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B. Gas Phase Vibrational Dynamics

B. Gas Phase Vibrational Dynamics

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638



Myers et al.



Figure 5 Vibrational relaxation rate data for the CO asymmetric stretch of

W(CO)6 in various supercritical fluids at low densities. Extrapolation of the data to

zero density gives an estimate of the gas phase (collisionless) lifetime of ¾1.1 ns.



showing that the pump-probe signal is due solely to gaseous W(CO)6 and

not W(CO)6 crystallized on the window surfaces. Experiments with the

probe at the magic angle and a variety of other angles showed no change

either in the decay times or the relative amplitudes of the components.

Therefore, orientational dynamics are not contributing to the decays. (Orientational dynamics should occur on a time scale much faster than that of the



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Figure 6 Typical pump-probe data for the CO asymmetric stretch of W(CO)6 in

the gas phase at 326 K. The fit (barely discernible) is a tri-exponential with components 113 ps, 1.26 ns, and 157 ns. The values reported in the text are averages of

many different data sets taken on many samples.



experiments.) The only significant gas in the cell is W(CO)6 itself. W(CO)6

is at a very low concentration of 10 5 mol/L, which dictates a hard sphere

time between collisions of ¾10 6 s. The possibility of very long range interactions was tested. By varying the temperature a few degrees, the W(CO)6

concentration (vapor pressure) can be varied by more than an order of

magnitude. The change in concentration did not affect the tri-exponential

decay.



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Myers et al.



The explanation for the slowest component in the tri-exponential

decay is relatively straightforward. The frequencies of all of the modes

of W(CO)6 are known (77–79). Using these frequencies, the average total

internal vibrational energy (all modes) was determined. For T D 326 K, the

average energy is 2900 cm 1 . When the high-frequency CO stretch relaxes,

2000 cm 1 of additional energy are deposited into the low-frequency

modes. Because the molecules are collision-free on the time scale of

the experiments, the extra 2000 cm 1 cannot leave the molecule. The

deposition of energy raises the vibrational temperature for the average

molecule to ¾450 K. The data in Fig. 7 show the measured temperature

dependence of the Q-branch peak position as a function of temperature. The

figure shows that as the temperature is increased, the spectrum red shifts

slightly. By 450 K, we estimate that the peak of the Q-branch has shifted

¾1.1 cm 1 from its position at 326 K, the initial sample temperature.



Figure 7 Absorption frequency (Q-branch) vs. temperature for the CO asymmetric stretch of W(CO)6 in the gas phase. A representative error bar is shown.

Extrapolation to 450 K (internal vibrational temperature following relaxation of the

2000 cm 1 CO stretch) yields a temperature-dependent shift of ¾1.1 cm 1 from

the peak position at 326 K, the initial sample temperature.



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641



Following a pump pulse that produces vibrational excitations, the

probe pulse transmission through the sample is increased. A pump-probe

signal has two contributions, stimulated emission and ground state depletion. These two contributions are equal in magnitude; each contribution is

50% of the signal. As the excited state population relaxes into a combination

of low-frequency modes, the stimulated emission is reduced. Normally, it

is expected that relaxation of the excited state population fills in the ground

state depletion. Thus, the stimulated emission and the ground state depletion

decay together.

However, in the system under study here, relaxation of the 2000 cm 1

CO stretch into a combination of low-frequency modes heats the molecule

substantially and shifts the absorption spectrum to the red. First, consider

the case in which the absorption shift is very large. Although the CO stretch

is no longer excited, the ground state absorption at the laser frequency

does not recover because the molecules that have undergone excitation and

relaxation do not absorb at their initial frequency. The stimulated emission

contribution to the signal is lost, but the ground state depletion at the

probe frequency still exists. Thus, the signal will decay to 50%, and, in

the absence of other processes, the signal will remain at 50%. Examining

Fig. 6, it is seen that the slow component of the decay is somewhat below

50%. The decay to a value of less than 50% occurs because the heatinginduced spectral shift is relatively small. The shift of ¾1.1 cm 1 does not

completely shift the relaxed population off of the probe wavelength because

the probe bandwidth is ¾0.8 cm 1 . Thus the ground state depletion within

the probe bandwidth recovers to some extent, and the signal decays to less

than 50%. The signal will eventually recover as infrequent collisions cool

the molecules and cold molecules from other parts of the cell move into

the probe volume. We estimate that the time scale for these processes is

approximately the same, i.e., ắ1 às. Thus, over the 12 ns of delay used in

the experiments, the long component appears essentially flat.

To more fully understand the nature of this long component, we

performed experiments with various argon pressures in the cell. Figure 8a

shows a decay curve with a moderate pressure (18 psia, ¾0.05 mol/L) of

Ar. The slow decay component is now noticeably faster, with a decay time

of 8.1 ns. Collisions with Ar cool the low-frequency modes, allowing the

spectrum to return to its initial wavelength, which results in ground state

recovery. Figure 8b shows data with an even higher Ar pressure (1100 psia,

¾3 mol/L). The decay is now a single exponential. The collision-induced

cooling is fast compared to T1 , and the stimulated emission and ground

state depletion decay together, eliminating the long-lived component. It



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Copyright © 2001 by Taylor & Francis Group, LLC



Myers et al.



