Molecular spectra

optical spectra of emission and absorption, as well as Raman scattering of light (See Raman scattering of light) , belonging to free or weakly interconnected Molecule m. M. s. have a complex structure. Typical M. with. - striped, they are observed in emission and absorption and in Raman scattering as a combination more or less narrow lanes in the ultraviolet, visible and near infrared regions, which, with sufficient resolving power of the spectral instruments used, break up into a set of closely spaced lines. The specific structure of M. s. is different for different molecules and, generally speaking, becomes more complicated with an increase in the number of atoms in a molecule. For highly complex molecules, the visible and ultraviolet spectra consist of a few broad continuous bands; the spectra of such molecules are similar to each other.

hν = E‘ - E‘’, (1)

Where hν is the energy of the emitted absorbed Photon and the frequency ν ( h- The bar is constant). For Raman scattering hν is equal to the difference between the energies of the incident and scattered photons. M. s. much more complicated than the line atomic spectra, which is determined by the greater complexity of internal motions in a molecule than in atoms. Along with the movement of electrons relative to two or more nuclei in molecules, there is an oscillatory movement of the nuclei (together with the internal electrons surrounding them) around the equilibrium positions and a rotational movement of the molecule as a whole. These three types of motions - electronic, vibrational and rotational - correspond to three types of energy levels and three types of spectra.

According to quantum mechanics, the energy of all types of motion in a molecule can only take on certain values, that is, it is quantized. The total energy of the molecule E can be approximately represented as the sum of the quantized values ​​of the energies of the three types of its motion:

E = E email + E count + E rotation (2)

In order of magnitude

Where m is the mass of the electron, and the quantity M has the order of the mass of the nuclei of atoms in the molecule, i.e. m/M Molecular spectra 10 -3 -10 -5, therefore:

E email >> E count >> E rotation (4)

Usually E el of the order of several ev(several hundred kJ/mol), E col Molecular spectra 10 -2 -10 -1 ev, E rotation Molecular spectra 10 -5 -10 -3 ev.

In accordance with (4), the system of energy levels of a molecule is characterized by a set of electronic levels far apart from each other (different values E email at E count = E rotation = 0), vibrational levels located much closer to each other (different values E count at a given E l and E rotation = 0) and even more closely spaced rotational levels (different values E rotation at given E email and E count). On rice. 1 the scheme of levels of a diatomic molecule is given; for polyatomic molecules, the system of levels becomes even more complicated.

Electronic energy levels ( E el in (2) and on the diagram rice. 1 correspond to the equilibrium configurations of the molecule (in the case of a diatomic molecule characterized by the equilibrium value r 0 internuclear distance r, cm. rice. 1 in Art. Molecule). Each electronic state corresponds to a certain equilibrium configuration and a certain value E el; smallest value corresponds to the basic energy level.

The set of electronic states of a molecule is determined by the properties of its electron shell. Basically the values E el can be calculated by quantum chemistry methods (See Quantum Chemistry) , however, this problem can be solved only with the help of approximate methods and for relatively simple molecules. The most important information about the electronic levels of a molecule (the arrangement of the electronic energy levels and their characteristics), which is determined by its chemical structure, is obtained by studying its molecular structure.

A very important characteristic of a given electronic energy level is the value of the quantum number (See Quantum numbers) S, characterizing the absolute value of the total spin moment of all electrons of the molecule. Chemically stable molecules have, as a rule, an even number of electrons, and for them S= 0, 1, 2... (for the main electronic level, the value S= 0, and for excited - S= 0 and S= 1). Levels from S= 0 are called singlets, with S= 1 - triplet (because the interaction in the molecule leads to their splitting into χ = 2 S+ 1 = 3 sublevels; see Multiplicity) . Free radicals usually have odd number electrons, for them S= 1 / 2 , 3 / 2 , ... and the value S= 1 / 2 (doublet levels splitting into χ = 2 sublevels).

For molecules whose equilibrium configuration has symmetry, the electronic levels can be further classified. In the case of diatomic and linear triatomic molecules having an axis of symmetry (of infinite order) passing through the nuclei of all atoms (see Fig. rice. 2 , b) , electronic levels are characterized by the values ​​of the quantum number λ, which determines the absolute value of the projection of the total orbital angular momentum of all electrons onto the axis of the molecule. Levels with λ = 0, 1, 2, ... are denoted respectively by Σ, П, Δ..., and the value of χ is indicated by the index at the top left (for example, 3 Σ, 2 π, ...). For molecules with a center of symmetry, such as CO 2 and C 6 H 6 (see. rice. 2 , b, c), all electronic levels are divided into even and odd, denoted by indices g And u(depending on whether the wave function retains its sign when reversing at the center of symmetry or changes it).

Vibrational energy levels (values E kol) can be found by quantizing the oscillatory motion, which is approximately considered harmonic. In the simplest case of a diatomic molecule (one vibrational degree of freedom corresponding to a change in the internuclear distance r) it is considered as a harmonic oscillator ; its quantization gives equidistant energy levels:

E count = hν e (υ +1/2), (5)

where ν e is the fundamental frequency harmonic vibrations molecules, υ is a vibrational quantum number that takes the values ​​0, 1, 2, ... On rice. 1 vibrational levels for two electronic states are shown.

