Molecular spectra. Vibrational spectra of diatomic molecules See what “vibrational spectra” are in other dictionaries

Infrared spectroscopy (IR) belongs to a broad group of molecular spectroscopy methods and is based on selective absorption of radiation in the infrared region (0.8 - 1000 μm) of the spectrum

Only those molecules of substances and compounds whose dipole moment changes during atomic vibrations can absorb infrared (IR) radiation

IR radiation is spent only on changing the vibrational and rotational energy of the molecule, without causing electronic transitions due to a 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 vibrational processes

IR spectra of molecules are characterized by high information content

Usually, to depict IR spectra, the abscissa axis is plotted frequency , wave number , less often - wavelength .

Wavelength () and frequency () are related to each other by the relation:

where C is the speed of radiation propagation in a certain environment.

To characterize electromagnetic radiation, the wave number ( ,  /) – 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 wave number is [cm–1]. The wave number is often called frequency, although it should be recognized that this is not entirely correct. They are proportional to each other.

The 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 IR radiation is usually expressed by the transmittance value (T):

where I is the intensity of radiation passing through the sample;

I 0 – intensity of incident radiation.

Infrared spectroscopy is a universal method for determining important functional groups, as well as structural fragments in small quantities of a substance in any state of aggregation.

The range of issues related in one way or another to the use of IR spectroscopy is extremely wide.

Using IR spectroscopy, it is possible to carry out identification of substances, structural group analysis, quantitative analysis, study of intra- and intermolecular interactions, determination of configuration, study of reaction kinetics, etc. Modern automatic IR spectrophotometers allow you to very quickly obtain an absorption spectrum, and the operator requires a minimum of special knowledge and skills. Let us consider the reasons for the absorption of IR radiation by molecules.

Vibrations of atoms in a molecule

Absorption of infrared radiation by a substance causes transitions between vibrational levels of the ground electronic state. At the same time, the rotational levels also change. Therefore, the IR spectra are vibrational-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 the molecule. For a harmonic oscillator, the restoring force is proportional to the magnitude of the displacement of the nuclei from the equilibrium position and is directed in the direction opposite to the displacement:

where K is the proportionality coefficient, which is called force constant and characterizes the rigidity of the connection (elasticity of the connection).

From the laws of classical mechanics it is known that the oscillation frequency of such a system is related to the force constant K and the atomic masses (m 1 and m 2) by the following relationship:

, (8.1)

where  – reduced mass,
.

The strength constants of single, double and triple bonds are in a ratio of approximately 1:2:3.

From relation (8.1) it follows that the vibration frequency increases with increasing bond strength (bond multiplicity) and with decreasing atomic masses.

Those. the frequency depends on the mass of the atoms: a lighter atom means a higher 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: (stronger bond – higher frequency)

C≡O (2143 cm -1), C=O (1715 cm -1), C-O (1100 cm -1).

If we assume that, to a first approximation, vibrations for a diatomic molecule are harmonic, and thus such a molecule is likened to a harmonic oscillator, then the value of the total vibration energy obeys the basic quantum condition:

, (8.2)

where  is a vibrational quantum number that takes the values ​​of integers: 0, 1, 2, 3, 4, etc.;

 0 – frequency of the fundamental 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 = 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 is clear that the frequency of absorbed radiation () is equal to the main vibrational frequency ( 0), determined by equation (8.1).

Thus, the spectrum of a harmonic oscillator consists of one line or band with frequency  0, which is the natural frequency of the oscillator (Fig. 8. 1).

Typically, at room temperature, most molecules are in the lower vibrational state, since the energy of thermal excitation is much less than the energy of transition from the ground state to the excited state.

Therefore, experimentally it is easiest to 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 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).

Due to the superposition of rotational transitions, the vibrational spectral line 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 energy difference between the rotational sublevels is less than that of the vibrational ones. Of all vibrational transitions, the most probable is the transition to the nearest vibrational sublevel. It corresponds to 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, but the intensity is much less. The main line is denoted by , and the overtones are denoted by 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 bond lengths, and the angles between the bonds change little, then such a vibration is called valence and is denoted by . Stretching vibrations can be symmetrical ( s) and asymmetrical ( as).

