Lecture # 11- The Raman Effect & Rovibrational Band Structure

8/3/2019 Lecture # 11- The Raman Eff�ect & Rovibrational Band Structure

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30Jan2009Chemistry 21b – Spectroscopy

Lecture # 11 – The Raman Effect & Rovibrational Band Structure

Although the Raman effect has been known of for over fifty years, its employment as a

routine spectroscopic method was quite limited until the development of high power laserswhich provided a source of highly monochromatic radiation. We’ll first look at Ramanscattering from a classical perspective.

Unlike pure rotational or rovibrational spectroscopy, the Raman effect does not involvethe direct absorption or emission of radiation. Instead, it involves the scattering of incidentradiation which has been modified by some internal change in the system. The scatteringcan either be elastic, in which case there is no change in the internal state of the molecule,or inelastic, in which the molecule either gains energy from or contributes energy to theelectromagnetic radiation field. The Raman effect is straightforward to understand whenone considers the electrical nature of matter.

Atoms and molecules consist of collections of oppositely charged particles whose

relative positions can be altered by the application of external electric fields. This alterationleads to an eletric dipole moment being introduced ino the system. These ease withwhich a molecule or atom may be distorted by an electric field is measured by the electricpolarizability α. We have already considered static polarizabilities in the context of theStark effect, as analyzed by perturbation theory.

For atoms, where the symmetry is spherical, the polarizability will be the same in alldirections and it can be expressed by a single scalar quantity. For molecules with lower-than-spherical symmetry, the polarizability will not be the same along all directions, and

just as is true for the moment(s) of inertia, the full polarizability is described by a tensor.In matrix form, the induced dipole is given by

µI = α · E or

µIx

µIy

µIz

=

αxx αxy αxz

αyx αyy αyz

αzx αzy αzz

E xE yE z

The polarizability tensor is clearly symmetric, and just as the principal axesdiagonalize the moment-of-inertia tensor so too can the polarizability tensor bediagonalized and a polarizability ellipsoid defined. The isotropic polarizability, which iswhat matters in many applications, is then

< α > =1

3(αxx + αyy + αzz) ,

where the primes denote the principal polarizability axes.If the molecule is undergoing some sort of internal motion, such as vibration or

rotation, this can alter the isotropic polarizability such that

α = α + Asin 2πν it ,

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where ν i is the frequency of the internal motion. If a time-varying electric fieldcharacterized by E = E ◦sin(2πν Lasert) is applied, the induced moment is then given by

µI = αE = E ◦(α + Asin 2πν it)sin 2πν Lasert

or

µI = αE ◦sin 2πν Lasert +1

2AE ◦cos 2π(ν Laser + ν i)t +

1

2AE ◦cos 2π(ν Laser − ν i)t ,

and the emitted radiation will contain inelastic components at ν Laser ± ν i as well as theelastic scattering component at ν Laser.

Thus, a prerequisite to the observation of the Raman effect is not only the occurrenceof a periodic internal motion, but also the association with this motion of a change inthe polarizability of the molecule. This change can either be in the magnitude of thepolarizability tensor components or in the direction of the principal axis system, since thelatter will also lead to the presence of a changing dipole. The rotation of a molecule thatdoes not have spherical symmetry will produce a change in the spatial orientation of theprinciple axis system. Hence, the rotation of most molecules will provide a type of motionwhich gives rise to a Raman effect. The stretching or bending of a bond as a result of thevibration of a molecule can also result in a change in the polarizability. To go any further,we’ll need to look at the quantum mechanics of the Raman effect.

Raman Selection Rules in Diatomics

As always, what we need to investigate is the matrix element of the induced dipole

moment with the upper and lower states under consideration. Let’s look first at a diatomicmolecule, for which the eigenstates are labeled |J M > and for which the appropriate matrixelement is

< J M |µI E |J M > .

Unlike the previous case of pure rotational selection rules in which there was only a singledirection cosine within the matrix element integral, now there are two (one for the fieldand one for the induced moment). Each gives rise to selection rules like ∆J = ±1, and sothe overall selection rule for the rotational Raman effect for a diatomic (or linear) moleculeis:

∆J Raman = 0,±2 .

The ∆J = 0 term corresponds to elastic scattering, and occurs at the laser frequencyµLaser. This is called the Rayleigh peak . For ∆J = −2, the inelastic scattering occurs atlower frequencies than the laser frequency, and the collection of lower frequency transitionsare called the Stokes lines. Similarly, the ∆J = +2 occur to higher frequencies than thelaser radiation, and are collectively called the anti-Stokes lines. Figure 11.1 presents apictorial view of rotational Raman scattering.

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01

2

3

4

5

Low J = 3 2 1 0 0 1 2 3

Anti-StokesStokes

νL

Figure 11.1 An outline of the rotational Raman selection rules in diatomic/linearmolecules, and the definition of the Stokes/anti-Stokes branches.

