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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

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Infrared Vibrational Echo Experiments


and an exponential fit [Equation (1)]. Within experimental uncertainty, the

decay shown in Fig. 2 is a single exponential, indicating that a Lorentzian

homogeneous line shape. The T2 time is 95.2 ps, yielding a homogeneous linewidth of 0.11 cm 1 . For comparison, the absorption spectrum

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

absorption line is massively inhomogeneously broadened. The absorption

spectrum measures the inhomogeneous width and cannot provide information on the underlying homogeneous dephasing. The vibrational echo

experiments show that the absorption line is inhomogeneously broadened

at all temperatures studied, including 250 K 1/ T2 D 1.5 cm 1 . Above

Tg , the sample is a liquid, but the vibrational spectrum is still inhomogeneously broadened.

Figure 4 displays the temperature-dependent vibrational echo

(triangles) and pump-probe (squares) experimental results. The pump-probe

experiments measure T1 . The data is plotted as 2T1 , since this is the

Figure 4 Vibrational echo (triangles) and pump-probe (squares) data for the

asymmetrical CO-stretching mode of Rh(CO)2 acac in DBP. The pump-probe

results are plotted as 2T1 , for use with Equation (2). The solid line through

the T1 data is the best fit to the temperature dependence. Using these results,

the temperature-dependent pure dephasing times, TŁ2 , can be calculated from

Equation (2).

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Rector and Fayer

relevant quantity [see Equation (2)]. As often is the case, the temperature

dependence of 2T1 is very mild, and the temperature dependence of T2 is

much steeper. The pure dephasing, TŁ2 is obtained using Equation (2) and

the 2T1 and T2 values obtained from the experiments.

Figure 5 displays the values of the pure dephasing width versus

temperature on a log plot (17,33). The solid line through the data is a

Figure 5 Pure dephasing widths, 1/ TŁ2 , of the asymmetrical CO-stretching

mode of Rh(CO)2 acac in DBP versus temperature on a log plot. The solid line

through the data is a fit to Equation (4), the sum of a power law and an exponentially

activated process. The inset is an Arrhenius plot at higher temperatures showing

that the process is activated. Note that there is no break at the experimental glass

transition temperature, 169 K. The best fit has the power law exponent, ˛ D 1.0,

and the activation energy, E D 385 cm 1 .

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Infrared Vibrational Echo Experiments

fit to the form


D a 1 T˛ C a 2 e





with ˛ D 1.0 š 0.1 and E D 385 š 50 cm 1 . The inset is an Arrhenius

(time vs. inverse temperature on semilog) plot of the high-temperature data

showing that the data are exponentially activated at higher temperatures

and that there is no break in the temperature dependence at Tg D 169 K.

B. Liquid/Glass Dephasing Mechanisms

1. Low-Temperature Pure Dephasing of Rh CO 2 acac

Pure dephasing of the form T˛ where ˛ ³ 1 has been observed for

homogeneous pure dephasing of electronic transitions of molecules in

low-temperature glasses using photon echoes (5–7) and hole-burning

spectroscopy (7,8,34–37). The electronic dephasing has been described

using the two-level system (TLS) model of low-temperature glass

dynamics (7,34,38,39).

The TLS theory was originally developed in the early 1970s to explain

the anomalous heat capacity of low-temperature glasses, which is approximately linear in temperature (40,41). Glasses are continuously undergoing

structural changes, even at low temperatures. The complex potential surface

on which local structural dynamics occur is modeled as a collection of

double wells. Only the lowest energy levels are involved, at low temperatures, so these are referred to as TLS. The mechanism of TLS-induced

pure dephasing is illustrated in Fig. 6. Phonon-assisted tunneling can cause

transitions between the two energy levels. At very low temperatures, the

uptake of energy in going from a lower energy structure to a higher energy

structure dominates the heat capacity. A glass is modeled as having many

TLS with a broad distribution of tunnel splittings, E. If the probability,

P E , of having a splitting E is constant, P E D C, (all Es are equally

probable), then the heat capacity is T1 .

