J . Phys. Chem. 1988, 92, 3363-3314
3363
Raman Spectroscopy: Time-Dependent Pictures
Stewart 0. Williams and Dan G. Imre*
Department of Chemistry, University of Washington, Seattle, Washington 981 95
(Received: September 9, 1987)
In an effort to better illustrate the link between nuclear dynamics and spectroscopy, we use a time-dependenttheory to investigate
the Raman process. Two-photon phenomena are presented for continuous-wave excitation of diatomic molecules. Exact
numerical results are shown for both a bound excited state and a repulsive excited state. In particular, we introduce the
Raman wave function and demonstrate its equivalence to the virtual state-the virtual state is the term used to describe
the intermediate state produced in the Raman process. The paper is a pictorial overview of two-photon processes.
Introduction
The increased interest in polyatomic vibrational spectroscopy
has sprouted many two-photon spectroscopies. We find when
dealing with polyatomic molecules that the density of states is
usually such that it is hopeless to try to assign each spectroscopic
line. Instead we resort to a dynamic interpretation of the spectrum.
In the two-photon experiment, the dynamics on the excited
electronic surface is probed in a two-step process. A laser prepares
the molecule in some vibrational state (not necessarily an eigenstate) of the electronically excited state. The vibrational
properties (dynamics) of that state are examined by recording the
projection of the prepared state onto vibrational levels of another
surface-in most cases the ground electronic surface (fluorescence/Raman spectroscopy).
Whether an experiment should be treated in the framework
of Raman spectroscopy or fluorescence spectroscopy is not always
obvious. Indeed we find that some authors prefer to describe their
experiments in the language of fluorescence spectroscopy while
others prefer the language of resonance Raman spectroscopy. 1-3
What has been established is that for a given molecule the spectra
that are obtained strongly depend on the incident laser f r e q ~ e n c y . ~
That is, qualitatively different results are obtained if the laser
frequency is in resonance with transitions to discrete states, or
is in resonance with transitions to a continuum of states, or is far
from resonance with any state.
It seems that if we are going to use these spectroscopies to study
dynamics we should address some very elementary questions
relating to two-photon transitions, and if we are going to interpret
the experiments from a dynamic point of view, it is appropriate
to cast the twoLphoton process in a time-dependent formalism.
This paper attempts to give further insights into the link between
nuclear dynamics and spectroscopy. We utilize a time-dependent
theory and provide a consistent picture which treats the two-photon
process (absorption and subsequent emission) from short time to
long time, from off resonance to on resonance, from Raman to
fluorescence. This paper is a pictorial presentation of the twophoton process from three time-dependent points of view.
We address a very simple question-what is emitting in each
case and thus what can be learned from the spectrum. The model
that we examine assumes a continuous-wave excitation source,
and the molecules are diatomics. Section I is a simple overview
of two-photon processes presented from four points of view: the
standard time-independent KHD theory and the three time-dependent theories we will use. In the following sections we present
results for the three time-dependent pictures. Section I1 presents
the time evolution of the overlaps between the intermediate
one-photon state and the final probing state. The time evolution
is observed for two cases: a repulsive excited state and a bound
excited state. We use the time evolution of the overlaps to obtain
Raman excitation profiles. In section 111 we address the question
of what is prepared by the laser. We provide a graphic presentation of the virtual state for two cases-a bound and a repulsive
excited state. Here we use the third picture presented in section
I in which there is no need to know the eigenstates of the excited-state surface. The ground state, on the other hand, is treated
in the stationary picture and we do need to know the eigenstates
of that surface to calculate the spectrum. Section IV adopts the
fourth point of view in which there is no need to know the eigenstates of either surface. The dynamics of the virtual state on
the ground-state surface determines the spectrum. This approach
will be most useful for on-resonance Raman spectroscopy, where
very often high vibrational levels are observed. In section V we
address a long outstanding question concerning the connection
or differences between Raman and fluorescence.
( I ) Behringer, J. J . Raman Spectrosc. 1974, 2, 275.
(2) Mingardi, M.; Siebrand, W. J . Chem. Phys. 1975, 62, 1074.
