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2. Symmetric tops (
MOLECULAR SPECTROSCOPY
)
Born Oppenheimer Approximation: A typical diatomic molecule has rotational energy separations of Vibrational energy separation of nearly = 3000 Since the energies of the motions are so different we may as a first approximation, consider a diatomic molecule can execute rotations and vibrations quite independently. This approximation is called Born Oppenheimer Approximation This approximation is strictly includes electronic spectra
1. Rotation about the bond axis (C-F axis) ( is not negligible) 2. end over end rotation in the plane of paper 3. end over end rotation out of the plane of paper
Type of symmetric tops 1) Prolate symmetric top Ex : methyl fluoride: 2) oblate symmetric top :
Microwave Spectroscopy The rotation of a three dimensional body may be quite complex and it is convenient to resolve it into rotational component about the mutually perpendicular directions through the centre of gravity – the principal axes of rotation. Thus a body has three principal moments of inertia one about each axis are
DOOZY STUDY Classification of molecules principal moments of inertia 1.Linear Molecules (
Ex: Boron trichloride (
)
= <
3)Spherical tops
about
)
Example: 1. Hydrogen Chloride 2.O-C-S Carbon Oxy sulphide Three rotations are: 1. About the bond axis 2. End-over-end rot in the plane of the paper 3. End-over-end rot at right angles to the plane
It has no dipole moment Example: Methane 4)Asymmetric tops Ex; water( H2O ); Vinyl chloride (
)
Moments for (2) and (3) are same and is very small thus take
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Majority of the molecules have this character
Where
is called reduced mass
ROTATIONAL SPECTRA Rotational energy is quantized (i.e., a molecule cannot have any arbitrary amount of rotational energy ) Its energy is limited to certain definite values depending on the shape and size of the molecule concerned The permitted energy values are called rotational energy levels
If the molecule rotates with angular velocity then classically it would have energy According to Bohr quantization condition
from (1) and (2)
Diatomic molecule as a rigid rotator i.e., rotational energy where number
termed as rotational quantum
By the use of Schrodinger equation allowed rotational energy levels for diatomic molecules are
DOOZY STUDY Masses and (the bond) Whose length is
are joined by a rigid bar
Where inertia
( either
and is the moment of since both are equal.)
The molecule rotates end –over-end about C
In spectroscopy, Energy is expressed in terms of wave number.
Balancing equ :
Wavenumber
=
The moment of inertia about C is
where C is velocity of light
in
Where
=
Or B is called
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Allowed rotational energy levels
Frequency of spectral line
About rotational spectroscopy: Only hetero nuclear molecules rotational spectrum. Ex :
gives
= Homonuclear molecules rotational spectrum Ex:
=
do
not
give
DOOZY STUDY If the molecule in the state where
state is raise to =2B
= 4B
In general to raise the molecule from
The intensities of spectral lines
Intensities of spectral lines is directly proportional to the initial number of molecules in each level. i.e., population of the rotational levels
For maximum population J= Intensity will be maximum at or near the value given by above equation.
The effect of Isotope substitution: Thus a stepwise raising of the rotational energy results in an absorption spectrum consisting of lines at Thus on the frequency scale (wave number), the lines are equidistant Selection rule :
i.e.,
changes by one unit only
When a particular atom in a molecule is replaced by its isotope – the resulting substance is identical chemically with the original. There is no appreciable change in internuclear distance on isotopic substitution. However a change in total mass and hence the moment of inertia and B value changed. Thus rotational energy levels of the molecule changes.
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Example: For carbon monoxide (CO)
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In simple harmonic motion a molecular bond is compressed and extended an equal amount on each side of the equilibrium distance. But the average distance is unchanged. The average value of Where
Where and Thus separation between the lines For heavier atom For lighter atom
The difference b/w r and though small is not negligible compared with the precession of B can be measured spectroscopically.
Non Rigid Rotator
But real vibrations are not simple harmonic since a real bond may be stretched more easily than it may be compressed and thus
From experimental data the separation between successive lines are not equidistant Separation decreases slightly with increasing values Thus B decreases and internuclear distance ( ) increases Thus bond length increases with values Thus our assumption a rigid bond is only on approximation Ofcourse all bonds are elastic to some extent. Increase in length with shows if a diatomic molecule rotates fast resulting centrifugal force is very high. Thus high centrifugal force tending to move the atoms apart.