For each electronic state of a polyatomic molecule consisting of N atoms ( N≥ 3) and having f vibrational degrees of freedom ( f = 3N- 5 and f = 3N- 6 for linear and non-linear molecules, respectively), it turns out f so-called. normal oscillations with frequencies ν i ( i = 1, 2, 3, ..., f) and a complex system of vibrational levels:

Where υ i = 0, 1, 2, ... are the corresponding vibrational quantum numbers. The set of frequencies of normal vibrations in the ground electronic state is a very important characteristic of a molecule, depending on its chemical structure. All the atoms of the molecule or part of them participate in a certain normal vibration; atoms in this case make harmonic vibrations with one frequency v i , but with different amplitudes that determine the shape of the oscillation. Normal vibrations are divided according to their shape into valence (at which the lengths of bond lines change) and deformation (at which the angles between chemical bonds change - valence angles). The number of different vibrational frequencies for molecules of low symmetry (having no symmetry axes of order higher than 2) is 2, and all vibrations are non-degenerate, while for more symmetrical molecules there are double and triple degenerate vibrations (pairs and triplets of vibrations coinciding in frequency). For example, for a nonlinear triatomic molecule H 2 O ( rice. 2 , A) f= 3 and three nondegenerate vibrations are possible (two valence and one deformation). A more symmetrical linear triatomic CO 2 molecule ( rice. 2 , b) has f= 4 - two non-degenerate vibrations (valence) and one doubly degenerate (deformation). For a planar highly symmetric molecule C 6 H 6 ( rice. 2 , c) it turns out f= 30 - ten non-degenerate and 10 doubly degenerate vibrations; of these, 14 vibrations occur in the plane of the molecule (8 valence and 6 deformation) and 6 non-planar deformation vibrations - perpendicular to this plane. An even more symmetrical tetrahedral CH 4 molecule ( rice. 2 , d) has f = 9 - one non-degenerate vibration (valence), one doubly degenerate (deformation) and two three times degenerate (one valence and one deformation).

Rotational energy levels can be found by quantizing the rotational motion of a molecule, considering it as a solid body with certain moments of inertia (See moment of inertia). In the simplest case of a diatomic or linear polyatomic molecule, its rotational energy

Where I is the moment of inertia of the molecule about an axis perpendicular to the axis of the molecule, and M- rotational moment of momentum. According to the quantization rules,

where is the rotational quantum number J= 0, 1, 2, ..., and, therefore, for E rotation received:

where is the rotational constant rice. 1 rotational levels are shown for each electronic-vibrational state.

Various types of M. with. arise during various types of transitions between the energy levels of molecules. According to (1) and (2)

Δ E = E‘ - E‘’ = Δ E el + Δ E count + Δ E rotation, (8)

where changes Δ E el, Δ E count and Δ E rotation of electronic, vibrational and rotational energies satisfy the condition:

Δ E email >> Δ E count >> Δ E rotation (9)

[distances between levels of the same order as the energies themselves E el, E ol and E rotation satisfying condition (4)].

At Δ E el ≠ 0, electronic M. s are obtained, observed in the visible and in the ultraviolet (UV) regions. Usually at Δ E el ≠ 0 simultaneously Δ E count ≠ 0 and Δ E rotation ≠ 0; different Δ E count for a given Δ E el correspond to different vibrational bands ( rice. 3 ), and different Δ E rotation for given Δ E el and Δ E count - separate rotational lines into which this band breaks up; a characteristic striped structure is obtained ( rice. 4 ). The set of bands with a given Δ E el (corresponding to a purely electronic transition with a frequency v el = Δ E email / h) called the system of bands; individual bands have different intensities depending on the relative transition probabilities (see Quantum transitions), which can be approximately calculated by quantum mechanical methods. For complex molecules, the bands of one system, corresponding to a given electronic transition, usually merge into one wide continuous band, and several such broad bands can overlap each other. Characteristic discrete electronic spectra are observed in frozen solutions of organic compounds (see the Shpol'skii effect). Electronic (more precisely, electronic-vibrational-rotational) spectra are studied experimentally using spectrographs and spectrometers with glass (for the visible region) and quartz (for the UV region) optics, in which prisms or diffraction gratings are used to decompose light into a spectrum (see Fig. Spectral instruments).

At Δ E el = 0, and Δ E col ≠ 0, vibrational M. s are obtained, observed in a close (up to several micron) and in the middle (up to several tens micron) infrared (IR) region, usually in absorption, as well as in Raman scattering of light. As a rule, at the same time Δ E rotation ≠ 0 and for a given E If this is done, an oscillatory band is obtained, which breaks up into separate rotational lines. The most intense in vibrational M. s. bands corresponding to Δ υ = υ ’ - υ '' = 1 (for polyatomic molecules - Δ υ i = υ i'- υ i ''= 1 at Δ υ k = υ k'- υ k '' = 0, where k≠i).

For purely harmonic oscillations, these selection rules , forbidding other transitions are performed strictly; bands appear for anharmonic vibrations, for which Δ υ > 1 (overtones); their intensity is usually small and decreases with increasing Δ υ .

Vibrational (more precisely, vibrational-rotational) spectra are studied experimentally in the IR region in absorption using IR spectrometers with prisms transparent to IR radiation, or with diffraction gratings, as well as Fourier spectrometers and in Raman scattering using high-aperture spectrographs ( for the visible region) using laser excitation.

At Δ E el = 0 and Δ E col = 0, purely rotational M. s., consisting of individual lines, are obtained. They are observed in absorption in the distant (hundreds micron) IR region and especially in the microwave region, as well as in the Raman spectra. For diatomic and linear polyatomic molecules (as well as for sufficiently symmetric nonlinear polyatomic molecules), these lines are equally spaced (in the frequency scale) from each other with intervals Δν = 2 B in absorption spectra and Δν = 4 B in Raman spectra.

