Vibrational spectroscopy. Vibrational spectra of diatomic molecules See what "vibrational spectra" are in other dictionaries
Infrared spectroscopy (IRS) belongs to a large group of molecular spectroscopy methods and is based on the selective absorption of radiation in the infrared region (0.8 - 1000 μm) of the spectrum
Absorb infrared (IR) radiation can only those molecules of substances and compounds in which the dipole moment changes during vibrations of atoms
IR radiation is spent only on changing the vibrational and rotational energy of the molecule, without causing electronic transitions due to the lack of absorbed energy (hν)
IR spectra are more complex than electronic spectra in the visible region, since most of the absorbed energy is spent on oscillatory processes
IR spectra of molecules are characterized by high information content
Usually, for the image of IR spectra, the abscissa is set aside frequency , wave number , less often wavelength .
The wavelength () and frequency () are related by the relationship:
where C is the speed of propagation of radiation in a certain medium.
To characterize electromagnetic radiation, the wave number is also used ( 
, /) is the reciprocal of the wavelength:
It shows how many waves fit in a unit of length, most often 1 cm; in this case, the dimension of the wavenumber is [cm–1]. Often the wave number is called the frequency, although it should be recognized that this is not entirely correct. They are proportional to each other.
IR - region in the general electromagnetic spectrum occupies the wavelength range from 2 to 50 microns (wave number 5000 - 200 cm -1).
The intensity of absorption of infrared radiation, as a rule, is expressed by the value of transmission (T):
where I is the intensity of radiation transmitted through the sample;
I 0 is the intensity of the incident radiation.
Infrared spectroscopy is a universal method for determining important functional groups, as well as structural fragments in small amounts of a substance in any of its state of aggregation.
The range of questions connected in one way or another with the use of IR spectroscopy is extremely wide.
With the help of IR spectroscopy, it is possible to carry out the identification of substances, structural group analysis, quantitative analysis, the study of intra- and intermolecular interactions, the establishment of a configuration, the study of reaction kinetics, etc. Modern automatic IR spectrophotometers make it possible to obtain an absorption spectrum very quickly, and the operator requires a minimum of special knowledge and skills. Let us consider the reasons for the absorption of infrared radiation by molecules.
Vibrations of atoms in a molecule
The absorption of infrared radiation by a substance causes transitions between the vibrational levels of the ground electronic state. In this case, the rotational levels also change. Therefore, the IR spectra are oscillatory-rotational.
A chemical bond in a diatomic molecule can be simplified as an elastic spring. Then its stretching and compression will simulate the vibration of atoms in a molecule. For a harmonic oscillator, the restoring force is proportional to the displacement of the nuclei from the equilibrium position and is directed in the direction opposite to the displacement:
where K is the coefficient of proportionality, which is called force constant and characterizes the stiffness of the bond (the elasticity of the bond).
It is known from the laws of classical mechanics that the oscillation frequency of such a system is related to the force constant K and to the masses of atoms (m 1 and m 2) by the following relationship:
, (8.1)
where - reduced mass,
.
The force constants of single, double and triple bonds are related approximately as 1: 2: 3.
It follows from relation (8.1) that the oscillation frequency increases with increasing bond strength (bond multiplicity) and with decreasing atomic masses.
Those. the frequency depends on the mass of the atoms: the lighter the atom, the higher the frequency.
C-H (3000 cm -1), C-D (2200 cm -1), C-O (1100 cm -1), C-Cl (700 cm -1).
The frequency depends on the bond energy: (the bond is stronger - the frequency is higher)
С≡O (2143 cm -1), C=O (1715 cm -1), C-O (1100 cm -1).
If we assume that, in the first approximation, for a diatomic molecule, the vibrations are harmonic, and thus such a molecule is likened to a harmonic oscillator, then the value of the total vibration energy obeys the main quantum condition:
, (8.2)
where is a vibrational quantum number that takes the values of integers: 0, 1, 2, 3, 4, etc.;
0 - the frequency of the main vibration (fundamental tone), determined by equation (8.1).
Expression (8.2) corresponds to a system of equally spaced energy levels (Fig. 8.1).
It should be noted that at = 0 E count 0 (E = 1/2 h 0).
