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Excitation profiles of resonance raman spectra in condensed media
Volume
80, number 2
EXClTATfON
CHEMICAL
PROFZLES
OF RESONANCE
PHYSICS
LETTERS
RA;vIAN SPECTRA
1 June
n?r CONDENSED
1981
MEDIA
Kenji KODAMA and And& D. BANDRAW Dgpartenzent de &true, Fuculte’ des sciences, Universite’ de Sherbrooke, Sherbrookeie, P Q.> Car&a JIK 2RI Received
29 December
1980,
in final form
19 februark
1981
A simple model, obtained by reducmg a multunode system to a sunpfe coordmate, is proposed to ekplak the broadened of RRS in liquid or sohd medxa. It is shown that the coherent excitation of the near continuous background of low-frequency modes can cause homogeneous broadening of RRS profiles. The model is applied to the excitation profile of p-carotene.
1. Introduction In the theory of resonance Rantan scattering one usually introduces a constant linewidth or damping f_acrGr to represent the homogeneous broadenmg of the vlbronic excited states. It is not uncommon to come 3cfoss very large r values, i.e. > 100 cm-l to explain excitation profiles and depolarization ratios. Such Iarge lmewrdths give unusually short !ifetlmes [much less than h picosecond) for resonant excited states so that one may we1I questlon the nature of the broadening. We sbafl show that part of the broadening, which is not a lifetune effect, comes from coherent scattering from the background continuum of low-frequency modas of large molecules and their medium (phonons), be it liquid or sohd_ One may look at the system as a supermolecute, i e. molecule plus solvent, The high-frequency modes will correspond to the internal modes of the molecules, whereas the how-frequency modes will correspond to the external modes, such as Vibrations of the molecule and also solvent modes. The need to incorporate external modes such as solvent modes is well known for absorption and emission spectra E1 f and electron transfer reactions [2] . It would seem natural to try to explain the dependence of RRS excitation profiles on the difference of configuration of these external modes in the ground and excited states. These low-frequency modes may also involve some intramolecular modes. Champion and Albrecht [3--51 have recently considered contnbutions from 248
other internal modes to the excitation profiles of highfrequency vibrations by assuming displaced harmonic oscillators and introducing also damping factors. Our model thou& similar in spirit will focus on the contribution of the quasi-continuous background of low-frequency external and internal modes to the excitation profiles of high-frequency modes.
2. Model In order to describe resonance Raman scattering in a medium we assume one resonance intermediate electronic state, we shall ignore vibronic couplings such as Herzberg-Teller, Born-Oppenheimer couplings [6], Duschinsky effect f7] etc. and shall adopt the displaced harmonic oscillator model for excited internal f8,9] as weil as external modes (21. The only perturbation present is the radiative transition between the ground and intermediate states. Under the above assumptions the hamiltonians for the ground and excited states, HP and He, are given as follows:
0 009-2614/81/0000-00~0/$02.50
(1) 0 North-Holland
publishing Company
Volume 80, number 2
CiiEMiCAL
PHYSICS
where MI, Pi, Qj and S’Zjfii, pi2 (7i and w,) are the reduced mass, momentum, coordinate and angular frequency for jth phonon mode (ith vibrational mode), QF(& is the displacement of ith phonon (ith vlbrational) mode; hv, .Eg and Ee are the mcrdent photon energy and electronic energy for the ground and excited states. We assumed same frequencies in both electronic states. Then the scattering tensor ot~,~ for a transition {Oj]--{Oj] vfa _ intermediate states I t..$}for the $th phonon mode (external1 and a transition (Oi)- {Vi>via intermediates state I$> for the ith vibrational (internal) mode is proportional to the following expression:
n-coIv’) (UlfO.)
