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On the determination of short lifetimes of vibronic states from resonance Raman excitation profiles
Volume 6.5, number 3
ON THE DETERMINATION
CHEMICAL
PHYSICS LETTERS
I September
1979
OF SHORT LIFETIMES OF VIBRONIC STATES
FROM RESONANCE RAMAN EXCITATION PROFILES A-V_ LUKASHIN Institute of hfotectdar Genenks. Acade:n_v of Sciexces of USSR. bfoscow 123182, USSR
Received 5 April I979
The rela~ational width (IT) of an excited state and the soIventinduced broadening (o) produce a considerable and nonadditire effect on the intendty distribution inside the resonance Raman excitation profiles. An approach based on the generating function method is proposed which makes it possible to separate these effects and determine the r and G vah%s. It Is shown that the fun moment .Wr of an excitation profiR depends on P alone_ The wrriance of the profde is a sum of o2 and a function of J? alone. A theoreticaI dependence of&f,(p) derived under rery general assumptions is shown to be a sharp and monotonous function of p. This dependence makes it possible to determine the value of P from e\perimentaIIy avaikrble value of Mr. The requirements are formulated that shouId be met by e\perimentaI data for a practical reahzaticn of the proposed approach_
The intensity distribution inside the resonance Raman (RR) spectra and the shape of the Raman excitation profile (REP) contain a lot of information about excited states. First of all these data chancterize the changes in the potential surface under electronic excitation (nuclear equilibrium positions, frequencies and fomrs of modes) and the lifetimes of excited vibronic states. The determination of these characteristics from experimental data is not a simple task (see, e.g. refs. [I-6] ) because the expressions determining the RR spectra as well as REP have a complex structure even under the simplest assumptions_ Allowance for the solvent effects entails an additional difficulty [2,6] _ Denote by F the relaxation width of a vibronic state end by II the value of solvent-induced broadening. In contrast with the case of absorption spectra the shape of REP depends on both F and CTin a compIex way [2,6] _ This fact complicates the task of interpreting the experimental REP data but at the same time it may prove to be of crucial importance for the determination of the F values, as was pointed out originally by Penner and Siebrand [2] _ These authors have proposed to determine the F values by adjusting the parameters to achieve the “best tit” of the theoretical REP to the experimental one.
In the present paper a somewhat different approach to the determination of F values from experimental REP data is proposed. First of all assume that the parnmeters characterizing the changes in equilibrium nuclear positions under electronic excitation have been determined_ For this the method proposed in our previous paper [6] seems to be most suitabIe_ This method makes it possible tc determine these parameters only from experimental RR spectra obtained for the case when the frequency of external radiation tie is close to the frequency of pure electronic transition w,s_ It is important that in many cases the frequencies of active vibrations are much higher than the F and CTvalues_ & shown in ref_ [6], in such cases the vaiues of the determined changes in the equilibrium nuclear positions do not depend on the particular I’ and CJv&es. Suppose the former parameters are known. Then the F and u values may be detemrined from the experimental REP using the method of moments. Let Aw be the mean value of the difference w. - wuo for an ensemble of molecules in solution. The REP, y-e_ the dependence of the intensity of a line in RR spectrum on Aw is characterized by (see refs.