A necessary condition for an oscillatory transition is a change dipole moment molecules during atomic vibrations. A symmetrical molecule that does not have a dipole moment cannot absorb infrared radiation. The ability of a substance to absorb the energy of IR 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 vibration and rotation, can absorb infrared radiation. The dipole moment means the mismatch between the centers of gravity of positive and negative charges in a 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 lack a dipole moment and do not acquire it during 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 stretching vibrations of atoms, etc.).

If, during vibrations of a molecule, the angle between the bonds changes without changing the length of the bonds, then such vibrations are called deformation.

These fluctuations are indicated by:  or  . They can also be symmetrical ( s,  s) and asymmetrical ( a s,  a s).

Deformation vibrations are divided into fan, torsional, scissor and pendulum. The same categories are acceptable for describing the vibrations of individual groups.

Each type of vibration is characterized by a certain excitation energy. Stretching vibrations correspond to higher energies than bending vibrations, and, therefore, the bands of stretching vibrations lie in the shorter wavelength region (or at higher frequencies).

Rotational spectra

Let's consider the rotation of a two-atomic molecule around its axis. A 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... the rotation speed is 2,3,4... times greater than at j=0. The internal energy of a molecule increases with increasing rotation speed and the distance between levels increases. The energy difference between neighboring levels increases all the time by the same amount E 1 . In this regard, the rotational spectrum consists of individual lines; for the first line ν 1 = E 1 /ħ, and the next 2ν 1, 3 ν 1, etc. The energy difference between the rotational levels is very small, so even at room temperature the kinetic energy of molecules upon their collision is sufficient to excite the rotational levels. The 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 toward longer wavelengths.

Rotational spectra can be observed for substances in the gaseous state. In liquid and solid bodies there is practically no figurative rotation. The need to transform the analyte into a gaseous state without destroying it greatly limits the use of rotational spectra (as well as the difficulty of working in the far-IR region).

If the molecule is given additional energy less than the energy of breaking the E chemical bond, then the atoms will vibrate around the equilibrium position, and the amplitude of the vibrations will have only certain values. In vibrational spectra, bands are observed, 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. Liquids and solids do not have specific rotational levels. So there is one wide stripe in them. The vibrations of polyatomic molecules are much more complex than those of 2-atomic molecules, because the number of possible types of vibrations increases rapidly with the number of atoms in the molecule.

For example, a linear CO 2 molecule has 3 types of vibrations.

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 valence bonds, the length of which remains almost constant. Such vibrations are called deformation vibrations. 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 vibrations are called skeletal. They are characteristic only of a given molecule and the corresponding bands do not coincide even with substances with a similar structure.

Complex molecules also exhibit vibrations 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. Thus, 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 identical spectra.

As was established in the previous section, when transitioning between rotational levels, the rotational quantum number can change by one. If we limit ourselves to the first term in formula (11.15) and take , then the expression for the frequencies of rotational transitions will take the form:

, (13.1)

i.e. with increasing per unit, the distance between rotational levels increases by
.

In this case, the distance between adjacent rotational lines in the spectrum is:

. (13.2)

The slide shows allowed transitions between rotational levels and an example of the observed rotational absorption spectrum.

However, if we take into account the second term in expression (11.15), it turns out that the distance between adjacent spectral lines with increasing number J decreases.

As for the intensities of rotational spectral lines, it should first of all be said that they depend significantly on temperature. Indeed, the distance between adjacent rotational lines of many molecules is significantly less than kT. Therefore, when the temperature changes, the populations of the rotational levels change significantly. As a consequence, the intensities of the spectral lines change. It is necessary to take into account that the statistical weight of rotational states is equal to
. Expression for the population of a rotational level with number J therefore it looks like:

The dependence of the populations of rotational levels on the number of rotational quantum number is illustrated on the slide.

When calculating the intensity of a spectral line, it is necessary to take into account the populations of the upper and lower levels between which the transition occurs. In this case, the average value from the statistical weights of the upper and lower levels is taken as the statistical weight:

Therefore, the expression for the intensity of the spectral line takes the form:

This dependence has a maximum at a certain value J, which can be obtained from the condition
:

. (13.6)

For different molecules of size J max have a wide spread. Thus, for a CO molecule at room temperature the maximum intensity corresponds to the 7th rotational level, and for an iodine molecule – to the 40th.

