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B. Force Correlation Function Approach

B. Force Correlation Function Approach

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computed. But there are two significant problems. First, it is presently

impossible to do the fully quantum mechanical simulation on a system of

any reasonable size. Classical mechanics may be used instead, but then a

“quantum correction” must be introduced (52,60,61) to Equation (4). The

quantum correction may be quite large (orders of magnitude) (52), and

computing it accurately for anharmonic baths is problematic. Second, we

often need to know the VER rate constant for vibrations with frequencies

× ωD . At higher frequencies far above the fundamental characteristic

bath frequency, the Fourier transform is very small. For example, in liquid

O2 , the vibrational frequency D 1552 cm 1 , whereas ωD ¾ 50 cm 1 , so

the Fourier transform is required at a frequency about 30 times greater than

the highest fundamental bath frequency (52). In such cases, numerical errors

in the calculation of the correlation function lead to large inaccuracies in

the Fourier transform.

C. Perturbation Approach



An alternative approach widely used in polyatomic molecule studies is

based on the Golden Rule and a perturbative treatment of the anharmonic

coupling (57,62). This approach is not much used for diatomic molecules.

In the liquid O2 example cited above, the Hamiltonian must be expanded

to 30th order or so to calculate the multiphonon emission rate. But for

vibrations of polyatomic molecules, which can always find relatively loworder VER pathways for each VER step, perturbation theory is very useful.

In the perturbation approach, the molecule’s entire ladder of vibrational

excitations is the “system” and the phonons are the “bath.” Only lowerorder processes are ordinarily needed (57) because polyatomic molecules

have many vibrations ranging from higher to lower frequencies and only

a small number of phonons, usually one or two, are excited in each VER

ˆ q ,Q

step. The usual practice is to expand the interaction Hamiltonian V

in Equation (2) in powers of normal coordinates (57,62):

ˆ D

V

˛



C



C



ˆ

∂V

∂Q˛



Q˛ C

fQD0g



1

2!



ˆ

∂3 V

∂Q˛ ∂Q ∂Q∂



˛,



1

3! ˛,







1

4! ˛,



ˆ

∂4 V

∂Q˛ ∂Q ∂Q∂ ∂Qε

,∂,ε



ˆ

∂2 V

∂Q˛ ∂Q



Q˛ Q

fQD0g



Q˛ Q Q∂

fQD0g



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Q˛ Q Q∂ Qε C Ð Ð Ð

fQD0g



5



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549



For polyatomics, usually only the last two terms in Equation (5),

the cubic and quartic anharmonic terms, need be considered (57). The

lowest-order process involving cubic anharmonic coupling involves excited

vibration relaxing by interacting with two other states, say another vibration ω and one phonon ωph , or alternatively a pair of phonons. For example,

the total rate constant for energy loss from for cubic coupling was given

by Fayer and coworkers as (57)

K D



[nω 1 C n







Cω C Cω



ω



C 1 C nω ˛ C nj



ωj



ω Cj



ωj ]



6



where nω is the thermal occupation number,

nω D eh¯ ω/kT



1



1



7



ω is the density of phonon states at ω, Cω is a product of coupling constants

that contains factors such as h¯ /2 ω and the derivatives of V in Equation (5),

and ˛ D 1 if

> ω or ˛ D 0 if

< ω. When T ! 0, all the thermal

occupation factors in Equation (6) vanish, but the VER rate does not vanish.

VER is then said to occur via spontaneous emission of phonons. As T is

increased, two new thermally activated processes turn on. One involves

stimulated phonon emission and the other phonon absorption. Spontaneous

and stimulated emission processes convert

to a lower energy vibration

ω (down-conversion). Phonon absorption converts

to a higher energy

vibration ω (up-conversion).

Some representative examples of common zero-temperature VER

mechanisms are shown in Fig. 2b–f. Figures 2b,c describe the decay of

the lone vibration of a diatomic molecule or the lowest energy vibrations

in a polyatomic molecule, termed the “doorway vibration” (63), since it

is the doorway from the intramolecular vibrational ladder to the phonon

bath. In Fig. 2b, the excited doorway vibration

lies below ωD , which

can be the case for large molecules or macromolecules. In the language of

Equation (4), fluctuating forces of fundamental excitations of the bath at

frequency are exerted on the molecule, inducing a spontaneous transition

to the vibrational ground state plus excitation of a phonon at ωph D .

