INTERATOMIC INTERACTION IN DIATOMIC MOLECULES WITH TAKING INTO ACCOUNT THE REPULSION OF IONS IN A POSITIVELY CHARGED CORE

Introduction/purpose: Different types of interactions in diatomic molecules of complex atoms are analysed. Methods: The empirical formulas of Lennard-Jones, Buckingham, Buckingham-Corner, Morse, Danem, Gulbert-Hirschfelder, Klein and their combinations without their clear physical justification are used to take into account the repulsive and attractive forces in the molecule. To improve the situation of the binary interaction inside condensed matter, Gretchikhin and his associates proposed applying the Heitler-London quantum theory, but only at distances greater than equilibrium. At distances less than equilibrium between atoms in the binary interaction, the Lennard-Jones formula was still used. Using various kinds of fitting coefficients, in each case we obtained a match with the experimental data on the dissociation energy. A more general idea of all possible types of interactions was completely absent. In this connection, the need arose to reveal all possible types of interactions inside diatomic molecules and theoretically obtain dissociation energy, activation energy, and standard atomization enthalpy. The application of quantum mechanics methods in the Heitler-London theory allowed to take into account not only the Coulomb deterrence during exchange interaction, but also the Coulomb repulsion of nuclei. Results: The electric dipoles for neutral atoms and for positively charged ions of the core of diatomic molecules were calculated. This made it possible to calculate the electron-dipole and dipole-dipole interactions. A theory of the repulsion of positively charged nuclei of complex atoms in diatomic molecules has been developed. The interaction potentials for the molecules of carbon, nitrogen, oxygen, aluminum, silicon, and sodium are


INTERATOMIC INTERACTION IN DIATOMIC MOLECULES WITH TAKING INTO ACCOUNT THE REPULSION OF IONS IN A POSITIVELY CHARGED CORE
Leonid I. Gretchikhin

Introduction
The interaction of atoms is a rather complicated process. This mechanism is understood qualitatively under the assumption that there are attractive and repulsive forces between the interacting atoms. However, these forces are extremely entangled and that is why creating a model of their interaction appeared to be a difficult task. Therefore, a simpler way of applying the regression analysis to describe the interaction potential between atoms was used. As a result, Lennard-Jones, Buckingham, Buckingham-Corner, Morse, Dunham, Hulbert-Hirschfelder, and Klein potentials were obtained as well as their combinations (Hirschfelder et al, 1954). The first, quite widespread formula, representing the interaction potential, was proposed by Lennard and Jones in the following form (Gretchikhin, 2018), (Yelyashevich, 1962): where a and b constants are fitting coefficients. The formula proposed by Dunham turned out to be more universal: where: V 0 is the depth of a potential well; r 0 is the equilibrium value of the radius corresponding to the minimum interaction potential V(r 0 ); and b 0 , b n and n are the fitting parameters. For diatomic molecular systems the potential proposed by Morse is usually used (Gretchikhin, 2018): ( The initial parameters for constructing the Morse potential are the dissociation energy D e , the distance difference relative to the location of one of the nuclei ∆r, and β, which is determined as follows: In turn, the force constant  is found from the vibrational spectrum and was obtained for most diatomic molecules. To exclude any fitting coefficients, the paper (Gretchikhin, 2018) proposed a slightly different approach presented in the following form for the condensed systems: Here V res. (r) is the interaction potential of particles depending on the distance between the particles; V 0 = D e is the energy of interaction of the particles at the equilibrium distance which is equal to the energy of dissociation; r e is the equilibrium distance between the interacting particles; N k(l) is the number of particles within the k(l)-th energy state; κ i is the visibility coefficient; is the distribution of the electron density ρ e of the particles around the power centers A and B of the interacting particles with the energies ε k and ε l in the k-th and l-th energy states; Н 1,1 , Н 1,2 и S are the Coulomb, exchange, and overlap integrals, respectively.
This formation of the interaction potential between atoms in a diatomic molecule involves two problems. The first one is that the equilibrium distance is always greater than the value obtained from the analysis of the vibrational spectrum of the resulting molecule, since the repulsion energy of ions in a positively charged core is not taken into account.
The second problem arose in that the overlap integral at the equilibrium distance turned out to be greater than the unity which does not correspond to the physical nature of this integral. The integral should always be less than one.
