Spectroscopic Characteristics of Triply Vibrationally Enhanced Four

J. Phys. Chem. B 2004, 108, 10493-10504

10493

Spectroscopic Characteristics of Triply Vibrationally Enhanced Four-Wave Mixing Spectroscopy† Daniel M. Besemann, Kent A. Meyer, and John C. Wright* Department of Chemistry, 1101 UniVersity AVenue, UniVersity of Wisconsin, Madison, Wisconsin 53706 ReceiVed: January 29, 2004; In Final Form: March 8, 2004

Triply vibrationally enhanced (TRIVE) four-wave mixing is a fully resonant, frequency domain spectroscopy that is capable of coherent multidimensional vibrational spectroscopy. TRIVE has 12 different coherence pathways that differ in their time ordering and resonances. The pathways are the coherent analogue to twocolor pump-probe pathways. Specific pathways or sets of pathways can be chosen by appropriate selection of time delays and resonance conditions. The pathways have characteristic positions and line shapes in threedimensional frequency space and their coherent interference has consequences in interpreting the spectra. The line shapes and the relative intensities of different pathways are dependent on the population relaxation and dephasing rates. The different pathways also have different capabilities for line-narrowing inhomogeneously broadened transitions. The narrowing is controlled by the interference between pathways and the quantum level interference between different parts of the inhomogeneously broadened envelope. We also show that selection of the output frequency in two-color TRIVE methods constrains the selection rules that control the relative transition probabilities of the four transitions.

Introduction This paper is dedicated to Gerry Small’s pioneering work on the applications of lasers to chemical measurement, including hole burning,1-13 fluorescence line narrowing,9,14-29 laserexcited fluorescence in supersonic jets,30,31 and nonlinear line narrowing with multiresonant four-wave mixing.32 These applications have opened up new research areas, because they have provided us with high-resolution methods that can probe the molecular- and condensed-phase dynamics and resolve spectral features that are often obscured by inhomogeneous broadening and spectral congestion.33 All of these methods rely on an electronic excitation of a subset of molecules within an inhomogeneous envelope and an optical probe that is sensitive to the excited molecules. The probe observes a narrowing of the inhomogeneously broadened line and can often reveal the homogeneous profile for the subset of excited molecules. Recent work has extended these methods to coherent multidimensional vibrational spectroscopy (CMDVS) and created a new family of nonlinear spectroscopies that are multidimensional and probe the coupling between different quantum states.34-56 CMDVS can be performed in the frequency or time domain. Frequency-domain experiments measure the output intensity as a function of the frequency of each excitation field. Time-domain experiments heterodyne the output electric field with a local oscillator and measure the signal as a function of the time delays between each input field and the output signal. Fourier transforms into the frequency domain then provide multidimensional spectra. Most CMDVS methods are based on four-wave mixing (FWM), where three excitation beams (with frequencies ω1, ω2, ω3) can resonantly excite three different, sequential coherences. The final coherence launches an output †

Part of the special issue “Gerald Small Festschrift”. * Author to whom correspondence should be addressed. E-mail address: [email protected].

Figure 1. (a) Photon echo pathways and (b) DOVE-IR and DOVERaman pathways. The letters indicate the two states that represent the coherences or populations, and the numbers label the photon fields. The circles indicate the initial conditions, and the squares indicate the final coherences.

beam at an output frequency ω4 in the phase-matching direction that is defined by the wave vector B k4. Doubly vibrationally enhanced four-wave mixing (DOVEFWM) and heterodyned stimulated photon echo (HSPE) are two commonly used CMDVS methods that are vibrational analogues of multidimensional heteronuclear and homonuclear NMR.34,36-38,45,46 Homonuclear methods require an ultrafast pulse whose bandwidth simultaneously excites all coherences, whereas heteronuclear methods require excitation frequencies that exceed the pulse bandwidth.57,58 DOVE-FWM uses two independently tunable infrared beams (frequencies ω1 and ω2) and a third visible/near-visible beam (frequency ω3) to create the ω4 output beam. As shown in Figure 1, there are several coherence pathways that lead to the same output coherence and their contributions interfere.59 Each pathway has vibrational transitions that involve two infrared absorption/emission resonances and a single Raman resonance. Cross peaks can only occur in DOVE methods if one of the resonances involves a

10.1021/jp049597l CCC: $27.50 © 2004 American Chemical Society Published on Web 04/17/2004

10494 J. Phys. Chem. B, Vol. 108, No. 29, 2004 combination (or overtone) band that is created by intramolecular or intermolecular coupling. HSPE experiments are fully resonant FWM methods that are analogous to multidimensional homonuclear NMR methods such as COSY and NOESY.45-47,60-65 HSPE experiments require reproducible phase relationships between the excitation beams, so it is important that the excitation pulses be derived from a single ultrafast pulse with sufficient bandwidth to excite all the different vibrational states simultaneously. The ultrafast pulse k 2, B k3 beams that create the output B k4 is divided into the B k1, B beam. As shown in Figure 1, there are two final coherences that are responsible for creating the output. The first coherence involves only the fundamental vibrational modes, whereas the second coherence also involves the combination band or overtone of the fundamental modes. The two coherences have opposite signs and can cancel if they have the same amplitudes, frequencies, and dephasing rates. If the fundamental modes are coupled by intramolecular or intermolecular interactions, the two coherences will not cancel completely and cross peaks between the modes appear in the CMDVS spectra. This sensitivity to mode coupling is the signature of multidimensional methods. There is interest in a fully resonant FWM method that can provide CMDVS measurements on modes that are separated by frequencies beyond the bandwidth of an ultrafast laser system.57,58 Such experiments are also vibrational analogues of heteronuclear NMR. There have been several studies that have produced two-color two-dimensional (2-D) vibrational spectra. A two-color pump-probe experiment can excite one mode and probe the effects on a separate mode to create a 2-D IR spectrum.66,67 A coherent two-color heterodyned photon echo method has also been used to excite two different vibrational modes.68 The phase relationships were maintained in this latter approach, because the mode separation was smaller than the bandwidth of a typical ultrafast pulse, so one could derive the two colors from a single pulse. A frequency-domain triply vibrationally enhanced (TRIVE) FWM method has demonstrated CMDVS using three IR excitation beams (with frequencies of ω1, ω2, ω2′) to create an output at ω4 ) ω1 - ω2 + ω2′, using both temporally overlapped and temporally resolved pulses.69,70 The frequency-domain method does not require constant phase relationships for successive excitation pulses. In this paper, we explore the spectroscopic and line-narrowing characteristics of TRIVE methods. TRIVE methods have 12 different coherence pathways, and each pathway has characteristic resonances and interferences that affect the intensity, shape, and width of spectral features, depending upon the transition moments, dephasing rates, and inhomogeneous broadening effects on different molecular states. The pathways are the coherent analogue of the pathways for pump-probe methods. Quantum Interference Effects and Spectroscopic Characteristics The twelve TRIVE-FWM coherence pathways differ in regard to their time ordering and the resonant states.69,70 The pathways are shown in Figure 2 and summarized in Table 1. The pathways are labeled by subscripts that indicate the time ordering of the first two excitation beams and superscripts that indicate whether there are an even or odd number of bra side interactions (denoted as e and o, respectively). There are six even and six odd pathways for coupling between two IR fundamentals, labeled modes b and c (state d is the combination band of states b and c). The even and odd pathways create nonlinear polarizations that have opposite signs and, in the absence of coupling, they

Besemann et al.