EW31): 527
V01ume 65. number 3
CHEMICAL
I September1979
PHYSICS LJZITERS
FI;“(t- JA)= exp(+iKV)
+ P))1In) (71
X
(1) This equation is valid within the Condon approximation- C is 5 we& function of 5w. 10) and In>are wavefunctions of the initial and final states of the mo:ecuk They are eigenfunctions for the ground state hamiltonian HP;_The wavefunctions Ik’) and Im’> correspond to the upper state hamikonkm H,_ It should be recalled :hat the parameters of hamiltonians Hg and H, are supposed to be known_ The 2 function is:
Defining the moments of intensity distribution within
RFP in the IISLI~~ way: mp=
2&.(4G)
r&W)
+gz(4w)l (3)
where
Using the integral representation for the gaussianexponential functions in eq. (4) and for the denominator in cq_ (3) as well as the rules of matrix summation one can rewrite the distribution S,(fiw) as: S,(4o)
=C /
d t exp(hf)
S&),
-_
(5)
where S,&) is a generatiag function and S,#)
= exp(- &?r3_) /
dp exp(--‘>r&
0 X [F;r(t, B)+- F,;k and 528
(6)
~11,
(4wY’S,z(4w)
d(h),
(8)
one can obtain for the three fundamental characteristics of the distribution, the integral intensity, nfo, the ccntre of band,fift =ml/mo, and the variance,Mz = m,/mo - Mt, the following expressions: mo=C
where ek. and E,. are the eigenvahrescorresponding to eigenfunctions IX-‘>and Im’}, p(v, 40) is the distribution function of the difference w. - wu,, relative to its mean vaiue 4~_ In ref_ [2] this distribution was assumed to be Iorentzian- As was discussed in ref- [71 it is more natural to approximate the p(~, 4w) function by a gausskm distribution wivith the variance o’In this case one obtains [61 I ani = %i - ek. - 2ir
s -a0
s dfl exp(-2l%)[Fi
+ F; 1 rsOt
0
00)
M
s
dj.texp(-3,rfl)
0 IV, = V
X
o2+ &
jdp
exp(-2Q)
0 anr;;:a+,-
_--C at2
(9)
at2 It=O
01) a;_
It is essential that the centre of REP (Mt) is a function of I’ alone and the varianceM? is the sum of two terms, uz and a function of r alone. The principal idea of the present paper is to expIoit these facts. First, within the framework of a harmonic approximation using the method of coherent states [S] one can obtain an analytical expression for the F1:(t, p) functions_ Secondly, all parameters entering the expression could be determined from independent data as mentioned above_ So one can overtly perform an integration in eqs. (9) and (10) and obtain theMt value as a function of r_ FmaiIy, one can determine the real r v&e comparing the theoretical Mt (r) function with the experimental M1 value- Calculating the second and third terms of eq. (1 I) from the determined r value and comparing the result with experimental M2 value one can determine the a value_ Note that the simpiicity of the above equations is largely due to the assumption that all vibronic states
Volume 6.5, number 3
CHEMICAL PHYSICS LEl-lXRS
have the same lifetimes_ It is not the case under any real conditions_ However one can expect that variatiops in these values are not large as compared with their absolute values and the proposed method provides a reasonable estimation of some “mean” I? value. Our approach could be regarded as practical only if Ml(r) is a sufficientIy smooth and monotonous function of J?. To clarify this point Iet us consider a simple model. We neglect the changes in vibrational frequencies under electronic excitation and the Dushinshy effect, i-e. electronic excitation is assumed to cause changes (4di) only in normal mode equilibrium positions. Moreover we restrict ourselves to the simplest case of REP for one of the fundamental lines of the RR spectrum i-e_ we will assume that 1~) = I Itt) = IO, ___,la, .__X Then one has: [F;a c FJ
X exp
X
t=O = 4Y;(l
-2 1
C Yf(1 i
[qa+FJ,=,,
--OS
52&
1
- cos X+.C)
(12)
,
1 September 1979
which the experimental data should meet to make an application of our approach possible. (1) The intensity distribution in the RR spectrum in the region of w. z o;, should be measured at least within the fundamental r;gion. An extension of the data to the overtone and combination region is also desirable. If this is not the case, the fluorescence and/ or absorption spectra should be determined_ All this is needed to determine parameters characterizhig the change in the potential surface of a molecule under electronic excitation (for details see ref. [6]). (2) Accurate measurements of REP for one or several of the most intense (at wO = wUg) lines in the fundamental region are required_ It is essential to vary the w. value over the whoIe range where the molecule has a nonzero absorption. Only if this requirement is met can one obtain a reliable estimation of the Ai, and M, values. Altl;ough contemporary experin:entaI techniques make it possible to perform measurements meeting the above requirements, such measurements have not been performed for any molecule. The results of Inagaki et al_ [I] remain the most informative in this respect.