For inelastic scattering off a molecular vibration of a diatomic molecule, the rotationalselection rules are still ∆J Raman = 0,±2. For ∆J = 0, or Q-branch lines, the offsetfrom the laser is simply the vibrational frequency ν ◦. The Stokes and anit-Stokes O and Sbranches (or ∆J = −2, ∆J = 2 transitions) occur to either side, and Figure 11.2 presentsa pictorial view of rovibrational Raman scattering.

1

3

4

5

1

4

5

v = 0

v = 13

2

0

2

0

Stokes

Rayleigh

Anti-Stokes

O Q S O Q S

Figure 11.2 The vibrational Raman effect in diatomic/linear molecules.

Raman Selection Rules in Polyatomics

For polyatomic species, the best recourse for determining whether or not a normalmode is Raman active or inactive is to use group theory. The character tables at the backsof most textbooks, including Atkins & Friedman, include the irreducible representationsof the polarizability operators (which are sometimes listed as α’s, but may also be listed as

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x2, y2, xy, ...). As before, we demand that the overall matrix element have A1 symmetry.Going through the C 2v and C 3v tables, we see that all of the vibrational modes in moleculeslike water or ammonia are Raman active (and also electric dipole allowed).

The interesting difference of absorption spectroscopy and Raman spectroscopy comeswhen considering molecules like CO2 or acetylene which have a center of inversion (and

hence which support g and u states). Acetylene, for example, has 3(4)−5 = 7 modes. Threeare non-degenerate and four are degenerate, and are summarized in Figure 11.3 below. Wesee that two of the modes are infrared active and Raman inactive (the antisymmetric C–Hstretch and the cis-bend), while three of the normal modes are IR inactive and Ramanactive (the C–C stretch, the symmetric C–H stretch, and the trans-bend). This is a specificexample of the exclusion rule, in which it can be shown via group theory that for moleculeswith a center of inversion modes which are Raman active are electric dipole forbidden andvice versa. Thus, to fully characterize the force field of such polyatomic systems, both theRaman and IR absorption spectra must be acquired.

- + -+

+ +- -

ν

ν

ν

ν

ν

1

2

3

4

5

3374

1974

3287

612

729

No Yes

No Yes

No Yes

Yes No

Yes No

Frequency IR? Raman?

Figure 11.3 The normal modes of acetylene and their IR/Raman activity.

So far, we’ve only examined the symmetry properties and propensities of the vibrationsthemselves. Of course, molecules vibrate and rotate at the same time, and so the nextsections briefly outline the rotational envelopes that are associated with the different kindsof vibrational modes in linear, symmetric top, and asymmetric top molecules.

Rotational Profiles of Vibrations in Linear Molecules

The point groups of linear molecules are either C ∞v (if they are polar, and hence

have a rotational spectrum) or D∞v (if they are symmetric, and hence have no rotationalspectrum). The vibrations associated with such molecules are either of Σ or Π symmetry,and transitions to these kinds of states are called parallel and perpendicular transitions,respectively. The latter have vibrational angular momenta associated with the vibrationaldegeneracy, and so have different rotational selection rules associated with them that arereflected in the rotational envelopes of the vibrational bands. From the ground state of a closed shell linear molecule, the parallel transitions are of Σ → Σ symmetry, while theperpendicular bands are of Σ → Π symmetry.

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Figure 11.4 An overview of the mid-IR spectrum of HCN.

The most important difference between these types of bands is that the former obey∆J = ±1 selection rules, while the latter obey ∆J = 0,±1 selection rules. Thus, at coarseresolution, parallel bands have only P and R branches while perpendicular bands have P,

Q, and R branches. Physically, the parallel bands have the dipole derivative along theinternuclear axis, while the perpendicular bands have the dipole derivative normal to theinternuclear axis. An overview of the appearance of these kinds of bands for the HCNmolecule is shown in Figure 11.3. At higher resolution, the parallel bands for a C ∞v linearmolecule such as HCN look just like a vibrational transition in a heteronuclear diatomicmolecule, as Figure 11.4 shows. The major difference, especially as the molecule getslarger, is the presence of hot bands in the linear molecule spectrum that are associated

with low lying vibrational states that can be significantly populated at room temperature.For perpendicular bands, the angular momentum of vibration leads to a doublydegenerate state in the absence of rotation. In the rotating principal axis reference frame,however, there are Coriolis forces which lift the degeneracy. This is called l-type doubling ,and modifies the term expression for linear molecules to

F v(J ) = BvJ (J + 1) ±qi

2J (J + 1) ,

where qi is the l-type doubling constant. The most obvious effect of the perpendicularnature of the vibration is that the Q-branch is now allowed, as is shown for the ν 1 + ν 5combination band of acetylene in Figure 11.4 (in which the ortho/para alternative is clearly

visible). Upon closer inspection of the actual rotation levels involved, it can be seen that theP and R branches terminate on the lower l-type doublet, while the Q-branch terminates onthe upper half of the l-type doublet. Thus, in order to measure the qi values from infraredspectra, both the P/R- and Q-branches must be rotationally resolved! This can be quitedifficult for larger molecules, and so in many cases the l-type doubling is in fact measuredvia the pure rotational spectra arising from the thermally populated upper vibrationalstates. In this case, the l-type doublets appear weakly to either side of the intense groundstate rotational transition, and the splitting grows with frequency (or J ).

