**4. Vibration of polyatomic molecules**

In the case the vibration diatomic molecules atoms can oscillate just in the direction of connection covalent) binding atoms. In the case of molecules consisting of several atoms (N atoms) the description of the assembly oscillations, even in harmonic approximation, is significantly more complicated. In principle, each atom in the structure molecule can execute, independently of the other atoms of the same molecule, the three-way oscillation linear independent (after three axes orthogonal coordinate attached each of atoms).

Therefore, the N atoms can run periodic shifts after 3N directions. In other words, a polyatomic molecule, consisting of N atoms, has 3N degrees of freedom of movement of constituent atoms. From this number, not all directions of movement of individual atoms correspond to deformations of real three-dimensional structure of the molecule, as oscillations in which each constituent atom is moving at the same time in the same direction, with the same amplitude and phase. They are equivalent to move whole molecules (translation molecule) without deforming them.

Also, movements that the N atoms can be synchronized in such a way that the assembly rotation movements correspond to atomic molecule as a whole, without causing structural deformation. Whereas entire molecule can be translated into 3 directions linear independent of each other and can rotate around a three axis-oriented perpendicular to each other. Of the total number of 3N directions of movement are deleted atomic 6 shifts (that is 6 degrees of freedom) in order to obtain the number (3N - 6) for detailed rules for the movement of atoms, therefore the same number of modes of oscillation actual molecular structure. .

It must be remembered, however that if molecule is not free, but is linked to a structure with crystalline comparable forces with those which act between atoms of molecule, then moving molecule as a whole from its position of equilibrium means a deformation, But, in this case is not of the molecule, instead of the structure supra molecular (crystalline lattice) in which molecule is a constituent.

In the particular case of a molecules polyatomic (with N atoms) having structure linear (all the atoms constituents are willing co-linear), the number of actual oscillation of the molecule is equal to (3N - 5). In this case is deleted of the total number of 3N possible directions of displacement 3. Degrees of freedom correlated with translation no deformation of the entire molecule and only 2 degrees of freedom corresponding to rotation molecule around two mutually orthogonal axis and perpendicular to the longitudinal axis of the molecule.

The explanation lies in the fact that in this case of the inertia of the molecule to the third axis of rotation (the flush to the longitudinal axis of the molecule) is vertical, so that rotation around this axis is virtually builds up kinetic energy.

At each of the (3N - 6) (i.e. (3N - 5)) possible ways of real oscillation, in principle each of constituents of atoms vibrating molecule running around their own positions of equilibrium. But in a particular mode of oscillation, individual atoms oscillate after other directions and with other amplitudes, but with the same frequency (feature mode of vibration in question). Directions and the amplitudes individual travel, and the frequency (common) of oscillation are characteristic mode of vibration.

In the case the vibration diatomic molecules atoms can oscillate just in the direction of connection covalent) binding atoms. In the case of molecules consisting of several atoms (N atoms) the description of the assembly oscillations, even in harmonic approximation, is significantly more complicated. In principle, each atom in the structure molecule can execute, independently of the other atoms of the same molecule, the three-way oscillation

Therefore, the N atoms can run periodic shifts after 3N directions. In other words, a polyatomic molecule, consisting of N atoms, has 3N degrees of freedom of movement of constituent atoms. From this number, not all directions of movement of individual atoms correspond to deformations of real three-dimensional structure of the molecule, as oscillations in which each constituent atom is moving at the same time in the same direction, with the same amplitude and phase. They are equivalent to move whole molecules

Also, movements that the N atoms can be synchronized in such a way that the assembly rotation movements correspond to atomic molecule as a whole, without causing structural deformation. Whereas entire molecule can be translated into 3 directions linear independent of each other and can rotate around a three axis-oriented perpendicular to each other. Of the total number of 3N directions of movement are deleted atomic 6 shifts (that is 6 degrees of freedom) in order to obtain the number (3N - 6) for detailed rules for the movement of atoms, therefore the same number of modes of oscillation actual molecular structure. .

