Molecular vibration

A molecular vibration occurs when the Atome S of a Molécule are in a periodic movement while the molecule as a whole undergoes a rotation and translatory movement. The frequency of the periodic movement is called frequency of vibration. A non-linear molecule made up of N atoms has 3 N − 6 normal modes of vibration, whereas a linear molecule has only 3 of them N − 5 normal modes of vibration, since rotation around its molecular axis cannot be observed. A diatomic Molécule does not have as well as a normal mode of vibration. The normal modes of the polyatomic molecules are independent from/to each other, each one of them implying simultaneous vibrations of the various parts of the molécule.
A molecular vibration is produced when the molecule absorbs a quantum of energy, E, corresponds to a vibration of frequency, ν , according to the well-known relation E=hν , where H is the Planck's constant. A fundamental vibration is excited when such a quantum of energy is absorbed by the molecule in its fundamental state. When two quanta are absorbed the first harmonic is excited, and so on for the harmonics suivantes.
At first approximation, the normal movement of vibration can be described like a kind of harmonic Mouvement simple. In this approximation, the energy of vibration is a quadratic function (parabola) atomic displacements and the first harmonic is of twice the frequency of the fundamental frequency. Actually, the vibrations are anharmonic and the first harmonic has a frequency which is slightly lower than twice the fundamental one. The excitation of the higher harmonics gradually less and less requires energy additional and led to the dissociation of the molecule, the potential energy of the molecule resembling more a Potentiel of Morse.
The states vibrationnels of a molecule can be studied according to several ways. Most direct is the infra-red Spectroscopie, the vibrationnelles transitions requiring a quantity from energy which corresponds typically to the infra-red area of the spectrum. The Raman Spectroscopie, which uses the visible light typically, can also be used to measure the frequencies of vibration directement.
The excitation of vibration can occur in a way combined with the electronic excitation (vibrationnelle Transition), giving a fine vibrationnelle structure to the electronic transitions, particularly with the molecules with the state gazeux.
The excitation simultaneous of a vibration and rotations gives rise to the spectrum of rotation-vibration.

Vibrationnelles coordinates

The coordinate of a normal vibration is a combination of changes in the positions of the atoms of a molecule. When the vibration is excited, the coordinate changes in a sinusoidal way with a frequency ν, the frequency of vibration.

Internal coordinates

The internal '' coordinated '' are following types, as illustrated starting from the plane molecule of ethene, and very often indicated by the anglophone terms:

• Stretching (stretching): variation in the length of a liaision, like CH or DC.
• Bending (folding): variation in the angle formed by two connections, such as for example angle HCH in a group Methylene.
• Rocking (swinging): variation in the angle between a group of atoms, such as for example a group methylene and the remainder of the molecule.
• Wagging (agitation): variation of the angle between the plan of a group of atoms, like a group methylene and a plan passing by the remainder of the molecule.
• Twisting (torsion): variation in the angle formed by the respective plans of two groups of atoms, such as for example between the two methylenes groups.
• Out of the plan: movement nonpresent in the ethene, but which can meet for example in BF₃ when the boron atom leaves and returns in the plan formed by the three fluorine atoms.

In coordinated rocking, wagging or twisting, the angles and lengths of connections in the groups concerned do not change. The rocking can be distinguished from the wagging by the fact that the atoms in the group remain in same the plan.
In ethene, there are twelve internal coordinates: 4 stretchings CH, 1 streching DC, 2 bendings H-C-H, 2 rockings CH₂, 2 wagging CH₂, 1 twisting. It will be noted that angles H-C-C cannot be used like internal coordinates, the angles with each carbon which cannot vary at the same time.

Coordinates adapted to symmetry

Coordinates adapted to symmetry can be created by applying a projector to a whole of internal coordinates. The projector is built with the assistance of the Table of characters of the specific Groupe of molecular symmetry. Thus, the four coordinates of stretching CH (not-standardized) of the molecule of ethene are given by:

Q_s1 = q₁ + q₂ + q₃ + q₄

Q_s2 = q₁ + q₂ - q₃ - q₄

Q_s3 = q₁ - q₂ + q₃ - q₄

Q_s4 = q₁ - q₂ - q₃ + q₄

where q₁ - q₄ is the internal coordinates for the stretching of each of the four C-H.
connections Illustrations of coordinates adapted to symmetry for the majority of the small molecules can be found in the work of Nakamoto.

Normal conditions

A normal coordinate, Q , can sometimes be built directly like uen coordinated adapted to symmetry. It is possible when the normal coordinate belongs only to a irreducible representation particular of a specific group of molecular symmetry. Thus, the coordinates adapted to symmetry for the stretching of connection of the linear molecule of Carbon dioxide, O=C=O, are both of the normal coordinates:

• symmetrical stretching: the sum of the two coordinates of CO stretching, the two lengths of connections CO are modified identically and the carbon atom is stationary. Q = q₁ + q₂ .
• asymmetrical stretching: the difference of the two coordinates of CO stretching; a connection CO grows when the different one decrease. Q = q₁ - q₂ .