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is interesting to note that while Ar is effective in causing the energy in

the low-frequency modes to relax into the bath, the value of T1 remains

unchanged. This is in contrast to the experiments in polyatomic supercritical

fluids, where T1 becomes shorter as the solvent density is increased. The

difference between Ar and polyatomic solvents will be discussed further

below.

We propose that the fast component of the tri-exponential decay

(140 ps) can be explained by the influence of the low-frequency modes

of the molecule on the high-frequency CO stretch. In the gas phase vibrational experiment, prior to application of the pump pulse, the initial state

of the molecule is prepared by its last collision with the wall of the cell or

another molecule. The initial state is a complex superposition of eigenstates.

Each mode, , will have some occupation number, n . Under collision-free

conditions, for a given molecule the n are fixed. At the 326 K sample

temperature, the average total internal vibrational energy of a molecule is

2900 cm 1 , and the density of states at this energy (calculated with the

harmonic approximation) is 5 ð 105 states/cm 1 (78,80). Thus, there are

a vast number of initial states of the molecules that comprise the experimental ensemble. Absorbing a photon, which takes a molecule from the 0

to 1 state of the high-frequency CO stretch, changes the potential of the

low-frequency modes. The change in potential produces a time evolution

that influences the CO stretch optical transition probability measured by

the probe pulse.

As a first approach to describing the fast time evolution of the system

quantitatively (71), a description using a harmonic basis set with anharmonic coupling of the high-frequency CO stretch and the other modes can



Figure 8 Pump-probe scans on the CO asymmetric stretch of W(CO)6 in argon at

two different pressures at 333 K 60° C . (a) Pressure D 18 psia (¾1 atm). Argon

density of ¾0.05 mol/L and inverse collision frequency of ¾0.2 ns. The decay is fit

with tri-exponential components 107 ps, 1.29 ns, and 8.1 ns. The long component

has become substantially faster compared to collision-free conditions. Collisions

with argon cause vibrational cooling of the low-frequency modes of W(CO)6 . The

cooling reduces the spectral shift, which is responsible for the long decay component, and the decay becomes faster. (b) Pressure D 1100 psia. Argon density of

¾3 mol/L. The decay is now a single exponential with lifetime 1.20 ns. The fastest

and slowest tri-exponential components observed in the gas phase (Fig. 6) are eliminated by rapid collisions with argon at this high pressure (inverse collision frequency

of ¾3 ps). The lifetime is unchanged from the gas phase value, within experimental

error.



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Myers et al.



be employed. The Hamiltonian for the system is

1 2 1 2 2

P C ω q

2

2



HD



1 2 1

P C

2

2



C



2



Q2 C V Q, fq g



1



where the sum is over the harmonic part of all modes, q , other than

the high-frequency asymmetric CO stretching mode, the second term in

brackets is for the CO stretching mode, Q, and the last term is the potential

for the anharmonic interactions among modes. Since the experiments only

involve the ground and first excited vibrational states of the CO stretch, the

Hamiltonian can be restricted to that space. Then,

1 2 1 2 2

P C ω q

2

2



HD



C E0 j0ih0j C E0 C



j1ih1j C V0 j0ih0j C V1 j1ih1j



2



where

V0 D h0jV Q, fq g j0i Á V0 fq g

V1 D h1jV Q, fq g j1i Á V1 fq g



3

4



V0 is the anharmonic interaction when the CO stretch is in the ground

vibrational state, and V1 is the anharmonic interaction when the CO stretch

is in its first excited vibrational state. To simplify the analysis, take E0 D 0,

and transform into the interaction representation with respect to all of the

fq g. Furthermore, take V0 D 0, which restricts the analysis to the excited

vibrational state but does not fundamentally change the nature of the results.

Then, the effective Hamiltonian for the CO stretch is

Heff t D



j1ih1j C V1 fq t g j1ih1j



5



Initially the CO stretch is in the ground state, so  D j0i. The laser prepares

the state j1i, and the system evolves as

j t i D U(t)j1i D e



i t



e0



i



t

0



V1 q



d



j1i



6



where U t is the time propagator and eo is the time-ordered exponential.

For a pump-probe experiment, we need the probability, P t , of making a

transition back to j0i. P t depends on the dynamics of the low-frequency

modes and is obtained through an average over the initial states of fq g

2



P(t) / jh0j j t ij / jh1j t ij



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7



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where

is the transition dipole operator and the bar over the expression indicates the average over initial states. Substituting Equation (6) into

Equation (7) gives

P(t) D A h1je0



i



t

0



V1 q



d



2



j1i



8



with A a constant. As an initial approximation, the average in Equation (8)

was replaced by a second cumulant approximation, and a simple model was

used in which the potential V1 depends on the coupling between modes as

V1 fq g D



c q q



9



,



where c is the coupling constant and

expressions yields (71)



P(t) D A exp  2

,



6D . Evaluation of the resulting



jc0 , j2 [n n C n C 1 n C 1 ]



