Prerequisites for the emergence of the method

Chronological method molecular orbitals appeared later than the method of valence bonds, since there were questions in the theory of covalent bonds that could not be explained by the method of valence bonds. Let's consider some of them.

The main position of the method of valence bonds is that the bond between atoms is carried out due to electron pairs (binding two-electron clouds). But it is not always the case. In some cases, individual electrons are involved in the formation of a chemical bond. So, in a molecular ion H 2+ one-electron bond. The method of valence bonds cannot explain the formation of a one-electron bond, it contradicts its basic position.

The method of valence bonds also does not explain the role of unpaired electrons in a molecule. Molecules with unpaired electrons are paramagnetic, i.e. are drawn into the magnetic field, since the unpaired electron creates a permanent magnetic moment. If there are no unpaired electrons in the molecules, then they are diamagnetic - they are pushed out of the magnetic field. The oxygen molecule is paramagnetic, it has two electrons with parallel spins, which contradicts the method of valence bonds. It should also be noted that the method of valence bonds could not explain a number of properties complex compounds- their color, etc.

To explain these facts, the molecular orbital method was proposed.

The main provisions of the method

According to the molecular orbital method, electrons in molecules are distributed in molecular orbitals, which, like atomic orbitals, are characterized by a certain energy (energy level) and shape. Unlike atomic orbitals, molecular orbitals cover not one atom, but the entire molecule, i.e. are two- or multicenter. If in the method of valence bonds the atoms of molecules retain a certain individuality, then in the method of molecular orbitals the molecule is considered as a single system.

The most widely used in the molecular orbital method is a linear combination of atomic orbitals. In this case, several rules are observed:

Schrödinger equation for a molecular system must consist of a kinetic energy term and a potential energy term for all electrons at once. But the solution of one equation with such a large number of variables (indices and coordinates of all electrons) is impossible, so the concept is introduced one-electron approximation.

The one-electron approximation assumes that each electron can be considered as moving in the field of the nuclei and the averaged field of the remaining electrons of the molecule. This means that every i th electron in a molecule is described by its own function ψ i and has its own energy E i. In accordance with this, for each electron in the molecule, one can compose its own Schrödinger equation. Then for n electrons need to be solved n equations. This is carried out by methods of matrix calculus with the help of computers.

When solving the Schrödinger equation for a multicenter and multielectron system, solutions are obtained in the form of one-electron wave functions - molecular orbitals, their energies and the electronic energy of the entire molecular system as a whole.

Linear combination of atomic orbitals

In the one-electron approximation, the molecular orbital method describes each electron with its own orbital. Just as an atom has atomic orbitals, so a molecule has molecular orbitals. The difference is that molecular orbitals are multicenter.

Consider an electron located in a molecular orbital ψ i neutral molecule, at the moment when it is near the nucleus of some atom m. In this region of space, the potential field is created mainly by the nucleus of the atom m and nearby electrons. Since the molecule is generally neutral, the attraction between the electron in question and some other nucleus n is approximately compensated by the repulsion between the electron in question and the electrons near the nucleus n. This means that near the nucleus the motion of an electron will be approximately the same as in the absence of other atoms. Therefore, in the orbital approximation, the molecular orbital ψ i near the core m should be similar to one of the atomic orbitals of that atom. Since the atomic orbital has significant values ​​only near its nuclei, one can approximately represent the molecular orbital ψ i as linear combination of atomic orbitals individual atoms.

For the simplest molecular system consisting of two nuclei of hydrogen atoms, taking into account 1s-atomic orbitals describing the motion of an electron in an atom H, the molecular orbital is represented as:

Quantities c 1i And c 2i- numerical coefficients, which are the solution Schrödinger equations. They show the contribution of each atomic orbital to a particular molecular orbital. In the general case, the coefficients take values ​​in the range from -1 to +1. If one of the coefficients prevails in the expression for a particular molecular orbital, then this corresponds to the fact that an electron, being in a given molecular orbital, is mainly located near that nucleus and is described mainly by that atomic orbital, whose coefficient is greater. If the coefficient in front of the atomic orbital is close to zero, then this means that the presence of an electron in the region described by this atomic orbital is unlikely. According to the physical meaning, the squares of these coefficients determine the probability of finding an electron in the region of space and energies described by a given atomic orbital.

In the LCAO method, for the formation of a stable molecular orbital, it is necessary that the energies of the atomic orbitals be close to each other. In addition, it is necessary that their symmetry does not differ much. If these two requirements are met, the coefficients should be close in their values, and this, in turn, ensures the maximum overlap of electron clouds. When adding atomic orbitals, a molecular orbital is formed, the energy of which decreases relative to the energies of the atomic orbitals. This molecular orbital is called binding. The wave function corresponding to the bonding orbital is obtained by adding wave functions with the same sign. In this case, the electron density is concentrated between the nuclei, and the wave function takes on a positive value. When atomic orbitals are subtracted, the energy of the molecular orbital increases. This orbital is called loosening. The electron density in this case is located behind the nuclei, and between them is equal to zero. The wave function in the two formed electron clouds has opposite signs, which is clearly seen from the scheme of formation of the bonding and loosening orbitals.

When the atomic orbital of one of the atoms, due to a large difference in energy or symmetry, cannot interact with the atomic orbital of another atom, it passes into the energy scheme of the molecular orbitals of a molecule with the energy corresponding to it in the atom. This type of orbital is called non-binding.

Orbital classification

Classification of orbitals on σ or π produced according to the symmetry of their electron clouds. σ -orbital has such a symmetry of the electron cloud, in which turning it around the axis connecting the nuclei by 180 ° leads to an orbital that is indistinguishable from the original in shape. The sign of the wave function does not change. When π -orbital, when it is rotated by 180°, the sign of the wave function is reversed. Hence it follows that s-electrons of atoms, when interacting with each other, can form only σ -orbitals, and three (six) p- orbitals of an atom - one σ- and two π -orbitals, and σ -orbital occurs when interacting p x atomic orbitals, and π -orbital - during interaction r y And pz. Molecular π -orbitals are rotated relative to the internuclear axis by 90°.

In order to distinguish bonding and antibonding orbitals from each other, as well as their origin, the following notation has been adopted. The bonding orbital is denoted by the abbreviation "sv", located at the top right after the Greek letter denoting the orbital, and loosening - respectively "razr". One more designation is adopted: antibonding orbitals are marked with an asterisk, and bonding orbitals without an asterisk are marked. After the designation of the molecular orbital, the designation of the atomic orbital is written, to which the molecular orbital owes its origin, for example, π bit 2 py. This means that the molecular orbital π -type, loosening, formed during the interaction of 2 r y- atomic orbitals.

The position of an atomic orbital on the energy scale is determined by the value of the ionization energy of the atom, which corresponds to the removal of an electron described by this orbital to an infinite distance. This ionization energy is called orbital ionization energy. So, for an oxygen atom, types of ionization are possible when an electron is removed from 2p- or with 2s-electronic subshell.

The position of the molecular orbital in energy diagrams is also determined on the basis of quantum chemical calculations of the electronic structure of molecules. For complex molecules, the number of energy levels of molecular orbitals in energy diagrams is large, but for specific chemical problems it is often important to know the energies and composition of not all molecular orbitals, but only the most “sensitive” to external influences. These orbitals are molecular orbitals that contain the highest energy electrons. These electrons can easily interact with the electrons of other molecules, be removed from a given molecular orbital, and the molecule will go into an ionized state or change due to the destruction of one or the formation of other bonds. Such a molecular orbital is the highest occupied molecular orbital. Knowing the number of molecular orbitals (equal to the total number of all atomic orbitals) and the number of electrons, it is easy to determine the serial number of the HOMO and, accordingly, from the calculation data, its energy and composition. Also, the lowest free molecular orbital, i.e., is most important for the study of chemical problems. next in line to the HOMO on the energy scale, but a vacant molecular orbital. Other orbitals that are adjacent in energy to the HOMO and LUMO are also important.

Molecular orbitals in molecules, like atomic orbitals in atoms, are characterized not only by relative energy, but also by a certain total shape of the electron cloud. Just as atoms have s-, R-, d-, ... orbitals, the simplest molecular orbital, providing a connection between only two centers (two-center molecular orbital), can be σ -, π -, δ -, ... type. Molecular orbitals are divided into types depending on what symmetry they have with respect to the line connecting the nuclei of atoms relative to the plane passing through the nuclei of the molecule, etc. This leads to the fact that the electron cloud of the molecular orbital is distributed in space in various ways.

σ -orbitals are molecular orbitals symmetrical with respect to rotation around the internuclear axis. Region of increased electron density σ -molecular orbital is distributed along the given axis. Such molecular orbitals can be formed by any atomic orbitals of atomic orbitals of any symmetry. In the figure, sections of wave functions with a negative sign are marked with filling; the rest of the segments have a positive sign.	π -orbitals are molecular orbitals that are antisymmetric with respect to rotation around the internuclear axis. Region of increased electron density π -molecular orbitals are distributed outside the internuclear axis. molecular orbitals π -symmetries are formed with a special overlap R-, d- And f-atomic orbitals.	δ -orbitals are molecular orbitals that are antisymmetric with respect to reflection in two mutually perpendicular planes passing through the internuclear axis. δ -molecular orbital is formed by a special overlap d- And f-atomic orbitals. The electron cloud of molecular orbital data is distributed mainly outside the internuclear axis.

