|
|
| Line 1: |
Line 1: |
| A '''molecular vibration''' occurs when [[atoms]] in a [[molecule]] are in [[Periodic function|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 [[Resonance Raman spectroscopy#Molecular Vibrations and Infrared Radiation|typical frequencies of molecular vibrations]] range from less than 10<sup>12</sup> to approximately 10<sup>14</sup> Hz.
| | The writer is known as Irwin. South Dakota is where me and my spouse live. One of the extremely very best issues in the globe for him is to gather badges but he is struggling to find time for it. Bookkeeping is what I do.<br><br>My web site - at home std test ([http://nuvem.tk/altergalactica/MarissaepDonnt go here]) |
| | |
| In general, a molecule with ''N'' atoms has 3''N'' – 6 [[vibrational mode|normal modes of vibration]], but a ''linear'' molecule has 3''N'' – 5 such modes, as rotation about its molecular axis cannot be observed.<ref>{{cite book |last=Landau |first=L. D. |last2=Lifshitz |first2=E. M. |year=1976 |title=Mechanics |edition=3rd |publisher=Pergamon Press |isbn=0-08-021022-8 }}</ref> 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'' = ''hν'' (where ''h'' is [[Planck constant|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 [[anharmonicity|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]].
| |
| | |
| 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 [[Centrosymmetry|centrosymmetric molecules]].
| |
| | |
| Vibrational excitation can occur in conjunction with electronic excitation ([[vibronic transition]]), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state.
| |
| | |
| Simultaneous excitation of a vibration and rotations gives rise to [[Rovibrational coupling|vibration-rotation]] spectra.
| |
| | |
| ==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===
| |
| [[Eckart conditions#Separation of external and internal_coordinates|''Internal coordinates'']] are of the following types, illustrated with reference to the planar molecule [[ethylene]], [[Image:Ethylene.svg|120px|Ethylene|center]]
| |
| * 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 BF<sub>3</sub> 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 CH<sub>2</sub> rocking, 2 CH<sub>2</sub> 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.
| |
| | |
| ===={{anchor|scissoring}}Vibrations of a Methylene group (-CH<sub>2</sub>-) in a molecule for illustration====
| |
| The atoms in a CH<sub>2</sub> group, commonly found in [[organic compound]]s, can vibrate in six different ways: '''symmetric and asymmetric stretching''', '''scissoring''', '''rocking''', '''wagging''' and '''twisting''' as shown here:
| |
| {| class="wikitable"
| |
| |-
| |
| ! Symmetrical<br>stretching
| |
| ! Asymmetrical<br>stretching
| |
| ! Scissoring (Bending)
| |
| |-
| |
| | [[Image:Symmetrical stretching.gif]]
| |
| | [[Image:Asymmetrical stretching.gif]]
| |
| | [[Image:Scissoring.gif]]
| |
| |-
| |
| ! Rocking
| |
| ! Wagging
| |
| ! Twisting
| |
| |-
| |
| | [[Image:Modo rotacao.gif]]
| |
| | [[Image:Wagging.gif]]
| |
| | [[Image:Twisting.gif]]
| |
| |}
| |
| (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.<ref name="Cotton">{{cite book |first=F. A. |last=Cotton |title=Chemical Applications of Group Theory |publisher=Wiley |edition=2nd |year=1971 |location=New York |isbn=0471175706 }}</ref> 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
| |
| :<math>Q_{s1} = q_{1} + q_{2} + q_{3} + q_{4}\!</math>
| |
| :<math>Q_{s2} = q_{1} + q_{2} - q_{3} - q_{4}\!</math>
| |
| :<math>Q_{s3} = q_{1} - q_{2} + q_{3} - q_{4}\!</math>
| |
| :<math>Q_{s4} = q_{1} - q_{2} - q_{3} + q_{4}\!</math>
| |
| | |
| where <math>q_{1} - q_{4}</math> 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.<ref name="Nakamoto">{{cite book |first=K. |last=Nakamoto |title=Infrared and Raman spectra of inorganic and coordination compounds, Part A |edition=5th |publisher=Wiley |location=New York |year=1997 |isbn=0471163945 }}</ref>
| |
| | |
| ===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 CO<sub>2</sub>, 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<sub>1</sub> + q<sub>2</sub>''
| |
| * asymmetric stretching: the difference of the two C-O stretching coordinates; one C-O bond length increases while the other decreases. ''Q = q<sub>1</sub> - q<sub>2</sub>''
| |
| | |
| 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; Q<sub>1</sub> = q<sub>1</sub> + a q<sub>2</sub> (a << 1)
| |
| # principally C-N stretching with a little C-H stretching; Q<sub>2</sub> = b q<sub>1</sub> + q<sub>2</sub> (b << 1)
| |
| The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson [[GF method]].<ref name="WilsonDeciusCross">{{cite book |first=E. B. |last=Wilson |first2=J. C. |last2=Decius |first3=P. C. |last3=Cross |title=Molecular Vibrations |location=New York |origyear=1955 |publisher=Dover |year=1995 |isbn=048663941X }}</ref>
| |
| | |
| ==Newtonian mechanics==
| |
| [[File:Anharmonic oscillator.gif|350px|thumb|The {{color|#FF0080|H}}{{color|green|Cl}} molecule as an anharmonic oscillator vibrating at energy level E<sub>3</sub>. D<sub>0</sub> is [[Bond-dissociation energy|dissociation energy]] here, r<sub>0</sub> [[bond length]], U [[potential energy]]. Energy is expressed in [[wavenumber]]s. 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.<ref name="Califano">{{cite book |first=S. |last=Califano |title=Vibrational States |publisher=Wiley |location=New York |year=1976 |isbn=0471129968 }}</ref>
| |
| :<math>\mathrm{Force}=- k Q \!</math>
| |
| By [[Newton's laws of motion|Newton’s second law of motion]] this force is also equal to a [[reduced mass]], ''μ'', times acceleration.