(3) Holzer, W.; Murphy, W. F.; Bernstein, H. J. J. Chem. Phys. 1970, 52,
(5) (a) Tannor, D. J.; Heller, E. J. J. Chem. Phys. 1982, 77, 202. (b) Lee,
S . Y.; Heller, E. J. J . Chem. Phys. 1979, 71, 4777.
(6) Heller, E. J. Potential Energy Surfaces and Dynamics Calculations;
Plenum: New York, 1981; p 103.
(7) Heller, E. J. Acc. Chem. Res. 1981, 14, 368.
399.
(4) Rousseau, D. L.; Williams, P. F. J . Chem. Phys. 1976, 64, 3519.
0022-36S4/88/2092-3363$0~.50/0
I. Raman Spectroscopy: Four Points of View
1 . The common way of expressing the Raman amplitudes is
given by the Kramers-Heisenberg-Dirac (KHD)4-7 expression
where Ix,)’s are the excited-state vibrational eigenstates; Idl) =
p 1 2 1 x l )p12
, is the transition moment between surfaces 1 and 2;
Ixi)is the initial ground state (normally u” = 0);
= p2,1xf),
Ix,) is the final vibrational eigenstate of the ground state; w’ =
wll wI,toll is the zero-point energy and wIis the incident radiation
frequency; w, is the energy of the excited-state eigenstate; and
r is the phenomenological lifetime.
The KHD expression is derived from second-order perturbation
theory* and presents the two-photon event from a stationary state
point of view. As such it provides a static picture of the two-photon
event. It implies that the first photon prepares a coherent superposition of the excited-state vibrational eigenstates. This superposition is weighted by two factors: the Franck-Condon (FC)
overlap of each state with the initial vibrational level and the
amount of detuning (Aw = w’ - a,) from that state. The Raman
amplitude into the final state is then a sum of overlaps between
the individual states that make up the superposition of states and
the final vibrational state
While it is true that any timedependent approach must deal with the same set of states, it is
not necessary to know all the states.
To interpret the Raman spectrum in this picture requires
knowledge of all or many of the excited-state eigenfunctions. In
essence, knowing all the eigenstates amounts to knowing the
dynamics to infinite time. This is in sharp contrast to the nature
of a typical Raman experiment, which provides information only
+
0 1988 American Chemical Society
3364 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
about very short time dynamics. Moreover, for most polyatomic
molecules it is impossible to obtain the many eigenfunctions that
are needed for implementation of eq 1. The stationary-state
approach takes no advantage of the fact that only a very small
region of the potential energy surface is probed by this experiment.
It leads one to conclude that Raman spectroscopy is useful mainly
for obtaining ground-state vibrational frequencies. As a consequence, Raman spectroscopy has been treated as a vibrational
spectroscopy. In summary, this approach requires knowledge of
eigenfunctions for two surfaces and it makes no connection between the spectrum and the dynamics it probes.
2. An equivalent e x p r e s ~ i o ncast
~~'~
in a time-dependent formulation sheds some light on the connection between the experimental results and the dynamics. Here the Raman amplitude
is
where 14,) = cL1zIxI),I+J) = K21bJ)? and I4(t)) = e'H"'*14,). Ix,)'s
are ground-state vibrational wave functions. w' = wI E, where
wI is the laser frequency and E, is the energy of the initial
ground-state wave function. r is a phenomenological lifetime
which can represent collisions or intersystem crossings.
If we examine this expression we find, as before, two factors
control the Raman amplitude: the detuning frequency and an
overlap function. Let us assume for the moment the incident laser
frequency is on resonance with an excited electronic state or that
Aw = 0; hence, e'awr= 1 and the Raman amplitude depends only
on the correlation function ($A$(?)). This part of the integral
contains the dynamical information. It says that the first photon
transfers Iq),the initial wave function, to the electronically excited
state where it is not an eigenstate. It becomes a moving wavepacket, Id(?)), with its motion governed by the excited-state
Hamiltonian, Hex. I4(t)) changes shape and moves away from
the Franck-Condon region. In doing so, it develops overlap with
other vibrational levels of the ground state. A second photon
transfers it back to that state. This point of view makes a simple
connection between the dynamics and the spectrum. For offresonance spectra we only need to know the dynamics for a very
short time, typically on the order of a few femtoseconds. There
is no need to know the excited-state eigenfunctions, but we still
need to know the ground-state vibrational eigenfunctions, ( ~ A ' s .