The study of rotational spectra is of interest for the experimental determination of the rotational constant B v, since measuring its value makes it possible to determine internuclear distances, which in turn is valuable information for constructing potential interaction curves.

Let us now turn to consideration of vibrational-rotational spectra. There are no pure vibrational transitions, since when transitioning between two vibrational levels, the rotation numbers of the upper and lower levels always change. Therefore, to determine the frequency of the vibrational-rotational spectral line, one must proceed from the following expression for the vibrational-rotational term:

. (13.7)

To obtain a complete picture of the vibrational-rotational spectra, proceed as follows. As a first approximation, we will neglect the presence of a rotational structure and consider only transitions between vibrational levels. As was shown in the previous section, there are no selection rules for changing vibrational quantum numbers. However, there are probabilistic properties, which are as follows.

Firstly, the statistical weight for the vibrational levels of molecules is equal to unity. Therefore, the populations of vibrational levels decrease with increasing V(picture on slide). As a consequence, the intensities of the spectral lines decrease.

Secondly, the intensities of spectral lines decrease sharply with increasing  V approximately in the following ratio:.

About transitions with  V=1 is spoken of as transitions at the fundamental frequency (1-0, 2-1), transitions with V>1 are called overtone ( V=2 – first overtone (2-0), V=3 – second overtone (3-0, 4-1), etc.). Transitions in which only excited vibrational levels (2-1, 3-2) participate are called hot, since to register them, the substance is usually heated in order to increase the population of excited vibrational levels.

The expression for the transition frequencies at the fundamental frequency, taking into account the first two terms in (h), has the form:

and for overtones:

These expressions are used to experimentally determine vibrational frequencies and constant anharmonicity
.

In fact, if you measure the frequencies of two adjacent vibrational transitions (picture on the slide), you can determine the magnitude of the vibrational quantum defect:

(13.10)

After this, using expression (12.8), the value is determined .

Now let's take into account the rotational structure. The structure of the rotational branches is shown on the slide. It is characteristic that, due to the selection rules for a change in the rotational quantum number, the first line in R-branch is a line R(0), and in P-branches – P(1).

Having designated
, let's write expressions for frequencies P- And R-branches.

Limiting ourselves to one term in (11.15), for the frequency R-branch we get the equation:

Where

Likewise, for P-branches:

Where

As stated above, as the number of the vibrational quantum number increases, the value of the rotational constant will decrease. Therefore always
. Therefore, the signs of the coefficients for For P- And R-the branches are different, and with growth J spectral lines R-the branches begin to converge, and the spectral lines P- branches - diverge.

The resulting conclusion can be understood even more simply if we use simplified expressions for the frequencies of both branches. Indeed, for neighboring vibrational levels, the probabilities of transitions between which are the greatest, we can, to a first approximation, assume that
. Then:

From this condition, in addition, it follows that the frequencies in each of the branches are located on different sides of . As an example, the slide shows several vibrational-rotational spectra obtained at different temperatures. An explanation of the patterns of intensity distribution in these spectra is given by considering purely rotational transitions.

Using vibrational-rotational spectra, it is possible to determine not only vibrational, but also rotational constants of molecules. So, the value of the rotational constant
can be determined from the spectrum consisting of the lines shown on the slide. It is easy to see that the quantity

directly proportional
:
.

Likewise:

Accordingly constant
And
determined from dependencies from the number of the rotational level.

After this, you can measure the values ​​of the rotational constants
And
. To do this you need to build dependencies

. (13.16)

In conclusion of this section, we will consider the electronic-vibrational-rotational spectra. In general, the system of all possible energy states of a diatomic molecule can be written as:

Where T e is the term of the purely electronic state, which is assumed to be zero for the ground electronic state.

Purely electronic transitions are not observed in the spectra, since the transition from one electronic state to another is always accompanied by a change in both vibrational and rotational states. The vibrational and rotational structures in such spectra appear in the form of numerous bands, and the spectra themselves are therefore called striped.