The physical meaning of the method

For any other system including k atomic orbitals, the molecular orbital in the approximation of the LCAO method will be written in general view in the following way:

For clarification physical sense such an approach, we recall that the wave function Ψ corresponds to the amplitude of the wave process characterizing the state of the electron. As you know, when interacting, for example, sound or electromagnetic waves, their amplitudes add up. As can be seen, the above equation for the decomposition of a molecular orbital into constituent atomic orbitals is equivalent to the assumption that the amplitudes of the molecular "electron wave" (i.e., the molecular wave function) are also formed by adding the amplitudes of the interacting atomic "electron waves" (i.e., adding the atomic wave functions ). In this case, however, under the influence of the force fields of the nuclei and electrons of neighboring atoms, the wave function of each atomic electron changes in comparison with the initial wave function of this electron in an isolated atom. In the LCAO method, these changes are taken into account by introducing the coefficients c iμ, where the index i defines a specific molecular orbital, and the index cm- specific atomic orbital. So when finding the molecular wave function, not the original, but the changed amplitudes are added - c iμ ψ μ.

Find out what form the molecular wave function will have Ψ 1, formed as a result of the interaction of wave functions ψ 1 And ψ 2 - 1s orbitals of two identical atoms. To do this, we find the sum c 11 ψ 1 + c 12 ψ 2. In this case, both considered atoms are the same, so that the coefficients from 11 And from 12 are equal in size ( from 11 = from 12 = c 1) and the problem is reduced to determining the sum c 1 (ψ 1 + ψ 2). Because the constant factor c 1 does not affect the form of the desired molecular wave function, but only changes its absolute values, we confine ourselves to finding the sum (ψ 1 + ψ 2). To do this, we place the nuclei of interacting atoms at the distance from each other (r) where they are located in the molecule, and depict the wave functions 1s-orbitals of these atoms (Figure A).

To find the molecular wave function Ψ 1, add the values ψ 1 And ψ 2: the result is the curve shown in (figure b). As can be seen, in the space between the nuclei, the values ​​of the molecular wave function Ψ 1 greater than the values ​​of the original atomic wave functions. But the square of the wave function characterizes the probability of finding an electron in the corresponding region of space, i.e., the density of the electron cloud. So the increase Ψ 1 compared to ψ 1 And ψ 2 means that during the formation of a molecular orbital, the density of the electron cloud in the internuclear space increases. As a result, a chemical bond is formed. Therefore, the molecular orbital of the type in question is called binding.

In this case, the region of increased electron density is located near the bond axis, so that the resulting molecular orbital belongs to σ -type. In accordance with this, the bonding molecular orbital obtained as a result of the interaction of two atomic 1s-orbitals, denoted σ 1s sv.

Electrons in a bonding molecular orbital are called bonding electrons.

Consider another molecular orbital Ψ 2. Due to the symmetry of the system, it should be assumed that the coefficients in front of the atomic orbitals in the expression for the molecular orbital Ψ 2 = c 21 ψ 1 + c 22 ψ 2 must be equal in modulus. But then they should differ from each other by a sign: from 21 = - from 22 = c 2.

Hence, except for the case where the signs of the contributions of both wave functions are the same, the case is also possible when the signs of the contributions 1s-atomic orbitals are different. In this case (fig. (A))contribution 1s-atomic orbital of one atom is positive, and the other is negative. When these wave functions are added together, the curve shown in Fig. (b). The molecular orbital formed during such an interaction is characterized by a decrease in the absolute value of the wave function in the internuclear space compared to its value in the initial atoms: even a nodal point appears on the bond axis, at which the value of the wave function, and, consequently, its square, turns into zero. This means that in the case under consideration, the density of the electron cloud in the space between the atoms will also decrease. As a result, the attraction of each atomic nucleus in the direction towards the internuclear region of space will be weaker than in the opposite direction, i.e. forces will arise that lead to mutual repulsion of the nuclei. Here, therefore, no chemical bond arises; the resulting molecular orbital is called loosening σ 1s *, and the electrons on it - loosening electrons.

Transfer of electrons from atomic 1s-orbitals to the bonding molecular orbital, leading to the appearance of a chemical bond, is accompanied by the release of energy. On the contrary, the transition of electrons from atomic 1s-orbitals per antibonding molecular orbital requires energy. Therefore, the energy of electrons in the orbital σ 1s sv below, but in orbital σ 1s * higher than nuclear 1s-orbitals. Approximately, we can assume that when passing 1s-electron is allocated to the bonding molecular orbital the same amount of energy as it is necessary to spend for its transfer to the loosening molecular orbital.

Communication order

In the molecular orbital method, to characterize the electron density responsible for the binding of atoms into a molecule, the value is introduced - communication order. The link order, in contrast to the link multiplicity, can take non-integer values. The bond order in diatomic molecules is usually determined by the number of bonding electrons involved in its formation: two bonding electrons correspond to a single bond, four bonding electrons to a double bond, etc. In this case, loosening electrons compensate for the action of the corresponding number of bonding electrons. So, if there are 6 binding and 2 loosening electrons in a molecule, then the excess of the number of binding electrons over the number of loosening electrons is four, which corresponds to the formation of a double bond. Therefore, from the standpoint of the molecular orbital method, a chemical bond in a hydrogen molecule formed by two bonding electrons should be considered as a simple bond.

For elements of the first period, the valence orbital is 1s-orbital. These two atomic orbitals form two σ -molecular orbitals - bonding and loosening. Consider the electronic structure of a molecular ion H2+. It has one electron, which will occupy a more energetically favorable s bonding orbital. In accordance with the rule for counting the multiplicity of bonds, it will be equal to 0.5, and since there is one unpaired electron in the ion, H2+ will have paramagnetic properties. The electronic structure of this ion will be written by analogy with the electronic structure of an atom as follows: σ 1s sv. The appearance of a second electron s-bonding orbitals will lead to an energy diagram describing the hydrogen molecule, an increase in the bond multiplicity to unity and diamagnetic properties. An increase in the multiplicity of bonds will also entail an increase in the dissociation energy of the molecule H2 and a shorter internuclear distance compared to that of the hydrogen ion.

diatomic molecule Not 2 will not exist, since the four electrons present in two helium atoms will be located on the bonding and loosening orbitals, which leads to a zero multiplicity of bonds. But at the same time the ion He2+ will be stable and the multiplicity of communication in it is equal to 0.5. Just like the hydrogen ion, this ion will have paramagnetic properties.

The elements of the second period have four more atomic orbitals: 2s, 2p x, 2p y, 2p z, which will take part in the formation of molecular orbitals. Energy Difference 2s- And 2p-orbitals are large, and they will not interact with each other to form molecular orbitals. This energy difference will increase as you move from the first element to the last. In connection with this circumstance, the electronic structure of diatomic homonuclear molecules of elements of the second period will be described by two energy diagrams that differ in the order of arrangement on them σ st 2p x And π sv 2p y,z. With relative energy proximity 2s- And 2p-orbitals observed at the beginning of the period, including the nitrogen atom, electrons located on σ res 2s And σ st 2p x-orbitals, repel each other. That's why π sv 2p y- And π sv 2p z orbitals are energetically more favorable than σ st 2p x-orbital. The figure shows both diagrams. Since participation 1s-electrons in the formation of a chemical bond is insignificant, they can be ignored in the electronic description of the structure of molecules formed by elements of the second period.

The second period of the system is opened by lithium and beryllium, in which the external energy level contains only s-electrons. For these elements, the scheme of molecular orbitals will not differ in any way from the energy diagrams of molecules and ions of hydrogen and helium, with the only difference that in the latter it is built from 1s-electrons, and Li 2 And Be 2- from 2s-electrons. 1s-electrons of lithium and beryllium can be considered as nonbonding, i.e. belonging to individual atoms. Here, the same patterns will be observed in changing the bond order, dissociation energy, and magnetic properties. And he Li2+ has one unpaired electron located on σ st 2s-orbitals - the ion is paramagnetic. The appearance of a second electron in this orbital will lead to an increase in the dissociation energy of the molecule Li 2 and an increase in the multiplicity of the bond from 0.5 to 1. The magnetic properties will acquire a diamagnetic character. Third s- the electron will be located on σ res-orbitals, which will help reduce the bond multiplicity to 0.5 and, as a consequence, lower the dissociation energy. Such an electronic structure has a paramagnetic ion Be 2+. Molecule Be 2, as well as He 2, cannot exist due to the zero order of the relationship. In these molecules, the number of binding electrons is equal to the number of loosening ones.

As can be seen from the figure, as the bonding orbitals are filled, the dissociation energy of molecules increases, and with the appearance of electrons in the antibonding orbitals, it decreases. The series ends with an unstable molecule Ne 2. The figure also shows that the removal of an electron from the antibonding orbital leads to an increase in the bond multiplicity and, as a consequence, to an increase in the dissociation energy and a decrease in the internuclear distance. The ionization of the molecule, accompanied by the removal of the binding electron, has the opposite effect.

The molecular orbital (MO) method has been abbreviated in the literature as the linear combination of atomic orbitals (LCAO) method. The molecule is considered as a whole, and not as a collection of atoms that retain their individuality. Each electron belongs to the entire molecule as a whole and moves in the field of all its nuclei and other electrons.

