Volume 203, number 2,3
CHEMICAL PHYSICS LETTERS
19 February 1993
Derivation of the reflection principle in continuum resonance Raman scattering Soo-Y.
Lee of Chemistry, National University of Singapore, 0511 Kent Ridge, Singapore
Department
Received 15 October 1992; in final form 14 December 1992
We provethe reflectionprinciplein continuum resonance Raman scattering as stated recently by Kolba, Manz, Schreier and Trisca. For a Stokes (or anti-Stokes) transition, the nodal pattern of the coordinate or momentum representation of the initial wavefunction 1i) (or final wavefunction Ifi ) is mapped on the time-dependent overlap 1(fl i( I) ) 1.An interesting aspect of the proof is the use of all three approaches to Raman scattering: the Kramer+Heisenberg-Dirac sum-over-states approach, the LeeHeller time-dependent approach, and the Hizhnyakov-Tehver transform method. We also derive new expressions for the resonance Raman scattering amplitude and associated correlation function in terms of the Rayleigh scattering amplitude and associated autocorrelation function. A new algorithm using a fast Fourier transform to implement the Hizhnyakov-Tehver transform method for fundamental Raman excitation profiles is also discussed.
1. Introduction Recently, Kolba et al. [ 1 ] proposed, and illustrated with model calculations, a new reflection principle for continuum resonance Raman scattering of diatomic molecules on continuous wave (cw) excitation. The principle states the following: anti-Stokes resonance Raman scattering from I i) to Ifi, cf< i) yields an oscillatory pattern of (J1 i(t) ) which reflects the nodal structure of (Q Ifi. Likewise, the invariance of the correlation function to real bra-ket interchange implies that in the Stokes resonance Raman transition (i
lem clearly, and the main purpose of this Letter is to provide the derivation of the principle. There have been three major theoretical approaches to resonance Raman scattering, and they are interrelated. The traditional sum-over-states, or energy-frame approach of Kramers, Heisenberg and Dirac [ 2-41 involves summation over all the vibrational levels of the resonant electronic state. This
summation over intermediate vibrational states is avoided in the equivalent time-frame approach of Lee and Heller [ 5-71, who also provided a physical picture of the Raman cross section formula in terms of a half-Fourier transform of an overlap
and suggeststhe use of the Lee-Heller time-dependent approach to Raman scattering [ 5-71. The mapping onto the overlap (fl i(t)) by the initial wavefunction Ii} or final wavefunction Ifi, depending on whether it is Stokes or anti-Stokes Raman scattering, rather than by the product ( Qlfi ( QI i} is somewhat surprising, but it reminds us of the reflection approximation for pho-
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toabsorption [ 12-151 where the mapping involves a single initial wavefunction Ii}. Since the transform method relates photoabsorption to Raman scattering, it should feature in the derivation. The results of the transform method that we require are more readily derived by the sum-over-states approach, so in the proof of the new reflection principle in section 2 we have employed aspects of all the three major theoretical approaches to Raman scattering.
2. Theory It is convenient to work with the dimensionless coordinate Q and momentum P, which are related to the corresponding quantities (q, p) with dimensions by Q=qm and P=p/&%, where ,u is the reduced mass and w is the oscillator frequency with dimension of inverse time. As our model, consider harmonic potentials of equal frequencies wh for both the ground and excited electronic states. Let the dimensionless displacement in the equilibrium position of the two potentials be d. The vibrational states of the ground state potential will be unprimed, but those of the excited state potential will be primed. We choose the harmonic model for ease of deriving the results that we want. On the face of it, the harmonic model would lead to bound state resonance Raman scattering with long-time vibrational dynamics on the excited state potential [ 16,17 1. However, the time-dependent theory assures us that the continuum resonance Raman scattering results can also be derived in this model by taking a large displacement d and considering short-time dynamics of much less than a vibrational period, such that the wavepacket 1i(t) ) propagated on the excited state surface makes just one pass over the width of the final state If> without bouncing off the opposite wall and giving rise to return overlaps in the correlation function (fl i(t) ) , as happens in bound state resonance Raman scattering [ 16,171. Moreover, the force on the propagated wavepacket in the Franck-Condon region of the excited state, which is an important quantity in continuum resonance Raman scattering, can be adjusted simply by varying the displacement A;hence the choice of equal frequencies or, for both 94
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CHEMICAL PHYSICSLETTERS
the ground and excited electronic states is not a limitation on the results below. To keep the derivation simple, we assume constant dipole transition moments. The resonance Raman scattering amplitude in the sum-over-states approach [ 2-41 is given by (flll’)(l’lm)
An,m(u)= CE m+7&3-E,,
+ir’
(1)
where E,,, and & are the energies of the vibrational states 1m) and 1l’}, respectively; #to is the incident photon energy, and r is the homogeneous damping constant. The Raman cross section is proportional to lA,,J o) 1’. Now, we can derive the following recursion relation [ 111 using phonon operator algebra: (m+k(l')(l'lm}=(m+k)-'/Z
+;?i(m+k-lll’)(l’lm) -$(m+k-11(‘-0(l’-lim)).
