**2. The vibration of a diatomic molecule**

For an understanding of the vibrations of a polyatomic molecule, should be first a preliminary analysis of the oscillations of a molecule composed of two atoms linked by covalent binding. Such a molecule, with N= 2 atoms, shows N = 3x2 - 5 = 1 modes of vibration. The steering as defined by the covalent binding of the two atoms is the only special steering, it is ordinary to accept that atoms will move (in a periodic motion) after direction of the covalent connection. Assembly oscillation may be considered in relation to several systems of reference. It may choose as origin of the system of reference the center of gravity of the diatomic assembly.

In this case the both atoms perform periodic shifts in relation to this reference point. The mathematical analysis of oscillations is advantageous to place the reference origin in one of atoms. In this case, however, in the place of mass mA and mB of the atoms of the molecule A-B are used *reduced mass* ( noted with μ ) of the assembly dimolecule. Reduced mass is calculated from the **mA** and **mB** of atoms of the dimolecule assembly in accordance with following relationship.

$$\frac{1}{\mu} = \frac{1}{m\_A} + \frac{1}{m\_B} \tag{1}$$

Rigorous deduction of the relationship (1) can be found in literature on the subject. During the oscillation, the kinetic energy, **E**c, and potential energy **E**p of the assembly are varying, periodically. If the system does not radiate energy to environment, or do not accept energy

different atoms of the molecule, but at a certain mode of vibration, each atom in the molecule oscillates with the same frequency. In other words, the frequency of oscillations of the atoms in molecule is characteristic of a particular mode of oscillation of the molecule.

A molecule composed of N atoms has several possible modes of oscillation. In each mode of oscillation (in principle) all the atoms of molecule perform periodic shifts around level position with a frequency of oscillation mode which is a feature of the assembly. Because each of the N atoms can run periodic shifts in 3 perpendicular directions each other, the assembly of N atoms can have 3N ways of motion. But, those displacements that correspond to moving molecule as a whole (not deform the geometry of the molecule) and movements, which correspond to entire molecules rotation about an axis (also without deforming the molecule's geometry), do not represent actual oscillation (associated with actual

These displacements (3 in number) and rotations around the three orthogonal axis (also 3 in number) are eliminated of the total number of atomic movements possible. Therefore, a molecule is, in general, (3N - 6) distinct modes of oscillation and in each of these (3N - 6) modes of oscillation each atom oscillates with frequencies characteristic individual modes of vibration. A special case represents molecules whose structures are linear, because in these cases the inertia of the molecule, in relation to the axis flush by molecule, it is practically zero. For this reason, in the case of a linear molecules consisting of N atoms, the number of

For an understanding of the vibrations of a polyatomic molecule, should be first a preliminary analysis of the oscillations of a molecule composed of two atoms linked by covalent binding. Such a molecule, with N= 2 atoms, shows N = 3x2 - 5 = 1 modes of vibration. The steering as defined by the covalent binding of the two atoms is the only special steering, it is ordinary to accept that atoms will move (in a periodic motion) after direction of the covalent connection. Assembly oscillation may be considered in relation to several systems of reference. It may choose as origin of the system of reference the center of

In this case the both atoms perform periodic shifts in relation to this reference point. The mathematical analysis of oscillations is advantageous to place the reference origin in one of atoms. In this case, however, in the place of mass mA and mB of the atoms of the molecule A-B are used *reduced mass* ( noted with μ ) of the assembly dimolecule. Reduced mass is calculated from the **mA** and **mB** of atoms of the dimolecule assembly in accordance with

11 1

*m m <sup>A</sup> <sup>B</sup>*

Rigorous deduction of the relationship (1) can be found in literature on the subject. During the oscillation, the kinetic energy, **E**c, and potential energy **E**p of the assembly are varying, periodically. If the system does not radiate energy to environment, or do not accept energy

= + (1)

μ

deformation of the molecule).

modes of vibration is 3N - 5.

gravity of the diatomic assembly.

following relationship.

**2. The vibration of a diatomic molecule** 

from the environment, then the amount of Ec and Ep remains constant during oscillation. Potential energy is dependent on the single variable of the diatomic system (namely, the deviation of the Δ**r** inter-atomic distance to **r**0) which is variable in time. Potential energy dependence of the Δ**r** (i.e. lengthening the deformation of the diatomic molecule) is expressed, in the harmonic approximation, of the relationship (2).

$$E\_p = \frac{1}{2} \cdot k \cdot \Delta r^2 = \frac{1}{2} \cdot k \cdot (r - r\_0)^2 \tag{2}$$

In the relationship (2) the coefficient 'k' is *constant of force*, size that characterises the strength of inter-atomic connection in the molecule. On the basis of expression (2) the potential energy of the diatomic assembly, using the mechanics in this quantum mechanics, may deduct quantified values ( 'allowed') of diatomic oscillator.

