Rotational Raman SpectroscopyIf EM radiation falls upon an atom or molecule, it may beabsorbed if the energy of the radiation corresponds to theseparation of two energy levels of the atoms or molecules.If the EM radiation is not observed, it may be eithertransmitted or scattered by the molecule.

RayleighScattering, hL

hLas

hLs

hL

Most of the scattered radiation has an unchangedwavelength and is the Rayleigh scattering (elasticscattering). The intensity of the scattered light is I % 8-4. Blue radiation is scattered preferentially by particles in theatmosphere, resulting in a blue sky.

It was shown theoretically by Smekal in 1923 and byRaman and Krishnan in 1928 that a small amount of theradiation scattered by all phases of matter is slightlyincreased or decreased in wavelength. This is called theRaman effect, and the decreased or increased frequencies(energies) are referred to as Stokes and anti-StokesRaman scattering (inelastic scattering), respectively.

L0 L0 + LvL0 - Lvincreasing L

Lv anti-Stokes lineStokes line

Centre of broad Rayleigh line

Raman spectraThe generalized form of a Raman spectrum is sketchedbelow, complete with vibrational and rotational Ramaneffects.

The peaks in Stokes and anti-Stokes rotational spectragenerally have similar intensities, since the levels aresimilarly populated

However, for Stokes and anti-Stokes vibrational peaks, thelatter are of lesser intensity at standard temperatures, sincethe excited states are generally much less populated than theground states (we will deal with the vibrational peaks in alater lecture).

The Raman effect is a two-photon process (as we shallsee) involving excitation to a virtual excited state followedby return to a lower energy state. The processes are similarto vibrations and rotations (just different energies).

Experimental Set UpScattered radiation is ideally collected at right angles tothe incident light and sample holder.

Incident monochromatic radiation is passed through asample cell, and must be very intense, since the Ramanscattering is very weak (multiple reflection gas cells used).This makes lasers the ideal source of incident radiation. Interestingly, lasers are usually in the optical or UV range,but rotational Raman spectra are observed in the µ-waveregion - in some cases fluorescence can overwhelm theweak Raman signal. IR lasers are also used, which overcomes the problem offluorescence, but then a grating cannot be used to dispersethe radiation. Interferometry and FT-IR techniquesrevolutionized Raman spectroscopy, so a FT-Ramanspectrometer is an FT-IR spectrometer adapted toaccomodate a laser, filters to remove laser radiation duringdetection, and a variety of IR detectors. The Nd-YAGlaser, operating at 1064 nm, is the most common.

Rotational Raman ScatteringElectronic, vibrational or rotational transitions may beinvolved in Raman scattering, but here we consider onlyrotational Raman scattering.The polarizability, ", of a molecule determines the degreeof scattering of incident radiation, and when the radiation isin the uv or visible region, it is a measure of the degree towhich electrons in the molecule can be displaced ordistorted relative to the nuclei.When monochromatic radiation falls upon a molecularsample and is not absorbed by it, the oscillating electricfield, E, of the EM radiation induces an electric dipole, µ,which is related to E by the polarizability:

µ ' "E

where µ and E are both vector quantities, and themagnitude of the vector E can be written as:

E ' A sin2BcL̄t

where A is the amplitude and L is the EMR wavenumber.The polarizability is not a scalar, but actually ananisotropic property, which means that at equal distancesfrom the centre of the molecule, " may have differentmagnitudes when measured in different directions.

Atoms and spherical rotors are isotropically polarizable,meaning that the same distortion is induced regardless ofthe direction of the applied field.

Anisotropic PolarizabilityNon-spherical molecules are anisotropically polarizable,meaning that the magnitude of polarization at differentpositions in the molecule equidistant from the centre aredifferent (orientation dependent polarizability).A surface drawn so that the distance from the origin to apoint on the surface is "-1/2, where " is the polarizability inthat direction, forms an ellipsoid. The most common wayto mathematically describe such an anisotropic property(including moments of inertia, electrical conductivity incrystals, chemical shielding in NMR and anisotropic stressand strain in solid materials) is using a second-rank tensor

" '

"xx "xy "xz

"yx "yy "yz

"zx "zy "zz

"zz

"yy"xx

where "xx, "yy and "zz are the principal components alongthe x, y and z axes of the molecule. The tensor issymmetric in the sense that "xz = "zx, "yz = "zy and "xy = "yx.

So there are six differentcomponents of ": "xx, "yy, "zz, "xz,"yz and "xy, each of which can beassigned to one of the symmetryspecies of the point group towhich the molecule belongs.

Polarizability Ellipsoid

The polarizability ellipsoidrotates with the molecule at afrequency of <rot. Thepolarizability appears tochange at twice thefrequency of rotation sincethe ellipsoid is symmetric toa rotation by B about anyCartesian axis.

" ' "0,r % "1,rsin2Bc(2L̄rot)t

where "0,r is the average polarizability and "1,r is theamplitude of the change of the polarizability duringrotation. Using the relations µ = "E and E = Asin 2BcLt,the magnitude of the induced dipole moment is:

µ ' "0,rAsin2BcL̄t & ½"1,rAcos2Bc(L̄ % 2L̄rot)t% ½"1,rAcos2Bc(L̄ & 2L̄rot)t

1st term: Rayleigh scattering of unchanged L2nd term: anti-Stokes scattering3rd term: Stokes scattering (L̄ & 2L̄rot )

(¯ + 2L̄rot )L

¯

The variation of " with rotation is given by:

The distortion induced bythe electric field dependsupon the orientation ofthe molecule w.r.t. E

L

Rotational Raman: Linear MoleculesSince the polarizability ellipsoid returns to its initial valueafter rotating only 180o, the selection rule for rotationalRaman spectroscopy of a linear molecule is:

)J ' 0, ±2The )J = 0 transitions correspond to Rayleigh scatteringand are very intense - however, this scattering isunimportant in considering Raman spectroscopy.Another selection rule is that the molecule must haveanisotropic polarizability (i.e., " not the same in alldirections) - just atoms and spherical rotors do not meetthis requirement while all other molecules do.

This means that all linear molecules, whether or not theyhave a centre of inversion, will be Raman active

Molecules initially in the J = 0 stateencounter intense, monochromaticradiation of frequency L. If hL doesnot correspond to a difference inenergy between J = 0 and any otherstate (rotational, vibrational orelectronic), it is not absorbed by themolecule. Rather, it produces aninduced dipole µ = "E.

The rotating molecule modulates thescattering of the light via rotation ofthe polarizability ellipsoid.

Origin of Rotational Raman Spectra

J2

10

V0

V1

The molecule with the induced dipole resulting from EMirradiation is said to be in a virtual state, V0, and whenscattering occurs the molecule returns to J = 0 (Rayleigh)or J = 2 (Stokes).A molecule initially in the J = 2 state goes to the virtualstate V1 and returns to J = 2 (Rayleigh) and J = 4 (Stokes)or J = 0 (anti-Stokes). The overall transitions are indicatedby the solid lines below.

Stokes anti-StokesRayleigh

For a symmetric rotor molecule the selection rules forrotational Raman spectroscopy are:

)J = 0, ±1, ±2; )K = 0resulting in R and S branches for each value of K (as wellas Rayleigh scattering). For asymmetric rotors, )J = 0, ±1,±2, but since K is not a good quantum number, spectrabecome quite complicated.

Ramanspectroscopy istherefore a two-photon process,unlike othertechniquesdiscussed so far.

Origin of Rotational Raman Spectra, 2More overall transitions are shown in the figure below,with overall transitions pointing downwards indicatinganti-Stokes lines (positive frequency shift) and thosepointing upwards indicating Stokes lines (negativefrequency shift).

012345

6

7J

Stokes lines anti-Stokes linesRayleighincreasing frequency

The intensities along the branches show a maximumbecause the populations of the initial levels of thetransitions quickly rise to a maximum and then fall off.In practice, the anti-Stokes lines are less intense than theStokes lines, since the former have their origin in anexcited rotational state.

Rotational Raman SpectraConventionally, )J = J(upper) - J(lower), so the realselection rule is just )J = +2. The magnitude of the Ramandisplacement from exciting radiation < is

|)<̄| ' F(J % 2) & F(J)

where and <L is the wavenumber of excitinglaser radiation.

)<̄ ' <̄ & <̄L

F(J) 'Er

hc'

h8B2cI

J(J % 1) ' BJ(J % 1)

The term symbol for rotation is the same as before:

so substitution yields|)<̄| ' 4B0 J % 6B0

where the subscript “0” indicates the ground vibrationalstate. So lines will be evenly spaced by 4B0, and the firstanti-Stokes and Stokes lines are separated by 12B0.

As for rotational spectroscopy, centrifugal distortion maybe taken into account, resulting in additional terms:

|)<̄| ' (4B0 J - 6D0) J %32

& 8D0 J %32

3

¯

Series of rotational transitions are referred to as branches:)J ..., -2, -1, 0, +1, +2, ...Branch ..., O, P, Q, R, S, ...Stokes and anti-Stokes are both S branches, though they aresometimes referred to as O and S branches, respectively.

Rotational Raman Spectrum of 15N2The rotational Raman spectrum of 15N2 is shown below,which was obtained with 476.5 nm radiation from an argonion laser.

From this spectrum a very accurate value of B0 is obtained:B0 = 1.857 672 ± 0.000 027 cm-1, from which a bond lengthof r0 = 1.099 985 ± 0.000 010 Å is calculated.One unusual feature of this spectrum is the intensityalternation of 1:3 when the J value of the transitions thathave even:odd - this is due to the nuclear spin of the 15N2nuclei (we will discuss this shortly).Another feature (marked with crosses) are grating ghosts,which are spectral artifacts resulting from dispersal of theradiation from the sample using a grating. Spectralartifacts occur in all forms of spectroscopy and do notcorrespond to any real features of the spectrum of themolecule (sometimes can severely hamper analysis!).

Nuclear Spin Statistical WeightsThe nuclear spin can be included in the total wavefunctionfor a molecule:

R ' ReRvRrRns

We focus upon the symmetry properties of Rr and Rns here.

The parity of the rotational state = (-1)J, J = 0, 1, 2, ...Thus, for even values of J, Rr is symmetric to exchange,and for odd values of J, Rr is antisymmetric to exchange.

For a symmetrical D4h diatomic or linear polyatomicmolecule with any number of identical nuclei having thesame nuclear spin quantum number, I = n + ½, where nis zero or an integer, exchange of any two nuclei results ina change of sign of R which is said to be antisymmetric tonuclear exchange. The nuclei are said to be fermions (orFermi particles), obeying Fermi-Dirac statistics.

If I = n, where n is an integer, R is symmetric to nuclearexchange and the nuclei are said to be bosons (Boseparticles) obeying Bose-Einstein statistics.

Consider the molecule 1H2, for which the ground state of Reand Rv for any vibrational state are symmetric with respectto nuclear exchange. Since I = ½, then RrRns must be anti-symmetric w.r.t. nuclear exchange.

If 2 fermions are exchanged: Rr Rns must be antisymmetricIf 2 bosons are exchanged: Rr Rns must be symmetric

Nuclear Spin Statistical Weights, 2For I = ½, space quantization gives mI = +1/2 or -1/2. Thenuclear spin wavefunction is normally written as " or $ tocorrespond to these values, respectively. Both 1H nuclei,labelled 1 and 2, can have either " or $ spin wavefunctions, so there are four possible forms of Rns:Rns ' "(1)"(2); $(1)$(2); "(1)$(2); or $(1)"(2)

The first two functions are symmetric w.r.t. exchange oflabels, but the other functions are neither symmetric norantisymmetric. So it is necessary to use the linearcombinations of the spin wave functions shown below:

"(1)"(2)(s) Rns ' 21/2["(1)$(2) % "(2)$(1)]

$(1)$(2)

(a) Rns ' 21/2["(1)$(2) & "(2)$(1)]

So 3 of the spin wave functions are symmetric (s) w.r.t.nuclear exchange and 1 is antisymmetric (a)For a homonuclear diatomic molecule, there are(2I+1)(I+1) symmetric and (2I+1)I antisymmetric spinwave functions, so:

Number of (s) functionsNumber of (a) functions

'I % 1

I

Nuclear Spin Statistical Weights, 3In order for RrRns to be antisymmetric for 1H2, theantisymmetric Rns are associated with even J states and thesymmetric Rns are associated with the odd J states.

So 1H2 is regarded as exisiting in two distinct forms:(1) para-hydrogen with Rns antisymmetric and anti-

parallel nuclear spins(2) ortho-hydrogen with Rns symmetric and parallel

nuclear spinsThe interchange of two identicalfermion nuclei results in a change inthe sign of the overall wavefunction. Re-labelling can be though of in twosteps: (1) a rotation of the molecule,(2) interchange of the unlike spins(differently coloured nuclei).

If the nuclei have anti-parallel spins,the wavefunction changes sign.

para-1H2 can only exist in even J states and ortho-1H2 inodd J states. At temperatures where there are populationsup to fairly high values of J, there will be roughly 3 timesas much ortho- as there is para-1H2. If J is small, there ismostly para-1H2. All other homonuclear diatomicmolecules with I = ½ for each nucleus (e.g., 19F2) also haveortho and para forms (see the diagram on the next page)

Nuclear Statistical Spin Weights, 4For I = 1 for each nucleus (e.g., 2H2 and 14N2) the totalwavefunction must be symmetric w.r.t. nuclear exchange. There are 9 nuclear spin wave functions, 6 of which aresymmetric and 3 of which are anti-symmetric.This means that molecules like 2H2 and 14N2 have orthoforms only when J is even and para forms only when J isodd, and there is roughly two times as much ortho as thereis para at normal temperatures. At low temperatures, thereis an excess of the ortho form.

Rr Rns o/p ns stat wt. Rns o/p ns stat wt. Rr Rns o/p ns stat wt.

a s o 3 a p 3 a s o 1

s a p 1 s o 6 a

a s o 3 a p 3 a s o 1

s a p 1 s o 6 a

a s o 3 a p 3 a s o 1s a p 1 s o 6 a

1H2 or 19F22H2 or 14N2

16O2

J

5

4

3

2

10

Nuclear spin statistical weights of rotational states forvarious diatomic molecules are shown below:

Nuclear Spin Statistic Weights, 5The effects of low temperatures on ortho:para ratio is moreimportant for light molecules (e.g., 1H2 and 2H2) than forheavy molecules (e.g., 19F2 and 14N2), since the separationof J = 0 and J = 1 energy levels is smaller for heaviermolecules than for light molecules.

The symmetrical linear polyatomic molecule acetylene,1H - 12C / 12C - 1H is very similar to 1H2, since the nuclearspin of 12C is I = 0. However, since the rotational energylevels are more closely spaced in acetylene, a very lowtemperature is required to produce the para formpredominantly, in comparison to 1H2

For 16O2, I = 0, and there are no antisymmetric nuclear spinwave functions. Since each 16O nucleus is a boson, thetotal wavefunction must be symmetric to nuclear exchange.Since this molecule has two unpaired electrons in itsground state (triplet state), Re is antisymmetric, so Rns issymmetric w.r.t. nuclear exchange.

So 16O2 only has levels with odd values of J. In therotational Raman spectrum, alternate lines with even valuesof NO are missing. (For molecules with unpaired electrons,the quantum number N, rather than J, distinguishes therotational energy levels).

Structure DeterminationMeasurement of the rotational constants of the zero pointvibrational state, B0, and one or more vibrational states, Bv,may allow for the determination of the rotational constantof the hypothetical equilibrium state, Be.

These rotational constants can be used to determine r0 andre. re will be independent of the isotopic species whereas r0will not, so re is typically used in discussing bond lengths ofthe highest accuracies.

Bv ' Be & "(v%1/2)

Molecule r0(Å) re (Å)14N2 1.100 105 ± 0.000 010 1.097 651 ± 0.000 03015N2 1.099 985 ± 0.000 010 1.097 614 ± 0.000 030

Note that re does not change withisotopic substitution since it refersto bond length at the minimum ofthe potential energy curve (whichdoes not change for either harmonicor anharmonic oscillators).

On the other hand, vibrational energy levels within thepotential energy curve change, thereby resulting in changesin r0 with isotopic subsitution.For non-linear molecules, determination of rotationalconstants becomes increasingly difficult - substitutionstructures and isotropic substitution are used for largermolecules (e.g., aniline, formaldehyde, etc.).

Key Concepts1. EM radiation that is not absorbed by the molecule is

scattered. Most of the scattered radiation isunchanged in wavelength (Rayleigh scattering) and asmall percentage of the light is increased anddecreased in wavelength (referred to as Stokes andanti-Stokes Raman scattering, respectively).

2. The polarizability, ", of the molecule determines thedegree of scattering of the incident radiation. Polarizability is an anisotropic property, whichmeans that " may have different magnitudes whenmeasured in different directions. The polarizabilityellipsoid is a described by a second-rank tensor.

3. Due to the symmetry of the polarizability ellipsoid,when a molecule rotates the frequency of rotation ofthe ellipsoid appears to proceed at twice the frequencyof rotation.

4. The selection rule for rotational Raman spectroscopyof a linear molecule is )J = 0, ±2. Only atoms andspherical rotors are Raman inactive, all othermolecules are Raman active.

5. The Rayleigh line is in the centre and intense, theweaker Stokes and anti-Stokes lines occur at lowerand higher frequencies, respectively.

6. Nuclei which are fermions or bosons result indifferent peak intensities in rotational Raman spectra. Nuclear spin statistics change with temperature andmay alter the appearance of the spectrum.