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 contain 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, we 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 you can see, 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 initial 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 bond formed by one common electron pair is considered simple, a bond formed by two common electron pairs is considered to be a double bond, 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 the 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, 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.
When using the method of molecular orbitals, it is considered, in contrast to the method of valence bonds, that each electron is in the field of all nuclei. In this case, the bond is not necessarily formed by a pair of electrons. For example, the H 2 + ion consists of two protons and one electron. Between two protons there are repulsive forces (Fig. 30), between each of the protons and an electron - forces of attraction. A chemical particle is formed only if the mutual repulsion of protons is compensated by their attraction to the electron. This is possible if the electron is located between the nuclei - in the binding region (Fig. 31). Otherwise, the repulsive forces are not compensated by the forces of attraction - they say that the electron is in the region of antibonding, or loosening.
Two-center molecular orbitals
The molecular orbital method uses the idea of a molecular orbital to describe the electron density distribution in a molecule (similar to the atomic orbital for an atom). Molecular orbitals are the wave functions of an electron in a molecule or other polyatomic chemical particle. Each molecular orbital (MO), like the atomic orbital (AO), can be occupied by one or two electrons. The state of an electron in the binding region is described by the bonding molecular orbital, in the loosening region - by the loosening molecular orbital. The distribution of electrons in molecular orbitals follows the same rules as the distribution of electrons in atomic orbitals in an isolated atom. Molecular orbitals are formed by certain combinations of atomic orbitals. Their number, energy, and shape can be derived from the number, energy, and shape of the orbitals of the atoms that make up the molecule.
In the general case, the wave functions corresponding to molecular orbitals in a diatomic molecule are represented as the sum and difference of the wave functions of atomic orbitals multiplied by some constant coefficients that take into account the proportion of atomic orbitals of each atom in the formation of molecular orbitals (they depend on the electronegativity of atoms):
φ(AB) = s 1 ψ(A) ± s 2 ψ(B)
This method of calculating the one-electron wave function is called "molecular orbitals in the approximation of a linear combination of atomic orbitals" (MO LCAO).
So, when an H 2 + ion or a H 2 hydrogen molecule is formed from two s-orbitals of hydrogen atoms form two molecular orbitals. One of them is binding (it is denoted by σ st), the other is loosening (σ *).
The energies of the bonding orbitals are lower than the energies of the atomic orbitals used to form them. The electrons that populate the bonding molecular orbitals are predominantly located in the space between the bonded atoms, i.e. in the so-called binding region. The energies of the antibonding orbitals are higher than the energies of the initial atomic orbitals. The population of loosening molecular orbitals with electrons contributes to the weakening of the bond: a decrease in its energy and an increase in the distance between atoms in a molecule. The electrons of the hydrogen molecule, which have become common to both bonded atoms, occupy the bonding orbital.
Combination R-orbitals leads to two types of molecular orbitals. Of the two R-orbitals of interacting atoms directed along the bond line, bonding σ St - and loosening σ*-orbitals are formed. Combinations R-orbitals perpendicular to the bond lines give two bonding π- and two loosening π*-orbitals. Using the same rules when populating molecular orbitals with electrons as when filling atomic orbitals in isolated atoms, one can determine the electronic structure of diatomic molecules, for example, O 2 and N 2 (Fig. 35).
From the distribution of electrons in molecular orbitals, the bond order (ω) can be calculated. From the number of electrons located in the bonding orbitals, subtract the number of electrons located in the antibonding orbitals, and the result is divided by 2 n(based on n connections):
ω = / 2 n
It can be seen from the energy diagram that for the H 2 molecule ω = 1.
The molecular orbital method gives the same chemical bond order values as the valence bond method for O 2 (double bond) and N 2 (triple bond) molecules. At the same time, it allows non-integer values of the link order. This is observed, for example, when a two-center bond is formed by one electron (in the H 2 + ion). In this case, ω = 0.5. The magnitude of the bond order directly affects its strength. The higher the bond order, the greater the bond energy and the shorter its length:
Regularities in changes in the order, energy and length of the bond can be traced on the examples of the molecule and molecular ions of oxygen.
The combination of the orbitals of two different atoms with the formation of a molecule is possible only if their energies are close, while the atomic orbitals of an atom of higher electronegativity in the energy diagram are always located lower.