If in expression (13.17) we first omit the rotational terms, that is, in fact, limit ourselves to electronic-vibrational transitions, then the expression for the position of the frequencies of electronic-vibrational spectral lines will take the form:

Where
– frequency of purely electronic transition.

The slide shows some of the possible transitions.

If transitions occur from a certain vibrational level V'' to different levels V’ or from various V’ to the same level V'', then the series of lines (strips) obtained in this case are called progressions By V’ (or by V''). Series of bars with constant value V’- V'' are called diagonal in series or sequences. Despite the fact that the selection rules for transitions with different values V does not exist, a fairly limited number of lines are observed in the spectra due to the Franck-Condon principle discussed above. For almost all molecules, the observed spectra contain from several to one to two dozen systems of bands.

For the convenience of representing electronic vibrational spectra, the observed systems of bands are given in the form of so-called Delandre tables, where each cell is filled with the value of the wave number of the corresponding transition. The slide shows a fragment of Delandre's table for the BO molecule.

Let us now consider the rotational structure of electronic vibrational lines. To do this, let's put:
. Then the rotational structure is described by the relation:

In accordance with the rules of selection by quantum number J for frequencies P-,Q- And R-branches (limiting ourselves to quadratic terms in formula (11.15)) we obtain the following expressions:

Sometimes for convenience the frequency P- And R-branches are written with one formula:

Where m = 1, 2, 3… for R-branches ( m =J+1), and m= -1, -2, -3... for P-branches ( m = -J).

Since the internuclear distance in one of the electronic states can be either greater or less than in the other, the difference
can be either positive or negative. At
<0 с ростомJ frequencies in R-the branches gradually stop growing and then begin to decrease, forming the so-called edging (highest frequency R-branches). At
>0 edge is formed in P-branches

Dependence of the position of the lines of the rotational structure on the quantum number J called a Fortra diagram. As an example, such a diagram is shown on the slide.

To find the quantum rotational number of the vertex of the Fortr diagram (corresponding to the edge), it is necessary to differentiate expression (13.23) with respect to m:

(13.24)

and set it equal to zero, after which:

. (13.25)

Distance between edge frequency and in this case:

. (13.26)

To conclude this section, we will consider how the position of the energy states of a molecule is affected by isotopic substitution of nuclei (a change in the mass of at least one of the nuclei without a change in charge). This phenomenon is called isotopic shift.

First of all, you should pay attention to the fact that the dissociation energy (see figure on slide) is a purely theoretical value and corresponds to the transition of a molecule from a hypothetical state corresponding to the minimum potential energy , into a state of two non-interacting atoms located at an infinite distance from each other. The quantity is measured experimentally , since the molecule cannot be in a state lower than the ground state with
, whose energy
. From here
. A molecule dissociates if the sum of its own potential energy and the communicated one exceeds the value .

Since the interaction forces in a molecule are of an electrical nature, the influence of the mass of atoms with the same charge during isotopic substitution should not affect the potential energy curve, dissociation energy and on the position of the electronic states of the molecule.

However, the position of vibrational and rotational levels and the magnitude of the dissociation energy should change significantly. This is due to the fact that the expressions for the energies of the corresponding levels include the coefficients
And , depending on the reduced mass of the molecule.

The slide shows the vibrational states of a molecule with a reduced mass (solid line) and a heavier isotopic modification of the molecule (dashed line) with reduced mass . The dissociation energy for a heavier molecule is greater than for a light one. Moreover, with an increase in the vibrational quantum number, the difference between the vibrational states of isotope-substituted molecules gradually increases. If you enter the designation
, then it can be shown that:

<1, (13.27)

since constant
for isotopically substituted molecules is the same. For the ratio of anharmonicity coefficients and rotational constants we obtain:

,. (13.28)

It is obvious that with an increase in the reduced mass of molecules, the magnitude of isotopic effects should decrease. So, if for light molecules D 2 and H 2
0.5, then for the isotopes 129 I 2 and 127 I 2
0.992.