Figure 2. WMEL diagrams for TRIVE pathways. The numerical subscripts refer to the order of the first two laser interactions. Each column is an even/odd (e/o) pair. State d is the combination band of states b and c.

destructively interfere. The interference eliminates the cross peaks between the two modes. Intramolecular and intermolecular coupling changes the frequencies, amplitudes, and/or dephasing rates of the pathways, so that the interference is incomplete and cross peaks appear. The letter pairs in the table represent coherences between the indicated states; the coherence represented by mn corresponds to the time-dependent wave function ψ(x,t) ) cmψm(x)eiωmt + cnψn(x)eiωnt. The response functions in the frequency domain and the time domain are also given in Table 1. The even and odd pathways of any particular time ordering have opposite signs, and the resulting spectroscopic line shape is dependent on their interference. To determine the total response, it is helpful to compare the responses from even/odd pairs. As a specific example, the frequency-domain response function for the even/odd pair with time ordering ω1, ω2, ω3 (Re,o 12 ) is

Re,o 12 (ω1,ω2,ω3) ∝

(

)

µgcµgb µbgµcg µcdµdb ∆cg∆cb ∆cg ∆db

(1)

where ∆mn ≡ δmn - iΓmn with δmn ≡ ωmn - ωL. The parameters ωmn, Γmn, and µmn respectively represent the frequency difference, dephasing rate, and transition moment for the m f n transition, δmn is the resonant detuning, and ωL represents the linear combination of input lasers that created the nm coherence. The summed time-domain response function for the even and odd pathways of the same time ordering is 3 -i(ωcg-iΓcg)t12 -i(ωcb-iΓcb)t23 e × Re,o 12 (t12,t23,t34) ∝ i µgcµgbe

(µbgµcge-i(ωcg-iΓcg)t34 - µcdµdbe-i(ωdb-iΓdb)t34) (2) Here, tij is the time delay between the interactions with the ωi and ωj pulses. These expressions can be rewritten to highlight the effects of the interference:

Re,o 12 (ω1,ω2,ω3) ∝

(

)

µgcµgb (µbgµcg - µcdµdb)∆cg - µbgµcg[ξ - i(Γdb - Γcg)] ∆cg∆cb ∆cg∆db

(3) Re,o 12 (t12,t23,t34) ∝ i3µgcµgbe-i(ωcg-iΓcg)t12e-i(ωcb-iΓcb)t23e-i(ωcg-iΓcg)t34 × (µbgµcg - µcdµdbei[ξ-i(Γcg-Γdb)]t34) (4)

Spectroscopic Characteristics of TRIVE-FWM

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10495

TABLE 1: TRIVE-FWM Coherence Pathways response function

frequency domain

pathway +1

-2

+3

-4

+1

+3

-2

-4

-2

+1

-2

+3

+1

-4

+3

+1

-2

-4

+3

-2

+1

-4

+1

-2

+3

-4

+1

+3

-2

-4

-2

+1

+3

-4

-2

+3

+1

-4

+3

+1

-2

-4

+3

-2

+1

-4

Re12

gg 98 cg 98 cb 98 cg 98 gg

Re13

gg 98 cg 98 dg 98 cg 98 gg

Re21

gg 98 gb 98cb 98 cg 98 gg

Re23

gg 98 gb 98 gg 98 cg 98 gg

Re31

gg 98 bg 98 dg 98 cg 98 gg

Re32

gg 98 bg 98 gg 98 cg 98 gg

Ro12

gg 98 cg 98 cb 98 db 98 bb

Ro13

gg 98 cg 98 dg 98 db 98 bb

Ro21

gg 98 gb 98 cb 98 db 98 bb

Ro23

gg 98 gb 98 bb 98 db 98 bb

Ro31

gg 98 bg 98 dg 98 db 98 bb

Ro32

gg 98 bg 98 bb 98 db 98 bb

+3

-4

µgcµgbµbgµcg ∆cg∆cb∆cg µgcµcdµdcµcg ∆cg∆dg∆cg µgbµgcµbgµcg ∆gb∆cb∆cg µgbµbgµgcµcg ∆gb∆gg∆cg µgbµbdµdcµcg ∆bg∆dg∆cg µgbµbgµgcµcg ∆bg∆gg∆cg -µgcµgbµdcµdb ∆cg∆cb∆db -µgcµcdµgbµdb ∆cg∆dg∆db -µgbµgcµcdµdb ∆gb∆cb∆db -µgbµbgµbdµdb ∆gb∆bb∆db -µgbµbdµgbµdb ∆bg∆dg∆db -µgbµbgµbdµdb ∆bg∆bb∆db

where ξ is the anharmonicity of the combination band (state d), i.e., ξ ) ωcg - ωdb ) ωbg - ωdc. Equations 3 and 4 show that any of three inequalities will prevent complete destructive interference between even and odd pathways:

µbgµcg * µcdµdb

(5)

ξ*0

(6)

Γdb * Γcg

(7)

All of these inequalities are manifestations of mode coupling.47 The three resonances in each TRIVE pathway create threedimensional (3-D) spectra with characteristic shapes. Figure 3a

Figure 3. Three-dimensional (3-D) frequency-domain (a) natural and (b) laser spaces for |Re12|. The surface is 10% of the maximum of |Re12|, which occurs at the center of the shape.