Purely rotational spectra are studied in absorption in the far infrared region using IR spectrometers with special diffraction gratings (echelettes) and Fourier spectrometers, in the microwave region using microwave (microwave) spectrometers (see Microwave spectroscopy) , and also in Raman scattering with the help of high-aperture spectrographs.

Methods of molecular spectroscopy, based on the study of molecular weight, make it possible to solve various problems in chemistry, biology, and other sciences (for example, to determine the composition of petroleum products, polymeric substances, and so on). In chemistry according to M. s. study the structure of molecules. Electronic M. with. make it possible to obtain information about the electron shells of molecules, to determine the excited levels and their characteristics, to find the dissociation energies of molecules (by the convergence of the vibrational levels of the molecule to the dissociation boundaries). Study of vibrational M. s. allows you to find the characteristic vibration frequencies corresponding to certain types of chemical bonds in a molecule (for example, simple double and triple C-C connections, C-H connections, N-H, O-H for organic molecules), various groups of atoms (for example, CH 2, CH 3, NH 2), determine the spatial structure of molecules, distinguish between cis- and trans-isomers. For this, both infrared absorption spectra (IRS) and Raman spectra (RSS) are used. The IR method has become especially widespread as one of the most effective optical methods for studying the structure of molecules. Most full information it gives in combination with the TFR method. The study of rotational molecular forces, as well as the rotational structure of electronic and vibrational spectra, makes it possible, from the values ​​of the moments of inertia of molecules found from experience [which are obtained from the values ​​of rotational constants, see (7)], to find with great accuracy (for simpler molecules, for example H 2 O) parameters of the equilibrium configuration of the molecule - bond lengths and bond angles. To increase the number of parameters to be determined, the spectra of isotopic molecules (in particular, in which hydrogen is replaced by deuterium) are studied, which have the same parameters of equilibrium configurations, but different moments of inertia.

As an example of M.'s application with. to determine the chemical structure of molecules, consider a benzene molecule C 6 H 6 . The study of her M. s. confirms the correctness of the model, according to which the molecule is flat, and all 6 C-C bonds in the benzene ring are equivalent and form a regular hexagon ( rice. 2 , b), which has a sixth-order symmetry axis passing through the center of symmetry of the molecule perpendicular to its plane. Electronic M. with. absorption C 6 H 6 consists of several systems of bands corresponding to transitions from the ground even singlet level to excited odd levels, of which the first is triplet, and the higher ones are singlets ( rice. 5 ). The system of bands is most intense in the region of 1840 Å (E 5 - E 1 = 7,0 ev), the system of bands is weakest in the region of 3400 Å (E 2 - E 1 = 3,8ev), corresponding to the singlet-triplet transition, which is forbidden by the approximate selection rules for the total spin. Transitions correspond to the excitation of the so-called. π electrons delocalized throughout the benzene ring (see Molecule) ; level diagram derived from electronic molecular spectra rice. 5 is in agreement with approximate quantum mechanical calculations. Vibrational M. s. C 6 H 6 correspond to the presence of a center of symmetry in the molecule - the vibrational frequencies that appear (active) in the ICS are absent (inactive) in the SKR and vice versa (the so-called alternative prohibition). Of the 20 normal vibrations of C6H6, 4 are active in the ICS and 7 are active in the TFR, the remaining 11 are inactive both in the ICS and in the TFR. The values ​​of the measured frequencies (in cm -1): 673, 1038, 1486, 3080 (in the ICS) and 607, 850, 992, 1178, 1596, 3047, 3062 (in the TFR). Frequencies 673 and 850 correspond to out-of-plane vibrations, all other frequencies correspond to plane vibrations. Particularly characteristic for plane vibrations is the frequency 992 (corresponding to the stretching vibration of C-C bonds, which consists in periodic compression and tension benzene ring), frequencies 3062 and 3080 (corresponding to stretching vibrations of C-H bonds) and frequency 607 (corresponding to the bending vibration of the benzene ring). The observed vibrational spectra of C 6 H 6 (and similar vibrational spectra of C 6 D 6) are in very good agreement with theoretical calculations, which made it possible to give a complete interpretation of these spectra and find the forms of all normal vibrations.

Similarly, with the help of M. s. determine the structure of various classes of organic and inorganic molecules, up to very complex ones, such as polymer molecules.

Lit.: Kondratiev V.N., Structure of atoms and molecules, 2nd ed., M., 1959; Elyashevich M. A., Atomic and molecular spectroscopy, M., 1962; Herzberg G., Spectra and structure of diatomic molecules, trans. from English, M., 1949; his, Vibrational and rotational spectra of polyatomic molecules, trans. from English, M., 1949; his, Electronic spectra and the structure of polyatomic molecules, trans. from English, M., 1969; Application of spectroscopy in chemistry, ed. V. Vesta, trans. from English, M., 1959.

M. A. Elyashevich.

Rice. 4. Rotational splitting of the 3805 Å electron-vibrational band of the N 2 molecule.

Rice. 1. Scheme of energy levels of a diatomic molecule: a and b - electronic levels; v" And v" - quantum numbers of vibrational levels. J" And J" - quantum numbers of rotational levels.