They represent a model of two interacting point masses m 1 and m 2 with an equilibrium distance r e between them (bond length), and oscillations. the movement of nuclei is considered harmonic and is described by the unity coordinate q=r-r e, where r is the current internuclear distance. Dependence of potential energy of oscillations. the movements of V from q are determined in the harmonic approximation. oscillator [oscillating material point with reduced mass m =m 1 m 2 /(m 1 +m 2)] as a function V= l / 2 (K e q 2), where K e =(d 2 V/dq 2) q=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 with the equilibrium value of r); horizontal straight lines show oscillations. levels (0, 1, 2, ... values ​​of vibrational quantum number), vertical arrows - certain vibrations. transitions; D 0 - molecule dissociation energy; The shaded area corresponds to the continuous spectrum. molecules (dashed curve in Fig. 1). According to the classic mechanics, harmonic frequency fluctuations Quantum Mech. consideration of such a system gives a discrete sequence of equally spaced 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 transitioning between adjacent levels, according to the selection rule D v=1, a photon with energy hv= is absorbed 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 hesitation. Therefore v e is called. also harmonious frequency. For real molecules, the potential energy curve is not the indicated quadratic function q, i.e., a parabola. Oscillation the levels become increasingly closer as they approach 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 transitions between adjacent 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 called. fundamental, 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 there are vibrations. frequencies are observed only in heteronuclear molecules (HCl, NO, CO, etc.), and the selection rules are determined by changes in their electrical. dipole moment during vibrations. In the Raman spectra there are vibrations. 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 spectra of harmonics. constants K e and v e , anharmonicity constants, as well as dissociation energy D 0 are important characteristics of the molecule, necessary, in particular, for thermochemical processes. calculations. Study of vibration-rotation. spectra of gases and vapors allows you to determine rotation. constants B v (see Rotational spectra), moments of inertia and internuclear distances of diatomic molecules. Polyatomic molecules are considered as systems of connected point masses. Oscillation the movement of nuclei relative to equilibrium positions with a stationary 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, bond and dihedral angles of spaces, molecule model. A molecule consisting of N atoms has n=3N - 6 (a linear molecule has 3N - 5) vibrations. degrees of freedom. In the space of natural coordinates q i complex oscillation. the movement of 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 natural frequencies change. coordinates q i at amplitudes q 0 i and phases defined for a given oscillation. Such fluctuations are called normal. For example, a triatomic linear molecule AX 2 has three normal vibrations:


Oscillation v 1 called. symmetric stretching vibration (stretching bonds), v 2 - deformation vibration (change in bond angle), v 3 antisymmetric stretching vibration. In more complex molecules, other normal vibrations also occur (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, a system of oscillations. energy levels:
where v ek - harmonic. oscillate constant, v k - oscillation. quantum numbers, d k - degree of degeneracy of the energy level over the kth oscillations. quantum number. Basic frequencies in the vibrational spectra are due to transitions from the zero level [all v k =0, oscillations. energy to 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. approaching, 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 - speed of light) of the H 2 O molecule and certain transitions; v 1, v 2. v 3 - 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 may be significantly less than their possible number, theoretically corresponding to the predicted set of levels. Usually basic 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, because related respectively with electrical changes dipole moment and polarizability of the molecule at 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 arising as a result of vibrations of the nuclei to the symmetry operations characterizing its equilibrium configuration). Some of the bands of 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, everything is basic. frequencies are observed with different intensities in both spectra; for molecules with an inversion center, none of the observed frequencies are repeated in the IR and Raman spectra (alternative exclusion rule); Some frequencies may be absent in both spectra. Therefore, the most important application 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 molecules with different symmetries, it is possible to theoretically calculate in advance for each of the models how many frequencies should be observed in the IR and Raman spectra, and based on comparison with experiment. data to make the appropriate choice of model. Although every normal fluctuation, by definition, is an oscillation. by the movement of the entire molecule, some of them, especially in large molecules, can most of all affect only the cl. fragment of a molecule. The displacement amplitudes of nuclei not included in this fragment are very small during such a normal oscillation. On this basis is widely used in structural analysis. research concept of the so-called. group, or characteristic, frequencies: certain functions. groups or fragments repeated in decomposition molecules. conn., are characterized by approximately the same frequencies in the vibrational spectra, according to the Crimea m.b. their presence in the molecule of a given substance has been established (though not always with an equally high degree of reliability). 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 substance under study. At the same time, the presence of k.-l. bands in the indicated area is not yet unambiguous evidence of the presence of a carbonyl group in the molecule, because the frequencies of other vibrations of the molecule may accidentally appear in this region. Therefore, structural analysis and determination of conformations based on vibrations. function frequencies groups should rely on several. characteristic frequencies, and the proposed structure of the molecule must be confirmed by data from other methods (see Structural chemistry). There are directories containing numerous. structural-spectral correlations; There are also data banks and corresponding programs for information retrieval systems and structural analysis. research using computers. The correct interpretation of vibrational spectra is helped by isotope. substitution of atoms, leading to a change in vibrations. frequency Yes, replacement

