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The dushinsky effect in resonance Raman spectroscopy
CHEMICAL
VoIume 62. number 1
PHYSICS LETIERS
THE DUSHINSKY EFFECT IN RESONANCE Wlem
SIEBRAND
and hlarek Z. ZGIERSKI
I.5 March 1979
RAMAN SPECTROSCOPY
*
*
Dirisron of Chernisny. National Research CorazciI of Canada. Qrtaw, Canada, KIA OR6
ReceRed 22 October
1978; in tinaI form I I December
1975
The effect of normal-coordinate rot&ion on resonttnce Rxnrtn euxtation profiles and depolarization is mvestrgated theoretically for two nontotally symmetric modes of the ame symmetr) -
dispersion cur\es
In a moIecuIe with more than one normal mode of the same symmetry species (irreducible point group representation), these modes can mix under electronic excitation_ This mixing which can be depicted zs 3 norrx~lcoordinate rotation, is known as the Dushinsky effect [I--&] _ In this note we investigate its manifestation in resonance Raman scattering for a simple vibronic coupling model involving two nondegenerate, nontotally symmetric modes of the same symmetry_ These modes are assumed to derive their Raman activity by weakly coupling two electronic states that are energeticaIIy well separated_ Normal-coordinate rotation between the ground and resonant electronic states only is considered; that between the two coupled electronic states is neglected since the vibronic coupling is assumed to be weak. In the absence of normal-coordinate rotation, the model reduces to Sonnich LMortensen’s model [5,6] for independent nontotally symmetric modes except that nonadiabatic interactions [7,8] are included_ In that case the Raman spectrum is thus dominated by fundamentals and their excitation profiles consist of pairs of Iorentzians corresponding to the O-O uld O-l absorption bands, such that the depolarization ratio forms another Iorentzian which peaks between two such bands. We are looking for deviations from this pattern caused by the Dushinsky effect. The adiabatic ground-state potentials of the two modes labeledi = I,2 are taken to be harmomc: Eg(Q) = f
c ci$Q;
_
i
Here w,. and Qi denote frequencies responding excited state potentials
Ek(Q) = E,(O) -I- f Fc$Qf
+
and mass-weighted are
fQ1Q2,
k = tn.
Hence frequency shifts between the three electronic decoupled by the normal-coordinate rotation
Q = V(Q)a
V(Q) =
I:,”
-sm ’ ),
cos Q
normal coordmatep
(in umts tl = l), respectively.
The cor-
II.
states are excluded
tan 20 =
2f/(wz - a:),
from the model. The potentAs
(2) can be
0)
l Issued as NRCC No. I71 89. ‘CL NRCC Research Assocczte.
3
Volume
62, number
1
CHEMICAL
PHYSICS
15 hfarch 1979
LJS-I-ERS
(4) The ground and excited state wavefunctions *g = Qgl’l*U&
are written in the form
*i = *,n Ei f *tr rli,
(5)
where lu,) represents the uth harmonic oscillator wavefunction of mode I in the electronic and vi represent solutions of the coupled Schrodinger equations
ground state, and &
r(~~IT~l~,~)+E,(P)--jl~j~+~,IT~I’P,l)~~=o.
