numbers, P(r) = ∑ i12 r ∫ 0 ⟨i12 � i12⟩dx is the probability of an electron getting into the interatomic space (located between vertical planes with coordinates 0 and r). In accordance with Eq. (6), the FCB has the meaning of the probability of electron exchange between the atoms. To create a stationary interatomic bond, it is necessary that from a pair of electrons of different atoms, at least one electron occurs in the bond zone. Therefore, if the probability of electronic exchange is < 1∕2 , then the participation of two electrons in the formation of a bond will not be enough. Thus, to create a homopolar covalent bond, the following condition must be fulfilled: ≥ 1∕2.
In the stationary process of the transition of electrons between atoms during the formation of a homopolar bond, the number of electrons in each atom is conserved, i.e., electronic transitions are accompanied by an equivalent exchange. The latter can be interpreted as the transfer of exchanged electrons along closed lines passing inside each of the atoms and through potential jumps at the boundaries of atoms. The FCB as the probability of an electronic transition between atoms, regardless of the direction of the transition, is equal to the product of the probability v a for an arbitrary electron to be on the boundary of its own atom and the conditional probability of crossing the boundary to another atom. It is expressed as: where f is the tunneling parameter that by the order of magnitude is equal to the ratio between the half-width of the potential barrier (i.e., the difference between atomic and covalent radii) and the screening radius. This ratio usually has a small value. For instance, the hydrogen molecule has the half-width of the barrier Δr ≡ a 0 − r c ≈ 0.3 ( a 0 = 1 is the Bohr radius, r c is the covalent radius of the hydrogen atom) and r s ≈ 6 , whence f ≈ 0.05 . Below, this value will be corrected ( f ≈ 0.08 ). For inert gases, according to [25], the tunneling parameter is by one order of magnitude higher, which greatly reduces the FCB value from (7) and excludes the possibility of covalent bonding. Below, inert gases are not considered. Taking in the first approximation f ≈ 0 , and expressing the probabilities through characteristics of atoms: v a = V a Z a , where V a is the EV of an atom "a," we find As follows from (8), the FCB of a homonuclear compound is related to the GC of its atom by expression: Within a rigid molecular segment, the additivity of EV takes place: V = ∑ , where are weights of the segment electrons. In expressions of the TGC, such as Eqs. (1) or (8), the electron volumes of interacting bodies exist in the form of their product ( V 1 V 2 ). It was shown [22] that the contribution to this product for a pair of electrons with the same orbital state, 2 , is equal to the degeneracy multiplicity of this state. For electrons of the π-bond, the magnetic quantum number is equal to +1 or −1; therefore, = √ 2 , whereas for electrons of the σ-bond in accordance with the hydrogen normalizing, = 1 . In general case, the electron weight is equal e = √ d , when d is the degeneracy degree. Thus, being an additive characteristic of electrons, the EV of an atom takes the form: where N is defined as the number of electrons of an atom getting into the interaction region (i.e., valence).
An example of the validity of form (9) is the expression for the EV of the element in diatomic hydride [30,31]: , where Z is the atomic number of the element, n ≥ 2 is the principal quantum number of its outer shell.