For example, in the formation of a hydrogen fluoride molecule, the combination 1 s-AO of the hydrogen atom and 1 s-AO or 2 s-AO of the fluorine atom, since they differ greatly in energy. Closest in energy 1 s-AO of the hydrogen atom and 2 p-AO of the fluorine atom. The combination of these orbitals causes the appearance of two molecular orbitals: bonding σb and loosening σ*.
Remaining 2 R-orbitals of the fluorine atom cannot be combined with 1 s-AO of the hydrogen atom, since they have different symmetry relative to the internuclear axis. They form non-bonding π 0 -MOs having the same energy as the original 2 R-orbitals of the fluorine atom.
Not participating in LCAO s-orbitals of the fluorine atom form non-bonding σ 0 -MO. Population of nonbonding orbitals by electrons does not promote or prevent bond formation in the molecule. When calculating the link order, their contribution is not taken into account.
Multicenter molecular orbitals
In multicenter molecules, molecular orbitals are multicenter, as they are a linear combination of the orbitals of all the atoms involved in the formation of bonds. In the general case, molecular orbitals are not localized, that is, the electron density corresponding to each orbital is more or less evenly distributed throughout the entire volume of the molecule. However, with the help of mathematical transformations, it is possible to obtain localized molecular orbitals of a certain shape, corresponding to individual two- or three-center bonds or lone electrons.
The simplest example of a three-center bond is the molecular ion H 3 + . Of the three s-orbitals of hydrogen atoms, three molecular orbitals are formed: bonding, non-bonding and loosening. A pair of electrons populates a bonding orbital. The resulting bond is a two-electron three-center bond; the bond order is 0.5.
Chemical particles containing unpaired electrons have paramagnetic properties (in contrast to the diamagnetic properties of chemical particles, in which all electrons are paired). Paramagnets are all substances consisting of chemical particles with an odd number of electrons, such as NO. The method of molecular orbitals makes it possible to identify paramagnets among substances consisting of chemical particles with an even number of electrons, for example, O 2, in the molecule of which two unpaired electrons are located on two loosening π * orbitals.
Chemical species with unpaired electrons in outer orbitals are called free radicals. They are paramagnetic and highly reactive. Inorganic radicals with localized unpaired electrons, for example . H, . NH 2 are usually short lived. They are formed during photolysis, radiolysis, pyrolysis, electrolysis. Low temperatures are used to stabilize them. Short-lived radicals are intermediate species in many reactions.
CHEMICAL BOND
MOLECULAR ORBITAL METHOD.
The molecular orbital (MO) method is the most universal and widely used method for describing the nature of a chemical bond. This metope is based on the latest achievements in the field of quantum mechanics and requires the involvement of a complex mathematical apparatus. This section discusses the main qualitative conclusions about the nature and properties of the chemical bond.
3.1. Main goals.
The MO method makes it possible to describe the most important properties of molecular systems:
1. The fundamental possibility of the formation of molecular systems.
2. Saturation of a chemical bond and composition of molecules.
3. Energy stability of molecules and (corresponding molecular ions) chemical bond strength.
4. Distribution of electron density and polarity of chemical bonds.
5. Donor-acceptor properties of molecular systems.
3.2. The main provisions of the method.
The main provisions of the molecular orbital method are as follows:
1. All electrons belong to the molecule as a whole and move in the field of its nuclei and electrons.
2. An increased electron density is created in the space between the nuclei due to the quantum mechanical effect of the exchange interaction of all socialized (delocalized) electrons. Note that in reality the main contribution is made by the delocalized valence electrons of atoms.
3. The formation of a chemical bond is considered as the transition of electrons from atomic orbitals to molecular habitations, covering all nuclei, with a gain in energy. If the transition to molecular orbitals is associated with the clamped energy, then the molecule is not formed.
4. The solution of the problem is reduced to finding possible MOs, distributing electrons to them in accordance with quantum mechanical principles (principle of minimum, energy, Pauli prohibition, Hund's rule) and making a conclusion based on the properties of the resulting (or not) molecular system.
Molecular orbitals are obtained by combining atomic orbitals (AO), hence the name of the MO LCAO method (MO Linear Combination of Atomic Orbitals).
Rules for describing molecules
The rules for finding MO from AO and the conclusion about the possibility of forming molecules are as follows:
1. Only AOs that are closest in energy interact with each other (usually with a difference of no more than 12 eV) 1 .
The required set of interacting AOs under consideration (basic set of atomic orbitals) for s- and p-elements of period 2 includes valence 2s- and 2p-AOs. It is this AO basis that makes it possible to conclude that there is an energy gain in the transition of electrons to MO.