The state of an electron in a molecule is described by a one-electron wave function i (i means i th electron). This function is called the molecular orbital (MO) and is characterized by a certain set quantum numbers. It is found as a result of solving the Schrödinger equation for a molecular system with one electron. Unlike a single-center atomic orbital (AO), a molecular orbital is always multicenter, since the number of nuclei in a molecule is at least two. As for an electron in an atom, the square of the modulus of the wave function | i | 2 determines the probability density of finding an electron or the density of an electron cloud. Each molecular orbital i characterized by a certain value of energy E i. It can be determined by knowing the ionization potential of a given orbital. The electronic configuration of a molecule (its lower unexcited state) is given by the set of MOs occupied by electrons. The filling of molecular orbitals with electrons is based on two main assumptions. An electron in a molecule occupies a free orbital with the lowest energy, and one MO cannot contain more than two electrons with antiparallel spins (Pauli principle). If the molecule contains 2 n electrons, then to describe its electronic configuration it is required n molecular orbitals. True, in practice, a smaller number of MOs is often considered, using the concept of valence electrons, i.e., those electrons that enter into a chemical bond.

When one electron of a molecule passes from an occupied MO to a higher free MO, the molecule as a whole passes from the ground state (Ψ) to an excited state ( * ). For a molecule, there is a certain set of allowed states, which correspond to certain energy values. Transitions between these states with absorption and emission of light give rise to the electronic spectrum of the molecule.

To find the energy spectrum of a molecule, it is necessary to solve the Schrödinger equation of the form

Ĥ = E , (5.15)

if the molecular wave function is known. However, the difficulty of solving equation (5.35) lies in the fact that we often do not know. Therefore, one of the main problems of quantum mechanics is to find the molecular wave function. The most common way to write a molecular orbital is to use a specific set of atomic orbitals obtained for the atoms that make up the molecule. If the molecular orbital is denoted as i, and atomic - through φ k, then the general relation for MO has the form

i.e. MO is a linear combination of atomic orbitals φ k with their coefficients Cik. Number of independent solutions for i is equal to the number φ k in the original basis. to reduce the number of atomic wave functions, only those AOs are chosen that contribute to the chemical bond. The MO symmetry properties can be determined from the signs and numerical values ​​of the coefficients Cik(LCAO coefficients) and symmetry properties of atomic orbitals. The filling of molecular orbitals with electrons is carried out by analogy with atomic ones. The most accurate calculations for molecules are performed by the self-consistent field method (SFC). Molecular orbitals calculated by the SSP method are closest to the true ones and are called Hartree-Fock orbitals.

5.3.3 Application of the molecular orbital method
to describe the chemical bond in the H 2 + ion

The simplest diatomic molecule is the hydrogen molecule H 2, in which the chemical bond is formed by two electrons (type 1 s) belonging to hydrogen atoms. If we remove one electron, we get even more simple system H 2 + is a molecular hydrogen ion, in which the chemical bond is carried out by one electron. This stable particle with internuclear distance r e(H 2 +) = 0.106 nm dissociation energy D 0 (H 2 +) = 2.65 eV. From the point of view of quantum mechanics, this problem is multicenter, one electron revolves around nuclei (Fig. 5.10).

The Schrödinger equation for such a system is written in the form (5.15), where is the wave function of the molecular ion H 2 + , which is composed of the wave functions of the hydrogen atom in the form

= with 1 j 1 + with 2 j 2 , (5.17)

where j 1 and j 2 are atomic wave functions (1 s atomic orbitals of hydrogen); With 1 and With 2 – coefficients to be determined; Ĥ is the Hamilton operator, which has the form

The last three terms give the value of the potential energy of nuclear and electron-nuclear interactions, R 12 - distance between nuclei, r 1 and r 2 are the distances from the electron to the corresponding nuclei.

As follows from Fig. 5.10, one electron moves around two nuclei, which are assumed to be stationary. Such a task in quantum mechanics cannot be solved exactly, so we will consider its approximate solution by the MO method. This will allow us to get to know the most characteristic features method. The physical picture of the formation of a chemical bond will be revealed qualitatively, despite the approximate values ​​of the parameters With 1 and With 2 when recording the wave function. Fundamentals of the theory of the method for the simplest ion H 2 + will serve as a starting point for understanding the nature of the chemical bond in more complex molecules.

The problem of finding the coefficients With 1 and With 2 and the energies of the H 2 + system will be solved using the variational method. The essence of the method is as follows. We multiply both sides of equation (5.15) by the complex conjugate wave function Ψ * and integrate over the entire range of variables. As a result, we get the expression:

Where dτ is the elementary volume (in the Cartesian coordinate system dτ = dx dy dz).

If the wave function is known (in our case it is given with coefficients With 1 and With 2) and the Hamiltonian Ĥ , then we can calculate the energy of the system E. in a state of stable equilibrium ( r e(H 2 +) = 0.106 nm), the energy of the H 2 + system should be minimal.

Substituting the value of function (5.17) into the expression for energy (5.19), we obtain

Having performed the appropriate transformations, we obtain

To simplify the notation of (5.21), we introduce the notation for integrals:

It follows from the properties of the overlap integrals that S 12 = S 21 . taking into account the commutation properties of the Hamilton operator, we can show that H 21 = H 12 .

Substituting into (5.21) the values ​​of the integrals (5.22), we obtain

It is possible to calculate the energy value according to (5.23) if the values ​​of the coefficients are known With 1 and With 2. However, they are not known under the conditions of our problem. To find them, the variational method is used, according to which the function Ψ (5.17) must correspond to the minimum energy E. Minimum condition E as a function With 1 and With 2 will be equal to zero partial derivatives: and

Let us first find the partial derivative of E By from 1 and set it equal to zero.

After transformation we get

Comparing (5.23) and (5.25), we can write

Grouped by variables With 1 and With 2 , we rewrite (5.26) as follows:

Differentiating the energy value (5.24) with respect to With 2 , similarly we get

Expressions (5.27) and (5.28) represent linear system equations in two unknowns With 1 and With 2. For this system to be solvable, it is necessary that the determinant consisting of the coefficients of the unknowns be equal to zero, i.e.

Since the MO is formed from two atomic functions, we got a second-order determinant, with a combination of three atomic wave functions we would get a third-order determinant, etc. The numbers in the indices coincide with the row number (first) and with the column number (second). This correspondence can be generalized to functions that are linear combinations n atomic orbitals. We then get the determinant n th order type

Where i And j have n values.

The determinant can be simplified by setting the integrals S 11 = S 22 = 1 if the atomic wave functions are normalized. Integral S 12 denote by S. In our case H 11 = H 22 because the atomic wave functions φ 1 and φ 2 are the same. Denote the integrals H 11 = H 22 = α , A H 12 through β. Then the determinant (5.29) will have the form

Expanding this determinant, we get

Solving equation (5.33) with respect to E, we obtain two energy values

So, when solving the Schrödinger equation with a known wave function, up to coefficients With 1 and With 2 we obtain two energy eigenvalues. Let us determine the values ​​of the coefficients With 1 and 2, or rather their ratio, since from two equations (5.27) and (5.28) it is impossible to obtain three unknowns - E, s 1 and With 2. Knowing the meaning E s from (5.33) one can find the relation With 1 /With 2 of (5.28)

Substituting the values E s from (5.34) into the last equation, we obtain

where With 1 =With 2 = with s.

Similarly, substituting in (5.28) instead of E meaning E as , we get the second possible relation:

With 1 /With 2 = -1 or With 1 = - with 2 = with as. (5.38)

Substituting (5.37) and (5.38) into (5.17) leads to two solutions of the Schrödinger equation for H 2 + , to two molecular orbitals:

To determine the numerical value of the coefficients With s and With as we use the normalization condition for the molecular function:

Substituting for s its value from (5.39) gives the following expression:

The first and second terms on the right side are equal to one, since φ 1 and φ 2 are normalized. Then

Similarly, the coefficient with as:

If the overlap integral S neglect compared to unity (although for the H 2 + ion and the H 2 molecule it is comparable to unity, but for the sake of generality it is neglected), then we will have:

From (5.39) and (5.40) we obtain two molecular wave functions corresponding to two energy values E s And E as,

Both MOs are approximate solutions of the Schrödinger equation obtained by the variational method. One of them with lower energy (Ψ s) corresponds to the main one, the second (Ψ as) to the nearest higher state.

Based on the obtained wave functions (5.46) and (5.47), one can determine the electron density distribution in the H 2 + molecular ion corresponding to the energies E s And E as.

As can be seen, the symmetric function leads to an increase in the electron charge density in the region of overlapping atomic wave functions (in the internuclear space A And IN) in comparison with the charge density described by the functions φ 1 2 and φ 2 2 . The antisymmetric wave function leads to a decrease in the charge density. On fig. 5.11 this is shown graphically. The dotted lines represent the charge density of individual atoms separated from one another by an infinitely large distance, and the solid line shows the electron density distribution in the molecular hydrogen ion along the internuclear axis. Obviously, the symmetric wave function (5.46) favors such a distribution of charge, in which it is concentrated between the nuclei. Such MO is called binding. And vice versa, asymmetric MO (5.47) leads to a decrease in the charge density in the internuclear space and its concentration near individual atomic nuclei.

Such MO is called antibonding or loosening. Therefore, only the symmetric function causes the formation of a stable molecule (H 2 +). On the curve of dependence of potential energy on the distance between the nuclei ( RAB) (see Fig. 5.11) at some of these distances there will be a minimum. We get two potential curves: one for the bonding orbital, and the second for the loosening orbital (Figure 5.12).

In energy values E s(5.34) and E as(5.35) the same integrals α, β and S, however, the energy values ​​are not the same due to the difference in signs on the right-hand sides.

Let us analyze the integrals in more detail. We substitute the Hamilton operator (5.34) into the first integral. Then we get:

the integral can be simplified if we take into account that is the Hamiltonian operator for a hydrogen atom with an electron near the nucleus A. It gives the value of energy E 0 in the hydrogen atom. the Hamilton operator for the molecular hydrogen ion can be written as follows:

Where E 0 is the energy of the ground state of the hydrogen atom.