(13
where summation is performed over all active modes, Yi = 4cl,(MiS2i/2fi)112, Mi and Qi are the modes’ masses and frequencies_ To perform the integration in eqs. (9) and (10) using the functions from (12) and (I 3) one needs the YF and -0-i values. Here we turn to a specific example of o-carotene which has three active modes with frequencies -O-l = 1125 cm-I_ Q7 = 1155 cm-1 and -0-3 = 1005 cm-* [ I] and with dimensionless changes of equilibrium positions Yf = O-6, Y$ = O-6, Yz = 0.3 [6] _ The calculated M,(r) functions for three lines situated in the fundamental region are shown in fig. 1_ One can see that theMI functions for 0, and !2-, are steep and monotonous for all the r values. By contrast, the Ml(r) function for X23 is not monotonous_ This is due to the comparatively small value of Yg_ In general one can expect the steepest Ml(r> function for a mode with the largest Y2 value, i.e. (see ref. [6]) the mode having the highest intensity in the fundamental region of the RR spectrum in the case of w. = w,,. In conclusion let us formulate the requirgments
IO00
1
0
I 250
500
750
r(clll-l
Fig. I. Drpendznce of the resonmce R.m~m ewitation profde’s c*ntre position JI, on the r v.due calculated by eqs. ;9). (10). (12) md (13) for the three-osctitor model of$cxotene (see the text) for fundanentai lines R, (a), 1?2 (b1 and fi3
Cd-
529
Volume 65, number 3
CHEMICAL
Pff YSICS LETTERS
lhey meet the first requirement {rre ref_ E6J)_However their REP data are too scanty to make ;i Aiabie estimation of the bands’ moments possibfeReferences [ 11 F. Imgaki, 31. Tasumi aud T_ &liy?lu\tn.J. 5101.Spectry50 (1974) +S6. 131 A_ Penner amI \Y_SIebf;md, Chem_ Phys. Letters 39 (19763 II-
1 September 1979
(31 A. Nushei snd P. Dauber. J. Chem_ Phys_ 66 (1977) 5477. 141 S. Sufr;i. G, Detlepiane. G_ Masetti and G_ Zerbi, J- Ramm Spectry. 6 (1977) 267_ 151 _ R_ Liang, 0. Schnepp and A_ Wu-shel. Chem. Phys. 34 (1978) ii. __ (6J A-V- Lukashin and M-D_ Frank-Kamenetskii, Chem.. Pbys. 35 (1978) 469.. 171 31.d_ l&k-Kamenetskii and A-V_ Lukashin. Usp. I%. Nwk I16 (1975) 193 [English tmusl. Soviet P&s. Us~ekhi I8 (1976) 39I1_ [Si RJ.CIauber.Phys- Rev_ 13lf1963) 2766_
ON THE DETERMINATION
CHEMICAL
PHYSICS LETTERS
I September
1979
OF SHORT LIFETIMES OF VIBRONIC STATES
FROM RESONANCE RAMAN EXCITATION PROFILES A-V_ LUKASHIN Institute of hfotectdar Genenks. Acade:n_v of Sciexces of USSR. bfoscow 123182, USSR
Received 5 April I979
The rela~ational width (IT) of an excited state and the soIventinduced broadening (o) produce a considerable and nonadditire effect on the intendty distribution inside the resonance Raman excitation profiles. An approach based on the generating function method is proposed which makes it possible to separate these effects and determine the r and G vah%s. It Is shown that the fun moment .Wr of an excitation profiR depends on P alone_ The wrriance of the profde is a sum of o2 and a function of J? alone. A theoreticaI dependence of&f,(p) derived under rery general assumptions is shown to be a sharp and monotonous function of p. This dependence makes it possible to determine the value of P from e\perimentaIIy avaikrble value of Mr. The requirements are formulated that shouId be met by e\perimentaI data for a practical reahzaticn of the proposed approach_
The intensity distribution inside the resonance Raman (RR) spectra and the shape of the Raman excitation profile (REP) contain a lot of information about excited states. First of all these data chancterize the changes in the potential surface under electronic excitation (nuclear equilibrium positions, frequencies and fomrs of modes) and the lifetimes of excited vibronic states. The determination of these characteristics from experimental data is not a simple task (see, e.g. refs. [I-6] ) because the expressions determining the RR spectra as well as REP have a complex structure even under the simplest assumptions_ Allowance for the solvent effects entails an additional difficulty [2,6] _ Denote by F the relaxation width of a vibronic state end by II the value of solvent-induced broadening. In contrast with the case of absorption spectra the shape of REP depends on both F and CTin a compIex way [2,6] _ This fact complicates the task of interpreting the experimental REP data but at the same time it may prove to be of crucial importance for the determination of the F values, as was pointed out originally by Penner and Siebrand [2] _ These authors have proposed to determine the F values by adjusting the parameters to achieve the “best tit” of the theoretical REP to the experimental one.