| |
| :<math> \mathrm{Force} = \mu \frac{d^2Q}{dt^2}</math>
| |
| Since this is one and the same force the [[ordinary differential equation]] follows.
| |
| :<math>\mu \frac{d^2Q}{dt^2} + k Q = 0</math>
| |
| The solution to this equation of [[simple harmonic motion]] is
| |
| :<math>Q(t) = A \cos (2 \pi \nu t) ;\ \ \nu = {1\over {2 \pi}} \sqrt{k \over \mu}. \!</math>
| |
| ''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, ''m<sub>A</sub>'' and ''m<sub>B</sub>'', as
| |
| :<math>\frac{1}{\mu} = \frac{1}{m_A}+\frac{1}{m_B}.</math>
| |
| 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.
| |
| :<math>k=\frac{\partial ^2V}{\partial Q^2}</math>
| |
| | |
| 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''<sub>i</sub> are obtained from the [[eigenvalues]],''λ''<sub>i</sub>, of the [[matrix product]] '''''GF'''''. '''''G''''' is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.<ref name="WilsonDeciusCross"/> '''''F''''' is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.<ref name="Gans">{{cite book |first=P. |last=Gans |title=Vibrating Molecules |publisher=Chapman and Hall |location=New York |year=1971 |isbn=0412102900 }}</ref>
| |
| | |
| ==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
| |
| :<math>E_n = h \left( n + {1 \over 2 } \right)\nu=h\left( n + {1 \over 2 } \right) {1\over {2 \pi}} \sqrt{k \over m} \!</math>,
| |
| 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''.<ref>{{cite book |first=J. M. |last=Hollas |title=Modern Spectroscopy |edition=3rd |publisher=John Wiley |year=1996 |page=21 |isbn=0471965227 }}</ref><ref>{{cite book |first=P. W. |last=Atkins |first2=J. de |last2=Paula |title=Physical Chemistry |edition=8th |publisher=W. H. Freeman |location=New York |year=2006 |pages=291 and 453 |isbn=0716787598 }}</ref>
| |
| | |
| The difference in energy when ''n'' (or ''v'') changes by 1 is therefore equal to <math>h\nu</math>, 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 <math>\nu</math> (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,
| |
| :<math>\Delta n = \pm 1</math>
| |
| 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 transition|hot band]].
| |
| | |
| ===Intensities===
| |
| In an infrared spectrum the [[intensity (physics)|intensity]] of an absorption band is proportional to the derivative of the [[molecular dipole moment]] with respect to the normal coordinate.<ref name="Steele">{{cite book |first=D. |last=Steele |title=Theory of vibrational spectroscopy |publisher=W. B. Saunders |location=Philadelphia |year=1971 |isbn=0721685803 }}</ref> The intensity of Raman bands depends on [[polarizability]].
| |
| | |
| ==See also==
| |
| *[[Near infrared spectroscopy]]
| |
| *[[Resonance Raman spectroscopy]]
| |
| *[[Coherent anti-Stokes Raman spectroscopy]]
| |
| *[[Eckart conditions]]
| |
| *[[FG method]]
| |
| *[[Fermi resonance]]
| |
| *[[Lennard–Jones potential]]
| |
| *[[Transition dipole moment]]
| |
| *[[infrared spectroscopy of metal carbonyls]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==Further reading==
| |
| *{{cite book |first=P. M. A. |last=Sherwood |title=Vibrational Spectroscopy of Solids |publisher=Cambridge University Press |year=1972 |isbn=0521084822 }}
| |
| | |
| ==External links==
| |
| * [http://www.evtsz.bme.hu/web/staff/szabo/web_molecular_vibration/molec_vib_code.html Free Molecular Vibration code developed by Zs. Szabó and R. Scipioni]
| |
| * [http://www.lsbu.ac.uk/water/vibrat.html Molecular vibration and absorption]
| |
| * [http://hyperphysics.phy-astr.gsu.edu/Hbase/molecule/vibspe.html small explanation of vibrational spectra and a table including force constants].
| |
| * [http://symmetry.jacobs-university.de/ Character tables for chemically important point groups]
| |
| | |
| [[Category:Chemical physics]]
| |
| [[Category:Spectroscopy]]
| |
| [[Category:Molecular vibration]]
| |