The value of the integral (5.50) is rewritten as follows:

Quantities E 0 and RAB are constants and can be taken out of the integral sign:

Since the wave function φ 1 is normalized, i.e., then

Where I denotes the integral, called the Coulomb

which is not very easy to calculate, but nevertheless it makes a significant contribution to the total energy of the system.

So the integral H 11 = H 22 = α , as can be seen from (5.54), consists of three parts and conveys the classical Coulomb interaction of particles. It includes the energy of an electron in the ground state hydrogen atom ( E 0), Coulomb repulsion of nuclei ( e 2 /RAB) and energy I Coulomb interaction of the second proton ( IN) with an electron cloud surrounding the first proton ( A). at distances of the order of the equilibrium internuclear one, this integral is negative, and at large distances, where the repulsion of nuclei is small, it is practically equal to the energy of an electron in an atomic orbital, therefore, in the zeroth approximation, it is taken equal to the energy of an electron in a hydrogen atom ( E 0). Only at distances much smaller than the equilibrium one does it become positive and increase indefinitely.

Integral H 12 = H 21 = β is called exchange or resonant. The energy expressed by the integral β has no analogue in classical physics. It describes an additional decrease in the energy of the system, which occurs due to the possibility of an electron moving from the nucleus A to the core IN, as if exchanging the states φ 1 and φ 2 . This integral is equal to zero at infinity, and is negative at all other distances (except for very short, smaller internuclear ones). Its contribution determines the energy of the chemical bond (the larger this integral, the stronger the bond). By analogy with (5.53), this integral can be written as follows:

Taking the constant terms out of the integral sign, we obtain

the atomic orbital overlap integral (denoted S 12 = S 21 = S) forming a molecular orbital is a dimensionless quantity and is equal to unity at RAB = 0 drops to zero as the internuclear distance increases. At distances between atoms close to or equal to the equilibrium ones, the exchange integral H 12 the greater in absolute value, the greater the overlap integral.

Indeed, equality (5.57) can be rewritten as follows, if we introduce the notation S 12 and K

Where K denotes an integral of type

called the exchange integral.

The last integral in (5.57) gives the main negative addition to the general exchange integral H 12 .

If the values ​​of all obtained integrals are substituted into the equations for the energy (5.34) and (5.35) of the symmetric and asymmetric states, then we obtain

For the antisymmetric state, we obtain the following value

Calculating integrals I And K are quite complex, but it is possible to estimate their dependence on the distance between the nuclei of hydrogen atoms. The results of this dependence are shown by the potential energy curves in Figs. 5.12.

As can be seen from fig. 5.12, a symmetric energy state leads to a minimum of potential energy, so a stable particle H 2 + is formed. The antisymmetric state corresponds to an unstable energy state. in this case, the electron will be in an antisymmetric orbital and the molecular ion H 2 + will not be formed. Hence, E s corresponds to the ground state, and As– the first excited state of the molecular ion H 2 + .

If we assume approximately that S 12 = 0 and keep the notation for H 11 and H 12, respectively, through α and β, then the expressions for the wave functions of an electron in a molecule and its energy take on a simple form:

Since the integral β is negative, then E 1 < E 2 .

Thus, the MO method shows that when two atoms are combined into a molecule, two states of an electron are possible: – two molecular orbitals 1 and 2 , one of them with a lower energy E 1 , the other - with more high energy E 2. Since the presence of both two and one electron is possible on the MO, the MO method makes it possible to estimate the contribution to the chemical bond not only of electron pairs, but also of individual electrons.

The MO LCAO method for the H 2 + ion gives the values E 0 = 1.77 eV and r 0 = 0.13 nm, and according to experimental data E 0 = 2.79 eV and r 0 = 0.106 nm, i.e., the calculation is in qualitative agreement with the experimental data.

If, during the formation of a molecule from atoms, an electron occupies the lower orbital, then the total energy of the system will decrease - a chemical bond is formed.

Therefore, the wave function 1 (corresponding to s) is called a bonding orbital. The transition of an electron to the upper orbital 2 (corresponding to as) will increase the energy of the system. the connection is not formed, the system will become less stable. Such an orbital is called an antibonding orbital. The binding and loosening action of electrons is determined by the form of wave functions 1 and 2 .

In the H 2 hydrogen molecule, two electrons are placed in the lower bonding orbital, which leads to an increase in the bond strength and a decrease in the energy of the bonding orbital. The results of calculations by the MO method for the hydrogen molecule H2 lead to the value E 0 = 2.68 eV and r 0 = 0.085 nm, and the experiment gives the values E 0 = 4.7866 eV and r 0 = 0.074 nm. The results agree in order of magnitude, although the energy of the lowest state differs by almost a factor of two from the value obtained experimentally. Similarly, molecular orbitals are formed for other diatomic molecules consisting of heavier atoms.

5.4. Types of chemical bonds
in diatomic molecules.
σ -and π-connections

The most common types of bonds in molecules are σ- and π-bonds, which are formed as a result of overlapping electron clouds of external (valence) electrons. There are other types of chemical bonds that are characteristic of complex compounds containing atoms of the heaviest elements.

On fig. 5.13 and 5.14 show typical options for overlapping s-, R- And d- electron clouds during the formation of chemical bonds. Their overlap occurs in such a way that for a given bond length, the area of ​​overlap is the largest, which corresponds to the maximum possible strength of the chemical bond.

Under the σ-bond in a molecule, we mean such a bond, which is formed due to the overlap of external s- or p-electrons. with this overlap, the electron cloud in the space between atoms has cylindrical symmetry about the axis passing through the nuclei of atoms (see Fig. 5.13). The region of overlap of clouds with a cylindrically located electron density lies on the bond axis. The wave function is determined by the value of the electron density in the internuclear space (see Fig. 5.13). The maximum electron density is described by the σ-bonding MO orbital, and the minimum by the σ*‑antibonding one. In bonding MOs, the electron density between nuclei is greatest and the repulsion of nuclei decreases. The energy of the molecule is less than the energy of the AO, the molecule is stable, the overlap integral S > 0. In antibonding (or loosening) MOs, the electron density between nuclei is zero, the repulsion of nuclei increases, and the MO energy is greater than the AO energy. The state of the molecule is unstable, the overlap integral S< 0.

Each pair of AOs forming an MO gives two molecular orbitals (bonding and antibonding), which is reflected in the appearance of two energy levels and, accordingly, potential curves (see Fig. 5.12). In the normal state, bonding orbitals are filled with electrons.

In addition to bonding and antibonding orbitals, there are nonbonding orbitals. Usually this is the AO of an atom that does not form chemical bonds. The overlap integral in this case is equal to zero. What happens if the AOs belong to different types of symmetry.

Along with σ-bonds, π-bonds can also exist in the molecule, which are formed as a result of overlapping atomic p-orbitals or d- And R-orbitals (Fig. 5.14).

The π-bond electron cloud does not have axial symmetry. It is symmetrical with respect to the plane passing through the axis of the molecule. The density of the electron cloud vanishes in this plane. On fig. 5.15 shows the formation of a π bond and the electron density for
π s-orbitals. The π-bond is weaker than the σ-bond, and the energy of the π-bond is depicted on the level diagram above the energy of the σ-bond. The electronic configurations of the molecule and the filling of various shells with electrons is carried out in the same way as for atoms. Electrons are placed in series in twos, taking into account the Pauli principle (starting from a lower MO and ending with a higher one), with opposite spins per energy level (without degeneracy).

Consider the chemical bonds in the simplest diatomic molecules, their energy levels and their filling with electrons.

It is known that in the ion of the H 2 + molecule, the chemical bond is carried out by one 1 s-electron, and it is located on the bonding orbital σ s . This means that from 1 s-atomic orbital, a bonding molecular σ-orbital is formed. for a hydrogen molecule H 2 there are already two 1 s electron form a similar orbital - (σ s) 2 . We can assume that two bonding electrons correspond to a single chemical bond. Let us consider the electronic structure of the He 2 molecule. The helium atom contains two valence (1 s-electron) of an electron, therefore, when considering a molecule, we must place four valence electrons in molecular orbitals. According to the Pauli principle, two of them will be located on the bonding σ s -orbital, and the other two on the loosening σ s * -orbital. The electronic structure of this molecule can be written as follows:

Not 2 [(σ s) 2 (σ s *) 2 ].

Since one loosening electron destroys the action of the bonding electron, such a molecule cannot exist. It has two bonding and two loosening electrons. The order of a chemical bond is zero. But the He 2 + ion already exists. for him, the electronic structure will have the following form:

Not 2 + [(σ s) 2 (σ s *) 1 ].

One loosening electron does not compensate for two bonding electrons.

Consider the formation of molecules from atoms of elements of the second period of the periodic table. For these molecules, we will assume that the electrons of the filled layer do not take part in the chemical bond. The Li 2 molecule has two binding (2 s) electron - Li 2 (σ s) 2 . The Be 2 molecule must have an electronic configuration

Be 2 [(σ s) 2 (σ s *) 2 ],

in which four electrons are located in molecular orbitals (two 2 s-electron from each atom). The number of binding and loosening electrons is the same, so the Be 2 molecule does not exist (here there is a complete analogy with the He 2 molecule).

In a B 2 molecule, six electrons have to be placed in molecular orbitals (four 2 s-electron and two 2 R-electron). The electronic configuration will be written as follows:

B 2 [(σ s) 2 (σ s *) 2 (π x) (π y)].

Two electrons in a B 2 molecule are located one per π x- and π y orbitals with the same energy. According to Hund's rule, they have parallel spins (two electrons with the same spins cannot be located on the same orbital). Indeed, the experiment shows the presence of two unpaired electrons in this molecule.

