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A. One-Dimensional Measurements: Raman Line Shape and Free Induction Decays

A. One-Dimensional Measurements: Raman Line Shape and Free Induction Decays

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Vibrational Dephasing in Liquids



387



Fourier transform of CFID :

Iiso



1



D



0



CFID



e



i



0



d



2



0



Although experiments are most directly expressed in terms of the

oscillator coordinate q , the quantity of physical relevance is the instantaneous oscillator frequency ω t , or, expressed more conveniently, the

instantaneous deviation from the mean frequency, υω t D ω t

hωi. This

frequency deviation is the direct consequence of time-dependent forces

exerted on the oscillator by its environment. At a semiclassical level, the

FID correlation function is related to the time-dependent frequency by

CFID



D



exp



υω t dt



i



3



0



Unfortunately, the conversion from υω t to CFID t involves a significant

averaging and loss of information.

This loss of information is easier to see after applying a cumulant

approximation (61). This approximation is equivalent to assuming that the

distribution of υω is always Gaussian and simplifies Equation (3) to

CFID



t0



dt0



D exp



0



0



Cω t00 dt00



Cω t D hυω t υω 0 i



4

5



where

indicates operator multiplication. The FID correlation function

is derived from the frequency correlation function Cω t by averCFID

aging in time.

If we also assume that the vibrational frequency is modulated by

a single process, υω t is fully characterized by two parameters: ω , the

rms magnitude of υω, and ω , the correlation time of υω t . In the KuboAnderson model (61–63), the frequency correlation decay is taken to be

exponential:

Cω t D 2ω e



t/



ω



6



and the FID correlation function becomes

CFID



D exp



2ω



2

ω



e



/



ω



1C



ω



7



In the limit ω ω − 1, called the fast modulation or homogeneous limit,

the FID correlation function reduces to

CFID



De



/T2



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8



388



Berg



where T2 D 2ω ω , whereas in the limit ω

lation or inhomogeneous limit, it becomes

CFID



De



2



/22ω



ω



× 1, called the slow modu9



In each limit, the experimental decay is characterized by a single parameter,

whereas the underlying frequency correlation function is characterized by

at least two parameters. This problem exists for other models as well. In

general, the problem of inverting CFID

to find Cω t is underdetermined

and requires model-dependent assumptions to complete an analysis. This

problem reflects the loss of information caused by the time averaging in

Equation (4).

If we restrict ourselves to the Kubo-Anderson model, the problem

does not seem entirely intractable. The shape of the decay distinguishes the

limits: the fast limit has an exponential decay; the slow limit has a Gaussian

decay. In the intermediate regime [Equation (7)], the details of the decay

shape can be used to find both ω and ω .

However, these results are highly model dependent (64–66). For

example, a model based on a random distribution of dipole-dipole interactions gives an exponential decay in the slow modulation limit (64).

As another example, a nonexponential frequency correlation decay gives

different intermediate results than the Kubo-Anderson model (67,68).

The situation becomes even more complicated when multiple

dephasing processes are considered. Figure 2c shows FID curves from two

distinctly different modulation processes. In case #1 the are two processes

modulating the vibrational frequency: one fast and one slow. In case

#2, there is a single process at an intermediate rate. Real experimental

data contain both random noise and systematic errors from tails of

other transitions, weak overtones, hot bands, and isotope lines and from

background emissions from the sample. In the face of these factors,

the subtle differences between CFID

in case #1 and in case #2 are

almost impossible to distinguish, despite the significant differences in the

underlying physics.

In the fast modulation limit [Equation (8)], the loss of information is

fundamental. The lifetime of the frequency perturbations is short compared

to their magnitude, and the uncertainty principle precludes a full characterization of υω t by any experimental technique. However, in the slow

modulation limit and in the intermediate regimes, the loss of information in

the FID is not fundamental. The next section shows that the Raman echo

contains additional information about the rate of the frequency fluctuations

that is not present in the FID. By using a combination of Raman echo and



Copyright © 2001 by Taylor & Francis Group, LLC



Vibrational Dephasing in Liquids



389



FID measurements, a much more robust and more nearly unique model of

Cω t can be extracted.

B. A Two-Dimensional Measurement: The Raman Echo



Loring and Mukamel pointed out that additional information can be obtained

by a higher order spectroscopy, the Raman echo (26). The corresponding

observed correlation function is [see Equation (3)]

CRE



1,



3



D



1C 3



exp i



1



υω t dt



i



υω t dt



.



10



0



1



The Raman echo correlation function has two time variables and provides

direct information on the rate of frequency modulation (26,69,70). In

essence, the echo tests whether υω t differs between two different time

intervals, whereas the FID only examines υω t during a single time

interval [Equation (3)]. If υω t changes slowly, i.e., ω × 1 , 3 , the two

integrals in Equation (10) cancel, with complete cancellation at 1 D 3 .

More specifically,

CRE



1, 3



D CFID j



3j



1



11



in the slow modulation limit. On the other hand, if υω t randomizes

quickly, the change in sign in the integrals is unimportant, no cancellation

occurs, and

CRE



1, 3



D CFID



1



C



12



3



in the fast modulation limit. Thus, a comparison of the Raman echo to the

FID provides qualitative and model-independent discrimination between

fast and slow modulation dephasing.

For more complex systems with intermediate modulation or multiple

modulation time scales, a cumulant approximation is useful [see Equation

(4)] (34):

CRE



1, 3



1



D exp



2 2

0

t0

0



dt00 C0ω t00



1C 3



dt0

0



.



3



dt0 C 2



dt0



0



13



The echo and FID decays derive from the same frequency correlation function, but the echo dissects Cω t with a more complex and informative

operator.



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390



Berg



Figure 1 Two-dimensional contour plots of the log of the Raman-echo correlation

function, ln CRE 1 , 3 , showing the effect of changing the rate of solvent-induced

perturbations. All cases give a Raman line with the same FWHM 5 cm 1 and

FIDs with similar decay times but give very different Raman echo results. (a) Fast

modulation; (b) intermediate modulation (ω D 3.32 cm 1 , ω D 1.60 ps); (c) slow

modulation. Calculations are based on a single Kubo-Anderson process [Equations

(7)–(9)].



This idea is illustrated in Fig. 1, which shows the behavior of the

Raman echo signal for fast, intermediate, and slow modulation dephasing.

In each of the cases shown, the Raman linewidth is the same, and the FID

decays times are very similar. Nonetheless, the Raman echo decays are

qualitatively different, depending on the rate of frequency modulation.

The additional information in the Raman echo is even more important

when multiple dephasing mechanisms might be operating simultaneously.



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