shows the magnitude of the Re12 pathway, as a function of the frequency combinations that characterize each resonance. The peak is centered at (ω1 ) ωcg, ω1 - ω2 ) ωcb, ω1 - ω2 + ω3 ) ωcg) and constant intensity contours form 3-D Lorentzian shells that are somewhat ellipsoidal nearer the maximum

time domain µgcµgbµbgµcge-i(ωcg-iΓcg)t12e-i(ωcb-iΓcb)t23e-i(ωcg-iΓcg)t34 µgcµcdµdcµcge-i(ωcg-iΓcg)t13e-i(ωdg-iΓdg)t32e-i(ωcg-iΓcg)t24 µgbµgcµbgµcge-i(ωgb-iΓbg)t21e-i(ωcb-iΓcb)t13e-i(ωcg-iΓcg)t34 µgbµbgµgcµcge-i(ωgb-iΓgb)t23e-i(ωgg-iΓgg)t31e-i(ωcg-iΓcg)t14 µgbµbdµdcµcge-i(ωbg-iΓbg)t31e-i(ωdg-iΓdg)t12e-i(ωcg-iΓcg)t24 µgbµbgµgcµcge-i(ωbg-iΓbg)t32e-i(ωgg-iΓgg)t21e-i(ωcg-iΓcg)t14 -µgcµgbµcdµdbe-i(ωcg-iΓcg)t12e-i(ωcb-iΓcb)t23e-i(ωdb- iΓdb)t34 -µgcµcdµgbµdbe-i(ωcg-iΓcg)t13e-i(ωdg-iΓdg)t32e-i(ωdb-iΓdb)t24 -µgbµgcµcdµdbe-i(ωgb-iΓbg)t21e-i(ωcb-iΓcb)t13e-i(ωdb-iΓdb)t34 -µgbµbgµbdµdbe-i(ωgb-iΓgb)t23e-i(ωbb-iΓbb)t31e-i(ωdb- iΓdb)t14 -µgbµbdµgbµdbe-i(ωbg-iΓbg)t31e-i(ωdg-iΓdg)t12e-i(ωdb-iΓdb)t24 -µgbµbgµbdµdbe-i(ωbg-iΓbg)t32e-i(ωbb-iΓbb)t21e-i(ωdb- iΓdb)t14

intensity in the center and starlike farther from the peak intensity. Figure 3b shows the skewed Lorentzian when the intensity contours are plotted as a function of the excitation frequencies (ω1, ω2, ω3). The skewing results because ω1, ω2, and ω3 affect different numbers of resonances. The peak is centered at (ω1 ) ωcg, ω2 ) ωbg, ω3 ) ωcg). Similar plots characterize the other pathways. The anharmonicity (ξ) created by the mode coupling shifts the peak position, which is characteristic of each pathway. The twelve TRIVE pathways give peaks at five corners of the cube with dimension ξ, which are illustrated in Figure 4 and

Figure 4. Laser space positions for TRIVE pathways. The origin of the coordinate system is located at (ω1 ) ωcg, ω2 ) wbg, ω3 ) ωbg). ξ is the anharmonicity of the combination band d ) b + c. The positions of each pathway are listed in Table 2.

summarized in Table 2. Notice that the peaks designated by D and E have opposite signs from those designated by A, B, and C. The extent of interference between the pathways is dependent

10496 J. Phys. Chem. B, Vol. 108, No. 29, 2004

Besemann et al.

TABLE 2: Laser Space Peak Positions for All TRIVE-FWM Pathways pathway positiona pathway positiona a

Re12 A Ro12 D

Re13 B Ro13 D

Re21 A Ro21 D

Re23 A Ro23 E

Re31 C Ro31 E

Re32 A Ro32 E

Letters A-E refer to features noted in Figure 4.

on the ratio ξ/Γ. In the limit where ξ ) 0, the pathways all collapse into a single peak, which could disappear if the µ values and Γ values are equal. Practical limitations often prevent acquisition of true 3-D spectra with three independently tunable beams such that ω4 ) ω1 - ω2 + ω3. If two tunable beams are available, two of the three excitation beams must have the same frequency so the spectroscopy is constrained to ω2 ) ω3 or ω1 ) ω3 (the choice ω1 ) ω2 is equivalent to ω2 ) ω3) and the resulting spectra are 2-D slices through the ω2 ) ω3 or ω1 ) ω3 planes in Figure 3. The two choices have very different selection rules for the possible transitions. If we assume that our two tunable frequencies are resonant with separate modes, the choice ω2 ) ω3 requires an emission and an absorption transition that involves the same mode and the transitions can each be fundamentals with ∆V ) (1. In this choice, the output frequency is identical to an excitation frequency (in the steady-state limit). The choice ω1 ) ω3 prevents all the transitions from being fundamentals and at least one transition must involve a combination band with lower transition probabilities. In this choice, spectral discrimination is possible, because the output frequency falls at ω4 ) 2ω1 - ω2 and is different from the excitation beams. Figure 5 schematically shows the changes in vibrational quantum numbers that are required for each choice. Here, the ket and bra side transitions are indicated by solid and dotted arrows, respectively, and the axes label the quantum numbers of the two coupled modes. Formally forbidden transitions are represented by diagonal arrows. Anharmonicity can prevent a fully resonant process from being achieved for some pathways. Figures 6 and 7 show the real and imaginary contributions from each even/odd pair to the 2-D ω2 ) ω3 response. The figures assume that all µ values and Γ values are equal, with ξ ) 5Γ. There are three possible peak positions for a fully resonant pathway, at (ω1 ) ωdb; ω2,ω3 ) ωbg), (ω1 ) ωcg; ω2,ω3 ) ωbg), and (ω1 ) ωcg; ω2,ω3 ) ωdc), and an additional peak for doubly resonant pathways, at (ω1 ) ωdb; ω2,ω3 ) ωdc). We designate these positions by the labels 1-4, respectively, and indicate them in the figures. From Figure 4, the ω2 ) ω3 plane passes through peaks A, B, and E, so peaks A (position 2), B (position 3), and E (position 1) have a tendency to dominate the 2-D TRIVE spectrum for ω2 ) ω3. Similarly, pathways centered at positions C and D cannot be triply resonant, and the corresponding pathways are reduced in their intensity. These pathways include Ro12, Ro13, Ro21, Re31. The dominant peaks at positions 1, 2, and 3 have relative magnitudes of approximately 3:4:1 if all µ values and Γ values are equal. Figure 8 shows the 2-D spectrum when all 12 TRIVE pathways are temporally overlapped under the conditions of equal µ values, Γij ) 1 cm-1, ωbg ) 2000 cm-1, ωcg ) 3000 cm-1, ξ ) 5 cm-1, and positive transition moments. Peaks 2 and 3 are dominated by even pathways, and the overall sign is positive, whereas peak 1 results from odd pathways and is negative. Because the transition moments for all TRIVE pathways are quite similar, it is expected that the transition moment signs for all TRIVE pathways will be identical. The net signs of the three peaks then will, generally, be such that