[Of?,,I TN I+,$ •t-E,(s)
-
ei]vi
-I-
(6)
= 0,
where TX E $Zp,? is the nuclear kinetic-energy operator_ We now transform the adiabatic subset {ip, ) a,] into a diabatic subset c(I?* &I$) by a unitary transformation V($) of the form (3) where the vibronic coupling paramits Q-independent compoeter 9 represents the Q-dependent component of {G,,*, @,I) and the subset {@E,@i] nent [9,iO]. We assume weak coupling, i.e. small ti such that sin $ = $ and cos I$ = I_ Accordingly we set (7)
The zeroth-order wmefunctions &G)
= Ior+
solutions of (6) $(G)
= IX’1 r.$,
obtained by setting Xl = X2 = 0 in (7)
are products of harmonic-oscillator (8)
where u and rv are vibrational quantum numbers referring to ~b,~ and anY respectively. Star:ing from this basis, we now formulate first-order corrected wavefunctions using standard Herzberg-Teller theory corrected for nonadiabatic coupling_ Thus for the lowest vibronic state of the +,n manifoId, we have [9]
(9) where E. = EJO) - Enz(O)-Ingene&,-the zeroth-order wavefunction Gm! uI u-)) will acquire correction terms ~Jw1rv7) such that ‘v1,2 = IJ~,~ -i- I ) since the coupling is assumed linear_ As a result fundamentals will dominate the F&tman spectrum_ Consider now the fundamental corresponding to IOO), + I IO&_ This scattering process involves the intermediate states 100),,r,,. IlO), n, 10X),n.,z and II I)m.lz; the 101) and 111) contributions vanish forf= 0 and thus are characteristic of the Dushinsky effect_ In the @m (or an) band region, we thus expect four rather than two peaks in the excitation profile. corresponding to the .u~LL~-u~u~ (or ula,-rv,rv,)= 00-00, 00-lO,OO-01 and 00-l 1 absorption bands. The Iast of these bands is a combination bandiocatedbetween the overtone bands 00-20 and 00-02 For consistency, we therefore include these overtone channels in the calculation. We assume symmetries such that the oniy nonvanishing matrix elements of the electric dipole operator bl arexng E <@z ~M'l@,) and yIne E (@riA$[+& The nonvanishing resonance components of the Raman scattering tensor are then
Volume 62, number 1
*oo,lO = -v
x,*gvm.g
CHEMICAL
I
coo -
+2*“A~(10112)+A~(10)10)
eoo+O1 3”‘A‘;(lOl30)
+W?
+2”13_Ai(10110)
fzo0+2WI
-a-iir +A~(l0121)
-SX-iir
s1~A‘;(10103)+A~(10i12) -I-(OOlO2j
+ 2”2A,(lOlOl) _
E~~+ZO~ -S2-iir
OOJO -X,rgYn*g
+ (IOlOl)
1.5 hfarch 1979
ir
Q -
+Ai(lOlOl)
2”‘A~(10121)
gVX
LETTERS
A~(10110) +A;(IoIOl)
(00 100)
+(00111)--
+ (00 120)
PHYSICS
17
(102)
21”Af(OO~20)+A~(OO(OO)+A~(OOI11) [
UOilO)
Af(OOll1)
coo+ 01 -Q-ir
•i-2r’~A;(oo102)
tA~(00100)
eoO+WZ--CL--C
(lob)
17
where eOOis the energy of coo, S2 the incident light frequency, and r the lorentzian linewidth, assumed to be the same for all bands. The vrbrational overlap integrals are all written in the form (u, 11, I u1 u7) and the coefficients A,: are given by for Wi 4 E. _
(11)
For harmonic oscillators, the overlap integrals in eq. (10) can be evaluated analytically [ 111; the results are listed in the appendix_ For semiquantitative purposes, convenient low-order approximations to eq. (10) can be obtained if some of the model parameters are small. Iff;lol 6 ol - w7 e wl , terms of order 0’ or higher and overIap integrais involving oscrllators differing by more than one quantum are negligrble. We can then reduce eq. (10)
to Af - AfO ,OO.lO xx y xy - *tg m.5 coo _ Q _ ir
-
A-6
Ai
00.10 _ a_VX
-‘cngYnzg
t
E
oo+Gl
(12a)
’
-Q-ir-eOOtO,-R-ir
1’
(12b)
where the difference between AT and A,r represents the nonadiabatic contribution_ It follows that nonconventional results are obtained only when both h2 and 0 differ from zero. If h, = 0. i.e., if the mode observed derives all its intensity from the Dushinsky effect, the excitation profde will show maxima for ulu2 = 00 and 01, but not for 10. The profile of the inactive mode thus reflects the vibrational spacing of the active mode. If Xl and h2o are of comparable magnitude, strong interference effects will be observed in the ul u, = 01 and 10 band regions. in the region For h, = X2@, the 00 band will be weak and the 01 and 10 bands will interfere conskrctively ~~~ + WZ < SL < coo + WI, whereas for Xl = -h,Q the reverse will hold_ Thus if wl - o2 d r”, the 01 and 10 bands will tend to merge into a single broad band for hl * X,0, but to form a well-resolved doublet for hl Z= --X&I; in the latter case, the depolarization ratio will peak between the members of this doublet. If wl - w2 is not very small, additional terms arise which turn eq. (12) into