In this connection, it is vital to set the following goal: to develop a model of the interaction of atomic structures that would allow, by applying the initial data on the ionization potentials experimentally obtained for all atoms of the periodic table, to fully obtain the interaction potential between the atoms. To achieve this goal, it is essential to solve the following tasks: -to find out the conditions of applicability of different types of chemical bonds that arise between the interacting atoms; -to formulate a model of the interaction between the ions of a positively charged core; -to develop a model of the interaction between different energy states of a positively charged core; -to make a comparative analysis of the interaction potentials using a number of examples and taking into account the presence of covalent, ionic, induced, electron-dipole, and dipole-dipole bonds, as well as the repulsion energy of atoms and positive core ions in the initial atomic structures.
Let us consider, one after another, these problems.
А Analysis of different types of chemical bonds between interacting atoms A fairly detailed analysis of different chemical bonds was performed in the paper (Gretchikhin, 2018). The paper shows how the following interactions are formed: covalent, ionic, induced, electron-dipole and dipole-dipole ones. However, the interactions resulting from the presence of a positive core inside the molecule were not considered.
A covalent bond is formed by the exchange of electrons between interacting particles. Heitler and London, using quantum mechanics, developed the theory of this type of interaction and applied it to the analysis of the hydrogen molecule. Figure 1 shows a schematic diagram of the electron exchange. Based on the general ideology considered in the paper (Gretchikhin et al, 1990), the exchange interaction potential in the notation shown in Figure 1 is represented in the following form: where 0  is the dielectric constant of the vacuum while 2 , 1 r is the distance between the electrons.
In the case of the interaction of identical atoms, the exchange of electrons occurs in the zone of overlapping wave functions, which is located at a distance of ~2 / e r from both atoms. Taking into account the probability a P of a valence electron being in the overlap zone, an electron that passes from the atom A to the atom B and is held by the electric field of the atom A has the following energy: A similar electron holding potential arises for the atom B: The resulting potential that holds the exchange of electrons for identical atoms at relatively small distances e r is equal to and therefore amounts to: Based on the form of the resulting confining potential, it was thought that the potential takes into account the repulsion of the nuclei of interacting atoms. Therefore, it was believed that the Heitler-London theory already took into account the interaction of the nuclei. This mistake costs us dearly. The first term in the interaction potential (5) does not take into account the repulsion of the nuclei, but confines the electrons during the transition from one atom to another. The repulsion of the nuclei should be taken into account separately and in a completely different way. Potential (5) was written for a hydrogen molecule in which identical atoms interact. For diatomic molecules composed of different atoms, potential (5) has a different form and, therefore, the Heitler-London theory in this case needs to be further developed.
The direct exchange of valence electrons between the interacting particles creates the bond exchange energy in dynamics. In the process, each electron of the atom A independently interacts with all electrons of the atom B. The resulting energy of the bond was obtained in the paper (Gretchikhin, 2004) Here, N А is the number of energy levels of the particle A and N В is the number of energy levels of the particle involved in the interaction, while the other values are as follows: is the overlap integral. Accordingly, a  and b  are the wave functions of the atom A and the atom B in the ground state. The probability of a valence electron being in the contact zone of the first and second interacting particles is determined as follows: and for the atom В Here the angles are: In addition, it is necessary to take into account the mutual shading of each internal energy state by external electrons. This process is taken into account by the share in the exchange interaction of each electronic state, which should be determined as follows: In the process of electron exchange, a situation arises when both electrons are near one of the interacting atoms. In this situation, an ionic bond occurs, i.e., the interaction of a negative ion with a singly ionized atom is realized.
Ionic bond. The theory of ionic bonding which makes it possible to determine quantitatively the magnitude of ionic bonding based on the residence time of valence electrons near interacting particles has been developed in the papers (Gretchikhin, 2004) and (Gretchikhin, 2008) not only for neutral particles, but also for each multiplicity of ionization.
