Molecular vibration

As molecular vibrational a is periodic motion of neighboring atoms in a molecule understood. These vibrations occur in every molecule. You can use the supply of energy are stimulated, for example, by the absorption of electromagnetic radiation . The absorption and emission of radiation form the basis for the methods of vibration spectroscopy , such as infrared or Raman spectroscopy . The frequency of the periodic movement is called the oscillation frequency.

Normal vibrations

The simplest case of a molecule is the diatomic molecule, for example molecular oxygen (O ₂ ) and nitrogen (N ₂ ) or carbon monoxide (CO). These molecules only have one degree of freedom of vibration along the bond axis, because the two atoms can only vibrate with respect to their one distance. It is therefore this is a stretch or stretching . In the case of polyatomic molecules, further independent types of vibration result in a more complicated vibration behavior, the so-called normal vibrations.

The number of possible normal oscillations results from the movement possibilities of the atoms in space. Each individual atom has three degrees of freedom of movement that correspond to the three spatial axes. Accordingly, atoms have degrees of freedom. In a molecule made up of atoms, each atom is "connected" to at least one other atom. that is, it cannot move in any direction at will. However, since a molecule is not completely rigid, the atoms can move a little towards each other in all directions, so that the number of degrees of freedom in the molecule is not reduced. These degrees of freedom can now be divided as follows: ${\ displaystyle N}$ ${\ displaystyle 3N}$ ${\ displaystyle N}$ ${\ displaystyle 3N}$ ${\ displaystyle 3N}$

There are three degrees of freedom for translations of the entire molecule in the x, y and z directions;
two (in the case of linear molecules where the rotation around the axis cannot be observed) or three (in the case of angled molecules) relate to molecular rotations in space.

For the vibration degrees of freedom (also referred to as vibration modes , normal vibration or natural vibration ), or degrees of freedom remain. For example, the linear 3-atom carbon dioxide molecule (CO ₂ ) exhibits normal vibrations , which are referred to as antisymmetric and symmetrical stretching vibrations (ν _as and ν _s ) as well as deformation vibrations (δ) parallel and perpendicular to the plane of the drawing. The kinked water molecule (H ₂ O), on the other hand, has vibration modes. ${\ displaystyle 3N-5}$ ${\ displaystyle 3N-6}$ ${\ displaystyle 3 \ times 3-5 = 4}$ ${\ displaystyle 3 \ times 3-6 = 3}$

Waveforms

For diatomic and triatomic molecules there are only a few normal vibrations (at most 5), which can be classified into two groups and are often referred to in the so-called McKean symbols :

Stretching vibrations (stretching vibrations): vibrations along the bond axis of two atoms in a molecule as a result of stretching or compressing the bond
Deformation vibrations : vibrations under the deformation of the bond angle
- in the plane (bending / flexing vibrations)
- out of plane (torsional / tilting vibrations) - mostly perpendicular to the plane of the bond

These vibrations can also be determined on molecular groups of 3 or 4 atoms, for example the methylene (> CH ₂ ) or methyl groups (-CH ₃ ) typical of organic compounds .

Stretching vibrations using the example of the methylene group
Symmetric stretch
(Engl. Symmetrical stretching )
Antisymmetric stretch
(Engl. Antisymmetrical stretching )

Deformation vibrations using the example of the methylene group
"Rocking vibration"
(Engl. Rocking )
Shear / deformation
vibration
(English scissoring or bending )
Torsional vibration
(English twisting or torsing )
"Rocking vibration"
(English wagging )

Example: ethene

Structure of ethene

In this section, the normal vibrations that occur are shown again using the example of the organic compound ethene . Ethene consists of 6 atoms and has a planar, angled structure. This results in normal vibrations that can be assigned as follows: ${\ displaystyle 3 \ times 6-6 = 12}$

5 stretching vibrations due to the change in length of the C – H (4 ×) or C = C bonds (1 ×)
2 shear vibrations of the two H – C – H groups, due to the change in the angle in the methylene group
2 rocking vibrations by changing the angle between a group of atoms, for example the methylene group, and the rest of the molecule.
2 Rocking oscillation due to the change in the angle between the plane in which a group of atoms, for example the methylene group, lies and the plane which is spanned by the other constituents.
1 torsional vibration due to the change in the angle in the two methylene groups.

Note that the H-C = C angles cannot be used as internal coordinates, since the angles on each carbon atom cannot all be increased at the same time.