This means that the vibrations of the nuclei in the molecule do not stop, and even in the lowest vibrational state the molecule has a certain reserve of vibrational energy.
When absorbing a quantum of light h the molecule will move to higher energy levels. It is known that the energy of an absorbed quantum is equal to the difference between the energies of two states:
h \u003d E + 1 - E (8.3)
In turn, the energy difference for two energy levels, as follows from equation (8.1.2), is:
E + 1 – E = h 0 (8.4)
When comparing relations (8.3) and (8.4), it can be seen that the frequency of absorbed radiation () is equal to the fundamental vibrational frequency ( 0) determined by equation (8.1).
Thus, the spectrum of a harmonic oscillator consists of one line or band with a frequency 0 , which is the natural frequency of the oscillator (Fig. 8. 1).
Usually, at room temperature, most molecules are in the lower vibrational state, since the energy of thermal excitation is much less than the energy of the transition from the ground state to the excited state.
Therefore, it is easiest to experimentally observe the absorption corresponding to the transition from the ground vibrational state ( = 0) to the first excited state ( = 1).
For a harmonic oscillator, other transitions are also possible with a change in the quantum number by one, i.e. transitions between neighboring levels:
= 1 (8.5)
The experimentally observed infrared absorption band of molecules in the gas phase has a complex structure, since each vibrational state of an isolated molecule is characterized by its own system of rotational sublevels (Fig. 8.2).
The vibrational spectral line, due to the superposition of rotational transitions, turns into a band consisting of many lines, and the IR spectrum is a set stripes absorption (similar to how an electronic transition is necessarily accompanied by vibrational and rotational transitions, and the electronic spectrum consists of absorption bands). The width of the vibrational bands is smaller than the electronic ones, since the difference between the energies of the rotational sublevels is smaller than that of the vibrational ones. Of all vibrational transitions, the transition to the nearest vibrational sublevel is the most probable. It matches main spectral line.
Rice. 8.1. Potential curves, energy levels and schematic spectra of harmonic (1) and anharmonic (2) oscillators
Less probable transitions to higher vibrational sublevels correspond to spectral lines called overtones. Their frequency is 2, 3, etc. times greater than the frequency of the main line, and the intensity is much less. The main line is denoted by , and the overtones are 2, 3, etc.
All vibrations in a molecule can be divided into two types: valence and deformation. If during the vibration under consideration there is mainly a change in the lengths of bonds, and the angles between the bonds change little, then such a vibration is called valence and is denoted by . Stretching vibrations can be symmetric ( s) and asymmetric ( as).
A necessary condition for an oscillatory transition is a change dipole moment molecules during vibrations of atoms. A symmetrical molecule that does not have a dipole moment cannot absorb infrared radiation. The ability of a substance to absorb the energy of infrared radiation depends on the total change in the dipole moment of the molecule during rotation and vibration, i.e. only a molecule that has an electric dipole moment, the magnitude or direction of which changes during oscillation and rotation, can absorb infrared radiation. The dipole moment means the mismatch between the centers of gravity of positive and negative charges in the molecule, i.e., the electrical asymmetry of the molecule.
Thus, not all molecules are able to absorb infrared radiation. Molecules that have a center of symmetry are devoid of a dipole moment and do not acquire it in the process of vibration and, therefore, are not active in the infrared spectrum. Examples of such molecules are diatomic molecules with a covalent bond (H 2 , N 2 , halogens, a CO 2 molecule with symmetrical valence vibrations of atoms, etc.).
If the angle between the bonds changes during vibrations of the molecule without changing the length of the bonds, then such vibrations are called deformational.
These fluctuations are referred to as or . They can also be symmetrical ( s, s) and asymmetric ( a s, a s).
Deformation oscillations are divided into fan, torsional, scissor and pendulum. The same categories are acceptable for describing the fluctuations of individual groups.
Each type of oscillation is characterized by a certain excitation energy. Stretching vibrations correspond to higher energies than bending vibrations, and, consequently, the stretching vibration bands lie in a shorter wavelength region (or at higher frequencies).
When a sufficiently high energy is imparted to the molecule, the nuclei in the molecule begin to oscillate about the equilibrium position. If a polyatomic molecule, which can be represented in the form of material points in which the masses of atoms connected by springs are concentrated, is shaken, then it will perform a complex movement, the so-called Lissajous movement . Such a motion can be decomposed into a relatively small number of normal vibrations, in which all nuclei oscillate in the same phase and at the same frequency, and which can be described by a set of normal coordinates.