Figure 5. Representative changes in vibrational quantum numbers for two coupled modes (ν1 and ν2) for the four transitions in the different TRIVE pathways. Fundamental transitions appear as horizontal or vertical lines, and combination band transitions appear as diagonal lines. The axes are indicated in the upper left-hand diagram. The solid and dotted lines indicate ket and bra-side transitions, respectively. Note that the phase-matching geometry for ω4 ) ω1 - ω2 + ω2′ involves only fundamentals, whereas that for ω4 ) ω1 - ω-1 + ω2 requires a combination band.

the upper left peak is the negative of the upper and lower righthand peaks, as is observed in the real and imaginary components of Figure 8. Although Figure 8 highlights the three TRIVE peaks, it is probably not a realistic simulation of the response from a condensed-phase system. In a typical condensed-phase system, the Γ values are not likely to be identical and, in fact, may differ significantly.59 Figure 9 shows how the response in Figure 8 changes if all parameters remain unchanged except Γgg ) Γbb ) 0.1. The heights of peaks 1 and 2 increase, relative to peak 3, which significantly reduces the effect of the latter on the spectrum. Figures 9a and 9b show the effects of Γgg * Γbb near the pure dephasing limit.71 If Γgg < Γbb, the height of peak 2 increases, relative to peak 1, because the even pathways that contribute to peak 2 involve an intermediate gg coherence. The opposite is true if Γbb < Γgg, because the odd pathways that contribute to peak 1 pass through a bb coherence. Most theoretical treatments of triply vibrationally resonant FWM processes assume Γgg ) Γbb.47,72,73 Although it is not clear if this approximation is valid for multilevel systems, the remaining discussion of TRIVE assumes Γgg ) Γbb. The effects of changing the Γgg/Γbb ratio on the relative intensities (see Figure 9) is interesting. If two (or more) TRIVE pathways have identical transition moments, their integrated contributions to the complete FWM response should be identical. In fact, it is easy to demonstrate that, in the pure dephasing



Figure 6. Real R for TRIVE pathways in laser space with ω2 ) ω3. Solid contours are positive, dashed contours negative in intervals of 10% of the maximum of |R|. For some pathways, the maximum is not in the ω2 ) ω3 plane.

limit, the TRIVE line shape is independent of the homogeneous frequency correlations between modes b and c. Their difference results from the 3-D nature of the integrated contribution from each TRIVE pathway. Although the 3-D integral of the TRIVE response is independent of the relative dephasing rates, the integrated response over the ω2 ) ω3 plane is dependent on the relative dephasing rates. In the example, reduction of Γgg (or Γbb) significantly flattens the Re23 and Re32 (or Ro23 and Ro32) responses such that almost all of their integrated response is in the ω2 ) ω3 plane. Consequently, in the pure dephasing limit, the Ro32 and Ro23 pathways dominate peak 1 and the Re32 and Re23 pathways dominate peak 2. Because two pathways dominate each peak, the ratio of the peak 1/peak 2 magnitudes changes from 3:4 (Figure 8) to 1:1 when Γgg ) Γbb. Figure 9 shows that the magnitude ratio is quite sensitive to the ratio Γgg/Γbb. Therefore, TRIVE should be able to determine the ratio of ground-state to excited-state population relaxation rates for systems near the pure dephasing limit. The TRIVE line shapes are dependent on the relative dephasing rates. Figure 10 illustrates the range of line shapes for different choices of a single Γij, relative to the other Γ values. Increasing the value of Γij would have the opposite effect on the spectrum. Notice that, although Γgg and Γbb affected the relative integrated sizes of peaks 1-3, the remaining Γij largely cause various forms of peak narrowing. These effects can be intuitively visualized by considering the specific pathways that

are affected by each Γij. It has already been established that four TRIVE pathwayssRo12, Ro13, Ro21, and Re31sdo not contribute significantly to the 2-D TRIVE response in the presence of an anharmonicity (see Figure 10). The remaining contributing pathways each show Lorentzian tails that are oriented in different directions in the ω2 ) ω3 plane (see Figures 6 and 7). The directions are determined by the resonant detunings, δij, inside each resonant denominator, ∆ij. The width of each tail is determined by the Γij contained within each ∆ij. The subfigures in Figure 10 show that each Γij controls the width of tails that point in a single specific direction and pass through specific peaks. This control is the result of changes in the shapes of the TRIVE pathways that contain the relevant ∆ij. The effects of the Γij on the 2-D TRIVE line shapes are summarized in Table 3. It is important to appreciate that the simulations in Figures 9 and 10 are not intended to represent realistic TRIVE spectra. Experimental TRIVE line shapes will consist of features that are a convolution of the effects of all the inequalities discussed earlier; instead, Figures 9 and 10 are a guide for qualitative spectral interpretation of dephasing effects. Inhomogeneous Broadening Vibrational transition frequencies fluctuate on different time scales, because of thermal motion in the bath.72,74,75 The effects of the fluctuations have been treated by Tokmakoff et al.76 for time-domain 2-D IR spectroscopy. The fluctuations are char-


Besemann et al.

Figure 7. Imaginary R for TRIVE pathways in laser space with ω2 ) ω3. Solid contours are positive, dashed contours negative, shown in intervals of 10% of the maximum of |R|. For some pathways, the maximum is not in the ω2 ) ω3 plane.

Figure 8. Example line shape resulting from interference between the 12 TRIVE pathways. Parameters are given in the text. Solid contours are positive, dashed contours are negative, shown in intervals of 10% of the maximum of |R|.

acterized by the system-bath correlation function, which fluctuates with an amplitude σ and correlation time τc. Very long time scales (where στc . 1) are characteristic of inhomogeneous broadening and very short time scales (where στc , 1) are characteristic of homogeneous broadening. The correlation between the fluctuations of different states is important in CMDVS because the nonlinearities involve multiple transitions that occur within the dephasing time of the coherence. The transitions provide a snapshot of the instantaneous structure of the bath over the coherence time. Inhomogeneous broadening

results from the different transition frequencies of molecules in different microenvironments. The frequency shifts for two states can be positively or negatively correlated or uncorrelated, as illustrated in Figure 11.76 Perfectly positively correlated shifts have a constant difference in frequencies between the states, whereas perfectly negatively correlated shifts have a constant sum. The spectral features in CMDVS methods can be narrower than the inhomogeneous widths for the individual transitions, because of two mechanisms that we have labeled natural line



Figure 9. Changes in spectra of Figure 8 near the pure dephasing limit. Parameters are as given in the text, except (a) Γgg ) Γbb ) 0.1 cm-1; (b) Γgg ) 0.02 cm-1, Γbb ) 0.1 cm-1; and (c) Γgg ) 0.1 cm-1, Γbb ) 0.02 cm-1. Contours are shown in intervals of 10% of the maximum of |R|.