A(E’)=T;,T;;& J
,
JJ
E=hv+Eg-Ee_
(4)
J
Assuming that the low-frequency modes can be regrouped around a mean frequency SZ (Einstein model) 22 ] , we introduce a recurrent transformation of coordinates:
x
(&f;/zQr-l)
+ 3f1f2Q_) J J’
i-1
tan@j=Dj
lI.k FIB: -
Q; =Q,t
‘)i = ~M~~~J2~~l~~Q~,
+P~N~2JvM1 +MlS’Z2(Q~)
- Q;‘j2/2,
(8)
with
For large D, we can replace the displaced coordinate by a continuous Iinear potential surface which is tangent to the original potential surface at the equilibrium position of the ground state. The other modes being undisplaced, will integrate to unity in the FC elements in the scattering amplitude (4) (i.e. only O-O transitions allowed). The ~troduction of the linear potent&l to mimic the repulsive wall of the largely displaced oscillator enables us to reduce the phonon Raman amplitude A@‘) to a previously derived analytic expression [lo-121 A@‘)
= Aoo(E”)
(5)
W&I E” = hv _ Eoo - Zi~~$wi and EOo now becomes the energy for a O-O transition, i.e.
-112
I
1 June 1981
LElTERS
,
j = 2, .__, N.
6%
Thus after N - I iterations of eq. (51, Le. from1 I=2 to IV, one can reduce the N-dimensional mtdtiphonon problem in eq. (l), to that of one displaced and N - 1 undisplaced coordinates_ Thus the phonon part of eq. (1) becomes:
The Zero point energy is now necessary to defme the zero energy of the linear potential_ The scattering amplitude which now includes scattering from the cont&urn of photon modes via A,, simplifies to
249
Volume 80. number 2
CHEMICAL PHYSICS LEXTERS
1
June 1981
The characteristics of the single mode amplitude A,,, (lo), have been previously described [lO--X2]_ Thus AoO is comprised of an imaginary part Im(E”) which is lorentzian about E”, and a real part Re(E”) which is anti-~ort~tz&r, i-e_ it is zero at E” = CLHence one can rewrite the amplitude,AoO, asfi(E”)/[E” + ir,f,E”)], withp(E”) a monotonic function of energy, The total scattering amphtude now becomes:
From eqs. (12) and (13) we see that the background continuum of low-frequency modes introduces an energy-dependent hnewidth I?&!?‘) since these contribute to the resonant scattering also if the excited-state geometry is much different from the ground state. This change in geometry is represented by the total effective displacement of one mode, i.e. D = (X,-D/-)112 where l)i is the displacement of each individual low-frequency mode, see eqs. (S)-(9). We emphasize that the appearance of ri in the new expression for the amplitude (13) is not a lifetime effect, but is rather the coherent RR scattenng from the background modes.
3. Application Resonance Raman scattering of p-carotene in several solvents, which has been investigated by many authors theoreticaIly [13-l 61 and experimentaliy [I 7-221, makes it possible to compare our model with these experimental and theoretical results. Excitation profiles have been caIeuIated for the three main strong lines 1005 (v3), 1157 (~2) and 1525 (vt ) cm-1 of that moie&e. The comparisons of our caIcuIations and the abovementioned theoretical and experimental results is carried out for the excitation profiles of the ZQ , ~2, ~3,2vI, 2~2, “I + v2 and YI + ~3 transitions in isopentane solvent 1181, pl, v2 and v3 in carbon disulptide solvent (203, v2 in acetone, cyclohexane, ethyl alcohol and toluene solvents f22]. These results are shown in figs. 14. in each series of RRS profiles, the relative scale is fured by fitting the calculated profile for Q to the experimental resulr. The profiles for ul, vz, v3 and combinations and overtones were calculated using essentialIy the parameters of Srebrand and Zgierski [ 165 _Thus the frequen250
Fig. 1. Absorption and RRS excitation profiIes of p-carotene in isopentane. calculated, ---observed absorption, o e+ p&mental RRS. Relative SC&S absorption (arbitrary), vl.v~:v3=1_1_o_5_
ties were Rwl = 1525,fiw2 = 115S,fiw3 = 1005 cm-l, and the corresponding excited displacement parameters dfare reported in table 1, where we define di as:
d.I = @iiw .? i /2)“‘2wqo l i’
(14)
The sum over virtual states involved 32 states formed from the manifoId (pvI, v2, v3 ) so that satisfactory convergence was always obtained. The best values for D
Fig. 2. RRS profiles - continuation pf f@ 21)t:2v~:Y~+v~:v~+Y~=1:0_5:1:0_5_
1. Relative scale:
Volume SO, number 2
1 June 1981
CHEMICAL PHYSICS LETTERS Tab& 1 List of parametersa1 so1vcnt
Phonon displacements D (cm-‘)
transition energy Eoo (cm-r)
isopentane
240 520
20100 19100 20400
CSZ cyclohexane
540
acetone toluene ethyl alcohol
590 600 620
a’dl(wl)= cm+.