For s- and p-elements of 3 periods, in many cases it turns out to be sufficient to restrict ourselves to the 3s- and 3p- basis of the AO, due to the relatively large difference in the energies of the 3p- and 3d-states.
2. The number of molecular orbitals is equal to the number of atomic orbitals from which they are formed. Moreover, in the space between the nuclei, the ARs overlapped and had the same symmetry about the bond axis (the x axis coincides with the bond axis). Molecular orbitals that have a lower energy (energetically more favorable state) than the combined AO are called bonding, and higher energy (energetically less favorable state) - loosening. If the energy of the MO is equal to the energy of the combined AO, then such MO is called nonbonding.
For example, period 2 atoms nitrogen and fluorine have 4 basic AOs: one 2s- three 2p-AOs. Then a diatomic molecule formed by two identical atoms of elements of period 2 (N 2 , F 2) has eight MO. Of these, 4 orbitals - type in symmetry about the bond axis ( S, P - bonding and loosening s * , p * and 4 orbitals - type in symmetry about the bond axis ( y and Z - bonding and loosening and ).
3. The formation of MO and the distribution of electrons is represented using energy diagrams. The horizontal lines along the edges of the diagrams correspond to the energy of each of the AOs of an individual atom, in the middle - to the energies of the corresponding MOs. The energies of the basic AO ns and np - elements of 1,2,3 periods are presented in Table 1.
The energy diagram for the oxygen molecule O 2 is shown in Figure 1.
When constructing energy diagrams, one should take into account the mutual influence of MOs close in energy. If the energy difference between the combined AOs of a given atom is small (less than 12 eV) and they have similar symmetry with respect to the bond axis, for example, 2s and 2p AOs from lithium to nitrogen, then an additional, i.e., MO configuration interaction. Such an interaction leads to the fact that in the energy diagram the connecting
P - MO are located higher than the binding - and - MO, for example, for diatomic molecules from Li 2 to N 2.
4. According to the MO method, a molecular system can be formed if the number of electrons on bonding MOs exceeds the number of electrons on antibonding MOs. Those. there is a gain in energy in comparison with the isolated state of the particles. The bond order (TS) in a diatomic particle, defined as the half-difference in the number of bonding and loosening electrons, must be greater than zero. Thus, PS = 2 for the oxygen molecule O 2 .
The presence of electrons in molecules on nonbonding MOs does not change the PS, but leads to some weakening of the binding energy due to an increase in the interelectronic repulsion. Indicates an increased reactivity of the molecule, a tendency for the transition of non-bonding electrons to binding MOs.
The shortcomings of the MVS considered above contributed to the development of another quantum mechanical method for describing the chemical bond, which was called molecular orbital method (MMO). The basic principles of this method were laid down by Lenard-Jones, Gund and Mulliken. It is based on the idea of a polyatomic particle as a single system of nuclei and electrons. Each electron in such a system experiences attraction from all nuclei and repulsion from all other electrons. Such a system can be conveniently described using molecular orbitals, which are formal analogues of atomic orbitals. The difference between atomic and molecular orbitals is that some describe the state of an electron in the field of a single nucleus, while others describe the state of an electron in the field of several nuclei. Considering the similarity of the approach to the description of atomic and molecular systems, we can conclude that the orbitals of an n-atomic molecule must have the following properties:
a) the state of each electron in the molecule is described by the wave function ψ, and the value ψ 2 expresses the probability of finding an electron in any unit volume of a polyatomic system; these wave functions are called molecular orbitals (MO) and, by definition, are multicenter, i.e. describe the motion of an electron in the field of all nuclei (the probability of being at any point in space);
b) each molecular orbital is characterized by a certain energy;
c) each electron in the molecule has a certain value of the spin quantum number, the Pauli principle in the molecule is fulfilled;
d) molecular orbitals are constructed from atomic orbitals by a linear combination of the latter: ∑c n ψ n (if the total number of wave functions used in the summation is k, then n takes values from 1 to k), with n are coefficients;
e) the MO energy minimum is reached at the maximum AO overlap;
f) the closer in energy are the initial ARs, the lower is the energy of MOs formed on their basis.