CHEMICAL PHYSICS LE-ITERS
Volume 62, number 1
Wb) where f.f and Y are vibrational 0verIap factors such that
v/i-l = Gq - +/h=+
++
04)
< l-
of the ut u2 = 00 and 10 bands in the profile and give rise to a (usuaIIy weak) Ii band. tensor elements. They are obtained from eqs. (12) and (13) Similar formuIas can be derived for the 1,cr“*‘t} by iqterchanging the subscripts I and 2, and changing the sign of 9. These conclusions are illustrated in figs. 1 and 2 for a system characterized by f = OS, Q = x/8 (wl = 21D, w2 = I)_ The “cross-sectinns” O(a) and depolarization ratios pr(,O_) for right ang!e scattering are calculated by substi&ing the results ofeq_ (ICI) into [13-l The v terms thus affect the relative intensity
00
F&_ I_ Fz_xeitationprcf3es(soIid curves)and depokuizGion ratios(broken curves)for the high-frequency fundamental in theQ,b~dre~on_Thepodtionoftheuluz -u;r;== 00-00,00-01,00-10,00-02,00--I1and00-20absoorg tion rnzxima isindicatedby arrows_ Parameter v&es (in units w2 = 1 ~IIC&C,,Y, = I): r = 0.2. wr = 2l’*, E,, = 1% f=O&and(a)X1 =O,+ =(202)'f2;(b)& =(2&)"*,+ O;(c)x-, =(2&)1'5~2
-(23*) 6
=(Z2)"*;(d)&
‘I*, where h = h/R with R < 1.
= (23r)“*,;i;
‘0123
=
=
Q10021120
00
01 lOO2rl20
0123
Q-E00 Fig_ 2. Same as tig. 1 for the low-frequency fundamental-
Volume 62, number 1
15 March 1979
CHEMICAL PHYSICS LmERS
and using the overlap integrak Iisted in the appendix_ The 2 X 4 profiles labeled a-d differ in the relative magnitude of ‘he vibronic coupling parameters Xi and cover the cases X, = 0, X, = 0 and h, (Gl)-“’ = kX2(ZZ)-fk_ Althoughfand w1 - w2 are not very small, the results agree reasonably well with the predictions of eqs. (12) and (13). Thus fig. la (ul u2 = 10 for X, = 0) and fig. 2b (ul uZ = 01 for X, = 0) show b asicaliy Sonnich Mortensen-type doublets for modes 1 and 2, respectively, whereas fig_ lb (ul u2 = 10 for Xl = 0) and fig. 2a (uIuZ = 01 for h2 = 0) show that the forbidden mode borrows the excitation profde of the allowed mode, as predicted by eq. (12). Similarly, fig. lc (uIuZ = 10 for X1 =h2,Q>O)andfig.2d(ulu2=01 for‘A1 = +, d < 0) show weak 00 bands together with constructively interfering 10 and 01 bands, whereas fig. Id (ul u2 = 10 for h, = -X2, Q > 0) and fig. 2c (I+ u2 = 0 1 for X, % AZ, Q < 0) show strong 00 bands together with destructively interfering 10 and 01 bands, again as predicted by eq. (12). In addition, eq. (13) accounts for the intensity of the 11 band and the re!ative intensities of the CO, 01 and 01 bands. in summary, we have shown that resonance Raman excitation profiIes and their depolarization ratios provide a powerfui tool to investigate normal coordinate rotation accompanying the electronic excitation of polyatomic molecules.
Appendix Analytical expressions for the vibrational overlap iqtegrak appearing in eq_ (10) and used in figs. 1 and 2 are readily derived from the well-known recursion formulas for harmonic-oscdlator wavefunctions [ 1 l] _Together with these results, we fist the low-order approximations, valid forf,/w, < o1 - w1 < wl, which are used in eq. (12). (00l00)
= 2J5v=
1,
(OOI11)=4FLN
(10101)
= 4UV3(o,02)112(Fcos
cosQ-Fsin@)==
= 2”‘LN(2G,N’?02
-
I),
(00120)
(10121)=23’2Gi1~V3(~1,,)1’2[4F~l (10112)
*(a1
= 23’2G,1L_W(o,Z,)‘/2
[(2G,N’+
- G~)c
w$l/7-,
+ w~)~(w~w~)-‘~,
= 2”2&V(2G2.&Zl
-
-
l),
sin 0 - F cos &)(G, - 6FZ_N%5
1 ) cos Q - FN’(GI
-2GlN’OZ>(Gl
(10[30)=2Gi’LN(6~~W~)‘/7-{(2W~
- w&5@+
1,
cos 0 f (Cl
(10~03)=~V3(6+2)11’N3(l
+ 20,
lr_,i
d - Gl sin 9) = -$(wl
(10~10)=4UV3(o101)1~“(G2 (00102)
3-(f.+oz)
sin 0 -
- Z,)].