Number of normal vibrations(or vibrational degrees of freedom ) of a molecule consisting of N atoms can be calculated as 3N− 5 for linear molecules and 3N− 6 for non-linear molecules. In this calculation, from the total number of degrees of freedom of an N-atomic molecule, equal to 3N, three degrees of freedom of the translational motion of the molecule as a whole and two (for linear) or three (for nonlinear) degrees of freedom of rotational motion are subtracted.
For example, for a diatomic molecule, the number of vibrational degrees of freedom is 3 ∙ 2 − 5 = 1. This is stretching vibration associated with a change in the bond length.
| H |
| Cl |
Figure 4.7 - Stretching vibration on the example of the HCl molecule
In a triatomic linear molecule of the XY 2 type (for example, CO 2), the number of normal vibrations is 3 ∙ 3 − 5 = 4. For such a molecule, in addition to two valence (symmetric ν s and asymmetric ν as) there are two more deformation oscillations δ associated with a change in the bond angle in the molecule in two mutually perpendicular planes.
The simplest model of a diatomic vibrating molecule is harmonic oscillator - a system of two balls connected by a spring and oscillating at a constant frequency (Figure 4.8). Oscillations in such a molecule can be considered as oscillations of the mass, equal to the reduced mass of the molecule, occurring with the same frequency, relative to the fixed wall to which it is attached by the same spring.
Figure 4.8 - Harmonic oscillator model
The oscillation frequency of the harmonic oscillator depends on the elastic force
This dependence corresponds to the parabola equation. By substituting the potential energy into the Schrödinger equation and solving it, we can obtain the equation for the energy of a harmonic oscillator
where is the reduced mass.
The vibrational quantum number can take the values \u003d 0, 1, 2 ...
Selection rule for the harmonic oscillator model 
.
Energy of the zero vibrational level in the harmonic oscillator model:
The vibrational spectrum of a harmonic oscillator is a single line with a wave number equal to the wave number of natural oscillations (Figure 4.9 b):
Figure 4.9 - Potential energy curve of a harmonic oscillator ( a)
b)
Figure 4.11 - Potential energy curve of an anharmonic oscillator ( a)
and a schematic view of the vibrational spectrum ( b)
As a result of substitution into the Schrödinger equation, the Morse potential energy equations
we obtain an equation for the vibrational energy of an anharmonic oscillator:
Selection rule for vibrational transitions:
Designations accepted. The absorption of a light quantum, which leads to the transition of the molecule from zero to the first vibrational level, leads to the appearance of a line in the spectrum, called the fundamental tone, but the second - the first overtone, to the third - the second overtone, etc. (Figure 4.11 b). The distances between lines in the vibrational spectrum of a molecule naturally decrease as the wave number increases. Since the majority of molecules at low temperatures are at the zero vibrational level, transitions from this level appear in the spectrum. Only with a significant increase in temperature can the so-called "hot" frequencies corresponding to transitions of the molecule from higher energy levels be detected in the spectrum.
The energy of the zero vibrational level in the anharmonic oscillator model can be calculated using the equation:
and the wave number of the line corresponding to this transition in the spectrum according to equation (4.65):
| (4.65) |
An analysis of a schematic representation of the potential energy curve and the vibrational spectrum of an anharmonic oscillator using the example of a diatomic molecule (Figure 4.11) indicates that with an increase in the vibrational quantum number, the distances between the energy levels decrease, and the lines in the spectrum corresponding to the absorption of a quantum upon transition to the level 
, are approaching. Last transition 
, corresponding to the dissociation of a molecule into atoms, corresponds to the boundary of the discrete and continuous spectrum. The relationship between the dissociation energy and the depth of the potential well is illustrated in Figure 4.12 and Equation (4.66)
Figure 4.12 - Determination of the dissociation energy from the potential curve
anharmonic oscillator energy
Substituting the resulting expression into equation (4.62) makes it possible to calculate the energy of the maximum vibrational level:
According to the second boundary condition, a different equation for the dissociation energy is obtained, which, however, gives close calculated results:
If the force constants are close, then the equation is further simplified:
for the first overtone 
Example: In the infrared absorption spectrum of a diatomic molecule, the positions of the first two strongest lines at 3962 and 7743 cm -1 were determined. Find the wave number of natural vibrations, the anharmonicity coefficient and the anharmonicity of the molecule.
