Molecular vibration

A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 1013 to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1.

In general, a non-linear molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 such modes, because rotation about its molecular axis cannot be observed.[1] A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E = (where h is Planck's constant). A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.

To a first approximation, the motion in a normal vibration can be described as a kind of simple 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 are anharmonic and 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 a Morse potential or more accurately, a Morse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared 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 the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Number of vibrational modes

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear 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 3N minus 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

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

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule 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 BF3 when 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 ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH2 rocking, 2 CH2 wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

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

The atoms in a CH2 group, commonly found in organic compounds, can vibrate in six different ways: symmetric and asymmetric stretching, scissoring, rocking, wagging and twisting as shown here:

Scissoring (Bending)
Symmetrical stretching Asymmetrical stretching Scissoring
Rocking Wagging Twisting
Modo rotacao 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

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates.[5] The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by

where are 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

The normal coordinates, denoted as Q, 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 advantage of working in normal modes is that they diagonalize the matrix governing the molecular vibrations, so each normal mode is an independent molecular vibration, associated with its own spectrum of quantum mechanical states. If the molecule possesses symmetries, it will belong to a point group, and the normal modes will "transform as" an irreducible representation under that group. The normal modes can then be qualitatively 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 = q1 - q2

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 determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

  1. principally C-H stretching with a little C-N stretching; Q1 = q1 + a q2 (a << 1)
  2. principally C-N stretching with a little C-H stretching; Q2 = b q1 + q2 (b << 1)

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

Newtonian mechanics

Anharmonic oscillator
The HCl molecule as an anharmonic oscillator vibrating at energy level E3. D0 is dissociation energy here, r0 bond length, U potential energy. Energy is expressed in wavenumbers. 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 obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere.[8]

By Newton’s second law of motion this force is also equal to a reduced mass, μ, times acceleration.

Since this is one and the same force the ordinary differential equation follows.

The solution to this equation of simple harmonic motion is

A is the maximum amplitude of the vibration coordinate Q. 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, mA and mB, 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 (see GF method). The vibration frequencies,νi are obtained from the eigenvalues,λi, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.[7] F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.[9]

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by


where n is 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, this vibrational quantum number is often designated as v.[10][11]

The difference in energy when n (or v) changes by 1 is therefore equal to , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes 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 states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of anharmonic oscillator, Dunham expansion is used.


In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate.[12] Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the wavelength of laser used.

See also


  1. ^ Landau, L. D.; Lifshitz, E. M. (1976). Mechanics (3rd ed.). Pergamon Press. ISBN 0-08-021022-8.
  2. ^ a b Hollas, J. M. (1996). Modern Spectroscopy (3rd ed.). John Wiley. p. 77. ISBN 0471965227.
  3. ^ a b Banwell, Colin N.; McCash, Elaine M. (1994). Fundamentals of Molecular Spectroscopy (4th ed.). McGraw Hill. p. 71. ISBN 0-07-707976-0.
  4. ^ a b Atkins, P. W.; Paula, J. de (2006). Physical Chemistry (8th ed.). New York: W. H. Freeman. p. 460. ISBN 0716787598.
  5. ^ Cotton, F. A. (1971). Chemical Applications of Group Theory (2nd ed.). New York: Wiley. ISBN 0471175706.
  6. ^ Nakamoto, K. (1997). Infrared and Raman spectra of inorganic and coordination compounds, Part A (5th ed.). New York: Wiley. ISBN 0471163945.
  7. ^ a b Wilson, E. B.; Decius, J. C.; Cross, P. C. (1995) [1955]. Molecular Vibrations. New York: Dover. ISBN 048663941X.
  8. ^ Califano, S. (1976). Vibrational States. New York: Wiley. ISBN 0471129968.
  9. ^ Gans, P. (1971). Vibrating Molecules. New York: Chapman and Hall. ISBN 0412102900.
  10. ^ Hollas, J. M. (1996). Modern Spectroscopy (3rd ed.). John Wiley. p. 21. ISBN 0471965227.
  11. ^ Atkins, P. W.; Paula, J. de (2006). Physical Chemistry (8th ed.). New York: W. H. Freeman. pp. 291 and 453. ISBN 0716787598.
  12. ^ Steele, D. (1971). Theory of vibrational spectroscopy. Philadelphia: W. B. Saunders. ISBN 0721685803.

Further reading

External links

Aroma compound

An aroma compound, also known as an odorant, aroma, fragrance, or flavor, is a chemical compound that has a smell or odor. A chemical compound has a smell or odor when it is sufficiently volatile to be transported to the olfactory system in the upper part of the nose.