F cosd)(6GIN’&
-
l)],
sin@ - Fcosp), OS
0
+
FlV2 [(Cl
-
3_F2Ar2Wl)(G1
sin Q - F cos d)
(2F cos 0 - G, sin O)] 1,
where L = (01 F=
f(ot
w,o+p, - o,)sin
Gi = wi + 0,. + (-l)i(,, 29,
IV= (G,G,
- c+)sin’Q.
- F2)-“2.
7
VoIume 62, number I
CHEMICAL
PHYSICS LETTERS
References [ I] F. Dushinsky, Acta Physicochim. URSS I (1937) 551. [?I EL Sharf and B. Hot@ Chem. Phys. Letters 7 (1970) I37[3j GJ- Small. J- Chem- Phys54 (1971) 33CO. i41 F_ hfetz. hfJ. Robey, E-W_ ScbIag and F_ Diirr. Chem_ Phys- Letters 51 (1977) 8. [S] O_ Sonnich hforrensen. Chem. Phys. Letters 30 (1975) 106. [6] 0. Sonnich hfortenscn. Chem_ Phls_ Letters 43 (1916) 576_ [?I GJ_ SmaU and ES_ Yeung.Chem_ Phys_ 9 (1975) 379_ [S] Bf.Z_ Zgierski. Chem- Phys_ Letters 36 (1975) 390. [9] A.R Gregory, W.H. Henneker. W_ Siebrand and &f-Z_ Zgierski. J. Chem. Phys. 65 (1976) 2071. [ 101 W-H. Henneker. A-P. Penner, \V_Siebrand and hf.2. Zgierski. J. Chem. Phys. 69 (1978) 1704. [II] C_Jhnnebxk,Ph~sicaI7(195I) lOOI[ I1 J Xf.bf_ Sushchinskii. Raman spectra of moIecuIes and crystals Keter. New York. 1972).
15 &larch 1979
VoIume 62. number 1
PHYSICS LETIERS
THE DUSHINSKY EFFECT IN RESONANCE Wlem
SIEBRAND
and hlarek Z. ZGIERSKI
I.5 March 1979
RAMAN SPECTROSCOPY
*
*
Dirisron of Chernisny. National Research CorazciI of Canada. Qrtaw, Canada, KIA OR6
ReceRed 22 October
1978; in tinaI form I I December
1975
The effect of normal-coordinate rot&ion on resonttnce Rxnrtn euxtation profiles and depolarization is mvestrgated theoretically for two nontotally symmetric modes of the ame symmetr) -
dispersion cur\es
In a moIecuIe with more than one normal mode of the same symmetry species (irreducible point group representation), these modes can mix under electronic excitation_ This mixing which can be depicted zs 3 norrx~lcoordinate rotation, is known as the Dushinsky effect [I--&] _ In this note we investigate its manifestation in resonance Raman scattering for a simple vibronic coupling model involving two nondegenerate, nontotally symmetric modes of the same symmetry_ These modes are assumed to derive their Raman activity by weakly coupling two electronic states that are energeticaIIy well separated_ Normal-coordinate rotation between the ground and resonant electronic states only is considered; that between the two coupled electronic states is neglected since the vibronic coupling is assumed to be weak. In the absence of normal-coordinate rotation, the model reduces to Sonnich LMortensen’s model [5,6] for independent nontotally symmetric modes except that nonadiabatic interactions [7,8] are included_ In that case the Raman spectrum is thus dominated by fundamentals and their excitation profiles consist of pairs of Iorentzians corresponding to the O-O uld O-l absorption bands, such that the depolarization ratio forms another Iorentzian which peaks between two such bands. We are looking for deviations from this pattern caused by the Dushinsky effect. The adiabatic ground-state potentials of the two modes labeledi = I,2 are taken to be harmomc: Eg(Q) = f
c ci$Q;
_
i
Here w,. and Qi denote frequencies responding excited state potentials
Ek(Q) = E,(O) -I- f Fc$Qf
+
and mass-weighted are
fQ1Q2,
k = tn.