The rate of this transition is proportional to the Fourier transform of the

force-force correlation function at frequency , denoted C

.

In Fig. 2c, the vibration

lies well above the phonon cut-off ωD ,

as for example the 379 cm 1 doorway vibration in ACN (46), where ωD

is in the 100–150 cm 1 range. Fluctuating forces exerted by the bath at

frequency

cause the doorway vibration to decay. In the language of



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Iwaki et al.



Equation (5), VER involves a higher-order anharmonic coupling matrix

element, which gives rise to decay via simultaneous emission of several

phonons nωph (multiphonon emission). In the ACN case, three phonons

must be emitted simultaneously via quartic anharmonic coupling (or four

phonons via fifth-order coupling, etc.).

In Fig. 2d, is one vibrational fundamental of a polyatomic molecule,

whose relaxation involves exciting ω1 , a lower energy vibration of the same

molecule. In the language of Equation (5), , ω1 , and other intramolecular

vibrations are part of the system and the phonons are the bath. Fluctuating

forces exerted by the bath at frequency ωph D

ω1 induce a transition

from to ω1 via cubic anharmonic coupling. This mechanism, which ought

to be most important in polyatomic molecules where the spacings between

adjacent vibrations are generally less than ωD (4), is the “ladder relaxation”

(57) process mentioned previously. The name derives from the motion of

the vibrational excitation, which hops downward from one vibration to

another, which are the “rungs” of the “ladder” of vibrational fundamentals

of the polyatomic molecule.

In Fig. 2e,

decays by an intramolecular vibrational redistribution

(IVR) process, involving lower energy vibrations ω1 , ω2 , ω3 , . . . , via a

higher-order anharmonic coupling which causes to decay by spontaneous

emission of several lower energy vibrations ω1 , ω2 , ω3 , . . . . In condensed

phases, phonons may play a role in IVR as well, which is analogous

to the role of rotations in gas-phase IVR, by dynamically modulating

and broadening the vibrational energy levels, making it more likely for

a resonance to occur.

D. Vibrational Cascade



The “vibrational cascade” (64) illustrated in Fig. 2f is widely believed to

be a prominent mode of vibrational cooling (VC). The general properties of

vibrational cascades in large molecules at finite temperature were studied

theoretically by Hill and Dlott (4,5). The vibrational cascade occurs when

the lowest-order ladder processes dominate. In a large molecule, the rungs

of the vibrational ladder are on average closely spaced. Here closely spaced

means the average energy difference is less than ωD , so that a step from

one rung to another can occur with just one phonon. In this case, energy

loss from a polyatomic molecule at zero temperature, with vibration

initially excited, occurs by a sequence of VER processes, each involving the

emission of just one phonon via cubic anharmonic coupling. A vibrational

cascade in an intermediate-size molecule such as ACN or NM might involve



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Ultrafast IR-Raman Spectroscopy



551



some steps where two or three phonons are emitted via quartic or fifthorder coupling. In a vibrational cascade at zero temperature, a vibrational

excitation descends the ladder, losing a small amount of energy in each step.

At the bottom rung of the ladder, the doorway vibration, the final VER step

cannot occur by a ladder process since there are no more lower energy rungs

(4). The final VER step occurs by a single or multiphonon emission process

(e.g., Fig. 1b or 1c). At finite temperature, each step along the ladder might

go up or down, depending on whether phonons are absorbed or emitted.

However the net motion of excess vibrational energy is always downward,

to states of lower energy (4,5).

Hill and Dlott (5) illustrated the properties of vibrational cascades in

model calculations of VC in crystalline naphthalene. Naphthalene C10 H8

has 48 normal modes. Forty of these vibrations (all except the eight C–H

stretching vibrations) lie in the frequency range 1627–180 cm 1 . In the

calculation, one unit of excitation is input to the highest vibration in this

range, 1627 cm 1 . The ensemble-averaged population of the ith mode is

determined by a master equation:

dP t

DKÐP t ,

8

dt

where P t is a vector of vibrational populations and K is a matrix of transition rate constants. The elements of the rate matrix were computed using

Equation (6), which assumes that VER occurs solely by cubic anharmonic

coupling. In naphthalene, the phonon density of states ωph is accurately

known from neutron scattering measurements (65,66). The coupling factors

C in Equation (6) were determined using the density of states and VER

lifetimes determined by low-temperature vibrational lineshape measurements (5,67). Where VER measurements were not available, C was taken

to be the average of the known values.

