A molecular vibration is a
periodic motion
Periodicity or periodic may refer to:
Mathematics
* Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups
* Periodic function, a function whose output contains values tha ...
of the
atoms
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, an ...
of a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
relative to each other, such that the center of mass of the molecule remains unchanged. The
typical
Typical may refer to:
* ''Typical'' (album), Peter Hammill
* "Typical" (song), song by MuteMath
*"Typical", song by Frazier Chorus from ''Sue'', 1987
*''Typical'', story collection by Padgett Powell, 1991
See also
*''Typical Rick
''Typical R ...
vibrational frequencies range from less than 10
13 Hz to approximately 10
14 Hz, corresponding to
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
s of approximately 300 to 3000 cm
−1 and
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s of approximately 30 to 3 µm.
For a diatomic molecule A−B, the vibrational frequency in s
−1 is given by
, where k is the
force constant
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
in dyne/cm or erg/cm
2 and μ is the
reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
given by
. The vibrational wavenumber in cm
−1 is
where c is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in cm/s.
Vibrations of polyatomic molecules are described in terms of
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s, 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 with ''N'' atoms has 3''N'' – 6
normal modes of vibration, but a ''linear'' molecule has 3''N'' – 5 modes, because rotation about the molecular axis cannot be observed. A
diatomic molecule
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
has 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 Δ''E'' = ''hν'', where ''h'' is
Planck's constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
. When multiple quanta are absorbed, the first and possibly higher
overtone
An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s are excited.
To a first approximation, the motion in a normal vibration can be described as a kind of
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
. 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
The Morse potential, named after physicist Philip M. Morse, is a convenient
interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the qua ...
or more accurately, a
Morse/Long-range potential
The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Due to the simplicity of the regular Morse potential (it only has three adjustable parameters), it is very limit ...
.
The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through
infrared spectroscopy
Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or function ...
, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum.
Raman spectroscopy
Raman spectroscopy () (named after Indian physicist C. V. Raman) is a spectroscopic technique typically used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be observed. Raman sp ...
, 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 The rule of mutual exclusion in molecular spectroscopy relates the observation of molecular vibrations to molecular symmetry. It states that no normal modes can be both Infrared and Raman active in a molecule that possesses a centre of symmetry. T ...
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
Vibronic spectroscopy is a branch of molecular spectroscopy concerned with vibronic transitions: the simultaneous changes in electronic and vibrational energy levels of a molecule due to the absorption or emission of a photon of the appropriate ...
, giving vibrational fine structure to
electronic transitions, particularly for molecules in the
gas state
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or com ...
.
Simultaneous excitation of a vibration and rotations gives rise to
vibration-rotation spectra.
Number of vibrational modes
For a molecule with atoms, the positions of all nuclei depend on a total of 3
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
, so that the molecule has 3
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
including
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3
cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a
linear molecule
In chemistry, the linear molecular geometry describes the geometry around a central atom bonded to two other atoms (or ''ligands'') placed at a bond angle of 180°. Linear organic molecules, such as acetylene (), are often described by invoking ...
, 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.
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.
The number of vibrational modes is therefore 3 minus the number of translational and rotational degrees of freedom, or 3–5 for linear and 3–6 for nonlinear molecules.
[
]
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 , the frequency of the vibration.
Internal coordinates
''Internal coordinates'' are of the following types, illustrated with reference to the planar molecule ethylene
Ethylene (IUPAC name: ethene) is a hydrocarbon which has the formula or . It is a colourless, flammable gas with a faint "sweet and musky" odour when pure. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds).
Ethylene i ...
,
* 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 ethylene 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 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
Within the CH2 group, commonly found in organic compound
In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen or carbon-carbon bonds. Due to carbon's ability to catenate (form chains with other carbon atoms), millions of organic compounds are known. The ...
s, the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1. symmetric and asymmetric stretching, 2. scissoring, and rocking, 3. wagging and twisting. These are shown here:
(These figures do not represent the "recoil
Recoil (often called knockback, kickback or simply kick) is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, as according to Newton's third law the force requ ...