Hence frequency shifts between the three electronic decoupled by the normal-coordinate rotation
Q = V(Q)a
V(Q) =
I:,”
-sm ’ ),
cos Q
normal coordmatep
(in umts tl = l), respectively.
The cor-
II.
states are excluded
tan 20 =
2f/(wz - a:),
from the model. The potentAs
(2) can be
0)
l Issued as NRCC No. I71 89. ‘CL NRCC Research Assocczte.
3
Volume
62, number
1
CHEMICAL
PHYSICS
15 hfarch 1979
LJS-I-ERS
(4) The ground and excited state wavefunctions *g = Qgl’l*U&
are written in the form
*i = *,n Ei f *tr rli,
(5)
where lu,) represents the uth harmonic oscillator wavefunction of mode I in the electronic and vi represent solutions of the coupled Schrodinger equations
ground state, and &
r(~~IT~l~,~)+E,(P)--jl~j~+~,IT~I’P,l)~~=o.
[Of?,,I TN I+,$ •t-E,(s)
-
ei]vi
-I-
(6)
= 0,
where TX E $Zp,? is the nuclear kinetic-energy operator_ We now transform the adiabatic subset {ip, ) a,] into a diabatic subset c(I?* &I$) by a unitary transformation V($) of the form (3) where the vibronic coupling paramits Q-independent compoeter 9 represents the Q-dependent component of {G,,*, @,I) and the subset {@E,@i] nent [9,iO]. We assume weak coupling, i.e. small ti such that sin $ = $ and cos I$ = I_ Accordingly we set (7)
The zeroth-order wmefunctions &G)
= Ior+
solutions of (6) $(G)
= IX’1 r.$,
obtained by setting Xl = X2 = 0 in (7)
are products of harmonic-oscillator (8)
where u and rv are vibrational quantum numbers referring to ~b,~ and anY respectively. Star:ing from this basis, we now formulate first-order corrected wavefunctions using standard Herzberg-Teller theory corrected for nonadiabatic coupling_ Thus for the lowest vibronic state of the +,n manifoId, we have [9]
(9) where E. = EJO) - Enz(O)-Ingene&,-the zeroth-order wavefunction Gm! uI u-)) will acquire correction terms ~Jw1rv7) such that ‘v1,2 = IJ~,~ -i- I ) since the coupling is assumed linear_ As a result fundamentals will dominate the F&tman spectrum_ Consider now the fundamental corresponding to IOO), + I IO&_ This scattering process involves the intermediate states 100),,r,,. IlO), n, 10X),n.,z and II I)m.lz; the 101) and 111) contributions vanish forf= 0 and thus are characteristic of the Dushinsky effect_ In the @m (or an) band region, we thus expect four rather than two peaks in the excitation profile. corresponding to the .u~LL~-u~u~ (or ula,-rv,rv,)= 00-00, 00-lO,OO-01 and 00-l 1 absorption bands. The Iast of these bands is a combination bandiocatedbetween the overtone bands 00-20 and 00-02 For consistency, we therefore include these overtone channels in the calculation. We assume symmetries such that the oniy nonvanishing matrix elements of the electric dipole operator bl arexng E <@z ~M'l@,) and yIne E (@riA$[+& The nonvanishing resonance components of the Raman scattering tensor are then
Volume 62, number 1
*oo,lO = -v
x,*gvm.g
CHEMICAL
I
coo -
+2*“A~(10112)+A~(10)10)
eoo+O1 3”‘A‘;(lOl30)
+W?