In a C 2 carbon molecule, eight valence electrons must be placed in molecular orbitals (two 2 s-electron and two 2 R electrons of one and the other atoms). The electronic structure will look like this:

С 2 [(σ s) 2 (σ s *) 2 (π x) 2 (π y) 2 ].

There are two loosening electrons in the C 2 molecule, and six bonding electrons. The excess of bonding electrons is four, so the bond in this molecule is double. The bond in the nitrogen molecule N 2 is carried out by electrons 2 s 2 and 2 R 3 . Consider only participation in the connection of three unpaired p-electrons. 2 s-electron form a filled shell and their participation in bond formation is close to zero. clouds of three p x,py,pz electrons extend in three mutually perpendicular directions. Therefore, only an s-bond is possible in a nitrogen molecule due to the concentration of electron density along the axis z(Fig. 5.16), i.e. s is formed due to the pair pz-electrons. The remaining two chemical bonds in the N 2 molecule will be only p-bonds (due to overlapping p x–p x , p y–py electrons. in fig. 5.16, b this overlap is shown separately.

Thus, three common electron pairs in a nitrogen molecule form one s- and two p-bonds. In this case, we speak of a triple chemical bond. Two atoms cannot be linked by more than three electron pairs. The electronic configuration of the N 2 molecule has the following form:

N 2 [(σ s) 2 (σ x*) 2 (π x ,y) 4 (σ z) 2 ].

The highest occupied orbital is σ z-orbital formed by overlapping two R-orbitals, the lobes of which are directed along the bond axis (axis z). This is due to the regularity of energy change 2 s- and 2 R-electrons with increasing atomic number of the element.

In the oxygen molecule O 2, twelve valence electrons should be distributed along molecular orbitals, two of which, in comparison with the N 2 molecule, should occupy loosening orbitals. The general electronic structure will be written as:

О 2 [(σ s) 2 (σ s *) 2 (σ z) 2 (π x) 2 , (π y) 2 (π x*) 1 (π y *) 1 ].

As in the B 2 molecule, two electrons with parallel spins occupy two different π orbitals. This causes a couple magnetic properties oxygen molecules, which is consistent with experimental data. An excess of four bonding electrons provides a bond order in the molecule equal to two.

In the F 2 molecule following oxygen, it is necessary to additionally place 2 valence orbitals in orbitals R-electron, so the fluorine molecule will have the following electronic structure:

F 2 [(σ s) 2 (σ s *) 2 (σ z) 2 (π x) 2 (π y) 2 (π x*) 2 (π y *) 2 ].

The excess of two bonding electrons characterizes a single chemical bond in the F 2 molecule.

It is easy to show that the Ne 2 molecule does not exist, since the number of bonding electrons in it is equal to the number of loosening ones.

Let us consider the electronic structure of individual diatomic molecules consisting of dissimilar atoms using the CO molecule as an example. In a CO molecule, ten valence electrons are located in molecular orbitals. Its electronic structure is similar to that of N 2 , which also has ten valence electrons in the same molecular orbitals. This explains the proximity of chemical and physical properties these molecules. On fig. 5.17 is a diagram of the energy levels of MO in a CO molecule.

It can be seen from the diagram that the energy levels 2 s-electrons of carbon and oxygen are significantly different, so their linear combination cannot correspond to the real MO in this molecule, as it could follow from simplified combinations. 2 s-electrons of oxygen remain in the molecule at the same energy level as in the atom, forming a non-bonding molecular orbital (s H). 2 s– AO of carbon in a linear combination with the corresponding symmetry 2 R- AO oxygen (2 pz) form a bonding s and an antibonding s* molecular orbital. With linear combination 2 p x and 2 r y– AO carbon and oxygen form molecular orbitals p x(connecting) and π x* (loosening) and similarly p y and p y *. 2pz– AO of carbon, to which one s-electron as a result of the reaction will be the second non-bonding
p H -orbital. One of the R- electrons of oxygen. Thus, ten valence electrons in a CO molecule fill three bonding and two nonbonding MOs. The electronic configuration of the outer electrons of the CO molecule will look like this:

(σ Н) 2 (σ) 2 (π x,y) 4 (π H)].

In the NO molecule, eleven electrons must be placed in orbitals, which will lead to the structure of the electron shell of the type:

NO [(σ s) 2 (σ s*) 2 (π x) 2 (π y) 2 (σ z) 2 (π x *)].

As can be seen, the number of excess binding electrons is five. From the point of view of the order of the chemical bond, one must introduce a fractional number, equal to 2.5, for its characteristics. If one electron is removed from this molecule, then an NO + ion with a stronger interatomic bond will be obtained, since the number of binding electrons here will be six (one electron with loosening π is removed x* -orbitals).

If two atoms can only be bonded by one common pair of electrons, then a σ-bond is always formed between such atoms. A π bond occurs when two atoms share two or three electron pairs. A typical example is the nitrogen molecule. chemical bond in it is carried out due to three unpaired p x, py, And pz-electrons. The angular lobes of their orbitals extend in three mutually perpendicular directions. If we take the axis for the communication line z, then the overlap pz-atomic orbitals will give one σ z-connection. Other orbitals p x And py will give only π-bonds. Thus, three pairs of bonding electrons give one σ-bond and two π-bonds. So, all single chemical bonds between atoms are σ-bonds. In any multiple bond, there is one σ-bond, and the rest are π-bonds.

5.5. Systematics of electronic states
in a diatomic molecule

For the systematics of electronic states in diatomic molecules, just as in atoms, certain quantum numbers are introduced that characterize the orbital and spin motion of electrons. The presence of electric and magnetic fields both in molecules and in atoms leads to the vector addition of the orbital and spin moments of momentum. However, in a diatomic molecule, valence electrons move not in a spherically symmetric electric field, which is typical for an atom, but in an axially symmetric one, which is typical for diatomic or linear polyatomic molecules. All diatomic molecules belong to two types of symmetry: D ∞h or WITH∞ u . Molecules consisting of identical atoms belong to the first type, and from opposite atoms to the second. The axis of infinite order is directed along the chemical bond. the electric field also acts in the same direction, which strongly affects the total orbital momentum, causing its precession around the field axis. As a result, the total orbital momentum ceases to be quantized, and only the quantization of its projection is preserved Lz on the axis of the molecule:

L z = m L ħ,(5.65)

Where mL is a quantum number that takes the values mL= 0, ±1, ±2, etc. In this case, the energy of the electronic state depends only on the absolute value mL, which corresponds to the fact that from a visual point of view, both rotations of an electron (right and left) around the axis of the molecule lead to the same energy value. Let us introduce some value Λ, which characterizes the absolute value of the projection of the total orbital momentum onto the axis of the molecule. Then the values ​​of Λ will be positive integers differing by one unit Λ = ê mLê = 0, 1,2,...

To classify the electronic states of a diatomic molecule, the numbers Λ play the same role as the orbital quantum number l for classifying the electronic states of atoms. The total total quantum number for atoms is usually denoted , where the summation is performed over all the electrons of the atom. If L= 0, then such electronic states are denoted by the letter s; If L= 1, then the electronic states are denoted by the letter R., i.e.

The VS method is widely used by chemists. Within the framework of this method, a large and complex molecule is considered as consisting of separate two-center and two-electron bonds. It is assumed that the electrons that cause the chemical bond are localized (located) between two atoms. The VS method can be successfully applied to most molecules. However, there are a number of molecules to which this method is not applicable or its conclusions are in conflict with experiment.

It has been established that in a number of cases the decisive role in the formation of a chemical bond is played not by electron pairs, but by individual electrons. The existence of the H 2 + ion indicates the possibility of chemical bonding with the help of one electron. When this ion is formed from a hydrogen atom and a hydrogen ion, an energy of 255 kJ is released. Thus, the chemical bond in the H 2 + ion is quite strong.

If we try to describe a chemical bond in an oxygen molecule using the VS method, we will come to the conclusion that, firstly, it must be double (σ- and p-bonds), and secondly, all electrons in an oxygen molecule must be paired, i.e., .e. the O 2 molecule must be diamagnetic (for diamagnetic substances, the atoms do not have a permanent magnetic moment and the substance is pushed out of the magnetic field). A paramagnetic substance is that whose atoms or molecules have a magnetic moment, and it has the property of being drawn into a magnetic field. Experimental data show that the energy of the bond in the oxygen molecule is indeed double, but the molecule is not diamagnetic, but paramagnetic. It has two unpaired electrons. The VS method is powerless to explain this fact.

The molecular orbital (MO) method is most visible in its graphical model of a linear combination of atomic orbitals (LCAO). The MO LCAO method is based on the following rules.

1) When atoms approach each other to the distances of chemical bonds, molecular orbitals (AO) are formed from atomic orbitals.

2) The number of obtained molecular orbitals is equal to the number of initial atomic ones.

3) Atomic orbitals that are close in energy overlap. As a result of the overlap of two atomic orbitals, two molecular orbitals are formed. One of them has a lower energy compared to the original atomic ones and is called binding , and the second molecular orbital has more energy than the original atomic orbitals, and is called loosening .

4) When atomic orbitals overlap, the formation of both σ-bonds (overlap along the chemical bond axis) and π-bonds (overlap on both sides of the chemical bond axis) is possible.

5) A molecular orbital that is not involved in the formation of a chemical bond is called non-binding . Its energy is equal to the energy of the original AO.

6) On one molecular orbital (as well as atomic orbital) it is possible to find no more than two electrons.

7) Electrons occupy the molecular orbital with the lowest energy (principle of least energy).