Rice. 2. Equilibrium configurations of molecules: a - H 2 O; b - CO 2; in - C 6 H 6; d - CH 4 . Numbers indicate bond lengths (in Å) and bond angles.

Rice. 5. Scheme of electronic levels and transitions for the benzene molecule. The energy levels are given in ev. C - singlet levels; T - triplet level. The level parity is indicated by the letters g and u. For systems of absorption bands, the approximate wavelength ranges in Å are indicated; more intense systems of bands are indicated by thicker arrows.

Rice. 3. Electronic-vibrational spectrum of the N 2 molecule in the near ultraviolet region; groups of bands correspond to different values ​​of Δ v = v" - v ".

Big soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what "Molecular Spectra" is in other dictionaries:

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spectrum called the sequence of energy quanta of electromagnetic radiation, absorbed, released, scattered or reflected by a substance during the transitions of atoms and molecules from one energy state to another.

Depending on the nature of the interaction of light with matter, the spectra can be divided into absorption (absorption) spectra; emissions (emission); scattering and reflection.

For the objects under study, optical spectroscopy, i.e. spectroscopy in the wavelength range 10 -3 ÷10 -8 m subdivided into atomic and molecular.

atomic spectrum is a sequence of lines, the position of which is determined by the energy of the transition of electrons from one level to another.

The energy of an atom can be represented as the sum of the kinetic energy of translational motion and electronic energy :

where - frequency, - wavelength, - wave number, - speed of light, - Planck's constant.

Since the energy of an electron in an atom is inversely proportional to the square of the principal quantum number , then for the line in the atomic spectrum we can write the equation:

.

(4.12)

Here - electron energies at higher and lower levels; - Rydberg constant; - spectral terms, expressed in units of wave numbers (m -1 , cm -1).

All lines of the atomic spectrum converge in the short-wavelength region to a limit determined by the ionization energy of the atom, after which there is a continuous spectrum.

Molecule energy in the first approximation can be considered as the sum of translational, rotational, vibrational and electronic energies:

(4.15)

For most molecules, this condition is satisfied. For example, for H 2 at 291 K, the individual components of the total energy differ by an order of magnitude or more:

309,5 kJ/mol,

=25,9 kJ/mol,

2,5 kJ/mol,

=3,8 kJ/mol.

The values ​​of photon energies in different regions of the spectrum are compared in Table 4.2.

Table 4.2 - Energy of absorbed quanta various areas optical spectrum of molecules

The concepts of "oscillations of nuclei" and "rotation of molecules" are conditional. In fact, such types of motion only very approximately convey ideas about the distribution of nuclei in space, which is of the same probabilistic nature as the distribution of electrons.

A schematic system of energy levels in the case of a diatomic molecule is shown in Figure 4.1.

Transitions between rotational energy levels give rise to rotational spectra in the far IR and microwave regions. Transitions between vibrational levels within the same electronic level give vibrational-rotational spectra in the near-IR region, since a change in the vibrational quantum number inevitably entails a change in the rotational quantum number . Finally, transitions between electronic levels cause the appearance of electronic-vibrational-rotational spectra in the visible and UV regions.

In the general case, the number of transitions can be very large, but in fact, far from all appear in the spectra. The number of transitions is limited selection rules .

Molecular spectra provide a wealth of information. They can be used:

For the identification of substances in a qualitative analysis, as each substance has its own unique spectrum;

For quantitative analysis;

For structural group analysis, since certain groups, such as, for example, >C=O, _ NH 2 , _ OH, etc., give characteristic bands in the spectra;

To determine the energy states of molecules and molecular characteristics (internuclear distance, moment of inertia, natural vibration frequencies, dissociation energies); a comprehensive study of molecular spectra makes it possible to draw conclusions about the spatial structure of molecules;

In kinetic studies, including for the study of very fast reactions.

- energies of electronic levels;

Energy of vibrational levels;

Energy of rotational levels

Figure 4.1 - Schematic arrangement of energy levels of a diatomic molecule

Bouguer-Lambert-Beer law

Quantitative molecular analysis using molecular spectroscopy is based on Bouguer-Lambert-Beer law , relating the intensity of the incident and transmitted light with the concentration and thickness of the absorbing layer (Figure 4.2):

or with a proportionality factor:

Integration result:

(4.19)

.

(4.20)

When the intensity of the incident light decreases by an order of magnitude

.

(4.21)

If \u003d 1 mol / l, then, i.e. the absorption coefficient is equal to the reciprocal thickness of the layer in which, at a concentration equal to 1, the intensity of the incident light decreases by an order of magnitude.

The absorption coefficients and depend on the wavelength. The type of this dependence is a kind of “fingerprint” of molecules, which is used in qualitative analysis to identify a substance. This dependence is characteristic and individual for a particular substance and reflects the characteristic groups and bonds included in the molecule.

Optical density D

expressed in %

4.2.3 Rotation energy of a diatomic molecule in the rigid rotator approximation. Rotational spectra of molecules and their application to determine molecular characteristics

The appearance of rotational spectra is due to the fact that the rotational energy of the molecule is quantized, i.e.

0

A

Energy of rotation of a molecule around the axis of rotation

Since the point O is the center of gravity of the molecule, then:

Introduction of the reduced mass notation:

(4.34)

leads to the equation

.

(4.35)

Thus, a diatomic molecule (Figure 4.7 A) rotating around the axis or , passing through the center of gravity, can be simplified as a particle with mass , describing a circle with a radius around the point O(Figure 4.7 b).