It must be remembered, however that if molecule is not free, but is linked to a structure with crystalline comparable forces with those which act between atoms of molecule, then moving molecule as a whole from its position of equilibrium means a deformation, But, in this case is not of the molecule, instead of the structure supra molecular (crystalline lattice) in which

In the particular case of a molecules polyatomic (with N atoms) having structure linear (all the atoms constituents are willing co-linear), the number of actual oscillation of the molecule is equal to (3N - 5). In this case is deleted of the total number of 3N possible directions of displacement 3. Degrees of freedom correlated with translation no deformation of the entire molecule and only 2 degrees of freedom corresponding to rotation molecule around two

The explanation lies in the fact that in this case of the inertia of the molecule to the third axis of rotation (the flush to the longitudinal axis of the molecule) is vertical, so that rotation

At each of the (3N - 6) (i.e. (3N - 5)) possible ways of real oscillation, in principle each of constituents of atoms vibrating molecule running around their own positions of equilibrium. But in a particular mode of oscillation, individual atoms oscillate after other directions and with other amplitudes, but with the same frequency (feature mode of vibration in question). Directions and the amplitudes individual travel, and the frequency

mutually orthogonal axis and perpendicular to the longitudinal axis of the molecule.

linear independent (after three axes orthogonal coordinate attached each of atoms).

**4. Vibration of polyatomic molecules** 

(translation molecule) without deforming them.

around this axis is virtually builds up kinetic energy.

(common) of oscillation are characteristic mode of vibration.

molecule is a constituent.

The vibrating atoms in a molecule polyatomic can be described as a function of internal coordinates instead of cartesian coordinates. Such are eliminated those movements which correspond translations to atoms and rotation molecule without deformation. Internal coordinates of a molecules polyatomic can be defined in different ways. How to define the most frequently involves the covalent connection between pairs of atoms connect (lab for atoms a and b), Angles between connections covalent) binding centered on an atom common (αabc for atoms a and c bound by common atom (b) and diedral angles θabcd (the angle of the plans P and R containing three connections between four **atoms covalent) binding) (Figure 2).**

Fig. 2. Define Internal coordinates l , α and θ

Each vibration mode of the molecule can be described as a time-dependent periodic variation of all the internal coordinates. For a specific molecular structure, consisting of N atoms, the whole structure runs ((3N-6) or (3N-5)) modes of preferential oscillations, involving in (3N-6) (or (3N-5)) ways the internal coordinates of the molecule. These "preferential" types of oscillations represent the normal oscillations of the molecule. In each normal type of oscillation, all the internal coordinates of the molecule oscillate at a common frequency (in principle), specific for that type of oscillation.

In each normal mode of oscillation, the internal coordinates are involved in varying degrees. For a normal oscillation is characteristic of the internal coordinate of the molecule (e.g., a covalent bond length) is more involved than the others, then it may be said (with some tolerance) that normally oscillation that the respective oscillation type is characteristic for the respective internal coordinate (e.g., the length of the covalent bond).

Figure 3 represents the fine vibration structure of the fundamental electronic state and of the first excited electronic states, for the case of a hypothetical triatomic molecule. This type of molecule has 3·3-6=3 normal vibration modes. Oscillation modes are also represented in Figure 3, indicating the direction and the direction of the relative shift of the individual atoms in one of the half period of oscillation. Each electronic state consists in a number of vibration states characterized by the vibration quantum numbers **nvib**. For each normal way