The description of electronic dephasing in low temperature glasses is

based on the TLS dynamics (7,34,38,39). We propose that identical considerations can apply to the vibrational dephasing of Rh(CO)2 acac in DBP at

low temperature. For those TLS with E not too large E < ¾2 kT , the TLS

are constantly making transition between the levels with a rate dependent

upon E and the tunneling parameter (42). This is illustrated in the bottom

part of Fig. 6. Transitions from one side of the double well to the other

correspond to changes in the local glass structure. The molecular oscillator

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Rector and Fayer

Figure 6 A schematic of the two-level system dynamics dephasing mechanism

proposed to explain the observed T1 temperature-dependent vibrational dephasing of

the asymmetrical CO-stretching mode of Rh(CO)2 acac in DBP at low temperature.

Phonon-assisted tunneling of the two-level systems (TLS) causes local structural

fluctuations in the glass. The molecular oscillator, M, is coupled to many TLS. The

structural fluctuations of the TLS produces fluctuating forces at M, resulting in pure


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Infrared Vibrational Echo Experiments


is coupled to many TLS. The structural changes produce fluctuating strains,

resulting in fluctuating forces on the CO oscillator. Thus, the vibrational

pure dephasing can be caused by TLS dynamics. The uncorrelated sudden

jump model predicts that for P E D CE , the temperature dependence of

the pure dephasing is T1C (7,8,43). Therefore, for the flat distribution,

D 0, the pure dephasing temperature dependence is T1 . T1 and somewhat steeper temperature dependences, e.g., T1.3 , have been observed in

electronic dephasing experiments in low-temperature glasses (5,7,8,37,43).

Recent theoretical work, which has examined the problem in more detail,

suggests that even the apparent superlinear temperature dependences may

arise from an energy distribution P E D C (28). Other theoretical work has

investigated the influence of coupled TLS (44). Regardless of the theoretical approach, the qualitative results are the same. Coupling of a transition

to a distribution of tunneling TLS can produce pure dephasing, which is

essentially T1 .

The success of the TLS model in describing a large variety of

distinct experiments adds weight to the proposition that the vibrational pure

dephasing is produced by coupling to TLS. In electron excited-state photon

echo experiments and hole-burning experiments, TLS dynamics have been

observed a temperatures ¾

D10 K. At higher temperatures, other processes

with steeper temperature dependences dominated the pure dephasing, as

well as other observables, such as heat capacities. In most systems,

manifestations of TLS dynamics cannot be observed above a few K. In

the vibrational dephasing experiments, the T1 temperature dependence

manifests itself to ¾80 K. Additional experiments on this system and

other low-temperature glassy systems are currently in progress. These will

add additional information on the nature of vibrational dynamics at low


2. High-Temperature Pure Dephasing of Rh CO 2 acac

Above ¾80 K, the T1 vibrational pure dephasing is dominated by the

exponentially activated process. Electronic dephasing experiments have

also shown power law temperature dependences that go over to activated

processes at higher temperatures (7,8,45). However, in the electronic

experiments, power law behavior is observed only to a few degrees K

because in the electronic dephasing experiments it is found that E D

¾15–30 cm 1 . Therefore, the activated process [arising from coupling

of the electronic transition to low-frequency modes of the glass (46,47).]

begins to dominate the power law pure dephasing at lower temperatures

than is observed for the CO vibrational pure dephasing of Rh(CO)2 acac. In

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Rector and Fayer

the vibrational pure dephasing experiments, the E D ¾400 cm 1 . Thus,

the power law component of the temperature dependence is not masked

until higher temperature.

In the Rh CO 2 acac in DBP system, the temperature dependence of

the pure dephasing changes rapidly above ¾80 K. By 100 K the temperature dependence is well described by the activated process alone (see inset

in Fig. 5). There is no break in the pure dephasing data as the sample passes

through Tg .

The activation energy, E D ¾400 cm 1 , is well above the phonon

modes of organic solids (48,49). Furthermore, the far-IR absorption spectra

of neat DBP show no significant transitions in the region around 400 cm 1 ,

indicating that there is no specific mode of the solvent that might couple

strongly to the CO mode. These facts suggest that the high-temperature

Arrhenius pure dephasing process is not caused by a motion associated

with the glass/liquid solvent, but rather that the pure dephasing arises from

coupling of the CO mode to another internal mode of Rh(CO)2 acac. If an

internal low-frequency mode is excited, the combination band frequency

can be different from the sum of the two vibrational frequencies. Therefore,

excitation of a low-frequency mode can shift the frequency of the CO mode

by an amount ω. The proposed mechanism is shown schematically in

Fig. 7.

