On the vibronic theory of resonance Raman scattering

~

Solid State Communications, Vol.32, pp.7-12. Pergamon Press Ltd. 1979. Printed in Great B£1taln.

ON THE VIBRONICTHEORYOF RESONANCERAMANSCATTERING P.M. Champion, G.M. Korenowski+and A.C. Albrecht Chemistry Department, Cornell University Ithaca, New York 14853

Resonance Raman e x c i t a t i o n p r o f i l e s of t o t a l l y symmetric vibrational modes are investigated using a model that a n a l y t i c a l l y includes the complete subspace of Franck-Condon active vibrations associated with each intermediate electronic state. This model i s used to f i t data obtained in resonance with the motet band of cytochrome c. The exc i t a t i o n p r o f i l e s are asymmetric and peak d i s t i n c t i v e l y to the blue of the Soret absorption maximum. Large damping factors and/or i n homogeneous s i t e d i s t r i b u t i o n s by themselves cannot account for the observed data. The theoretical results imply that the excited state l i f e t i m e associated with the Soret band of ferrocytochrome c has a lower l i m i t on the order of SO fs and t h a t , compared to the ferrous form, the f e r r i c cytochrome has a larger x-y s p l i t t i n g and a shorter lifetime. In contrast to the large multidimensional heme system, we also present the results of a simple model calculation applicable to small e r molecules. A two dimensional subspace is explored, where Raman Franck-Condon (RFC) and Franck-Condon (FC) factors are calculated for d i f f e r e n t potential energy surface parameters. Under certain conditions the RFC based scattering p r o f i l e s of one v i b r a t i o n are strongly coupled to the FC based behavtour of the other v i b r a t i o n . Rather complex p r o f i l e s are then predicted even for the simple twodimensional case.

The vlbrontc theory of Raman scattering natu r a l l y i d e n t i f i e s two basic sources of scattering. These have been termed "A-term" and "B-term" actt v t t y 1. The a c t i v i t y associated with an A-term involves resonances with individual zero-order electronic states and derives i n t e n s i t y from Raman Franck-Condon (RFC) overlaps a r i s i n g from s h i f t s in the excited state equilibrium position and/or changes in the excited state potential energy surface. Reman a c t i v i t y associated with modes i n v o l ved in vtbrontc mixing of two zero order electronic states i s c l a s s i f i e d as B-term a c t i v i t y . The amplitude o f the B-term modes depends upon the product of the t r a n s i t i o n dipole moments o f the electronic states being mixed, while the amplitude o f the A-term modes depends upon the square of the t r a n s i t i o n moment of the individual e l e c t ronic state in resonance. In the following discussion, we focus on the resonance Raman scattering i n t e n s i t y associated wtth A-term a c t i v i t y , paying special attention to the multidimensional nuclear subspace that must be included in the sum over the resonant e l e c t - ' ~ i c states. In order to demonstrate the influence of t h i s subspace on the theoretical Raman e x c i t a t i o n prof i l e , we attempt to f i t data obtained in resonance with the intense n e a r - u l t r a v i o l e t "Soret" absorption band of cytochrome c ~. Neglect of the nuc-

lear coordinate space generates theoretical excit a t i o n p r o f i l e s and absorption bands that are highly symmetric and that do not compare well with the data; inclusion of t h i s subspace, however, results in very good agreement with the data. In contrast to the large multidimensional heme chromophore of cytochrome c, we also present the results of a simple,model calculation applicable to smaller molecules °. We begin our discussion with the well-known 1 expression for the t o t a l l i g h t scattered by the mn'th t r a n s i t i o n in a molecule, averaged over a l l o r i e n t a t i o n s , into a Raman band centered at frequency ~: Im, n = I o ~ e 2 / , c )

2 v 4 ~ol(=pa)m,n 12.

(1)

Thg quantity I 0 is the incident power flux(Wcm2i, (eZ/Bc) is the fine structure constant and ~Qo represents the p,oth (p,o=x,y,z) component o~ the molecular p o l a r i z a b i l i t y tensor. The subscripts in eq. (1) r e f e r to the i n i t i a l and f i n a l states of the molecule ( i . e . hv=hvo-(En-Em); vo=incident laser frequency). The molecular p o l a r i z a b i l i t y tensor i s often expressed as a sum over v i r t u a l molecular eigenstates" and, t f the e x c i t a t i o n f r e quency is near resonance with a set of molecular states le> Width, r e) we can w r i t e :

Research supported through the National I n s t i t u t e s of Health AM2037g and the Materials Science Center of Cornell University. Present address Chemistry Department, Columbia Untverslty, New York, New York 10027.