The Spectrum of a Non Rigid Rotator: Rotational energy levels are: in Joules
DOOZY STUDY Consequence of Elasticity
in
Where
The above equation applies for simple harmonic only. For Anharmonic motion
1. When the bond is elastic the molecule may have vibrational energy (i.e., the bond will stretch and compress periodically ) 2.
If the motion is simple harmonic the force constant is given by . The variation of with is determined by the force constant
Where are small constants dependent upon the geometry of the molecule. Also are negligible. Relation B/W B and D
3. During vibration r and B changes B= Thus since all other quantities are independent of vibration.
The value of B is of the order of The value of D is of the order of i.e., D is very small compared with B
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For small , is negligible , For J it may become appreciable.
Since ., as per diatomic molecules the energylevels are identical with diatomic molecules.
Figure 2.9 whole of the discussion on diatomic molecules applies for all linear molecule. But Following two differences: I.
The MOI of the end –over – end rotation of the Polyatomic linear molecule is greater than diatomic and the B value will be smaller and the spectral lines more closely spaced
II.
A non cyclic polyatomic molecule contains N atoms has bond length to be determined. However only one bond length can be determined from the spectrum.
DOOZY STUDY Selection Rule
2. Symmetric top molecule
Example: methyl fluoride
Thus spectral lines are not equidistant . But their separation decreases slightly with increasing (Fig.2.9) The knowledge information:
of
give
two
useful
1. To determine the value of lines in an observed spectrum 2. To determine the vibrational frequency of a diatomic molecule inaccurately.(
Two direction of rotations are
1. About main symmetry axis C-F axis 2. Perpendicular to this axis Here two quantum numbers to describe the degree of rotation, one for and one for )
For rigid symmetric top Allowed energy levels:
)
Poly atomic Molecules
Where
1.Linear molecules
it has (2J+1) values
Examples: 1. Carbon oxysulphide OCS 2. Chloro Acetalene
and
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Energy depends on so that it is immaterial whether the top spins clockwise or anticlockwise. For all the rotational energy levels are doubly degenerate
selection rule: and SPECTRUM
Thus the spectrum is independent of Hence rotational changes about the symmetry axis do not give rise to a rotational spectrum 3.Asymmetric Top molecule (
)
These molecules having three different , also have more complicated spectra. No simple expression can be derived for them. They are usually treated by approximation methods
ygrenE
1 2
This model of vibrating diatomic molecule the so called simple harmonic oscillator model The simple harmonic oscillator: If the compression and extension is greater the energy of the is increased from (fig 3.1). But vibrational frequency will not change.
DOOZY STUDY Additional Data
An elastic bond like a spring has a certain intrinsic vibrational frequency. It is depend on the mass of the system, and force constant, but is independent of the amount of distortion.
INFRA RED SPECTROSCOPY
Classically the oscillation frequency is
Vibrating diatomic molecule: In fig the atom is fixed at one end and imagine pushing and fulling the atom closer to or further from the –a bigger push or pull result in rising the energy more. Energy is minimum at the internuclear distance is called equilibrium distance or bond length We assume that , the bond like a spring ,obey Hooke’s law
In wave number scale
Vibrational energies are quantized and the allowed vibrational energy is for any system may be calculated from the Schrodinger equation 𝑣
123
f restoring force k force constant
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vibrational quantum number
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Frequecy
(or) (for 𝑣
)
Where or is called zero point energy. It depends only the classical vibration frequency and hence the strength of the chemical bond and the atomic masses. The above equation shows that the diatomic molecule can never have zero vibrational energy. the atom never be completely at rest relative to each other.
For a harmonic oscillator the frequency of the radiation of emitted or absorbed should be equal to the mechanical frequency of the vibration of the system The Anharmonic Oscillator Real molecules do not obey exactly the law of simple harmonic motion Real bonds although elastic, are not so homogeneous as to obey Hooke’s law
Selection Rule: 𝑣 Applying the selection rule we have
Fig 3.3 shows, the shape of the energy curve for a typical diatomic molecule ,together with ( dashed) the ideal ,simple harmonic parabola.
DOOZY STUDY for emission
for absorption
Since the vibrational levels are equally spaced, transition between any two neighboring states will give rise to the same energy change (fig 3.2)
To explain this curve P.M.Morse gave a purely empirical expression called Morse function as
Where a
constant for particular molecule
dissociation energy
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1. 𝑣
Allowed vibrational energy levels 𝑣
𝑣
𝑣
𝑣
2.
