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B. Intermolecular Motions in CS2

B. Intermolecular Motions in CS2

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Fifth-Order 2D Raman Spectroscopy


Figure 8 Experimental phase matching geometries. The circles represent the five

incoming laser pulse positions. The open circles with an S inside are the fifth-order

signal positions. The squares are the parallel cascade intermediate positions, and

the diamonds are the sequential cascade intermediates. The phase matching magnitudes and effective path lengths are listed in Table 1. (a)–(c) Homodyne detected

geometries described in the text. (d) Heterodyne detection geometry. Note that the

k1 C k2 C k5 parallel intermediate is located at the same position as the signal,

thus serving as a local oscillator for the heterodyne detected signal.

Copyright © 2001 by Taylor & Francis Group, LLC


Fleming et al.

Although our sample cell has a path length of 1 mm, the actual path

length is determined by the crossing of the incoming beams in the sample.

To estimate the effective path length as determined by the beam crossing,

we calculate the overlap volume of all five beams as a function of position

in the sample. Our beams had diameters of 3 mm at the focusing lens. The

crossing volume was weighted by a Gaussian transverse beam amplitude

profile with a FWHM of 40% of the beam diameter. We assign the FWHM

of the Gaussian fit to the crossing volume distributions as the effective path

length. The resulting effective path lengths are listed in Table 1 for each of

the phase matching geometries employed along with the resulting values

of the magnitude of the phase matching factors for the overall fifth-order

process and the possible third-order cascade intermediates. The positions

of the cascade intermediates and overall fifth-order signals are shown in

Fig. 8.

The spectra obtained in phase matching geometries 8a and 8b are

shown in Fig. 9a and 9b. The spectrum in Fig. 9a shows a high degree

of asymmetry, while the spectrum in Fig. 9b appears quite symmetrical

in the behavior along the two adjustable time variables. The symmetrical

Table 1 The Magnitude of the Wavevector Matching Factor for the Overall

Fifth-Order Response and the Cascade Intermediates, sinc kl/2

sinc kl/2

Phase matching

k2 k3 C k4 k5

k1 C k2 k3 C k4 k5

k 1 k2 k3

k1 C k2 k3

k1 k2 C k4

k1 C k2 C k4


k1 k2 C k5


k1 C k2 C k5

k3 C k4 C k5

k3 C k4 C k5

Effective path length (mm)

Overall fifth-order

































The wavevector matching conditions in parenthesis refer to geometry 8(d); others refer to

geometries 8a–8c. The beam geometries are shown in Fig. 8. The effective path lengths are

the FWHM of the Gaussian fits to the crossing volume distributions.

a The FWHM of the crossing volume distribution for geometry 8d was 1.2 mm, therefore,

the effective path length was set to the smaller sample cell path length of 1.0 mm.

Copyright © 2001 by Taylor & Francis Group, LLC

Fifth-Order 2D Raman Spectroscopy


spectrum, Fig. 9b, has the same time-dependent behavior along each of

the two time variables as the one dimension time domain spectrum shown

in Fig. 3. The response at early times rises to a maximum at ¾ 180 fs

and then evolve into a single exponential decay at long times with a time

constant of tc ¾ 0.8 ps. In the asymmetrical spectrum (Fig. 9a), there is a

rapid decay along 2 from a maximum at 2 D 0 that at long times becomes

the 0.8 ps decay seen along both dimensions in Fig. 9b. Along 4 the initial

behavior is similar to the inertial behavior at early times in the third-order

response, but the signal then decays much more rapidly, tc ¾ 0.25 ps, than

along 2 . There is a prominent ridge along both time axes in and along

Figure 9 Measured 2D fifth-order time domain nonresonant Raman spectra for

CS2 at room temperature: (a) using the phase matching geometry shown in Fig. 8a

and (b) using the phase matching geometry shown in Fig. 8b. Figures (c) and

(d) are the cascading fifth-order 2D Raman spectra simulated directly from the

measured 1D Raman spectrum using the phase matching magnitude values listed

in Table 1 for the cascade intermediates and Equation (28) for the same phase

matching geometries used in (a) and (b), respectively.

Copyright © 2001 by Taylor & Francis Group, LLC


Fleming et al.

the 4 D 0 axis in Fig. 9a that is pulse width limited in one dimension and

shows nearly identical time-dependent behavior to the third-order response

along the other. Both spectra also contain a strong peak at the time origin

that is pulse width limited in both time dimensions, similar to the electronic

response at the time origin in the third-order spectrum.