Fig. 3. Absorption and RRS excitationprofdes of p-carotene calculated,o experimental.Relativescares: in C&f ~r-v~:~a=l:l-O25.
20
22
=w'=*
18
20
.:&xn"
Fig_4_ RRS excitationprofiles in different solvents: (a) acetone; (h) cyclohexane;(c) ethyl alcohol; (d) toluene.
O-O
20300 19800
20500
1086 crri 1,~~(~~)=760cm-1,d3(~3)=411
and Eoo as reported in table 1 were found by fitting the calculated spectra as well as possibie to the experimental ones. Ali peculations were done with eq. (12) at zero temperature_ The absorption profile was calculated from the usual Fermi golden rule formula inviilving squares of bound and continuum FC factors
and assuming D/&T2 = 2 as this seemed to give the best reasonable fit for intensities, especially for the highorder overtone intensities [I 1 f _ First of all, we notrce &at the absorption spectra, figs. 1 and 2, are broadened by the presence of the continuum background as expected. We find that there is generally marked agreement between our model and the experimental results. We must reemphasize that exact agreement is not expected since our calculations, contrary to the experimental results, correspond to 0 K experiments, inhomogeneous broadening has nor been included and we have assumed a constant frequency for all low-frequency modes as in models of electron transfer reactions [2 3 . Nevertheless, we see that the coherent scattering from the background continuum has broadened the ex&ation profrie, giving rise as mentioned above, to an ener~~epend~nt-pseudo-bends. The effective phonon displacement values derived from fitting the experimental results are repprted in table 1. The broadening increases with the larger displacement D, in agreement with multimode calculations of Champion and Albrecht f3-4 _ As we have already shown [ 121, fhe limit D -+*O gives rise to sharp peaks whereas D + OQgives rise to an~infiitely broad spectrum.
251
Volume 80, number 2
CHEMICAL PHYSICS LET-PERS References
4. Discussion Reducing a multiphonon system to a single displaced mode with an effective drsplacement D [es_ (9)] enabled us to show analytically and numerically that concomrtant excttation of low-frequency modes ~111 broaden RRS spectra. We have thus been able to satisfactorily fit the excitatton profile in carotene (figs. l-4). The effective displacement parameters are given in table 1 and fall reasonably well with values obtained from electron transfer reactions in solution (D/ha = M2. see ref [2] )_ Our somewhat large values for-D would mdicate that we are overestimatmg this contribution to the broadening. This must be so since we have neglected thermal excitations of the phonons. assumed a mean frequency (Einstem model) and neglected also inhomogeneous broadening. For mstance, different displacements t‘or different molecufes wil! aIso cause mhomogeneous broadening so thai such broadening can also be trwted by an extra mode approach as ours *_ fn spite of these simple calculations, we must conclude that coherent excitation of the quasi-continuous background of low-frequency modes must be an mlportant broadening mechanism. We have shown that this is indeed a homogeneous type of broadening but not a hfetlme broademng since no electromc perturbations (non-adiabatic) are present m the model. The latter wrll increase the homogeneous broadening by shortening the lifetime of the resonant electronic state. The transitrons moments and consequently the scattermg intensities will strongly depend on this broadening, whereas the phonon broadening will only affect FC factors
by redrstrrbutrng
the intensxtres to the Wings of
the escitation profile Furthermore due to thermal excitations, the phonon broadenmg should be temperature dependent. Further experimental and theoretica work is therefore necessary to elucidate the difference between the above two mechanisms, which as we emphasize are both homogmecxts and will affect simultaneously RRS excitation profiles. * Referee’s comment.