8) The filling of degenerate (with the same energy) orbitals occurs sequentially with one electron for each of them.

Let us apply the MO LCAO method and analyze the structure of the hydrogen molecule.

Let's mentally overlap two atomic orbitals, forming two molecular orbitals, one of which (bonding) has a lower energy (located below), and the second (loosening) has a higher energy (located above)

Rice. 8 Energy diagram of the formation of the H 2 molecule

The MO LCAO method makes it possible to visually explain the formation of H 2 + ions, which causes difficulties in the method of valence bonds. One electron of the H atom passes to the σ-bonding molecular orbital of the H 2 + cation with energy gain. A stable compound is formed with a binding energy of 255 kJ/mol. The multiplicity of the connection is ½. The molecular ion is paramagnetic. The ordinary hydrogen molecule already contains two electrons with opposite spins in σ cv 1s orbitals: The binding energy in H 2 is greater than in H 2 + - 435 kJ / mol. The H 2 molecule has a single bond, the molecule is diamagnetic.

Rice. 9 Energy diagram of the formation of the H 2 + ion

Using the MO LCAO method, we consider the possibility of the formation of the He 2 molecule

In this case, two electrons will occupy the bonding molecular orbital, and the other two will occupy the loosening orbital. Such a population of two orbitals with electrons will not bring a gain in energy. Therefore, the He 2 molecule does not exist.

Rice. 10 Energy diagram illustrating the impossibility of forming a chemical

bonds between He atoms

The filling of molecular orbitals occurs in compliance with the Pauli principle and Hund's rule as their energy increases in the following sequence:

σ1s< σ*1s < σ2s < σ*2s < σ2p z < π2p x = π2p y < π*2p x =π*2p y < σ*2p z

The energy values ​​σ2p and π2p are close and for some molecules (B 2 , C 2 , N 2) the ratio is the opposite of the above: first π2p then σ2p

Table 1 Energy and bond order in molecules of elements of period 1

Molecules and molecular ions	Electronic configuration	Bond energy	Communication order
	(σ s) 2 (σ s *) 1
	(σ s) 2 (σ s *) 1
	(σ s) 2 (σ s *) 1
	(σ s) 2 (σ s *) 1
	(σ s) 2 (σ s *) 2

According to the MO method communication procedure in a molecule is determined by the difference between the number of bonding and loosening orbitals, divided by two. The bond order can be zero (the molecule does not exist), an integer or a positive fractional number. When the bond multiplicity is zero, as in the case of He 2 , no molecule is formed.

Figure 11 shows the energy scheme for the formation of molecular orbitals from atomic orbitals for diatomic homonuclear (of the same element) molecules of elements of the second period. The number of binding and loosening electrons depends on their number in the atoms of the initial elements.

Fig.11 Energy diagram for the formation of diatomic molecules

elements 2 periods

The formation of molecules from atoms of elements of period II can be written as follows

(K - internal electronic layers):

Li 2

The Be 2 molecule was not detected, as was the He 2 molecule

B 2 molecule is paramagnetic

C2

N 2

O 2 molecule is paramagnetic

F2

Ne 2 molecule not detected

Using the MO LCAO method, it is easy to demonstrate the paramagnetic properties of the oxygen molecule. In order not to clutter up the figure, we will not consider overlap 1 s-orbitals of oxygen atoms of the first (inner) electron layer. We take into account that p-orbitals of the second (outer) electron layer can overlap in two ways. One of them will overlap with a similar one with the formation of a σ-bond.

Two others p-AO overlap on both sides of the axis x with the formation of two π-bonds.

Rice. 14 Energy diagram illustrating, using the MO LCAO method, the paramagnetic properties of the O 2 molecule

The energies of molecular orbitals can be determined from the absorption spectra of substances in the ultraviolet region. So, among the molecular orbitals of the oxygen molecule formed as a result of overlapping p-AO, two π-bonding degenerate (with the same energy) orbitals have less energy than the σ-bonding one, however, like π*-loosening orbitals, they have less energy compared to the σ*-loosening orbital.

In the O 2 molecule, two electrons with parallel spins ended up on two degenerate

(with the same energy) π*-antibonding molecular orbitals. It is the presence of unpaired electrons that determines the paramagnetic properties of the oxygen molecule, which will become noticeable if oxygen is cooled to a liquid state. So, the electronic configuration of O 2 molecules is described as follows:

О 2 [КК(σ s) 2 (σ s *) 2 (σ z) 2 (π x) 2 (π y) 2 (π x *) 1 (π y *) 1 ]

The letters KK show that four 1 s-electrons (two bonding and two loosening) have practically no effect on the chemical bond.

Since three hydrogen atoms have only three 1 s-orbitals, then the total number of formed molecular orbitals will be equal to six (three bonding and three loosening). Two electrons of the nitrogen atom will be in a non-bonding molecular orbital (lone electron pair).

The method of molecular orbitals (MO) is currently considered to be the best method for the quantum mechanical interpretation of a chemical bond. However, it is much more complicated than the VS method and is not as clear as the latter.

The existence of bonding and loosening MOs is confirmed by the physical properties of the molecules. The MO method makes it possible to foresee that if, during the formation of a molecule from atoms, the electrons in the molecule fall into bonding orbitals, then the ionization potentials of the molecules must be greater than the ionization potentials of atoms, and if the electrons fall into loosening orbitals, then vice versa. Thus, the ionization potentials of hydrogen and nitrogen molecules (bonding orbitals), 1485 and 1500 kJ/mol, respectively, are greater than the ionization potentials of hydrogen and nitrogen atoms, 1310 and 1390 kJ/mol, and the ionization potentials of oxygen and fluorine molecules (loosening orbitals) are 1170 and 1523 kJ/mol - less than that of the corresponding atoms - 1310 and 1670 kJ/mol. When molecules are ionized, the bond strength decreases if the electron is removed from the bonding orbital (H 2 and N 2), and increases if the electron is removed from the loosening orbital (O 2 and F 2).

Communication polarity

Between different atoms, a pure covalent bond can occur if the electronegativity (EO) of the atoms is the same. Such molecules are electrosymmetric, i.e. The "centers of gravity" of the positive charges of the nuclei and the negative charges of the electrons coincide at one point, therefore they are called non-polar.

If the connecting atoms have different EC, then the electron cloud located between them shifts from a symmetrical position closer to the atom with a higher EC:

The displacement of the electron cloud is called polarization. As a result of one-sided polarization, the centers of gravity of positive and negative charges in the molecule do not coincide at one point, a certain distance (l) appears between them. Such molecules are called polar or dipoles, and the bond between the atoms in them is called polar. For example, in the HCl molecule, the binding electron cloud is shifted towards the more electronegative chlorine atom. Thus, the hydrogen atom in hydrogen chloride is positively polarized, while the chlorine atom is negatively polarized.

A positive charge appears on the hydrogen atom δ= +0.18, and on the chlorine atom - a negative charge δ=-018. hence the bond in the hydrogen chloride molecule is 18% ionic.

A polar bond is a kind of covalent bond that has undergone a slight one-sided polarization. The distance between the "centers of gravity" of positive and negative charges in a molecule is called the dipole length. Naturally, the greater the polarization, the greater the length of the dipole and the greater the polarity of the molecules. To assess the polarity of molecules, a constant dipole moment µ is usually used, which is the product of the value of the elementary electric charge q and the length of the dipole (l), i.e. µ =q∙l. Dipole moments are measured in coulometers.

table 2 Electric moment of the dipole µ of some molecules

The total dipole moment of a complex molecule can be considered equal to the vector sum of the dipole moments of individual bonds. The dipole moment is usually considered to be directed from the positive end of the dipole to the negative. The result of the addition depends on the structure of the molecule. The dipole moent of highly symmetric BeCl 2 ,BF 3 ,CCl 4 molecules is equal to zero, although the Be-Cl,B-F,C-Cl bonds are highly polar. In the corner H 2 O molecule, the polar O-H bonds are located at an angle of 104.5 o. So the molecule is polar

(µ = 0.61∙10 -29 C∙m)

With a very large difference in electronegativity, the atoms have a clear unilateral polarization: the electron cloud of the bond shifts as much as possible towards the atom with the highest electronegativity, the atoms pass into oppositely charged ions, and an ionic molecule appears. The covalent bond becomes ionic. The electrical asymmetry of molecules increases, the length of the dipole increases, and the dipole moment increases.

The polarity of a bond can be predicted using the relative EO of atoms. The greater the difference between the relative EOs of atoms, the more pronounced the polarity. It is more correct to speak about the degree of ionicity of a bond, since bonds are not 100% ionic. Even in the CsF compound, the bond is only 89% ionic.

If we consider compounds of elements of any period with the same element, then as we move from the beginning to the end of the period, the predominantly ionic nature of the bond is replaced by a covalent one. For example, in fluorides of the 2nd period LiF, BeF 2 , CF 4 , NF 3 , OF 2 , F 2 the degree of ionicity of the bond from lithium fluoride gradually weakens and is replaced by a typically covalent bond in the fluorine molecule.

The electronegativity of sulfur is much less than the EO of oxygen. Therefore, the polarity of the H–S bond in H 2 S is less than the polarity of the H–O bond in H 2 O, and the length of the H–S bond (0.133 nm) is greater than H–O (0.56 nm) and the angle between the bonds approaches a straight line . For H 2 S it is 92 o, and for H 2 Se it is 91 o.

For the same reasons, the ammonia molecule has a pyramidal structure and the angle between the H–N–H valence bonds is greater than a straight one (107.3 o). In the transition from NH 3 to PH 3 , AsH 3 and SbH 3 the angles between the bonds are respectively 93.3 about; 91.8 o and 91.3 o.