(2)
This is shown in the Appendix. Combining this with eq. (l), we therefore have the following recursion relation for the resonance Raman scattering amplitude: A ,+k,,(~)=(m+k)-‘/2
+ -$
(
m’~2A,+~-l,m--I(~)
[A~+~-,,,(~)-Am+*-1,m(~-~h)l).
(31 In the case of Raman scattering from the lowest vibrational state, i.e. m=O, eq. (3) simplifies to /c-1,0(o) -A&-,,o(w-%)
1>
(4) = (k!)-‘1’
xLio
(-$
G-U’(5~o,obw~)]. (5)
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Furthermore, in the case of fundamental Raman scattering, i.e. k=l, eq. (4) gives A,,,(w)=
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CHEMICAL PHYSICS LETTERS
d2 [A0,0(~)-A0,o(~--%) Y=
1>
(6)
which is the expression used by the transform method with the standard assumptions [ 8-101, It relates the fundamental Raman scattering amplitude to the Rayleigh scattering amplitude; the latter can be deduced from the absorption profile, as discussed below. Let us now make the assumption that rn’/‘< A, i.e. a low initial vibrational level in comparison with the dimensionless displacement. In the time-dependent picture, this ensures one complete pass between the propagated initial state 1m(t) > and the final vibrational wavefunction 1m+ k).We can then neglect the first term on the right-hand side of eq. (3 )_ By iteration, this leads to a new expression which is an extension of the usual transform method where the Stokes Raman scattering amplitude from an initial excited vibrational state 1m) is given in terms of the Rayleigh scattering amplitude for 1m},
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given by the half-Fourier transform of the autocorrelation function C,,,(t). At this point, it is instructional to digress to consider the case of m = 0, where it has been shown that the autocorrelation function Co,o(t) can be deduced by the inverse Fourier transform of the single molecule, low temperature absorption profile [ 16,171 ew(-~tl~)CO,0(t) m =
exp[ -i(E,+ho)t/h]o(w) --oo s
d(ho)
(11)
and the absorption profile G(W) is related to the observed absorption intensity I(o) as follows: b(W) =
I(o)
/w
_I’“, [I(w)/01 d(fiw) ’
(12)
The current transform method of calculating the fundamental Raman excitation profile (REP) from the absorption spectrum begins with a Hilbert transform of the absorption profile a(w), essentially to determine the Rayleigh excitation profile [ 181,
A m+k,m(~)= (m+k)-“2 O” a(
x 5
[Am+!+m(W)
-Am+!+-l,m(W-WJ
I> (7)
(8) In the time-dependent approach, the resonance Raman scattering amplitude eq. ( 1) is written as the half-Fourier transform [ 5-7 ] A,,(u)=-~~exp[i(E,+~w)rla-n/h] 0
xG,m(t)
dt
>
(9)
where the correlation function C,,,(t) is defined as C,,,(t)~(nlexp(-~H,~tlA)Im) =(nlm(l)>
9
w’)
A,o(w)=/15~dW’+ilr~(w),
(13)
0
wherefi denotes the principal part. The accuracy is not high because of the singularity in the Hilbert transform, and this shows up as jitter particularly in the REPS of low-frequency modes where it is necessary to take the difference between two nearby Rayleigh excitation profiles that are relatively shifted by the low frequency [ 191 #l. The time-dependent approach offers a better, more stable algorithm devoid of singularities: the autocorrelation function is first determined from the absorption profile by a fast inverse Fourier transform, eq. ( 11), followed by a fast half-Fourier transform to obtain the Rayleigh excitation profile, eq. (9 ), with n = m = 0,which can then be used in eq. (6) to give the desired fundamental REP [ 201. Returning to eq. ( 9 ) and inserting it in eq. ( 8 ) , we deduce that the correlation function C,+,,(t) for
(10)
with H,, as the Hamiltonian for the excited state. The Rayleigh scattering amplitude where 12= m would be
*I Notice the jitter in the Raman excitation profile of the 168 cm-’ mode in fig. 3, which does not occur with the time-dependent approach.
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Raman scattering is related to the autocorrelation function C,,,(t) by
Using eq. ( 19) in eq. ( 16), we therefore have
x~kev(-4Qf)IMQf)I
X [l-exp(-iw,t)lkC,,,(t) ,
(14)
This relation is exact for m=O. In continuum resonance Raman scattering, the duration of the autocorrelation function C,,,(t) is short, typically less than a quarter of the vibrational period 2n/w,,, in which case we can make the expansion 1-exp( -iwhl) =:iOht
(15)
and therefore
.
