Oscillation types
The energy required to excite vibrations of atoms in a molecule corresponds to the energy of light quanta with a wavelength of 1-15 μm or a wave number of 400÷4000 cm -1, i.e. electromagnetic radiation of the middle infrared (IR) region. The vibrational levels of molecules are quantized, the energy of transitions between them and, consequently, the vibrational frequencies can only have strictly defined values. By absorbing a quantum of light, the molecule can move to a higher vibrational level, usually from the ground vibrational state to an excited one. The absorbed energy is then transferred to the excitation of rotational levels or is converted into the kinetic energy of molecules. Vibrations of molecules appear in two types of spectra: absorption spectra in the infrared region (IR spectra) and spectra of Raman scattering of light (Raman spectra).
The mathematical model of vibrations of polyatomic molecules is complex. The calculations were carried out only for the simplest diatomic molecules. Vibrational spectroscopy is mainly empirical in nature; the main vibration frequencies were obtained by comparing the spectra of many compounds of the same class. This, however, does not detract from the value of the method.
The main types of vibrations are valence and deformation.
Valence oscillations are called vibrations of the nuclei of atoms along the communication line, they are denoted by the letter n (nC=C, nC=O etc.).
An approximate mechanical model of stretching vibrations can be a system of two balls connected by a rigid spring (here the balls represent atoms, and the spring represents a chemical bond) (see Fig., A).
A, B - stretching vibrations in molecules;
C - deformation vibrations: I, II - scissor; III, IV - pendulum; V - fan; VI - torsion.
When the spring is stretched or compressed, the balls will begin to oscillate around the equilibrium position, i.e., a harmonic oscillation will be carried out, described by the equation
Where n - oscillation frequency; F - force constant, which characterizes the strength of the connection, or the force that returns the balls to the equilibrium position; m r - reduced mass of balls (atoms), calculated by the formulas
The frequencies of stretching vibrations are determined by the mass of atoms and the strength (energy) of the bond. The larger the mass, the lower the frequency, for example:
n C - C » 1000 cm -1; n C -H» 3000 cm -1
The stronger the bond, the higher the oscillation frequency, for example:
Perhaps the appearance of overtones - oscillations, the frequency of which is an integer number of times greater than that of the main ones ( 2n, 3n etc.). Usually the intensity of the overtones is much less: for the first overtone it is 1-10% of the intensity of the fundamental vibration; the third overtone is usually not found.
In a system of three or four atoms, two types of stretching vibrations are possible - in-phase (in one phase, or symmetrical, n s ) and antiphase (in different phases, or antisymmetric, as ) (Fig. B), although the terms are fully applicable to symmetrical molecules. The frequency of the out-of-phase oscillation is always higher than the in-phase one.
deformation vibrations are associated with a change in the bond angle formed by the bonds of a common atom; they are marked with the letter d . The types of some bending vibrations are shown in Fig., C. To excite bending vibrations, less energy is required than in the case of stretching vibrations, and, therefore, they have a lower frequency.
As the number of atoms in a molecule increases, the number of possible vibrations increases rapidly. In a real molecule, the vibrations of atoms are closely related to each other and interact with each other. The spectra of molecules are a complex set of different vibrations, each of which manifests itself in a narrow frequency range.
The absorption intensity is determined, as in UV spectroscopy, by the molar absorption coefficient, but in this case the measurement accuracy is much lower. Typically, the intensity of the bands is expressed as absorption (A) or transmission (T) of the light flux in percentages. The bands are also rated by intensity as strong ( With.), average ( cf.) and weak ( sl.).
Obtaining IR spectra
The basis for obtaining IR spectra is the direct absorption of radiation when passing through a layer of matter. From the wide range of IR radiation, the middle region (400-4000 cm -1) is usually used. In the near infrared region (4000÷14300 cm -1), where overtones appear mainly, sometimes a quantitative analysis is carried out. In the far IR region (100÷400 cm -1) get almost only vibrations of carbon-metal bonds.
The scheme of the IR spectrometer is similar to that of the UV spectrometer, but the design of the instruments is more complex. IR radiation is thermal; its source is usually a ceramic rod heated by a passing electric current. Using a system of mirrors, the luminous flux is divided into two identical beams, one of which is passed through the cell with the substance, the other through the reference cell. The radiation passed through the cuvettes enters the monochromator, which consists of a rotating prism, a mirror, and a slit, which makes it possible to isolate radiation with a strictly defined frequency and smoothly change this frequency. Given that in the IR region, most substances are opaque, prisms are made from single crystals of salts. In high-end instruments, three prisms are used: from LiF(2000÷3800 cm -1), NaCl(700÷2000 cm -1) and KVR(400÷700 cm -1). Each of the prisms in a different wavenumber range gives a much lower resolution. In a number of devices, the dispersion of radiation is carried out using diffraction gratings. The intensities of the two light fluxes (main and comparison beam) that have passed through the monochromator are automatically subtracted from one another. The electrical signal generated when the resulting light flux hits a thermocouple-type detector is amplified and recorded by a self-recording potentiometer. The record is an IR spectrum as a dependence of absorption or transmission (in %) on frequency (in cm -1) or wavelength (in µm). A typical view of the spectrum is shown in fig.
IR spectra are most often obtained as follows:
1. Solutions of substances most convenient for obtaining spectra, since in this case there are no intermolecular interactions. Due to the fact that any substance absorbs in the IR region, compounds of the simplest structure are used as solvents, the spectrum of which has the simplest form (the minimum number of bands), and most often carbon tetrachloride, which is transparent above 1300 cm -1, as well as carbon disulfide , practically transparent and below 1300 cm -1 . By successively dissolving a substance in both solvents, it is possible to record the entire IR spectrum.
For solutions, cylindrical cuvettes with a thickness of 0.1 ÷ 1.0 mm with windows made of salt plates are used. The volume of solution required to fill the cuvette is 0.1 ÷ 1.0 ml at a concentration of 0.05 ÷ 10%.