When two normal coordinates or more belong to the same irreducible representation of a group of specific symmetry molecular (i.e. have same symmetry), there is mix and the coefficients of the combination cannot be given a priori . Thus for example, in the linear molecule of Cyanide of hydrogen, HCN, the two vibrations of stretching are:

1. streching majority CH with stretching weak CN; Q₁ = q₁ + has q₂ (<< 1 has).
2. streching majority CN with stretching weak CH; Q₁ = B q₁ + q₂ (B << 1).

The coefficient has and B are determined by complete analysis of the normal coordinates by means of the Méthode GF of Wilson.

In traditional mechanics (Newtonian)

In a way perhaps surprising, the molecular vibrations can be treated by using traditional mechanics in order to calculate the frequencies of correct vibrations. The basic postulate is that each vibration can be treated as if it corresponded to a spring. In the harmonic approximation, a spring obeys the Loi of Hooke: the necessary force to stretch the spring is proportional to its extension. The proportionality factor is known under the name of constant of force F . The anharmonic oscillator will be considéŕe in addition.

$\backslash \; mathrm\; \{F\}\; =\; -\; F\; Q\; \backslash !$

According to the second law of the movement of Newton, this force is equal to the bulk product m by acceleration has :

$\backslash \; mathrm\; \{F\}\; =\; m\; \backslash \; frac\; \{d^2Q\}\; \{dt^2\}$

By identity, one obtains the following equation:

$m\; \backslash \; frac\; \{d^2Q\}\; \{dt^2\}\; +\; F\; Q\; =\; 0$

The solution with this equation of the simple harmonic Mouvement is:

$Q\; (T)\; =\; has\; \backslash \; cos\; (2\; \backslash \; pi\; \backslash \; naked\; T);\; \backslash \; \backslash \; \backslash \; naked\; =\; \{1\; \backslash \; over\; \{2\; \backslash \; pi\}\}\; \backslash \; sqrt\; \{F\; \backslash \; over\; m\}\; \backslash !$

has is the maximum of amplitude of the coordinate of vibration Q . It remains to define the mass, m . In a diatomic molecule homonucléaire like N₂, it is simply about the mass of the two atoms. In the case of a diatomic molecule hétéronucléaire, AB, it corresponds to the reduced Masse, μ defined by:

$\backslash \; frac\; \{1\}\; \{\backslash \; driven\}\; =\; \backslash \; frac\; \{1\}\; \{m\_A\}\; +\; \backslash \; frac\; \{1\}\; \{m\_B\}$

The use of the reduced mass makes it possible to ensure that the center of mass of the molecule is not affected by the vibration. In the harmonic approximation, the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the constant of force is equal to the derived second of this potential energy.

$f=\; \backslash \; frac\; \{\backslash \; partial\; ^2V\}\; \{\backslash \; partial\; Q^2\}$

When two normal vibrations or more have same symmetry, an analysis of the normal coordinates complete must be carried out (see Méthode GF). The frequencies of vibrations, ν _i, are obtained starting from the eigenvalues, λ _i, of the matric Produit GF . G is the matrix of numbers derived from the masses of the atoms and the geometry from the molecule.

In quantum mechanics

In the harmonic approximation, the potential energy is a quadratic function of the normal conditions. By solving the equation of Schrödinger, the energy states for each normal coordinate are given by:

$E\_n\; =\; \backslash \; left\; (N\; +\; \{1\; \backslash \; over\; 2\}\; \backslash \; right)\; \{1\; \backslash \; over\; \{2\; \backslash \; pi\}\}\; \backslash \; sqrt\; \{F\; \backslash \; over\; m\}\; \backslash !$,

where N is a quantum number being able to take as values 0,1,2… the difference in energy when N varies from 1 is thus equal to the derived energy used in traditional mechanics. One will be able to refer to the article Oscillateur quantum harmonic for more precise details. Knowing the functions of waves, some Règles of selection can be formulated. Thus, for a harmonic oscillator, the transitions are allowed only when the quantum number is modified of a unit,

$\backslash \; Delta\; N\; =\; \backslash \; pm\; 1$

but that does not apply to an anharmonic oscillator; the observation of the harmonics is only possible because the vibrations are anharmonic. Another consequence of the anharmonicity is that the transitions as between the states N =2 and N =1 have very small little less energy than the transitions between the fundamental state and the first excited state. Such a transition gives a growth towards a hot band.

Intensities

In an infra-red spectrum, the Intensité of an absorption band is proportional to derived to the Dipole moment molecular compared to the normal coordinate. The intensity of the Raman bands depends on the Polarisibilité. See also dipole Moment of transition.

References

Random links:

Monroe | Saint-Germain-of-Livet | European convention of the human rights | Battle of Varna | Best off The Corrs | Conky | Luna,_union_de_La

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org