P,xAaht.
(21)
It is therefore more convenient to write eq. (20) in the following form appropriate for continuum Raman scattering: l
q$!y2
X2-k/ZPfexp( - fQ:> I-UQ?) I , (16)
Eqs. ( 14) and ( 16) are new expressions giving the Raman correlation function in terms of the Rayleigh autocorrelation function. It has been shown that the one-dimensional harmonic oscillator wavefunctions retain the same form when propagated in a harmonic potential [ 2 1,221, and it can be deduced that I <~lew(-i&~l~)Im)
=(Q-Q,lm)
I = I
3
I
(17)
where the classical coordinate Q, for the center of the wavepacket on the excited state potential is given by Q,=A[ l-cos(w,t)]
.
(18)
It can then be shown [23-261 that
IG,,(~)I=I<~I~(~)>l E 1 7 WlQ>(Q-QAW -m =exp(--1.Q?) lLdQf)I where t,(x) 96
de/ ,
(19)
is the mth Laguerre polynomial [27].
(20)
For continuum Raman scattering which involves a repulsive excited state potential, both ah and A can be taken to be time-dependent quantities which are determined using a local harmonic approximation to the repulsive potential about the classical coordinate Qt [ 281. However, we also observe that at short times, the classical momentum is given by
1Cm+k,m(t)
x ~kcn,*(~)
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(22)
where only the classical quantities Q, and P, enter the expression. The classical coordinate Qt and the classical momentum P, are obtained by solving Hamilton’s equations of motion for the excited state potential with the initial conditions
Q,=,=o >
PC=,=o ,
(23)
where the equilibrium position in the ground state potential is taken to be zero, and Qt and P, would be increasing monotonically with t. The factor Pf in eq. (22) would induce a node in Cm++(t) at t =O and the Gaussian term exp( - jQ:) would dampen it for large values of Ql. In addition, the function L,(x) would have m zeros between x=0 and x=co, resulting in the correlation function IC,,,+k,m(t) 1, k> 1, ta 0, having exactly the nodal structure of the spatial representation I (Ql m) 1,or the equivalent momentum representation 1(PI m} I. This is the reflection principle for continuum resonance Raman scattering as stated by Kolba et al. [ 11. It is useful to note that the mapping is to a damped Laguerre polynomial, rather than a damped Hermite polynomial where Qt>O. This is not surprising, as the Hermite polynomial H,(x) has only [m/2] integral number of
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CHEMICAL PHYSICS LETTERS
nodes for XE(0, co ), which is half, or less than half, that for the Laguerre polynomial L,(x), and it also shows double degeneracy in the number of nodes for consecutive values of m. The argument of the Laguerre polynomial in eq. (22) goes as Q:, and, because the velocity of the propagated wavepacket is slow at the beginning and accelerates with time, this results in the nodes for ICm+k,m(t) 1being somewhat evenly spaced in time. These properties of the mapping are well illustrated in figs. 1, 2 and 3Rof ref. [ 11. Kolba et al. [ 1] have considered two limiting cases where the final, ground vibrational state is embedded in a narrow or a wide potential energy curve. These yielded structures in the correlation function (0 / 4(t) > which were washed-out as the &type coordinate or momentum final wavefunction /0) was broadened, as shown in fig. 3 of ref. [ 11. Our model above only deals with the “normal” case where the initial and final vibrational states in Raman scattering come from the same potential. Nevertheless, it is important to note that the illustration by Kolba et al. [ 1 ] is suggestive that the phase-space time-dependent overlap diagrams of Heller [ 131 can be used to provide a qualitative understanding of the reflection principle in continuum Raman scattering involving low initial or final vibrational quantum numbers. For vibrational states of low quantum number such as 0 or 1, the Wigner phase space distribution of the wavefunction has 0 or 1 nodal curve, and it is easier to gauge the sign and magnitude of the time-dependent overlap with it. The phase-space overlap diagrams for the cases illustrated in fig. 3 of ref. [ 1 ] are particularly straightforward.