The rotation of the molecule around the axis gives the moment of inertia, which is practically equal to zero, since the atomic radii are much smaller than the internuclear distance. Rotation about the axes or , mutually perpendicular to the bond line of the molecule, leads to equal moments of inertia:

where is a rotational quantum number that takes only integer values

0, 1, 2…. In accordance with selection rule for the rotational spectrum of a diatomic molecule, a change in the rotational quantum number upon absorption of an energy quantum is possible only by one, i.e.

transforms equation (4.37) into the form:

20 12 6 2

wavenumber of the line in the rotational spectrum corresponding to the absorption of a quantum upon transition from j energy level per level j+1, can be calculated by the equation:

Thus, the rotational spectrum in the rigid rotator model approximation is a system of lines at the same distance from each other (Figure 4.5b). Examples of rotational spectra of diatomic molecules estimated in the rigid rotator model are shown in Figure 4.6.

A b

Figure 4.6 - Rotational spectra HF (A) And CO(b)

For hydrogen halide molecules, this spectrum is shifted to the far IR region of the spectrum; for heavier molecules, to the microwave.

Based on the obtained patterns of the occurrence of the rotational spectrum of a diatomic molecule, in practice, first determine the distance between adjacent lines in the spectrum, from which they then find, and according to the equations:

,

(4.45)

Where - centrifugal distortion constant , is related to the rotational constant by the approximate relationship . The correction should be taken into account only for very large j.

For polyatomic molecules, in the general case, the existence of three different moments of inertia is possible . In the presence of symmetry elements in the molecule, the moments of inertia can coincide or even be equal to zero. For example, for linear polyatomic molecules(CO 2 , OCS, HCN, etc.)

Where - position of the line corresponding to the rotational transition in an isotopically substituted molecule.

To calculate the isotopic shift of the line, it is necessary to sequentially calculate the reduced mass of the isotopically substituted molecule, taking into account the change atomic mass isotope, moment of inertia , rotational constant and position of the line in the spectrum of the molecule according to equations (4.34), (4.35), (4.39) and (4.43), respectively, or to estimate the ratio of the wave numbers of lines corresponding to the same transition in isotopically substituted and non-isotopically substituted molecules, and then determine the direction and magnitude of the isotopic shift using equation (4.50). If the internuclear distance is approximately constant , then the ratio of the wave numbers corresponds to the inverse ratio of the reduced masses:

Where - total number particles, is the number of particles per i- that level of energy at temperature T, k- Boltzmann's constant, - statistical ve forces degree of degeneracy i-th energy level, characterizes the probability of finding particles at a given level.

For a rotational state, the population of a level is usually characterized by the ratio of the number of particles j- that energy level to the number of particles at the zero level:

,

(4.53)

Where - statistical weight j-th rotational energy level, corresponds to the number of projections of the momentum of a rotating molecule on its axis - the communication line of the molecule, , energy of the zero rotational level . The function goes through a maximum when increasing j, as Figure 4.7 illustrates with the CO molecule as an example.

The extremum of the function corresponds to the level with the maximum relative population, the value of the quantum number of which can be calculated by the equation obtained after determining the derivative of the function in the extremum:

.

(4.54)

Figure 4.7 - Relative population of rotational energy levels

molecules CO at temperatures of 298 and 1000 K

Example. In the rotational spectrum of HI, the distance between adjacent lines is determined cm -1. Calculate the rotational constant, the moment of inertia, and the equilibrium internuclear distance in the molecule.

Solution

In the approximation of the rigid rotator model, in accordance with equation (4.45), we determine the rotational constant:

cm -1.

The moment of inertia of the molecule is calculated from the value of the rotational constant according to equation (4.46):

kg . m 2.

To determine the equilibrium internuclear distance, we use equation (4.47), taking into account that the masses of hydrogen nuclei and iodine expressed in kg:

Example. In the far IR region of the spectrum of 1 H 35 Cl, lines were found whose wavenumbers are:

Determine the average values ​​of the moment of inertia and the internuclear distance of the molecule. Attribute the observed lines in the spectrum to rotational transitions.

Solution

According to the rigid rotator model, the difference between the wave numbers of adjacent lines of the rotational spectrum is constant and equal to 2 . Let us determine the rotational constant from the average value of the distances between adjacent lines in the spectrum:

cm -1 ,

cm -1

We find the moment of inertia of the molecule (equation (4.46)):

We calculate the equilibrium internuclear distance (equation (4.47)), taking into account that the masses of hydrogen nuclei and chlorine (expressed in kg):

Using equation (4.43), we estimate the position of the lines in the rotational spectrum of 1 H 35 Cl:

We correlate the calculated values ​​of the wave numbers of the lines with the experimental ones. It turns out that the lines observed in the rotational spectrum of 1 H 35 Cl correspond to the transitions:

N lines
, cm -1	85.384	106.730	128.076	149.422	170.768	192.114	213.466
	3 4	4 5	5 6	6 7	7 8	8 9	9 10

Example. Determine the magnitude and direction of the isotopic shift of the absorption line corresponding to the transition from energy level, in the rotational spectrum of the 1 H 35 Cl molecule when the chlorine atom is replaced by the 37 Cl isotope. The internuclear distance in 1 H 35 Cl and 1 H 37 Cl molecules is considered to be the same.