Figure 10. Illustration of individual dephasing signatures in 2-D frequency-domain TRIVE. Parameters are as given in the text, except (a) Γbg ) 0.25 cm-1; (b) Γcg ) 0.25 cm-1; (c) Γdg ) 0.25 cm-1; (d) Γdb ) 0.25 cm-1; and (e) Γcb ) 0.25 cm-1. Contours given in intervals of 10% of the maximum of |R|.

TABLE 3: Summary of Individual Dephasing Signatures in 2-D Frequency-Domain TRIVE-FWM for Decreasing Γija Γij

affected ∆ij

major affected pathways

spectral effect(s)

Γgg Γbb Γbg Γcg Γdg Γcb Γdb

∆gg ∆bb ∆bg, ∆gb ∆cg ∆dg ∆cb ∆db

Re23, Re32 Ro23, Ro32 Re21, Re23, Re32, Ro23, Ro31, Ro32 Re12, Re21, Re23, Re13, Re32 Re13, Ro31 Re12, Re21 Ro23, Ro31, Ro32

increase in size of peak 2 increase in size of peak 1 vertical narrowing of peaks 1 and 2 horizontal narrowing of peaks 2 and 3 diagonal narrowing of peaks 1 and 3 antidiagonal narrowing of peak 1 horizontal narrowing of peak 1

a Pathways not centered in the ω2 ) ω3 plane in the presence of an anharmonicity (Ro12, Ro13, Ro21, and Re31) are not included. Table assumes x ) ω1 and y ) ω2,3, as in Figure 9.

narrowing and traditional line narrowing. Natural line narrowing occurs when there is an intermediate coherence that involves the sum or difference of two inhomogeneous distributions, and the line width of the sum or difference coherence is manifested in the multidimensional spectrum. Because perfectly positively correlated transitions have a constant frequency difference between the two correlated modes, a coherence that is dependent

on the frequency difference between the two states will not exhibit any inhomogeneous broadening. Similarly, a coherence that is dependent on the sum of frequencies from two different states will not exhibit inhomogeneous broadening for negatively correlated broadening. If the inhomogeneous shifts are different in the two states, there will be residual inhomogeneous broadening but the widths of the sum or difference coherences will be


Besemann et al. o ωig ) ωig + ∆ωig

(8)

and combination sum bands as

ω(i + j)g ) ωig + ωjg - ξ

(9)

where ξ is the anharmonicity. The JPDF between any two frequency distributions is given by61

P(∆ωig,∆ωjg) )

{

1 2πx1 - F2σigσjg

×

(

[

)]}

2 2 ∆ωig ∆ωig∆ωjg ∆ωjg 1 exp - 2F + 2 2 σjgσjg 2(1 - F2) σig σjg

(10)

where σig is the inhomogeneous width of mode i and F is the correlation coefficient, which is defined as the normalized covariance of the random variables ∆ωig and ∆ωjg:

F) Figure 11. Relationships between frequency distributions of the Lorentzian profiles of example molecules within two inhomogeneously broadened envelopes: (a) perfect positive correlation, (b) perfect negative correlation, and (c) no correlation.

TABLE 4: TRIVE-FWM Processes Capable of Natural Line Narrowing intermediate coherence

narrowed correlation

Fcb

positive

Re12, Ro12, Re21, Ro21

Fdg

negative

Re13, Ro13, Re31, Ro31

capable IRFWM processes

reduced, relative to the contributing fundamentals. Table 4 summarizes the TRIVE pathways where natural line narrowing occurs because of an appropriate intermediate coherence. Note that the input fields that create the sum or difference do not need to be resonant with either inhomogeneously broadened modesthe only requirement for natural line narrowing is that the sum or difference of the two input fields be equal to the constant sum or difference of the correlated inhomogeneously broadened transitions. Of course, if the input fields are not resonant with at least one of the individual inhomogeneously broadened modes, the overall intensity of the nonlinear process is significantly reduced. In addition, the presence of these resonances is dependent on the presence of pure dephasing interactions.71 Traditional line narrowing has been discussed in the Introduction. An analytical expression that treats inhomogeneous broadening of FWM spectra in the frequency domain can be obtained by assuming that the broadening is linked to a single broadening parameter that has a Lorentzian distribution.77,78 In the time domain, an analytical expression can be obtained by assuming the broadening has a Gaussian probability distribution.72 Both models are semiquantitative because they assume a strict relationship between the inhomogeneously broadened distributions that does not account for the uncorrelated effects in a real chemical system. A normal joint probability distribution function (JPDF) can account for any degree of correlation and, therefore, is quite useful for quantitatively describing the IRFWM responses of inhomogeneously broadened systems.61,73 Implementation of the JPDF proceeds by first defining the ground-state molecular vibrational transition frequencies as

〈∆ωig∆ωig〉 σigσjg

(11)

where 〈‚‚‚〉 denotes a joint average of the individual transitions over the ensemble. The value of F can range from -1 in the case of perfect negative correlation, to 0 in the case of no correlation, to +1 for perfect positive correlation. The complete inhomogeneously broadened response function is obtained by integrating the homogeneously broadened response function over the JPDF:

RTOT(ω1,ω2) )

∫-∞∞ d(∆ωig)∫-∞∞ d(∆ωjg)P(∆ωig,∆ωjg)R(ω1,ω2,∆ωig,∆ωjg) (for the frequency domain) (12) or

RTOT(t1,t2) )

∫-∞∞ d(∆ωig)∫-∞∞ d(∆ωjg)P(∆ωig,∆ωjg)R(t1,t2,∆ωig,∆ωjg) (for the time domain) (13) The parameters t1 and t2 represent the delay times that have been chosen for the 2-D data acquisition. Although it is possible, at least in some cases, to obtain a solution that involves complex error functions, in this work, all integrations are performed numerically by replacing the integration limits with smaller, realistic values. Figure 12 shows simulations of frequency-domain spectra for the Re23 and Ro23 pathways to illustrate the TRIVE-FWM traditional line narrowing for positively correlated inhomogeneous broadening between two vibrational modes. The inhomogeneous envelope is 10 times larger than the homogeneous width of both modes. The frequency for the ω1 excitation beam is stepped to the different positions that are indicated in the figure, whereas the ω2 excitation frequency probes a shifted frequency for the second mode. When the ω1 frequency is detuned by ωocg - ω1 ) δ from the center of the g f c transition, the Re23 pathway produces a narrowed peak at ω2 ) ωobg - δ and the Ro23 pathway produces a narrowed peak at ω2 ) ωobg - δ + ξ. The peak of the Re23 pathway occurs when there is a triple resonance with the subset of molecules whose ωcg and ωbg frequencies match ω1 and ω2, whereas the Ro23



Figure 12. Simulations of line-narrowed spectra for the Re23 and Ro23 pathways for different selections of the ω1 frequency. The simulations assume the following: ωobg, ωocg, ωodb frequencies of 2000, 3000, and 2990 cm-1; homogeneous and inhomogeneous widths of 0.91 and 9.09 cm-1 (HWHM); and positively correlated inhomogeneous broadening. The absorption spectrum is shown for comparison.