3. Conclusion Using the sum-over-states, the time dependent, and the transform approaches to resonance Raman scattering, we have shown that for Stokes (or antiStokes) transitions, the nodal pattern of the coor-
19 February 1993
dinate or momentum representation of the initial wavefunction Ii) (or final wavefunction If) ) is mapped on the time-dependent overlap (fl i( t)}. This reflection principle is expected to deteriorate with increasing value of i; and for a fixed value of i, it deteriorates with increasing value ofJ: There is conservation of nodal structure in the Fourier transform of harmonic oscillator wavefunctions between the coordinate Q and momentum P representations [ 29 1. Since the correlation function c m+k,m(t) resembles ( Q 1m), we can therefore expect the topology to be preserved for the Fourier transform of Cm+k,m(t) to give the Raman amplitude Am+k,m(co). Now, both Stokes and anti-Stokes Raman scattering between the same two vibrational states have the same Raman amplitude; therefore, the Raman scattering cross section will have the topologyof /l or l(Plm)l,where Im} isthe lower of the two vibrational states. This is the predicted, observable reflection principle for continuum resonance Raman scattering. Levy, Shapiro and Yogev [ 301 have presented a complementary reflection principle for the case of pulse excitation in continuum Raman spectroscopy where all the emitted photons are collected. The mapping of the correlation function C,,,(t) onto I n ) or I m) discussed in this Letter also applies to Shapiro’s results once they are cast in the time-frame as shown above. Acknowledgement The author gratefully acknowledges useful conversations with Professor E.J. Heller, and the stimulation that he received on this problem from the XIIIth International Conference on Raman Spectroscopy held from 3 1 August-4 September 1992 in Wiirzburg, Germany. He would also like to thank the referee for useful comments. This work was supported by a grant RPl 11/82 from the National University of Singapore.
Appendix Assume harmonic potentials of equal frequencies @ for both the ground and excited electronic states, with displacement d in the equilibrium positions. Let excited state quantities be denoted by primes. If B+and a are phonon creation and annihilation operators, then 97
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Volume 203, number 2,3
=(m+k)-‘/*1-‘I*
P*(m+k-1
19 February 1993
l/‘-l)+
m’/*(I’-llm-l)(m+k-llI’--l)(I’-llm-l)+
-fs; (m-t!+-lII’)(I’Im)
--$(m+k-l~Ir-l)(i'-l~m)). This is eq. (2) of the text.
References
[ 151S.-Y. Lee, J. Chem. Phys. 82 (1985) 4588.
[ 161E.J. Heller, AccountsChem. Res. 14 (1981) 368. [ 1] E. Kolba, J. Manz, H.-J. Schreier and I. Trisca, Chem. Phys. Letters I89 ( 1992) 505. [2] H.A. Kramers and W. Heisenberg, Z. Physik 31 ( 1925) 681. [3] P.A.M. Dirac, Proc. Roy. Sot. 114 (1927) 710. [4] A.C. Albrecht, J. Chem. Phys. 34 (1961) 1476. [ 5 ] S.-Y. Lee and E.J. Heller, J. Chem. Phys. 71 ( 1979) 4777. [ 61 E.J. Heller, R.L. Sundberg and D. Tannor, J. Phys. Chem. 86 (1982) 1822. [ 71 D. Tannor and E.J. Heller, J. Chem. Phys. 77 (1982) 202. [ 81 V. Hizhnyakov and I. Tehver, Phys. Stat. Sol. 21 (1967) 755. [9] D.L.. Tonks and J.B. Page, Chem. Phys. Letters 66 ( 1979) 449. IO] J.B. PageandD.L. Tonks, J. Chem. Phys. 75 (1981) 5694. LII] S. Hassing and O.S. Mortensen, J. Chem. Phys. 73 ( 1980) 1078. [ 121 G. Henberg,Spectra of diatomic molecules, 2nd Ed. (Van Nostrand, Princeton, 1950) p. 393. [ 131 E.J. Heller, J. Chem. Phys. 68 (1978) 2066. [ 141 S.-Y. Lee, R.C. Brown and E.J. Heller, J. Phys. Chem. 87 (1983)2045.
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[ 171E.J. Heller, in: Potential energy surfaces and dynamics calculations, ed. DC. Truhlar (Plenum Press, New York, 1981) pp. 103-131. [ 181C.K. Chan and J.B. Page, J. Chem. Phys. 79 (1983) 5234. [ 191B.M. Britt, H.B. Lueck and J.L. McHale, Chem. Phys. Letters 190 (1992) 528. [20] S.-Y. Lee, to be published. [21] H.D. Meyer, Chem. Phys. 61 (1981) 365. [22] S.-Y. Lee, Chem. Phys. 108 (1986) 451. [23] S. Waldenstrom and K.R. Naqvi, Chem. Phys. Letters 85 (1982) 581. [24] M. Wagner, Z. Naturforsch. 14a ( 1959) 81. [25] T.H. Keil, Phys. Rev. A 140 (1965) 501. [26] J.K. Lewis and J.T. Hougen, J. Chem. Phys. 48 ( 1968) 5329. [ 27 ] M. Abramowitz and LA. Stegun, Handbook ofmathematical functions (Dover, New York, 1977) p. 780. [28] S.-Y. Lee and E.J. Heller, J. Chem. Phys. 76 (1982) 3035. [29] C. Cohen-Tannoudji, B. Diu and F. Lalo&, Quantum mechanics, Vol. 1 (Wiley, New York, 1977) pp. 542-546. [ 301 I. Levy, M. Shapiro and A. Yogev,J. Chem. Phys. 96 ( 1992) 1858.