Solution

To determine the isotopic shift of the line corresponding to the transition , we calculate the reduced mass of the 1 H 37 Cl molecule, taking into account the change in the atomic mass of 37 Cl:

then we calculate the moment of inertia, the rotational constant and the position of the line in the spectrum of the 1 H 37 Cl molecule and the value of the isotopic shift according to equations (4.35), (4.39), (4.43) and (4.50), respectively.

Otherwise, the isotopic shift can be estimated from the ratio of the wave numbers of lines corresponding to the same transition in molecules (we assume that the internuclear distance is constant) and then the position of the line in the spectrum using equation (4.51).

For 1 H 35 Cl and 1 H 37 Cl molecules, the ratio of the wave numbers of a given transition is:

To determine the wave number of the line of an isotopically substituted molecule, we substitute the value of the transition wave number found in the previous example j → j+1 (3→4):

We conclude: the isotopic shift to the low-frequency or long-wave region is

85.384-83.049=2.335 cm -1 .

Example. Calculate the wave number and wavelength of the most intense spectral line of the rotational spectrum of the 1 H 35 Cl molecule. Match the line to the corresponding rotational transition.

Solution

The most intense line in the rotational spectrum of the molecule is associated with the maximum relative population of the rotational energy level.

Substituting the value of the rotational constant found in the previous example for 1 H 35 Cl ( cm -1) into equation (4.54) allows you to calculate the number of this energy level:

.

The wave number of the rotational transition from this level is calculated by equation (4.43):

We find the transition wavelength from the equation (4.11) transformed with respect to:

4.2.4 Multivariant task No. 11 "Rotational spectra of diatomic molecules"

1. Write a quantum mechanical equation to calculate the rotational energy of a diatomic molecule as a rigid rotator.

2. Derive an equation for calculating the change in the rotation energy of a diatomic molecule as a rigid rotator when it passes to the next, higher quantum level .

3. Derive an equation for the dependence of the wave number of rotational lines in the absorption spectrum of a diatomic molecule on the rotational quantum number.

4. Derive an equation for calculating the difference between the wavenumbers of adjacent lines in the rotational absorption spectrum of a diatomic molecule.

5. Calculate the rotational constant (in cm -1 and m -1) of a diatomic molecule A by the wave numbers of two adjacent lines in the long-wavelength infrared region of the rotational absorption spectrum of the molecule (see Table 4.3) .

6. Determine the rotational energy of the molecule A at the first five quantum rotational levels (J).

7. Draw schematically the energy levels of the rotational motion of a diatomic molecule as a rigid rotator.

8. Plot on this diagram the rotational quantum levels of a molecule that is not a rigid rotator.

9. Derive an equation for calculating the equilibrium internuclear distance based on the difference between the wavenumbers of adjacent lines in the rotational absorption spectrum.

10. Determine the moment of inertia (kg. m 2) of a diatomic molecule A.

11. Calculate the reduced mass (kg) of the molecule A.

12. Calculate the equilibrium internuclear distance () of a molecule A. Compare the resulting value with the reference data.

13. Assign the observed lines in the rotational spectrum of the molecule A to rotational transitions.

14. Calculate the wavenumber of the spectral line corresponding to the rotational transition from the level j for a molecule A(see table 4.3).

15. Calculate the reduced mass (kg) of an isotopically substituted molecule B.

16. Calculate the wave number of the spectral line associated with the rotational transition from the level j for a molecule B(see table 4.3). Internuclear distances in molecules A And B consider equal.

17. Determine the magnitude and direction of the isotopic shift in the rotational spectra of molecules A And B for the spectral line corresponding to the rotational level transition j.

18. Explain the reason for the nonmonotonic change in the intensity of absorption lines as the rotational energy of the molecule increases

19. Determine the quantum number of the rotational level corresponding to the highest relative population. Calculate the wavelengths of the most intense spectral lines of the rotational spectra of molecules A And B.

MOLECULAR SPECTRA, spectra of emission and absorption of electromagnet. radiation and combinat. scattering of light belonging to free or weakly bound molecules. They have the form of a set of bands (lines) in the X-ray, UV, visible, IR and radio wave (including microwave) regions of the spectrum. The position of the bands (lines) in the spectra of emission (emission molecular spectra) and absorption (absorption molecular spectra) is characterized by frequencies v (wavelengths l \u003d c / v, where c is the speed of light) and wave numbers \u003d 1 / l; it is determined by the difference between the energies E "and E: those states of the molecule, between which a quantum transition occurs:

(h is Planck's constant). When combined scattering, the value of hv is equal to the difference between the energies of the incident and scattered photons. The intensity of the bands (lines) is related to the number (concentration) of molecules of a given type, the population of the energy levels E "and E: and the probability of the corresponding transition.

The probability of transitions with the emission or absorption of radiation is determined primarily by the square of the matrix element of the electric. dipole moment of the transition, and with a more accurate consideration - and the squares of the matrix elements of the magn. and electric quadrupole moments of the molecule (see Quantum transitions). When combined In light scattering, the transition probability is related to the matrix element of the induced (induced) dipole moment of the transition of the molecule, i.e. with the matrix element of the polarizability of the molecule .

states of the pier. systems, transitions between to-rymi are shown in the form of these or those molecular spectra, have the different nature and strongly differ on energy. The energy levels of certain types are located far from each other, so that during transitions the molecule absorbs or emits high-frequency radiation. The distance between the levels of other nature is small, and in some cases, in the absence of external. field levels merge (degenerate). At small energy differences, transitions are observed in the low-frequency region. For example, the nuclei of atoms of certain elements have their own. magn. torque and electric spin-related quadrupole moment. Electrons also have a magnet. the moment associated with their spin. In the absence of external magnetic orientation fields moments are arbitrary, i.e. they are not quantized and the corresponding energetic. states are degenerate. When applying external permanent magnet. field, degeneracy is lifted and transitions between energy levels are possible, which are observed in the radio-frequency region of the spectrum. This is how NMR and EPR spectra arise (see Nuclear magnetic resonance, Electron paramagnetic resonance).