From the latter position, we can conclude that the inner orbitals of atoms, which have a very low energy, will practically not take part in the formation of MOs, and their contribution to the energy of these orbitals can be neglected.
Taking into account the properties of MOs described above, let us consider their construction for a diatomic molecule of a simple substance, for example, for an H 2 molecule. Each of the atoms that make up the molecule (H A and H B) have one electron per 1s orbital, then MO can be represented as:
Ψ MO = c A ψ A (1s) + c B ψ B (1s)
Since in the case under consideration the atoms that form the molecule are identical, the normalizing factors (c), showing the share of participation of AO in the construction of the MO, are equal in absolute value and, therefore, two options are possible Ψ MO at c A \u003d c B and c A \u003d - c B:
Ψ MO(1) = c A ψ A (1s) + c B ψ B (1s) and
Ψ MO(2) = c A ψ A (1s) - c B ψ B (1s)
molecular orbital Ψ MO(1) corresponds to a state with a higher electron density between atoms compared to isolated atomic orbitals, and electrons located on it and having opposite spins in accordance with the Pauli principle have a lower energy compared to their energy in an atom. Such an orbital in the MMO LCAO is called linking.
At the same time, the molecular orbital Ψ MO(2) is the difference between the wave functions of the initial AO, i.e. characterizes the state of the system with reduced electron density in the internuclear space. The energy of such an orbital is higher than that of the initial AO, and the presence of electrons on it leads to an increase in the energy of the system. Such orbitals are called loosening. Figure 29.3 shows the formation of bonding and antibonding orbitals in the hydrogen molecule.
Fig.29.3. Formation of σ - bonding and σ-loosening orbitals in a hydrogen molecule.
Ψ MO(1) and Ψ MO(2) have cylindrical symmetry with respect to the axis passing through the centers of the nuclei. Orbitals of this type are called σ - symmetrical and are written: bonding - σ1s, loosening - σ ٭ 1s. Thus, the configuration σ1s 2 corresponds to the hydrogen molecule in the ground state, and the configuration of the He 2 + ion, which is formed in the electric discharge, in the ground state can be written as σ1s 2 σ ٭ 1s (Fig. 30.3).
Rice. 30.3. Energy diagram of the formation of bonding and antibonding orbitals and the electronic structure of molecules and ions of elements of the first period.
In the H 2 molecule, both electrons occupy a bonding orbital, which leads to a decrease in the energy of the system compared to the initial one (two isolated hydrogen atoms). As already noted, the binding energy in this molecule is 435 kJ/mol, and the bond length is 74 pm. The removal of an electron from the bonding orbital increases the energy of the system (reduces the stability of the reaction product compared to the precursor): the binding energy in H 2 + is 256 kJ/mol, and the bond length increases to 106 pm. In the H 2 - particle, the number of electrons increases to three, so one of them is located on a loosening orbital, which leads to destabilization of the system compared to the previously described: E (H 2 -) = 14.5 kJ / mol. Consequently, the appearance of an electron in an antibonding orbital affects the chemical bond energy to a greater extent than the removal of an electron from the bonding orbital. The above data indicate that the total binding energy is determined by the difference between the number of electrons in the bonding and loosening orbitals. For binary particles, this difference, divided in half, is called the bond order:
PS \u003d (ē St - ē Not St.) / 2
If PS is zero, then no chemical bond is formed (He 2 molecule, Figure 30.3). If the number of electrons in loosening orbitals in several systems is the same, then the particle with the maximum PS value has the greatest stability. At the same time, at the same PS value, a particle with a smaller number of electrons in antibonding orbitals (for example, H 2 + and H 2 - ions) is more stable. Another conclusion follows from Figure 30.3: a helium atom can form a chemical bond with an H + ion. Despite the fact that the energy of the 1s orbital of He is very low (-2373 kJ/mol), its linear combination with the 1s orbital of the hydrogen atom (E = -1312 kJ/mol) leads to the formation of a bonding orbital, the energy of which is lower than the helium AO. Since there are no electrons on the loosening orbitals of the HeH + particle, it is more stable than the system formed by helium atoms and hydrogen ions.