" 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
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
to a set of internal coordinates. The projection operator is constructed with the aid of the character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of character ...
of the molecular point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
. For example, the four (un–normalized) 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.
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 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" an irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
under its point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
. 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'' = ''q''1 + ''q''2
* asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. ''Q'' = ''q''1 − ''q''2
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
Hydrogen cyanide, sometimes called prussic acid, is a chemical compound with the formula HCN and structure . It is a colorless, extremely poisonous, and flammable liquid that boils slightly above room temperature, at . HCN is produced on an ...
, HCN, The two stretching vibrations are
* principally C–H stretching with a little C–N stretching; ''Q''1 = ''q''1 + ''a'' ''q''2 (''a'' << 1)
* principally C–N stretching with a little C–H stretching; ''Q''2 = ''b'' ''q''1 + ''q''2 (''b'' << 1)
The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Norm ...
.
Newtonian mechanics
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
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
: 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.
By Newton's second law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
this force is also equal to a reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
, ''μ'', times acceleration.
Since this is one and the same force the ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
follows.
The solution to this equation of simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
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 GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Norm ...
). The vibration frequencies, ''ν''i, are obtained from the eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, ''λ''i, of the matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
''GF''. ''G'' is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. ''F'' is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.
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''.
The difference in energy when ''n'' (or ''v'') changes by 1 is therefore equal to , the product of the Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
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 for graphs of the first 5 wave functions, which allow certain selection rules
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in ...
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 an anharmonic oscillator, Dunham expansion In quantum chemistry, the Dunham expansion is an expression for the rotational-vibrational energy levels of a diatomic molecule:
:
E(v,J,\Omega) = \sum_ Y_ (v+1/2)^k (J+1) - \Omega^2l,
where v and J are the vibrational and rotational quantum ...
is used.
Intensities
In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability
Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.
See also
*Coherent anti-Stokes Raman spectroscopy Coherent anti-Stokes Raman spectroscopy, also called Coherent anti-Stokes Raman scattering spectroscopy (CARS), is a form of spectroscopy used primarily in chemistry, physics and related fields. It is sensitive to the same vibrational signatures of ...
(CARS)
*Eckart conditions The Eckart conditions, named after Carl Eckart, simplify the nuclear motion (rovibrational) Hamiltonian that arises in the second step of the Born–Oppenheimer approximation. They make it possible to approximately separate rotation from vibra ...
* Fermi resonance
*GF method The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Norm ...
*Infrared spectroscopy of metal carbonyls
Metal carbonyls are coordination complexes of transition metals with carbon monoxide ligands. Metal carbonyls are useful in organic synthesis and as catalysts or catalyst precursors in homogeneous catalysis, such as hydroformylation and Reppe ch ...
*Lennard–Jones potential
The Lennard-Jones potential (also termed the LJ potential or 12-6 potential) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studie ...
*Near infrared spectroscopy
Near-infrared spectroscopy (NIRS) is a spectroscopic method that uses the near-infrared region of the electromagnetic spectrum (from 780 nm to 2500 nm). Typical applications include medical and physiological diagnostics and research incl ...
*Nuclear resonance vibrational spectroscopy
Nuclear resonance vibrational spectroscopy is a synchrotron-based technique that probes vibrational energy levels. The technique, often called NRVS, is specific for samples that contain nuclei that respond to Mössbauer spectroscopy, most commonl ...
*Resonance Raman spectroscopy Resonance Raman spectroscopy (RR spectroscopy) is a Raman spectroscopy technique in which the incident photon energy is close in energy to an electronic transition of a compound or material under examination. The frequency coincidence (or ''resonan ...
*Transition dipole moment
The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the tra ...
References
Further reading
*
External links
Free Molecular Vibration code developed by Zs. Szabó and R. Scipioni
* ttp://symmetry.jacobs-university.de/ Character tables for chemically important point groups
{{Branches of Spectroscopy
Chemical physics
Spectroscopy