Non-Rigid Molecules

In addition to normal vibrations, in which either the bond length or the bond angle is changed, intra-molecular vibrations can occur in which the configuration of the molecule is changed. Such a molecule is called non-rigid . For example, in the ammonia molecule, the three hydrogen atoms can oscillate around the nitrogen atom, causing an inversion . The energy required for this is about 24.3 kJ / mol . These configuration changes can occur in molecules with at least three flexible bonds. In the case of molecules with four bonds ( tetrahedron configuration ), such as methane , inversion cannot usually occur. Such a molecule is rigid. In the case of five flexible bonds, however, an inversion is possible again. The phosphorus pentafluoride molecule can be converted from a trigonal bipyramid into a square pyramid . The energy barrier for this is around 25 kJ / mol. Octahedral molecules with six bonds, on the other hand, are again particularly rigid.

Excitation by electromagnetic radiation

A molecular oscillation is excited when the molecule absorbs a quantum with the energy . Here is the frequency of the oscillation and Planck's quantum of action. The molecule is excited to its normal oscillation when such a quantum is absorbed by the molecule in the ground state. When another quantum is absorbed, the first "overtone" is stimulated. Further absorbed quanta stimulate the molecule to produce higher “overtones”. ${\ displaystyle E = h \ nu}$ ${\ displaystyle \ nu}$ ${\ displaystyle h}$

As a first approximation, the movement of a normal oscillation can be described as a kind of simple harmonic movement . In this approximation, the vibration energy is a quadratic function ( parabola ) in relation to the spatial displacements of the atoms and the first “overtone” (higher vibration mode) has twice the frequency of the “fundamental”. In reality, vibrations are anharmonic and the first "overtone" has a frequency that is slightly lower than twice the fundamental. The excitation of the higher "overtones" contains less and less additional energy and ultimately leads to the dissociation of the molecule, since the potential energy of the molecule tends to become flatter with increasing distance (see e.g. Morse potential ).

The vibrational states of a molecule can be examined by a variety of methods. The most direct way is the investigation with the help of infrared spectroscopy , because the energy necessary to excite vibrations corresponds to the energy of a photon in the infrared range for most compounds. Another frequently used method to measure the vibrations directly is Raman spectroscopy , which usually uses visible light.

The vibration can also be excited by electronic excitation ( vibronic transition ). In this way, the vibration fine structure can be examined. This is especially true for molecules in the gas state.

With simultaneous excitation of a molecular oscillation and rotation, rotation-oscillation spectra are created .

Normal coordinates

The normal coordinates relate to the normal modes of the oscillations of the atoms out of their equilibrium positions. Each normal mode is assigned to a single normal coordinate. Formally, normal vibrations are determined by diagonalizing a matrix (see eigenmode ), so that each normal mode is an independent molecular vibration that is connected to its own spectrum of quantum mechanical states. If the molecule has symmetries, it belongs to a point group and the normal modes correspond to an irreducible representation of the point group. The normal oscillation can then be qualitatively determined by applying group theory and projecting the irreducible representation onto the representation in Cartesian coordinates. An example of this is the application to the CO ₂ molecule. It is found that the C = O stretching vibrations can be broken down into a symmetrical and an asymmetrical O = C = O stretching vibration:

symmetrical stretching oscillation: the sum of the two C – O stretching coordinates. The length of the two C – O bonds changes by the same amount and the carbon atom is fixed: Q = q ₁ + q ₂
asymmetrical stretching oscillation: the difference between the two C – O stretching coordinates. The length of one C – O bond increases while the other decreases: Q = q ₁ - q ₂

If two or more normal coordinates have the same non-reducible representation of the molecular point group (colloquially, they have the same symmetry), then there is a "mixture" and the two coefficients of the combination can no longer be determined a priori . An example of this are the two stretching vibrations in the linear hydrogen cyanide (HCN) molecule

mainly a C – H stretch with a small C – N stretch; Q ₁ = q ₁ + a q ₂ ( a ≪ 1)
mainly a C – N stretch with a small C – H stretch; Q ₂ = b q ₁ + q ₂ ( b ≪ 1)

The coefficients a and b were found by a complete analysis of the normal coordinates using the GF method by Edgar Bright Wilson .

