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 a covalent bond, and gives a satisfactory description of the structure and properties of a large number of 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 are 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 the electron orbitals of the interacting atoms be 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 is one greater than the number of loosening ones. 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, the number of bonding 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.
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 of 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 , the chemical bond in which is formed by two electrons (type 1 s) belonging to hydrogen atoms. If one electron is removed, then we get an even simpler system H 2 + - 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 problem cannot be solved exactly in quantum mechanics, so we will consider its approximate solution by the MO method. This will allow us to get acquainted with the most characteristic features of the 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 a linear system of equations with 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 one with higher 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 px,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 px–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 determines the paramagnetic properties of the oxygen molecule, which corresponds to the 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 closeness of the chemical and physical properties of 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, it is necessary to introduce a fractional number equal to 2.5 to characterize it. 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. The chemical bond in it is carried out due to three unpaired px, 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 px 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 m L is a quantum number that takes the values m L= 0, ±1, ±2, etc. In this case, the energy of the electronic state depends only on the absolute value m L, 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 m Lê = 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.
1. As a result of a linear combination, two atomic orbitals (AO) form two molecular orbitals (MO) - a bonding one, the energy of which is lower than the energy of AO, and a loosening one, the energy of which is higher than the energy of AO
2. Electrons in a molecule are located in molecular orbitals in accordance with the Pauli principle and Hund's rule.
3. The negative contribution to the chemical bonding energy of an electron located in an antibonding orbital is greater than the positive contribution to this energy of an electron in a bonding MO.
4. The multiplicity of bonds in a molecule is equal to the difference by two of the number of electrons located on the binding and loosening MOs.
5. With an increase in the multiplicity of bonds in molecules of the same type, its binding energy increases and its length decreases.
If, during the formation of a molecule from atoms, an electron occupies a bonding MO, then the total energy of the system will decrease, i.e. a chemical bond is formed. When an electron passes to a loosening MO, the energy of the system will increase, the system will become less stable (Fig. 9.1).
Rice. 9.1. Energy diagram of the formation of molecular orbitals from two atomic orbitals
Molecular orbitals formed from s-atomic orbitals are denoted s s . If the MOs are formed by p z -atomic orbitals - they are denoted by s z . Molecular orbitals formed by p x - and p y -atomic orbitals, denoted by p x and p y respectively.
When filling molecular orbitals with electrons, one should be guided by the following principles:
1. Each MO corresponds to a certain energy. Molecular orbitals are filled in order of increasing energy.
2. No more than two electrons with opposite spins can be in one molecular orbital.
3. The filling of molecular quantum cells occurs in accordance with the Hund rule.
An experimental study (study of molecular spectra) showed that the energy molecular orbitals increases in the following sequence:
s 1s< s
*1s < s
2s Asterisk ( *
) in this row, antibonding molecular orbitals are marked. For B, C, and N atoms, the energies of the 2s- and 2p-electrons are close and the transition of the 2s-electron to the molecular orbital s 2p z requires energy. Therefore, for molecules B 2 , C 2 , N 2 orbital energy s 2pz becomes higher than the energy of the p 2p x and p 2p y orbitals: s 1s< s
*1s < s 2s< s
*2s < p
2р х
= p
2р у
< s
2p z < p
*2р х
= p
*2р у
< s