𝑣 3. 𝑣
Vibrational levels crowd more closely together with increasing 𝑣 .i.e., energy levels of an anharmonic oscillator are not equidistant. 𝑣
𝑣
Thus
Since very close to
lines lie
The line near absorption. The
is called fundamental
line
near 𝑣
to
For 𝑣
Thus
DOOZY STUDY ecnis
.
𝑣
Thus the anharmonic oscillator behaves like the harmonic oscillator but with an oscillation frequency which decreases steadily with increasing 𝑣 For ground state Zero point energy
The (ground state energy) zero point energy differs slightly from the harmonic oscillator.
Selection Rule: ,… But normally only the lines have observable intensity. The population of the 𝑣 state is 0.01 or 1 of the ground state population. Thus we may ignore all transitions originating at 𝑣 or more.
3
For the ideal harmonic oscillator the spectral absorption occurred exactly at the classical vibration frequency.
For an anharmonic oscillator, fundamental absorption frequency and equilibrium frequency are different. Although, we ignored transitions from 𝑣 to higher states, if the temperature is raised or if the vibration has particularly low frequency , the population of the 𝑣 is become appreciable. Thus: 𝑣
𝑣
𝑣
𝑣
Such weak absorption are called hot bands since a high temperature is one condition for their occurrence.
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The Diatomic Vibrating Rotator A typical diatomic molecule has rotational energy separations of 1-10 Vibrational energy separations of Since the energies of the two motions are so different As a first approximation (Born Oppenheimer) consider that a diatomic molecule can execute rotations and vibrations quite independently. i.e., The combine rotational – vibrational energy is simply the sum of the separate energies
We may also have but this corresponds to the purely rotational transitions. However, a diatomic molecule except under very special and rare circumstances, may not have in other words a vibrational change must be accompanied by simultaneous rotational change.
DOOZY STUDY In fig 3.6, energy levels and transitions, for the rotational quantum numbers in the 𝑣 state as and in the 𝑣 state as .
𝑣
We ignore the small centrifugal distortion constants etc, and hence
conventional notations in spectroscopy.
𝑣 The rotational levels are sketched in for the two lowest vibrational 𝑣 𝑣 .
levels
The rotational levels are filled in any molecular population , so the transitions shown will occur with varying intensities. This is shown in foot of
Selection rule:
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m has +ve values for for
and is negative
m cannot be zero since this would imply The frequency centre
or
is called band orgin or band
P,R BRANCH Eqn (3 ) represens the combined vibrations rotation spectrum. *It will consist of equally spaced lines (=2B) on each side of the band origin *Spacing b/w the lines immediately on each side of the centre is 4B Consider only the 𝑣 have
𝑣
transition we
*Since , the line at appear (Q branch is absent)
itself willnot
DOOZY STUDY *Lines to the low frequency side of correspondig to negative m are reffered to as P branch. *High frequency side +ve m,
*We shall concerned with values of in addition to considered here. For Line arising from: 1.
o
-
2.
(1)
*The inclusion of the centrifugal distortion constant D in the eqn (3 ) we get
)
(2) But Dis only
The two expressions may conveniently be combined into (3)
Where m replacing in 2nd equation
in
of B , D is negligible
The Vibrations of Polyatomic Molecules Fundamental Vibrations and Symmetry:
eqn and
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Consider a molecule having N atoms. It has degrees of freedom 1)Translational movement uses 3 of the degrees of freedom 2)The rotation of the non linear molecule about three perpendicular axes Thus for Non linear molecules , vibrational degrees of freedom i.e,
funtamental vibrations occur.
For the linear molecule, there is no rotation about the bond axis hence only two degrees of rotational freedom are required. Thus vibrational degrees of freedom for linear molecule = In both cases, N-atomic molecule has bonds b/w its atoms
DOOZY STUDY (N-1) of the vibrations are bond – stretching motions , (non linear) or 2N-4(linear) are bending motions. Examples 1)H2O
Non linear and triatomic vibrational modes. stretching mode bending mode
No of bond
It has
These three vibrational motions are also referrred to as the normal modes of vibration of the molecule.
*Normal vibration is defined as a molecular motion in which all the atoms oscillate with the same frequency and pass through their equilibrium position simutaneously.