When the certainty of the electronic states required for calculation by Eq. (9) is absent, it is convenient to use approximative statistical regularities. Taking into account the periodicity of the electronic structure of chemical elements, the repeatability of their group chemical properties within different rows of the Periodic table, it is possible to estimate the average degree of degeneracy of the electron states as Z∕ , where is the average length of the row (we assume ≈ 118 7 ≈ 17 ). The average length of the period in the statistical limit obviously agrees with the capacitance a for electrons of the cell of the GC phase space. According to Eqs. (8) and (9), for d i ≅ Z∕a , N = a , we obtain ≈ 1 √ Z∕a that is correct for Z >> a ; an asymptotic condition, → 1 , for Z → 0 yields: Formula (10) expresses the contribution of statistical ("classical") factor to FCB of homonuclear compound. Due to the EV additivity, the classical factor is supplemented by the quantum factor: = classic + quantum . The quantum factor is added due to the presence of discrete states of electrons and the symmetry of the homonuclear bond, which provides the coherence of the wave functions of the binding electrons. Among the pair combinations of electrons of two identical atoms, the fraction of pairs of coherent electrons is equal to 1∕Z (the fraction of diagonal terms in the square Z × Z -matrix). When the wave vectors coincide, the Pauli Exclusion Principle reduces to the difference in the spin function of the The spin function is the coefficient in the superposition of two electron states in bonded atoms: a n or b n , at that a 2 n + b 2 n = 1 . These coefficients for homonuclear system are a n = sin n + 2 n , b n = cos n + 2 n , where n = Z Z n is the phase of the spin function, which depends linearly on the total spin, i.e., number of electrons Z; Z n = n(n + 1)(2n + 1)∕3 = 2, 10, 28, 60... is the number of possible states for an electron with the principal quantum number n. The second term in parentheses reflects the oddness property of the Fermi particles. So, the sum of contributions from different variants gives the second term in the formula for the connection function ( Fig. 1): where l is the maximum orbital number of an electron in the outer shell of an atom ( l = n − 1 ). Note that in the proposed model, there is no quantum effect for s-elements. Figure 1 shows the dependence of the bond function on the atomic number calculated for atoms with filling p-, d-, and f-shells. According to the values of Z n , the first curve is for Z > 2 , the second one -for Z > 10 , the third one -for Z > 28 . Relating an estimate according to Eq. (11) with the condition ≥ 0.5 for a covalent bond, then we find that it is violated for the elements of the sixth and more higher rows of the periodic table. In addition, according to the traces in Fig. 1,

Equation for the covalent bond
Expressions (1), (5) and (8) imply an equation, relating the bond length and energy to atomic numbers (see [29] for more detail): is the "dimensionless potential" (DP) of the covalent bond. Substituting inequality ≥ 0.5 to Eq. (5), we obtain the limitation y ≤ 2.599 , which yields x ≥ 0.3453 in accordance with Eq. (12). By means of expressions (13) and (14), these inequalities are transformed into the lower limit for the covalent bond length: r ≥ 0.3453 Z 1 Z 2 1∕3 , and the upper limit for the potential: E 1 ≤ 10.90 Z 1 Z 2 −2∕3 . For example, for the iodine molecule with atomic number 53, the bond length must be no less than 2.578 Å, and the energy of a two-electron bond is no more than 2E 1 ≤ 21.80 × 53 −4∕3 = 2.978 eV. Experimental values for the bond length (2.666 Å) and dissociation energy of the molecule (151 kJ/mol) do not contradict these inequalities.
On the other hand, there is a quantum-mechanical limitation of the increment of momentum along the bond axis when an electron is localized in the internuclear region, which has the Weyl form of the uncertainty relation: r ⋅ p r ≥ 1∕2 , transformed into the dependence of the modulus of the minimum potential energy of an electron on the bond length: In the derivation, the well-known relations for the average values were used: in the isotropic case, the average square of the momentum increment is three times greater than the average square of the increment of the momentum projection in any of the three directions ( p) 2 = 3 p r 2 ; according to the virial theorem, the average kinetic energy of an electron p 2 2 ≥ ( p) 2 8 in a finite electrostatic system is half the modulus of the average potential energy.
Expression (15) is equivalent to inequality for DP: y ≥ 3∕8 , which gives restrictions x ≤ 1.367 and r ≤ 1.367 Z 1 Z 2 1∕3 in accordance with (12) and (13). Noteworthy, the extreme bond length calculated at Z 1 = Z 2 = 1 (0.7235 Å) corresponds to the hydrogen molecule (0.7414 Å) with an accuracy of 2%. Equation (12) is written in parametric form by using (5): Substituting the dependence (11) into (16), allows us to indicate the coordinates of chemical elements on the field of the graph y(x). Figure 2 shows the graph of Eq. (12) with plotted boundaries of the existing area and the points of homonuclear molecules. The points O l and O r correspond to conditions = 1∕2 ( y = 2.599 ) and y = 3∕8 ( = 0.9219 ). Atomic numbers are shown accordingly to formulas (11) and (16).