252
1 June 1981
[II N. hfataga and T. Kubota, Molecular interactions and
electronic spectra (Dekker, New York, 1970). [2] 3. Ufstrup and J. Jortner, J. Chem. Phys_ 63 (1975) 4358; J. Jortner, J. Chem. Phys 64 (1976) 4860. [3] P.kf. Champion, G M Korenowski and A.C. Albrecht, Solid State Commun 32 (1979) 7. [4] P.M. Champion and A.C. Albrecht, J. Chem. Phys 71 (1979) 1110. [S] P.M. ChamPion and A.C. Albrecht, J. Chem. Phys. 72 (1980) 6498. [6] W Srebrand and M Z Zgierskr, in: Excited states, Vol 4, ed. E.C Lrm (Aeademm Press, New York, 1980). f71 W. Siebrand and %fZ. Zgierski, Chem. Phys. Letters 62 (1979) 3. 181 L.A. Natie, P. Stein and W-L. Peticolas, Chem. Phys. Letters 12 (1971) 13i PI S. Kobmata, Bull. Chem. Sot Japan 46 (1973) 3636. 1101 M.L. Smk and A.D. Bandrauk, Chem. Phys. 33 (1978) 205, Chem. Phys. Letters 49 (1977) 508, K. Kodama and Y. Mori, hfof. Phys. 38 (1979) 1015. Llll K. Kodama and Y _hfori, Chem. Phys. 46 (1980) 37 I. [I21 K. Kodama and A D. Bandrauk, Chem. Phys. 57 (1981) 461. 1131 M. Tasumi, F. Inag& and T. Miyazawa. Chem. Phys. Letters 22 (1973) 515. [I41 A Warrhel and P. Dauber, J. Chem. Phys. 66 (1977) 5477. [ISI A.V. Lukashin and M D Frank-Kamenetskii, Chem. Phys. 35 (1978) 469. t161 W. Srebrand and hf.Z. Zgrerski, J. Chem. Phys 71 (1979) 3.561. [I71 L. Rfrnai, R-G. Kdponen and D. Gill, J. Am. Chem. Sot 92 (1970) 3824 IW F. Inagaki, M. Tasumi and T. Miyazawa, J. Mol. Spectry. 50 (1974) 286. rig1 L-A. Carreira, T C. Maguire and T B. Malloy Jr.. I. Chem. Phys 66 (1977) 2621. t201 S. Sufrj, G. Dellepiane, G. Masetti and G. Zerbi, 3. Raman Spectry. 6 (1977) 267. 1211 L A. Carreira, L P. GOSSand T.B. Malloy Jr., J. Chem. Phys. 69 (1978) 855. [22] L C. Hoskins, J. Chem Phys. 72 (1980) 4487.
80, number 2
EXClTATfON
CHEMICAL
PROFZLES
OF RESONANCE
PHYSICS
LETTERS
RA;vIAN SPECTRA
1 June
n?r CONDENSED
1981
MEDIA
Kenji KODAMA and And& D. BANDRAW Dgpartenzent de &true, Fuculte’ des sciences, Universite’ de Sherbrooke, Sherbrookeie, P Q.> Car&a JIK 2RI Received
29 December
1980,
in final form
19 februark
1981
A simple model, obtained by reducmg a multunode system to a sunpfe coordmate, is proposed to ekplak the broadened of RRS in liquid or sohd medxa. It is shown that the coherent excitation of the near continuous background of low-frequency modes can cause homogeneous broadening of RRS profiles. The model is applied to the excitation profile of p-carotene.
1. Introduction In the theory of resonance Rantan scattering one usually introduces a constant linewidth or damping f_acrGr to represent the homogeneous broadenmg of the vlbronic excited states. It is not uncommon to come 3cfoss very large r values, i.e. > 100 cm-l to explain excitation profiles and depolarization ratios. Such Iarge lmewrdths give unusually short !ifetlmes [much less than h picosecond) for resonant excited states so that one may we1I questlon the nature of the broadening. We sbafl show that part of the broadening, which is not a lifetune effect, comes from coherent scattering from the background continuum of low-frequency modas of large molecules and their medium (phonons), be it liquid or sohd_ One may look at the system as a supermolecute, i e. molecule plus solvent, The high-frequency modes will correspond to the internal modes of the molecules, whereas the how-frequency modes will correspond to the external modes, such as Vibrations of the molecule and also solvent modes. The need to incorporate external modes such as solvent modes is well known for absorption and emission spectra E1 f and electron transfer reactions [2] . It would seem natural to try to explain the dependence of RRS excitation profiles on the difference of configuration of these external modes in the ground and excited states. These low-frequency modes may also involve some intramolecular modes. Champion and Albrecht [3--51 have recently considered contnbutions from 248
other internal modes to the excitation profiles of highfrequency vibrations by assuming displaced harmonic oscillators and introducing also damping factors. Our model thou& similar in spirit will focus on the contribution of the quasi-continuous background of low-frequency external and internal modes to the excitation profiles of high-frequency modes.