In section I1 we examine the correlation functions, and the excitation profiles afi(w).
3. In a continuous-wave experiment such as the one we are
treating here, it is reasonable to describe the two-photon process
as follows: pieces of the ground-state wave function are continuously transferred to the excited state; some of these remain on
the excited state for a while, while others return to the ground
state. How long these pieces survive on the excited surface depends
on the laser frequency, the point being that new pieces are constantly arriving while other parts are leaving. At any time these
pieces form a steady-state-like "population" on the excited state;
this population forms the virtual state. We can go back to eq
2 to get some insight into what the nature of this state is. Equation
2 can be rewritten to yield
+
since (4jis time independent. Now define the Raman wave
dt.lo This is the virtual state.
function )R,,,q,)2 J"~e'"''-'lrld(t))
The Raman amplitude is then given by
(+jR(w))
(3)
which is a simple overlap between the Raman wave function and
the ground-state vibrational eigenfunctions. What this equation
(8) Lin, S. H.; Fujimura, Y.;Neusser, H. J.; Schlag, E. W. Multiphoton
Spectroscopy of Molecule; Academic: New York, 1984.
(9) Heller, E. J.; Sundberg, R. L.; Tannor, D. J. J . Phys. Chem. 1982,86,
1822.
(10) Sundberg, R. L.; Heller, E. J. Chem. Phys. Lett. 1982, 93, 586.
Williams and Imre
implies is that the laser prepares the Raman wave function which
is neither time dependent nor an eigenstate of either of the two
Hamiltonians involved. All the dynamical information is contained
within IR(w,q)). It extends in phase space over all the regions
that I+(?))
visits.9 ( R ( w , q ) )is a function of the incident laser
frequency and thus is an experimentally controlled wave function.
The laser frequency can be used to determine which part of the
dynamics of I + ( t ) ) will contribute to the Raman wave function.
This point of view is very similar to the second one in that the
excited state is treated dynamically without the need to know the
eigenfunctions, while the ground-state wave functions are used
to compute overlaps. From a computational point of view, this
is the preferred method for calculating Raman spectra for a few
excitation frequencies, because a single trajectory generates I+(?) )
which can be half Fourier transformed at those frequencies. The
shape of IR(w,q)) and its extent in coordinate and momentum
space for a given laser frequency indicate the region of the surface
that the experiment probes. Section I11 examines these wave
functions and the effect of the laser frequency on their shape.
4. The third point of view implies that the total Raman
spectrum is a sum of Franck-Condon overlaps9
I(%)
a
CI(+/iR(w4))126(0- E / / h )
J
where = w1 - w, + E J h , and w, is the frequency of the scattered
light. E, and E, are the energy of the initial and final eigenstates.
This is analogous to an absorption spectrum except that in this
case the initial state is JR(w,q)).We can then derive an expression
for the total Raman spectrum":
^m
(4)
where R(w,q,?))= e'H~f/hp211R(w,q))
is an evolving packet on the
ground electronic surface.
Equation 4 views the Raman process from yet another angle.
It implies that the two-photon process prepares a nonstationary
state (JR(w,q))on the ground electronic surface. The dynamics
of the Raman wave function on the ground state determines the
spectrum. When the dynamics on the ground electronic surface
are the primary interest, this would be the most appropriate picture
to adopt. It views the Raman process as a probe of the dynamics
of a displaced oscillator on the ground surface. The advantage
it offers over many other techniques which are used to study
ground-state dynamics is that it provides a very specific displacement. How much excitation and along which coordinates
depends on the laser frequency and the dynamics on the excited
state.
The fourth point of view provides a treatment that does not
make any reference to stationary states and thus will be most
appropriate for spectra where high vibrational levels are observed.
We will demonstrate a model calculation using eq 4 in section
IV.