While both phase matching geometries 8a and 8b are reasonably well

phase matched for the overall fifth-order response, they are very differently phase matched for the cascade intermediates (Table 1). Geometry 8a

is well phase matched for the parallel cascade intermediates and poorly

phase matched for the sequential cascade intermediates, and geometry 8b

presents the opposite case. If the measured signals were predominantly

generated by the direct fifth-order process, we would expect the response

to be identical between the two experiments. However, the clear difference between the two spectra provides a strong indication that the signals

are dominated by the fifth-order cascading processes. We can simulate the

fifth-order cascaded spectra directly from the measured third-order spectrum in Fig. 3 using Equation (27). Figure 9c shows the simulation for the

parallel cascaded response, and Fig. 9d shows the simulation for the sequential cascaded response. There is excellent agreement between Figs. 9a and

9c and Figs. 9b and 9d consistent with the relative values of the phase

matching factors in Fig. 8.

With the evidence from phase matching geometries 8a and 8b

demonstrating the measured fifth-order signals to be dominated by

cascaded processes, phase matching geometry 8c was designed to provide

a maximum level of discrimination against all four of the cascade

intermediates while leaving the direct overall fifth-order process well phase

matched. Using the same experimental conditions as employed for the data

in Fig. 9, there was no detectable signal with phase matching geometry 8c.

This confirms the domination by cascaded responses in Figs. 9a and 9b.

Although it is always difficult to apply a negative result and quantify the

lack of a signal, consideration of the observed signal levels in figs. 9(a)

and 9(b), and the failure to detect a signal in geometry 8c, allows us to

estimate an upper limit on the direct fifth-order signal at 2% of the cascaded


In the case of nonlinear polarizability coupling, the ratio of the

cascaded to direct fifth-order response can be expressed in terms of the

ratio of the first and second derivative of the polarizability [Equation (32)].

Using instantaneous normal mode simulations, Murry et al. (40) have

calculated the relative ratios of ˛ 1 and ˛ 2 for 5000 intermolecular modes

in CS2 . Although their results show this ratio to be somewhat randomly

Copyright © 2001 by Taylor & Francis Group, LLC

Fifth-Order 2D Raman Spectroscopy


distributed among the 5000 modes, the majority of the modes have a ratio

of ˛ 1 :˛ 2 > 100. This result implies that for the homodyne measurement,

where the signal is proportional to jR 5 j2 , the cascaded signal will be

102 –104 times larger than the fifth-order signal from NP coupling. This

is a value well above our lower limit for this ratio of 50. For anharmonic

coupling, even though one might expect a high degree of anharmonicity for

intermolecular motions at room temperature, it is the magnitude, and not

degree, of the cubic anharmonicity that will be reflected in the magnitude

of the direct fifth-order signal. The magnitude of the anharmonicity will be

quite small for intermolecular vibrations simply due to the low frequencies

of the modes involved. In addition, Murry and Fourkas (18) have suggested

that a cancellation among the continuous distribution of frequencies for

the anharmonic coupling will result in a much lower contribution to the

fifth-order response than NP coupling.

C. Intramolecular Vibrations in Carbon Tetrachloride and


To provide the additional sensitivity needed to probe the intramolecular

vibrations in liquid CCl4 and CHCl3 , time domain Raman spectra were

taken using an intrinsic heterodyne technique. As a heterodyne detection

technique, the measured signal is the cross-term between a local oscillator field and a signal field such as the fifth-order signal [Equation (34)].

Intrinsic refers to the fact that the local oscillator field is derived from

one of the third-order responses using a phase matching geometry designed

to generate both the third- and fifth-order signals along the same phase

matched direction. The phase matching geometry is shown in Fig. 8d. In

this geometry the overall signal is phase matched along ks5 D k1 C k2

k3 C k4 C k5 . In addition to the fifth-order signal, the third-order signal

ks3 D k1 C k2 C k5 is also generated along the same direction and serves

as the local oscillator. Assuming R 3 × R 5 and using Equation (34), the

resulting signal intensity will be proportional to the cross-term between the

third- and fifth-order responses,


2, 4

/ 2 RebR 3





2, 4



Figure 10 shows the Fourier transformations of intrinsic heterodyne

detected time domain two-dimensional Raman spectra of (a) CCl4 , (b)

CHCl3 , and (c) a 50:50 molar ratio CCl4 :CHCl3 mixture. Only quadrants

I and II are shown, since the other two quadrants are equal by

inversion symmetry. Aside from the diagonal and axial peaks, there

Copyright © 2001 by Taylor & Francis Group, LLC


Fleming et al.

Figure 10 2D fifth-order Raman spectra of (a) CCl4 , (b) CHCl3 , and (c) a 1:1

mole ratio CCl4 :CHCl3 mixture.

Copyright © 2001 by Taylor & Francis Group, LLC

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