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A. The Vibrational Echo Method

A. The Vibrational Echo Method

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Infrared Vibrational Echo Experiments


Vibrational echo experiments permit the use of optical coherence

methods to study the dynamics of the mechanical degrees of freedom

of condensed phase systems. Because vibrational transitions are relatively

narrow, it is possible to perform vibrational echo experiments on welldefined transitions and from very low temperature to room temperature

or higher. Further, vibrational echoes probe dynamics on the ground state

potential surface. Therefore, the excitation of the mode causes a minimal

perturbation of the solvent.

For experiments on vibrations, a source of ps IR pulses is tuned to

the transition of interest. The vibrational echo experiment involves a twopulse excitation sequence. The experiment is illustrated schematically in

Fig. 1A. Initially, all of the vibrations are in the ground state, j0i. This is

represented by an arrow pointing down in the first circle. The first pulse

excites each solute molecule’s vibration into a coherent superposition state

of the molecule’s ground vibrational state and the first excited vibration,

the j0i and j1i vibrational states. This is represented by an arrow in the

plane shown in the second circle. Each molecule in a superposition state

has associated with it a microscopic electric dipole, which oscillates at the

vibrational transition frequency. Immediately after the first pulse, all of the

microscopic dipoles in the sample oscillate in phase. Because there is a

distribution of vibrational transition frequencies, the dipoles will precess

with some distribution of frequencies. Thus, the initial phase relationship

is very rapidly lost. This is represented in the third circle by the arrows

fanning out. The molecules with lower transition frequencies fall behind

the average, and the molecules with higher frequencies get ahead of the

average. This effect is the free induction decay and occurs on a time scale

related to the inhomogeneous line width. After a time, , a second pulse,

traveling along a path making an angle, Â (see Fig. 1B), with that of the

first pulse, passes through the sample. This second pulse changes the phase

factors of each vibrational superposition state in a manner that initiates a

rephasing process. This is illustrated in the fourth circle. The fan of arrows

flips over so that the arrows that were moving apart are now moving toward

each other. At time 2 , the ensemble of superposition states is rephased.

This is shown in the fifth circle as the reformed single arrow. The phased

array of microscopic electric dipoles behaves as a macroscopic oscillating

electric dipole, which acts as a source term in Maxwell’s equations and

gives rise to an additional IR pulse of light, the vibrational echo. A free

induction decay again destroys the phase relationships, so only a short

pulse of light is generated. As shown in Fig. 1B, the vibrational echo pulse

propagates along a path that makes an angle, 2Â, with that of the first pulse.

Copyright © 2001 by Taylor & Francis Group, LLC


Copyright © 2001 by Taylor & Francis Group, LLC

Rector and Fayer

Infrared Vibrational Echo Experiments


The signal intensity is proportional to the intensity of the first pulse and

the intensity of the second pulse squared.

The rephasing at 2 removes the effects of inhomogeneous broadening

(25). The spread in frequencies responsible for the inhomogeneous linewidth

and, in the time domain, the free induction decay is eliminated by the

rephasing that gives rise to the echo pulse. However, fluctuating forces generated by interactions of the vibrational oscillator with the dynamical solvent

environment produce fluctuations in each oscillation’s frequency. At 2 the

rephasing is imperfect. As is increased, the fluctuations produce increasingly large accumulated phase errors among the microscopic dipoles at 2 ,

and the vibrational echo signal amplitude is reduced. Thus, the vibrational

echo decay is related to the homogeneous linewidth, i.e., the fast vibrational

frequency fluctuations, not the inhomogeneous spread in frequencies.

A plot of the vibrational echo intensity change with delay between

the pulses is a vibrational echo decay curve. Vibrational echo decays

are frequently exponential, although intrinsically nonexponential dynamics

(non-Lorentzian homogeneous line shapes) are also seen (26–28). In

all of the data presented below, the echo decays are exponential or

exponentials modified by laser pulse duration affects. At low temperature,

when the dynamics are slow, the data can be fit well with a simple

exponential. At high temperatures, the dynamics approach the time scale

of the laser pulses. For these data, a more complex fitting routine is

employed that takes into account the finite duration pulses. The signal is

calculated from the three time-ordered interactions of the sample with the

radiation fields. An example of a low-temperature vibrational echo decay

curve measured on CO asymmetrical stretching mode 2010 cm 1 of

(acetylacetonato)dicarbonylrhodium(I) (Rh(CO)2 acac) in dibutylphthalate

at 3.4 K and a fit to an exponential are shown in Fig. 2. This measurement

was performed at the Stanford free electron laser, as discussed below.