+2”13_Ai(10110)
fzo0+2WI
-a-iir +A~(l0121)
-SX-iir
s1~A‘;(10103)+A~(10i12) -I-(OOlO2j
+ 2”2A,(lOlOl) _
E~~+ZO~ -S2-iir
OOJO -X,rgYn*g
+ (IOlOl)
1.5 hfarch 1979
ir
Q -
+Ai(lOlOl)
2”‘A~(10121)
gVX
LETTERS
A~(10110) +A;(IoIOl)
(00 100)
+(00111)--
+ (00 120)
PHYSICS
17
(102)
21”Af(OO~20)+A~(OO(OO)+A~(OOI11) [
UOilO)
Af(OOll1)
coo+ 01 -Q-ir
•i-2r’~A;(oo102)
tA~(00100)
eoO+WZ--CL--C
(lob)
17
where eOOis the energy of coo, S2 the incident light frequency, and r the lorentzian linewidth, assumed to be the same for all bands. The vrbrational overlap integrals are all written in the form (u, 11, I u1 u7) and the coefficients A,: are given by for Wi 4 E. _
(11)
For harmonic oscillators, the overlap integrals in eq. (10) can be evaluated analytically [ 111; the results are listed in the appendix_ For semiquantitative purposes, convenient low-order approximations to eq. (10) can be obtained if some of the model parameters are small. Iff;lol 6 ol - w7 e wl , terms of order 0’ or higher and overIap integrais involving oscrllators differing by more than one quantum are negligrble. We can then reduce eq. (10)
to Af - AfO ,OO.lO xx y xy - *tg m.5 coo _ Q _ ir
-
A-6
Ai
00.10 _ a_VX
-‘cngYnzg
t
E
oo+Gl
(12a)
’
-Q-ir-eOOtO,-R-ir
1’
(12b)
where the difference between AT and A,r represents the nonadiabatic contribution_ It follows that nonconventional results are obtained only when both h2 and 0 differ from zero. If h, = 0. i.e., if the mode observed derives all its intensity from the Dushinsky effect, the excitation profde will show maxima for ulu2 = 00 and 01, but not for 10. The profile of the inactive mode thus reflects the vibrational spacing of the active mode. If Xl and h2o are of comparable magnitude, strong interference effects will be observed in the ul u, = 01 and 10 band regions. in the region For h, = X2@, the 00 band will be weak and the 01 and 10 bands will interfere conskrctively ~~~ + WZ < SL < coo + WI, whereas for Xl = -h,Q the reverse will hold_ Thus if wl - o2 d r”, the 01 and 10 bands will tend to merge into a single broad band for hl * X,0, but to form a well-resolved doublet for hl Z= --X&I; in the latter case, the depolarization ratio will peak between the members of this doublet. If wl - w2 is not very small, additional terms arise which turn eq. (12) into
CHEMICAL PHYSICS LE-ITERS
Volume 62, number 1
Wb) where f.f and Y are vibrational 0verIap factors such that
v/i-l = Gq - +/h=+
++
04)
< l-
of the ut u2 = 00 and 10 bands in the profile and give rise to a (usuaIIy weak) Ii band. tensor elements. They are obtained from eqs. (12) and (13) Similar formuIas can be derived for the 1,cr“*‘t} by iqterchanging the subscripts I and 2, and changing the sign of 9. These conclusions are illustrated in figs. 1 and 2 for a system characterized by f = OS, Q = x/8 (wl = 21D, w2 = I)_ The “cross-sectinns” O(a) and depolarization ratios pr(,O_) for right ang!e scattering are calculated by substi&ing the results ofeq_ (ICI) into [13-l The v terms thus affect the relative intensity
00
F&_ I_ Fz_xeitationprcf3es(soIid curves)and depokuizGion ratios(broken curves)for the high-frequency fundamental in theQ,b~dre~on_Thepodtionoftheuluz -u;r;== 00-00,00-01,00-10,00-02,00--I1and00-20absoorg tion rnzxima isindicatedby arrows_ Parameter v&es (in units w2 = 1 ~IIC&C,,Y, = I): r = 0.2. wr = 2l’*, E,, = 1% f=O&and(a)X1 =O,+ =(202)'f2;(b)& =(2&)"*,+ O;(c)x-, =(2&)1'5~2
-(23*) 6
=(Z2)"*;(d)&
‘I*, where h = h/R with R < 1.