2. Model In order to describe resonance Raman scattering in a medium we assume one resonance intermediate electronic state, we shall ignore vibronic couplings such as Herzberg-Teller, Born-Oppenheimer couplings [6], Duschinsky effect f7] etc. and shall adopt the displaced harmonic oscillator model for excited internal f8,9] as weil as external modes (21. The only perturbation present is the radiative transition between the ground and intermediate states. Under the above assumptions the hamiltonians for the ground and excited states, HP and He, are given as follows:
0 009-2614/81/0000-00~0/$02.50
(1) 0 North-Holland
publishing Company
Volume 80, number 2
CiiEMiCAL
PHYSICS
where MI, Pi, Qj and S’Zjfii, pi2 (7i and w,) are the reduced mass, momentum, coordinate and angular frequency for jth phonon mode (ith vibrational mode), QF(& is the displacement of ith phonon (ith vlbrational) mode; hv, .Eg and Ee are the mcrdent photon energy and electronic energy for the ground and excited states. We assumed same frequencies in both electronic states. Then the scattering tensor ot~,~ for a transition {Oj]--{Oj] vfa _ intermediate states I t..$}for the $th phonon mode (external1 and a transition (Oi)- {Vi>via intermediates state I$> for the ith vibrational (internal) mode is proportional to the following expression:
n-coIv’) (UlfO.)
A(E’)=T;,T;;& J
,
JJ
E=hv+Eg-Ee_
(4)
J
Assuming that the low-frequency modes can be regrouped around a mean frequency SZ (Einstein model) 22 ] , we introduce a recurrent transformation of coordinates:
x
(&f;/zQr-l)
+ 3f1f2Q_) J J’
i-1
tan@j=Dj
lI.k FIB: -
Q; =Q,t
‘)i = ~M~~~J2~~l~~Q~,
+P~N~2JvM1 +MlS’Z2(Q~)
- Q;‘j2/2,
(8)
with
For large D, we can replace the displaced coordinate by a continuous Iinear potential surface which is tangent to the original potential surface at the equilibrium position of the ground state. The other modes being undisplaced, will integrate to unity in the FC elements in the scattering amplitude (4) (i.e. only O-O transitions allowed). The ~troduction of the linear potent&l to mimic the repulsive wall of the largely displaced oscillator enables us to reduce the phonon Raman amplitude A@‘) to a previously derived analytic expression [lo-121 A@‘)
= Aoo(E”)
(5)
W&I E” = hv _ Eoo - Zi~~$wi and EOo now becomes the energy for a O-O transition, i.e.
-112
I
1 June 1981
LElTERS
,
j = 2, .__, N.
6%
Thus after N - I iterations of eq. (51, Le. from1 I=2 to IV, one can reduce the N-dimensional mtdtiphonon problem in eq. (l), to that of one displaced and N - 1 undisplaced coordinates_ Thus the phonon part of eq. (1) becomes:
The Zero point energy is now necessary to defme the zero energy of the linear potential_ The scattering amplitude which now includes scattering from the cont&urn of photon modes via A,, simplifies to
249
Volume 80. number 2
CHEMICAL PHYSICS LEXTERS
1
June 1981
The characteristics of the single mode amplitude A,,, (lo), have been previously described [lO--X2]_ Thus AoO is comprised of an imaginary part Im(E”) which is lorentzian about E”, and a real part Re(E”) which is anti-~ort~tz&r, i-e_ it is zero at E” = CLHence one can rewrite the amplitude,AoO, asfi(E”)/[E” + ir,f,E”)], withp(E”) a monotonic function of energy, The total scattering amphtude now becomes:
From eqs. (12) and (13) we see that the background continuum of low-frequency modes introduces an energy-dependent hnewidth I?&!?‘) since these contribute to the resonant scattering also if the excited-state geometry is much different from the ground state. This change in geometry is represented by the total effective displacement of one mode, i.e. D = (X,-D/-)112 where l)i is the displacement of each individual low-frequency mode, see eqs. (S)-(9). We emphasize that the appearance of ri in the new expression for the amplitude (13) is not a lifetime effect, but is rather the coherent RR scattenng from the background modes.