Kinetic distribution energies of electrons emitted by the pier. systems as a result of irradiation with X-ray or hard UV radiation, gives X-rayspectroscopy and photoelectron spectroscopy. Additional processes in the mall. system, caused by the initial excitation, lead to the appearance of other spectra. Thus, Auger spectra arise as a result of relaxation. electron capture from ext. shells to.-l. atom per vacant ext. shell, and the released energy turned into. in the kinetic energy other electron ext. shell emitted by an atom. In this case, a quantum transition is carried out from a certain state of a neutral molecule to a state they say. ion (see Auger spectroscopy).

Traditionally, only the spectra associated with the optical properties are referred to as molecular spectra proper. transitions between electronic-vibrational-rotate, energy levels of the molecule associated with three main. energy types. levels of the molecule - electronic E el, vibrational E count and rotational E vr, corresponding to three types of ext. movement in a molecule. For E el take the energy of the equilibrium configuration of the molecule in a given electronic state. The set of possible electronic states of a molecule is determined by the properties of its electron shell and symmetry. Swing. the motion of the nuclei in the molecule relative to their equilibrium position in each electronic state is quantized so that at several vibrations. degrees of freedom, a complex system of vibrations is formed. energy levels E col. Rotation of a molecule as a whole as a rigid system related nuclei characterized by rotation. the moment of the number of motion, which is quantized, forming a rotation. states (rotational energy levels) E temp. Usually the energy of electronic transitions is of the order of several. eV, vibrational -10 -2 ... 10 -1 eV, rotational -10 -5 ... 10 -3 eV.

Depending on between which energy levels there are transitions with emission, absorption or combinations. electromagnetic scattering. radiation - electronic, oscillating. or rotational, distinguish between electronic, oscillating. and rotational molecular spectra. The articles Electronic spectra , Vibrational spectra , Rotational spectra provide information about the corresponding states of molecules, selection rules for quantum transitions, methods of pier. spectroscopy, as well as what characteristics of molecules can be. obtained from molecular spectra: St. Islands and symmetry of electronic states, vibrate. constants, dissociation energy, molecular symmetry, rotation. constants, moments of inertia, geom. parameters, electrical dipole moments, data on the structure and ext. force fields, etc. Electronic absorption and luminescence spectra in the visible and UV regions provide information on the distribution

chemical bonds and structure of molecules.

Molecule - the smallest particle of a substance, consisting of the same or different atoms connected to each other chemical bonds, and being the carrier of its main chemical and physical properties. Chemical bonds are due to the interaction of external, valence electrons of atoms. There are two types of bonds most often found in molecules: ionic and covalent.

Ionic bond (for example, in molecules NaCl, KVR) is carried out by the electrostatic interaction of atoms during the transition of an electron from one atom to another, i.e. in the formation of positive and negative ions.

A covalent bond (for example, in H 2 , C 2 , CO molecules) is carried out when valence electrons are shared by two neighboring atoms (the spins of valence electrons must be antiparallel). The covalent bond is explained on the basis of the principle of indistinguishability of identical particles, such as electrons in a hydrogen molecule. The indistinguishability of particles leads to exchange interaction.

The molecule is a quantum system; it is described by the Schrödinger equation, which takes into account the motion of electrons in a molecule, the vibrations of the atoms of the molecule, and the rotation of the molecule. The solution of this equation is a very complex problem, which is usually divided into two: for electrons and nuclei. Energy of an isolated molecule:

where is the energy of motion of electrons relative to nuclei, is the energy of vibrations of nuclei (as a result of which the relative position of nuclei periodically changes), is the energy of rotation of nuclei (as a result of which the orientation of the molecule in space periodically changes). Formula (13.1) does not take into account the translational energy of the center of mass of the molecule and the energy of the nuclei of atoms in the molecule. The first of them is not quantized, so its changes cannot lead to the appearance of a molecular spectrum, and the second can be ignored if the hyperfine structure of the spectral lines is not considered. It is proved that eV, eV, eV, so >>>>.

Each of the energies included in expression (13.1) is quantized (it corresponds to a set of discrete energy levels) and is determined by quantum numbers. During the transition from one energy state to another, energy is absorbed or emitted D E=hv. During such transitions, the energy of electron motion, the energy of vibrations and rotation change simultaneously. It follows from theory and experiment that the distance between rotational energy levels D is much less than the distance between vibrational levels D, which, in turn, is less than the distance between electronic levels D. Figure 13.1 schematically shows the energy levels of a diatomic molecule (for example, only two electronic levels are considered are shown in bold lines).

The structure of molecules and the properties of their energy levels are manifested in molecular spectra– emission (absorption) spectra arising from quantum transitions between the energy levels of molecules. The emission spectrum of a molecule is determined by the structure of its energy levels and the corresponding selection rules.

Thus, different types of transitions between levels give rise to different types of molecular spectra. The frequencies of the spectral lines emitted by molecules can correspond to transitions from one electronic level to another (electronic spectra) or from one vibrational (rotational) level to another ( vibrational (rotational) spectra). In addition, transitions with the same values ​​are also possible And to levels having different values ​​of all three components, resulting in electronic-vibrational and vibrational-rotational spectra.