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THE VIBRONIC THEORY OF RESONANCE RAHAN SCATTERING

8

(2) (%o)m,n= ~ Ee-Em'hvo-lre Within the zeroth order Born-Oppenheimer approximation the molecular eigenstate is written as a product of functions in electronic and nuclear subspaces. The l a t t e r is factored into the eigenstate of the Ramanactive mode, Iv>, and the eigenstate, Iv'>, reflecting the remaining multidimensional nuclear coordinate space. Since we l i m i t our discussion to (RFC allowed) A-term scattering, we evaluate the electronic eigenfunctions at equilibrium nuclear configurations. Thus, for fundamental (Av=l) scattering from the cold ~T=OK, Em=O) ground state we can rewrite eq.(2) as::

where p(¢) Is the density of states in v' space. The summation over v' space Is now completed by performing the integral over ¢ from 0 to =. Thus we have: = KP°(e'v)F~"(¢)P="(¢)dc (=Pa)O,l=~v ~ Aev + c : - i t e -"

(7)

where we have been careful to note how the average FC factors and densities of states in v' space, in general, can depend on the virtual parent states leO)Iv>. The product of FC factors with the density of states function In eq.(7) is quite similar to the product one might describe for constructing

(Mo)~,#(Mp)eOgO Il z Aev + cv, - irevv, In eq.(3) we have s p l i t the sum over intermediate molecular states into two parts. The f i r s t part contains the usual sum over electronic and Raman vibrational states; the second part contains the sum over excitations of the remaining modes in the intermediate state. Here we also associate an energy, t v ' , with each configuration of the remaining (v') vibrations. We set ev,=O when Iv'>=IO'>. The quantity Aev is simply the electronic plus Ramanvibrational energy in the Intermedlatestate, minus the incident laser energy:

an absorption band except that the Ramanmode under study has been removed for special treatment in eq.(7) and is being summed over explic i t l y . This fact allows us to make some reasona~le assumptions about the functional form of Fev(¢)Pev(¢). For the multidimensional hamesystom a truncated Lorentztan appears to work quite well in f i t t i n g the absorption band shape~. We l e t the function m a x l ~ be at ¢=% above the parent state zero-point ener~LY and l e t the HWHM o f the Lorentztan equal ~. The Lorentztan ts truncated because there are no states below the

Aev = E~ + v(hv R) - hvo

(4)

where v is the occupation number of the Raman mode in the intermediate state and v R is the excited state Raman frequency. We now reduce eq.(3) to a final functional form: KPa(e'v) l<°'Iv'>l 2 (5) (=po)O,l = ~ v ' Aev + tv' " i~e where we l e t revv' be independent of vibronic

zero-point energy. We assume that the same Lorentzian applies f o r each term in the ~v" Eq.(7) then becomes: (~po~j=N -I Z KP°(e~)f'='

ev

dE

I

o ~ e

~-Nz+~2 (8)

where N(6,¢o) is the normalization factor for t h i s function and ts equal to the area o f the truncated Lorentzian. Evaluation o f the Integrals ( y i e l d s the f i n a l r e s u l t :

6 (~pa)O,l =~vKP~(e,v) (Aev+eo-iTe) + 2 - ' ~

~} tn{

16("::)*^) +

~ u+

(9)

(Aev+¢o-ire)(Aev+¢o-ire) + 82 state. I t is important to emphasize at t h i s point that Iv'> is multidimensional and that any given energy in t h i s subspace is many-fold degenerate, i f only accidentally. We also note that in many treatments the role of the Iv'> subspace is eliminated by setting = 60,, v' y i e l d i n g a simple sum over molecular states: KP°(e,v)

(%o)°'1= ~v Aev - ire

(6)

where, as before, the KP°(e,v) are products of electronic moments (M~)gO,eO(Mp)eOgO and RFC factors, . We next propose to complete the summation over v' space in eq.(5) by integration. We proceed by f i x i n g c v, at a value e and sum over only those v' corresponding to states in v' space having energy between ¢ and c+d~. This sum is replaced by the average of a l l Franc~-Condon (FC) factors for these states )~IZ=F'(~), times the number of states in this interval, p(¢)dc,

where ee is the phase angle for the pole at -Aev+tP e and e+ is the phase angle for the pole at co+t6. Thus, to asses the effects of the multidimensional subspace on the Raman excitation p r o f i l e s , we need only specify the Lorentzfan function (via ~o and 8). In the case of the nearly x-y degenerate home chromophore we hay( four terms In the z which are simply related c. These terms can bee~alculated and summed easily with a small computer. The expression for the absorption intensity is quite analogous to^eq.(9) and can be calculated in a similar fashion c so that we are required to fit both the Raman excitation profile and the absorption band shape with the same ~odel parameters. The ~esults of some model calculations for various fitting parameters are shown in figures l and 2. Figure l contains examples where the non-Raman subspace has been neglected; in this case highly symmetric profiles are predicted. If the subspace is included as outlined above,

Vol.