Generally molecules meeting this specification have molecular weights of less than 300. Flavors affect both the sense of taste and smell, whereas fragrances affect only smell. Flavors tend to be naturally occurring, and fragrances tend to be synthetic.Aroma compounds can be found in food, wine, spices, floral scent, perfumes, fragrance oils, and essential oils. For example, many form biochemically during the ripening of fruits and other crops. In wines, most form as byproducts of fermentation. Also, many of the aroma compounds play a significant role in the production of flavorants, which are used in the food service industry to flavor, improve, and generally increase the appeal of their products.

An odorizer may add a detectable odor to a dangerous odorless substance, like propane, natural gas, or hydrogen, as a safety measure.

Electromagnetic spectrum

The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies.

The electromagnetic spectrum covers electromagnetic waves with frequencies ranging from below one hertz to above 1025 hertz, corresponding to wavelengths from thousands of kilometers down to a fraction of the size of an atomic nucleus. This frequency range is divided into separate bands, and the electromagnetic waves within each frequency band are called by different names; beginning at the low frequency (long wavelength) end of the spectrum these are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays at the high-frequency (short wavelength) end. The electromagnetic waves in each of these bands have different characteristics, such as how they are produced, how they interact with matter, and their practical applications. The limit for long wavelengths is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length. Gamma rays, X-rays, and high ultraviolet are classified as ionizing radiation as their photons have enough energy to ionize atoms, causing chemical reactions. Exposure to these rays can be a health hazard, causing radiation sickness, DNA damage and cancer. Radiation of visible light wavelengths and lower are called nonionizing radiation as they cannot cause these effects.

In most of the frequency bands above, a technique called spectroscopy can be used to physically separate waves of different frequencies, producing a spectrum showing the constituent frequencies. Spectroscopy is used to study the interactions of electromagnetic waves with matter. Other technological uses are described under electromagnetic radiation.

Jacobi coordinates

In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows:

For the N-body problem the result is:


The vector is the center of mass of all the bodies:

The result one is left with is thus a system of N-1 translationally invariant coordinates and a center of mass coordinate , from iteratively reducing two-body systems within the many-body system.

Libration (molecule)

Libration (from the Latin verb librare "to balance, to sway"; cf. libra "scales") is a type of reciprocating motion in which an object with a nearly fixed orientation repeatedly rotates slightly back and forth. In physics and chemistry, a molecule (or other group of atoms) can undergo libration if it is subject to external forces or constraints that restrict its orientation.

For example, in liquid water, any given water molecule is attracted to neighboring molecules, so that it has a preferred orientation and cannot freely rotate. (Of course, over time, the neighboring molecules move around and the preferred orientation changes.) However, it can undergo librational motions, which are measureable in an infrared absorption spectrum and contribute to motional narrowing of other peaks, for instance the OH stretch.

Another example is a molecular crystal: Each molecular unit has a preferred orientation due to interactions with the nearby molecules, but they have librational modes corresponding to small rotations about this preferred orientation.

Lindemann index

The Lindemann index is a simple measure of thermally driven disorder in atoms or molecules. The local Lindemann index is defined as:

Where angle brackets indicate a time average. The global Lindemann index is a system average of this quantity.

In condensed matter physics 
a departure from linearity in the behaviour of the global Lindemann index or an increase above a threshold value related to the spacing between atoms (or micelles, particles, globules, etc.) is often taken as the indication that a solid-liquid phase transition has taken place. See Lindemann melting criterion.
often possess separate regions with different order characteristics. In order to quantify or illustrate local disorder, the local Lindemann index can be used.

Care must be taken if the molecule possesses globally defined dynamics, such as about a hinge or pivot, because these motions will obscure the local motions which the Lindemann index is designed to quantify. An appropriate tactic in this circumstance is to sum the rij only over a small number of neighbouring atoms to arrive at each qi. A further variety of such modifications to the Lindemann index are available and have different merits, e.g. for the study of glassy vs crystalline materials.

List of infrared articles

This is a list of infrared topics.

Luca Turin

Luca Turin (born 20 November 1953) is a biophysicist and writer with a long-standing interest in the sense of smell, perfumery, and the fragrance industry.


Mechanosynthesis is a term for hypothetical chemical syntheses in which reaction outcomes are determined by the use of mechanical constraints to direct reactive molecules to specific molecular sites. There are presently no non-biological chemical syntheses which achieve this aim. Some atomic placement has been achieved with scanning tunnelling microscopes.

Melting point

The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at a standard pressure such as 1 atmosphere or 100 kPa.

When considered as the temperature of the reverse change from liquid to solid, it is referred to as the freezing point or crystallization point. Because of the ability of some substances to supercool, the freezing point is not considered as a characteristic property of a substance. When the "characteristic freezing point" of a substance is determined, in fact the actual methodology is almost always "the principle of observing the disappearance rather than the formation of ice", that is, the melting point.