*2p z. When a molecule is formed, electrons are placed in orbitals with lower energy. When constructing an MO, one usually confines oneself to using valence AO(orbitals of the outer layer), since they make the main contribution to the formation of a chemical bond. Electronic structure of homonuclear diatomic molecules and ions The process of formation of the particle H 2 + H + H + H 2 +. Thus, one electron is located on the bonding molecular s-orbital. The multiplicity of the bond is equal to the half-difference of the number of electrons in the bonding and loosening orbitals. Hence, the multiplicity of the bond in the particle H 2 + equal to (1 - 0): 2 = 0.5. The VS method, in contrast to the MO method, does not explain the possibility of bond formation by one electron. The hydrogen molecule has the following electronic configuration: H 2 [(s 1s) 2]. In the H 2 molecule there are two bonding electrons, which means that the bond in the molecule is single. Molecular ion H 2 - has an electronic configuration: H 2 - [(s 1s) 2 (s *1s) 1]. Multiplicity of bonds in H 2 - is (2 - 1): 2 = 0.5. Let us now consider homonuclear molecules and ions of the second period. The electronic configuration of the Li 2 molecule is as follows: 2Li (K2s) Li 2 . Li 2 molecule contains two bonding electrons, which corresponds to a single bond. The process of formation of the Be 2 molecule can be represented as follows: 2 Be (K2s 2) Be 2 . The number of binding and loosening electrons in the Be 2 molecule the same way, and since one loosening electron destroys the action of one bonding electron, the Be molecule 2 not found in the ground state. In a nitrogen molecule, 10 valence electrons are located in orbitals. The electronic structure of the N 2 molecule: N2. Since in the molecule N 2 eight bonding and two loosening electrons, then this molecule has a triple bond. The nitrogen molecule is diamagnetic because it does not contain unpaired electrons. On the orbitals of the O 2 molecule 12 valence electrons are distributed, therefore, this molecule has a configuration: O 2 . Rice. 9.2. Scheme of the formation of molecular orbitals in the O 2 molecule (only 2p electrons of oxygen atoms are shown) In the O 2 molecule , according to Hund's rule, two electrons with parallel spins are placed one at a time in two orbitals with the same energy (Fig. 9.2). According to the VS method, the oxygen molecule does not have unpaired electrons and should have diamagnetic properties, which is inconsistent with the experimental data. The molecular orbital method confirms the paramagnetic properties of oxygen, which are due to the presence of two unpaired electrons in the oxygen molecule. The multiplicity of bonds in an oxygen molecule is (8–4):2 = 2. Let us consider the electronic structure of the O 2 + and O 2 ions - . In the O 2 + ion 11 electrons are placed in its orbitals, therefore, the configuration of the ion is as follows: O2+ O 2 + . The multiplicity of bonds in the O 2 + ion equals (8–3):2 = 2.5. In the O 2 ion - 13 electrons are distributed in its orbitals. This ion has the following structure: O2- O 2 - . The multiplicity of bonds in the ion O 2 - equal to (8 - 5): 2 = 1.5. Ions O 2 - and O 2 + They are paramagnetic because they contain unpaired electrons. The electronic configuration of the F 2 molecule has the form: F2. The multiplicity of bonds in the molecule F 2 is equal to 1, since there is an excess of two bonding electrons. Since there are no unpaired electrons in the molecule, it is diamagnetic. In the series N 2, O 2, F 2 energies and bond lengths in molecules are: An increase in the excess of binding electrons leads to an increase in the binding energy (bond strength). In the transition from N 2 to F 2 the bond length increases, due to the weakening of the bond. In the series O 2 -, O 2, O 2 + the bond multiplicity increases, the bond energy also increases, and the bond length decreases. Electronic structure of heteronuclear molecules and ions Isoelectronic According to the MO method, the electronic structure of the CO molecule is similar to the structure of the N 2 molecule: The orbitals of a CO molecule contain 10 electrons (4 valence electrons of the carbon atom and 6 valence electrons of the oxygen atom). In the CO molecule, as in the N 2 molecule , triple bond. Similarities in the electronic structure of N molecules 2 and CO determines the closeness of the physical properties of these substances. In the NO molecule, 11 electrons are distributed in orbitals (5 electrons of the nitrogen atom and 6 electrons of the oxygen atom), therefore, the electronic configuration of the molecule is as follows: NO or The bond multiplicity in the NO molecule is (8–3):2 = 2.5. The configuration of molecular orbitals in the NO - ion: NO- The bond multiplicity in this molecule is (8–4):2 = 2. NO+ ion has the following electronic structure: NO + . The excess of binding electrons in this particle is 6, therefore, the bond multiplicity in the NO + ion is three. In the series NO - , NO, NO + the excess of binding electrons increases, which leads to an increase in the strength of the bond and a decrease in its length. Tasks for independent solution 9.1.Using the MO method, set the order of decreasing chemical bond energy in particles: 9.3.Based on the MO method, determine which of the listed particles do not exist: 9.4.Distribute electrons in molecular orbitals for the B 2 molecule. Determine the multiplicity of the connection. 