According to relations (13) and (16 1 ), the expression for the bond length, equivalent to Eq. (12), has the form: To calculate the bond length of a homonuclear compound, it is necessary to substitute into expression (17): (11), which was first demonstrated in our previous contribution [31]. The resulting theoretical dependence is shown in Fig. 3.
The boundaries of the existence area of the covalent bond seen in the figure are denoted by line a and b, restricting the bond, respectively, from up (it corresponds to bonds with an extremely high kinetic energy of valence electrons; the point O r in Fig. 2) and from down (it is built up on the assumption of a minimum excess electron density corresponding to one electron permanently located in the bond region; the point O l ). Reference data (see [32][33][34]) on bond lengths of homonuclear diatomic molecules and doubled covalent radii of elements are presented in the figure by dots for comparison with the calculation. The characteristic of the hydrogen molecule is shown separately, on the line a. The data of the figure clearly demonstrate how the features of the electronic configuration of atoms are reflected in the properties of the resultant molecules. For example, the difference between the regularities for d-and p-elements is clearly obvious. Just as in the case for FCB, we can see that in the presence of the necessary states, the values corresponding to the uppermost sections of the graph are realized. It seems interesting that d-states appear during the formation of a bond between chlorine (and, probably, sulfur) atoms, despite the fact that arccos isolated atoms do not have them. Summarizing, the comparison between the theoretical and reference values allows us to point out that a rather complex regularity was found in Ref. [31] for the length of a homonuclear covalent bond.

Potential of a covalent bond
The potential of covalent bond is expressed through the FCB using formulas (13,14,16): Substituting here the function of homonuclear compounds from (11), the dependence of the homonuclear bond potential on the atomic number and type of valence electrons E 1 (Z, l) can be detailed as shown in Fig. 4.
Similarly to the bond length dependence, the value of the potential corresponds to the trace with the smallest its value for which there is a suitable electron state. The trace marked as "3" lies higher than other traces, so that none of its parts is realized. In the range of atomic numbers from 11 to 60, traces "1" and "2" intersect at points 14 and 42 with the supposed transition of valence electrons from the p-to the d-shell and with values of 28.5 and 56 for the reverse transition. Adjusted to the possible structure of the electron shells, the p-d transitions correspond to points 16 and 42, while the d-p transitions correspond to points 29 and 51. This dependence has a big maximum at 4.5 eV, near to which the potentials of C, N, and O are located. Approximately two times lower, but still quite large are the values for F, S, and Cl. If hydrogen with a binding potential of about 2.5 eV is added to these elements, then we come to group of basic elements of organic chemistry. The potentials of homonuclear bonds of other elements are noticeably lower. The average value of the potential ("0" line) for heavy elements reaches a horizontal region near to 1.60 eV with a weak maximum at 1.609 eV for atomic number 49.

Dissociation energy of homonuclear molecule
The binding energy can be found from the solution of the stationary Schrödinger equation −∇ 2 2 = (E − V) with a known potential when the wave function of the binding electron is substituted to. However, in this case, I confine myself to a more illustrative semi-classical description, according to which the dissociation energy of a diatomic molecule E D differs from the potential energy modulus of z binding electrons ( zE 1 ) by rotation and vibrational terms.
In the ground state at zero temperature, the motions of the whole molecule and its atoms can be neglected, and the kinetic energy can be taken into account only for electrons. The kinetic energy of a binding electron contains the contributions of rotation and oscillations with non-negative integer quantum numbers l and v , respectively, according to: where 0 is the angular frequency of oscillations, which is related to the energy of zero-point oscillations along the axis with amplitude r : 0 2 = 1 32r 2 (see the derivation of (15)).