2. Thin films (<0,01 мм) жидкого вещества, помещенные между солевыми пластинами, удерживаемыми капиллярными силами.
3. Pastes prepared by carefully triturating a solid sample with vaseline oil and placed in a thin layer between salt plates. Vaseline oil itself, which is a mixture of hydrocarbons, absorbs intensively in the region of »2900 cm -1 and »1400 cm -1 . Sometimes hexachlorobutadiene is used to prepare pastes, which is transparent above 1600 cm -1 and in the region of 1250÷1500 cm -1, i.e. in those frequency ranges in which vaseline oil absorbs.
4. Solids in the form of a fine powder(0.5÷1.0 mg), thoroughly mixed with potassium bromide powder (~100 mg) and then compressed in a special device under pressure up to "4.5 × 10 8 Pa into a thin plate.
5. Method frustrated total internal reflection(ATI):
The amount of substance required to obtain an IR spectrum, regardless of the method of sample preparation, is 0.5÷2 mg.
Since the cuvette material is salt plates, the sample must not contain water. The method of IR spectroscopy is one of the most accessible in laboratory practice. The instruments are easy to use and take only a few minutes to acquire the spectrum.
Another type of spectra that carry information about oscillations in this range are Raman spectra (RS).
Their main feature is the fixation of wavelengths mainly in the visible range.. The condition for their production is the presence of a high-intensity source of highly monochromatic radiation, more often a laser, and initially, individual lines of the atomic spectrum of a low-pressure fluorescent mercury lamp.
The spectrum appears as a result of inelastic interactions of light beam photons with substance molecules. A photon, colliding with an electron of a molecule, can transfer it to a higher molecular energy level, while losing some of the energy. The lines that appear are called Stokes . It is possible that a photon collides with an electron at a high molecular energy level and transfers it to a lower orbit, capturing some of its energy. Lines appear that are symmetrical to Stokes with respect to the main line (incident photon) and are called anti-Stokes . Stokes lines, i.e. less energetic, more intense, because the process of energy transfer by a photon to an electron is more probable. However, all the lines of the Raman spectra are of low intensity compared to the exciting line (only about 10 -7 of the total scattered light intensity). Therefore, Raman spectra are recorded perpendicular to the direction of the exciting beam. Spectrum registration is carried out as usual. At the same time, near the main excitation line n 0 a series of narrow lines is formed, corresponding to n i . According to the distances between n 0 And n i values are determined Dn .
The form of the spectrum is similar to that obtained in IR spectroscopy. In modern devices, scattered light is excited by a monochromatic laser beam, which makes it possible to dispense with 1–10 mg of a substance when obtaining a spectrum. The sample can be administered as a pure liquid or solution, or as a solid powder.
Electromagnetic oscillations arise due to the movement of charges. Accordingly, their absorption is associated with a displacement of charges. Obviously, direct absorption in the IR region will occur with sufficient intensity if the bond is polar. In the Raman spectra, intense bands give rise to symmetric vibrations of nonpolar bonds, since in this case the dipole moment arising in the process of oscillation is important. Therefore, in the simplest cases vibrations that are inactive in the Raman spectra should appear in the IR spectra, and, accordingly, vice versa. For symmetrical molecules, antiphase vibrations are active in the IR spectra, while in-phase vibrations are active in the Raman spectra. As the symmetry of the molecule decreases, many vibrations are quite intense in both spectra. Consequently, the IR and Raman spectra complement each other, and the combined use of these methods can provide maximum information on the vibrational frequencies of the substance under study.
Bands in vibrational spectra are divided into two types. Characteristic(mainly valence) bands, the presence of which in the spectrum proves the presence of certain structural elements in the substance under study.
Characteristic oscillations are those that, in at least one parameter, (m r or F ) significantly different from the main fluctuations S-S (these are vibrations of light atoms: S-N, O-N, N-N or multiple bonds).
The characteristic oscillation belongs to a certain relation and hence has a fairly constant frequency in various substances, which changes only slightly due to interaction with the rest of the molecule.
Non-characteristic bands occupying the region 400÷1000 cm -1 , where numerous non-attributable stretching vibrations of bonds are manifested C-C, C-N, N-O and deformation vibrations. This is the region of vibrations of the carbon skeleton of the molecule, which reacts sharply to the slightest changes in the structure of the molecule. Non-characteristic vibrations make up the main part of the spectrum and for each substance they form their own, unique set of bands. No two compounds, with the exception of enantiomers (optical antipodes), have the same IR spectra (and Raman spectra). This is often used to establish the identity of substances, since the coincidence of the IR spectra is a convincing proof of the identity of the studied samples.
Considering that in the spectrum of one substance a band can always be found that is absent in the spectrum of another, qualitative analysis of mixtures is possible if the spectra of the components are known.
On the same basis, a quantitative analysis can be performed by measuring the intensities of the respective bands. When the structure of the substance has already been established, in the non-characteristic region of the spectrum, some bands can be attributed to certain vibrations.
However, the researcher is usually faced with the opposite task - to establish the structure from the spectrum. In this respect the possibilities of IR spectroscopy should not be overestimated, only absolutely reliable criteria should be used.
In particular, the data obtained from the examination of the area below 1500 cm -1 cannot be regarded as evidence, but only as evidence in favor of the presence of one or another structural element. The use of small changes in the value of the characteristic frequency for structural assignments (in particular, for determining the conformation and the nearest environment) can lead to erroneous conclusions.
In other words, one should not try to extract information from vibrational spectra, the reliability of which is doubtful.
To describe vibrational spectra, the following information is most often used:
Communication fluctuations S-N. Stretching vibrations C-H at a saturated carbon atom appear in the region of 2800÷3000 cm -1 . For acyclic and unstrained cyclic structures, n CH have the following values (in cm -1):
| CH 3 | 2962 cm -1 | 2972 cm -1 |
| CH 2 | 2853 cm -1 | 2926 cm -1 |
| CH | 2890 cm -1 |
The bands are characteristic, but not very informative, since various vibrations are usually observed in the substance. S-N , which, moreover, can interact with each other. Separate oscillation bands are superimposed on each other, forming a band in the region of 2800÷3000 cm -1, which has separate weakly expressed maxima. These bands can be useful for determining the structure of a substance only if there are few hydrogen atoms in the compound, as, for example, in polyhaloalkanes. The absence of bands in this region is a convincing proof of the absence of hydrogen atoms in the substance at saturated carbon atoms.
