NationMaster - Encyclopedia: 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. In general, a molecule with N atoms has 3N-6 normal modes of vibration but linear molecules have only 3N-5 normal modes of vibration as rotation about its molecular axis cannot be observed. 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. Properties For alternative meanings see atom (disambiguation). ... In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... Various normal modes in a 1D-lattice. ... Diatomic molecules are molecules formed of two atoms of the same element. ...

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, which corresponds to a vibration frequency, ν, according to the well-known relation E=hν, where h is Planck's constant. A fundamental vibration is excited when one 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. In physics, the ground state of a quantum mechanical system is its lowest-energy state. ...

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 dispacements and the first overtone would have 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 as the potential energy of the molecule is more like a Morse potential. Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... Anharmonicity is the deviation of a system from being a harmonic oscillator. ... The Morse potential (blue) and harmonic oscillator potential (green). ...

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is infrared spectroscopy because vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. However, Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. Infrared spectroscopy (IR Spectroscopy) is the subset of spectroscopy that deals with the infrared region of the electromagnetic spectrum. ... Raman spectroscopy is a spectroscopic technique used in condensed matter physics and chemistry to study vibrational, rotational, and other low-frequency modes in a system. ...

Vibrational excitation can occur with electronic excitation (vibronic transition) to give vibrational fine structure to electronic transitions, particularly with molecules in the gas state. A vibronic transition denotes the simultaneous change of vibrational and electronic quantum number in a molecule. ...

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra. Rovibrational coupling stands for coupled rotational and vibrational excitation of a molecule. ...

1 Vibrational coordinates
2 Newtonian mechanics
3 Quantum mechanics
- 3.1 Intensities
4 References
5 See also
6 External links

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 ethene, The Eckart conditions,[1] named after Carl Eckart, sometimes referred to as Sayvetz conditions,[2] simplify the nuclear motion (rovibrational) SchrÃ¶dinger equation that arises in the second step of the Born-Oppenheimer approximation. ... Ethylene or ethene is the simplest alkene hydrocarbon, consisting of two carbon atoms and four hydrogens. ...

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: Not present in ethene, but an 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 angles and bond lengths within the groups involved do not change. Rocking may be distinguished from wagging by the fact that the atoms in the group stay in the same plane. Image File history File links No higher resolution available. ...

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.

See infrared spectroscopy for some animated illustrations of internal coordinates. Infrared spectroscopy (IR Spectroscopy) is the subset of spectroscopy that deals with the infrared region of the electromagnetic spectrum. ...

Symmetry-adapted coordinates

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates.^[1] 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 Template:Unite See also projection (linear algebra). ... In mathematics, the character of a group representation ρ : G → GLn is the function χ : G -> C which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ...

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 are the internal coordinates for stretching of each of the four C-H bonds.

Illustrations of symmetry-adapred coordinates for most small molecules can be found in Nakamoto.^[2]

Normal coordinates

A normal coordinate, Q, may sometimes be constructed directly as a symmetry-adapted coordinate. This is possible when the normal coordinate belongs uniquely to a particular irreducible representation of the molecular point group. For example, the symmetry-adapted coordinates for bond-stretching of the linear carbon dioxide molecule, O=C=O are both normal coordinates: Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... Carbon dioxide is a chemical compound composed of one carbon and two oxygen atoms. ...

symmetric stretching: the sum of the two C-O stretching coordinates; the two C-O bond lengths change by the same amount and the cabon 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 Hydrogen cyanide is a chemical compound with chemical formula HCN. A solution of hydrogen cyanide in water is called hydrocyanic acid or Prussic acid. ...

principally C-H stretching with a little C-N stretching; Q₁ = q₁ + a q₂ (a << 1)
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.^[3] Wilsons GF method, sometimes referred to as FG method, is a classical mechanical method to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Qk. ...

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: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, f. The anharmonic oscillator is considered elesewhere.^[4] Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

$Force=- f Q !$

By Newton’s second law of motion this force is also equal to a "mass", m, times accelaration. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

$Force = m frac{d^2Q}{dt^2}$

Since this is one and the same force the ordinary differential equation follows. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

$m frac{d^2Q}{dt^2} + f Q = 0$

The solution to this equation of simple harmonic motion is Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ...

$Q(t) = A cos (2 pi nu t) !$ ; $nu = {1over {2 pi}} sqrt{f over m} !$

A is the maximum amplitude of the vibration coordinate Q. It remains to define the "mass", m. In a homonuclear diatomic molecule such as N₂ it is simply the mass of the two atoms. In a heteronuclear diatomic molecule, AB, it is the reduced mass, μ given by Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ...

$frac{1}{mu} = frac{1}{m_A}+frac{1}{m_B}$

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.

$f=frac{partial ^2V}{partial Q^2}$

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,ν_i are obtained from the eigenvalues,λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.^[3] F is a matrix derived from force-constant values. Details concering the determination of the eigenvalues can be found in ^[5]. Wilsons GF method, sometimes referred to as FG method, is a classical mechanical method to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Qk. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... This article gives an overview of the various ways to multiply matrices. ...

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 In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

$E_n = left( n + {1 over 2 } right) {1over {2 pi}} sqrt{f over m} !$ ,

where n is a quantum number that can take values of 0, 1, 2 ... The difference in energy when n changes by 1 are therefore equal to the energy derived using classical mechanics. See quantum harmonic oscillator for graphs of the first 5 wave functions. Knowing the wave functions, certain selection rules can be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In physics, especially in the context of quantum mechanics, a selection rule is a condition constraining the physical properties of the initial system and the final system that is necessary for a process to occur with a nonzero probability. ...

$Delta n = pm 1$

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. // In molecular vibrations a transition between two states of a single normal mode of vibration, neither of which is the ground state is known as a hot transition. ...

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the devative of the molecular dipole moment with respect to the normal coordinate.^[6] The intensity of Raman bands depends on polarizability. See also transition dipole moment. In physics, intensity is a measure of the time-averaged energy flux. ... This article is about the electromagnetic phenomenon. ... Polarizability is the relative tendency of the electron cloud of an atom to be distorted from its normal shape by the presence of a nearby ion or dipole--that is, by an external electric field. ... The Transition dipole moment or just Transition moment, is a term usually denoted . ...

References

^ F.A. Cotton Chemical applications of group theory, Wiley, 1962, 1971
^ K. Nakamoto Infrared and Raman spectra of inorganic and coordination compounds, 5th. edition, Part A, Wiley, 1997
^ ^a ^b E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations, McGraw-Hill, 1955. (Reprinted by Dover 1980)
^ S. Califano, Vibrational states, Wiley, 1976
^ P. Gans, Vibrating molecules, Chapman and Hall, 1971
^ D. Steele, Theory of vibrational spectroscopy, W.B. Saunders, 1971

External links

Molecular vibration and absorption
small explanation of vibrational spectra and a table including force constants.
Character tables for chemically important point groups

Categories: Physical chemistry stubs | Chemical physics | Spectroscopy

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