Organic Compounds FT-IR Spectroscopy 153

number (the energy gap is maximum), the second absorption band corresponding to the normal mode of vibration no. "2", at an intermediate wave number (the energy gap is intermediate) and the third absorption band corresponding to the normal mode of vibration no."3", at the lowest value number of the wave number (the energy gap is the smallest). The superior indices of the vibration quantum numbers that are present in Figure 3 ( ) <sup>123</sup> n ,n ,n vib vib vib refers to the serial number of the respective normal vibration mode. The above made statement made, that each normal vibration type generates an absorption band in the IR spectrum of the analyzed substance, is true only in principle. It is common that not all the normal vibration types of a molecule (3N-5 or 3N-6, for an N atoms molecule) to

If the covalent bonds in a molecule (or a molecular fragment) have comparable force constants or the masses of the atoms involved in the covalent bonds are close, then those atoms are involved, to some extent, in all possible normal modes of vibration of the molecule (or molecular fragment). In these situations, the individual internal coordinates (bond lengths or individual angles) do not independently vibrate; their vibrations are

If one of the atoms linked by a covalent bond has a much smaller mass than the other atom (for example, the C-H, N-H, O-H, S, H, P-H bonds), the reduced mass of the ensemble of bound atoms is almost equal to the mass of the lighter atom and in this case the oscillation of the bond length is quasi-independent from the oscillation of the rest of the molecule. It may be said, in this case, that the absorption band associated with the normal mode of vibration affects the covalent bond, and it is characteristic of the presence of that chemical

On the other hand, if two atoms in a molecule are connected by a significantly stronger covalent bond then the other covalent bonds of the molecule (e.g., the isolated double or triple bond, as in one of the cases: >C=C<, >C=O, >C=N-, -N=N-, -C-C-), than the vibration of the respective bond can be also considered quasi-independent from the oscillation of the rest of the molecule. In this case, we can say that in one of the normal modes of vibration of the molecule practically occurs only the elongation oscillation of that bond, so the optical absorption band, generated in the IR spectrum by the normal mode of vibration, is

Often, a group of atoms (or even a functional group) have their "own" normal ways of vibrations, quasi-independent of the rest of the molecule vibrations. In these situations, the normal "own" modes of the group of atoms is manifested in the IR absorption spectrum as a corresponding number of "group characteristic bands". An example is the characteristic

The IR absorption spectrum is the graphical representation of a measure of energy depending on a measure of wave of the involved radiation. IR practice has established the use of the wave number (reciprocal of the wave length and proportional to the frequency of the radiation) and of the percentage transmission (T%) or absorbance (A), as related to the

**5. Practical aspects of infrared spectrophotometric analysis** 

coupled, generating the appropriate number (N) of normal vibrations.

present absorbtion bands.

bond in the molecule structure.

characteristic for the presence of that covalent bond

band of the amidic functional group.

energy of the radiation.

of vibration (indicated by "1", "2" and "3" in Figure 3), the vibrations in the electronic states may be differently arranged. In other words, for different normal types of vibrations, the fundamental electronic state is differently "split" in substrates of vibration. In practice, the most probable vibrational transition occurs between the vibration states corresponding to the quantum numbers nvib=0 and nvib=1 for each normal vibration (transitions represented in Figure 3 by bold arrows). There is only a diminished probability to also occur, for every type of normal vibration, transitions between the vibration states corresponding to the quantum numbers nvib=0 and nvib=2 (transitions depicted in Figure 3 by dashed arrows). This kind of transition at a normal vibration type generates its first superior harmonic.

Fig. 3. Fine vibration structure of the fundamental and first excited electronic states

The transition between the vibrational states is initiated by the absorption of radiation with proper frequency (or wave number). The transition is visualized, in principle, by the appearance of the absorption band in the absorption spectrum of the analyzed substance. The incident radiation on the sample. Probability that is able produce transitions between the vibrational states of the fundamental electronic state), is located in the infrared (IR) domain. In principle, the IR absorption spectrum of a molecule contains a number of absorption bands that is equal to the normal modes of vibration of the molecule in question. Thus, in the case illustrated in Figure 3, the molecule would present the first absorption band corresponding to the normal mode of vibration no. "1" at the highest value of the wave present absorbtion bands.