Deformation oscillations dCH , located in the region of 1350÷1470 cm -1, are of little characteristic, but can usually be found in the spectrum:
| CH 3 | 1375 cm -1 | 1450 cm -1 |
| CH 2 | 1475 cm -1 |
The absorption of two methyl groups at one carbon atom (geminal substitution) is considered quite characteristic, forming two close maxima (doublet) of approximately equal intensity in the region of 1370÷1385 cm -1 .
The information content of the spectra can be increased by using differences in the vibrational frequencies of bonds containing various isotopic modifications of atoms. In particular, deuterated compounds enriched in deuterium instead of protium are often used.
In the analysis of compounds labeled with deuterium, the band n CD 2100÷2160 cm -1 located in the area where there are practically no other bands.
Fluctuations in the C=C bond. In compounds with an isolated double bond v c = c is at 1600-1680 cm -1 .
In cyclic systems, especially in stressed ones, the value of this frequency is somewhat lower. The vibrational frequency of the double bond increases markedly with an increase in the degree of its substitution, for example:
In the IR spectra of symmetrically substituted alkenes (non-polar double bond) n С=С manifests itself as a band of negligible intensity, as, for example, in the spectra of compounds (I) and (III); for an unsymmetrically substituted double bond (for example, in compound II), this band is rather intense. In the Raman spectra, the oscillation C=C in any case more active than in the IR spectrum, and any double bond gives a powerful (usually the most intense in the spectrum) line. The presence of a double bond in the substance may additionally be indicated by the characteristic band(s) n=CH located in the region 3000÷3100 cm -1 .
Deformation oscillations d=CH can be useful for determining the configuration of substituents on a double bond: for cis-isomers they are located in the region of 650÷750 cm -1 , and for trans-isomers - in the region of 960÷970 cm -1 .
Thus, based on the data of vibrational spectra (especially the Raman spectrum), the presence of an isolated double bond in the substance can be detected and certain conclusions can be drawn about the nature of its substitution.
Band n = C D very characteristic (2200÷2300 cm -1) and allows you to confidently distinguish the deuterium atom, located at the double bond, from the D atom at a saturated carbon atom.
Vibrations of conjugated diene systems.
Conjugated diene systems in the region of 1500÷1650 cm -1 have two bands corresponding to two types of stretching vibrations - in-phase and anti-phase, for example: 
On the whole, the bands of vibrations of diene systems in the IR and Raman spectra are much more intense than the bands of isolated double bonds, especially if the diene system has a transoidal configuration. In the IR spectrum, the vibration is more active, while in the Raman spectrum, the vibration is more active. In the IR spectrum of symmetrical dienes (for example, butadiene), the intensity of the band can be vanishingly small. When alkyl substituents are introduced into the diene system, the values of the frequencies and naturally increase. fluctuations n=CH in dienes they appear in the same region as in alkenes (3000÷3100 cm -1).
Thus, the presence of a diene system in a substance is relatively easy to determine from the data of vibrational spectra. When a double bond is conjugated with an aromatic nucleus, the frequency of its vibration is shifted to the low-frequency region (by »30 cm -1), while the absorption intensity increases. With an increase in the length of the conjugation chain (in the spectra of polyenes), the total number of bands increases n С=С , and the frequencies of their oscillations decrease, and the intensity increases significantly.
Vibrations of aromatic systems. Stretching vibrations of C-C bonds of the benzene nucleus give bands of moderate intensity at 1585÷1600 cm -1 and 1400÷1500 cm -1 , which makes them inconvenient for identification, since this region is close to vibrations n C=C. Vibrations n CH arenes lie in the region of 3020÷3100 cm -1 ; they usually appear as a group of medium-intensity bands, slightly greater than those of alkenes absorbing in the same n = CH region.
The spectra of aromatic compounds contain intense bands of out-of-plane bending vibrations S-N in the region of 650÷900 cm -1 . This area provides some opportunities for determining the number and arrangement of substituents in the aromatic nucleus, as well as the mutual arrangement of benzene rings in polynuclear aromatic compounds. As a rule, the absence of strong bands in the region of 650÷900 cm -1 indicates the absence of an aromatic nucleus in the substance. In addition, vibrations of carbon-halogen bonds are manifested in this region, and the bands usually have a high intensity: C-Cl (550÷850 cm -1), C-Br (515÷690 cm -1), C-I (500÷600 cm -1). Communication fluctuations C-F appear in the region of skeletal vibrations of bonds S-S so it is very difficult to observe them. It makes no sense to use vibrations of carbon-halogen bonds to determine halogens in the composition of a substance (there are many methods that are faster and more accurate), but for the observation of intermediate products and interactions in the study of reaction mechanisms, the appearance of bands can provide useful information.
To establish the position of substituents in the aromatic nucleus, the region 1650÷2000 cm -1 is sometimes used, where overtones and tones of a more complex origin appear in exceptionally weak bands. The bands in this region, depending on the nature of the substitution, have a different contour. The reliability of this feature is low, and, in addition, this area is completely covered in the presence of a carbonyl group in the substance.
The vibrational spectra of the most important heterocyclic systems have much in common with the spectra of benzene derivatives: for example, for furan, thiophene, pyrrole, and pyridine n CH 3010÷3080 cm -1 and n C -C (ring) 1300÷1600 cm -1 , and the position of the strip v C-C essentially depends on the type of heterocycle and the nature of the substitution. Two to four bands may appear in this area. Below are the main frequencies in the spectra of the most important heterocycles (in cm -1)
Communication fluctuations СºС. The presence of a bond is usually established by the band of stretching vibrations 2100÷2250 cm -1 , because in this region, other bands are practically absent. The band of medium intensity, with symmetric substitution in the IR spectrum, it can become almost imperceptible, in the Raman spectrum the band is always active and its intensity is the greater, the less symmetrical the alkyne is.