As can be seen, high-quality data can be obtained in vibrational echo


Figure 1 (A) Semiclassical Bloch picture of a vibrational echo in a frame rotating

at the center frequency of the vibrational line. Vertical axis in circles represents

the population axis of the j0i to j1i vibrational transition. The other two axes

represent the coherence plane. The relationship of the diagram to the vibrational

echo experiment is discussed in the text. (B) Schematic of the vibrational echo

pulse sequence. The two excitation pulses are crossed and focused in a sample at

a small angle, Â. The vibrational echo is emitted from the sample at an angle, 2Â,

from that of the first pulse.

Copyright © 2001 by Taylor & Francis Group, LLC


Rector and Fayer

Figure 2 Vibrational echo decay data for the asymmetrical CO-stretching mode of

Rh(CO)2 acac in DBP ¾2000 cm 1 at 3.4 K and a fit to a single exponential function. The data were taken using the Stanford Free Electron Laser. The decay constant

is 23.8 ps, which yields a homogeneous linewidth of 0.11 cm 1 . The absorption

spectrum has a linewidth of ¾15 cm 1 at this temperature, demonstrating that the

line is massively inhomogeneously broadened.

Copyright © 2001 by Taylor & Francis Group, LLC

Infrared Vibrational Echo Experiments

The vibrational echo decay signal, S


D S0 e

4 /T2


, is given by


where T2 is the homogeneous dephasing time. The Fourier transform of

the echo decay is directly related to the homogeneous lineshape (25). For

systems in which orientational relaxation is not significant,









where TŁ2 is the homogeneous pure dephasing time and T1 is the vibrational

lifetime. T2 is determined from the echo decay constant. T1 is measured

with pump-probe experiments. Measurements of T2 and T1 permit the

determination of TŁ2 , the pure dephasing contribution to the linewidth. An

exponential vibrational echo decay corresponds to a Lorentzian lineshape

with a linewidth, , given by










2 T1


Pure dephasing describes the adiabatic modulation of the vibrational energy

levels of a transition caused by fast fluctuations of its environment (29,30).

Measurement of this quantity, and how this quantity changes with temperature, solvent, viscosity, or other experimental parameter, provides detailed

insight into the dynamics of the system.

In addition to vibrational pure dephasing and vibrational population

relaxation (lifetime), another contribution to the homogeneous dephasing

time is orientational relaxation. The role of orientational relaxation in vibrational echo experiments of W(CO)6 has been previously discussed in detail

(21). In the experiments presented below on Rh(CO)2 acac in dibutyl phthalate (DBP) and on myoglobin and hemoglobin proteins, orientational relaxation does not occur on the time scale of the vibrational echo experiments

because of the samples’ high viscosities (16). This fact was confirmed by

magic angle pump probe experiments. Therefore, orientational relaxation

is not discussed further.

B. Experimental Procedures

The vibrational echo experiments require tunable IR pulses with durations

of ¾1 ps, energies of ắ1 àJ. Most of the experiments described below

were performed using IR pulses of wavelength near ắ5 àm generated by

Copyright © 2001 by Taylor & Francis Group, LLC


Rector and Fayer

the Stanford superconducting-accelerator-pumped free electron laser (FEL).

The FEL has been described in detail elsewhere (14,31,32). As stated above,

an example vibrational echo scan performed using the FEL is shown in

Fig. 2, which required approximately 15 minutes of averaging time. Signalto-noise ratios of this quality are typical for these experiments and enable

the resolution of dephasing mechanisms, as detailed below.

More recently, a commercial table top Ti:sapphire-based OPA system

has been used to perform vibrational echo experiments. Several years

ago, the FEL made it possible to perform the first ultrafast vibrational

echo experiments. The advent of tabletop systems now makes it possible

to perform vibrational echo experiments more routinely. Briefly, the

Ti:sapphire-based system uses 5 W from a diode pumped doubled Nd:VO4

laser to pump a Ti:sapphire oscillator, which produces fast, high-repetitionrate, 1 W, ¾800 nm pulses. These pulses are used as a seed for a Ti:sapphire