For the proposed mechanism to account for the observed hightemperature pure dephasing, a mode of ¾400 cm 1 must couple

nonnegligibly to the asymmetric CO stretch so that ω is significant. The

Rh-C asymmetric stretching mode has an transition energy of 405 cm 1

(50). The closest other modes of Rh(CO)2 acac are outside of the error

bars on the activation energy (50). Rh-C stretch couples more strongly to

the CO mode than modes of lower frequency, which become populated at

lower temperature. Rh(CO)2 acac has significant back donation of electron

density from the Rh d to the CO p Ł antibonding orbital (back bonding)

that weakens the CO bond and red-shifts the transition energy. Thus, the

magnitude of back bonding plays a significant role in determining the

transition frequency. When the Rh-C mode is thermally excited from the

v D 0 state to the v D 1 state, the average bond length will increase. The

increase in the sigma bond length will decrease the Rh d -CO p Ł orbital

overlap and, therefore, decrease the magnitude of the back bonding. Thus,

excitation of the Rh-C mode causes a blue shift of the CO-stretching

frequency by decreasing the back bonding (33).

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Infrared Vibrational Echo Experiments


Figure 7 A schematic intramolecular low-frequency vibrational dynamics mechanism proposed to explain the observed exponentially activated temperature-dependent vibrational dephasing of the asymmetric CO-stretching mode of

Rh(CO)2 acac in DBP at high temperature. Excitation and relaxation of the

low-frequency Rh-C stretching mode causes the CO stretch frequency to shift ω.

The Rh-C mode is strongly coupled to the CO stretch through back bonding. Excitation of the Rh-C stretch increases the bond length, reducing and back bonding,

and causing a blue shift of the CO stretch by ω.

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Rector and Fayer

The proposed mechanism is thermal excitation of the Rh-C stretch

mode causes the CO-stretching mode transition frequency to shift a small

amount, ω, as shown in Fig. 7A. During the time period in which the

Rh-C mode is excited, the initially prepared CO superposition state precesses at a higher frequency, as indicated by the dashed arrow in Fig. 7A.

Thus, a phase error develops. For a small ω and a short , the phase error

is on the order of ω < 1. In the slow-exchange, weak coupling limit, the

pure dephasing contribution to the linewidth from repeated excitation and

relaxation of the low-frequency mode is (51,52):



ω 2



1 C ω





Equation (5) shows that the contribution to the homogeneous linewidth

from the excitation of the low-frequency mode will be exponentially activated. The right-hand side of Equation (5) is consistent with Equation (4)

which was used to fit the data. The factor multiplying the exponential

is the constant a2 in Equation (4). This term dominates the temperature

dependence at high temperature.

Although data is not available for Rh(CO)2 acac, IR absorption

measurements on M(CO)6 (M D Mo, Cr) support this mechanism (53).

The frequency of the combination band of the M-C asymmetrical stretch

and the CO asymmetrical stretch is ¾20 cm 1 higher than the sum of

the two fundamental energies alone (53). Thus, the transition of the CO

stretch is 20 cm 1 higher in energy when the M-C mode is excited

ω ¾

D C20 cm 1 . The change in back bonding upon excitation of the

Rh-C mode provides a direct mechanism for coupling excitation of the

Rh-C stretch to the CO stretch transition frequency.

It should be possible to estimate , the lifetime of the low-frequency

Rh-C stretch using a2 and ω in Equation (5). The values for a2 D 1.2 THz

and ω D 20 cm 1 yield a value for of 0.75 ps. For a low-frequency

mode that has a number of lower-frequency internal modes and the

continuum of solvent modes to relax into, 0.75 ps is not an unreasonable

value for the lifetime. A measurement of the Rh-C stretching mode lifetime

would provide the necessary information to determine if the proposed

dephasing mechanism is valid. In principle, the same mechanism will

produce a temperature-dependent absorption line shift. However, other

factors, particularly the change in the solvent density with temperature

strongly influence the line position. Therefore, temperature-dependent line

shifts cannot be used to test the proposed model.