Newtonian mechanics

Molecule between two uncharged capacitor plates
Molecule aligns itself in the electromagnetic field and increases its bond distance

A simple model for explaining the excitation of molecular vibrations and rotations is the behavior of a permanent electric dipole in an electric field according to classical Newtonian mechanics. Thereby forces act on the dipole through the external field. The following reactions therefore result for the model of dipole excitation by the (static) electric field of a plate capacitor shown in the picture opposite.

The dipole or the molecule aligns itself along the electric field
The binding distance increases due to the forces acting.

If alternating voltage is now applied or if the molecule is excited with an electromagnetic wave, the functional groups “hanging” on the bonds begin to vibrate and rotate.

It is based on the fact that attractive and repulsive forces act between bound atoms in a molecule. The optimal bond distance in the molecule is in a minimum of the associated potential function. Put simply, the bond between the atoms can be viewed as a kind of spring and the molecular oscillation can therefore be viewed as a spring oscillator. Assuming a harmonic approximation - the anharmonic oscillator is considered in Califano (1976), among others - the spring obeys Hook's law , i.e. the required force is proportional to the change in length of the spring. The constant of proportionality k is called the force constant . ${\ displaystyle F}$

{\ displaystyle F = -kQ \!}

According to Newton's second law , this force is equal to the product of the reduced mass μ and the acceleration.

{\ displaystyle F = \ mu {\ frac {d ^ {2} Q} {dt ^ {2}}}}

Since it is one and the same force, we get the ordinary differential equation :

{\ displaystyle \ mu {\ frac {d ^ {2} Q} {dt ^ {2}}} + kQ = 0}

The solution to this equation for simple harmonic motion is:

{\ displaystyle Q (t) = A \ cos (2 \ pi \ nu t); \ \ \ nu = {1 \ over {2 \ pi}} {\ sqrt {k \ over \ mu}}. \!}

wherein A is the maximum amplitude of the vibration component Q is. It remains to define the reduced mass μ . In general, the reduced mass of a diatomic molecule AB with the respective atomic weights m _A and m _{B is} :

{\ displaystyle {\ frac {1} {\ mu}} = {\ frac {1} {m_ {A}}} + {\ frac {1} {m_ {B}}}.}

Using the reduced mass ensures that the center of mass of the molecule is not affected by the vibration. In the case of the harmonic approximation, the potential energy of the molecule is a quadratic function of the respective normal coordinates. It follows that the force constant is equal to the second derivative of the potential energy:

{\ displaystyle k = {\ frac {\ partial ^ {2} V} {\ partial Q ^ {2}}}}

If two or more normal vibrations have the same symmetry, a full normal coordinate analysis must be carried out ( GF method ). The oscillation frequencies ν _i can be determined from the eigenvalues λ _{i of} the matrix product GF . G is a matrix of numbers determined from the masses of the atoms and the geometry of the molecule. F is a matrix that is determined from the values of the force constants. Details about the determination of the eigenvalues can be found in Gans (1971), among others.

This mechanistic model can only be used to a limited extent, because firstly it cannot describe molecules without a permanent dipole moment and secondly it cannot explain why only discrete energies are allowed for the excitation.

Quantum mechanical description

Parabola of the quantum mechanical model

As in the model of classical mechanics, the basis of the quantum mechanical model of oscillation and rotation excitation is the potential function. In the case of the harmonic approximation, the potential function can be approximated in its minimum by a quadratic function (such a parabola results from the integration of Hooke's law of springs.) By absorbing electromagnetic radiation, the molecule begins to oscillate, which causes the oscillation from the Basic in the first excited oscillation state is raised. To determine the energy required for this , the Schrödinger equation must be solved for this potential. After separating the relative motion of atomic nuclei and electrons ( Born-Oppenheimer approximation ), the solution of the Schrödinger equation results in a relationship between the required energy, the bond strength ( ) and the reduced mass ( ). In contrast to the classical harmonic oscillator, in the quantum mechanical case the vibration energy is quantized by the vibration quantum number: ${\ displaystyle k}$ ${\ displaystyle \ mu}$ ${\ displaystyle n}$

{\ displaystyle - {\ frac {\ hbar ^ {2}} {2 \ mu}} {\ frac {d ^ {2} \ Psi} {dx ^ {2}}} + V \ Psi = E \ Psi}

with the reduced Planck constant and the potential (see harmonic oscillator ). The solution of the Schrödinger equation gives the following energy states: ${\ displaystyle \ hbar = {\ frac {h} {2 \ pi}}}$ ${\ displaystyle V = {\ frac {k} {2}} x ^ {2}}$