Molecular geometry

Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom.

Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of molecule, i.e. they can be understood as approximately local and hence transferable properties.

Molecular spring

A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration.

Any molecule can be deformed in several ways - A-A bond length, A-A-A angle, A-A-A-A torsion angle.

Deformed molecule store energy, which can be released and cause mechanical work as the molecule return into its optimal geometrical conformation.

The term molecular string is usually used in nano-science and molecular biology, however theoretically also macroscopic molecular springs can be considered, if it is manufactured. Such a device composed for example of arranged ultra-high molecular mass polymer fibres (Helicene, Polyacetylene) could store extraordinary (0.1-10MJ/kg in comparison to 0.0003MJ/kg of clockwork spring) amount of energy which can be stored and released almost instantly, with high energy conversion efficiency. The amount of energy storable in molecular spring is limited by the value of deformation the molecule can withstand until it undergoes chemical change. Manufacturing of such macroscopic device is however out of reach of contemporary technology, because of difficulties of synthesis and molecular arrangement of such long polymer molecules. In addition, the force needed to draw molecular string to its maximum length could be impractically high - comparable to the tensile strength of particular polymer molecule (~100GPa for some carbon compounds)

Olfactory receptor

Olfactory receptors (ORs), also known as odorant receptors, are expressed in the cell membranes of olfactory receptor neurons and are responsible for the detection of odorants (i.e., compounds that have an odor) which give rise to the sense of smell. Activated olfactory receptors trigger nerve impulses which transmit information about odor to the brain. These receptors are members of the class A rhodopsin-like family of G protein-coupled receptors (GPCRs). The olfactory receptors form a multigene family consisting of around 800 genes in humans and 1400 genes in mice.

Reaction intermediate

A reaction intermediate or an intermediate is a molecular entity that is formed from the reactants (or preceding intermediates) and reacts further to give the directly observed products of a chemical reaction. Most chemical reactions are stepwise, that is they take more than one elementary step to complete. An intermediate is the reaction product of each of these steps, except for the last one, which forms the final product. Reactive intermediates are usually short lived and are very seldom isolated. Also, owing to the short lifetime, they do not remain in the product mixture.

For example, consider this hypothetical stepwise reaction:

A + B → C + DThe reaction includes these elementary steps:

A + B → X*

X* → C + DThe chemical species X* is an intermediate.

Rovibronic coupling

Rovibronic coupling denotes the simultaneous interactions between rotational, vibrational, and electronic degrees of freedom in a molecule. When a rovibronic transition occurs, the rotational, vibrational, and electronic states change simultaneously, unlike in rovibrational coupling. The coupling can be observed spectroscopically and is most easily seen in the Renner-Teller effect in which a linear polyatomic molecule is in a degenerate electronic state and bending vibrations will cause a large rovibronic coupling.

Scintillation (physics)

Scintillation is a flash of light produced in a transparent material by the passage of a particle (an electron, an alpha particle, an ion, or a high-energy photon). See scintillator and scintillation counter for practical applications.

Simple harmonic motion

In mechanics and physics, simple harmonic motion is a special type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. For simple harmonic motion to be an accurate model for a pendulum, the net force on the object at the end of the pendulum must be proportional to the displacement. This is a good approximation when the angle of the swing is small.

Simple harmonic motion provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis (the definition can be restated as "the periodic motion of a body along a straight line, such that the acceleration is directed towards the center of the motion and also proportional to the displacement from that point").

Structural isomer

A structural isomer, or constitutional isomer (per IUPAC), is a type of isomer in which molecules with the same molecular formula have different bonding patterns and atomic organization, as opposed to stereoisomers, in which molecular bonds are always in the same order and only spatial arrangement differs. There are multiple synonyms for structural isomers.

Three categories of structural isomers are skeletal, positional, and functional isomers. Positional isomers are also called regioisomers.

Triatomic molecule

Triatomic molecules are molecules composed of three atoms, of either the same or different chemical elements. Examples include H2O, CO2 (pictured) and HCN.

Vibronic coupling

Vibronic coupling (also called nonadiabatic coupling or derivative coupling) in a molecule involves the interaction between electronic and nuclear vibrational motion. The term "vibronic" originates from the combination of the terms "vibrational" and "electronic", denoting the idea that in a molecule, vibrational and electronic interactions are interrelated and influence each other. The magnitude of vibronic coupling reflects the degree of such interrelation.

In theoretical chemistry, the vibronic coupling is neglected within the Born–Oppenheimer approximation. Vibronic couplings are crucial to the understanding of nonadiabatic processes, especially near points of conical intersections. The direct calculation of vibronic couplings is not common due to difficulties associated with its evaluation.

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