For the total energy of the binding electron, having a negative value, the equality is valid: The dissociation energy of a diatomic molecule is the sum over z binding electrons: Atomic number is the semi-square of angular momentum is the parameter of vibration energy (PVE). The vibrational number can be different from zero only for the case of the triplet state and is equal to 1 [17,18], which gives for two electrons in the triplet state = 1 ∕ 8 . The indicator of the molecular state with a nonzero PVE is the electronic paramagnetism.
It should be noted that the angular momentum of the binding electron is not an integral of motion. In particular, the orbital number of an electron in the outer shell of an isolated atom does not have to coincide with its orbital number, which is realized in the compound of this atom. The first one is only an upper constraint on the second one. In addition to this limitation and the prohibition of identical states, the condition of the minimum energy of the molecule is imposed on the state of the electron. The latter leads to choosing options with the minimum sum + .
The condition for the bond parameters is followed from Eq. (19): for which it is convenient to use the graph in Fig. 2.
At an equilibrium state, the system of electrons, being at the minimum energy, tends to uniformly distribute the total angular momentum between the particles. Hence, the total square of the angular momentum of an equilibrium system must be sought after averaging this momentum over particles, more precisely, over pairs of electrons that differ only in spins. The average value of the orbital number is determined by a formula that takes into account the degree of degeneracy of the states:from which we find the average semi-square of the angular momentum for pair of electrons where l m is the maximum value of the orbital number for the system of two electrons under consideration. According to inequality (23) (see below), the angular momentum of an electron of the covalent bond is not greater than 1. Consequently, the magnitude of l m for the system of two electrons (20)  The symmetry of diatomic molecules determines the preservation of the projection of angular momentum on the axis of the molecule (the magnetic number m). According to Eq. (20) , where the summation is over the bond electrons. Each the bracketed term included into the sum must be positive; otherwise, the corresponding electron will leave the bond region. This condition is expressed by inequality: After substituting y max = 2.599 , we obtain that the magnetic number has limitation |m| < 1.169 , i.e., the covalent bond can be established with -and -electrons, but not with -electrons.
According to the Pauli Exclusion Principle, conservation of the magnetic number of binding electron is possible only in a singlet state. For -electrons, the entire DP range is available: 0.375 ≤ y ≤ 2.599 (see Fig. 2). For -electrons with the constant magnetic number |m| = 1 and l(l + 1) ≥ |m|(|m| + 1) = 2 , only a narrower interval 2.062 ≤ y ≤ 2.599 is available. Such high dimensionless potentials are not characteristic for the elements of the second period, which are known to not prevent the participation of -electrons in "classical" multiple bonds of carbon or nitrogen. The reason is in the averaging of the squared angular momentum of electron between states with different orbital numbers. In "classical" multiple bonds (with -electrons), the states have orbital numbers 0 and 1. The first one relates to -electrons, but the second one can have three various magnetic numbers (−1, 0, +1). For the case of a double bond, the variant with two quantities (0 and ± 1) is realized, while for a triple bond, the variant possesses all three quantities. The average square of angular momentum ( 0+2 2 and 0+2+2 3 ) is the contribution of -electron to the squared angular momentum of multiple bond, i.e., 1 and 4/3 for the double and triple bond, respectively. Owing to formula (21), the lower limit of Eq. (12) corresponds to ordinate 1.062 for double and 1.396 for triple bond (see the graph in Fig. 2).
Accounting for formulas (17,18), the expression (20) for dissociation energy of diatomic homonuclear molecule takes the form: Here, atomic numbers Z and FCB from (11) are substituted; parameters z, , are taken according to the rules given below.
Calculation rules for the dissociation energy
Arsenic molecule is calculated like the nitrogen molecule (see Table 1).