Similar considerations apply to linear combinations of atomic p-orbitals. If the z-axis coincides with the axis passing through the centers of the nuclei, as shown in Figure 31.3, then the bonding and antibonding orbitals are described by the equations:
Ψ MO(1) = c A ψ A (2p z) + c B ψ B (2p z) and Ψ MO (2) \u003d c A ψ A (2p z) - c B ψ B (2p z)
When MOs are constructed from p-orbitals whose axes are perpendicular to the line connecting the atomic nuclei, then the formation of π-bonding and π-loosening molecular orbitals (Fig. 32.3) occurs. The molecular π at 2p and π at ٭ 2p orbitals are similar to those shown in Fig. 32.3, but rotated relative to the first by 90 about. Thus the π2p and π ٭ 2p orbitals are doubly degenerate.
It should be noted that a linear combination can be built not from any AO, but only from those that have a fairly close energy and whose overlap is possible from a geometric point of view. Pairs of such orbitals suitable for the formation of σ-bonding σ-loosening orbitals can be s - s, s - p z, s - d z 2, p z - p z, p z - d z 2, d z 2 - d z 2, while with a linear combination p x - p x , p y – p y , p x – d xz , p y – d yz , molecular π-bonding and π-loosening molecular orbitals are formed.
If you build an MO from AO of the type d x 2- y 2 - d x 2- y 2 or d xy - d xy, then δ-MOs are formed. Thus, as noted above, the division of MO into σ, π and δ is predetermined by their symmetry with respect to the line connecting the atomic nuclei. Thus, for a σ-MO, the number of nodal planes is zero, a π-MO has one such plane, and a δ-MO has two.
To describe homoatomic molecules of the second period within the framework of the MMO LCAO, it is necessary to take into account that a linear combination of atomic orbitals is possible only if the AO orbitals are close in energy and have the same symmetry.
Fig.31.3. Formation of σ-bonding σ-antibonding orbitals from atomic p-orbitals
Fig.32.3. Formation of π-bonding and π-antibonding molecular orbitals from atomic p-orbitals.
Of the orbitals of the second period, the 2s and 2p z orbitals have the same symmetry about the z axis. The difference in their energies for Li, Be, B, and C atoms is relatively small, so the wave functions 2s and 2p can mix in this case. For O and F atoms, the differences in energy 2s and 2p are much larger, so their mixing does not occur (Table 4.3)
Table 4.3.
∆E energies between 2s and 2p orbitals of various elements
According to the data of Table 4.3, as well as the calculations performed, it is shown that the relative energy of MO is different for Li 2 - N 2 molecules on the one hand and for O 2 - F 2 molecules on the other. For molecules of the first group, the order of increase in the MO energy can be represented as a series:
σ2sσ ٭ 2sπ2p x π2p y σ2p z π٭2p x π ٭ 2p y σ ٭ 2p z , and for O 2 and F 2 molecules in the form:
σ2sσ ٭ 2sσ2p z π2p x π2p y π٭2p x π ٭ 2p y σ ٭ 2p z (Figure 33.3).
Orbitals of type 1s, which have a very low energy compared to the orbitals of the second energy level, pass into the molecule unchanged, that is, they remain atomic and are not indicated on the energy diagram of the molecule.
Based on the energy diagrams of molecules and molecular ions, one can draw conclusions about the stability of particles and their magnetic properties. Thus, the stability of molecules, the MOs of which are constructed from the same AO, can be roughly judged by the value of the bond order, and the magnetic properties - by the number of unpaired electrons per MO (Fig. 34.3).
It should be noted that AO orbitals of non-valence, internal levels, from the point of view of the MMO of LCAO, do not take part in the formation of MO, but have a noticeable effect on the binding energy. For example, when passing from H 2 to Li 2, the binding energy decreases by more than four times (from 432 kJ/mol to 99 kJ/mol).
Fig.33.3 Energy distribution of MO in molecules (a) O 2 and F 2 and (b) Li 2 - N 2.
Fig.34.3 Energy diagrams of binary molecules of elements of the second period.
Detachment of an electron from an H 2 molecule reduces the binding energy in the system to 256 kJ/mol, which is caused by a decrease in the number of electrons in the bonding orbital and a decrease in PS from 1 to 0.5. In the case of detachment of an electron from the Li 2 molecule, the binding energy increases from 100 to 135.1 kJ / mol, although, as can be seen from Figure 6.9, the electron, as in the previous case, is removed from the bonding orbital and PS decreases to 0.5. The reason for this is that when an electron is removed from the Li 2 molecule, the repulsion between the electrons located on the bonding MO and the electrons occupying the inner 1s orbital decreases. This pattern is observed for the molecules of all elements of the main subgroup of the first group of the Periodic system.