Communication fluctuations O-N. In highly dilute solutions, ensuring the absence of intermolecular interactions, hydroxyl groups appear as a high-intensity band of stretching vibrations 3200÷3600 cm -1 . If the hydroxo group participates in hydrogen bonding, then the position and nature of the band begins to depend strongly on the degree of involvement, since the force constant begins to change. If the bond is intermolecular, a wide unstructured band appears covering the entire range of 3200÷3600 cm -1 . If an intramolecular hydrogen bond is observed, then this is evidenced by an intense band at about 3500 cm -1 , shifted to the region of low frequencies compared to free groups. To avoid the possibility of formation of intermolecular bonds, low-polarity solvents (hydrocarbons, CCl 4 ) and a concentration of less than 5×10 -3 mol/l should be used. Free phenolic hydroxyl is manifested by a band of stretching vibration 3600÷3615 cm -1 of high intensity.
Deformation vibrations of hydroxogroups are located in the region of 1330÷1420 cm -1 and are of little use for identification. Dimers of carboxylic acids appear as a broad intense band in the region of 1200÷1400 cm -1 , but the assignment of the band can only be made with certainty after proof that the substance is indeed a carboxylic acid.
Fluctuations in the C-O connection. The bond appears in ethers and alcohols as an intense band in the region 1000÷1275 cm -1 . Esters in the spectra contain two bands due to the C-O-C group: symmetrical vibration 1020÷1075 (weaker in the IR spectrum) and antisymmetric at 1200÷1275 cm -1 (weaker in the Raman spectrum). Bands of different groups appear in this range and the bands are of little characteristic, but most often they are the most intense.
Vibrations of the C=O bond. The stretching vibrations of the carbonyl group are present in the spectra of various compounds: aldehydes, ketones, carboxylic acids, anhydrides, etc. This is always a highly active peak in the region of 1650÷1680 cm -1 where other bands are practically absent. This is one of the most characteristic bands, its presence or absence can serve as a convincing argument for the presence or absence of carbonyl groups. The specific range of manifestation of the band depends on the neighboring groups and the group in which the carbonyl is included, the induction effect (-I) reduces the length of the C=O bond and, consequently, the force constant and frequency increase. For aldehydes and ketones, the band is about 1710÷1750, carboxylic acids - 1750÷1770 (monomers) and 1706÷1720 (dimers), esters - 1735÷1750, acid amides - 1650÷1695, acid chlorides - 1785÷1815, acid fluorides - 1865÷1875, acid anhydrides - 1740÷1790 and 1800÷1850 cm -1 . The effect of conjugation of p-electrons reduces the vibration frequency: in the systems C=C-C=O and C 6 H 5 -C=O the band is located around 1665÷1685 cm -1 .
Thus, the spectra of carbonyl compounds make it possible to obtain a large amount of completely unambiguous information, especially considering other bands: for esters and anhydrides, the band C-O , amides - band N-H , the spectra of aldehydes often contain a band of the group DREAM about 2695÷2830 cm -1 . For many aldehydes and ketones, the spectrum is the sum of the base and enol forms.
A summary of the spectral manifestations of various groups in the IR and Raman spectra is given in Table No. 2, however, there are special tables containing a larger set of frequencies and allowing one to study practical sets of bands from various samples.
Table No. 2 Basic vibration frequencies in IR spectroscopy
| Frequency, cm -1 | Intensity | The nature of vibrations | Connection type | |
| 3620- 3600 | s., cf. | n OH (free) | diluted alcohol solutions | |
| 3600- 3500 | s., cf. | n OH (bond) | Intramolecular hydrogen bonds in alcohols | |
| s., cf. | (free) | Dilute solutions of primary amides | ||
| 3400- 3350 | cf. | n NH (free) | Secondary amines, N-substituted amides | |
| 3550- 3520 | s., cf. | n OH (free) | dilute acid solutions | |
| 3500- 3400 | s., cf. | n NH2 (free) | Primary amines, amides | |
| With. | (free) | Dilute solutions of amides | ||
| 3330- 3260 | cf. | n ºCH | Monosubstituted alkynes | Alkanes |
| 1370- 1390 | s., cf. | Nitro compounds | ||
| 1280- 1200 | With. | n SOS | Esters | |
| 1250- 1180 | cf. | n C-N | Tertiary amines (ArNR 2 , (RCH 2) 3 N) | |
| 1220- 1125 | With. | n C-O | Secondary, tertiary alcohols | |
| 1200- 1160, 1145- 1105 | s., cf. | n C-O | Ketals, acetals | |
| 1150- 1050 | With. | Ethers | ||
| 1085- 1050 | s., cf. | n C-O | Alcohols | |
| 970- 950 | cf. | dCH | Trans-alkenes | |
| 900-650 | With. | dCH | Arenas | |
| 750- 650 | cf. | d=CH | cis-dienes |
| Type of communication and connections | Frequency, cm -1 | ||
| -C=C- | |||
| alkenes | 1680- 1620 | ||
| cis derivatives | 1665- 1635 | ||
| trans derivatives | 1675- 1660 | ||
| cyclic | 1650- 1550 | ||
| conjugate | 1660- 1580 | ||
| -C=C=C- | |||
| allenes | 1970- 1940 (as) | ||
| 1070- 1060 (ns) | |||
| -CºC- | |||
| Alkynes | 2270- 2190 | ||
| -CºC-H | 2140- 2100 | ||
| cC=O | |||
| Ketones | aliphatic | 1725- 1700 | |
| unlimited | 1690- 1660 | ||
| aryl ketones | 170- 1680 | ||
| diarylketones | 1670- 1660 | ||
| cyclic | 1780- 1700 | ||
| Diketones | a | 1730- 1710 | |
| b | 1640- 1635 | ||
| Aldehydes | aliphatic | 1740- 1720 | |
| unlimited | 1705- 1650 | ||
| aromatic | 1715- 1685 | ||
| carboxylic acids | monomer | ||
| dimer | 1725- 1700 | ||
| unlimited | 1715- 1680 | ||
| aromatic | 1700- 1680 | ||
| lactones | 1850- 1720 | ||
| anhydrides | |||
Absorption in the region of 10 2 - 10 3 cm -1 (IR - region) is usually due to vibrational transitions with a constant electronic state of the molecule; the corresponding spectra are called vibrational. More precisely, they should be called vibrational-rotational, since a change in the vibrational energy of a molecule upon absorption in this region is, as a rule, accompanied by a change in the rotational energy.
h \u003d E′ - E ″ \u003d (E vr ′ + E count ′) - (E vr ″ + E count ″) . (2.104)
The vibrational spectrum consists of a number of bands that are quite far apart from each other, the intensity of which sharply decreases with increasing wave number (Fig. 2.22). The first, most intense band is called the main band, or the main tone. Next are the 1st and 2nd overtones. The intensity of the subsequent bands decreases so sharply that even the 3rd and 4th overtones cannot be observed for most molecules.