9.5.Distribute electrons in molecular orbitals for the N 2 molecule. Determine the multiplicity of the connection. N 2 ; 9.9. Distribute electrons in molecular orbitals for the CN ion 9.10.Using the MO method, determine how the bond length and bond energy change in the series CN + , CN, CN - . © Faculty of Natural Sciences of the Russian Chemical Technical University named after. DI. Mendeleev. 2013 We already know that electrons in atoms are in allowed energy states - atomic orbitals (AO). Similarly, electrons in molecules exist in allowed energy states − molecular orbitals (MO). molecular orbital much more complicated than the atomic orbital. Here are a few rules that will guide us when building MO from AO: We introduce the concept communication order. In diatomic molecules, the bond order indicates how much the number of bonding electron pairs exceeds the number of antibonding electron pairs: Now let's look at an example of how these rules can be applied. Let's start with formation of a hydrogen molecule from two hydrogen atoms. As a result of interaction 1s orbitals each of the hydrogen atoms form two molecular orbitals. During the interaction, when the electron density is concentrated in the space between the nuclei, a bonding sigma - orbital(σ). This combination has a lower energy than the original atoms. In the interaction, when the electron density is concentrated in the outside of the internuclear region, a antibonding sigma - orbital(σ*). This combination has a higher energy than the original atoms. Electrons, according to Pauli principle, occupy first the orbital with the lowest energy σ-orbital. Now consider formation of the He 2 molecule, when two helium atoms approach each other. In this case, the interaction of 1s-orbitals also occurs and the formation of σ * -orbitals, while two electrons occupy the bonding orbital, and the other two electrons occupy the loosening orbital. The Σ * -orbital is destabilized to the same extent as the σ -orbital is stabilized, so two electrons occupying the σ * -orbital destabilize the He 2 molecule. Indeed, it has been experimentally proven that the He 2 molecule is very unstable. Next, consider formation of the Li 2 molecule, taking into account that the 1s and 2s orbitals differ too much in energy and therefore there is no strong interaction between them. The energy level diagram of the Li 2 molecule is shown below, where the electrons in the 1s-bonding and 1s-antibonding orbitals do not contribute significantly to bonding. Therefore, the formation of a chemical bond in the Li 2 molecule is responsible 2s electrons. This action extends to the formation of other molecules in which the filled atomic subshells (s, p, d) do not contribute to chemical bond. Thus, only valence electrons
. As a result, for alkali metals, the molecular orbital diagram will have a form similar to the diagram of the Li 2 molecule considered by us. Communication order n in the Li 2 molecule is 1 Let us consider how two identical atoms of the second period interact with each other, having a set of s- and p-orbitals. It should be expected that 2s orbitals will only connect with each other, and 2p orbitals will only connect with a 2p orbitals. Because 2p orbitals can interact with each other in two different ways, they form σ and π molecular orbitals. Using the summary diagram below, you can set electronic configurations of diatomic molecules of the second period
which are given in the table. Thus, the formation of a molecule, for example, fluorine F 2 of atoms in the notation molecular orbital theory can be written like this: 2F =F 2 [(σ 1s) 2 (σ * 1s) 2 (σ 2s) 2 (σ * 2 s) 2 (σ 2px) 2 (π 2py) 2 (π 2pz) 2 (π * 2py) 2 ( π * 2pz) 2 ]. Because Since the overlap of 1s clouds is negligible, the participation of electrons in these orbitals can be neglected. Then the electronic configuration of the fluorine molecule will be: F2, where K is the electronic configuration of the K-layer. Doctrine of MO allows you to explain and education diatomic heteronuclear molecules. If the atoms in the molecule are not too different from each other (for example, NO, CO, CN), then you can use the diagram above for elements of the 2nd period. With significant differences between the atoms