{\ displaystyle E_ {n} = h \ nu = \ left (n + {\ frac {1} {2}} \ right) \ hbar {\ sqrt {\ frac {k} {\ mu}}}}

With knowledge of the wave functions, certain selection rules can be formulated. For example, for a harmonic oscillator only transitions are allowed if the quantum number n only changes by 1:

{\ displaystyle \ Delta n = \ pm 1}

,

but these do not apply to an anharmonic oscillator; the observation of "overtones" is only possible because vibrations are anharmonic. Another consequence of this anharmonic character is that the transitions between the states and have slightly less energy than transitions between the ground state and the first excited state. Such a transition leads (engl. A "hot junction" hot transition ) or a "hot band " (eng. Hot Band ). ${\ displaystyle n = 2}$ ${\ displaystyle n = 1}$

literature

Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Vibrational Spectroscopy: An Introduction . Thieme, 1988, ISBN 978-3-13-625102-7 .
Claus Czeslik, Heiko Seemann, Claus Czeslik, Roland Winter, Heiko Seemann, Roland Winter: Basic knowledge of physical chemistry . Vieweg + Teubner, 2010, ISBN 978-3-8348-0937-7 .
Erwin Riedel: Inorganic Chemistry . Walter de Gruyter, 2004, ISBN 978-3-11-018168-5 .

Individual evidence

^ Lew D. Landau, Evgenij Michailovič Lifšic: Mechanics . In: Course of theoretical physics . 3. Edition. tape 1 . Pergamon, Oxford 1976, ISBN 0-08-021022-8 .
^ Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Schwingungsspektoskopie: An introduction . 2., revised. Edition. Thieme, Stuttgart 1988, ISBN 3-13-625102-4 , p. 42 .
↑ Entry on Ethylene (Vibrational and / or electronic energy levels). In: P. J. Linstrom, W. G. Mallard (Eds.): NIST Chemistry WebBook, NIST Standard Reference Database Number 69 . National Institute of Standards and Technology , Gaithersburg MD, accessed November 17, 2019.
↑ Ethylene, C2H4 (D2h) . Perdue University (requires MDL-Chime-Plugin!)
↑ Konrad Seppelt: Non-Rigid Molecules . In: Chemistry in Our Time . tape 9 , no. 1 , 1975, p. 10-17 , doi : 10.1002 / ciuz.19750090103 .
^ ^A ^b E. B. Wilson, JC Decius, PC Cross: Molecular vibrations. McGraw-Hill, 1955 (Reprinted by Dover 1980).
^ Salvatore Califano: Vibrational States. John Wiley & Sons, New York 1976.
↑ P. Gans: Vibrating molecules. An introduction to the interpretation of infrared and Raman spectra. Chapman and Hall, 1971.

[1] Lew D. Landau, Evgenij Michailovič Lifšic: Mechanics . In: Course of theoretical physics . 3. Edition. tape 1 . Pergamon, Oxford 1976, ISBN 0-08-021022-8 .

[2] Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Schwingungsspektoskopie: An introduction . 2., revised. Edition. Thieme, Stuttgart 1988, ISBN 3-13-625102-4 , p. 42 .

[3] Entry on Ethylene (Vibrational and / or electronic energy levels). In: P. J. Linstrom, W. G. Mallard (Eds.): NIST Chemistry WebBook, NIST Standard Reference Database Number 69 . National Institute of Standards and Technology , Gaithersburg MD, accessed November 17, 2019.

[4] Ethylene, C2H4 (D2h) . Perdue University (requires MDL-Chime-Plugin!)

[5] Konrad Seppelt: Non-Rigid Molecules . In: Chemistry in Our Time . tape 9 , no. 1 , 1975, p. 10-17 , doi : 10.1002 / ciuz.19750090103 .

[WilsonDeciusCross-6] A ^b E. B. Wilson, JC Decius, PC Cross: Molecular vibrations. McGraw-Hill, 1955 (Reprinted by Dover 1980).

[Califano-7] Salvatore Califano: Vibrational States. John Wiley & Sons, New York 1976.

[Gans-8] P. Gans: Vibrating molecules. An introduction to the interpretation of infrared and Raman spectra. Chapman and Hall, 1971.