*If we rotate the molecule about the C2-axis by its appearance is idential with the initial. *C2 refers twice in every complete rotation the molecule presents an identical to an observer. *H2O molecule has only one rotational symmetry axis (i.e C2 axis) First vibration (symmetric stretching)(𝑣1) If we rotate the vibrating molecule by the vibration is quite unchanged – Thus we call this vibration is called symmetric vibration. 2nd vibration (Bending)(𝑣2) Also symmetric
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3rd vibration (Asymmetric stretching)(𝑣3)
Symmetric stretch ( 1)
Rotation about the C2 axis, produces a vibration which is in anti phase with the original. Thus we call this vibration is called Asymmetric vibration.
Symmetric stretching produces no change in the dipole moment which remains zero - so that this vibration is infra-red inactive
In order to be infra-red active , there must be a dipole change during the vibration and this change may take place either along the line of symmetry axis (IIl to it) or at right angles to the line .
2. Bending vibrations ( 2) It consists Two vibrations One is in the plane of the paper another is to it.[i.e, Oxygen atom move into and out of the plane] The above two is identical in all except direction , and are called degenerate. This vibration is infra-red active 3.Assymmetric vibrations( 3) IR Active
DOOZY STUDY Overtone and Combination Frequency If a fundamental bands are v1 and v2 Over tones are: 2v1,2v2 First overtone:2v1,2v2 2nd overtone:3v1,3v2 All vibrational models of water (H2O) are IR active 1.
linear triatomic molecule
Combination bands :v1+v2,v1+2v2,2v1+v2 etc Difference bands :v1-v2, v1-2v2, 2v1-v2 etc The intensities of overtones or combination and difference bands are very small .But are often to be found in complex spectra. Since the intensities of above band are considerable increased by a resonance phenomena. Resonance may happen that two vibrational modes in a particular molecule have frequincies very close to each other –they are described as accidently degenerate.
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Accidental degeneracy is found most often b/w a fundamental and overtone or combination. Example for : CO2 :
𝑣
Quantum mechanics shows that two such bands may interfere with each other in such a way that the higher is raised in frequency, the lower depressed. One of these bands arises from a fundamental mode ( , the other from the overtone thus we expect the fundamental is more intense than overtone.
causing a dipole change either (1) parallel (or) (2) perpendicular to the major axis of rotational symmetry. The selection rules for rotational transitions of complex molecules depends on the type of vibration and shape of the molecule also. 1. Linear molecules Parallel vibrations: Selection rule: 𝑣
But really , they have almost same intensity 𝑣 Since overtone has gained intensity expense of the fundamental.
at the
Fermi Resonance
The spectra will thus be similar appearance, consisting of P and R branches with lines about equally spaced on each side, no occurring at the band centre. However the MOI may be larger, the B value correspondingly smaller, and the P or R line spacing will be less.
DOOZY STUDY Two close molecular vibrational frequencies resonate and exchange energy – the phenomenon known as Fermi resonance when fundamental resonates with an overtone In the spectrum of a complex molecule exhibiting many fundamentals and overtones , there is a good chance of accidental degeneracy and fermi resonance occuring However not all such degeneracies lead to resonance. Since resonance also depends the molecular symmetry and type of degenerate vibrations. THE INFLUENCE OF RADIATION ON THE SPECTRA OF POLY ATOMIC MOLECULES: The selection rule for vibrating rotation:
Perpendicular vibrations: Selection rule: 𝑣
Which implies that , a vibrational change can take place with no simultaneous rotational transition If the oscillation is taken as SHM the energy levels are identical with equation (3.18) and the P and R branch lines are given by Equation (3.20) or 3.21 Transition with , correspond to a Q branch whose lines may be derived from the equations
𝑣 We showed that the vibrations of complex molecules could be subdivided into those
.
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Thus Q branch consists of lines super imposed upon each other at the band centre The resultant line is usually very intense If B values differ slightly in the upper and lower vibrational levels.
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In the case of acetelene,alternate levels have populations which differ by a factor of so that the P and R branch lines show a strong, weak, st ong, weak,… alte nation in intensity Symmetric Top Molecules:
If branch line would become split into a series of lines on the low frequency side of .
𝑣
𝑣 2
It should remembered that polyatomic molecules with zero dipole moment do not give rise to pure rotation spectra (For Ex: CO2, HC=CH, CH4) such molecules do however show vibrational spectra (or Raman).
In this case centrifugal distortion is negligible
The influence of Nuclear Spin
=0 means K is same in the upper and lower state. Thus spectral frequencies is independent of K. The spectrum will certain P,Q and R branches with a P,R line spacing of 2B and a strong central Q branch. This situation is identical with perpendicular vibrations of a linear molecule.
