Amolecular vibrationis aperiodic motionof theatomsof amoleculerelative to each other, such that the center of mass of the molecule remains unchanged. Thetypicalvibrational frequenciesrange from less than 1013Hzto approximately 1014Hz, corresponding towavenumbersof approximately 300 to 3000 cm−1andwavelengthsof approximately 30 to 3 μm.

For a diatomic molecule A−B, the vibrational frequency in s−1is given by,where k is theforce constantin dyne/cm or erg/cm2and μ is thereduced massgiven by.The vibrational wavenumber in cm−1iswhere c is thespeed of lightin cm/s.

Vibrations of polyatomic molecules are described in terms ofnormal modes,which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule. In general, a non-linear molecule withNatoms has 3N– 6normal modes of vibration,but alinearmolecule has 3N– 5 modes, because rotation about the molecular axis cannot be observed.[1]Adiatomic moleculehas one normal mode of vibration, since it can only stretch or compress the single bond.

A molecular vibration is excited when the molecule absorbs energy,ΔE,corresponding to the vibration's frequency,ν,according to the relation ΔE=,wherehisPlanck's constant.A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in itsground state.When multiple quanta are absorbed, the first and possibly higherovertonesare excited.

To a first approximation, the motion in a normal vibration can be described as a kind ofsimple harmonic motion.In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations areanharmonicand the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like aMorse potentialor more accurately, aMorse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is throughinfrared spectroscopy,as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum.Raman spectroscopy,which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of therule of mutual exclusionforcentrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in theultraviolet-visibleregion. The combined excitation is known as avibronic transition,giving vibrational fine structure toelectronic transitions,particularly for molecules in thegas state.

Simultaneous excitation of a vibration and rotations gives rise tovibration–rotationspectra.

Number of vibrational modes

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For a molecule withNatoms, the positions of allNnuclei depend on a total of 3Ncoordinates,so that the molecule has 3Ndegrees of freedomincludingtranslation,rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For alinear molecule,rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates.[2][3]

An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate.[4]

The number of vibrational modes is therefore 3Nminus the number of translational and rotational degrees of freedom, or 3N–5 for linear and 3N–6 for nonlinear molecules.[2][3][4]

Vibrational coordinates

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The coordinate of a normal vibration is a combination ofchangesin the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequencyν,the frequency of the vibration.

Internal coordinates

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Internal coordinatesare of the following types, illustrated with reference to the planar moleculeethylene,

Ethylene
  • Stretching: a change in the length of a bond, such as C–H or C–C
  • Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
  • Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule.
  • Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule,
  • Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups.
  • Out-of-plane: a change in the angle between any one of the C–H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF3when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethylene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH2rocking, 2 CH2wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as well as the H-C-H angle because the angles at each carbon atom cannot all increase at the same time.

Note that these coordinates do not correspond to normal modes (see#Normal coordinates). In other words, they do not correspond to particular frequencies or vibrational transitions.

Vibrations of a methylene group (−CH2−) in a molecule for illustration

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Within the CH2group, commonly found inorganic compounds,the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1.symmetricandasymmetric stretching,2.scissoringandrocking,3.waggingandtwisting.These are shown here:

Symmetrical
stretching
Asymmetrical
stretching
Scissoring (Bending)
Rocking Wagging Twisting

(These figures do not represent the "recoil"of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

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Symmetry–adapted coordinates may be created by applying aprojection operatorto a set of internal coordinates.[5]The projection operator is constructed with the aid of thecharacter tableof the molecularpoint group.For example, the four (un–normalized) C–H stretching coordinates of the molecule ethene are given by whereare the internal coordinates for stretching of each of the four C–H bonds.