These values of energy 'allowed' shall be calculated on the basis of the expression (3) by substituting for the number of quantum vibration (**n**vib) integers numbers (0, 1, 2, . .

$$E\_{vib}(n\_{vib}) = E\_c + E\_p = \hbar \cdot \nu\_0 \cdot \left(n\_{vib} + \frac{1}{2}\right) \tag{3}$$

The expression (3) shows that the energy **E**vib (the sume of the kinetic energy **E**c and potential energy **E**p) has a state of vibration allowed to diatomic system depends on the number of vibration quantum **n**vib.

The lower value of energy (in the fundamental vibration's state diatomic system) is obtained by replacing **n**vib**=** 0 in the relationship (3). In the relationship (3) h is the size Planck constant. (6,626075 x 10-34 Js). If diatomic molecule fundamental changes from the vibration (**n**vib = 0) in the state of vibration excited immediately above (**n**vib = 1), then change of energy Δ**E**vib**(01)** is expressed by the relationship (4).

$$
\Delta E\_{\text{vib}}(0 \twoheadrightarrow 1) = \text{h.v0} \tag{4}
$$

This value to change the vibration energy determines how often (or the number of wavelength) at which diatomic molecule shows preferential absorption of radiation.

In principle, diatomic molecule can pass from the fundamental (**n**vib = 0 ) in a excited state (for example, corresponding **n**vib = 2) but, those quantum transitions in which the number is changing more than one establishment are prohibited by the rules of selection.

Rigorous justification of the rules of selection is treated in detail in literature on the subject.

Preferred frequency (ν0 ), the favorite number of wave **n**vib = 0 to which a small diatomic molecule absorbs radiation (hence to which generates a strip of absorption) as the transition (01), is expressed quantitatively the relationship (5)

$$\nu\_0 = \frac{1}{2 \cdot \pi} \cdot \sqrt{\frac{k}{\mu}} \quad ; \quad \bar{\nu}\_0 = \frac{1}{2 \cdot \pi \cdot c} \cdot \sqrt{\frac{k}{\mu}} \tag{5}$$

Organic Compounds FT-IR Spectroscopy 149

The horizontal lines, arranged on the inside of the cavity Morse curve, shows the values allowed (quantifiable) of the energy of vibration of the assembly diatomic. Advanced to a deformation of the length of connection inter atomic, the energy potential of deformation tends toward a limit value (**D**0) over which the energy of deformation of the assembly shall

In an approximation more accurate, 'anharmonic', in the phenomenon of vibration, the amounts permitted of the energy of oscillation are expressed a relationship similar to (3), with the difference that the anarmonic approximation. Status of vibration energy depends

1 1 ( ) ν ν

The coefficient 'x' in the relationship (8) characterized quantitatively anarmonicity of molecule diatomic vibration, i.e. the drift behavior system from the model of harmonic

In inharmonic approximation of the vibration of diatomic molecules of the selection rule, relating to the variation in **n**vib allowed for the quantum number, it is not so strict as in the case described harmonics. The model does not exclude the possibility inharmonic transitions between the status of vibration to which variation **n**vib quantum number to be

Transitions associated with variations in higher than the unit are called harmonics of the upper fundamental transition (i.e., the transition that starts at the same lower status, but for

The appearance of the absorption bands assigned to upper harmonics inherent in spectra are observed frequently in IR (especially in the case polyatomic molecules), but as a rule occur

Strips of the upper harmonics associated with fundamental tape appear at frequencies (or wave numbers) which are approximately multiples whole frequency (or the wave-number)

Another practical consequence of the inharmonicity of vibration of molecules is the rise of

These bands of absorptions are observed at frequencies equal to the sum or the difference between two frequencies or fundamental frequency of a fundamental and a harmonic one. By cause of bands of combination appear various normal modes of oscillation of the molecule. The high harmonics and the bands of combination in IR absorption spectra cause

with intensities that are smaller than corresponding fundamental bands.

the inter-combination bands in the IR absorption spectra.

considerably complications in their interpretation.

**3. Potential energy dependence of the inter atomic distance of a diatomic** 

0 0

= + =⋅ ⋅ + −⋅ ⋅⋅ +

2 2

2

(8)

on the binomial **n**vib quantum number after an expression of the degree 2 in relation:.

*E n E E h n h xn vib vib c p vib vib*

cease to be quantified.

vibration.

which Δ**n**vib = 1 ).

fundamental.