Molecular orbital method based on the assumption that electrons in a molecule are located in molecular orbitals, similar to atomic orbitals in an isolated atom. Each molecular orbital corresponds to a certain set of molecular quantum numbers. For molecular orbitals, the Pauli principle remains valid, i.e. Each molecular orbital can contain no more than two electrons with antiparallel spins.

In the general case, in a polyatomic molecule, the electron cloud belongs simultaneously to all atoms, i.e. participates in the formation of a multicenter chemical bond. Thus, all electrons in a molecule belong simultaneously to the whole molecule, and are not the property of two bonded atoms. Hence, the molecule is viewed as a whole, and not as a collection of individual atoms.

In a molecule, as in any system of nuclei and electrons, the state of an electron in molecular orbitals must be described by the corresponding wave function. In the most common version of the molecular orbital method, the wave functions of electrons are found by representing molecular orbital as a linear combination of atomic orbitals(the variant itself received the abbreviated name "MOLCAO").

In the MOLCAO method, it is assumed that the wave function y , corresponding to the molecular orbital, can be represented as a sum:

y = c 1 y 1 + c 2 y 2 + ¼ + c n y n

where y i are wave functions characterizing the orbitals of interacting atoms;

c i are numerical coefficients, the introduction of which is necessary because the contribution of different atomic orbitals to the total molecular orbital can be different.

Since the square of the wave function reflects the probability of finding an electron at some point in space between interacting atoms, it is of interest to find out what form the molecular wave function should have. The easiest way to solve this problem is in the case of a combination of wave functions of 1s-orbitals of two identical atoms:

y = c 1 y 1 + c 2 y 2

Since for identical atoms with 1 \u003d c 2 \u003d c, one should consider the sum

y = c 1 (y 1 + y 2)

Constant With affects only the value of the amplitude of the function, therefore, to find the shape of the orbital, it is enough to find out what the sum will be y 1 And y2 .

By placing the nuclei of two interacting atoms at a distance, equal to the length connection, and having depicted the wave functions of 1s-orbitals, we will add them. It turns out that, depending on the signs of the wave functions, their addition gives different results. In the case of adding functions with the same signs (Fig. 4.15, a), the values y in the internuclear space is greater than the values y 1 And y2 . In the opposite case (Fig. 4.15, b), the total molecular orbital is characterized by a decrease in the absolute value of the wave function in the internuclear space compared to the wave functions of the original atoms.

y2

y 1

Rice. 4.15. Scheme of addition of atomic orbitals during formation

binding (a) and loosening (b) MO

Since the square of the wave function characterizes the probability of finding an electron in the corresponding region of space, i.e. the density of the electron cloud, which means that in the first version of the addition of wave functions, the density of the electron cloud in the internuclear space increases, and in the second it decreases.

Thus, the addition of wave functions with the same signs leads to the appearance of attractive forces of positively charged nuclei to the negatively charged internuclear region and the formation of a chemical bond. This molecular orbital is called binding , and the electrons located on it - bonding electrons .

In the case of the addition of wave functions of different signs, the attraction of each nucleus in the direction of the internuclear region weakens, and repulsive forces prevail - the chemical bond is not strengthened, and the resulting molecular orbital is called loosening (electrons located on it - loosening electrons ).

Similar to atomic s-, p-, d-, f-orbitals, MO denote s- , p- , d- , j orbitals . Molecular orbitals arising from the interaction of two 1s-orbitals denote: s-linking And s (with an asterisk) - loosening . When two atomic orbitals interact, two molecular orbitals are always formed - a bonding and a loosening.

The transition of an electron from the atomic 1s-orbital to the s-orbital, leading to the formation of a chemical bond, is accompanied by the release of energy. The transition of an electron from the 1s orbital to the s orbital requires energy. Consequently, the energy of the s-bonding orbital is lower, and the s-opening orbital is higher than the energy of the original atomic 1s-orbitals, which is usually depicted in the form of corresponding diagrams (Fig. 4.16).

JSC MO JSC

Rice. 4.16. Energy diagram of the formation of the MO of the hydrogen molecule

Along with the energy diagrams of the formation of molecular orbitals, it is interesting appearance molecular clouds obtained by overlapping or repulsing the orbitals of interacting atoms.

Here it should be taken into account that not any orbitals can interact, but only those satisfying certain requirements.

1. The energies of the initial atomic orbitals should not differ greatly from each other - they should be comparable in magnitude.

2. Atomic orbitals must have the same symmetry properties about the axis of the molecule.

The last requirement leads to the fact that they can combine with each other, for example, s - s (Fig. 4.17, a), s - p x (Fig. 4.17, b), p x - p x, but they cannot s - p y, s - p z (Fig. 4.17, c), because in the first three cases, both orbitals do not change when rotating around the internuclear axis (Fig. 3.17 a, b), and in the last cases they change sign (Fig. 4.17, c). This leads, in the latter cases, to the mutual subtraction of the formed areas of overlap, and it does not occur.

3. Electron clouds of interacting atoms should overlap as much as possible. This means, for example, that it is impossible to combine p x – p y , p x – p z or p y – p z orbitals that do not have overlapping regions.

(a B C)

Rice. 4.17. Influence of the symmetry of atomic orbitals on the possibility

formation of molecular orbitals: MOs are formed (a, b),

not formed (in)

In the case of the interaction of two s-orbitals, the resulting s- and s-orbitals look like this (Fig. 3.18)

1s

s 1

1s

+

Rice. 4.18. Scheme for combining two 1s orbitals

The interaction of two p x -orbitals also gives an s-bond, because the resulting bond is directed along a straight line connecting the centers of atoms. The emerging molecular orbitals are designated respectively s and s, the scheme of their formation is shown in fig. 4.19.

Rice. 4.19. Scheme for combining two p x orbitals

With a combination of p y - p y or p z - p z -orbitals (Fig. 4.20), s-orbitals cannot be formed, because the regions of possible overlapping orbitals are not located on a straight line connecting the centers of atoms. In these cases, degenerate p y - and p z -, as well as p - and p - orbitals are formed (the term "degenerate" means in this case "the same in shape and energy").

Rice. 4.20. Scheme for combining two p z orbitals

When calculating the molecular orbitals of polyatomic systems, in addition, there may appear energy levels midway between bonding and loosening molecular orbitals. Such mo called non-binding .

As in atoms, electrons in molecules tend to occupy molecular orbitals corresponding to the minimum energy. So, in a hydrogen molecule, both electrons will transfer from the 1s orbital to the bonding s 1 s orbital (Fig. 4.14), which can be represented by the formula:

Like atomic orbitals, molecular orbitals can hold at most two electrons.

The MO LCAO method does not operate with the concept of valency, but introduces the term "order" or "link multiplicity".

Communication order (P)is equal to the quotient of dividing the difference between the number of bonding and loosening electrons by the number of interacting atoms, i.e. in the case of diatomic molecules, half of this difference. The bond order can take integer and fractional values, including zero (if the bond order is zero, the system is unstable and no chemical bond occurs).

Therefore, from the standpoint of the MO method, the chemical bond in the H 2 molecule, formed by two bonding electrons, should be considered as a single bond, which also corresponds to the method of valence bonds.

It is clear, from the point of view of the MO method, and the existence of a stable molecular ion H . In this case, the only electron passes from the atomic 1s orbital to the molecular s 1 S orbital, which is accompanied by the release of energy and the formation of a chemical bond with a multiplicity of 0.5.

In the case of molecular ions H and He (containing three electrons), the third electron is already placed on the antibonding s-orbital (for example, He (s 1 S) 2 (s) 1), and the bond order in such ions, according to the definition, is 0.5. Such ions exist, but the bond in them is weaker than in the hydrogen molecule.

Since there should be 4 electrons in a hypothetical He 2 molecule, they can only be located 2 in s 1 S - bonding and s - loosening orbitals, i.e. the bond order is zero, and diatomic molecules of helium, like other noble gases, do not exist. Similarly, Be 2 , Ca 2 , Mg 2 , Ba 2 etc. molecules cannot be formed.

Thus, from the point of view of the molecular orbital method, two interacting atomic orbitals form two molecular orbitals: bonding and loosening. For AO with principal quantum numbers 1 and 2, the formation of MOs presented in Table 1 is possible. 4.4.

As shown in the previous paragraphs, the VS method makes it possible to understand the ability of atoms to form a certain number of covalent bonds, explains the direction of the covalent bond, and gives a satisfactory description of the structure and properties a large number molecules. However, in a number of cases the VS method cannot explain the nature of the formed chemical bonds or leads to incorrect conclusions about the properties of molecules.

Thus, according to the VS method, all covalent bonds carried out by a common pair of electrons. Meanwhile, at the end of the last century, the existence of a fairly strong molecular hydrogen ion was established: the bond breaking energy is here. However, no electron pair can be formed in this case, since only one electron is included in the composition of the ion. Thus, the VS method does not provide a satisfactory explanation for the existence of the ion.

According to this description, the molecule contains no unpaired electrons. However, the magnetic properties of oxygen indicate that there are two unpaired electrons in the molecule.

Each electron, due to its spin, creates its own magnetic field. The direction of this field is determined by the direction of the spin, so that the magnetic fields formed by the two paired electrons cancel each other out.

Therefore, molecules containing only paired electrons do not create their own magnetic field. Substances consisting of such molecules are diamagnetic - they are pushed out of the magnetic field. On the contrary, substances whose molecules contain unpaired electrons have their own magnetic field and are paramagnetic; such substances are drawn into a magnetic field.