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.[6]

Normal coordinates

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The normal coordinates, denoted asQ,refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" anirreducible representationunder itspoint group.The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO2,it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch:

  • symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary.Q=q1+q2
  • asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases.Q=q1q2

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determineda priori.For example, in the linear moleculehydrogen cyanide,HCN, The two stretching vibrations are

  • principally C–H stretching with a little C–N stretching;Q1=q1+aq2(a<< 1)
  • principally C–N stretching with a little C–H stretching;Q2=bq1+q2(b<< 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the WilsonGF method.[7]

Newtonian mechanics

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TheHClmolecule as an anharmonic oscillator vibrating at energy level E3.D0isdissociation energyhere, r0bond length,Upotential energy.Energy is expressed inwavenumbers.The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeysHooke's law:the force required to extend the spring is proportional to the extension. The proportionality constant is known as aforce constant, k.The anharmonic oscillator is considered elsewhere.[8] ByNewton's second law of motionthis force is also equal to areduced mass,μ,times acceleration. Since this is one and the same force theordinary differential equationfollows. The solution to this equation ofsimple harmonic motionis Ais the maximum amplitude of the vibration coordinateQ.It remains to define the reduced mass,μ.In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses,mAandmB,as The use of the reduced mass ensures that the centre 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 force-constant is equal to the second derivative of the potential energy.

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (seeGF method). The vibration frequencies,νi,are obtained from theeigenvalues,λi,of thematrix productGF.Gis a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.[7]Fis a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.[9]

Quantum mechanics

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In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving theSchrödinger wave equation,the energy states for each normal coordinate are given by wherenis a quantum number that can take values of 0, 1, 2... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, thisvibrational quantum numberis often designated asv.[10][11]

The difference in energy whenn(orv) changes by 1 is therefore equal to,the product of thePlanck constantand the vibration frequency derived using classical mechanics. For a transition from levelnto leveln+1due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency(in the harmonic oscillator approximation).

Seequantum harmonic oscillatorfor graphs of the first 5 wave functions, which allow certainselection rulesto be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum numbernchanges by one,

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between statesn=2 andn=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to ahot band.To describe vibrational levels of an anharmonic oscillator,Dunham expansionis used.

Intensities

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In an infrared spectrum theintensityof an absorption band is proportional to the derivative of themolecular dipole momentwith respect to the normal coordinate.[12]Likewise, the intensity of Raman bands depends on the derivative ofpolarizabilitywith respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.

See also

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References

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  1. ^Landau, L. D.; Lifshitz, E. M. (1976).Mechanics(3rd ed.).Pergamon Press.ISBN0-08-021022-8.
  2. ^abHollas, J. M. (1996).Modern Spectroscopy(3rd ed.). John Wiley. p. 77.ISBN0471965227.
  3. ^abBanwell, Colin N.; McCash, Elaine M. (1994).Fundamentals of Molecular Spectroscopy(4th ed.).McGraw Hill.p.71.ISBN0-07-707976-0.
  4. ^abAtkins, P. W.; Paula, J. de (2006).Physical Chemistry(8th ed.). New York:W. H. Freeman.p.460.ISBN0716787598.
  5. ^Cotton, F. A. (1971).Chemical Applications of Group Theory(2nd ed.). New York: Wiley.ISBN0471175706.
  6. ^Nakamoto, K. (1997).Infrared and Raman spectra of inorganic and coordination compounds, Part A(5th ed.). New York: Wiley.ISBN0471163945.
  7. ^abWilson, E. B.; Decius, J. C.; Cross, P. C. (1995) [1955].Molecular Vibrations.New York: Dover.ISBN048663941X.
  8. ^Califano, S. (1976).Vibrational States.New York: Wiley.ISBN0471129968.
  9. ^Gans, P. (1971).Vibrating Molecules.New York:Chapman and Hall.ISBN0412102900.
  10. ^Hollas, J. M. (1996).Modern Spectroscopy(3rd ed.). John Wiley. p. 21.ISBN0471965227.
  11. ^Atkins, P. W.; Paula, J. de (2006).Physical Chemistry(8th ed.). New York: W. H. Freeman. pp.291 and 453.ISBN0716787598.
  12. ^Steele, D. (1971).Theory of vibrational spectroscopy.Philadelphia:W. B. Saunders.ISBN0721685803.

Further reading

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