ð



1



cos[ ω C ω t]

ω Cω 2



ð



1



cos[ ω

ω t]

2

ω

ω



C [n n C 1 C n n C 1 ]

10



Even for the simple model of cubic anharmonic coupling Qq q of the

high-frequency mode Q and the low-frequency modes fq g, Equation (10)

demonstrates that the pump-probe signal will have a time dependence in

addition to the vibrational lifetime, T1 . (Note that for W(CO)6 , ω D ω

will not appear in the sum since the coupling coefficient for such terms will

vanish. Q is an antisymmetric mode. For ω D ω , the modes q and q

are members of a degenerate mode. Therefore, the direct product of their

irreducible representations is symmetric, and the matrix element vanishes

by symmetry.) Equation (10) describes the time dependence of a single

molecule. The time dependence depends on the frequencies of the lowfrequency harmonic modes, the coupling strengths of the modes, and the

occupation numbers of the modes. While the harmonic mode frequencies

and the coupling strengths will be identical for all molecules, the occupation

numbers will differ. Each molecule will display a complex time dependence.

W(CO)6 has 27 low-frequency modes ranging in frequency from 60 to

590 cm 1 . Thus, the time dependence will be complicated and recurrences



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Myers et al.



will occur. An ensemble average of Equation (10) was performed numerically by properly selecting a large number of sets of occupation numbers

and evaluating Equation (10) for each set and summing the results. An

initial partial decay of P(t) was observed followed by oscillations. The oscillations occur because of the harmonic nature of the theoretical model. The

occupation number of a given mode varies from one molecule to another,

but because the modes are harmonic, the frequency does not change with the

occupation number. It is anticipated that a more detailed theory that includes

anharmonicity will eliminate the recurrences. Nonetheless, the theoretical

treatment illustrates that even in the gas phase in the absence of collisions,

a fast component can appear in the pump-probe signal.

The decay of the CO stretch is a single exponential when W(CO)6

has substantial interactions with a solvent. A single exponential (aside from

orientational relaxation in liquids) is observed even when very fast pulses

are used in the experiments (81). In the gas phase, the transition frequency

of the CO stretch evolves over a range of frequencies because of its timedependent interaction with the low-frequency modes. When a buffer gas

or solvent is added, collisions cause the coherent evolution of the slow

modes to be interrupted frequently, possibly averaging away the perturbation responsible for the observed fast time dependence. Thus, the fastest

and slowest components of the tri-exponential decay are inherently lowpressure, gas phase phenomena.



IV. THEORY OF T1 IN SUPERCRITICAL FLUIDS



Using a Fermi’s Golden Rule approach, if the coupling between the oscillator and the bath modes is weak, then, to first order, the transition rate

from the first excited vibrational level to the ground state is given by (3)

k , T D T1 1 D



1

h¯ 2



1

1



dt eiωt hV10 t V01 0 i



11



where h¯ is Planck’s constant divided by 2 . For a diatomic, ω is the oscillator frequency. However, for a polyatomic, in which some fraction of the

energy of the initially excited mode can be transferred to other internal

modes of the molecule, ω is the frequency corresponding to the amount of

energy that is transferred to the bath (solvent). Vij is the ij matrix element

of the perturbation coupling the bath and oscillator. The brackets denote

the thermally weighted average over all the quantum states of the bath. The



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perturbation V is usually assumed to be linear in the solute coordinate, i.e.,

VD



qF



12



so that F is the solvent force on the oscillator. If one makes the additional

assumption that the oscillator is harmonic, q can be rewritten in terms of

raising and lowering operators and Equation (11) can be reduced to

T1 1 D



1

2m¯hω



1

1



dt eiωt hF t F 0 iqm



13



where m is the oscillator mass. The “qm” subscript is a reminder that this

is a quantum mechanical trace over the solvent. This force-force correlation

function is difficult to calculate quantum mechanically, so that the relaxation

rate is often calculated by

T1 1 D



1

2m¯hω



1

1



dt eiωt hF t F 0 icl



14



where cl indicates that the force-force correlation function is classical.

However, as several authors have pointed out (5,7,82), it is incorrect

to directly replace the quantum mechanical correlation function with its

classical analog because the detailed balance condition will not be met.

Therefore, the correct expression is

T1 1 D



Q

2m¯hω



1

1



dt eiωt hF t F 0 icl



15



where Q is called the quantum correction factor and accounts for the limitations of the semi-classical approximation. The nature of the correction

factor is a topic of considerable recent interest (82). Several forms of Q

have been suggested, and they are generally functions of temperature and

oscillator frequency. The choice of an appropriate correction factor will be

discussed in greater detail below.

The calculation of the lifetime is thus reduced to the problem of calculating hF t F 0 i. This is a problem that has had a fairly long association

with studies of solvation dynamics, where it usually appears in the context

of efforts to model friction coefficients. A great deal of activity in this

field has been directed at using the methods of density functional theory

(83) to derive expressions for the correlation function that involve the thermodynamic parameters of the system (72,84), which themselves are often

amenable to further analytical treatment or else may be determined experimentally or through simulations. In the treatment of vibrational relaxation



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