Each band in the spectrum is complex and, when recorded on a device with a high resolution of the device, breaks up into a number of separate lines. The appearance of such a fine structure is characteristic of substances in the gaseous state. The position of the bands in the spectrum is determined by vibrational transitions, and the fine structure of each band is determined by rotational transitions.
In order to understand the origin of such a spectrum, let us first consider only vibrational motion and vibrational transitions, abstracting from the rotation of molecules, i.e., we take
h \u003d E count ′– E count ″. (2.105)
From the point of view of classical mechanics, the oscillatory motion of a diatomic molecule can be represented as a periodic change in the distance between the nuclei.
According to Hooke's law, which describes harmonic oscillations, the force that returns the nuclei to the equilibrium position is proportional to the displacement of the nuclei from the equilibrium position:
f = – kq , (2.106)
where k is the force constant;
q is vibrational coordinate; q \u003d r a + r b \u003d r - r e.
Hooke's equation is valid only for small displacements of nuclei, i.e., when q >> r e , in the limit at q = 0.
The force constant of a diatomic molecule is a value that characterizes the elasticity of the bond and is numerically equal to the force that forms (stretching or compressing) the bond per unit length f = k at q = 1.
Elementary work of elastic force:
dA = – f dq . (2.107)
Taking into account equation (2.106), we obtain:
dA = -kq dq . (2.108)
After integration within
(2.109)
for the potential energy of a diatomic molecule, we obtain:
u = A = 1/2 kq 2 . (2.110)
Equation (2.110) implies that
k \u003d (d 2 u / dq 2) q \u003d 0. (2.111)
Thus, for small displacements, the potential energy is a quadratic function of q = r – r e . The u–q or u–r curve is a parabola, and the force constant k characterizes the curvature of the parabola near the minimum.
When the expression (2.110) is substituted into the Schrödinger equation
2 count + (8 2 / h 2) (E count - u) count \u003d 0 (2.112)
and solving this equation, the following equation is obtained for the eigenvalues of the vibrational energy of a diatomic molecule as a harmonic oscillator:
E count \u003d h about (v + 1/2) , (2.113)
where v is vibrational quantum number, which takes the values of positive integers starting from zero (v = 0, 1, 2, 3.......);
0 is the natural vibration frequency of the vibrator.
Equation (2.113) can be represented in another form:
E count \u003d hc e (v + 1/2) , (2.114)
where e is the eigenwave number (oscillatory constant), which characterizes the oscillation frequency referred to the minimum of the potential curve, i.e. the frequency that, according to classical mechanics, a molecule would have for an infinitely small oscillation amplitude (q = 0, r = r e) . The value of e is expressed in m -1 or cm -1. It is a molecular constant. Any diatomic molecule is characterized in each electronic state by some constant value e .
Equation (2.114) indicates the quantization of vibrational energy and the existence of zero energy of the oscillator at v = 0:
E 0 count \u003d hc e / 2. (2.115)
This energy is not zero. The oscillation energy of a harmonic oscillator increases in direct proportion to the quantum number v, which corresponds to a system of equally spaced quantum levels. According to the quantum mechanical selection rules for a harmonic oscillator, transitions with v = 1 are possible. When light is absorbed, v changes by +1, and the energy and amplitude of oscillations increase.
However, the harmonic oscillator model leads to statements that contradict the experimental data:
1) E count within this model can be arbitrarily large. In this case, the chemical bond in the molecule would be infinitely elastic and its breaking would be impossible. We know it's not;
2) for a harmonic oscillator, only one band should be observed in the absorption spectrum, which follows from the selection rules and the equivalence of vibrational levels (Fig. 2.23 a). However, several bands are observed in the spectrum of a real diatomic molecule.
Rice. 2.23 Potential energy curves (a) and the dependence of the vibrational energy E col on V col (b) for a harmonic oscillator
All this means that real molecules are not harmonic oscillators. The harmonic approximation for them can be used only for small displacements of the nuclei from the equilibrium position, i.e. at small values of the vibrational quantum number (v = 0; 1).
For real diatomic molecules, the function U(r) is not a parabola, and the restoring force is not strictly proportional to the displacement of the nuclei. This leads to the model anharmonic oscillator, for which the potential energy curve is depicted as shown in Fig. 2.24.
For an approximate description of the potential energy curve, the Morse function is used:
u = D e 2 , (2.116)
where D e is the dissociation energy;
is a constant for a given molecule.
Rice. 2.24 Potential energy curves (a) and the dependence of the vibrational energy Ecol on Vcol (b) for an anharmonic oscillator
When solving the Schrödinger equation for a diatomic molecule, when u(r) is expressed by the Morse function, the eigenvalues of the vibrational energy Ecol are described by the binomial:
E count \u003d hc e (v + 1/2) - hc e x e (v + 1/2) 2, (2.117)
where x e is the anharmonicity coefficient characterizing the deviation from harmony, this value is dimensionless, and
e >> e x e > 0. (2.118)
From equation (2.117) one can obtain an expression for the zero energy of an anharmonic oscillator (where v = 0):
E 0 \u003d 1/2 hc e - 1/4 hc e x e. (2.119)
From equation (2.117) the following conclusions follow:
the dependence of E col on v is not linear;
vibrational quantum levels converge as v increases.
Indeed, as the quantum number increases by one, the difference in the energy of vibrations decreases with increasing V:
Е v+1 v \u003d E (v + 1) - E (v) \u003d hc [ e - 2 e x e (v + 1)] . (2.120)
Let us find the first and second derivatives of the function (2.117):
E v = hc e (V + 1/2) – hc e x e (V + 1/2) 2 , (2.121)
dE V /dV = hc [ e - 2 e x e (V + 1/2)] , (2.122)
d 2 E V /dV = –2hc e x e< 0 . (2.123)
The expression indicates that the curve E v -V has a maximum (Figure 2.16, b) and the vibrational levels converge to a certain value V max. , which can be found from the maximum condition:
dE V /dV = 0 , (2.124)
dE V /dV = hc[ e - 2 e x e (V max + 1/2)] = 0 , (2.125)
V max \u003d ( e / 2 e x e) - 1/2, (2.126)
V max \u003d 1/2x e - 1/2
. (2.127)Thus, there is a finite number of discrete vibrational levels and the maximum energy of an anharmonic oscillator E V, max. If the vibrational energy E V > EV, max, is reported to the molecule, dissociation will occur, as can be seen from the potential energy curve (Fig. 2.16, a).