The average energy jump E for every step up or down the ladder in

a large molecule is approximately equal to the average phonon frequency,

Eavg D



h¯ ω



ph

ω dω

ph

ω dω



9



For naphthalene (5), Eavg D 95 cm 1 .

Figure 3 shows the results of a calculation (5) at T D 0 assuming

an initial condition of unit vibrational excitation at 1627 cm 1 . As time

progresses, the center of the vibrational population distribution moves

toward lower energy. The vibrational population distribution spreads out

because the size of each step is distributed over the range 0–¯hωD , with the

average step being 95 cm 1 . Vibrational cooling is essentially complete



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Iwaki et al.



Figure 3 Calculated vibrational cascade in crystalline naphthalene at T D 0, for

initial excitation at 1627 cm 1 . The calculation uses Equation (6), which assumes

that cubic anharmonic coupling dominates. From Ref. 5.



by about 200 ps. When the vibrational density of states is a constant, the

population distribution should eventually approach a Gaussian distribution

(4,5). The peak of the distribution will move toward lower energy at a

constant velocity (the “vibrational velocity,” with units of energy dissipated

per unit time) and the width will increase as the square root of time. The

dashed curves in Fig. 3 are the best fits to this Gaussian distribution.

The temperature dependence of VC discovered by Hill and Dlott (4,5),

as a consequence of the temperature dependence predicted by Equation (6),

is very interesting. Equation (6) shows that the lifetime, that is the rate of

leaving a particular state, decreases with increasing T. Equation (6) has

three parts: temperature-independent spontaneous emission (downward)

and temperature-dependent stimulated emission (down) and absorption

(up). It is the increase in the rates of the latter two temperature-dependent

processes that causes the lifetime to decrease with increasing T. The two

temperature-dependent processes by themselves do not cause vibrationally

hot molecules to cool, since they are as likely to drive an excited

vibration to higher energy states as to lower energy states. Increasing

the temperature only increases the rate of vibrational energy jumping

up and down. Any actual cooling, which is caused by the net motion

of vibrational excitation to lower energy, is driven by the temperatureindependent spontaneous emission processes. Thus the VC process is

largely independent of temperature when only lower order anharmonic

coupling processes dominate. This point is illustrated in Fig. 4, which plots



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Ultrafast IR-Raman Spectroscopy



553



Figure 4 Average energy of the nonequilibrium vibrational population distribution computed for the vibrational cascade in crystalline naphthalene in Fig. 3. At

T D 0, the peak moves toward lower energy at a roughly constant rate, the “vibrational velocity” of 8.9 cm 1 ps. The initial 1627 cm 1 of vibrational energy is

dissipated in ¾180 ps. The vibrational velocity is the same at 300 K. In the limit

that cubic anharmonic coupling dominates [Equation (6)], increasing the temperature increases the rates of up- and down-conversion processes, but has no effect on

the net downward motion of the population distribution. Although the lifetimes of

individual vibrational levels will decrease with increasing temperature, VC is not

very dependent on temperature in this limit. (From Ref. 5.)



the time dependence of the average energy of the population distribution

after 1627 cm 1 excitation at T D 0 and T D 300 K. At both temperatures

the population distribution moves down at an approximately constant rate,

with a vibrational velocity V0 D 9 cm 1 /ps. That is to say, the average

rate of energy lost from the molecule is 9 cm 1 per picosecond; losing

1627 cm 1 takes about 180 ps. This calculation shows that although the

rates of VER processes may increase dramatically with T, the overall rate

of VC ought not to be much affected by temperature.



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554



Iwaki et al.



III. THE IR-RAMAN TECHNIQUE

A. The Method



A tunable mid-IR pulse at frequency ωIR pumps vibrational excitations

in a polyatomic liquid (all work discussed here is at ambient temperature

¾295 K). A time-delayed visible probe pulse at frequency ωL generates

incoherent anti-Stokes Raman scattering. For an instantaneous pump pulse

arriving at time t D 0, the change in the anti-Stokes intensity of transition

i, with frequency ωi , the “anti-Stokes transient,” is (44)

IAS

t D constni t gi

i



Ri



ωL C ωi



4



10



where the constant depends on the experimental set up, ni t is the instantaneous change in vibrational population, gi is the degeneracy, and Ri is

the Raman cross section. Equation (10) shows that the intensity of an antiStokes transient is proportional to the population change in the vibrational

transition during the VER process induced by the pump pulse (2).