Despite the fact that the points of oxygen and nitrogen molecules in the graph of Fig. 2 coincide, one should take into account the impossibility of a triple bond for the oxygen molecule. In accordance with rule E, the triplet state causes the absence of a constant magnetic number for the electrons of the double bond of the oxygen molecule. With parameters Z = 8 , = 0.703 , z = 4 , = 2 × 68 ∕ 81 , and = 1 ∕ 8 , formula (24) gives an estimate 517.3 kJ/mol, which deviates from the experimental value (498.3 kJ/mol) by 4%.
Sulfur and selenium molecules with double bonds have identical SSAMs, but different PVEs (selenium molecule is paramagnetic).
Halogen molecules are calculated by formula (24) by substituting the corresponding Z, , z = 2 , = 0 , m . Due to the absence of vacant states at fluorine and bromine atoms, the SSAM takes the maximum value of 68/81. The presence of new d-states in chlorine molecule makes it possible to reduce the SSAM value to 3/8. These and similarly obtained results of a priori calculation of the dissociation energy of some homonuclear compounds using formulas (11,24) and the developed parametrization rules are shown in Table 1 and compared with reference data [36].
From the analysis of the values of the angular momentum presented in the table, we can conclude that -electrons participate in the bonds of homonuclear compounds only in the atoms C, N, Si, P, and As. Despite that the bonds in the homonuclear molecules of the elements B, O, Al, Ge, S, and Se are double, they do not contain -electrons.
Another important consequence of this analysis for homonuclear molecules is a strong correlation between the values of the dimensionless potential (DP) and the semi-square of electron angular momentum (SSAM), which will be investigated not before long.

Conclusions
Here, it was demonstrated that the approach developed on the basis of the theory of generalized charges offers an effective tool for describing interatomic interactions. More specifically, this approach allows for making a priori estimates of the characteristics of homonuclear compounds with sufficient accuracy and establishing analytical dependences of the bond parameters on the parameters of the electronic structure of atoms. The approach developed here can be extended to the class of heteronuclear compounds, which will be proved in future works.

Appendix 1.Basic statements of theory of generalized charges
The model of multicomponent electron gas expands statements of the Thomas-Fermi theory to pair states in the inhomogeneous electron gas (see Appendix 2 for more detail). Four state components are described: singlet, triplet, incomplete singlet, and incomplete triplet, -designated by indices i = 0, 1, 2, 3, respectively. The incomplete components differ from the first two ones by the presence of a vacancy instead of an electron paired. The "singlet" component has the properties of the densest Thomas-Fermi gas, and the "incomplete triplet" -the The electron gas components e i take part in reaction: The meaning of reaction (27) is in the interconversion of the components during the transfer of an electron of dense component (singlet with index "0" or triplet "1") to a vacancy of incomplete component (incomplete triplet "3" or incomplete singlet "2," respectively); then the left side of Eq. (27) is transformed into the right side and vice versa, and balance is achieved. The application of the equilibrium equation for the components in the form of the law of mass action leads to the relationship between the electron density n i ( ) of i-th component and potential ( ) (measured relative to the potential 0 at the zero-density surface): The dimensionless variables of the potential ( ) and radial coordinate (x) are expressed by the following: where A, B are positive constants associated with the nuclear charge Z due to the boundary condition at the atom nucleus: For neutral atom, x 0 = ∞ and 0 = 0 . The potential, combining the potentials of the electron gas components -solutions of Eq. (30) -with coefficients (25), describes quite precisely the intra-atomic field and the properties of chemical elements, such as total energy, ionization energy, and atomic size.