The values of Vmax calculated by formula (2.127) for most molecules are several tens, for some - up to one and a half hundred.
Selection rules:
if for a harmonic oscillation of an oscillator V = 1, then for an anharmonic oscillator the quantum mechanical selection rules allow any transitions: V = 1, 2, 3, etc.;
any substances (polar and non-polar) can be described.
By substituting the values of V, e , x e into equation (2.117), we can draw up a scheme of allowed vibration energy levels.
Rice. 2.25 Scheme of permitted levels of oscillation energies.
For most diatomic molecules, the 01 vibrational transition requires 10 – 100 kJ/mol. This is much more than the average energy of thermal motion of gas molecules at a temperature of 18 - 25 o C (RT = 2.5 kJ / mol at 298 o K). Therefore, we can assume that at the temperature of the experiment, the vast majority of molecules are at the lower energy level, i.e. V″=0.
The selection rule allows you to derive an equation for all frequencies observed in the spectrum and derive the vibrational spectrum:
\u003d E V / hc \u003d e (V + 1/2) - e x e (V + 1/2) 2. (2.128)
Substituting the quantities V′ and V″ into equation (2.128) and taking the difference of the wave numbers, we get:
V ″ 0 \u003d [ e (V′ + 1/2) - e x e (V "+ 1/2) 2] - [ e (V ″ + 1/2) - e x e (V ″ + 1 /2) 2 ] (2.129)
After conversion:
\u003d (V "- V ″) [ e - e x e (1 + V" + V ″)] . (2.130)
Considering that V’’=0, we obtain an expression for the wave numbers of the only experimentally observed series of transitions shown in the figure, namely, the V″ (0)V" transitions:
\u003d V "[ e - e x e (1 + V")] , (2.131)
where V" = 1, 2, 3,..... V max.
The smallest energy is required for the 01 transition. This corresponds to the appearance in the absorption spectrum of the first (low-frequency) band - the main band, or the main type. Transitions 02; 03, etc. give subsequent bands - overtones.
The wave numbers of the main band and overtones are determined in accordance with (2.131) as follows:
01 main band or overtone,
0 1 = e - 2 e x e = e (1 - 2x e), (2.132)
02 1st overtone,
0 2 = 2 e - 6 e x e = 2 e (1 - 3x e), (2.133)
03 2nd overtone,
0 3 = 3 e - 12 e x e = 3 e (1 - 4x e), (2.134)
In general, for the transition 0V":
0 V’ = V " e - V’(V’+1) e x e . (2.135)
It follows from the expressions obtained that the absorption bands in the vibrational spectrum converge, although, due to the fact that e x e<< e , эта сходимость для первых двух-трех полос выражена слабо. Величина e x e составляет обычно несколько см -1 , реже – десятки см -1 , в то время как e = 10 2 – 10 3 см -1 .
The probability of the 01 transition is the highest, which explains the intensity of the main absorption band. Transition probability 02; 03, etc. decreases sharply with increasing V", which is reflected in the absorption spectrum.
Determination of the vibrational constant e and anharmonicity coefficientx e .
The most important result of the experimental study of IR absorption spectra is the determination of molecular constants - the vibrational constant e and the anharmonicity coefficient x e .
the absorption bands are attributed to certain vibrational transitions.
determine the oscillation frequency of each transition: 1 , 2 , 3 .
each of the frequencies is compared with equations of the type (2.132) - (2.135) and, having solved them together, determine e and x e . For example:
0 1 = e (1–2x e)
0 2 = 2 e (1–3x e).
Determination of dissociation energy (chemical bond). The energy of a chemical bond is the energy that must be expended to transfer a molecule from zero to the maximum vibrational quantum level:
Recall equation (2.127):
V max \u003d 1/2x e - 1/2.
Substituting this equation into (2.127), we get:
E 0 Vmax = hc e (1/2x e - 1/2 + 1/2) - hc e x e (1/2x - 1/2 + 1/2) 2 , (2.136)
E 0 Vmax = hc e /2x e – hc e x e /4x e = hc e x e /4x, (2.137)
E max \u003d hc e / 4x e. (2.138)
Let's move on to molar values of energy in J/mol:
E max (mol) \u003d E max N A, (2.139)
E max (mol) \u003d hc e N A / 4x e. (2.140)
The dissociation energy, counted from the zero level and referred to 1 mole, is called the true dissociation energy and is denoted by D o:
E x.s. \u003d D o \u003d E max - E 0. (2.141)
If the dissociation energy is counted from the minimum of the potential curve, then it exceeds D 0 by the value of zero energy (Fig. 2.18):
D e \u003d D 0 + E 0. (2.142)
hc e N A
hc e
Recall that
E 0 \u003d 1/2 hc e - 1/4 hc e x e,
D 0 \u003d hc e / 4x e - (hc e / 2 - hc e x e / 4) , (2.143)
D 0 \u003d (1–x e) 2. (2.144)
Turning to molar values, we find the value of D 0 in J / mol:
D 0 \u003d (1–x e) 2. (2.145)
Thus: from the vibrational spectrum, the following molecular constants can be obtained:
Natural oscillation frequency e;
Anharmonicity coefficient x e;
The energy of vibrational motion of molecules;
The energy of a chemical bond.
Electronic spectra (basic concepts). When electrons are excited in molecules, radiation is observed in the ultraviolet and visible regions of the spectrum.
h \u003d E "" - E" \u003d (E "" vr - E " vr) + (E "" count - E " count) + (E "" el - E " el).
P 
At the same time, a combination of all types of energy changes takes place. The spectrum is complex and is called electronic-vibrational-rotational. The spectrum consists of absorption bands. The maximum of the absorption band corresponds to the most probable transition in the given wavelength region.