In the present paper a somewhat different approach to the determination of F values from experimental REP data is proposed. First of all assume that the parnmeters characterizing the changes in equilibrium nuclear positions under electronic excitation have been determined_ For this the method proposed in our previous paper [6] seems to be most suitabIe_ This method makes it possible tc determine these parameters only from experimental RR spectra obtained for the case when the frequency of external radiation tie is close to the frequency of pure electronic transition w,s_ It is important that in many cases the frequencies of active vibrations are much higher than the F and CTvalues_ & shown in ref_ [6], in such cases the vaiues of the determined changes in the equilibrium nuclear positions do not depend on the particular I’ and CJv&es. Suppose the former parameters are known. Then the F and u values may be detemrined from the experimental REP using the method of moments. Let Aw be the mean value of the difference w. - wuo for an ensemble of molecules in solution. The REP, y-e_ the dependence of the intensity of a line in RR spectrum on Aw is characterized by (see refs.
EW31): 527
V01ume 65. number 3
CHEMICAL
I September1979
PHYSICS LJZITERS
FI;“(t- JA)= exp(+iKV)
+ P))1In) (71
X
(1) This equation is valid within the Condon approximation- C is 5 we& function of 5w. 10) and In>are wavefunctions of the initial and final states of the mo:ecuk They are eigenfunctions for the ground state hamiltonian HP;_The wavefunctions Ik’) and Im’> correspond to the upper state hamikonkm H,_ It should be recalled :hat the parameters of hamiltonians Hg and H, are supposed to be known_ The 2 function is:
Defining the moments of intensity distribution within
RFP in the IISLI~~ way: mp=
2&.(4G)
r&W)
+gz(4w)l (3)
where
Using the integral representation for the gaussianexponential functions in eq. (4) and for the denominator in cq_ (3) as well as the rules of matrix summation one can rewrite the distribution S,(fiw) as: S,(4o)
=C /
d t exp(hf)
S&),
-_
(5)
where S,&) is a generatiag function and S,#)
= exp(- &?r3_) /
dp exp(--‘>r&
0 X [F;r(t, B)+- F,;k and 528
(6)
~11,
(4wY’S,z(4w)
d(h),
(8)
one can obtain for the three fundamental characteristics of the distribution, the integral intensity, nfo, the ccntre of band,fift =ml/mo, and the variance,Mz = m,/mo - Mt, the following expressions: mo=C
where ek. and E,. are the eigenvahrescorresponding to eigenfunctions IX-‘>and Im’}, p(v, 40) is the distribution function of the difference w. - wu,, relative to its mean vaiue 4~_ In ref_ [2] this distribution was assumed to be Iorentzian- As was discussed in ref- [71 it is more natural to approximate the p(~, 4w) function by a gausskm distribution wivith the variance o’In this case one obtains [61 I ani = %i - ek. - 2ir
s -a0
s dfl exp(-2l%)[Fi
+ F; 1 rsOt
0
00)
M
s
dj.texp(-3,rfl)
0 IV, = V
X
o2+ &
jdp
exp(-2Q)
0 anr;;:a+,-
_--C at2
(9)
at2 It=O
01) a;_
It is essential that the centre of REP (Mt) is a function of I’ alone and the varianceM? is the sum of two terms, uz and a function of r alone. The principal idea of the present paper is to expIoit these facts. First, within the framework of a harmonic approximation using the method of coherent states [S] one can obtain an analytical expression for the F1:(t, p) functions_ Secondly, all parameters entering the expression could be determined from independent data as mentioned above_ So one can overtly perform an integration in eqs. (9) and (10) and obtain theMt value as a function of r_ FmaiIy, one can determine the real r v&e comparing the theoretical Mt (r) function with the experimental M1 value- Calculating the second and third terms of eq. (1 I) from the determined r value and comparing the result with experimental M2 value one can determine the a value_ Note that the simpiicity of the above equations is largely due to the assumption that all vibronic states
Volume 6.5, number 3
CHEMICAL PHYSICS LEl-lXRS