Solution.
The first line corresponds to the fundamental tone (equation (4.75)), the second - to the first overtone (equation (4.76)). We compose and solve the system of equations:
Let's multiply the first equation by 3:
Subtract the second from the first equation:
cm -1
We substitute the obtained value, for example, into the first equation (the equation for the fundamental tone) and calculate the anharmonicity and then the anharmonicity coefficient:
Example: In the vibrational spectrum of the CO molecule in the free state and in cases where CO is adsorbed on the surface of metals, the wave numbers of the main absorption band are determined. The wave number of natural vibrations of CO in the free non-adsorbed state is 2169 cm -1 . Considering the anharmonicity coefficient to be the same in all cases, calculate the wavenumbers of natural oscillations of CO, the energy and force constant of the C=O coupling in all cases. Make a conclusion about the nature of adsorption.
Solution.
The main CO band or fundamental tone corresponds to equation (4.75). We use the data for the CO molecule in the free state to find the anharmonicity coefficient:
Let us transform equation (4.75) with respect to and calculate the wave numbers of natural vibrations for CO molecules adsorbed on metals:
The binding energy in a molecule is opposite in its physical meaning to the dissociation energy. We use, for example, equation (4.71) to evaluate it:
The dissociation energies for CO molecules adsorbed on metals are estimated similarly:
Finally Ni:
To find the force constant, we rewrite equation (4.57):
 | (4.78) |
Let us first estimate the reduced mass of the CO molecule:
, where the masses of carbon nuclei 
and oxygen 
(in grams).
Then we calculate the force constants:
for free CO:
and for a CO molecule adsorbed on metals:
An analysis of the performed calculations indicates a decrease in the binding energy and force constant and, consequently, a weakening of the C=O bond during adsorption on metals.
Solution.
From the previous example, we take the necessary data for a non-isotopically substituted CO molecule adsorbed on copper:
wave number of natural vibrations:
cm -1 ,
reduced mass of a CO molecule:
Calculate the reduced mass of the isotopically substituted 13 CO molecule:
Assuming that the force constants are close to , we estimate by equation (4.74) the ratio of the wave numbers of natural vibrations and the value for an isotopically substituted molecule:
The position of the fundamental tone band is determined by equation (4.75), borrowing from the previous example the value of the anharmonicity coefficient and the position of the fundamental tone band for a non-isotopically substituted CO molecule adsorbed on copper:
2128 cm -1 .
We conclude: the isotope shift to the long-wavelength (or low-frequency) region is 2128 - 2082 = 46 cm -1 .
4.2.6 Multivariant task No. 12 "Vibrational spectra of diatomic molecules"
1. Write a quantum mechanical equation to calculate the energy of the vibrational motion of a diatomic molecule as a harmonic oscillator.
2. Write a quantum mechanical equation to calculate the energy of the vibrational motion of a diatomic molecule as an anharmonic oscillator.
3. Calculate from determined from the IR absorption spectrum of a diatomic molecule A fundamental tone (), first overtone (), second overtone () or third overtone () (table 4.4) wave number of natural vibrations, anharmonicity coefficient and anharmonicity.
4. Calculate the wavenumbers of the missing lines in the IR absorption spectrum of a diatomic molecule A( , , or ).
5. Determine the energy of the vibrational motion of the molecule A at the zero vibrational quantum level E 0 (J) using the wavenumber of natural vibrations and anharmonicity (see item 3)
6. Derive an equation to calculate the maximum vibrational quantum number.
7. Determine the maximum vibrational quantum number for a molecule A.
8. Determine the energy of vibrational motion (J) at the maximum vibrational quantum level.
9. Determine the dissociation energy D 0 molecules A(kJ/mol).
10. Draw a dependency graph 
, choosing 3-4 values of the quantum number and calculating the values E count in the range from 0 to .
11. Indicate on the graph the energy of vibrational motion at the maximum vibrational quantum level E max (J) and dissociation energy D 0 .