3. Application Resonance Raman scattering of p-carotene in several solvents, which has been investigated by many authors theoreticaIly [13-l 61 and experimentaliy [I 7-221, makes it possible to compare our model with these experimental and theoretical results. Excitation profiles have been caIeuIated for the three main strong lines 1005 (v3), 1157 (~2) and 1525 (vt ) cm-1 of that moie&e. The comparisons of our caIcuIations and the abovementioned theoretical and experimental results is carried out for the excitation profiles of the ZQ , ~2, ~3,2vI, 2~2, “I + v2 and YI + ~3 transitions in isopentane solvent 1181, pl, v2 and v3 in carbon disulptide solvent (203, v2 in acetone, cyclohexane, ethyl alcohol and toluene solvents f22]. These results are shown in figs. 14. in each series of RRS profiles, the relative scale is fured by fitting the calculated profile for Q to the experimental resulr. The profiles for ul, vz, v3 and combinations and overtones were calculated using essentialIy the parameters of Srebrand and Zgierski [ 165 _Thus the frequen250
Fig. 1. Absorption and RRS excitation profiIes of p-carotene in isopentane. calculated, ---observed absorption, o e+ p&mental RRS. Relative SC&S absorption (arbitrary), vl.v~:v3=1_1_o_5_
ties were Rwl = 1525,fiw2 = 115S,fiw3 = 1005 cm-l, and the corresponding excited displacement parameters dfare reported in table 1, where we define di as:
d.I = @iiw .? i /2)“‘2wqo l i’
(14)
The sum over virtual states involved 32 states formed from the manifoId (pvI, v2, v3 ) so that satisfactory convergence was always obtained. The best values for D
Fig. 2. RRS profiles - continuation pf f@ 21)t:2v~:Y~+v~:v~+Y~=1:0_5:1:0_5_
1. Relative scale:
Volume SO, number 2
1 June 1981
CHEMICAL PHYSICS LETTERS Tab& 1 List of parametersa1 so1vcnt
Phonon displacements D (cm-‘)
transition energy Eoo (cm-r)
isopentane
240 520
20100 19100 20400
CSZ cyclohexane
540
acetone toluene ethyl alcohol
590 600 620
a’dl(wl)= cm+.
Fig. 3. Absorption and RRS excitationprofdes of p-carotene calculated,o experimental.Relativescares: in C&f ~r-v~:~a=l:l-O25.
20
22
=w'=*
18
20
.:&xn"
Fig_4_ RRS excitationprofiles in different solvents: (a) acetone; (h) cyclohexane;(c) ethyl alcohol; (d) toluene.
O-O
20300 19800
20500
1086 crri 1,~~(~~)=760cm-1,d3(~3)=411
and Eoo as reported in table 1 were found by fitting the calculated spectra as well as possibie to the experimental ones. Ali peculations were done with eq. (12) at zero temperature_ The absorption profile was calculated from the usual Fermi golden rule formula inviilving squares of bound and continuum FC factors
and assuming D/&T2 = 2 as this seemed to give the best reasonable fit for intensities, especially for the highorder overtone intensities [I 1 f _ First of all, we notrce &at the absorption spectra, figs. 1 and 2, are broadened by the presence of the continuum background as expected. We find that there is generally marked agreement between our model and the experimental results. We must reemphasize that exact agreement is not expected since our calculations, contrary to the experimental results, correspond to 0 K experiments, inhomogeneous broadening has nor been included and we have assumed a constant frequency for all low-frequency modes as in models of electron transfer reactions [2 3 . Nevertheless, we see that the coherent scattering from the background continuum has broadened the ex&ation profrie, giving rise as mentioned above, to an ener~~epend~nt-pseudo-bends. The effective phonon displacement values derived from fitting the experimental results are repprted in table 1. The broadening increases with the larger displacement D, in agreement with multimode calculations of Champion and Albrecht f3-4 _ As we have already shown [ 121, fhe limit D -+*O gives rise to sharp peaks whereas D + OQgives rise to an~infiitely broad spectrum.