The IR-Raman experiment is difficult. An ultrashort tunable mid-IR

pulse is needed that produces a substantial number of vibrational excitations. An intense visible pulse is also needed that generates enough

anti-Stokes signal photons despite the small magnitude of the Raman cross

section. Detecting small numbers of anti-Stokes photons is no problem

with today’s 90% quantum efficiency CCD detectors. The real problem

arises from optical background, as discussed below. The laser pulses must

be short enough to time-resolve VER processes of interest, but if they

are too short undesirable effects occur (45): (1) the spectral bandwidth of

the pulses becomes too broad to resolve individual vibrational transitions

and (2) the short pulses at fluence levels needed to generate and detect

transient vibrational populations will generate optical background in the

sample via nonlinear optical interactions. We designed our apparatus to

produce an ¾0.8 ps pulse, which is short enough to time-resolve most

VER processes of interest. The transform limited spectral bandwidth of

a 0.8 ps pulse is ¾20 cm 1 , but practical bandwidths in our system are

25–35 cm 1 (45). Figure 5 shows IR and Raman spectra of neat liquid

ACN, obtained using conventional spectrometers with resolution better than

the natural linewidths (46). In Fig. 5 we also show a Stokes Raman spectrum obtained with the ultrafast laser system. Most of the Raman transitions

can be resolved, although the C–H bending modes in particular tend to run

together.



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Ultrafast IR-Raman Spectroscopy



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Figure 5 Vibrational spectra of neat liquid acetonitrile (ACN): (top) mid-IR spectrum; (bottom) Stokes Raman spectrum using a conventional spectrometer (solid

line). The dashed spectrum obtained with the ultrafast laser system has somewhat

lower resolution (From Ref. [46].)



B. The Laser



Since the early IR-Raman experiments with Nd:glass lasers, Ti:sapphire

lasers with chirped-pulse amplification (CPA) (68,69) have revolutionized

ultrafast spectroscopy. Ti:sapphire lasers ordinarily run in femtosecond

mode (pulse duration ¾100 fs) where the spectral bandwidth >140 cm 1

is too large for Raman spectroscopy, so methods have to be found to

lengthen the Ti:sapphire pulses and reduce the spectral bandwidth. That

is a bit ironic, since so much work has been devoted to producing pulses

with ever-shorter duration. Efforts are being made today to produce longerduration, spectrally narrower bandwidth pulses from Ti:sapphire (45) or to

convert femtosecond pulses into spectrally narrower picosecond pulses in a

more efficient manner than simply removing energy with a narrow bandpass

filter (70).



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Iwaki et al.



The experimental setup used at the University of Illinois is

diagrammed in Fig. 6. A CPA laser from Clark-MXR Corp. was modified

by the manufacturer to run in picosecond mode by substituting more

dispersive gratings (2000 lines/mm for 1400 lines/mm) and by adding

birefringent filters (71) in the regenerative amplifier cavity. The system

outputs ¾0.8 ps duration pulses with 1.0 mJ energy at a repetition rate of

1 kHz.

The most common technology used today to produce intense pulses

at wavelengths other than 800 nm and its harmonics is optical parametric

amplification (OPA) (72). In an OPA, a “signal” laser pulse at frequency ωS

propagates through a nonlinear crystal along with an intense pump pulse at



Figure 6 Block diagram of the two-color optical parametric amplifier (OPA)

and IR-Raman apparatus. CPA D Chirped pulse amplification system; Fs

OSC D femtosecond Ti:sapphire oscillator; Stretch D pulse stretcher; Regen D regenerative pulse amplifier; SHGYAG D intracavity frequency-doubled Q-switched

Nd:YAG laser; YAG D diode-pumped, single longitudinal mode, Q-switched

Nd:YAG laser; KTA D potassium titanyl arsenate crystals; BBO D ˇ-barium

borate crystal; PMT D photomultiplier tube; HNF D holographic notch filter;

IF D narrow-band interference filter; CCD D charge-coupled device optical array

detector. (From Ref. 96.)