Simultaneously with the change in the vibrational state of the molecule, its rotational state also changes. Changes in vibrational and rotational states lead to the appearance of rotational-vibrational spectra. The vibrational energy of molecules is approximately one hundred times greater than its rotational energy, so rotation does not disturb the vibrational structure of molecular spectra. The superposition of rotational quanta, which are small in energy terms, on vibrational quanta, which are relatively large in energy, shifts the lines of the vibrational spectrum to the near infrared region of the electromagnetic spectrum and turns them into bands. For this reason, the rotational-vibrational spectrum, which is observed in the near infrared region, has a line-striped structure.

Each band of such a spectrum has a central line (dashed line), the frequency of which is determined by the difference in the vibrational terms of the molecule. The set of such frequencies represents the pure vibrational spectrum of the molecule. Quantum mechanical calculations related to the solution of the Schrödinger wave equation, taking into account the mutual influence of the rotational and vibrational states of the molecule, lead to the expression:

where and are not constant for all energy levels and depend on the vibrational quantum number.

where and are constants, smaller in magnitude than and . Due to the smallness of the parameters and , in comparison with the values ​​and , the second terms in these relationships can be neglected and the actual rotational-vibrational energy of the molecule can be considered as the sum of the vibrational and rotational energy of a rigid molecule, then the corresponding expression is:

This expression conveys the structure of the spectrum well and leads to distortion only at large values ​​of quantum numbers and . Let us consider the rotational structure of the rotational-vibrational spectrum. Thus, during radiation, a molecule moves from higher energy levels to lower ones, and lines with frequencies appear in the spectrum:

those. for the frequency of the line of the rotational-vibrational spectrum can be written accordingly:

the combination of frequencies gives a rotational-vibrational spectrum. The first term in this equation expresses the spectral frequency that occurs when only the vibrational energy changes. Let us consider the distribution of rotational lines in spectral bands. Within the boundaries of one band, its fine rotational structure is determined only by the value of the rotational quantum number. For such a band it can be written in the form:

According to Pauli's selection rule:

the entire band is divided into two groups of spectral series, which are located relatively on both sides. Valid if:

those. When:

then we get a group of lines:

those. When:

then we get a group of lines:

In the case of transitions, when a molecule moves from the th rotational level to a rotational energy level, a group of spectral lines with frequencies appears. This group of lines is called the positive or - branch of the spectrum band, starting with . During transitions, when a molecule moves from the th to the energy level, a group of spectral lines appears, with frequencies. This group of lines is called the negative or - branch of the spectrum band, starting with . This is explained by the fact that the value that corresponds has no physical meaning. - and - strip branches, based on equations of the form:

consist of lines:

Thus, each band of the rotational-vibrational spectrum consists of two groups of equidistant lines with a distance between adjacent lines:

for a real non-rigid molecule, given the equation:

for the frequency of lines - and - strip branches, we obtain:

As a result, the lines of - and - branches are curved and not equidistant lines are observed, but - branches that diverge and - branches that come closer to form the edge of the strip. Thus, the quantum theory of molecular spectra has proven capable of deciphering spectral bands in the near-infrared region, treating them as the result of simultaneous changes in rotational and vibrational energy. It should be noted that molecular spectra are a valuable source of information about the structure of molecules. By studying molecular spectra, one can directly determine the various discrete energy states of molecules and, based on the data obtained, make reliable and accurate conclusions regarding the movement of electrons, vibrations and rotation of nuclei in a molecule, as well as obtain accurate information regarding the forces acting between atoms in molecules, internuclear distances and geometric the location of nuclei in molecules, the dissociation energy of the molecule itself, etc.