We have briefly shown four mathematically equivalent ways
of looking at the two-photon process. The time-independent KHD
approach requires knowledge of the excited-state eigenfunctions
and does not provide the user with much insight concerning what
the molecule is doing (such as reaction dynamics on the excited
state, etc.). Therefore, since direct interpretation of the results
obtained from the KHD expression is very difficult, we will not
provide results using this time-independent approach. Instead,
the rest of the paper will concentrate on results from the three
time-dependent approaches to the two-photon process.
11. Raman Correlation Functions
In this section we adopt eq 2. According to this equation the
Raman amplitude for each band is obtained by a half Fourier
transform of the correlation function (@A@(?)). We examine the
ingredients that make up the equation. First we need to look at
the dynamics of I+(?)).Then the overlaps of the moving wave
(11) Coveleskie, R. A,; Dolson, D. A.; Parmenter, C. S. J. Phyr. Chem.
1985, 89, 645.
Raman Spectroscopy: Time-Dependent Pictures
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0.0
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2.0
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The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3365
l
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DISPLACEMENT
Figure 1. Moving wave packet on the excited-state surface a t time r =
0 ( t o ) and a t later times tl and t2. Shown are the magnitude (-) and
the real part (-) of the evolving wave packet.
packet with the various vibrational levels are obtained. Finally
these overlaps are used to calculate excitation profiles. We study
two cases: a repulsive excited state and a bound displaced excited
state.
a. Repulsiue Excited State. We start the study by examining
the simpler of the two cases. The dynamics on a 1-D repulsive
state consists of motion of the packet away from the FranckCondon region. I @ ( t ) )never returns to the FC region. In turn
we expect the correlation functions, ( @ b @ ( t ) )to, exhibit simple
behavior.
Figure 1 shows the excited-state potential and a few snapshots
of the time-evolving wave packet. In this calculation the ground
state is a harmonic oscillator with a vibrational period of 1000
cm-’. The propagation is done by solving the time-dependent
Schrodinger equation on a grid of 256 points, using the Fourier
transform method as developed by R. Kosloff et a1.18J9 We check
for convergence by doubling the grid size and decreasing the time
step, and K~~ is constant for these calculations. The magnitude
of I @ ( t ) ) and the real part are shown for each time frame. The
time evolution in this case is rather simple. At time t = 0, the
laser transfers the ground-state wave function, I@,),to the excited-state potential. [@,) is not an eigenstate of the excited-state
Hamiltonian and hence becomes an evolving wave packet I @ ( t ) )
on the excited-state surface. For a short time the real (and
imaginary) part develops nodes indicating acceleration and increasing momentum, and then I+(t)) moves away from the
Franck-Condon region, spreading as it proceeds toward separated
fragments. Figure 2 shows the time evolution of the overlaps with
I ) u” = 0 (self-overlap)
final Raman vibration levels ( I ( @ i @ ( t ) )for
and u” = 1, 2, 3, 4, 5, and 6. We note that the overlaps, and the
Raman spectrum for that matter, evolve in time in a simple
sequential manner; the self-overlap decays rapidly while the
(12) Coveleskie, R. A.; Dolson, D.A.; Parmenter, C. S . J . Phys. Chem.
1985, 89, 655.
(13) (a) Shibuya, K.; Stuhl, F. J . Chem. Phys. 1982, 76, 1184. (b) Krupenie, P. H. J . Phys. Chem. Ref: Dura 1972, 4, 423. (c) Julienne, P. S.;
Krauss, M. J . Mol. Spectrosc. 1975, 56, 270.
(14) (a) H o l m , W.; Murphy, W. F.; Bernstein, H. J. J. Chem. Phys. 1970,
52, 399. (b) Ghandour, F.; Jacon, M. J . Chem. Phys. 1983, 79, 2150.
(15) (a) Baierl, P.; Kiefer, W. J . Chem. Phys. 1975, 62, 306. (b) Coxon,
J. A. J . Mol. Spectrosc. 1971, 37, 39.
(16) Imre, D.G.; Kinsey, J. L.; Sinha, A.; Krenos, J. J. Phys. Chem. 1984,
88, 3955.
