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A. Vibrational Echo Spectroscopy Theory

A. Vibrational Echo Spectroscopy Theory

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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



(6)



i,j



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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(7)



Infrared Vibrational Echo Experiments



249



where ij is the transition dipole matrix element of the jth transition of the

ith species.

B. Model Calculation



To perform the VES calculations it is necessary to consider a finite duration pulse, which has a finite bandwidth. In addition, the actual shape of the

vibrational echo spectrum depends on the bandwidth of the laser pulse and

the spectroscopic line shape. Several species with different concentrations,

transition dipole moments, line shapes, and homogeneous dephasing times

can contribute to the signal. Therefore, VES calculations require determination of the nonlinear polarization using procedures that can accommodate

these properties of real systems.

To calculate the vibrational echo intensity as a function of laser wavelength, using the details of the sample and realistic laser pulses, an efficient

numerical algorithm for computing the vibrational echo signal is employed

(63). The vibrational echo spectrum is calculated by numerically evaluating

all of the rephasing and nonrephasing terms for the third-order nonlinear

polarization, P 3 , that contribute to the signal in the vibrational echo geometry (63).

To calculate the vibrational echo observable for a fixed laser

frequency, ω1 , P 3 must be integrated over the spectroscopic line, g ω , or

the laser bandwidth, whichever is narrower, and then the modulus square

of the result must be integrated over all time since the observable is the

integrated intensity of the vibrational echo pulse,

Is , ω



/



1

1



1



dts



2



dωg ω Ptot3 ω, ts , ω



(8)



0



is the separation between the two laser pulses. This is the situation

for a single transition of a single species. In general, there are two or

more spectroscopic lines with independent P 3 . The contribution from each

transition of each species must be summed at the polarization level and

squared

Is , ω



/



1

1



1



2



dω gi,j ω

i,j



dts

i,j



0



i,j



3

Ptot,i,j



ω , ts , ω

i,j



(9)



where i is the label for the species and j is the label for the jth transition of

the ith species. It is necessary to distinguish between transitions on different



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Rector and Fayer



species since the species may have different concentrations as well as the

transitions having distinct line shapes, gi,j ωi,j , and transition dipole matrix

elements, µi,j .

Calculations were performed using Equation (9). Figure 8 displays

model calculations for a system with a broad, high optical density

(OD) solvent absorption and a narrow, low OD solute absorption. The

abscissa is centered about the peak of the solute spectrum. Figure 8A

is a model absorption spectrum. The parameters have been selected so

that the broad solvent absorption has a 100 times larger optical density

than the solute absorption. The inset shows a magnified view of the

solute absorption. Figures 8B and 8C show background free vibrational

echo spectra calculated using Equation (9), which demonstrate the two

mechanisms for solvent background suppression. In Fig. 8B, the spectrum

is calculated with D 0, and the suppression occurs through transition

dipole selectivity because the solute has a large µ but low concentration

relative to the solvent. The suppression arises from the m2 µ8 dependence of

the vibrational echo spectrum versus the mµ2 dependence of the absorption

spectrum. This situation may be encountered frequently in real systems in

which the peak of interest is a solute fundamental while the background

consists of overtones and combination bands of the solvent. In Fig. 8C,

an example of T2 suppression is shown. The spectrum is calculated with

D 5 ps. The solvent has T2 D 1.0 ps and the solute has T2 D 10 ps.



Figure 8 Model calculations of vibrational echo spectroscopy (VES) for a system

with a broad, strongly absorbing solvent background and a narrow, weakly

absorbing solute. The abscissa is centered about the peak of the solute spectrum.

(A) Absorption spectrum. The parameters have been selected so that the broad

solute absorption has a 100 times larger optical density than the solute absorption.

The inset shows a magnified view of the solute absorption. The solute is a

strong absorber but low in concentration. The background is composed of weak

absorbers at high concentration. (B) The VES spectrum calculated with D 0. The

background suppression occurs because the solute has a large transition dipole

matrix element relative to the solvent even though it is low in concentration.

