Molecular vibration
A molecular vibration is a
For a diatomic molecule A−B, the vibrational frequency in s−1 is given by , where k is the
Vibrations of polyatomic molecules are described in terms of
A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = 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. When multiple quanta are absorbed, the first and possibly higher overtones are excited.
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
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 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 compounds, 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:
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
Symmetry–adapted coordinates may be created by applying a
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 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 under its point 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 = 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
- 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.[7]
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, k. The anharmonic oscillator is considered elsewhere.[8]
When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see
Quantum mechanics
In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the
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
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
Intensities
In an infrared spectrum the
See also
- Coherent anti-Stokes Raman spectroscopy (CARS)
- Eckart conditions
- Fermi resonance
- GF method
- Infrared spectroscopy of metal carbonyls
- Lennard-Jones potential
- Near-infrared spectroscopy
- Nuclear resonance vibrational spectroscopy
- Resonance Raman spectroscopy
- Transition dipole moment
References
- ISBN 0-08-021022-8.
- ^ ISBN 0471965227.
- ^ ISBN 0-07-707976-0.
- ^ ISBN 0716787598.
- ISBN 0471175706.
- ISBN 0471163945.
- ^ ISBN 048663941X.
- ISBN 0471129968.
- ISBN 0412102900.
- ISBN 0471965227.
- ISBN 0716787598.
- ISBN 0721685803.
Further reading
- Sherwood, P. M. A. (1972). Vibrational Spectroscopy of Solids. ISBN 0521084822.