(17) Rebane, K. K.; Tehver, I. V.; Hizhnyakov, V. V . Theory ofLight
Scorfering in Condensed Muffer;Plenum: New York, 1976; 393.
(18) Kosloff, D.;Kosloff, R. J. Compuf. Phys. 1983, 52, 35.
(19) Kosloff, R.; Kosloff, D. J . Chem. Phys. 1983, 79, 1823.
’
l
’
l
’
7.5
5
1
IO
’
l
11.5
’
l
~
15
l
.
i
XI
17.5
TIME(femto eeol
Figure 2. Time evolution of the correlation function (4&i(t)) for f =
0, 1, 2, 3, 4, 5 , and 6 and i = 0.
1
li
II
I 1
I 1
I 1
I I
d.M
0.01
I
I
1I
I
1
0.06
I
0.11
I
PREOUENCYLau)
Figure 3. Raman excitation profile for scattering into u” = 0 (- - -), u”
= 1(-), u ” = 2 (-*-),and u ” = 3 (---). Also shown is the absorption
spectrum (-).
overlaps with higher vibrational levels are increasing in order u”
= 1, then u” = 2, etc. This simple picture is true when the
transition moment is constant with respect to nuclear motion. The
Raman amplitudes are half Fourier transforms of these correlation
functions; Figure 3 shows the square of the Raman amplitudes,
lafiI2.
Also shown is the absorption spectrum for the repulsive
potential, which is the full Fourier transform of the self-overlap
(eq 5). Figure 3 represents the Raman excitation profiles. The
Raman excitation profile gives the Raman intensity into a particular ground vibrational level as a function of the incident laser
frequency. To obtain a Raman spectrum for a given laser frequency, one has to measure the height of each profile a t t h a t
frequency-the height is proportional to the spectral intensity.
As we have expected, the excitation profiles are smooth and
they peak at the same frequency as does the absorption spectrum.
We note also that the line shapes are not symmetric with respect
to the center of the band. Note also the long wings on the excitation profile for u’ = 1 as compared with the absorption
spectrum.
The dynamics of I @ ( t ) )determines the Raman amplitude. An
observed Raman line corresponding to
is an indication that
Williams and Imre
3366 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
1I
dispLocement
,
dispLacement
0
1
1
displacement
dispLocement
diapLocement
dispLocement
dispLacement
i
a
TIME(femto eecl
1
I I
Figure 4. Moving wave packet on the bound excited state. Time t = 0
and at later times-each frame represents 2.5 fs.
I$(t)) has visited a region spanned by the ground-state eigenfunction
Its intensity is a function of how much of I $ ( t ) )
visited the region of I@)and the length of time the wave packet
spent in that region. Moreover, the spectrum tells not only where
I$(t)) travels but also when, since the overlaps with the
ground-state vibrational levels develop in a simple sequential order
with higher vibrational levels indicating later times. These simple
ideas do not hold when the excited state is bound and the laser
is tuned into resonance with one of the eigenstates. Here I $ ( t ) )
may contribute to a given spectral line more than once since it
may revist regions spanned by I$,-) at later times.
b. Bound Excited State. Here, the ground state is the same
as in the previous section but the excited state is a Morse oscillator
with W , = 1000 cm-’ and De = 30000 cm-I. Figure 4 shows the
time evolution of 14,) on the Morse potential. Two vibrational
periods are shown. The initial motion in the bound potential is
very similar to that on the repulsive excited state. Once I$(t))
reaches the other wall it reflects back. It returns to the FC region.