Parallel vibrations: Selection rule: 𝑣
DOOZY STUDY Centre of symmetry
Identical atoms are symmetrically displaced with respect to the centre of gravity of the molecule. CO2 [O=C=O] Ethene [H-C=C-H] posses centre of symmetry HCN, N2O not symmetric
The centre of symmetry has an effect on the intensity of alternate lines in the P and R branches. The effect is due to the existence of nuclear spin and it is additional factor determining the population of rotational levels. In case of CO2 every alternate rotational level is completely unoccupied and so alternate lines in the P and R branches have zero intensity .Thus a line spacing is 4B (instead of 2B) The isotopic molecule is no longer a centre of symmetry nuclear spin does not affect the spectrum and the line spacing is equal to 2B
Perpendicular vibrations 𝑣 1.
2.
3.
This type of vibration gives many sets of P and R branch lines since each J value there are many allowed values of
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𝑣 line is always greater line.
The Q branch is also complex. Since it will consist of a series of lines on both sides of separated by
For The intensity of than the
Raman Spectroscopy
Classical Theory of Raman Theory
If monochromatic radiation (or) radiation of very narrow frequency band, is used, the scattering energy will consist almost entirely of radiation of the incident frequency [the so called Rayleigh scattering] but in addition certain discrete frequencies above and below that of the incident beam will be scattered – This scattering is called Raman scattering. Quantum Theory of Raman Effect Let the incident light consist of photons of energy . 1. If a photon strikes an atom or molecule in a liquid, part of the energy of the incident photon may be used to excite the atom of the liquid and the rest in scattered. The spectral line will have lower frequency and it is called stokes line. 2. If a photon strikes an atom or molecule in a liquid which is in an excited state, the scattered photons gains energy. The spectral line will have higher frequency and it is called Anti- Stoke lines. 3. In some cases, when a photon strikes atoms or molecules, photons may be scattered elasticity. Then the photons neither gains nor lose energy. The spectral line will have unmodified frequency – Rayleigh line. Raman shift or Raman frequency .
The Classical Theory of Raman effect, while not wholly adequate, is worth consideration since it leads to an understanding of a concept – the polarizability of a Molecule. When a molecule is put into a static electric field – the positively charged nuclei attracted towards the negative pole of the field, the electrons to the positive pole. This separation of charge centers causes an induced electric dipole moment to be set up in the molecules and the molecule is said to be polarized. The size of the induced dipole , depends to the both on the magnitude of the applied field E, and on the case with which the molecule can be distorted.
DOOZY STUDY
Raman shift does not depends upon the frequency of the incident light. It depends on the characteristic of the substance producing Raman effect. For line 𝑣
i.e., Where -polarizibily Consider the diatomic molecule H2, is placed in an electric field in fig 4.1(a) and 4.2(b) in end – on and sideways orientation respectively. The electrons forming the bond are more easily displaced by the field along the bond axis { fig 4.1(a)} than that across the bond. Thus the polarizability is anisotropic Fig 4.1 The polarizability of a molecule in various directions is represented by drawing a polarizability ellipsoid.
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For hydrogen its general shape is a squashed sphere, a tangerine [Fig 4.1 (c) and (d) In general a polarizability ellipsoid is defined as a three dimensional surface whose distance from the electrical centre of the molecule is proportional to ,
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then the oscillating dipole will have superimposed upon it the vibrational or rotational oscillation. Consider, a vibration of frequency polarizability, we can write,
changes
Where polarizability along the line joining a point i on the ellipsoid with the electrical centre.
Thus where the polarizability is greatest, the axis of the ellipsoid is least and vice versa. If we imagine applying an electric field across the bond axis, of H2 [fig 4.1(a)], a certain amount of polarization will occur. If we also imagine the molecule rotating about its bond axis, [it is obvious that it will present exactly the same aspect to the electric field at all orientations] –i.e., its polarizability will be exactly the same in any direction across the axis. This means that a section through the polarizability ellipsoid will be circular. [Fig 4.1 (c)]
Expanding we get
Thus the oscillating dipole has frequency components and the exciting frequency .
DOOZY STUDY If the field is applied along the bond axis, [Fig 4.1 (b)] polarizability is greater. Thus the crosssection of the ellipsoid is less. [Fig 4.1 (d)] All diatomic molecules have ellipsoids of the same general tangerine shape as H2, as do linear molecules such CO2 etc. They differ only in the relative sizes of their major and minor axis When a sample of such molecules is subjected to a beam of radiation of frequency the electric field experienced by each molecule various according to the eqn.