The probability of a valence electron being in the contact zone near the first and second interacting particles is determined as follows: for the atom A: and for the atom В (12) Given this, the probability that the valence electron of the atom A will go over to the atom B will amount to a P S while the probability that the valence electron of the second atom is not within the contact zone amounts to (1-P b ). The resulting probability of this event is equal to a P S (1b P ). A similar situation occurs when the valence electron of the second atom is within the contact zone and passes to the first atom while the electron of the first atom is not within the contact zone; the probability is equal to In this case, the probability of the event is equal to (1a P )·P b S. Both in the first and second cases, an ionic bond arises. The total probability of the ionic bond formation is the sum of these probabilities Therefore, in the binary interaction of two identical atoms, the ionic bond energy, in general, is determined by the following formula: (14) Table 1 shows, as an example, the shares of the ionic bond arising from the binary interaction of two atoms with the formation of a diatomic molecule. It turns out that, even for identical atoms, the fraction of the ionic bond energy is noticeable and not equal to zero.
In the process of electron exchange, an uncompensated charge arises in the region of overlapping wave functions. This charge, interacting with the positive core of the molecule, additionally increases the binding energy of the atoms in the diatomic molecule, i.e., the binding energy arises due to the induced electric charge. (Gombás, 1950) Induced bond. An induced bond is determined by a potential barrier occurring between the interacting particles when exchanging valence electrons.
The electron of the atom A and the electron of the atom B create a negative potential barrier independently of each other. The electron when traveling from one particle to the other is partially reflected and retarded at the boundary interacting particles. The probability of such an event is equal to the product . At the same time, the second electron is also reflected from the boundary with the probability of . Therefore, the probability of the formation of a negative potential barrier is equal to the sum of these probabilities. Thus, the value of the negative charge in the contact zone will amount to As a result, additional binding energy arises: are accordingly, the effective charges of atoms and ions of the first and second degrees of ionization of the interacting particles, respectively.
Most atoms and ions of complex atomic structures feature a built-in and induced dipole electric moment. Therefore, the electron-dipole and dipole-dipole interactions should be taken into account.
Electron-dipole interaction. In the process of the interaction of the atomic particles the built-in dipole electric moments are arranged so that they ensure maximum overlap of the wave functions of the interacting particles with their orbits. In the process of the exchange interaction between the interacting particles, a negative charge ∆Q arises; the value of the charge is determined above by formula (15). This negative charge interacts with the dipole moments of the first and second interacting particles, weakening the binary bond between the particles. The energy of this bond is as follows: where р э,1 , р э,2 , р э,3 , р э,4 , р э,5 и р э,6 are the built-in dipole electric moments of ions of the first, second and third ionization of interacting particles. Odd values refer to atom A while even values refer to atom B.
The presence of built-in electric moments in the interacting particles leads to the appearance of a dipole-dipole bond.
Dipole-dipole bond. A dipole-dipole bond in the binary interaction of complex particles is determined according to the following formula: In the formation of an ionic bond, a cross interaction of dipole electric moments is formed. Then, taking into account (18) .
In the process of the binary interaction, a particle with shared electron shell is formed due to sharing of outer electrons. The bond breaking energy for a newly formed particle is determined by all types of bonds combined. In this case, it is essential to know the effective radiuses of the interacting particles and the equilibrium distances between them. The interaction potential, even considered with taking into account all the above interactions at close distances, does not make it possible to obtain a significant decrease in the binding energies (D e ). In this regard, let us consider the process of the interaction of the ions of a positive core, which quite effectively reduces the binding energy of the interacting atoms in a diatomic molecule.
Coulomb repulsion of the particles of a positively charged ionic core In complex molecular systems, the valence electrons of individual atoms are shared, and a condensed medium appears in the form of a two-component mixture consisting of a generalized electron cloud and an ionic core of single ions. The electrons of neutral atoms are bound to a positively charged core with their effective charge. These effective charges interact with each other when they open due to the overlap of their wave functions. Based on the divergence theorem, the flux of the electric displacement vector is equal to the charge inside a closed surface. In the presence of a second positive charge, the flux of the electric displacement vector is deformed and a third one -negative charge -arises on the surface that separates two positive charges. The electric displacement vector is directed perpendicular to the surface.
Then the flux of the electric displacement vector through this surface amounts to: where  z is the effective charge of a positive ion, rdr dS Accordingly, the share of the charge involved in the mutual repulsion is: In addition, the mutual screening of the effective charge should be taken into account since each successive ion charge is screened by the previous energy levels of the atomic system, i.e.: ...
Then the repulsion energy amounts to: When negative ions are formed, a repulsion arises between the induced charge and the external electron of the negative ion, i.e.