that make up the molecule, the diagram changes. Consider HF molecule, in which the atoms differ greatly in electronegativity. The energy of the 1s-orbital of the hydrogen atom is higher than the energy of the highest of the valence orbitals of fluorine, the 2p-orbital. The interaction of the 1s-orbital of the hydrogen atom and the 2p-orbital of fluorine leads to the formation bonding and antibonding orbitals, as it shown on the picture. A pair of electrons located in the bonding orbital of the HF molecule form polar covalent bond. For the bonding orbital HF molecules The 2p orbital of the fluorine atom plays a more important role than the 1s orbital of the hydrogen atom. For an antibonding orbital HF molecules vice versa: the 1s orbital of the hydrogen atom plays a more important role than the 2p orbital of the fluorine atom 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, energy is released in 255 kJ (61 kcal). 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. The molecular orbital method proceeds from the fact that each molecular orbital is represented as an algebraic sum (linear combination) of atomic orbitals. For example, in a hydrogen molecule, only 1s atomic orbitals of two hydrogen atoms can participate in the formation of MO, which give two MO, which is the sum and difference of atomic orbitals 1s 1 and 1s 2 - MO ± = C 1 1s 1 ±C 2 1s 2. The electron density of these two states is proportional to |MO ± | 2 . Since interaction in a hydrogen molecule is possible only along the axis of the molecule, each of MO ± can be redesignated as σb = 1s 1 + 1s 2 and σ* = 1s 1 – 1s 2 and named, respectively, bonding (σb) and loosening (σ* ) molecular orbitals. From fig. 10 it can be seen that the electron density in the middle between the nuclei for σ is significant, and for σ* it is equal to zero. A negatively charged electron cloud concentrated in the internuclear space attracts positively charged nuclei and corresponds to the bonding molecular orbital σ St. A MO with zero density in the internuclear space corresponds to the loosening orbital σ*. The states σst and σ* correspond to different energy levels, and the molecular orbital σst has a lower energy compared to the initial AO of two noninteracting hydrogen atoms 1s 1 and 1s 2 . The transition of two electrons to MO σ st contributes to a decrease in the energy of the system; this energy gain is equal to the binding energy between atoms in the H–H hydrogen molecule. Even the removal of one electron from MO (σ st) 2 with the formation of (σ st) 1 in the molecular ion H 2 + leaves this system more stable than separately existing H atom and H + ion. Let us apply the MO LCAO method and analyze the structure of the hydrogen molecule. Let us depict on two parallel diagrams the energy levels of the atomic orbitals of the initial hydrogen atoms It can be seen (see Figs. 11 and 12) that there is a gain in energy compared to unbound atoms. Both electrons lowered their energy, which corresponds to the unit of valence in the method of valence bonds (a bond is formed by a pair of electrons). In the H 2 - anion, three electrons must already be placed in two molecular orbitals. If two electrons, having descended to the bonding orbital, give a gain in energy, then the third electron has to increase its energy. However, the energy gained by two electrons is greater than that lost by one. Such a particle may exist. Overlapping identical 1 s-orbitals of lithium atoms will give two molecular orbitals (bonding and loosening), which, according to the principle of minimum energy, will be completely populated by four electrons. The gain in energy resulting from the transition of two electrons to the bonding molecular orbital is not able to compensate for its losses during the transition of two other electrons to the antibonding molecular orbital. That is why only the electrons of the outer (valence) electron layer contribute to the formation of a chemical bond between lithium atoms. 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. 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 2 Energy and bond order in molecules of elements of period 1 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 17 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.17 Energy diagram for the formation of diatomic molecules elements 2 periods It should be noted that during the formation of B 2 , C 2 and N 2 molecules, the energy of the binding s 2 px-orbitals more energy binding p 2 py- and p 2 pz-orbitals, while in O 2 and F 2 molecules, on the contrary, the energy of the binding p 2 py- and p 2 pz-orbitals more energy binding s 2 px-orbitals. This must be taken into account when depicting the energy schemes of the corresponding molecules. Like electronic formulas showing the distribution of electrons in an atom along atomic orbitals, the MO method composes formulas of molecules that reflect their electronic configuration. By analogy with atomic s-,