Figure 2.25 shows the relative arrangement of the energy levels of MO molecular orbitals ( and are bonding MOs, * and * are loosening MOs)
In the ground state, the - and -orbitals are usually occupied by electrons (this is a more stable energy state with a lower potential energy).
The * transition requires the greatest energy - it manifests itself in the far UV region and is characteristic of saturated hydrocarbon molecules. The * transitions correspond to the visible and near UV regions and are typical for molecules of unsaturated compounds.
Rice. 2.26. Interaction potential energy curves for electronic transitions 
When large quanta of radiant energy are absorbed, an electronic jump can occur. The potential dissociation energy D 0 - decreases, and E - increases. With an increase in energy E, the interatomic distance r e increases as a result of oscillatory motion (Fig. 2.26).
Each type of bond has its own energy of electronic transitions and its own characteristic absorption band with a certain wavelength.
MOLECULAR SPECTRA- absorption, emission or scattering spectra arising from quantum transitions molecules from one energetic. states to another. M. s. determined by the composition of the molecule, its structure, the nature of the chemical. communication and interaction with external fields (and, consequently, with the surrounding atoms and molecules). Naib. characteristic are M. s. rarefied molecular gases, when there is no spectral line broadening pressure: such a spectrum consists of narrow lines with a Doppler width.
Rice. 1. Scheme of energy levels of a diatomic molecule: a And b-electronic levels; u" And u"" - oscillatory quantum numbers; J" And J"" - rotational quantum numbers.
In accordance with the three systems of energy levels in a molecule - electronic, vibrational and rotational (Fig. 1), M. s. consist of a set of electronic, vibrating. and rotate. spectra and lie in a wide range of e-magn. waves - from radio frequencies to x-rays. region of the spectrum. The frequency of transitions between rotation. energy levels usually fall into the microwave region (in the scale of wave numbers 0.03-30 cm -1), the frequency of transitions between oscillations. levels - in the IR region (400-10,000 cm -1), and the frequencies of transitions between electronic levels - in the visible and UV regions of the spectrum. This division is conditional, because they often rotate. transitions also fall into the IR region, oscillate. transitions - in the visible region, and electronic transitions - in the IR region. Usually, electronic transitions are accompanied by a change in vibrations. energy of the molecule, and when vibrating. transitions changes and rotates. energy. Therefore, most often the electronic spectrum is a system of electron oscillations. bands, and with a high resolution of the spectral equipment, their rotation is detected. structure. The intensity of lines and stripes in M. s. is determined by the probability of the corresponding quantum transition. Naib. the intense lines correspond to the transition allowed selection rules.K M. s. also include Auger spectra and X-rays. spectra of molecules(not considered in the article; see Auger effect, Auger spectroscopy, X-ray spectra, X-ray spectroscopy).
Electronic spectra. Purely electronic M. s. arise when the electronic energy of the molecules changes, if the vibrations do not change. and rotate. energy. Electronic M. with. are observed both in absorption (absorption spectra) and in emission (spectra ). During electronic transitions, the electric current usually changes. . Electrical dipole transition between electronic states of a G-type molecule "
and G ""
(cm. Symmetry of molecules) is allowed if the direct product Г "
G ""
contains the symmetry type of at least one of the components of the dipole moment vector d
. In absorption spectra, transitions from the ground (totally symmetric) electronic state to excited electronic states are usually observed. Obviously, for such a transition to occur, the types of symmetry of the excited state and the dipole moment must coincide. T. to. electric Since the dipole moment does not depend on the spin, then the spin must be conserved during an electronic transition, i.e., only transitions between states with the same multiplicity are allowed (inter-combination prohibition). This rule, however, is broken
for molecules with strong spin-orbit interaction, which leads to intercombination quantum transitions. As a result of such transitions, for example, phosphorescence spectra arise, which correspond to transitions from an excited triplet state to the main state. singlet state.
Molecules in various electronic states often have different geom. symmetry. In such cases, the condition D "
G ""
G d must be performed for a point group of a low-symmetry configuration. However, when using a permutation-inversion (PI) group, this problem does not arise, since the PI group for all states can be chosen the same.
For linear molecules of symmetry With hu dipole moment symmetry type Г d=S + (dz)-P( d x , d y), therefore, only transitions S + - S +, S - - S -, P - P, etc. are allowed for them with a transition dipole moment directed along the axis of the molecule, and transitions S + - P, P - D, etc. with the moment of transition directed perpendicular to the axis of the molecule (for the designations of states, see Art. Molecule).
Probability IN electric dipole transition from the electronic level T to the electronic level P, summed over all oscillatory-rotating. electronic level levels T, is determined by f-loy:
dipole moment matrix element for the transition n-m,y en and y em- wave functions of electrons. Integral coefficient. absorption, which can be measured experimentally, is determined by the expression
Where N m- the number of molecules in the beginning. able m, v nm- transition frequency TP. Often electronic transitions are characterized
Simultaneously with the change in the vibrational state of the molecule, its rotational state also changes. A change in vibrational and rotational states leads to the appearance of rotational-vibrational spectra. The vibrational energy of molecules is about a hundred times greater than its rotational energy, so rotation does not disturb the vibrational structure of molecular spectra. The imposition of small in energy rotational quanta on relatively high energy vibrational quanta 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 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 energies of a rigid molecule, then, respectively, the expression:
This expression conveys well the structure of the spectrum and leads to distortion only at large values of the quantum numbers and . Let us consider the rotational structure of the rotational-vibrational spectrum. So, during radiation, the molecule passes 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, respectively:
the set 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 the 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 as:
According to the Pauli selection rule:
the entire band is divided into two groups of spectral series, which are located relatively on both sides. Indeed, 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 the molecule passes from the -th rotational level to the 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 the molecule passes 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 to has no physical meaning. - and - strip branches, based on equations of the form:
are made up 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 get:
As a result, the lines - and - of the branches are bent and not equidistant lines are observed, but - branches that diverge and - branches that approach with the formation of the edge of the strip. Thus, the quantum theory of molecular spectra turned out to be capable of deciphering the spectral bands in the near infrared region, treating them as the result of a simultaneous change 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 various discrete energy states of molecules and, based on the data obtained, draw reliable and accurate conclusions regarding the motion 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 arrangement of nuclei in molecules, the dissociation energy of the molecule itself, etc.