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A. Vibrational Echo Spectroscopy Theory

In this section we present theoretical and experimental demonstrations of a

vibrational spectroscopic technique, vibrational echo spectroscopy (VES)

(54,55). The VES technique can generate a vibrational transition spectrum with background suppression using the nonlinear vibrational echo

pulse sequence. In contrast to the previous results, VES is a utilization of

vibrational echoes to measure spectra rather than dynamics. In a standard

vibrational echo experiment, the wavelength of the IR light is fixed, and

the delay, , between the excitation pulses is scanned. In VES, is fixed

and the wavelength is scanned.

Background suppression in VES is in some respects analogous to

NMR background suppression techniques (56,57). In both types of spectroscopy, coherent pulses sequences are used to remove unwanted spectral


Nuclear magnetic resonance and other magnetic resonance spectroscopies have had an enormous impact on the understanding of molecular

structure and dynamics in the last 50 years (2,58–60). IR spectroscopy

is inherently faster than NMR and can yield information about molecular

motions and interactions in the fs-ns time ranges, while NMR yields information on far longer time scales. Infrared absorption spectroscopy has a

much longer history, dating back to Newton’s discovery of infrared radiation in the early 1700s (61). However, because of the relative difficulty in

obtaining ultrafast IR pulses, coherent pulsed IR spectroscopy is a relatively

new field.

Infrared absorption spectroscopy is a powerful technique for obtaining

molecular structural information. In the midinfrared, all but the smallest

molecules have a large number of transitions, which arise from fundamental,

overtone, and combination modes. An absorption spectrum provides information about bonding, the shape of the molecular potential surface, solvent

interactions, and dynamics. However, even moderate-sized molecules can

generate spectra with a large number of peaks. For a large molecule, such

as a protein, a solute in a complex solvent, tissue, or cells, the spectrum

may become so crowded that clean observation of the spectral feature of

interest can become difficult. The FTIR measurements, like the NMR spectroscopy before the spin echo, is a useful technique, but its utility falls off

rapidly with molecular size. The VES technique may extend many of the

useful observations of FTIR to far larger and more complex structures by

improving line contrast and background suppression.

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Rector and Fayer

In VES, spectral selectivity can be achieved through two mechanisms:

transition dipole selectivity and homogeneous dephasing T2 selectivity.

If the background absorption, which can be a broad, essentially continuous absorption of undesired peaks, has homogeneous dephasing times, Tb2

(where the superscript b indicates background), short compared to the T2 of

the lines of interest, then VES can use the time evolution of the system to

discriminate against the unwanted features. The time, , between the pulses

in the vibrational echo sequence is set such that it is long compared to Tb2

but short compared to T2 . The VES signal from the background will have

decayed to zero while the signal from the desired peaks will be nonzero.

Scanning the IR wavelength of the vibrational echo excitation pulses and

detecting the vibrational echo signal versus frequency will generate a spectrum in which the background is removed. If the background is composed

of essentially a continuum of overtones and combination bands, while the

peak of interest is a fundamental, it is likely that Tb2 < T2 .

It is also possible to discriminate against the unwanted signals based

on the relative strengths of the transitions even when Tb2 ¾

D T2 . Absorbance

is proportional to m 2 while the vibrational echo signal is proportional to

m2 8 , where m is the concentration of the species and is the transition

dipole matrix element. When background is composed of a high concentration of weak absorbers (m large, small) and the spectral features of

interest are in low concentration but are strong absorbers (m small, large),

the background absorption can overwhelm the desired features while the

vibrational echo spectrum suppresses the background and reveals the relevant peaks.

Each spectral line can arise from a species with a particular

concentration and transition dipole moment matrix element and a particular

linewidth determined by the extent of homogeneous and inhomogeneous

broadening. The magnitude of absorption as a function of frequency is

given by Beer’s law:

εij ω mi l

Aω D



where A ω is the absorption at frequency ω and εij ω is the molar absorbtivity or the extinction coefficient of the jth transition of the ith species. ε

has units of M 1 cm 1 and is related to the transition dipole matrix element

squared (62). mi is the concentration of the ith species in the sample, and

l is the length of the sample. For the jth transition of the ith species, the

absorption is

A D εij mi l / j

j mi l

ij 2

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