p-,
d-,
f-
molecular orbitals are denoted by the Greek letters s, p, d, j. The formation of molecules from atoms of elements of the II period 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 O 2 molecule is paramagnetic 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. 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 in two degenerate (with the same energy) π*-loosening 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. The letters KK show that four 1 s-electrons (two bonding and two loosening) have practically no effect on the chemical bond. Among the diatomic molecules, one of the strongest is the CO molecule. The MO LCAO method makes it easy to explain this fact. The AO energies of the oxygen atom lie below the energies of the corresponding carbon orbitals (1080 kJ/mol), they are located closer to the nucleus. The result of the overlap p-orbitals of the O and C atoms is the formation of two degenerate π-bonding and one σ-bonding orbitals. These molecular orbitals will occupy six electrons. Therefore, the multiplicity of the bond is three. The electronic configuration is the same as for N 2: With the removal of one electron, which leaves the binding orbit (formation of the CO + ion), the bond strength decreases to 803 kJ/mol. The multiplicity of communication becomes equal to 2.5. The MO LCAO method can be used not only for diatomic molecules, but also for polyatomic ones. As an example, within the framework of this method, we will analyze the structure of the ammonia molecule. 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. So, the ionization potentials of hydrogen molecules and nitrogen (bonding orbitals) - 1485 and 1500 kJ / mol, respectively - more 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) - 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). Consider the formation of MO in a molecule of hydrogen fluoride HF. Since the ionization potential of fluorine (17.4 eV or 1670 kJ/mol) is greater than that of hydrogen (13.6 eV or 1310 kJ/mol), the 2p orbitals of fluorine have less energy than the 1s orbital of hydrogen. Due to the large energy difference, the 1s orbital of the hydrogen atom and the 2s orbital of the fluorine atom do not interact. Thus, the 2s orbital of fluorine becomes without changing the energy of the MO in HF. Such orbitals are called nonbonding. The 2p y and 2p z orbitals of fluorine also cannot interact with the 1s orbital of hydrogen due to the difference in symmetry about the bond axis. They also become nonbonding MOs. The binding and loosening MOs are formed from the 1s orbital of hydrogen and the 2p x orbital of fluorine. Hydrogen and fluorine atoms are linked by a two-electron bond with an energy of 560 kJ/mol.
NF + ; NF-; NF.
He 2 ; He2+; Be 2 ; Be 2 + .
N 2 -.
The multiplicity of bonds in N 2 is (8–2): 2=3;
The multiplicity of bonds in N 2 - is (8–3): 2 = 2.5.
The decrease in binding energy during the transition from the neutral molecule N 2 to the ion N 2 -
associated with a decrease in the multiplicity of communication.
Molecular orbital diagrams of the elements of the first period
MO diagrams of hydrogen and helium molecules 
MO diagram of a lithium molecule Molecular orbital diagrams of the elements of the second period
MO diagrams of diatomic molecules of elements 2 periods Molecular orbitals of polar diatomic molecules
Categories ,
The MO LCAO method makes it possible to visually explain the formation of H 2 + and H 2 - ions (see Figs. 13 and 14), 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.
It is known that alkali metals in the gaseous state exist in the form of diatomic molecules. Let's try to verify the possibility of the existence of a diatomic Li 2 molecule using the MO LCAO method (Fig. 15). The original lithium atom contains electrons at two energy levels - the first and second (1 s and 2 s).
Overlapping valence 2 s-orbitals of lithium atoms will also lead to the formation of one σ-bonding and one antibonding molecular orbitals. The two outer electrons will occupy the bonding orbital, providing an overall gain in energy (the bond multiplicity is 1).
Using the MO LCAO method, let us consider the possibility of the formation of the He 2 molecule.
The filling of molecular orbitals occurs in compliance with the Pauli principle and Hund's rule as their energy increases in the following sequence:
So, the electronic configuration of O 2 molecules is described as follows: O 2 [KK (σ s) 2 (σ s *) 2 (σ z) 2 (π x) 2 (π y) 2 (π x *) 1 (π y *) 1 ]
[KK(σ s) 2 (σ s *) 2 (σ z) 2 (π x) 2 (π y) 2 (σ z) 2 ] . The bond strengths in CO (1021 kJ/mol) and N 2 (941 kJ/mol) molecules are close.