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Ultrafast IR-Raman Spectroscopy



557



ωP . The pump pulse amplifies the signal while simultaneously producing

an “idler” pulse at frequency ωI D ωP ωS . For 800 nm Ti:sapphire

pump pulses, the idler will be tunable throughout the mid-IR (broadly

400–4000 cm 1 ) for signal pulses in the 820–1176 nm range, provided the

nonlinear crystal is transparent in all these spectral regions. We needed a

design that would produce tunable mid-IR pulses, but we wanted the probe

pulses to have a fixed wavelength so that readily available interference

filters and holographic Raman notch filters could be used in the detection

setup. In our development phase we took anti-Stokes spectra from various

liquids using ¾0.8 ps pulses at 800 and 400 nm. The longer wavelength

800 nm pulses had poor scattering efficiency due to the ω4 dependence in

Equation (10). The 400 nm second harmonic pulses, in principle 16 times

better than the fundamental, produced too much multiphoton fluorescence

and ionization. Believing that a probe pulse in the ¾500 nm region was

optimal, we designed at two-color OPA system (45) as in Fig. 6.

Our OPA is based on potassium titanyl arsenate (KTA) crystals (73).

KTA is quite similar to KTP which is widely used in OPAs, but KTA

has better mid-IR transparency than KTP. The two-color OPA is seeded

(the “seed” is the signal pulse) in a quasi-CW fashion (45,74) by a diodepumped Q-switched Nd:YAG laser running in a single longitudinal mode

at 1.064 àm, which generates ắ10 àJ in a 50 ns pulse (45). Only a ¾1 ps

duration slice of this seed pulse is amplified in the OPA. The seed power

is about 1 kW. Laser seeding produces narrower bandwidths than the more

usual white-light seeding, and it avoids the problems of generating a seed

supercontinuum with the picosecond pulses (75), which is more difficult

than with femtosecond pulses. With the Nd:YAG seed, the center frequency

of the amplified signal output will remain pinned at 1.064 µm. The signal

pulses are frequency doubled in a BBO crystal to produce the fixedfrequency Raman probe pulses at 532 nm, where off-the-shelf optics and

filters are readily available. The mid-IR idler pulses can be tuned over an

¾100 cm 1 range by simply tilting the crystals and over an ¾1000 cm 1

range by tuning the CPA laser. When the CPA pump pulses are tuned in

the 770–820 range, the mid-IR output ranges from 2800–3600 cm 1 . This

mid-IR range allows us to pump almost any molecule containing at least

one hydrogen atom, including common functional groups such as C–H,

O–H, S–H, N–H, etc. (76).

The performance of the two-color OPA (45) is illustrated in Fig. 7.

Mid-IR pulse energies of 40–50 µJ are typically obtained in a nominal

0.8 ps pulse with a 35 cm 1 spectral bandwidth. These are very large midIR energies for a kilohertz laser (45). It is easier to obtain high conversion



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Iwaki et al.



Figure 7 Performance of the mid-IR OPA. The insets show a cross-correlation

between mid-IR pulses and 532 nm pulses from the OPA, with FWHM of 0.8 ps,

and the 35 cm 1 FWHM spectrum of mid-IR pulses at 3000 cm 1 . (From Ref. 45.)



efficiency with our 0.8 ps pulses than with 100 fs pulses. The higher pump

power of femtosecond pulses is not really an advantage because the power

at the nonlinear crystal must be kept below a critical level, typically a

few hundred GW/cm2 , which is determined mainly by the onset of supercontinuum generation. For a 100 fs pulse with ¾150 cm 1 bandwidth,

our calculations have shown that the KTA crystal length is limited to

<2 mm. The crystal length is ordinarily limited either by group-velocity

mismatch or by the acceptance bandwidth (72). For a 100 fs pulse, the

acceptance bandwidth becomes limiting (i.e., it is ¾150 cm 1 ) at ¾2 mm

thickness, and the group-velocity mismatch becomes significantly limiting

at a slightly greater thickness. For 0.7–1.0 ps duration pulses, the bandwidth and the group-velocity mismatch becomes limiting in the 7–10 mm

thickness range. Because the OPA gain in the small-depletion limit is an

exponential function of the usable crystal length (72), the ability to use

longer (7–10 mm) crystals in our picosecond OPA provides us with enormous small-signal gain. With the gain so great, the OPA can be run deeply



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