Motion in this anharmonic potential can be rather complex as
I$(?))is composed of many eigenstates, each with a slightly
different frequency. When we observe the dynamics for a time
long enough for these frequency differences to be significant, we
will observe a break up of I $ ( t ) ) into many small pieces. As a
consequence we expect the correlation functions to also exhibit
rich structures. Figure 5 shows the correlation data for slightly
over one vibrational period. At early times the correlation data
behaves very much like the repulsive case (see insert in Figure
5), which is consistent with the above discussion about the dynamics of W t ) ) for early time. As 1$(t)) returns to the FC region,
we observe highly structured recurrences. The oscillatory behavior
is due to the anharmonicity of the potential. The overall envelopes
for v” = 1 and v” = 2 show two peaks. These are due to a simpler
effect and will be seen even in a harmonic potential. Overlap with
v” = 1 develops when I $ ( t ) ) returns to the FC region on its way
from large to small displacements and then again as it reflects
back from the steep wall on its way from small to large displacement, resulting in the structure observed in Figure 4. Later
in time I # ( t ) ) undergoes many recurrences which manifest
themselves in a somewhat complicated, highly structured Raman
excitation profile. The profiles for scattering into v” = 0, 1, 2,
and 3 are shown in parts e, d, c, and b, respectively, of Figure
6 while the absorption spectrum is shown in Figure 6a. Note
the nodal pattern in the excitation profiles reflects the level for
Figure 5. Time evolution of the correlation function ( 4,+$i(t))for f =
0 (-),f = 1 (- - -), andf = 2 (...), and i = 0. Insert shows short-time
evolution.
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0.01
0.m
I
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PRWUENCY Iau 1
Figure 6. (a) Absorption spectrum and Raman excitation profile for
scattering into (b) u” = 3, (c) u” = 2, (d) u” = 1, and (e) 0’’ = 1.
which the profile is shown-one node for v ” = 1; two nodes for
u” = 2, etc. One can easily see how sensitive the Raman intensity
into a final state as a function of incident laser frequency will be.
Small changes in incident laser frequency will lead to vastly
different intensities in v” = 1, 2, etc. This would be further
illustrated in section 111.
Excitation profiles have been used to map absorption spectra.
Figure 6 shows that for a molecule with a long-lived excited state
the excitation profiles for v” > 0 do not follow the absorption
spectrum. Moreover, the overall excitation profiles can easily be
mistaken for more than a single excited state contributing to the
absorption. Clearly, we show here that a single excited state can
produce an excitation profile for v“ = 1, for example, with two
maxima, neither corresponding to the maximum in the absorption
spectrum. Thus, extreme care must be exercised when using these
data to obtain information about the absorption spectrum. As
we have shown, it is a reasonable method for the repulsive state
Raman Spectroscopy: Time-Dependent Pictures
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3367
la1
Ib)
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I
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I
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PREWENCYlauI
Figure 7. Normalized absorption spectrum for harmonic oscillator excited state as a function of lifetime, r. (a) r = 13 vibrational periods,
(b) I? = 1.2 vibrational periods, (c) I? = 0.72 vibrational period, (d) r
= 0.24 vibrational period, and (e) r = 0.12 vibrational period.
case but not for a long-lived excited state, where the molecule lives
on the excited state for over a single vibrational period.
111. The Raman Wave Functions
C
I
I
D
The "virtual state" has been a common term which was introduced to explain the Raman phenomenon. There has been a
lot of speculation as to the nature of that state.14 The third point
of view we discussed in section I sheds light on that subject. We
have equated the virtual state with the Raman wave function. As
we have done before, let us assume for the moment that the laser
is tuned such that eiWf= 1 and r = a, then the Raman wave
function is given by
This equation implies that the first photon prepares a timeindependent wave function, whose form depends on the dynamics
of the initial wave function on the excited-state potential. It is
as if we add up the I d ( t ) ) to itself at every time step to account
for the contribution of I+(t)) to the spectrum over all time. The
term eiwfacts as a filter, and it projects out of I q j ( t ) ) components
with frequency w. On the other hand, the decay term e-r/r is a
phenomenological lifetime which can control the time scale,
contributing to IR(w,q)).We will examine the effect of these terms
on the virtual state for two cases: a repulsive and a bound excited
state.
a. Bound Excited State. Both ground and excited states are
harmonic with vibrational frequencies of 500 cm-'. The excited
state is displaced from the ground state by 0.6 A so that the vertical
transition peaks at u' = 7. We assume a constant transition
moment with respect to nuclear geometry, and the dynamics of
Ir$(t)) is obtained by semiclassical methods (in this case the method
is exact).