have the same lifetimes_ It is not the case under any real conditions_ However one can expect that variatiops in these values are not large as compared with their absolute values and the proposed method provides a reasonable estimation of some “mean” I? value. Our approach could be regarded as practical only if Ml(r) is a sufficientIy smooth and monotonous function of J?. To clarify this point Iet us consider a simple model. We neglect the changes in vibrational frequencies under electronic excitation and the Dushinshy effect, i-e. electronic excitation is assumed to cause changes (4di) only in normal mode equilibrium positions. Moreover we restrict ourselves to the simplest case of REP for one of the fundamental lines of the RR spectrum i-e_ we will assume that 1~) = I Itt) = IO, ___,la, .__X Then one has: [F;a c FJ
X exp
X
t=O = 4Y;(l
-2 1
C Yf(1 i
[qa+FJ,=,,
--OS
52&
1
- cos X+.C)
(12)
,
1 September 1979
which the experimental data should meet to make an application of our approach possible. (1) The intensity distribution in the RR spectrum in the region of w. z o;, should be measured at least within the fundamental r;gion. An extension of the data to the overtone and combination region is also desirable. If this is not the case, the fluorescence and/ or absorption spectra should be determined_ All this is needed to determine parameters characterizhig the change in the potential surface of a molecule under electronic excitation (for details see ref. [6]). (2) Accurate measurements of REP for one or several of the most intense (at wO = wUg) lines in the fundamental region are required_ It is essential to vary the w. value over the whoIe range where the molecule has a nonzero absorption. Only if this requirement is met can one obtain a reliable estimation of the Ai, and M, values. Altl;ough contemporary experin:entaI techniques make it possible to perform measurements meeting the above requirements, such measurements have not been performed for any molecule. The results of Inagaki et al_ [I] remain the most informative in this respect.
(13
where summation is performed over all active modes, Yi = 4cl,(MiS2i/2fi)112, Mi and Qi are the modes’ masses and frequencies_ To perform the integration in eqs. (9) and (10) using the functions from (12) and (I 3) one needs the YF and -0-i values. Here we turn to a specific example of o-carotene which has three active modes with frequencies -O-l = 1125 cm-I_ Q7 = 1155 cm-1 and -0-3 = 1005 cm-* [ I] and with dimensionless changes of equilibrium positions Yf = O-6, Y$ = O-6, Yz = 0.3 [6] _ The calculated M,(r) functions for three lines situated in the fundamental region are shown in fig. 1_ One can see that theMI functions for 0, and !2-, are steep and monotonous for all the r values. By contrast, the Ml(r) function for X23 is not monotonous_ This is due to the comparatively small value of Yg_ In general one can expect the steepest Ml(r> function for a mode with the largest Y2 value, i.e. (see ref. [6]) the mode having the highest intensity in the fundamental region of the RR spectrum in the case of w. = w,,. In conclusion let us formulate the requirgments
IO00
1
0
I 250
500
750
r(clll-l
Fig. I. Drpendznce of the resonmce R.m~m ewitation profde’s c*ntre position JI, on the r v.due calculated by eqs. ;9). (10). (12) md (13) for the three-osctitor model of$cxotene (see the text) for fundanentai lines R, (a), 1?2 (b1 and fi3
Cd-
529
Volume 65, number 3
CHEMICAL
Pff YSICS LETTERS
lhey meet the first requirement {rre ref_ E6J)_However their REP data are too scanty to make ;i Aiabie estimation of the bands’ moments possibfeReferences [ 11 F. Imgaki, 31. Tasumi aud T_ &liy?lu\tn.J. 5101.Spectry50 (1974) +S6. 131 A_ Penner amI \Y_SIebf;md, Chem_ Phys. Letters 39 (19763 II-
1 September 1979
(31 A. Nushei snd P. Dauber. J. Chem_ Phys_ 66 (1977) 5477. 141 S. Sufr;i. G, Detlepiane. G_ Masetti and G_ Zerbi, J- Ramm Spectry. 6 (1977) 267_ 151 _ R_ Liang, 0. Schnepp and A_ Wu-shel. Chem. Phys. 34 (1978) ii. __ (6J A-V- Lukashin and M-D_ Frank-Kamenetskii, Chem.. Pbys. 35 (1978) 469.. 171 31.d_ l&k-Kamenetskii and A-V_ Lukashin. Usp. I%. Nwk I16 (1975) 193 [English tmusl. Soviet P&s. Us~ekhi I8 (1976) 39I1_ [Si RJ.CIauber.Phys- Rev_ 13lf1963) 2766_