Oxygen is a paramagnetic substance, which indicates the presence of unpaired electrons in its molecule.

On the basis of the VS method, it is also difficult to explain that the detachment of electrons from certain molecules leads to the strengthening of the chemical bond. So, the bond breaking energy in a molecule is , and in a molecular ion - ; the analogous values ​​for molecules and molecular ions are 494 and , respectively.

The facts presented here and many other facts receive a more satisfactory explanation on the basis of the molecular orbital method (MO method).

We already know that the state of electrons in an atom is described by quantum mechanics as a set of atomic electron orbitals (atomic electron clouds); each such orbital is characterized by a certain set of atomic quantum numbers. The MO method proceeds from the assumption that the state of electrons in a molecule can also be described as a set of molecular electron orbitals (molecular electron clouds), with each molecular orbital (MO) corresponding to a certain set of molecular quantum numbers. As in any other many-electron system, the Pauli principle remains valid in a molecule (see § 32), so that each MO can have no more than two electrons, which must have oppositely directed spins.

A molecular electron cloud can be concentrated near one of the atomic nuclei that make up the molecule: such an electron practically belongs to one atom and does not take part in the formation of chemical bonds. In other cases, the predominant part of the electron cloud is located in a region of space close to two atomic nuclei; this corresponds to the formation of a two-center chemical bond. However, in the most general case, the electron cloud belongs to several atomic nuclei and participates in the formation of a multicenter chemical bond. Thus, from the point of view of the MO method, a two-center bond is only a special case of a multicenter chemical bond.

The main problem of the MO method is finding the wave functions that describe the state of electrons in molecular orbitals. In the most common version of this method, which has received the abbreviated designation "MO LCAO method" (molecular orbitals, linear combination of atomic orbitals), this problem is solved as follows.

Let electron orbitals interacting atoms are characterized by wave functions, etc. Then it is assumed that the wave function corresponding to the molecular orbital can be represented as the sum

where are some numerical coefficients.

To clarify the physical meaning of this approach, let us recall that the wave function corresponds to the amplitude of the wave process characterizing the state of the electron (see § 26). As you know, when interacting, for example, sound or electromagnetic waves, their amplitudes add up. As can be seen, the above equation is equivalent to the assumption that the amplitudes of the molecular "electron wave" (i.e., the molecular wave function) are also formed by adding the amplitudes of the interacting atomic "electron waves" (i.e., adding the atomic wave functions). In this case, however, under the influence of the force fields of the nuclei and electrons of neighboring atoms, the wave function of each atomic electron changes in comparison with the initial wave function of this electron in an isolated atom. In the MO LCAO method, these changes are taken into account by introducing coefficients, etc., so that when the molecular wave function is found, not the original, but the changed amplitudes are added, etc.

Let us find out what form the molecular wave function will have, formed as a result of the interaction of the wave functions ( and ) -orbitals of two identical atoms. To do this, we find the sum. In this case, both considered atoms are the same, so that the coefficients and are equal in value, and the problem is reduced to determining the sum. Since the constant coefficient C does not affect the form of the desired molecular wave function, but only changes its absolute values, we will restrict ourselves to finding the sum .

To do this, we place the nuclei of the interacting atoms at the distance from each other (r) at which they are in the molecule, and depict the wave functions of the orbitals of these atoms (Fig. 43, a); Each of these functions has the form shown in Fig. 9, a (p. 76). To find the molecular wave function , we add the quantities and : the result is the curve shown in Fig. 43b. As can be seen, in the space between the nuclei, the values ​​of the molecular wave function are greater than the values ​​of the initial atomic wave functions. But the square of the wave function characterizes the probability of finding an electron in the corresponding region of space, i.e., the density of the electron cloud (see § 26). This means that an increase in comparison with and means that during the formation of the MO, the density of the electron cloud in the internuclear space increases.

Rice. 43. Scheme of the formation of a binding MO from atomic -orbitals.

As a result, forces of attraction of positively charged atomic nuclei to this region arise - a chemical bond is formed. Therefore, the MO of the type under consideration is called binding.

In this case, the region of increased electron density is located near the bond axis, so that the formed MO is of the -type. In accordance with this, the binding MO, obtained as a result of the interaction of two atomic orbitals, is denoted .

The electrons on the bonding MO are called bonding electrons.

As indicated on page 76, the wave function of the -orbital has a constant sign. For a single atom, the choice of this sign is arbitrary: up to now we have considered it positive. But when two atoms interact, the signs of the wave functions of their -orbitals may turn out to be different. So, apart from the case shown in Fig. 43a, where the signs of both wave functions are the same, the case is also possible when the signs of the wave functions of the interacting -orbitals are different. Such a case is shown in Fig. 44a: here the wave function of the -orbitals of one atom is positive, and the other is negative. When these wave functions are added together, the curve shown in Fig. 44b. The molecular orbital formed during such an interaction is characterized by a decrease in the absolute value of the wave function in the internuclear space compared to its value in the original atoms: even a point appears on the bond axis at which the value of the wave function, and, consequently, its square, vanishes . This means that in the case under consideration, the density of the electron cloud in the space between the atoms will also decrease.

Rice. 44. Scheme of the formation of a loosening MO from atomic -orbitals.

As a result, the attraction of each atomic nucleus in the direction of the internuclear region of space will be weaker than in the opposite direction, i.e., forces will arise that lead to the mutual repulsion of the nuclei. Here, therefore, no chemical bond arises; the MO formed in this case is called loosening, and the electrons on it are called loosening electrons.

The transition of electrons from atomic orbitals to the bonding MO, leading to the formation of a chemical bond, is accompanied by the release of energy. On the contrary, the transition of electrons from atomic -orbitals to the antibonding MO requires the expenditure of energy. Consequently, the energy of electrons in the orbital is lower, and in the orbital is higher than in the atomic -orbitals. This ratio of energies is shown in Fig. 45, which shows both the initial -orbitals of two hydrogen atoms, and molecular orbitals and immediately. Approximately, it can be considered that during the transition of an -electron to a bonding MO, the same amount of energy is released as it is necessary to spend to transfer it to a loosening MO.

We know that in the most stable (unexcited) state of an atom, electrons occupy atomic orbitals characterized by the lowest possible energy. Similarly, the most stable state of the molecule is achieved when the electrons occupy the MO corresponding to the minimum energy. Therefore, when a hydrogen molecule is formed, both electrons will transfer from atomic orbitals to a bonding molecular orbital (Fig. 46); According to the Pauli principle, electrons in the same MO must have oppositely directed spins.

Rice. 45. Energy scheme for the formation of MO during the interaction of -orbitals of two identical atoms.

Rice. 46. ​​Energy scheme for the formation of a hydrogen molecule.

Using symbols expressing the placement of electrons in atomic and molecular orbitals, the formation of a hydrogen molecule can be represented by the scheme:

In the VS method, the bond multiplicity is determined by the number of common electron pairs: a simple bond is considered to be formed by one common electron pair, a double bond is a bond formed by two common electron pairs, etc. Similarly, in the MO method, the bond multiplicity is usually determined by the number of bonding electrons involved in its formation: two bonding electrons correspond to a single bond, four bonding electrons to a double bond, etc. In this case, the loosening electrons compensate for the action of the corresponding number of bonding electrons. So, if there are 6 binding and 2 loosening electrons in a molecule, then the excess of the number of binding electrons over the number of loosening electrons is four, which corresponds to the formation of a double bond. Therefore, from the standpoint of the MO method, a chemical bond in a hydrogen molecule formed by two bonding electrons should be considered as a simple bond.

Now it becomes clear the possibility of the existence of a stable molecular ion in its formation, the only electron passes from the atomic orbital to the bonding orbital, which is accompanied by the release of energy (Fig. 47) and can be expressed by the scheme:

A molecular ion (Fig. 48) has only three electrons. According to the Pauli principle, only two electrons can be placed on the bonding molecular orbital, therefore the third electron occupies the loosening orbital.

Rice. 47. Energy scheme for the formation of a molecular hydrogen ion.

Rice. 48. Energy scheme for the formation of the helium molecular ion.

Rice. 49. Energy scheme for the formation of a lithium molecule.

Rice. 50. Energy scheme for the formation of MO during the interaction of -orbitals of two identical atoms.

Thus, the number of bonding electrons here per unit more number loosening. Therefore, the ion must be energetically stable. Indeed, the existence of an ion has been experimentally confirmed and it has been established that energy is released during its formation;

On the contrary, a hypothetical molecule should be energetically unstable, since here, out of the four electrons that should be placed on the MO, two will occupy the bonding MO, and two - the loosening MO. Therefore, the formation of a molecule will not be accompanied by the release of energy. Indeed, the molecules have not been experimentally detected.

In molecules of elements of the second period, MOs are formed as a result of the interaction of atomic and -orbitals; the participation of internal -electrons in the formation of a chemical bond is negligible here. So, in fig. 49 shows the energy diagram of the formation of a molecule: there are two bonding electrons here, which corresponds to the formation of a simple bond. In a molecule, however, the number of binding and loosening electrons is the same, so this molecule, like the molecule, is energetically unstable. Indeed, the molecules could not be detected.

The scheme of MO formation during the interaction of atomic -orbitals is shown in fig. 50. As you can see, six MOs are formed from the six initial -orbitals: three binding and three loosening. In this case, one bonding () and one loosening orbitals belong to the -type: they are formed by the interaction of atomic -orbitals oriented along the bond. Two bonding and two loosening () orbitals are formed by the interaction of -orbitals oriented perpendicular to the bond axis; these orbitals belong to the -type.