Figure 14. Simulations of the magnitude and real and imaginary components of χ(3) for the Re23 and Ro23 pathways for ω1 ) 3000 cm-1. The simulations assume the same conditions as Figure 12, except the inhomogeneous broadening is negatively correlated and narrowing does not occur.

Figure 13. Simulations of the magnitude and real and imaginary components of χ(3) for the Re23 and Ro23 pathways for ω1 ) 3000 cm-1. The simulations assume the same conditions as Figure 12, so narrowing occurs.

peak occurs when there is a triple resonance with the subset of molecules whose ωdb and ωbg frequencies match ω1 and ω2. The relative intensity of the two peaks changes to reflect the Gaussian frequency distribution of vibrational energies. If the inhomogeneous broadening of the two modes is negatively

correlated, similar spectra have only one peak that is as broad as the absorption peak. There is no narrowing or shifting of the line positions as a function of ω1. The difference in the behavior for positively and negatively correlated broadening results from interference between the subsets of molecules within the inhomogeneous distribution. Figures 13 and 14 show the real and imaginary contributions to χ(3) from molecules that have their ωcg transitions below (ωocg - 2Γcg), between (ωocg - 2Γcg) and (ωocg + 2Γcg), and above (ωocg + 2Γcg), when ω1 is tuned to the center of the inhomogeneous distribution, ωocg. Figure 13 shows positively correlated broadening, and Figure 14 shows negatively correlated broadening. The figures assume that ω1 is fixed at ωocg and ω2 is tuned. Figure 13 shows the simulations for the same narrowing conditions as Figure 12. All three subsets contribute constructively to the net response. Figure 14 assumes the same parameters, except the inhomogeneous broadening of the two modes is negatively correlated. Note now that the contributions from the subsets that are above and below the central frequency destructively interfere with the contribution from the subset that is centered on ωocg. The destructive interference prevents the most-resonant subset from making a dominant contribution and creating the line-narrowed peak. Note also that the contributions from the two pathways create both asymmetries in the contributions from the subsets that are above and below resonance and more-complex line shapes for each subset.


Besemann et al.

Figure 15. Effects of inhomogeneous frequency correlations and line widths on frequency-domain TRIVE spectra in the pure dephasing limit. Inhomogeneous:homogeneous HWHM ratio is 10:1 for all simulations, ξ ) 5 cm-1, and all transition moments are equal. Rows indicate linear HWHM for states b and c of (1) 1 cm-1 and (2) 10 cm-1. Columns indicate (a) Fbc ) 0.99, (b) Fbc ) 0, and (c) Fbc ) -0.99. Contours given in intervals of 10% of the maximum of |R|.

In the time domain where ultrafast pulses excite the entire inhomogeneous envelope, the time dependence of a coherence o between states m and n is e-i(ωnm(∆ω)te-Γnmt (see Table 1), where ∆ω represents the shift induced in a microenvironment by inhomogeneous broadening. The distribution of ∆ω values causes dephasing of the microenvironment contributions. If the inhomogeneous broadening of two vibrational modes is positively correlated, the dephasing can be reversed for pathways such as Re23, which involve coherences with opposite dephasing characteristics. Table 1 shows that this pathway involves both gb and cg coherences, which have time dependences of o o e-i(ωgb-∆ω-iΓgb)t23 and e-i(ωcg(∆ω-iΓcg)t1, respectively. When t23 ) t14, the dephasing induced by the spread in frequencies (∆ω) of the first coherence is canceled by the rephasing induced by the second coherence. This rephasing is identical to photon echo methods.51,55,56,79 If the inhomogeneous broadening is negatively correlated, so positive shifts of one state are correlated with negative shifts in the other state, the dephasing can be reversed for pathways such as Re32. Table 1 shows that this pathway involves both bg and cg coherences, which have time dependo o ences of e-i(ωbg(∆ω - iΓbg)t23 and e-i(ωcg-∆ω-iΓcg)t14, respectively. Although ωbg and ωcg have the same signs, the frequency shifts of the b and c states are opposite, so the dephasing that results from the bg coherence will rephase after transformation to the cg coherence. Table 5 summarizes the traditional line-narrowing properties for each TRIVE pathway. As a rule of thumb, if a pathway first passes through a single quantum bra-side coherence (such as Fgb) and subsequently passes through a ket-side (i.e., the ket is higher in energy than the bra) single quantum coherence (such as Fcg or Fdb), line narrowing occurs for positively correlated inhomogeneous distributions. If a pathway first passes through a single quantum ket-side coherence (Fbg, Fcg) and subsequently passes through another ket-side single quantum coherence (Fag, Fcg, Fdb), that pathway is capable of narrowing negatively

TABLE 5: IR-FWM Processes Capable of Traditional Line Narrowinga probed correlation b-c b-c c-c

narrowed correlation positive negative not applicable

capable IRFWM processes Re21, Ro21, Re23, Ro23 Re31, Ro31, Re32, Ro32 Re12, Ro12, Re13, Ro13

a The last entry is not a candidate for narrowing, because the probed correlation involves only state c.

correlated inhomogeneous distributions. The Re12, Ro12, Re13, Ro13 pathways do not involve coherences that compensate dephasing, so they are not expected to exhibit traditional line narrowing; however, they do exhibit natural line narrowing, because of their intermediate coherences. Finally, it is important to note that, for TRIVE, even/odd pairs have the same narrowing properties and can destructively interfere, thereby reducing the magnitude of the line-narrowed signal. As was discussed previously, in the pure dephasing limit,71,80 e R23, Ro23, Re32, and Ro32 dominate the 2-D TRIVE spectrum. According to Table 5, Re23 and Ro23 narrow negatively correlated distributions and Re32 and Ro32 narrow positively correlated distributions. Therefore, frequency-domain TRIVE should be able to narrow both positively and negatively correlated distributions. Figure 15 shows that this expectation is true. The number of peaks (one or two) and the apparent line width of the narrowed diagonal (positive correlation) or antidiagonal (negative correlation) are dependent on the size of ξ, relative to the homogeneous line width. In the limit where ξ is significantly larger than Γ, and with perfect positive or negative inhomogeneous correlation, two distinct TRIVE peaks are observed, each with a narrowed line width that is approximately equal to the homogeneous line width. If ξ is not significantly greater than Γ, a single TRIVE peak is observed. The narrow line width of this single peak is greater than the homogeneous