= (23r)“*,;i;
‘0123
=
=
Q10021120
00
01 lOO2rl20
0123
Q-E00 Fig_ 2. Same as tig. 1 for the low-frequency fundamental-
Volume 62, number 1
15 March 1979
CHEMICAL PHYSICS LmERS
and using the overlap integrak Iisted in the appendix_ The 2 X 4 profiles labeled a-d differ in the relative magnitude of ‘he vibronic coupling parameters Xi and cover the cases X, = 0, X, = 0 and h, (Gl)-“’ = kX2(ZZ)-fk_ Althoughfand w1 - w2 are not very small, the results agree reasonably well with the predictions of eqs. (12) and (13). Thus fig. la (ul u2 = 10 for X, = 0) and fig. 2b (ul uZ = 01 for X, = 0) show b asicaliy Sonnich Mortensen-type doublets for modes 1 and 2, respectively, whereas fig_ lb (ul u2 = 10 for Xl = 0) and fig. 2a (uIuZ = 01 for h2 = 0) show that the forbidden mode borrows the excitation profde of the allowed mode, as predicted by eq. (12). Similarly, fig. lc (uIuZ = 10 for X1 =h2,Q>O)andfig.2d(ulu2=01 for‘A1 = +, d < 0) show weak 00 bands together with constructively interfering 10 and 01 bands, whereas fig. Id (ul u2 = 10 for h, = -X2, Q > 0) and fig. 2c (I+ u2 = 0 1 for X, % AZ, Q < 0) show strong 00 bands together with destructively interfering 10 and 01 bands, again as predicted by eq. (12). In addition, eq. (13) accounts for the intensity of the 11 band and the re!ative intensities of the CO, 01 and 01 bands. in summary, we have shown that resonance Raman excitation profiIes and their depolarization ratios provide a powerfui tool to investigate normal coordinate rotation accompanying the electronic excitation of polyatomic molecules.
Appendix Analytical expressions for the vibrational overlap iqtegrak appearing in eq_ (10) and used in figs. 1 and 2 are readily derived from the well-known recursion formulas for harmonic-oscdlator wavefunctions [ 1 l] _Together with these results, we fist the low-order approximations, valid forf,/w, < o1 - w1 < wl, which are used in eq. (12). (00l00)
= 2J5v=
1,
(OOI11)=4FLN
(10101)
= 4UV3(o,02)112(Fcos
cosQ-Fsin@)==
= 2”‘LN(2G,N’?02
-
I),
(00120)
(10121)=23’2Gi1~V3(~1,,)1’2[4F~l (10112)
*(a1
= 23’2G,1L_W(o,Z,)‘/2
[(2G,N’+
- G~)c
w$l/7-,
+ w~)~(w~w~)-‘~,
= 2”2&V(2G2.&Zl
-
-
l),
sin 0 - F cos &)(G, - 6FZ_N%5
1 ) cos Q - FN’(GI
-2GlN’OZ>(Gl
(10[30)=2Gi’LN(6~~W~)‘/7-{(2W~
- w&5@+
1,
cos 0 f (Cl
(10~03)=~V3(6+2)11’N3(l
+ 20,
lr_,i
d - Gl sin 9) = -$(wl
(10~10)=4UV3(o101)1~“(G2 (00102)
3-(f.+oz)
sin 0 -
- Z,)].
F cosd)(6GIN’&
-
l)],
sin@ - Fcosp), OS
0
+
FlV2 [(Cl
-
3_F2Ar2Wl)(G1
sin Q - F cos d)
(2F cos 0 - G, sin O)] 1,
where L = (01 F=
f(ot
w,o+p, - o,)sin
Gi = wi + 0,. + (-l)i(,, 29,
IV= (G,G,
- c+)sin’Q.
- F2)-“2.
7
VoIume 62, number I
CHEMICAL
PHYSICS LETTERS
References [ I] F. Dushinsky, Acta Physicochim. URSS I (1937) 551. [?I EL Sharf and B. Hot@ Chem. Phys. Letters 7 (1970) I37[3j GJ- Small. J- Chem- Phys54 (1971) 33CO. i41 F_ hfetz. hfJ. Robey, E-W_ ScbIag and F_ Diirr. Chem_ Phys- Letters 51 (1977) 8. [S] O_ Sonnich hforrensen. Chem. Phys. Letters 30 (1975) 106. [6] 0. Sonnich hfortenscn. Chem_ Phls_ Letters 43 (1916) 576_ [?I GJ_ SmaU and ES_ Yeung.Chem_ Phys_ 9 (1975) 379_ [S] Bf.Z_ Zgierski. Chem- Phys_ Letters 36 (1975) 390. [9] A.R Gregory, W.H. Henneker. W_ Siebrand and &f-Z_ Zgierski. J. Chem. Phys. 65 (1976) 2071. [ 101 W-H. Henneker. A-P. Penner, \V_Siebrand and hf.2. Zgierski. J. Chem. Phys. 69 (1978) 1704. [II] C_Jhnnebxk,Ph~sicaI7(195I) lOOI[ I1 J Xf.bf_ Sushchinskii. Raman spectra of moIecuIes and crystals Keter. New York. 1972).
15 &larch 1979