The property of Eq. (30) to describe a function that decreases exponentially at small distances, but relatively slowly (hyperbolically) at large distances, allows us to consider the total magnitude as the sum of two fields: intraatomic and interatomic ones. The interatomic field obeys the same equations as the intra-atomic field described above, but in the first case, the boundary condition (31) at the nucleus is insignificant. The resulting asymptotics is described by new field constants instead of (33) new charges. The new conditions do not require equality of new and old spatial coordinates; they only establish proportionality with a certain scale factor. In the absence of the boundary condition at the origin of coordinates, the exact solution of Eq. (30) for i = 0, 1, 2 has the form: where (28) The asymptotic relationship (34) has been noted for b = 3∕2 [11,12]. The exact solution of Eq. (30) for incomplete triplet ( i = 3 ) has not a hyperbolic component: The dimensionless function (x) was defined in (32) as the ratio of the potential of the electron gas field to the Coulomb potential ( Z∕r ). For the inverse procedure, the functions from (34) and (36) have to be multiplied by the "Coulomb potential" Q∕r -with the charge Q, which is called "a generalized charge" (GC). The potentials of the electronic components for the atomic field region far from the nucleus have the form: Local expressions for the fields of electron gas and generalized charges are identical, which, like (28), is reflected in the density n q dependence on potential u: × c(2c + 1)] c are determined by substitution of (37), (38) into equation: Δu = 4 Qa n q , where the GC density factor a is introduced, which is necessary for its normalization outside the atom: 4 a ∫ n q r 2 dr = 1 . This constant is equal to square of the scale factor between interatomic and intra-atomic spaces. Comparing it with the normalization of the electron density, we note, that the constant a is the limit number of electrons filling one cell of phase space of generalized charges. The condition of "hydrogen normalization" corresponds to the asymptotic similarity of the intra-and inter-atomic fields -the docking of the electrostatic field and the GC field for the hydrogen atom, the intra-atomic field of which has the potential close to the Coulomb form, 1/r, up to the Bohr orbit of the electron ( r = 1 ). A unit of generalized charge is postulated for the hydrogen atom. The equality of the fields in the electron "orbit" provides the first approximation: A q(i=0) = 1 . For the GC determined by this way, the coefficients are as follows: Separately, for the field of each component, the potential and interaction energy of two GCs take the form: Far from the atom nucleus, the potential of electron gas (with the density n e ) coincides up to a constant coefficient with the potential u of the GC field. Hence, according to (37), Q ∝ n e n q 1∕b , where n q is the GC density. The additivity of the electron density within a rigid molecular segment composed of a group of smaller segments leads to the rule of addition of generalized charges: or its equivalent representation: The additive quantity V, that is called "electron volume" (EV), linearly depends on the number of electrons in various states creating the GC field. The "hydrogen normalization" sets its unit as the contribution of one -electron.
The equilibrium constant of reaction (27), which is identically equal to 1 for intra-atomic field, depends on a distance in the case of interatomic field: Function K q (r) passes through the maximum K qm = 6.852 ⋅ 10 −7 for the argument r m = 3.354 , and then decreases to zero. Thus, outside an atom, the electron gas decays into pairs of the components: the pair of "singlet and incomplete triplet" (with indices "0" and "3") dominates in the peak region, but in the region of its tail, the pair of "triplet and incomplete singlet" (with indices "1" and "2") dominates. The first pair of components is involved in covalent interaction, and the second pair -in van der Waals one. In accordance with (41) and (43), using weights (26), we obtain the expression for energy of covalent interaction where the second term can be omitted due to its negligibility (< 1%). Similarly, TGC describes the energy of van der Waals interaction -it has shape of the Lennard-Jones potential with theoretical coefficients:

Appendix 2. Model of multicomponent electron gas
In the degenerate electron gas (the Thomas-Fermi gas), the cells of phase space are filled by electron pairs. In this dense gas, there are spin correlations between the electrons of different cells. This aspect is missing from the Thomas-Fermi theory.