32,

No.

VIBRONIC

I

THEORY OF RBSONANC~ ~

SCAT~RING

parameters used in the f i t t i n g are found in Table I . The good agreement between theory and experiment makes i t evident that the complete v i b r a t i o n a l subspace needs to be tncluded in the t h e o r e t i c a l c a l c u l a t i o n s . Moreover, in large Table I :

F-

Parameters Used tn Calculations (cm-1) 0

Z LU I--

1` = i 0 0 0 crn-I

_z

J

Ill

~

Jl/mV

>-"'l

=

~,=~=o

1" • 2 0 0 cm~

Jl

/

m

16.3

19.0

1" = 5 c m - I

Tml~=0 21.5 24.0 2 6 . 5 29.0 31.5 ENERGY (kK)

|

1

Figure I : Theoretical e x c i t a t i o n p r o f i l e s as a function of the homogeneous broadening parameter, r=. Other parameters used in th@ c a l c u l a t i o n of t~ese curves are: E°= 23800 cm"m, E°-E °= 100 cm-1, hVR= 1362 om-I. TheXcurves are displaced and normmaTtzed f o r ease of viewing.

>-

r = 2 O O c m -~ %=600cm ~ ~=800cm ~

Z bJ FZ

F =Scm - I %= eOOcm- I 8 =80Oc~ =

bJ > .J bJ n-

1` = 5 cm - I ~o= 6 0 0 c m -E B = 1 6 0 0 cm-=

1~5

1~o

21t5

24.0

26.5

1` = 5 c m - I %=0 8 =800cm -I I 29.0 31.5

ENERGY (kK) Figure 2: Theoretical e x c i t a t i o n p r o f i l e s as a function o f the Lorentzian d i s t r i b u t i o n function described tn the t e x t . Other parameters used tn the c a l c u l a t i o n are the same as f i g u r e l . The curves are displaced and normalized f o r ease o f viewing. the calculated p r o f i l e s are asymmetric and skewed d i s t i n c t i v e l y to the blue as shown in f i g . ( 2 ) . Figures 3 and 4 contain experimental r e s u l t s as well as t h e o r e t i c a l f i t s to the absorption and e x c i t a t i o n p r o f i l e s o f cytochrome c~. The model

0

0

Ex

Ey-Ex

h~R

r

co

6

Fe2+

23800

lO0

1362

50

400

800

Fe3+

23600

400

1374

300

400

800

molecules l t k e home, i t appears t h a t a substant i a l amount of the absorption l l n e w t d t h may be due to the large number o f v i b r a t i o n a l modes, each having a weak FC s t r u c t u r e . This view contrasts wtth the idea that the Soret absorpt i o n band is broadened due to an extremely fast (=5 fs) e l e c t r o n i c r e l a x a t i o n and/or phase changing process. We wish to stress the fact that the l i f e t i m e broadening mechanism r e s u l t s In absorption bands that are symmetric in shape whereas the density-of-states-FC broadening mechanism r e s u l t s in the observed asymmetric band shapes. Phonon broadening o f o p t i c a l §pectra is also observed in s o l i d state systems ~ and in many cases where the zero-phonon l i n e is not resolved, one also observes smooth, asymmetric absorption band shapes. In smaller molecules, however, the FC structure is often resolved due to the presence of fewer, more a c t i v e , modes. In orde~ to demonstrate t h i s case, model c a l c u l a t i o n s v have been performed for A-type resonance s c a t t e r i n g throughout a single electronic transltlon In which only two vibrations are both FC and RFC active. The two vibrational frequencies in the ground )tate {excited state) are_taken to be 950 cm -m (900 cm -m) and 1300 cm "l (1250 om-l). Thus, they each suffer a decrease in force~constant upon electronic excitation. The remaining parameters are the displacements {AQ's) of the potential energy surfaces along each of the two normal coordinates upon electronic excitation. These are given characteristic values for either "small" or "large" displacements corresponding, in absorption, to a very short FC progression (0-0 strong, 0-I weak) or an extensive FC progression, r e s p e c t i v e l y . Each i n d i v i d u a l v i b r a t i o n a l - e l e c t r o n i c t r a n s i t i o n is assigned a homogeneous band w~dth, r e, o f about 5 cm-1 . However, to simulate llbra)lonal and/or low f r e quency sequence broadening ~, the i n d i v i d u a l Lorentztan bands are given a Gausstan d i s t r i b u t i o n , r e s u l t i n g in band widths of about 400 cm-1 for every vtbrontc t r a n s i t i o n b u t l t from the two high frequency v i b r a t i o n s under e x p l i c i t cons i d e r a t i o n . Both the complete absorption (FC) band and the resonance Raman p r o f i l e for each v i b r a t i o n are computed f o r a l l four combinations o f "large" and "small" potential energy displacement parameters. (This involves e x p l i c i t comput a t i o n of the various FC and RFC i n t e g r a l s and the appropriate summation in v i b r a t i o n a l space indicated in e q . ( 3 ) ) . The most s t r i k i n g r e s u l t is the complex nature o f the Raman p r o f i l e predicted f o r a weakly FC active v i b r a t i o n when the second v i b r a t i o n is s t r o n g l y FC a c t i v e . Unlike in a simple one dimensional problem, in t h i s two dimensional case the s t r u c t u r e in the p r o f i l e does not r e f l e c t in any obvious way the size o f