12. Calculate the force constant of the chemical bond of a diatomic molecule A.
13. Calculate the wavenumber of natural vibrations of an isotopically substituted molecule B.
14. Determine the magnitude and direction of the isotopic shift for the pitch.
Table 4.4 - Task options
| Option | Molecule BUT | Sub-option |  , m -1 |  , m -1 |  , m -1 |  , m -1 | Molecule B |
| HI | - | - | 2HI | ||||
| - | - | 3HI | |||||
| - | - | 2 H 129 I | |||||
| - | - | 3 H 131 I | |||||
| - | - | H 129 I | |||||
| - | - | H 131 I | |||||
| HF | - | 2HF | |||||
| 3 H 18 F | |||||||
| 2 H 18 F | |||||||
| 3HF | |||||||
| H 18 F | |||||||
| 2HF | |||||||
| HCl | - | 2 HCl | |||||
| 3 HCl | |||||||
| H37Cl | |||||||
| 3 HCl | |||||||
| 2H37Cl | |||||||
| 3H37Cl | |||||||
| SO | - | S 18 O | |||||
| 34 SO | |||||||
| S 1 7 O | |||||||
| 36 SO | |||||||
| 33 S 1 7 O | |||||||
| 33 SO | |||||||
| BCl | B37Cl | ||||||
| 10 BCl | |||||||
| B37Cl | |||||||
| 10 B 37 Cl | |||||||
| 10 BCl | |||||||
| B37Cl | |||||||
| NO | - | N 18 O | |||||
| N 1 7 O | |||||||
| 15 N 18 O | |||||||
| 15 NO | |||||||
| 15 N 1 7 O | |||||||
| 13NO | |||||||
| CaF | Sa 18 F | ||||||
| 43 Ca 18 F | |||||||
| 44 Ca 18 F | |||||||
| 42 CaF | |||||||
| 43 CaF | |||||||
| 44 CaF |
Continuation of table 4.4
| Option | Molecule BUT | Sub-option | , m -1 | , m -1 | , m -1 | , m -1 | Molecule B |
| HBr | - | 3HBr | |||||
| 2HBr | |||||||
| H77Br | |||||||
| H85Br | |||||||
| H 84 Br | |||||||
| H 82 Br | |||||||
| ClI | - | 37 CLI | |||||
| 37 Cl 129 I | |||||||
| Cl 129 I | |||||||
| 37 Cl 125 I | |||||||
| Cl 125 I | |||||||
| 37 CLI | |||||||
| FCl | - | 18 FCl | |||||
| F37Cl | |||||||
| 18 F 37 Cl | |||||||
| F37Cl | |||||||
| 18 FCl | |||||||
| F37Cl | |||||||
| CO | - | 13CO | |||||
| C17O | |||||||
| 17CO | |||||||
| C18O | |||||||
| 14 CO | |||||||
| 15 CO | |||||||
| H2H | 3 HH | ||||||
| 3 H 2 H | |||||||
| H 3 H | |||||||
| 2 HH | |||||||
| 2 H 2 H | |||||||
| 3 H 3 H | |||||||
| SH | S 2 H | ||||||
| 33 S 2 H | |||||||
| 33 S 3 H | |||||||
| 34S2H | |||||||
| 34 S 3 H | |||||||
| S 3 H | |||||||
| NBR | 13 NBR | ||||||
| 15 NBR | |||||||
| N79 Br | |||||||
| 15N79Br | |||||||
| 13N79Br | |||||||
| N 80Br |
Continuation of table 4.4
| Option | Molecule BUT | Sub-option | , m -1 | , m -1 | , m -1 | , m -1 | Molecule B |
| NS | S 13 N | ||||||
| 34 SN | |||||||
| S 15 N | |||||||
| 36 SN | |||||||
| 33 S 15 N | |||||||
| 33 SN | |||||||
| SiF | Si 18 F | ||||||
| 30 SiF | |||||||
| 29 Si 18 F | |||||||
| 30 Si 18 F | |||||||
| 29 SiF | |||||||
| Si 18 F | |||||||
| Oh | - | O 2 H | |||||
| 17O2H | |||||||
| 17OH | |||||||
| O 3 H | |||||||
| 18O2H | |||||||
| 18OH | |||||||
| SiN | Si 13 N | ||||||
| 30SiN | |||||||
| 29 Si 13 N | |||||||
| 30 Si 15 N | |||||||
| 29 SiN | |||||||
| Si 13 N | |||||||
| CP | C 30P | ||||||
| 13CP | |||||||
| 13 C 30 P | |||||||
| C 30P | |||||||
| 13CP | |||||||
| 13 C 30 P | |||||||
| BeS | Be 36S | ||||||
| Be 34S | |||||||
| Be 33S | |||||||
| 7BeS | |||||||
| 7 Be 33 S | |||||||
| 7 Be 36S | |||||||
| BeO | Be 18O | ||||||
| Be 17O | |||||||
| Be 18O | |||||||
| 7BeO | |||||||
| 7 Be 18 O | |||||||
| 7 Be 17 O |
Continuation of table 4.4
| Option | Molecule BUT | Sub-option | , m -1 | , m -1 | , m -1 | , m -1 | Molecule B |
| BeI | 7BeI | ||||||
| 7 Be 125 I | |||||||
| 7 Be 129 I | |||||||
| 7 Be 131 I | |||||||
| Be 125 I | |||||||
| Be 129 I | |||||||
| CN | - | 13 C N | |||||
| 13 C 15 N | |||||||
| 14CN | |||||||
| C13N | |||||||
| C 15 N | |||||||
| 13 C 13 |
Rotational spectra