As the nuclear charge increases, the influence of the 1s orbital electrons on the energy of the MO decreases, therefore, in the B 2, C 2 and N 2 molecules, the detachment of an electron will increase the energy of the system (decrease in the PS value, decrease in the total bond energy) due to the fact that the electron is removed from the bonding orbitals. In the case of O 2 , F 2 and Ne 2 molecules, the removal of an electron occurs from the loosening orbital, which leads to an increase in PS and the total binding energy in the system, for example, the binding energy in the F 2 molecule is 154.8 kJ / mol, and in the ion F 2 + is almost twice as high (322.1 kJ / mol). The above reasoning is valid for any molecules, regardless of their qualitative and quantitative composition. We recommend the reader to carry out a comparative analysis of the stability of binary molecules and their negatively charged molecular ions, i.e. estimate the change in the energy of the system in process A 2 + ē = A 2 - .
It also follows from Figure 34.3 that only the B 2 and O 2 molecules, which have unpaired electrons, are paramagnetic, while the rest of the binary molecules of the elements of the second period are diomagnetic particles.
Proof of the fairness of the IMO, i.e. evidence of the real existence of energy levels in molecules is the difference in the values of the ionization potentials of atoms and molecules formed from them (table 5.3).
Table 5.3.
Ionization potentials of atoms and molecules
|
atom |
first ionization potential kJ/mol |
molecule |
first ionization potential kJ/mol |
|
H 2 | |||
|
N 2 | |||
|
O 2 | |||
|
C 2 | |||
|
F 2 |
The data presented in the table indicate that some molecules have higher ionization potentials than the atoms from which they are formed, while others have lower ionization potentials. This fact is inexplicable from the point of view of the MVS. Analysis of the data in Figure 34.3 leads to the conclusion that the potential of the molecule is greater than that of the atom in the case when the electron is removed from the bonding orbital (molecules H 2, N 2, C 2). If the electron is removed from the loosening MO (O 2 and F 2 molecules), then this potential will be less compared to the atomic one.
Turning to the consideration of heteroatomic binary molecules within the framework of the MMO LCAO, it is necessary to recall that the orbitals of atoms of various elements that have the same values of the main and side quantum numbers differ in their energy. The higher the effective charge of the atomic nucleus with respect to the considered orbitals, the lower their energy. Figure 35.3 shows the MO energy diagram for heteroatomic molecules of type AB, in which the B atom is more electronegative. The orbitals of this atom are lower in energy than the similar orbitals of the A atom. In this regard, the contribution of the orbitals of the B atom to the bonding MOs will be greater than to the loosening MOs. On the contrary, the main contribution to the antibonding MO will be made by the AO of the A atom. The energy of the inner orbitals of both atoms during the formation of the molecule practically does not change, for example, in the hydrogen fluoride molecule, the orbitals 1s and 2s of the fluorine atom are concentrated near its nucleus, which, in particular, determines the polarity of this molecule (µ = 5.8 ∙ 10 -30). Consider, using Figure 34, the description of the NO molecule. The energy of oxygen AO is lower than that of nitrogen, the contribution of the former is higher to the bonding orbitals, and the latter to the loosening orbitals. The 1s and 2s orbitals of both atoms do not change their energy (σ2s and σ ٭ 2s are occupied by electron pairs, σ1s and σ ٭ 1s are not shown in the figure). The 2p orbitals of oxygen and nitrogen atoms, respectively, have four and three electrons. The total number of these electrons is 7, and there are three bonding orbitals formed due to 2p orbitals. After they are filled with six electrons, it becomes obvious that the seventh electron in the molecule is located on one of the antibonding π-orbitals and, therefore, is localized near the nitrogen atom. PS in the molecule: (8 - 3) / 2 = 2.5 i.e. the total binding energy in the molecule is high. However, an electron located in an antibonding orbital has a high energy, and its removal from the system will lead to its stabilization. This conclusion makes it possible to predict that the activation energy of NO oxidation processes will be low; these processes can proceed even at s.u..
At the same time, the thermal stability of these molecules will be high, the NO + ion will be close to nitrogen and CO molecules in terms of total binding energy, and NO will dimerize at low temperatures.