0

2,3 , etc. , in practice IR spectrophotometry.

ν

1 2

= ⋅ <sup>⋅</sup>

π

*k*

 μ

**molecule in Morse potential energy approximation** 

In the relationship (5) 'C' is the speed of propagation of electromagnetic radiation in a vacuum. At the harmonic approximation, the dependency Δ**r** is sinusoidal. But in the case of molecules, the potential energy is dependent on the momentary deflection Δ**r** of the system in a manner more complicated, so the approximation describes successfully the harmonic oscillations limited to a diatomic molecules. As a result of difficulties with mathematical order but the description of molecular oscillations, especially in the case poliatomice molecules, it accepts harmonic approximation."

$$E\_p = D\_{\mathcal{C}} \cdot \left[1 - \mathcal{C}^{-\mathcal{B}\_{\cdot (r - r\_0)}}\right]^2 \tag{6}$$

Figure 1 represents the dependency of potential energy Ep of a diatomic molecules to the momentary distance (r) in a approximately more faithful than the harmonic (based on parabolic dependence). In a more or less accurate in the description diatomic vibration of molecules, energy dependence potential (Ep ) by the distance inter atomic (r) is described by a function of type Morse (6) in place of a relationship of type (2).

Fig. 1. Dependency of potential energy Ep of a diatomic molecule to the momentary distance

In the function (6.) the coefficient β depends on the mass reduced (μ) of the assembly diatomic, in accordance with relationship (7).

$$
\beta = \nu\_0 \cdot \sqrt{2 \cdot \pi^2 \cdot \mu \cdot D\_{\mathcal{C}}} \tag{7}
$$

Continuous curve in figure 1 graphically represents the function (6). Morse function curve is compared with the curve corresponding harmonic approximation (parable with the interrupted curve).

In the relationship (5) 'C' is the speed of propagation of electromagnetic radiation in a vacuum. At the harmonic approximation, the dependency Δ**r** is sinusoidal. But in the case of molecules, the potential energy is dependent on the momentary deflection Δ**r** of the system in a manner more complicated, so the approximation describes successfully the harmonic oscillations limited to a diatomic molecules. As a result of difficulties with mathematical order but the description of molecular oscillations, especially in the case poliatomice

( ) <sup>0</sup> <sup>1</sup> *r r*

β

*E D <sup>p</sup> <sup>e</sup> e* − ⋅−

Figure 1 represents the dependency of potential energy Ep of a diatomic molecules to the momentary distance (r) in a approximately more faithful than the harmonic (based on parabolic dependence). In a more or less accurate in the description diatomic vibration of molecules, energy dependence potential (Ep ) by the distance inter atomic (r) is described by

Fig. 1. Dependency of potential energy Ep of a diatomic molecule to the momentary distance

In the function (6.) the coefficient β depends on the mass reduced (μ) of the assembly

βν

2 <sup>0</sup> 2 *De*

= ⋅ ⋅ ⋅⋅ (7)

 π μ

Continuous curve in figure 1 graphically represents the function (6). Morse function curve is compared with the curve corresponding harmonic approximation (parable with the

2

= ⋅− (6)

molecules, it accepts harmonic approximation."

diatomic, in accordance with relationship (7).

interrupted curve).

a function of type Morse (6) in place of a relationship of type (2).

The horizontal lines, arranged on the inside of the cavity Morse curve, shows the values allowed (quantifiable) of the energy of vibration of the assembly diatomic. Advanced to a deformation of the length of connection inter atomic, the energy potential of deformation tends toward a limit value (**D**0) over which the energy of deformation of the assembly shall cease to be quantified.

In an approximation more accurate, 'anharmonic', in the phenomenon of vibration, the amounts permitted of the energy of oscillation are expressed a relationship similar to (3), with the difference that the anarmonic approximation. Status of vibration energy depends on the binomial **n**vib quantum number after an expression of the degree 2 in relation:.

$$\begin{aligned} E\_{vib}(n\_{vib}) &= E\_c + E\_p = h \cdot \mathbf{v}\_0 \cdot \left( n\_{vib} + \frac{1}{2} \right) - h \cdot \mathbf{v}\_0 \cdot \mathbf{x} \cdot \left( n\_{vib} + \frac{1}{2} \right)^2 \\ \nu\_0 &= \frac{1}{2 \cdot \pi} \cdot \sqrt{\frac{k}{\mu}} \end{aligned} \tag{8}$$

The coefficient 'x' in the relationship (8) characterized quantitatively anarmonicity of molecule diatomic vibration, i.e. the drift behavior system from the model of harmonic vibration.