Spectroscopic Characteristics of TRIVE-FWM line width, because it is actually the sum of two narrowed peaks that are separated by a distance ξ. Conclusions Triply vibrationally enhanced four-wave mixing (TRIVEFWM) is a fully resonant coherent analogue of two-color pump-probe methods. This paper has described the spectroscopic characteristics of these methods and how they are dependent on the different coherence pathways, dephasing and population relaxation rates, and inhomogeneous broadening characteristics. We have shown how quantum-level interference between pathways affects the spectra. We also have shown how the interference between different subsets of molecules within an inhomogeneously broadened distribution controls the linenarrowing characteristics of pathways. Finally, we have shown that the selection of resonances for two-color TRIVE-FWM methods has important consequences for the selection rules that control the four transitions in the method. Acknowledgment. This work was supported by the Chemistry Program of the National Science Foundation (under Grant No. CHE-0130947). References and Notes (1) Zazubovich, V.; Jankowiak, R.; Riley, K.; Picorel, R.; Seibert, M.; Small, G. J. J. Phys. Chem. B 2003, 107, 2862. (2) Zazubovich, V.; Jankowiak, R.; Small, G. J. J. Phys. Chem. B 2002, 106, 6802. (3) Zazubovich, V.; Jankowiak, R.; Small, G. J. J. Lumin. 2002, 98, 123. (4) Milanovich, N.; Suh, M.; Jankowiak, R.; Small, G. J.; Hayes, J. M. J. Phys. Chem. 1996, 100, 9181. (5) Reddy, N. R. S.; Picorel, R.; Small, G. J. J. Phys. Chem. 1992, 96, 6458. (6) Raja, N.; Reddy, S.; Lyle, P. A.; Small, G. J. Photosynth. Res. 1992, 31, 167. (7) Shu, L.; Small, G. J. Chem. Phys. 1990, 141, 447. (8) Jankowiak, R.; Tang, D.; Small, G. J.; Seibert, M. J. Phys. Chem. 1989, 93, 1649. (9) Jankowiak, R.; Cooper, R. S.; Zamzow, D.; Small, G. J.; Doskocil, G.; Jeffrey, A. M. Chem. Res. Toxicol. 1988, 1, 60. (10) Small, G. J.; Hayes, J. M.; Gillie, J. K.; Kenney, M. J.; Jankowiak, R. Abstr. Pap.sAm. Chem. Soc. 1987, 193th, 17. (11) Small, G. J. Persistent Nonphotochemical Hole-Burning and Dephasing of Impurity Electronic Transitions in Glasses. In Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, 1st ed.; Granovich, V. M., Hochstrasser, R. M., Eds.; North-Holland: New York, 1983; p 515. (12) Fearey, B. L.; Carter, T. P.; Small, G. J. J. Phys. Chem. 1983, 87, 3590. (13) Hayes, J. M.; Stout, R. P.; Small, G. J. J. Chem. Phys. 1981, 74, 4266. (14) Roberts, K. P.; Lin, C. H.; Singhal, M.; Casale, G. P.; Small, G. J.; Jankowiak, R. Electrophoresis 2000, 21, 799. (15) Duhachek, S. D.; Kenseth, J. R.; Casale, G. P.; Small, G. J.; Porter, M. D.; Jankowiak, R. Anal. Chem. 2000, 72, 3709. (16) Ariese, F.; Jankowiak, R.; Suh, M.; Small, G. J.; Chen, L.; Devanesan, P. D.; Li, K. M.; Todorovic, R.; Rogan, E. G.; Cavalieri, E. L. Polycyclic Aromat. Compd. 1996, 10, 227. (17) Devanesan, P. D.; Ramakrishna, N. V. S.; Padmavathi, N. S.; Higginbotham, S.; Rogan, E. G.; Cavalieri, E. L.; Marsch, G. A.; Jankowiak, R.; Small, G. J. Chem. Res. Toxicol. 1993, 6, 364. (18) Ramakrishna, N. V. S.; Cavalieri, E. L.; Rogan, E. G.; Dolnikowski, G.; Cerny, R. L.; Gross, M. L.; Jeong, H.; Jankowiak, R.; Small, G. J. J. Am. Chem. Soc. 1992, 114, 1863. (19) Jankowiak, R.; Small, G. J. Science 1987, 237, 618. (20) Jankowiak, R.; Small, G. J. Chem. Res. Toxicol. 1991, 4, 256. (21) Jankowiak, R.; Day, B. W.; Lu, P. Q.; Doxtader, M. M.; Skipper, P. L.; Tannenbaum, S. R.; Small, G. J. J. Am. Chem. Soc. 1990, 112, 5866. (22) Jankowiak, R.; Small, G. J.; Nishimoto, M.; Varanasi, U.; Kim, S. K.; Geacintov, N. E. J. Pharm. Biomed. Anal. 1990, 8, 113. (23) Zamzow, D.; Jankowiak, R.; Cooper, R. S.; Small, G. J.; Tibbels, S. R.; Cremonesi, P.; Devanesan, P.; Rogan, E. G.; Cavalieri, E. L. Chem. Res. Toxicol. 1989, 2, 29. (24) Jankowiak, R.; Small, G. J. Anal. Chem. 1989, 61, 1023A.