  
Therefore, positive ions inhibit the exchange of electrons within the ionic core, and this is precisely what is taken into account in the Heitler-London theory. In addition, they mutually repel according to Coulomb's law. Each valence electron can be, with a certain probability, in the region of overlapping wave functions of neutral atoms and positive ions inside the core. In this case, based on the principle of superposition of electric fields, it is necessary to take into account the interaction of each ion with all other ions inside the positive core of the molecule. Specifically, for the binary interaction, this situation is shown in Figure 2.
The overlap zone for the valence electron of the neutral atom A is formed by the angle  while the one for the atom B is formed by the angle  ; these angles are determined by formulas (11) Thus, the total energy of the interaction of atoms in a diatomic molecule is realized as the following sum: Thus, the interaction potential of atoms in a diatomic molecule is determined as such a complex aggregate. Below the interaction potentials for some diatomic molecules were calculated in comparison with the Lennard-Jones and Morse potentials as an example. 2. For distances at which the overlap integral S is greater than unity, the ionic, induced, and electron-dipole bonds are equal to zero.
3. At all internuclear distances, the bond exchange energy is always greater than the dipole-dipole bond.
4. The overlap integral is always less than one. Specific calculations of the interaction potentials have been performed for diatomic molecules of carbon, nitrogen, oxygen, silicon, aluminum and sodium, for which the dissociation energy varies from a maximum value of 9.76 eV to a minimum value of 0.76 eV and they are shown in Figures 3 to 8. This is enough to fully understand the dynamics of the formation of interaction potentials during the formation of diatomic molecules.  An important property of the interaction potentials is that for given ionization potentials and dipole electric moments, the experimental value of the dissociation energy is realized with a strict ratio of the effective energy radius of a neutral atom and the distance between nuclei in a diatomic molecule. In this case, the effective energy radius should not coincide with the radius of the atom, since the radius of the atom corresponds to the maximum of the electron density distribution function while the effective energy radius of the interaction is determined by the mutual compensation of the positive and negative binding energies at an equilibrium distance. Predominantly, the energy radius turned out to be greater than the radius of the atom obtained by calculations using the Hartree-Fock wave functions (Brattsev, 1966). Only for silicon, the energy radius turned out to be ~ 10% less while for sodium both radii completely coincided.
For most diatomic molecules, the calculated equilibrium distance between atoms turned out to be greater than the equilibrium distance obtained from the vibrational spectra. Only for silicon and sodium did these values turn out to be less than the equilibrium distance obtained from the vibrational spectra.
The specific calculated values of the energy radii and the equilibrium distances of the interacting atoms as well as the values of different binding energies in diatomic molecules are summarized in Table 2.
The internuclear distance for all molecules does not coincide with the values obtained from the vibrational spectra. These values in Table 2 are shown after slash.
The induced binding energy, which was substantiated in the work (Gretchikhin et al, 2015, pp.29-41), and the repulsion energy introduced in this article, turned out to be determining. Without taking them into account, the interaction potential in diatomic molecules cannot be determined.

Average effective radius of molecules and ionization energy of binary formations
In diatomic molecules, the valence electrons travel each around their atoms. When both electrons enter the overlap zone, an exchange of electrons occurs. The bond of one of the external electrons with the core of the molecule determines the ionization potential while the second electron screens the core of the molecule. Obviously, such screening should be different depending on the size and structure of the electron shells of the interacting atoms. Each of the external valence electrons in the diatomic molecule interacts with one of the nearest atomic nuclei in two positions shown in Figure 2. Thus, the effective radius of the molecule can be represented as follows: The probability of electrons being in the overlap zone is determined by formulas (11) and (12) while taking into account the fact that they do not exchange their states. Accordingly, for each electron, the resulting probability of staying in the overlap zone is Outside the overlap zone, the ionization energy of the atom A is equal to the single ionization of the atom A. In the overlap zone, if the electron does not transfer from the atom A to the atom B, the ionization energy still corresponds to the atom A. The electron in the atom A is either in the overlap zone or outside this zone. The resulting probability of this event for the atom A, taking into account the fact that the valence electron is located at a distance of 1 r from the nucleus of the atom, is Then, for the electron near the atom A, the ionization energy of the molecule will amount to: while that for the electron near the atom B will amount to: Here a  and b  are the ionization energies of the atom A and the atom B, respectively. The results of the calculations of the ionization energies and the radii of neutral molecules and their ions of the first as well as the second multiplicity of ionization of some diatomic molecules are presented in Table. 3 as an example. The reference data for the first multiplicity of ionization of some diatomic molecules is shown in Table 3 after slash.