(C) An example of T2 suppression. The solute and solvent transition dipole matrix

elements and concentrations were selected to give similar vibrational echo signals

at D 0. However, in this case, the solute T2 D 10 ps and the solvent T2 D 1 ps.

The pulse delay is D 5 ps. Because T2 for the solvent is fast compared to T2

for the solute, the solvent vibrational echo decay is essentially complete while the

solute vibrational echo signal is still significant. Background suppression occurs

because of differences in dynamics.



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Infrared Vibrational Echo Experiments



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251



252



Rector and Fayer



C. Experimental Demonstrations of VES



VES requires a tunable source of infrared pulses. In the experiments

presented below, vibrational echo spectra were taken using the Stanford

FEL employing the same experimental setup used to perform the vibrational

echo decay experiments discussed above (16).

Figure 9 displays the absorption and VES spectra of W(CO)6 and

Rh(CO)2 acac in DBP at room temperature. The upper trace is the absorption

spectrum. The peak at 1976 cm 1 is the asymmetric CO-stretching mode of



Figure 9 Absorption spectrum (upper trace) and VES spectrum (lower trace)

of the asymmetrical CO-stretching modes of W(CO)6 (centered at 1976 cm 1 )

and Rh(CO)2 acac (centered at 2010 cm 1 ) in the solvent dibutylphthalate at room

temperature. The vertical axis is in absorbance units for the upper trace. For the

VES spectrum, the vertical axis is in arbitrary units. In the VES spectrum, the

solvent background and solvent peaks are eliminated.



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Infrared Vibrational Echo Experiments



253



W(CO)6 and the peak at 2012 cm 1 is the asymmetric CO-stretching mode

of Rh(CO)2 acac. The Rh(CO)2 acac mode is the same as the one mentioned

in the previous section. The W(CO)6 mode has been studied previously,

although not specifically mentioned here (14,15,64). The vertical axis is in

absorbance units for the upper trace. The DBP has a very broad absorption

in the region giving a background absorbance of ¾0.5. In addition there is at

least one solvent peak at ¾1948 cm 1 , which is indicated by an arrow. The

lower trace is the VES spectrum. For the VES spectrum, the vertical axis is

in arbitrary units. Two features are immediately clear. First, the background

is zero, and second, the solvent peak at 1948 cm 1 is not visible.

As discussed above, there are two mechanisms by which VES can

eliminate background and spectral peaks, T2 selectivity, and transition

dipole matrix element selectivity. The VES scan in Fig. 9 was taken with

D 0. Nonetheless, both selectivity mechanisms can be active. The pulses

have finite duration of ¾1 ps. The vibrational echo signal arises from three

interactions with the fields:the first interaction is with the first pulse, and

the second and third interactions are with the second pulse. The interactions do not have to be time coincident, only time ordered, i.e., the

second interaction must come after the first, and the third interaction must

come after the second. A transition with a T2 that is longer than the

pulse will produce a polarization that involves the integral of the time

ordered interactions throughout the pulses. Since the intensity of the signal

is related to the absolute value squared of the polarization, the signal

grows dramatically during the pulse duration. However, if T2 is very short,

the three interactions must occur almost simultaneously, and the polarization does not increase integrally throughout the pulse, greatly reducing the

signal.

While T2 selectivity may contribute to the elimination of the solvent

background, it is clear that in this sample transition dipole matrix element

selectivity will eliminate the background. Using round numbers, the DBP

concentration is ¾10 M and its absorbance is ¾1. The metal carbonyl

concentrations are ¾10 3 M and their absorbances are ¾1. Therefore, the

metal carbonyl extinction coefficients are ¾104 larger than the solvents

and their concentrations are ¾104 smaller than the solvents. In terms of

the extinction coefficient, ε, and the concentration, m, the VES signal,

Is / m2 ε4 . Therefore, Is should be on the order of ¾108 greater for the

metal carbonyls than for the DBP solvent. The result is the observed zero

solvent background spectrum.

Figure 10 illustrates T2 selectivity between the metal carbonyl peaks.

Two scans were taken, with zero delay time and with 1 ps delay time



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