The choice of the model situation is based on the known properties of nonrelativistic Hamiltonian of the electron gas [1]. Electrostatic interaction of electrons with an external field and with other electrons of the same gas in the nonrelativistic approximation does not depend on spins. The wave function of the system is therefore equal to the product of the spin function and the coordinate function: jm ⋅ j , ′ (r, spatial coordinates); moreover, the part of the coordinate function responsible for electrostatics does not depend on the combination of spins. It follows that the consideration of the electron gas can be divided into two stages -when the electric field is "turned off" (stage 1) and after it is "turned on" (stage 2).

Components of electron gas (stage 1)
The spin function jm of a system of two particles characterized by spin numbers j 1 = j 2 = 1∕2 (according to the angular momentum addition rule, there is j ≡ j 1 + j 2 = 0 , 1 for the system) and spin projections: m 1 = ±1∕2, m 2 = ±1∕2 ( m ≡ m 1 + m 2 = −1, 0, 1 ) describes 4 states. These states jj � correspond to two spatial distributions j : singlet -for particles with anti-parallel spins ( j = 0 ; m = 0 ) and triplet -for particles with parallel spins ( j = 1 ; m = −1, 0, 1 ). In view of the antisymmetry of the wave function of a fermionic system with respect to the rearrangement of a pair of particles, the singlet coordinate function is symmetric, and the triplet function is antisymmetric. Within the framework of the first order of perturbation theory, the singlet state ensures the approach of particles to the same extent as the triplet state prevents it. In accordance with this conclusion, we obtain that four electrons of two neighboring cells form a convex tetragon with alternating signs of the spin projection along the perimeter (Fig. 5). Let us define this conformation as "regular." Let us write the equivalence condition for the probability of formation of the regular conformation through pair probabilities (the digitals denote the electrons along the perimeter, + and − signs indicate the direction of the spin projection): where the last equality is a consequence of the uniqueness of the singlet state. On the left side, the products in square brackets are mutually independent, since they are composed of variants of the triplet state with different spin projections. In view of the absence of intersection of the terms united on the left side, the algebraic equality follows: where w 0 , w 1 are the probabilities of singlet and triplet states, respectively.
From a mathematical point of view, due to the mirror symmetry of the possible location of the fourth vertex relative to the diagonal, 4 particles form both convex and concave tetragons with equal probability. However, the shape of concave tetragon, for which the distance between a pair of particles in the triplet state is less than between a pair of particles in the singlet state, is not stable. In such an "irregular" conformation, there is no particle falling inside the triangle, and a vacancy arises. Incomplete states of a singlet and a triplet, containing a particle and a vacancy, arise similarly to complete states. Thus, four components can be distinguished in the electron gas: a singlet, a triplet, an incomplete singlet, and an incomplete triplet.
The listed types of pair states in a dense gas of electrons create a lattice, in an arbitrary section of which there are quadrangular cells with diagonals. The electrons located at the lattice nodes simultaneously form two structures: the sides form a singlet lattice and the diagonals form a triplet lattice.
The components of a dense electron gas are conventionally depicted in Table 2. This is where the numbering of the components is introduced.
Note that the transfer of an electron from component "0" to component "3" gives a pair of components "1" and (48) Four particles 1, 2, 3, and 4 forming a convex tetragon with diagonals. The regular conformation corresponds to a sign-alternating order of spins, e.g., 1 + , 2 − , 3 + , 4 − "2" and vice versa, i.e., a quasi-chemical reaction takes place: So, according to (48), the ratio of fractions of regular conformations is obtained: The occupancy of regular conformations is twice as high as the occupancy of irregular conformations, i.e., the fraction of a singlet is twice the fraction of an incomplete singlet, and the fraction of a triplet is twice the fraction of an incomplete triplet: Due to the binary representation of the electron gas as a singlet or triplet structure, each particle counts twice in the total number of particles, so when adding the fractions of all components, we get 2: Joining (50)-(52), we obtain a formula: us complete the cell of dense electron gas by filling the vacancy of the incomplete singlet with an electron. Since the distribution of pair states in dense gas is i , then Substituting (53), we obtain The electron gas density has only two independent components, i.e., the densities of the four components, in addition to normalization, must be related by two relations, which is reflected in the formula (53) or (56). In this regard, the presence of a hole (a negative term with index 2) in the density vector i does not lead to the necessity for a real introduction of a negative mass.