We first examine the time evolution of IR(w,q)). This is done
by introducing a phenomenological lifetime, r, which can be
thought of as representing collisions, intersystem crossing, or even
intramolecular relaxation into many bath modes. The effect of
the lifetime on t h e absorption spectrum is shown in Figure 7 . In
this figure, we have computed the absorption spectrum using the
time-dependent formula5
in which we used r = 0.12, 0.24, 0.72, 1.2, and 13.0 vibrational
periods. We will correlate the absorption data with the Raman
data.
Figure 8. (a) Overlap of IR(w,q)) with excited-state wave functions, (b)
normalized Raman wave functions, and (c) emission spectrum ( ( R ( w , q)Ib,)). Results show how all three of the above develop as the lifetime
changes for r = 0.12, 0.24, 0.72, 1.2, and 13 vibrational periods. Results
presented are for resonant excitation to (A) u' = 1, (B)u' = 2, (C) u'=
7, and (D) u' = 11, respectively.
Figure 8 (parts A, B, C, and D) shows results for laser excitation
resonant with u' = 1, 2, 7, and 11, respectively. Part b of each
figure shows IR(w,q))I2with the intensity chosen such that each
Raman wave function is normalized. For each excitation five
Raman wave functions are shown. These correspond t o the same
lifetimes mentioned above with F = 0.12, 0.24,0.72, 1.2, and 13.0
vibrational periods. The trajectory in each case is run for 100
vibrational periods.
The bottom trace, (c), shows the Raman spectrum for each of
the Raman wave functions. The Rayleigh line, that is, the intensity
into u" = 0, is not included in these plots. The top plot, (a), shows
the overlap of each of the Raman wave functions with the excited-state vibrational wave functions or, in other words, the
superposition of excited eigenstates that make up IR(w,q)).
Williams and Imre
3368 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
(a1
Figure 10. Frequency dependence of (b) Raman wave function, R(w,q);
(c) emission spectrum; and (a) the superposition of excited eigenstates
that make up ( R ( w , q ) )for excitation below the excited-state potential
(Le., normal Raman scattering). From front to back results are for
incident laser frequencies wI = lOhw and 3hw below the bottom of the
well, and w1 = V, (i.e., bottom of well).
Figure 9. Frequency dependence of (b) Raman wave function IR(w,q)),
(c) emission spectrum, and (a) superposition of excited eigenstates that
make up JR(w,q))for laser excitation between u’ = 1 and v‘ = 2. Frequencies used are resonant to u ’ = 1, 1
and u ’ = 2.
+ 0.25wc, 1 + 0.5wc, 1 + 0.75w,,
In all four cases, we find that, regardless of the excitation
frequency, when the lifetime is short, the Raman wave function
is almost entirely the ground-state initial wave function. The short
time evolution produces slight overlap with u’’ = 1 and 2. As we
increase I‘,the Raman wave function, and thus the superposition
of states that make up JR(w,q)),collapses to the single eigenstate
to which the laser is tuned. The spectrum at the same time evolves
from one which would normally be termed Raman to a fluorescence s p e c t r ~ m . Note
~
also that the Raman wave function at
different excitation frequencies do not collapse to their corresponding eigenstates at the same rate. u’= 7 has the highest FC
factor, and consequently, when the laser is tuned to that level,
it produces u’ = 7 faster than any other vibrational state.
It is useful to compare the lifetime studies of the Raman wave
functions described above to the lifetime studies of the absorption
spectra in Figure 7 . When the lifetime is short, the absorption
spectrum is made up of overlapping bands (meaning the intensity
between bands is not zero), as shown in parts c, d, and e of Figure
7 for I’ = 0.72, 0.24, and 0.12, respectively. According to Figure
8b, for these same lifetimes, the Raman wave functions are not
made up of a single excited eigenstate but have significant contributions from numerous excited eigenstates. In fact, for r =
0.12 the absorption spectrum shows no band structure, while at
the same time, it is almost impossible to distinugish between the
Raman wave functions created by excitation to u‘ = 1, 2, and 7.
Consequently, when the lifetime is short compared to a vibrational
period, the Raman spectra are not very sensitive to the incident
frequency. From one point of view, it is easy to understand this
phenomenon. The molecule, in essence, does not have enough time
to determine the laser frequency. When r is large compared to
a vibrational period, we find, as expepcted, large variations in the
shape of IR(w,q))and consequently large variation in the Raman
spectrum with small changes in laser frequency.