Consider the rotation of two atomic molecules around its axis. The molecule has the lowest energy in the absence of rotation. This state corresponds to the rotational quantum number j=0. The nearest excited level (j=1) corresponds to a certain rotation speed. To transfer a molecule to this level, energy E 1 must be expended. At j=2,3,4… rotation speed is 2,3,4… times higher than at j=0. The internal energy of the molecule increases with increasing rotational speed and the distance between the levels increases. The energy difference between neighboring levels increases all the time by the same value E 1 . In this regard, the rotational spectrum consists of separate lines; for the first line ν 1 \u003d E 1 /ħ, and the next 2ν 1, 3 ν 1, etc. The energy difference between rotational levels is very small, so even at room temperature the kinetic energy of molecules during their collision is sufficient to excite rotational levels. A molecule can absorb a photon and move to a higher rotational level. This way you can study the absorption spectra.
The frequency depends on the mass of the molecule and its size. As the mass increases, the distance between the levels decreases and the entire spectrum shifts towards longer wavelengths.
Rotational spectra can be observed in substances in the gaseous state. There is practically no rotation in liquid and solid bodies. The need to transfer the analyte to the gaseous state without destroying it severely limits the use of rotational spectra (as well as the difficulty of working in the far IR region).
If additional energy is imparted to the molecule, less than the bond breaking energy E chem, then the atoms will vibrate around the equilibrium position, and the vibration amplitude will have only certain values. Bands are observed in vibrational spectra rather than individual lines (as for atoms or in rotational spectra). The fact is that the energy of a molecule depends both on the positions of individual atoms and on the rotation of the entire molecule. Thus, any vibrational level turns out to be complex and splits into a number of simple levels.
In the vibrational spectra of gaseous substances, individual lines of the rotational structure are clearly visible. There are no definite rotational levels in liquids and solids. So they have one wide band. Vibrations of polyatomic molecules are much more complicated than those of 2-atomic ones, because the number of possible types of vibrations grows rapidly with the increase in the number of atoms in the molecule.
For example, a linear CO 2 molecule has vibrations of 3 types.
The first 2 types are valence (one is symmetric, the other is antisymmetric). During vibrations of the third type, the bond angles change and the atoms are displaced in directions perpendicular to the bond bonds, the length of which remains almost constant. Such oscillations are called deformational. To excite bending vibrations, less energy is required than for stretching vibrations. The absorption bands associated with the excitation of deformation transitions have a frequency 2–3 times lower than the frequencies of stretching vibrations. Vibrations in CO 2 affect all atoms at once. Such oscillations are called skeletal. They are characteristic only for a given molecule and the bands corresponding to them do not coincide even for substances with a similar structure.
In complex molecules, vibrations are also distinguished in which only small groups of atoms participate. The bands of such vibrations are characteristic of certain groups, and their frequencies change little when the structure of the rest of the molecule changes. So in the absorption spectra of chemical compounds it is easy to detect the presence of certain groups.