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10503 (25) Cooper, R. S.; Jankowiak, R.; Hayes, J. M.; Lu, P. Q.; Small, G. J. Anal. Chem. 1988, 60, 2692. (26) Small, G. J.; Hayes, J. M.; Jankowiak, R.; Gillie, J. K.; Cooper, R. S.; Zamzow, D. Abstr. Pap.sAm. Chem. Soc. 1987, 194th, 67. (27) Sanders, M. J.; Cooper, R. S.; Jankowiak, R.; Small, G. J.; Heisig, V.; Jeffrey, A. M. Anal. Chem. 1986, 58, 816. (28) Brown, J. C.; Duncanson, J. A.; Small, G. J. Anal. Chem. 1980, 52, 1711. (29) Brown, J. C.; Edelson, M. C.; Small, G. J. Anal. Chem. 1978, 50, 1394. (30) Warren, J. A.; Hayes, J. M.; Small, G. J. Anal. Chem. 1982, 54, 138. (31) Hayes, J. M.; Small, G. J. Anal. Chem. 1982, 54, 1202. (32) Chang, T. C.; Johnson, C. K.; Small, G. J. J. Phys. Chem. 1985, 89, 2984. (33) Jankowiak, R.; Hayes, J. M.; Small, G. J. Chem. ReV. 1993, 93, 1471. (34) Wright, J. C. Int. ReV. Phys. Chem. 2002, 21, 185. (35) Murdoch, K. M.; Condon, N. J.; Zhao, W.; Besemann, D. M.; Meyer, K. A.; Wright, J. C. Chem. Phys. Lett. 2001, 335, 349. (36) Zhao, W.; Wright, J. C. Phys. ReV. Lett. 2000, 84, 1411. (37) Zhao, W.; Wright, J. C. J. Am. Chem. Soc. 1999, 121, 10994. (38) Zhao, W.; Wright, J. C. Phys. ReV. Lett. 1999, 83, 1950. (39) Chen, P. C.; Hamilton, J. P.; Zilian, A.; LaBuda, M. J.; Wright, J. C. Appl. Spectrosc. 1998, 52, 380. (40) Wright, J. C.; Labuda, M. J.; Zilian, A.; Chen, P. C.; Hamilton, J. P. J. Lumin. 1997, 72-74, 799. (41) Wright, J. C.; Chen, P. C.; Hamilton, J. P.; Zilian, A.; LaBuda, M. J. Appl. Spectrosc. 1997, 51, 949. (42) Labuda, M. J.; Wright, J. C. Phys. ReV. Lett. 1997, 79, 2446. (43) Zilian, A.; LaBuda, M. J.; Hamilton, J. P.; Wright, J. C. J. Lumin. 1994, 60&61, 655. (44) Wright, J. C.; Carlson, R. J.; Hurst, G. B.; Steehler, J. K.; Riebe, M. T.; Price, B. B.; Nguyen, D. C.; Lee, S. H. Int. ReV. Phys. Chem. 1991, 10, 349. (45) Zanni, M. T.; Asplund, M. C.; Hochstrasser, R. M. J. Chem. Phys. 2001, 114, 4579. (46) Golonzka, O.; Khalil, M.; Demirdoven, N.; Tokmakoff, A. Phys. ReV. Lett. 2001, 86, 2154. (47) Tokmakoff, A. J. Phys. Chem. A 2000, 104, 4247. (48) Hamm, P.; Lim, M.; DeGrado, W. F.; Hochstrasser, R. M. J. Chem. Phys. 2000, 112, 1907. (49) Asplund, M. C.; Lim, M.; Hochstrasser, R. M. Chem. Phys. Lett. 2000, 323, 269. (50) Hamm, P.; Lim, M.; DeGrado, W. F.; Hochstrasser, R. M. J. Phys. Chem. A 1999, 103, 10049. (51) Rector, K. D.; Zimdars, D.; Fayer, M. D. J. Chem. Phys. 1998, 109, 5455. (52) Lim, M.; Hamm, P.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. 1998, 95, 15215. (53) Tokmakoff, A.; Lang, M. J.; Larsen, D. S.; Fleming, G. R. Chem. Phys. Lett. 1997, 272, 48. (54) Tokmakoff, A.; Lang, M. J.; Larsen, D. S.; Fleming, G. R.; Chernyak, V.; Mukamel, S. Phys. ReV. Lett. 1997, 79, 2702. (55) Tokmakoff, A.; Fayer, M. D. Acc. Chem. Res. 1995, 28, 437. (56) Zimdars, D.; Tokmakoff, A.; Chen, S.; Greenfield, S. R.; Fayer, M. D.; Smith, T. I.; Schwettman, H. A. Phys. ReV. Lett. 1993, 70, 2718. (57) Scheurer, C.; Mukamel, S. J. Chem. Phys. 2002, 116, 6803. (58) Scheurer, C.; Mukamel, S. Bull. Chem. Soc. Jpn. 2002, 75, 989. (59) Wright, J. C.; Condon, N. J.; Murdoch, K. M.; Besemann, D. M.; Meyer, K. A. J. Phys. Chem. 2003, 107, 8166. (60) Asplund, M. C.; Zanni, M. T.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. 2000, 97, 8219. (61) Golonzka, O.; Khalil, M.; Demirdoven, N.; Tokmakoff, A. J. Chem. Phys. 2001, 115, 10814. (62) Khalil, M.; Tokmakoff, A. Chem. Phys. 2001, 266, 213. (63) Zanni, M. T.; Gnanakaran, S.; Stenger, J.; Hochstrasser, R. M. J. Phys. Chem. B 2001, 105, 6520. (64) Zanni, M. T.; Hochstrasser, R. M. Curr. Opin. Struct. Biol. 2001, 11, 516. (65) Ge, N. H.; Hochstrasser, R. M. PhysChemComm 2002, 17. (66) Rubtsov, I. V.; Wang, J.; Hochstrasser, R. M. J. Chem. Phys. 2003, 118, 7733. (67) Rubtsov, I. V.; Wang, J. P.; Hochstrasser, R. M. J. Phys. Chem. A 2003, 107, 3384. (68) Rubtsov, I. V.; Wang, J. P.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 5601. (69) Meyer, K. A.; Wright, J. C. Chem. Phys. Lett. 2003, 381, 642. (70) Meyer, K. A.; Thompson, D. C.; Wright, J. C. Phys. ReV. Lett., submitted.

10504 J. Phys. Chem. B, Vol. 108, No. 29, 2004 (71) Prior, Y.; Bogdan, A. R.; Dagenais, M.; Bloembergen, N. Phys. ReV. Lett. 1981, 46, 111. (72) Mukamel, S. Principles of Nonlinear Optical Spectroscopy, 1st ed.; Oxford University Press: New York, 1995. (73) Ge, N. H.; Zanni, M. T.; Hochstrasser, R. M. J. Phys. Chem. A 2002, 106, 962. (74) Yan, Y. J.; Mukamel, S. J. Phys. Chem. 1989, 93, 6991. (75) Mukamel, S. J. Chem. Phys. 1985, 82, 5398.

Besemann et al. (76) Khalil, M.; Demirdoven, N.; Tokmakoff, A. J. Phys. Chem. A 2003, 107, 5258. (77) Dick, B.; Hochstrasser, R. M. J. Chem. Phys. 1983, 78, 3398. (78) Carlson, R. J.; Wright, J. C. J. Mol. Spectrosc. 1990, 143, 1. (79) Rella, C. W.; Kwok, A.; Rector, K.; Hill, J. R.; Schwettman, H. A.; Dlott, D. D.; Fayer, M. D. Phys. ReV. Lett. 1996, 77, 1648. (80) Bloembergen, N.; Lotem, H.; Lynch, R. T. Indian J. Pure Appl. Phys. 1978, 70, 151.

Spectroscopic Characteristics of Triply Vibrationally Enhanced Four

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