If the vibration does not alter the polarizability, then and the dipole oscillates only at the incident frequency. The same is true for a rotation. In order to be Raman active a molecular rotation or vibration must cause some change in a component of the molecular polarizability. A change in polarizability is by change in either magnitude or orientation of the polarizability ellipsoid.
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Polarizability Molecules
ellipsoid
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of
Complicated
H2O (Bend Tri atomic Molecule)
Spherical Top Molecules: Ex: CH4, CCl4, SiH4 etc. It has spherical polarizability surfaces, since, they completely isotropic.
PURE ROTATIONAL RAMAN SPECTRA Linear Molecules The rotational molecules are
energy
levels
of
linear
But in Raman spectroscopy, the centrifugal distortion constant is neglected,
Selection rule: only
DOOZY STUDY Polarizability is different along all three axes of the molecule. Three of the ellipsoidal axes are also different (i.e., various orientation) H2S (or) SO2 Similar shaped ellipsoid but different dimensions. Symmetric molecules Because of their axial symmetry, polarizability ellipsoid of symmetric top molecules rather similar to those of linear molecules i.e., with a circular cross-section at right angles to their axis of symmetry. However, the sections in other planes are truly elliptical The polarizability of ellipsoid for chloroform is shown in Fig 4.3 (b) The polarizability of chloroform is greater across the symmetry axis, the minor axis of the ellipsoid in this direction
The fact that in Raman work the rotational quantum number changes by two units rather than one is connected with the symmetry of the polarizability ellipsoid. For a linear molecule, during end-over-end rotation the ellipsoid presents the same appearance to an observer twice in every complete rotation.
The rotation about the bond axis produce no change in polarizability and hence as in infrared and wave spectroscopy we need concern ourselves only with end-over-end rotations. In
usual
practice,
we define as . Thus we can ignore the selection rule since for a pure rotational change, the upper state quantum no necessarily be greater than lower state. Further is trivial, since this represents no change in the molecular energy and hence Rayleigh scattering only.
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Energy levels
Since: lines .
, these lines are called S branch
rotational quantum number in the lower state. If the molecule gains rotational energy from photons, we have a series of S branch lines to the low wave number side of the exciting line [Stokes]. If the molecule loses energy to the photon, S branch lines appear high wave number side (anti stokes) The wave numbers of the spectral lines are,
Homo nuclear molecules (O2 , H2, etc) give no IR or wave spectra whereas they give a rotational Raman spectrum If the molecule has a centre of symmetry (O2, H2, CO2) then the effect of nuclear spin will be observed in Raman as in infra-Red. For O2 and CO2 (since the spin of oxygen is zero) every alternate rotational level is absent. For O2 every level with even values is missed and thus every transition of are missing from the spectrum For H2 and other molecules with non zero spin, the spectral lines show an alternation of intensity.
DOOZY STUDY + sign - anti stokes ;- sign stokes line
wave number of the exciting radiation.
The allowed transitions and the Raman spectrum arising are shown in Fig 4.4
Symmetric Top Molecules EX: Thus the separation of the first line from the exciting line is 6B While the separation b/w the successive line is 4B
Polarizability ellipsoid of this molecule is shown in Fig 4.3(b) .Rotation about the top axis produces no change in polarizability. End-over-end rotations will produce a change
Selection rule:
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K- rotational quantum number for axial rotation. So the implies that, changes in the angular momentum about the top axis will not give Raman spectrum- Raman inactive rotation The restriction of forK=0 means that cannot be for transitions involving the ground state (J=0) since Thus for all J values other than zero, K also may be different from zero and transitions are allowed. 1. ( R branch )
(J
2.
}
Thus pure rotation of spherical top molecules are completely Raman inactive. Asymmetric Top Molecules Normally all rotations of this molecule is Raman active Raman spectra of this molecule is quite complicated VIBRATIONAL RAMAN SPECTRA If a molecule has little or no symmetry – very easy to decide whether its vibrational modes will be Raman active or inactive If a molecule has symmetry – not easy to decide Raman active or inactive Example 1.