Typical molecular spectra are banded, which are a combination of more or less narrow bands in the ultraviolet, visible and infrared regions.

Using high-resolution spectral instruments, it can be seen that the fringes are such closely spaced lines that they are difficult to resolve. The structure of molecular spectra is different for different molecules and becomes more complicated with an increase in the number of atoms in a molecule (only continuous broad bands are observed). Only polyatomic molecules have vibrational and rotational spectra, while diatomic ones do not have them. This is explained by the fact that diatomic molecules do not have dipole moments (during vibrational and rotational transitions, there is no change in the dipole moment, which is a necessary condition for the transition probability to differ from zero). Molecular spectra are used to study the structure and properties of molecules, are used in molecular spectral analysis, laser spectroscopy, quantum electronics, etc.

MOLECULAR SPECTRA- absorption, emission or scattering spectra arising from quantum transitions molecules from one energetic. states to another. M. s. determined molecular composition, 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 - to visible area, 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 (luminescence spectra). During electronic transitions, the electric current usually changes. dipole moment of the molecule. Electrical dipole transition between the electronic states of a molecule of type G symmetry " and G "" (cm. Symmetry of molecules) is allowed if direct product G " 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 by the strength of the oscillator

Where e And t e are the charge and mass of the electron. For intense transitions f nm~ 1. From (1) and (4) cf. excited state lifetime:

These f-ly are also valid for vibrations. and rotate. transitions (in this case, the matrix elements of the dipole moment should be redefined). For allowed electronic transitions, the coefficient is usually absorption for several orders more than for oscillating. and rotate. transitions. Sometimes the coefficient absorption reaches a value of ~10 3 -10 4 cm -1 atm -1, i.e. electron bands are observed at very low pressures (~10 -3 - 10 -4 mm Hg) and small thicknesses (~10-100 cm) layer of matter.

Vibrational spectra observed when the vibration changes. energy (electronic and rotational energies should not change). Normal vibrations of molecules are usually represented as a set of non-interacting harmonics. oscillators. If we confine ourselves to the linear terms of the expansion of the dipole moment d (in the case of absorption spectra) or polarizability a (in the case of combination scattering) along normal coordinates Qk, then the allowed vibrations. transitions are considered only transitions with a change in one of the quantum numbers u k per unit. Such transitions correspond to the main. oscillating stripes, they are oscillating. spectra max. intense.

Main oscillating bands of a linear polyatomic molecule corresponding to transitions from the main. oscillating states can be of two types: parallel (||) bands corresponding to transitions with a transition dipole moment directed along the molecular axis, and perpendicular (1) bands corresponding to transitions with a transition dipole moment perpendicular to the molecular axis. The parallel strip consists of only R- And R-branches, and in a perpendicular strip

resolved also Q-branch (Fig. 2). Main spectrum absorption bands of a symmetrical top molecule also consists of || And | stripes, but rotate. the structure of these bands (see below) is more complex; Q-branch in || lane is also not allowed. Allowed fluctuations. stripes represent vk. Band Intensity vk depends on the square of the derivative ( dd/dQ To ) 2 or ( d a/ dQk) 2 . If the band corresponds to the transition from an excited state to a higher one, then it is called. hot.

Rice. 2. IR absorption band v 4 SF 6 molecules, obtained on a Fourier spectrometer with a resolution of 0.04 cm -1 ; niche showing fine structure lines R(39) measured on a diode laser spectrometer with a resolution of 10 -4 cm -1.

When taking into account the anharmonicity of oscillations and nonlinear terms in the expansions d and a by Qk become probable and transitions forbidden by the selection rule for u k. Transitions with a change in one of the numbers u k on 2, 3, 4, etc. called. overtone (Du k=2 - first overtone, Du k\u003d 3 - second overtone, etc.). If two or more of the numbers u change during the transition k, then such a transition is called combinational or total (if all u To increase) and difference (if some of u k decrease). Overtone bands are denoted 2 vk, 3vk, ..., total bands vk + v l, 2vk + v l etc., and the difference bands vk - v l, 2vk - e l etc. Band intensities 2u k, vk + v l And vk - v l depend on the first and second derivatives d By Qk(or a by Qk) and cubic. coefficients of anharmonicity potent. energy; the intensities of higher transitions depend on the coefficient. higher degrees of decomposition d(or a) and potent. energy by Qk.

For molecules that do not have symmetry elements, all vibrations are allowed. transitions both in the absorption of excitation energy and in combination. scattering of light. For molecules with an inversion center (eg, CO 2 , C 2 H 4 , etc.), transitions allowed in absorption are forbidden for combinations. scattering, and vice versa (alternative prohibition). The transition between oscillation energy levels of symmetry types Г 1 and Г 2 is allowed in absorption if the direct product Г 1 Г 2 contains the symmetry type of the dipole moment, and is allowed in combination. scattering if the product Г 1

Г 2 contains the symmetry type of the polarizability tensor. This selection rule is approximate, since it does not take into account the interaction of vibrations. movements with electronic and rotating. movements. Accounting for these interactions leads to the appearance of bands that are forbidden according to pure oscillations. selection rules.

The study of fluctuations. M. s. allows you to set the harmonic. oscillation frequencies, anharmonicity constants. According to fluctuations spectra is carried out conformation. analysis