As was established in the previous section, the rotational quantum number can change by one in the transition between rotational levels. If we confine 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 an increase 
per unit distance between rotational levels increases by 
.
In this case, the distance between adjacent rotational lines in the spectrum:
.
(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), then it turns out that the distance between neighboring spectral lines with increasing number J decreases.
As regards the intensities of the rotational spectral lines, it should first of all be said that they depend essentially on the temperature. Indeed, the distance between neighboring rotational lines of many molecules is much less than the value kT. Therefore, as the temperature changes, the populations of the rotational levels change significantly. As a consequence, the intensities of the spectral lines change. In this case, it should be taken into account that the statistical weight of the rotational states is equal to 
. The expression for the population of the rotational level with the number J so it looks like:
The dependence of populations of rotational levels on the 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 of 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, the values J max have a large spread. Thus, for a CO molecule at room temperature, the intensity maximum 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 the measurement of 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 the consideration of vibrational-rotational spectra. There are no pure vibrational transitions, because when you go between two vibrational levels, the numbers of rotational 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. In the first approximation, we neglect the presence of a rotational structure and consider only transitions between vibrational levels. As shown in the previous section, there are no selection rules for changing vibrational quantum numbers. However, there are probabilistic properties, which are as follows.
First, the value of 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 in this case.
Secondly, the intensities of the spectral lines sharply decrease with increasing V approximately in the following ratio:
About transitions from V\u003d 1 they say, as about transitions at the main frequency (1-0, 2-1), transitions with V>1 are called overtones ( V=2 – first overtone (2-0), V\u003d 3 - second overtone (3-0, 4-1), etc.). Transitions in which only excited vibrational levels (2-1, 3-2) participate are called hot, since, as a rule, the substance is heated to register them in order to increase the population of the 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 the vibrational frequencies 
and anharmonicity constants 
.
Indeed, if we measure the frequencies of two neighboring vibrational transitions (figure on the slide), then we can determine the magnitude of the vibrational quantum defect:
(13.10)
After that, using the expression (12.8), the value is determined 
.
Now we take into account the rotational structure. The structure of the rotational branches is shown on the slide. Characteristically, 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).
Denoting 
, we write expressions for the frequencies P- And R-branches.
Restricting in (11.15) to one term, for the frequency R-branches we get the equation:
Where 
Similarly, for P- branches:
Where 
As mentioned above, with an increase in the number of the vibrational quantum number, the value of the rotational constant will decrease. Therefore, always 
. Therefore, the signs of the coefficients at 
For P- And R- 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 highest, we can assume in the first approximation that 
. Then:
From this condition, in addition, it follows that the frequencies in each of the branches are located on opposite sides of 
. As an example, the slide shows several vibrational-rotational spectra obtained at different temperatures. The regularities in the intensity distribution in these spectra are explained by considering purely rotational transitions.
With the help of vibrational-rotational spectra, it is possible to determine not only the vibrational, but also the rotational constants of molecules. So, the value of the rotational constant 
can be determined from the spectrum, consisting of the lines depicted on the slide. It is easy to see that the value
directly proportional 
:
.
The same way:
Accordingly, constant 
And 
determined from dependencies 
from the number of rotational level.
After that, you can measure the values of rotational constants 
And 
. for this you need to build dependencies
.
(13.16)
In concluding this section, we consider the electronic-vibrational-rotational spectra. In the general case, 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 as 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 the electronic-vibrational spectral lines will take the form:
Where 
is the frequency of the 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 (bands) obtained in this case are called progressions By V' (or by V''). Series of bands with constant value V’- V'' are called diagonal series or sequences. Despite the fact that the selection rules for transitions with different values V does not exist, a rather limited number of lines are observed in the spectra due to the Franck-Condon principle considered above. For practically all molecules, the observed spectra contain from several to one or two dozen systems of bands.
For the convenience of representing the electronic-vibrational spectra, the observed systems of bands are given in the form of the so-called Delandre tables, where each cell is filled with the value of the wavenumber of the corresponding transition. The slide shows a fragment of the Delandre table for the BO molecule.
Let us now consider the rotational structure of the electronic vibrational lines. For this we put: 
. Then the rotational structure is described by the relation:
In accordance with the selection rules for the 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 larger or smaller than in the other, the difference 
can be either positive or negative. At 
<0
с ростомJ frequencies in R-branches gradually stop increasing and then begin to decrease, forming the so-called edge (the highest frequency R-branches). At 
>0 edging is formed in P-branches.
Dependence of the position of the lines of the rotational structure on the quantum number J is called a Fortre diagram. An example of such a diagram is shown on the slide.
To find the quantum rotational number of the vertex of the Fortre diagram (corresponding to Kant), it is necessary to differentiate the expression (13.23) with respect to m:
(13.24)
and equate it to zero, after which:
.
(13.25)
The distance between the edge frequency and 
wherein:
.
(13.26)
In conclusion of this section, let us consider how the isotopic substitution of nuclei (a change in the mass of at least one of the nuclei without changing the charge) affects the position of the energy states of a molecule. This phenomenon is called isotopic shift.
First of all, it should be noted that the dissociation energy 
(see figure on the slide) is a purely theoretical value and corresponds to the transition of a molecule from a hypothetical state corresponding to a minimum of potential energy 
, into the state of two non-interacting atoms located at an infinite distance from each other. The quantity is experimentally measured 
, 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 electrical in nature, the influence of the mass of atoms with the same charge during isotopic substitution should not affect the potential energy curve, the dissociation energy 
and on the position of the electronic states of the molecule.
However, the position of the vibrational and rotational levels and the value 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 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. In this case, with an increase in the vibrational quantum number, the difference between the vibrational states of isotopically substituted molecules gradually increases. If we introduce the designation 
, then it can be shown that:
<1,
(13.27)
since the constant 
is the same for isotopically substituted molecules. For the ratio of anharmonicity coefficients and rotational constants, we obtain:
,
.
(13.28)
Obviously, 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 isotopes 129 I 2 and 127 I 2 
0.992.
