B. Gas Phase Vibrational Dynamics
Tải bản đầy đủ - 0trang
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
Copyright © 2001 by Taylor & Francis Group, LLC
VER of Polyatomic Molecules
639
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.
Copyright © 2001 by Taylor & Francis Group, LLC
640
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.
Copyright © 2001 by Taylor & Francis Group, LLC
VER of Polyatomic Molecules
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
Copyright © 2001 by Taylor & Francis Group, LLC
642
Copyright © 2001 by Taylor & Francis Group, LLC
Myers et al.
VER of Polyatomic Molecules
643
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.
Copyright © 2001 by Taylor & Francis Group, LLC
644
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
Copyright © 2001 by Taylor & Francis Group, LLC
2
7
VER of Polyatomic Molecules
645
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
Copyright © 2001 by Taylor & Francis Group, LLC
646
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
Copyright © 2001 by Taylor & Francis Group, LLC
VER of Polyatomic Molecules
647
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
Copyright © 2001 by Taylor & Francis Group, LLC