251
Volume 80, number 2
CHEMICAL PHYSICS LET-PERS References
4. Discussion Reducing a multiphonon system to a single displaced mode with an effective drsplacement D [es_ (9)] enabled us to show analytically and numerically that concomrtant excttation of low-frequency modes ~111 broaden RRS spectra. We have thus been able to satisfactorily fit the excitatton profile in carotene (figs. l-4). The effective displacement parameters are given in table 1 and fall reasonably well with values obtained from electron transfer reactions in solution (D/ha = M2. see ref [2] )_ Our somewhat large values for-D would mdicate that we are overestimatmg this contribution to the broadening. This must be so since we have neglected thermal excitations of the phonons. assumed a mean frequency (Einstem model) and neglected also inhomogeneous broadening. For mstance, different displacements t‘or different molecufes wil! aIso cause mhomogeneous broadening so thai such broadening can also be trwted by an extra mode approach as ours *_ fn spite of these simple calculations, we must conclude that coherent excitation of the quasi-continuous background of low-frequency modes must be an mlportant broadening mechanism. We have shown that this is indeed a homogeneous type of broadening but not a hfetlme broademng since no electromc perturbations (non-adiabatic) are present m the model. The latter wrll increase the homogeneous broadening by shortening the lifetime of the resonant electronic state. The transitrons moments and consequently the scattermg intensities will strongly depend on this broadening, whereas the phonon broadening will only affect FC factors
by redrstrrbutrng
the intensxtres to the Wings of
the escitation profile Furthermore due to thermal excitations, the phonon broadenmg should be temperature dependent. Further experimental and theoretica work is therefore necessary to elucidate the difference between the above two mechanisms, which as we emphasize are both homogmecxts and will affect simultaneously RRS excitation profiles. * Referee’s comment.
252
1 June 1981
[II N. hfataga and T. Kubota, Molecular interactions and
electronic spectra (Dekker, New York, 1970). [2] 3. Ufstrup and J. Jortner, J. Chem. Phys_ 63 (1975) 4358; J. Jortner, J. Chem. Phys 64 (1976) 4860. [3] P.kf. Champion, G M Korenowski and A.C. Albrecht, Solid State Commun 32 (1979) 7. [4] P.M. Champion and A.C. Albrecht, J. Chem. Phys 71 (1979) 1110. [S] P.M. ChamPion and A.C. Albrecht, J. Chem. Phys. 72 (1980) 6498. [6] W Srebrand and M Z Zgierskr, in: Excited states, Vol 4, ed. E.C Lrm (Aeademm Press, New York, 1980). f71 W. Siebrand and %fZ. Zgierski, Chem. Phys. Letters 62 (1979) 3. 181 L.A. Natie, P. Stein and W-L. Peticolas, Chem. Phys. Letters 12 (1971) 13i PI S. Kobmata, Bull. Chem. Sot Japan 46 (1973) 3636. 1101 M.L. Smk and A.D. Bandrauk, Chem. Phys. 33 (1978) 205, Chem. Phys. Letters 49 (1977) 508, K. Kodama and Y. Mori, hfof. Phys. 38 (1979) 1015. Llll K. Kodama and Y _hfori, Chem. Phys. 46 (1980) 37 I. [I21 K. Kodama and A D. Bandrauk, Chem. Phys. 57 (1981) 461. 1131 M. Tasumi, F. Inag& and T. Miyazawa. Chem. Phys. Letters 22 (1973) 515. [I41 A Warrhel and P. Dauber, J. Chem. Phys. 66 (1977) 5477. [ISI A.V. Lukashin and M D Frank-Kamenetskii, Chem. Phys. 35 (1978) 469. t161 W. Srebrand and hf.Z. Zgrerski, J. Chem. Phys 71 (1979) 3.561. [I71 L. Rfrnai, R-G. Kdponen and D. Gill, J. Am. Chem. Sot 92 (1970) 3824 IW F. Inagaki, M. Tasumi and T. Miyazawa, J. Mol. Spectry. 50 (1974) 286. rig1 L-A. Carreira, T C. Maguire and T B. Malloy Jr.. I. Chem. Phys 66 (1977) 2621. t201 S. Sufrj, G. Dellepiane, G. Masetti and G. Zerbi, 3. Raman Spectry. 6 (1977) 267. 1211 L A. Carreira, L P. GOSSand T.B. Malloy Jr., J. Chem. Phys. 69 (1978) 855. [22] L C. Hoskins, J. Chem Phys. 72 (1980) 4487.