regenerative amplifier (regen). The seed pulses are temporally stretched

using conventional techniques. The bandwidth of the pulses is limited

by slits in the stretching system. For the experiments described here, the

bandwidth was limited to ¾18 cm 1 . The seed is then injected into the

regen’s cavity. The regen is pumped with 9.5 W from an intercavity doubled

Nd:YLF laser. The regen cavity is triggered and amplifies the pulses to

>1 W at 1 kHz. The regen output is compressed back to ¾1 ps, 18 cm 1 ,

1 W at 1 kHz for pumping an OPA.

In the OPA, part of the incoming light from the regen is used to

generate a white light continuum. This light is mixed with the rest of

the regen beam in two passes through a BBO crystal. After the first pass

through the BBO, a grating is used to wavelength select and narrow the

broad bandwidth that is the output of the BBO. The narrowed idler from

the first pass is amplified in the second pass. The signal and idler output

of the BBO OPA become the pump and signal in a final AgGaS OPA,

which generates midinfrared light. At 5 µm, the OPA typically produces

6 7 µJ/pulse at 1 kHz. Substantially more energy can be obtained when

fs pulses rather than ps pulses are used. The IR output of the OPA is

directed into the experimental set up for performing vibrational echo and

vibrational pump-probe experiments.

One of the difficulties in performing the IR vibrational echo experiments is the fact that the IR beam is invisible. Unlike a UV beam, which can

be viewed with a fluorescing card, there is no really good simple method

for visualizing the IR beam. To overcome this problem, a coalignment

system is used. The coalignment system efficiently coaligns the IR beam

with a visible (HeNe) beam. The HeNe beam is brought into the system by

Copyright © 2001 by Taylor & Francis Group, LLC

Infrared Vibrational Echo Experiments


reflection off of a Ge plate set at Brewster’s angle for the IR. The IR beam

passes through the plate. The HeNe and IR beams are made collinear. All

subsequent optics are achromatic, e.g., off-axis parabolic reflectors are used

instead of lenses, so the visible and IR beams remain aligned. It is then

possible to align the experimental system using the visible HeNe beam.

The entire mid-IR part of the experiment is be enclosed in a purged

(with dry air or N2 ) compartment to eliminate the substantial atmospheric

water absorptions. Fifteen percent of the IR beam is split off with a ZnSe

beamsplitter and directed to the sample. The remaining 85% of the beam

passes down a computer-controlled 0.1 µm step stepper motor delay line

and is then sent into the sample. For the echo experiments, the probe beam is

chopped at 500 Hz; for the pump-probe experiments the pump is chopped.

Two matched 6” f.l. 90° off-axis parabolic reectors are used to focus to

ắ100 àm and then recollimate the IR beams. The sample is contained in

continuous flow cryostat. After the focused IR beams pass through the

sample and are recollimated, either the probe or echo beam is directed into

a HgCdTe detector. The signal from the detector is sampled by a gated

integrator, the output of which is measured using a lock in amplifier. The

500 Hz signal from the lock-in is digitized for storage by a computer. To

switch between a vibrational echo and pump probe experiments, only the

delay line scanning direction, the beam that is chopped, and the detector

that is sampled are changed.

A vibrational echo scan taken with the OPA system is shown in Fig. 3.

These data, on hemoglobin-CO in EgOH/H2 O at 40 K, decay exponentially

at 11.0 s. These echo data were taken on a sample in which the protein

has a very strong background absorption compared to the CO peak under

study. In addition, the sample is somewhat turbid to the eye. Nonetheless,

it is possible to obtain high-quality echo data. The data took approximately

10 minutes to acquire.



In this section, a detailed vibrational echo study of Rh(CO)2 acac in DBP

above and below the solvent’s glass transition temperature Tg D 169 K is

presented (17,18). The asymmetrical CO-stretching mode of Rh(CO)2 acac

near 2000 cm 1 is examined over a wide range of temperatures. The temperature dependence of the pure dephasing time, TŁ2 , which reflects the magnitude of the perturbations of the transition energy caused by fluctuations of

Copyright © 2001 by Taylor & Francis Group, LLC


Rector and Fayer

Figure 3 Vibrational echo decay data for the CO-stretching mode of the protein

hemoglobin-CO ¾1950 cm 1 at 40 K and a fit to a single exponential function. The data were taken using a Ti:sapphire-based optical parametric amplifier

system. The decay constant is 11.0 ps, corresponding to a homogeneous linewidth

of 0.24 cm 1 . In contrast, the absorption linewidth is ¾9 cm 1 .

the bath, shows two clear temperature ranges in which different dynamics

are responsible for the vibrational dephasing.

A. Liquid/Glass Results

Vibrational echo and vibrational pump-probe experiments were conducted

on the CO asymmetric stretching mode of Rh(CO)2 acac 2010 cm 1 in

DBP from 3.4 to 250 K. Figure 2 shows vibrational echo data taken at 3.4 K

Copyright © 2001 by Taylor & Francis Group, LLC

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A. The Vibrational Echo Method

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