( S branch )
Whose polarizability ellipsoid is shown in fig 4.6
DOOZY STUDY }
Thus we have two series lines in Raman spectrum
(J
}
}
In
the
R
branch lines appear at From exciting line. S b anch lines appea at 6B, 0B, 4B,… The complete spectrum shown in fig 4.5 ( c ) illustrates how every alternate R line is overlapped by an S line. Spherical Top Molecules Symmetric stretch The polarizability ellipsoid for such molecules in spherical surface. Rotation of this ellipsoid produce no change in polarizability.
)
During this motion the molecule as a whole increases and decreases in size
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When a band is stretched the electrons forming less firmly held by the nuclei and so the bond becomes more polarized. Thus polarizability ellipsoid is small and increase while they compress.
1. Symmetric stretch ) During this motion the molecule changes size and so there is corresponding fluctuation in the size of the ellipsoid Thus mode is Raman active
Bending motion
2.bending mode stretch )
)
Vibration is large at one extreme (on the left fig) while at other extreme it approximates to a diatomic molecule with a vertical axis ( right fig ) Symmetric stretch ) Both size and shape are constant but the direction changes Thus for – three modes of vibrations are Raman active 1.
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linear triatomic –centre of symmetric molecule
)
and
Asymmetric
The molecule changes shape during each vibration but these vibrations are Raman inactive Explanation To discuss the change of polarizability with some displacement coordinate For a stretching motion is a measure of extention (+ve ) and compression (-ve ) For a bending motion is displacement of the bond angle from its equilibrium value
DOOZY STUDY Consider symmetric stretching
Polarizability ellipsoid is shown in fig 4.7
if the equilibrium value of polarizability is when the bond stretch increases ( 𝑣 ) when the bond contract decreases 𝑣 we draw a graph between and (fig 4.7 a) at the curve has distinct slope i.e., . thus mode is Raman active Bending stretch
mode
)
and
Asymmetric
) thus
and
modes are
raman inactive i.e., for small displacements the polarizability is no change Generally , the polarizability curve has a large slope the Raman line will be strong. If the slope is small the Raman line is weak. If the slope is zero Raman line not allowed b y Khttp://doozystudy.blogspot.in/ . L a k s h m a n a n M . S c . , M . P h i l . , B . E d . , m o b i l e n o : 9 5 0 0http://doozystudy.blogspot.in/ 440393 Page 20
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General rule Symmetric vibrations give rise to intense Raman lines non symmetric ones are usually weak and sometimes unobservable Rule of Mutually exclusion
𝑣 𝑣
𝑣 𝑣
The intensity of Stoke’s lines is greater than antistoke’s For example – symmetric top
If a molecule has a centre of symmetry then Raman active vibrations are IR inactive and vice versa . if there is no centre of symmetry then some( but not necessarily all ) vibrations may be both Raman and IR active
This molecule contains 5 atoms Number of fundamental vibrations is 3N-6 =9 Three vibrations are degenerate . Thus different fundamental absorption is 6 All 6 vibrations are both Raman and IR active
Vibrational Raman spectra
Rotational fine structure
𝑣 (𝑣
𝑣
)
We need not consider in detail rotational fine structure of Raman spectra in general Because such fine structure is rarely resolved ( except for diatomic molecules ) For diatomic molecules the vibration – rotation energy is
DOOZY STUDY Selection Rule
𝑣
𝑣 Same for Raman and IR spectroscopy The probability of 𝑣 Decreasing rapidly 𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
In Raman , we ignore centrifugal distortion for diatomic molecule selection rule
Where we write Since the Raman scattered light is of low intensity we can ignore completely all the weaker effects 𝑣 Raman lines to appear at distances from the exciting line corresponding to each active fundamental vibrations
For Stoke’s lines lines to low frequency of the exciting radiation )
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The strong Q branch line is present in the Raman spectrum but in IR spectrum Q branch is absent .