Properties of density of electron gas
When the field is included in the gas description, the density vector, written as a set of partial densities of the gas components, becomes an explicit function of coordinates; its components are the product of constants (56) and functions n i ( ) , normalized to the number of particles in the gas N: where d is an element of volume. Then the gas density is defined by the sum: According to the Hohenberg-Kohn theorem [3], the density and potential of the ground state are uniquely related. It follows from the nature of the atomic field potential that the functions n i ( ) have no discontinuities and therefore can serve as a scale factor in the homeomorphic transformation of the lattice. The homeomorphism of the transformation consists in such a deformation of the grid, in which its lines have no discontinuities. A small section of the lattice remains practically homogeneous under this transformation, and relations (53) must be satisfied for it, which turn into equations for scale factors: It is easy to see that the constant of the quasi-chemical Eq. (49) according to (60) is equal to 1.
The relationship between the density of singlet n 0 and the electrostatic potential φ is derived by the Thomas-Fermi method for particles that move in finite motion in the discrete phase space with volume of cells (2 ℏ) 3 , each of which contains 2 electrons: where a 0 = ℏ 2 me 2 is the Bohr radius, a = − ∕e is the potential on the atom boundary, and is the chemical potential of electron gas (below we return to atomic units). (60) n 0 n 3 n 1 n 2 = 1 (61) n 0 ( ) = 2 When describing the field of an incomplete triplet, due to its sparseness, one can apply the Mott relation for the first order of perturbation theory: From expressions (60-62), the general formula follows for relation of the density of a component with the potential: In the general form, we have the formula:

Self-consistent field
Applying the superposition principle, we express the gas potential in terms of potentials of the components: The density of the electron gas is related to the electric potential by the Poisson equation: where Δ is the Laplace operator. Due to differences in the values of b i , the Poisson equation breaks down into partial equations: Equation (67) is not independent due to the relationship between the component density and the potential of the entire gas, so we obtain a closed system of four equations in the form: The method for solving system (68) is based on the lattice representation of electron gas. Each small region of space contains cells of different gas components. Let us group the cells (using the symmetry of space) in such a way that sectoral zones of uniform state are created: a singlet zone, a triplet zone, etc. For each zone, an independent equation can be written: The calculation by formula (75) for the roughest case on restriction (76) ( D(b = 3∕2) = 0.157 ) in the interval x ∈ (0, 1) deviates by about 3% from the exact function χ(x) obtained by numerically integrating Eq. (72) with boundary conditions (73) [1]. Having defined the boundary of the atom as the upper limit x Γ ∼ 1 , we obtain the interval: D(b) < x < x Γ corresponding to the entire practically important region inside the atom.
Substituting (75) into (70) gives the expression for the field potential of the electron gas component in a neutral atom:

Energy
For the energy of the component of a neutral atom, taking into account the virial theorem and the Euler equation, the formula is derived: and the ratios of the terms of electron-electron and electronnuclear interactions are found: From (79), for i = 0 , the well-known relations of the Thomas-Fermi theory follow; in particular, the energy of interelectronic interaction for the singlet is 1/7 of the absolute value of the energy of electron-nuclear interaction. The corresponding value for an incomplete triplet is 0, so the value E 3 can be interpreted as the energy of a screened nucleus. To determine the net energy of the electron gas, this value should be subtracted from the weighted sum of the components of electron gas: where In Table 3, all parameters of the electron gas components are given.
By substituting the expressions for density and potential, integration in (78) yields: where Г(x) is the Euler gamma function. Applying formula (80), we obtain a good approximation for dependence of the total ionization energy of atom on the atomic number: