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Determination of electronic symmetries from polarized Raman spectra
Volume
3. number
1
CHEMICAL
DETERMJNATEON FROM
OF
PKYSICS
LETTERS
ELECTRONIC
POLARIZED
RAMAN
January
SYMMETRIES SPECTRA
0. SONNICH MORTENSEN Chemical
Laboratory
IV,
.
H. C. Ousted Institute,
The Unioersicy of Copenhagen, 2100 Copenhagen, Received
1969
Universitetsparken Dentnark
6 November
5,
1968
From a consideration of the exact quantum field expressions for Raman intensities it is shown that it is possible, under specified conditions, to determine symmetries of electronically excited states from polarized rotational-and vibrational Raman spectra.
It is well known that it is not possible to determine symmetries of electronic states from ordinary emission oi’ absorption spectra in the gas phase. Essentially this is due to the fact that direct absorption or emission is a one-photon process and that a vector has only one invariant, its length. Only one piece of information is therefore obtained from the spectrum (besides of course the frequency of the. transition) the intensity of the transition, and this is not sufficient to.determine the symmetries of the electronic slates between which the transition takes place. Raman scattering, on the other hand, is a two-photon process, and determines a tensor quality. A general tensor has three invariants and it must in principle be possible to determine three quantities from the Raman spectrum. It can be shown that for ordinary Raman transitions where the exciting frequency is not too close to an electronic absorption band, one of these invariants is zero so that only two quantities can be derived from the spectra. As will be shown in the following this is just sufficient to determine the symmetries of excited electronic states in the very important case where the electronic ground state is non-degenerate. For the ordinary case, where the F&man scattered light is observed in a direction perpendicular to the propagation direction of the incoming light, two Raman spectra can be obtained, one in which the scattered light is polarized perpendicular to the polarization direction of the incident light, the other poIarized parallel to the incident light. The ratio of the intensities for one pariicular transition in the two directions is conven4
tionally denoted the depolarization ratio for linearly polarized light and is related to a molecular scattering tensor in the following way:
where the cues‘9 ’ are irreducible tensor-components deterrkined by the internal motions of the molecule. We have here summed over all values of initial and final rotational states, so that the depolarization ratio defined is the depolarization ratio taken over the whole rotational envelope of the internal transitioli. As mentioned previously, in the usual case the tensor components ct$ (corresponding to cartesian tensor components as ‘Y_~~- CY& are zero, so that 0 G p1 c : depending on the ratio
I CY; I %1
@!fi”.From
a measurement
of PI we
therefore determine this ratio as:
The scattering tensor is determined by the wave functions of the molecule in the following way [13
(2) Here k and n refer to the initial and final states respectively, and the summation over Y goes over all excited internal states of the molecule.
Volume 3, number 1
CHEMICAL PHYSICS
p and (T stand for Cartesian vector-components, v is the frequency of the incident light and yr is inversely proportional to the lifetime of the state r. The operator yp is the electric dipole operator of the electrons and we have neglected the part of the transition dipole moment that is due to the nuclei. (By direct evaluation it is easily seen that these terms do not contribute to Raman scattering.) We wish to emphasize that eq. (2), unlike the conventional polarizabiiity theory, is valid also in cases where the frequency of the incident light is close to the frequency of an electronic transition. In that case, in the summation over excited states r we can in first aljproximation neglect all states but the one close to the incident frequency, and the scattering tensor can in this way be calculated from a knowledge of only the initial, the final and ooze intermediate state. The condition for the non-vanishing of a matrix element like (J/n{Yp 1J/r> is of course that the product of the three functions transforms as the totally symmetric representation of the molecular point group. If we, for the moment, concentrate on rotational Raman scattering, the final state will be identical to the electronic ground state of the molecule and for most molecules it will be totally symmetric. The intermediate state I- must therefore transform as rp and if we know from a measurement of the depolarization ratio which vector components are active we can identify the symmetry of the intermediate state ‘Y. Let us illustrate this with a specific example that shows exactly what we mean by the above statement. Consider the pointgroup C4U and assume that the molecular ground state transforms as Al. The intermediate state T can then be of symmetry A1 correspondiig to the non-vanishing vector component r,, or of symmetry E corresponding to the components ^/x, Y,,. The former case corresponds to the tensor component oZZ being non-zero, while in the latter case we have oXX = o!,,,, f 0 with all other tensor components vanishing. The Cartesian tensor components are linked to the irreducible tensor components of eq. (1) in the following way [ZJ:
January
LETTERS
IS69
If we insert this into eq. (l), weget then for the depolarization ratio in the two cases: 0.125
p1 = 10.333
intermediate intermediate
state E state AZ
This result is not peculiar to the pointgroup C4* but holds (with trivial changes in symmetry-designations) for pointgroups with at least one axis of fold three or higher, though excluding the cubic groups where all vector components transform as one irreducible degenerate representation. So far, we have only considered Raman scattering corresponding to pure rotational transitions, but quite similar considerations hold for the depolarization ratios of bands corresponding to totalIy symmetric vibrations. Instead of the purely electronic matrix element @,I ~bj*& one deals in this case with the first derivative of this electronic matrix element with respect to the totally symmetric normal coordinate 131. Et is evident that for totally symmetric modes the same symmetry-considerations as for the purely electronic case must be valid and we arrive at the same conclusion as stated for the rotational Raman scattering. We can therefore state, quite generally, that measurement of depolarization ratios for rotational as well as vibrational Raman bands for totally symmetric modes, in region-s of ne?.r resonance determines the symmetry of the electronic state with energy close to the frequency of the incident light. For a more thorough discussion of the concepts mentioned in this note and for derivations of the pertinent equations, we refer to a forthcoming communication. REFERENCES [1] G. Placzek. Marx Handbuch der Radiologie. 2nd ed.. Vol. 6 (II) (1934) p. 206. [Z] 0. Sonnich Mortensen and J. A. Koningstein, J. Chem. Phys. 48 (1968) 3971. [3] A. C. Albrecht. J. Chem. Phys. 34 (1961) 1476.
5
3. number
1
CHEMICAL
DETERMJNATEON FROM
OF
PKYSICS
LETTERS
ELECTRONIC
POLARIZED
RAMAN
January
SYMMETRIES SPECTRA
0. SONNICH MORTENSEN Chemical
Laboratory
IV,
.
H. C. Ousted Institute,
The Unioersicy of Copenhagen, 2100 Copenhagen, Received
1969
Universitetsparken Dentnark
6 November
5,
1968
From a consideration of the exact quantum field expressions for Raman intensities it is shown that it is possible, under specified conditions, to determine symmetries of electronically excited states from polarized rotational-and vibrational Raman spectra.
It is well known that it is not possible to determine symmetries of electronic states from ordinary emission oi’ absorption spectra in the gas phase. Essentially this is due to the fact that direct absorption or emission is a one-photon process and that a vector has only one invariant, its length. Only one piece of information is therefore obtained from the spectrum (besides of course the frequency of the. transition) the intensity of the transition, and this is not sufficient to.determine the symmetries of the electronic slates between which the transition takes place. Raman scattering, on the other hand, is a two-photon process, and determines a tensor quality. A general tensor has three invariants and it must in principle be possible to determine three quantities from the Raman spectrum. It can be shown that for ordinary Raman transitions where the exciting frequency is not too close to an electronic absorption band, one of these invariants is zero so that only two quantities can be derived from the spectra. As will be shown in the following this is just sufficient to determine the symmetries of excited electronic states in the very important case where the electronic ground state is non-degenerate. For the ordinary case, where the F&man scattered light is observed in a direction perpendicular to the propagation direction of the incoming light, two Raman spectra can be obtained, one in which the scattered light is polarized perpendicular to the polarization direction of the incident light, the other poIarized parallel to the incident light. The ratio of the intensities for one pariicular transition in the two directions is conven4
tionally denoted the depolarization ratio for linearly polarized light and is related to a molecular scattering tensor in the following way:
where the cues‘9 ’ are irreducible tensor-components deterrkined by the internal motions of the molecule. We have here summed over all values of initial and final rotational states, so that the depolarization ratio defined is the depolarization ratio taken over the whole rotational envelope of the internal transitioli. As mentioned previously, in the usual case the tensor components ct$ (corresponding to cartesian tensor components as ‘Y_~~- CY& are zero, so that 0 G p1 c : depending on the ratio
I CY; I %1
@!fi”.From
a measurement
of PI we
therefore determine this ratio as:
The scattering tensor is determined by the wave functions of the molecule in the following way [13
(2) Here k and n refer to the initial and final states respectively, and the summation over Y goes over all excited internal states of the molecule.
Volume 3, number 1
CHEMICAL PHYSICS
p and (T stand for Cartesian vector-components, v is the frequency of the incident light and yr is inversely proportional to the lifetime of the state r. The operator yp is the electric dipole operator of the electrons and we have neglected the part of the transition dipole moment that is due to the nuclei. (By direct evaluation it is easily seen that these terms do not contribute to Raman scattering.) We wish to emphasize that eq. (2), unlike the conventional polarizabiiity theory, is valid also in cases where the frequency of the incident light is close to the frequency of an electronic transition. In that case, in the summation over excited states r we can in first aljproximation neglect all states but the one close to the incident frequency, and the scattering tensor can in this way be calculated from a knowledge of only the initial, the final and ooze intermediate state. The condition for the non-vanishing of a matrix element like (J/n{Yp 1J/r> is of course that the product of the three functions transforms as the totally symmetric representation of the molecular point group. If we, for the moment, concentrate on rotational Raman scattering, the final state will be identical to the electronic ground state of the molecule and for most molecules it will be totally symmetric. The intermediate state I- must therefore transform as rp and if we know from a measurement of the depolarization ratio which vector components are active we can identify the symmetry of the intermediate state ‘Y. Let us illustrate this with a specific example that shows exactly what we mean by the above statement. Consider the pointgroup C4U and assume that the molecular ground state transforms as Al. The intermediate state T can then be of symmetry A1 correspondiig to the non-vanishing vector component r,, or of symmetry E corresponding to the components ^/x, Y,,. The former case corresponds to the tensor component oZZ being non-zero, while in the latter case we have oXX = o!,,,, f 0 with all other tensor components vanishing. The Cartesian tensor components are linked to the irreducible tensor components of eq. (1) in the following way [ZJ:
January
LETTERS
IS69
If we insert this into eq. (l), weget then for the depolarization ratio in the two cases: 0.125
p1 = 10.333
intermediate intermediate
state E state AZ
This result is not peculiar to the pointgroup C4* but holds (with trivial changes in symmetry-designations) for pointgroups with at least one axis of fold three or higher, though excluding the cubic groups where all vector components transform as one irreducible degenerate representation. So far, we have only considered Raman scattering corresponding to pure rotational transitions, but quite similar considerations hold for the depolarization ratios of bands corresponding to totalIy symmetric vibrations. Instead of the purely electronic matrix element @,I ~bj*& one deals in this case with the first derivative of this electronic matrix element with respect to the totally symmetric normal coordinate 131. Et is evident that for totally symmetric modes the same symmetry-considerations as for the purely electronic case must be valid and we arrive at the same conclusion as stated for the rotational Raman scattering. We can therefore state, quite generally, that measurement of depolarization ratios for rotational as well as vibrational Raman bands for totally symmetric modes, in region-s of ne?.r resonance determines the symmetry of the electronic state with energy close to the frequency of the incident light. For a more thorough discussion of the concepts mentioned in this note and for derivations of the pertinent equations, we refer to a forthcoming communication. REFERENCES [1] G. Placzek. Marx Handbuch der Radiologie. 2nd ed.. Vol. 6 (II) (1934) p. 206. [Z] 0. Sonnich Mortensen and J. A. Koningstein, J. Chem. Phys. 48 (1968) 3971. [3] A. C. Albrecht. J. Chem. Phys. 34 (1961) 1476.
5