So, any molecule has its own specific absorption spectrum in the IR region of the spectrum. It is almost impossible to find 2 substances with the same spectra.
Represent the model of two interacting point masses m 1 and m 2 with an equilibrium distance r e between them (bond length), and fluctuate. the motion of the nuclei is considered harmonic and is described by unity, the coordinate q=r-r e , where r is the current internuclear distance. The dependence of the potential energy fluctuates. motion V from q is determined in the harmonic approximation. oscillator [oscillating material point with reduced mass m \u003d m 1 m 2 / (m 1 + m 2)] as a function V \u003d l / 2 (K e q 2), where K e \u003d (d 2 V / dq 2) q \u003d 0 - harmonic. force constant
Rice. 1. Dependence of the potential energy V of a harmonic oscillator (dashed curve) and a real diatomic molecule (solid curve) on the internuclear distance r (r is the equilibrium value of r); horizontal straight lines show oscillate. levels (0, 1, 2, ... values of the oscillatory quantum number), vertical arrows - some oscillatory. transitions; D 0 - dissociation energy of the molecule; the shaded area corresponds to the continuous spectrum. molecules (dashed curve in Fig. 1).
According to the classic mechanics, harmonic frequency. hesitation 
Quantum Mech. consideration of such a system gives a discrete sequence of equidistant energy levels E(v)=hv e (v+ 1 / 2), where v = 0, 1, 2, 3, ... - vibrational quantum number, v e - harmonic. vibrational constant of the molecule (h - Planck's constant). When moving between adjacent levels, according to the selection rule D v=1, a photon with energy hv= D E=E(v+1)-E(v)=hv e (v+1+ 1 / 2)-hv e (v+ 1 / 2)=hv e , i.e. the transition frequency between any two adjacent levels is always one and the same, and coincides with the classic. harmonic frequency. fluctuations. Therefore, v e is called. also harmonic. frequency.
For real molecules, the potential energy curve is not the indicated quadratic function q, i.e., a parabola. Swing. the levels converge more and more as one approaches the dissociation limit of the molecule and for the anharmonic model. oscillator are described by the equation: E(v)=, where X 1 is the first constant
anharmonicity. The frequency of transition between neighboring levels does not remain constant, and, in addition, transitions are possible that meet the selection rules D v=2, 3, .... Frequency of transition from level v=0 to level v=1 rev. the main, or fundamental, frequency, transitions from level v=0 to levels v>1 give overtone frequencies, and transitions from levels v>0 - the so-called. hot frequencies.
In the IR absorption spectrum of diatomic molecules, they vibrate. frequencies are observed only in heteronuclear molecules (HCl, NO, CO, etc.), and the selection rules are determined by changing their electric. dipole moment during vibrations. In the Raman spectra fluctuate. frequencies are observed for any diatomic molecules, both homonuclear and heteronuclear (N 2 , O 2 , CN, etc.), because for such spectra, the selection rules are determined by the change in the polarizability of molecules during vibrations. Determined from the vibrational harmonic spectra. constants K e and v e , anharmonicity constants, as well as the dissociation energy D 0 are important characteristics of the molecule, necessary, in particular, for thermochemical. calculations. The study of vibration-rotation. spectra of gases and vapors allows you to determine the rotation. constants B v (see rotational spectra), moments of inertia and internuclear distances of diatomic molecules.
Polyatomic molecules are considered as systems of bound point masses. Swing. the motion of nuclei relative to equilibrium positions with a fixed center of mass in the absence of rotation of the molecule as a whole is usually described using the so-called. internal natural coordinates q i , chosen as changes in bond lengths, valence and dihedral angles of spaces, molecular model . A molecule consisting of N atoms has n=3N - 6 (for a linear molecule 3N - 5) vibrations. degrees of freedom. In the space of nature. coordinates q i complex oscillating. the motion of the nuclei can be represented by n separate oscillations, each with a certain frequency v k (k takes values from 1 to n), with which all nature changes. coordinates q i at amplitudes q 0 i and phases determined for a given oscillation. Such fluctuations are called normal. For example, a triatomic linear AX 2 molecule has three normal vibrations:
Oscillation v 1 rev. symmetric stretching vibration (stretching of bonds), v 2 - deformation vibration (change in the valence angle), v 3 antisymmetric stretching vibration. More complex molecules also contain other normal vibrations (changes in dihedral angles, torsional vibrations, cycle pulsations, etc.).