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1

3

4

5

1

4

5

v = 0

v = 13

2

0

2

0

Σ

Σ+

+

J’

J’’

+-+

-

+

-

-

+

-

+-+

R(0) . . .. . . P(1)

Figure 11.5 Selection rules for a linear molecule parallel transition (top), and a specificexample, the C–H stretch of HCN.

+-

+

-

+

-+

+-

-+

+-

-

+

0

2

3

4

1

1

2

3

4

5

. . . R (0 ) Q(1 ). .. P(2 ) . . .

Σ+

Πu

g

Figure 11.6 Selection rules for a linear molecule perpendicular transition (top), and aspecific example, the 11

0510

combination band of acetylene.

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Rotational Profiles of Vibrations in Symmetric Top Molecules

With symmetric rotors we now have K = 0 to worry about. For symmetric rotors inthe C 3v group, which is a common occurrence, remember there are A and E symmetries. Aquick check of the character table reveals that only A1 → A1 and A1 → E transitions areallowed. Just as for the linear molecule case, these are called parallel and perpendiculartransitions, respectively. For the parallel bands the selection rules are

∆K = 0 and ∆J = ±1, for K = 0

∆K = 0 and ∆J = 0,±1, for K = 0

while for perpendicular bands the selection rules are

∆K = ±1 a n d ∆J = 0,±1.

Thus, unlike the case for a linear molecule, for a symmetric top there is a Q-branchfor both parallel and perpendicular bands. For the parallel bands, however, all of thedifferent K -values for which there is a Q-branch have transitions in essentially the sameplace – splittings are introduced only by the small change in rotational constant upstairsand downstairs and by the centrifugal distortion constants. Thus, for parallel bands thereis a single tight Q-branch with P - and R-branches to either side with a spacing betweenfeatures of roughly 2B. As J increases, a larger and larger number of K states becomeavailable, and even the P - and R-branch features begin to get smeared out as J increases.

For the perpendicular bands, life is even more complicated. Here, since K is allowedto change by ±1, the individual Q-branches are spread out according to K . Just asfor diatomic molecules, the change in angular momentum by one leads to a spacing of

approximately a rotational constant. For a prolate symmetric top the spacing between theQ-branches is approximately 2(A−B), for oblate tops the appropriate spacing is obviously2(C −B). Underlying these evenly spaced Q-branches are the assorted P - and R-branchesthat go along with them. These are smeared out at low resolution, and what is mostapparent are then the evenly spaced Q-branch features. Two examples of parallel andperpendicular bands are presented in Figures 11.7 and 11.8, which show the 11

0parallel

band of CH3F and the 610

perpendicular band of SiH3F. Here, the notation 110

should beread as meaning the 0 → 1 transition of the v1 mode, etc.

Rotational Profiles of Vibrations in Asymmetric Top Molecules

Finally, the most complex examples of rotational profiles in polyatomic molecules are

due to those of asymmetric tops. As before with rotational spectra, the only rigorousselection rule left is

∆J = 0,±1 ,

which leads to a widely scattered suite of P -, Q-, and R-branches. Just as was true forthe permanent dipole moment projections, the dipole derivative for a given vibration canbe broken down into it components along the a, b, c inertial axes that rotate with themolecule. By analogy with pure rotational spectroscopy, we’ll again label various bands as

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Figure 11.7 The 110 A1 → A1 parallel vibrational band of CD3F.

Figure 11.8 The 610 E → A1 perpendicular vibrational band of SiH3F.

type A, type B, or type C according to their dipole derivative projections. The different

projections lead to different overall shapes for the bands, as summarized in Hollas.For prolate tops, the type A bands resemble those of parallel bands from symmetric

tops – their is a strong central Q-branch feature flanked by compact P - and R-branch wingswhose feature spacings are close to 2B. For the type B bands, the change in both K p andK o spreads all of the features out, much as in a perpendicular band of a symmetric top. Atlow resolution, therefore, type B bands tend to lack a strong central peak, but instead canoften look like the parallel bands of linear molecules at low resolution and signal-to-noise(so watch out!). At somewhat higher resolution, the Q-branch features spaced by roughly2A begin to appear. At very high resolution these type B bands are very complex, but sinceso many transitions are permitted, once they are fully understood quite accurate valuesof the A , B , C rotational constants and centrifugal distortion constants can be extracted.

Example type A and type B bands from the ethylene molecule are presented in Figure11.9. Finally, for a type C band there is again a strong central feature, but one which ismore spread out than that of a type A band. In addition, the wings to higher and lowerfrequency are stronger relative to the central feature in a type C band than in a type A

band. Other examples of such spectra may be found in various texts (such as Spectra of Atoms and Molecules, Bernath).

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Figure 11.9 The 1110

type A band of ethylene (top), the the 910

type B band of ethylene(middle), and the 71

0type C band of ethylene.

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Lecture # 11- The Raman Effect & Rovibrational Band Structure

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