|0

THE VIBRDNIC THEORY OF RESONANCE RAMAN SCATTERING

4.8

4.2

>- 3.6 Z I- 3.0 z

tA

LU

_~ 2.4 _J w ~: 1.8

1.2

0.6

OL-~ IG5

19.0

21.5

24.0

26.5

29.0

31.5

ENERGY (kK)

Ftgure 3: Experimental e x c i t a t i o n p r o f i l e of the 1362 cm"1 mode of ferrocytochrome c. The soltd dark l t n e ts the measured absorption spectrum. The thtn soltd 1tries are the r e s u l t of a theoretical calcul a t i o n ustng the parameters tn Table I and assume a Boltzmann dist r i b u t i o n of ground state energtes (T=200 cm-1). The thtn upper curve ts the calculated Soret absorption band, displaced for ease of viewing. The measured absorption bands tn f t g s . ( 3 ) and (4) are scaled to represent the r e l a t t v e concentration of c~cochrome present In each sample. The e x c i t a t i o n p r o f f l e data have the ~ dependence factored out and are corrected for monochromator response and reabsorptton e f f e c t s .

Vol. 32, No. 1

Vol. 32, No. 1

THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING

4.8

4.2

>_ 3.6 I--z "' z

3.0

w

_~ 2.4 J w

~: 1.8

1.2

0.6

0

16.5

~

19.0

J

21.5

i

24.0 26.5 ENERGY (kK)

i

29.0

31.5

Figure 4: Experimental profile of the 1374 cm -I mode of ferric cytochrome c. The solid dark line is the measured absorption spectrum. The thin solid lines are the result of a theoretical calculation using the parameters in Table I and assume a Boltzmann distribution of ground state energies. The upper curve is the calculated Soret absorption band, displaced for ease of viewing. The theoretical excitation profiles in flgs.(3) and (4) are not scaled independently(i.e, the weaker scattering in fig. (4) is predicted using the larger value of Ye"

Il

12

THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING

Vol. 32, No. I

A (absorption)

ij

>I-Z LU

Iz W

_>

B (Roman)

LU

16.0

II~.O

20.0

22.0 24.0 ENERGY (kK)

26.0

28.0

Figure 5: The model vibrational-electronic absorption band (top) and the predicted resonance Raman scattering profile (bottom) for the case where the Raman scattered vibration (950 cm -I) is "weakly" (AQ~0.05(amu) I/2 k) Franck-Condon ~ctive and a second vibration (1300 cm -I) is "strongly" FC active (AQ=0.3 (amu) I/2 A). See the text and especially reference 3 for more details. the scattered vibrational quantum. In fact it more nearly reveals the presence of the second FC active vibration (the one not being viewed by Raman scattering), but even then not in any regular fashion. We present in figure 5 Just one

example ( of the very many posslble combinations) which emphasizes how in only two dimensions the resonance Raman profile of a given vibration may be quite complex, provided other vibrations are FC active in the same electronic transition.

REFERENCES 1.

J. Tang and A. C. Albrecht, tn Raman SpectVOl. 2, H.A. Sz~nanskt Ed.,Plenum Press (1970).

~k 2.

P.M. Champion and A. C. Albrecht, Journal of Chemical Physics (su~Itted).

3.

G.H. Korenowskt, Ph.D. Thesis, Cornell University (1979) unpublished.

4a. R.H. Stlsbee, In Opttcal Properties of SolIds, Sol Nudelman and S. M1tra Eds.,Plenum Press, New York (1969) b. D . B . Fttchen, tn Ph~stcs of Color Centers, W. B. Fowler,Ed., Academic, New York (1968). 5. G.N. Korenowskt and A. C. Albrecht,Chemical Physics 00, 0000(1979).