Quantization of oscillations. energy of a polyatomic molecule in the multidimensional harmonic approximation. oscillator leads to a trace, the system oscillates. energy levels:
where v ek - harmonic. oscillating constants, v k - fluctuating. quantum numbers, d k - the degree of degeneracy of the energy level in the k-th oscillatory. quantum number. Main frequencies in the vibrational spectra are due to transitions from the zero level [all v k =0, oscillatory. energy into levels characterized by 
such sets of quantum numbers v k, in which only one of them is equal to 1, and all the others are equal to 0. As in the case of diatomic molecules, in anharmonic. approximation, overtone and "hot" transitions are also possible, and, in addition, the so-called. combined, or
composite, transitions involving levels, for which two or more of the quantum numbers v k are nonzero (Fig. 2).
Rice. 2. System of vibrational terms E/hc (cm"; c is the speed of light) of the H 2 O molecule and certain transitions; v 1 , v 2 . v 3 are vibrational quantum numbers.
Interpretation and application. The vibrational spectra of polyatomic molecules are highly specific and present a complex picture, although the total number of experimentally observed bands can be significantly less than their possible number, theoretically corresponding to the predicted set of levels. Usually the main the frequencies correspond to more intense bands in the vibrational spectra. The selection rules and the probability of transitions in the IR and Raman spectra are different, since connected resp. with electrical changes dipole moment and polarizability of the molecule for each normal vibration. Therefore, the appearance and intensity of bands in the IR and Raman spectra depends differently on the type of vibrational symmetry (the ratio of the configurations of the molecule resulting from vibrations of nuclei to the symmetry operations , characterizing its equilibrium configuration). Some of the bands of the vibrational spectra can be observed only in the IR or only in the Raman spectrum, others with different intensities in both spectra, and some are not observed experimentally at all. So, for molecules that do not have symmetry or have low symmetry without an inversion center, all main. frequencies are observed with different intensities in both spectra; for molecules with an inversion center, none of the observed frequencies is repeated in the IR and Raman spectra (alternative exclusion rule); some of the frequencies may be absent in both spectra. Therefore, the most important of the applications of vibrational spectra is the determination of the symmetry of a molecule from a comparison of IR and Raman spectra, along with the use of other experiments. data. Given models of a molecule with different symmetry , it is possible to theoretically calculate in advance for each of the models how many frequencies in the IR and Raman spectra should be observed, and based on a comparison with experiments. data to make an appropriate choice of model. Although every normal vibration, by definition, is oscillatory. the movement of the entire molecule, some of them, especially in large molecules, can most of all affect only c.-l. fragment of a molecule. The displacement amplitudes of nuclei not included in this fragment are very small for such a normal vibration. This is the basis of the widely used structural analyte. research concept of the so-called. group, or characteristic, frequencies: certain funkts. groups or fragments repeating in molecules decomp. Comm., are characterized by approximately the same frequencies in the vibrational spectra, according to the Crimea can be. their presence in the molecule of a given substance has been established (although not always with an equally high degree of certainty). For example, the carbonyl group is characterized by a very intense band in the IR absorption spectrum in the region of ~1700(b 50) cm -1 related to the stretching vibration. The absence of absorption bands in this region of the spectrum proves that there is no group in the molecule of the studied in-va groups. At the same time, the presence of c.-l. bands in this area is not yet unambiguous evidence of the presence of a carbonyl group in the molecule, since frequencies of other vibrations of the molecule may accidentally appear in this region. Therefore, structural analysis and determination of conformations by vibrations. frequencies of the func. groups should be based on several. characteristic frequencies, and the proposed structure of the molecule must be confirmed by data from other methods (see Structural Chemistry). There are reference books containing numerous structural-spectral correlations; there are also data banks and corresponding programs for information retrieval systems and structural analysis. research using computers. Correct interpretation of the vibrational spectra helps isotopic. substitution of atoms, leading to a change in vibration. frequencies. Yes, replacement