The analysis of the NO molecule within the framework of this method leads to another important conclusion - the most stable will be binary heteroatomic molecules, which include atoms with a total number of electrons in the valence s and p orbitals equal to 10. In this case, PS = 3. An increase in or a decrease in this number will lead to a decrease in the value of PS, i.e. to the destabilization of the particle.
Polyatomic molecules in MMO LCAO are considered based on the same principles as described above for duatomic particles. Molecular orbitals in this case are formed by a linear combination of AO of all atoms that make up the molecule. Consequently, MOs in such particles are multicenter, delocalized, and describe the chemical bond in the system as a whole. The equilibrium distances between the centers of atoms in a molecule correspond to the minimum potential energy of the system.
Fig.35.3. Energy diagram of MO of binary heteroatomic molecules
(Atom B has a high electronegativity).
Fig.36.3. Energy diagrams of molecules of various types in
within the MMO. (the p x axis of the orbital coincides with the bond axis)
Figure 36.3 shows the MOs of various types of molecules. We will consider the principle of their construction using the example of the BeH 2 molecule (Fig. 37.3). The formation of three-center MOs in this particle involves the 1s orbitals of two hydrogen atoms, as well as the 2s and 2p orbitals of the Be atom (the 1s orbital of this atom does not participate in the formation of the MO and is localized near its nucleus). Let us assume that the p-axis of the Be z-orbital coincides with the communication line in the particle under consideration. A linear combination of s orbitals of hydrogen and beryllium atoms leads to the formation of σ s and σ s ٭ , and the same operation with the participation of s orbitals of hydrogen atoms and the p z orbital of Be leads to the formation of a bonding and loosening MO σ z and σ z ٭ , respectively.
Fig.37.3. MO in the Ven 2 molecule
Valence electrons are located in the molecule in bonding orbitals, i.e. its electronic formula can be represented as (σ s) 2 (σ z) 2 . The energy of these bonding orbitals is lower than the energy of the orbitals of the H atom, which ensures the relative stability of the molecule under consideration.
In the case when all systems of atoms have p-orbitals suitable for a linear combination, along with σ-MOs, multicenter bonding, non-bonding, and loosening π-MOs are formed. Consider such particles on the example of a CO 2 molecule (Fig. 38.3 and 39.3).
Fig.38.3 CO 2 molecules binding and loosening σ-MO
Fig.39.3. Energy diagram of MO in a CO 2 molecule.
In this molecule, σ-MOs are formed by combining 2s and 2p x orbitals of a carbon atom with 2p x orbitals of oxygen atoms. Delocalized π-MOs are formed due to the linear combination of p y and p z orbitals of all atoms,
included in the molecule. As a result, three pairs of π-MOs are formed with different energies: binding - π y c in π z sv, non-bonding - π y π z (corresponding in energy to the p-orbitals of oxygen atoms), and loosening - π y res π z res.
When considering molecules within the framework of the MMO LCAO, abbreviated schemes for describing particles are often used (Fig. 40.3). When forming an MO, for example, in the BCI 3 molecule, it is sufficient to indicate only those AOs that take part in the linear combination MO)
Fig.40.3. MO in the BCI 3 molecule
The energy diagram of MO in the CH 4 molecule is shown in Fig. 41.3. An analysis of the electronic structure of the carbon atom shows that due to the different directions of its 2p orbitals, the formation of five-center MOs in the CH 4 molecule with the participation of these AOs is impossible for geometric reasons. At the same time, the 2s orbital of carbon is equally capable of overlapping with the 1s orbitals of hydrogen atoms, resulting in the formation of five-center σ s and σ s ٭ MO. In the case of combinations of 2p and 1s orbitals, the number of atomic functions in a linear combination is only three, i.e. the energy of σ-MO in this case will be higher than that of the corresponding σ s and σ s ٭ .
Fig.41.3 .. Energy diagram of the MO of the CH 4 molecule.
The different energies of the five-center and three-center bonding orbitals are confirmed by experimental data on ionization potentials, which are different for electrons moving away from σ s and from σ x (σ y . σ z).
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. In this way, all electrons in a molecule belong simultaneously to the whole molecule, and are not the property of two bonded atoms. Consequently, 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 .
Having located the nuclei of two interacting atoms at a distance equal to the bond length, 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.
|
|
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 forces of attraction 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, the appearance of molecular clouds obtained by overlapping or repulsing the orbitals of interacting atoms is of interest.
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)
|
|
|
|
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 is, according to the definition, 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.