152 Macro to Nano Spectroscopy

of vibration (indicated by "1", "2" and "3" in Figure 3), the vibrations in the electronic states may be differently arranged. In other words, for different normal types of vibrations, the fundamental electronic state is differently "split" in substrates of vibration. In practice, the most probable vibrational transition occurs between the vibration states corresponding to the quantum numbers nvib=0 and nvib=1 for each normal vibration (transitions represented in Figure 3 by bold arrows). There is only a diminished probability to also occur, for every type of normal vibration, transitions between the vibration states corresponding to the quantum numbers nvib=0 and nvib=2 (transitions depicted in Figure 3 by dashed arrows). This kind of transition at a normal vibration type generates its first superior harmonic.

Fig. 3. Fine vibration structure of the fundamental and first excited electronic states

The transition between the vibrational states is initiated by the absorption of radiation with proper frequency (or wave number). The transition is visualized, in principle, by the appearance of the absorption band in the absorption spectrum of the analyzed substance. The incident radiation on the sample. Probability that is able produce transitions between the vibrational states of the fundamental electronic state), is located in the infrared (IR) domain. In principle, the IR absorption spectrum of a molecule contains a number of absorption bands that is equal to the normal modes of vibration of the molecule in question. Thus, in the case illustrated in Figure 3, the molecule would present the first absorption band corresponding to the normal mode of vibration no. "1" at the highest value of the wave number (the energy gap is maximum), the second absorption band corresponding to the normal mode of vibration no. "2", at an intermediate wave number (the energy gap is intermediate) and the third absorption band corresponding to the normal mode of vibration no."3", at the lowest value number of the wave number (the energy gap is the smallest). The superior indices of the vibration quantum numbers that are present in Figure 3 ( ) <sup>123</sup> n ,n ,n vib vib vib refers to the serial number of the respective normal vibration mode. The above made statement made, that each normal vibration type generates an absorption band in the IR spectrum of the analyzed substance, is true only in principle. It is common that not all the normal vibration types of a molecule (3N-5 or 3N-6, for an N atoms molecule) to

If the covalent bonds in a molecule (or a molecular fragment) have comparable force constants or the masses of the atoms involved in the covalent bonds are close, then those atoms are involved, to some extent, in all possible normal modes of vibration of the molecule (or molecular fragment). In these situations, the individual internal coordinates (bond lengths or individual angles) do not independently vibrate; their vibrations are coupled, generating the appropriate number (N) of normal vibrations.

If one of the atoms linked by a covalent bond has a much smaller mass than the other atom (for example, the C-H, N-H, O-H, S, H, P-H bonds), the reduced mass of the ensemble of bound atoms is almost equal to the mass of the lighter atom and in this case the oscillation of the bond length is quasi-independent from the oscillation of the rest of the molecule. It may be said, in this case, that the absorption band associated with the normal mode of vibration affects the covalent bond, and it is characteristic of the presence of that chemical bond in the molecule structure.

On the other hand, if two atoms in a molecule are connected by a significantly stronger covalent bond then the other covalent bonds of the molecule (e.g., the isolated double or triple bond, as in one of the cases: >C=C<, >C=O, >C=N-, -N=N-, -C-C-), than the vibration of the respective bond can be also considered quasi-independent from the oscillation of the rest of the molecule. In this case, we can say that in one of the normal modes of vibration of the molecule practically occurs only the elongation oscillation of that bond, so the optical absorption band, generated in the IR spectrum by the normal mode of vibration, is characteristic for the presence of that covalent bond

Often, a group of atoms (or even a functional group) have their "own" normal ways of vibrations, quasi-independent of the rest of the molecule vibrations. In these situations, the normal "own" modes of the group of atoms is manifested in the IR absorption spectrum as a corresponding number of "group characteristic bands". An example is the characteristic band of the amidic functional group.