Antistoke s line Much weaker anti lines will occur at the same distance from but high frequency side of exciting line For larger molecule we can ignore the rotational fine structure since it is not resolved. Even the O and S ( or ) bands are rarely observed since they are very week compared with Q branch
vibrational spectra require exchange of dipole ,but electronic spectra are given by all molecules , since changes in the electronic distribution are always accompanied by a dipole change. Thus homonuclear molecules ( gives electronic spectra and show vibration and rotation Structure in their spectra from which rotational constants and bond vibrational frequencies may be derived
Vibrational course structure : Progressions
Additional data For
DOOZY STUDY 𝑣
Electronic spectra Electronic spectra of diatomic molecule
Ignoring rotational changes we get
According to Born – Oppenheimer approximation Electronic, vibrational and rotational energies are independent of each other. i.e.,
=
𝑣
𝑣
Thus the change in the total energy is Energy levels of this equation are shown in fig 6.1
Thus vibrational changes will produce a coarse st uctu e and otational changes a “fine st uctu e” on the electronic spectra Pure rotation are shown only by molecules possessing a permanent dipole moment and
There is no selection rule for v when a molecule undergoes an electronic transition. However virtually all the molecules exist in the lowest vib ational state i.e., v’’ 0 and so 0,0 (1,0) (2,0) etc. transition only observed with appreciable intensity Such a set of transitions is called a band , since under low resolution each line of set appears
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somewhat b oad and diffuse and is called v’ p og ession, since the value of v’ inc eases by unity for each line in the set. The diagram shows that the lines in a band crowd together more closely at high frequencies due to anharmonicity There fore 𝑣
𝑣 +122
{𝑣 +12
𝑣 +122
And provided some half dozen lines can be observed in the band, values for and and can be calculated. → separations between electronic states From the band spectrum, the vibrational frequency and an harmonicity constant in the ground state ( and , but also the excited electronic ( are calculated.
transition, the vibrational lines in a progression are not all of the same intensity. In some spectra the (0,0) transition is the strongest, in others the intensity increases to a maximum at some value of v’, while in yet othe s only a few vib ational lines with high v’ followed by a continuum. All these types of spectrum are explained by Franck – Condon principle This principle states that an electronic transition takes place so rapidly that a vibrating molecule does not change its inter nuclear distance appreciably during the transition
DOOZY STUDY This latter information is valuable since such excited states are unstable and the molecule exist in them for very short time. Intensity of Vibrational – Electronic Spectra The Franck – Condon principle : According to classical theory the oscillating atom would spend most of its time on the curve at the turning point of its motion. According to quantum theory, for v=0 the atom is most likely to be found at the centre of its motion. i.e., at the equilibrium internuclear distance (req). For 𝑣 ,the most probable positions steadily approach the extremities until, for high v the quantal and classical pictures merge. Fig 6.2 shows the variation of with inter nuclear distance, where 𝑣 𝑣 Although quantum mechanics imposes no restrictions on the change in the vibrational quantum number during an electronic b y Khttp://doozystudy.blogspot.in/ . L a k s h m a n a n M . S c . , M . P h i l . , B . E d . , m o b i l e n o : 9 5 0 0http://doozystudy.blogspot.in/ 440393 Page 23
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Explanation : Fig 6.3shows four possibilities:
The resulting transitions and spectrum are similar in case(2)
Fig 6.3(a) case (i)
Case(4) fig 6.3(d)
The upper electronic states having the same equilibrium inter nuclear distance at the lower [Now the F-C principle suggests that a transition occurs vertically on this diagram]. Since the internuclear distance does not change, and so if we consider the molecule to be initially in the ground state both electronically ( and vib ationally v’’ 0 , then the most probable transition is that indicated by the vertical line.
The upper state is considerably greater than lower state The t ansition takes place has a high v’ value Further transitions can now occur to a state where excited molecule has energy in excess of its own dissociation energy. From such states the molecule will dissociates without any vibrations. Since the atoms are formed may take up any value of kinetic energy, the transitions are not quantized and a continuum results.
Thus the st ongest spect al line of the v’’ 0 progression will be the (0,0) However , the quantum theory only says that the probability of finding the oscillating atom is greatest at the equilibrium distance in the v=0 state- It allows some, although small chance of the atom being near extremities of its vibrational motion Hence there is some chance of the transition starting from 𝑣 𝑣 The ,0 , 2,0 ,…lines diminish rapidly in intensity, as shown in foot of fig 6.3 (a).
DOOZY STUDY Case (2) fig 6.3(b) Excited electronic state has a slightly smaller internuclear distance than the ground state A ve tical t ansition f om the v’’ 0 level will be most likely to occu into the uppe state v’ 2. T ansitions to lowe and highe v’ a e less probable. The probability of transitions to the upper state depend on the difference between the equilibrium separations in the lower and upper states. Case (3) fig 6.3(c) Excited state has large inter nuclear distance than ground state. b y Khttp://doozystudy.blogspot.in/ . L a k s h m a n a n M . S c . , M . P h i l . , B . E d . , m o b i l e n o : 9 5 0 0http://doozystudy.blogspot.in/ 440393 Page 24