Agreement with the data for diatomic molecules, given in the handbook (Radzig & Smirnov, 1985), for the energies of single ionization, is quite satisfactory. A number of molecules feature a relatively large dipole electric moment. This property of molecules is still not clear. When the molecules in which the initial atoms have a built-in dipole electric moment are considered, this property of molecules can be understood.
However, even in the case of simple diatomic molecules, such as BH and LiH, in which the initial atoms do not have a built-in dipole electric moment, their own dipole electric moments are quite substantial. This property of molecules needs further justification. In order to exclude the influence of intrinsic built-in dipole electric moments of the initial atoms, let us consider, as an example, the aforementioned simple diatomic molecules BH LiH. For these molecules, the dipole electric moments are respectively 4. 33·10-30 C·m and 19.6·10-30 C·m (Radzig & Smirnov, 1985). The first value was obtained with an error exceeding 10% while the second one -with an error not exceeding 1%, i.e., quite accurately.
The calculation of the dissociation energies of these molecules is presented in Table 4. The dissociation energy and the internuclear distance for the HV molecule are obtained from the equality of the theoretical value of the dissociation energy obtained experimentally with an error not exceeding 10%, which is quite an acceptable value. Therefore, the obtained value of the dipole electric moment by theoretical calculation presents quite a realistic value. The probability of the presence of both valence electrons near the atom of boron and hydrogen is 0.483 and 0.104, respectively.
For the LiH molecule, the situation is rather complicated since the radius of the lithium atom needs to be clarified. If the radius of the lithium atom calculated using the modified Slater wave functions and given in the handbook (Radzig & Smirnov, 1985) is used, then the minimum potential energy of the interaction between lithium and hydrogen atoms is realized at a distance between atoms of  Å (Radzig & Smirnov, 1985), (Slater, 1937). The probability of the presence of both valence electrons near the atoms of lithium and hydrogen is 0.640 and 0.040. Therefore, both for the HV molecule and for the LiH molecule, the boron and lithium atoms with respect to the hydrogen atom are more negative, which determines such a large value of the dipole electric moment of these molecules. For other molecules the situation is similar.
Thus, the interaction of atoms in a diatomic molecule is a rather complicated process, in which the exchange interaction of electrons of the single ionization of the interacting atoms should be taken into account, as well as the presence of built-in electric moments of the atoms that are directed inside the molecule relative to each other as well as the probability of stay of valence electrons relative to the interacting atoms. It should be specially noted that the positively charged nuclei of the interacting atoms impede the exchange of valence electrons and, in turn, repel each other according to Coulomb's law.

Conclusions
As a result of the studies of different types of chemical bonds between atoms, it was found: 1. The following types of interactions have been clarified: exchange, ionic, induced, electron-dipole, dipole-dipole and nuclear repulsion energies, and the probability of their occurrence has been determined.
2. The covalent bond is not decisive and is due only to the exchange of valence electrons of neutral atoms.
3. It is shown that the probability of valence electrons staying in the overlapping region of wave functions can be determined not only by the residence time of valence electrons near interacting atoms but also by the ratio of the area of the overlapping wave functions to the entire area of the sphere on which the electron travels. 4. A theory of the interaction of atomic nuclei in a diatomic molecule of identical atoms has been developed.
5. The interaction potentials for a number of diatomic molecules have been calculated in comparison with the Lennard-Jones and Morse potentials and the failure of empirical interaction potentials has been shown.
6. At the double energy radius of the molecules, a potential jump occurs with a transition from the negative to the positive region of binding energies and determines the activation energy of the formation of diatomic molecules.
7. The obtained atomic radii and the internuclear distances of diatomic molecules are effective values at which the experimental values of the binding energies are realized.
The development of various theories of chemical bonds, taking into account their probabilities of occurrence, is at the initial stage of their formation.
It seems possible, based on the obtained interaction potentials of atoms in diatomic molecules, to determine the activation energy, ionization energy, standard atomization enthalpy, and electron affinity for diatomic molecules. A lot of scientific work is ahead in this direction.