We conclude that when the absorption spectrum can be resolved
into individual bands, and the laser frequency is chosen to be on
resonance with one of these bands, the laser will produce, to a
good approximation, the eigenstate that corresponds to that band.
This will not be true, however, when intra/intermolecular dynamics
is on a fast time scale. This will be especially true for bands with
weak FC transitions in the absorption spectrum.
We now examine the effect of tuning off-resonance. Figure
9 shows results for the same system for a series of laser frequencies,
rather than lifetimes, between u’ = 1 and u’ = 2 (actual frequencies, u ’ = 1, 1 0.25we, 1 0.5we, 1 O.75we, and u ’ = 2).
The lifetime is on the order of 13 vibrational periods, so that we
remove any effects due to I’. The top trace, as before, shows the
square of the coefficients of the individual excited eigenstates that
make up the Raman wave functions. Note that when the laser
is tuned exactly in between u’ = 1 and 2, we do not produce a wave
function which is a superposition of 1:l u’ = 1 and u ’ = 2. The
top trace shows that in this case, the u’ = 2 component is 5 times
larger than the u ’ = 1; furthermore, other vibrational levels are
represented. This is due to the fact that the overlap of u’ = 2 with
the initial wave function is greater than that for u’ = 1. The
bottom trace shows the Raman spectra for each of the Raman
wave functions. Note the reflection of the nodal pattern in llR(w,q))12 in the Raman spectra. There is a one-to-one mapping
of peaks in the Raman wave function and peaks in the spectrum.
That is, the spectra exhibit the structure of the Raman wave
function as predicted by the reflection principle.20 Excitation
+
+
+
(20) Tellinghuisen, J. J . Mol. Specfrosc. 1984, 103, 455.
(21) Hale, M. 0.;Galica, G . E.; Glogover, S. G.; Kinsey, J. L. J . Phys.
Chem. 1986, 90, 4997.
( 2 2 ) Sunberg, R. L.; Imre, D.G.; Hale, M . 0.;
Kinsey, J. L.; Coalson, R.
D. J . Phys. Chem. 1986, 90, 5001.
Raman Spectroscopy: Time-Dependent Pictures
between vibrational levels is an intermediate case which cannot
be classified as normal Raman or fluorescence. To create a nodal
pattern, I4(t)) has to undergo a recurrence. Therefore, the fact
that (R(w,q))exhibits nodes is an indication that the time scale
that contributes to the Raman wave function is longer than half
a vibrational period.
Figure 10 shows results for excitation below the excited-state
potential (Le., in the region of normal Raman scattering). Three
wave functions are shown: for the first (from the back), the laser
is tuned such that wI= Vo(i.e. bottom of the well); then wI= 3hw
and lOhw below the bottom of the well. The lifetime is 13
vibrational periods. The Raman spectra show a simple pattern.
u” = 1 has the most intensity in all cases. As the laser is detuned,
the spectra exhibit fewer overtones. For the first trace the laser
is off resonance with a transition to u’ = 0 by only ‘ / 2 h w and
,
inspection of Figure 10a shows that indeed the projection of
IR(w,q))onto u‘ = 0 is larger here than in any of the other states
represented in Figure 10. The same trajectory as in Figure 9 is
used here. The fact that we see no nodes in the emission spectra
or Jd(w,q)) means that the time scale contributing to the Raman
wave functions and to the Raman spectra is much shorter than
one vibrational period. Note also the similarity between the
off-resonance Raman spectra and wave functions and those obtained with the short lifetime on resonance in Figure 8. It demonstrates that, by detuning the laser, it is possible to choose which
part of the dynamics will contribute to the spectrum. Thus we
have the option to time resolve fast events without resorting to
very short pulsed lasers.
Previous calculations by Heller et al.1° have revealed the effect
of detuning on the time scale of the emission process or on the
time scale of the observable dynamics. Again it is demonstrated
that only short-time dynamics is observed for large detuning
frequencies. That is, the Raman wave function remains in the
local FC region for large detuning frequencies.
These concepts are nicely illustrated in numerous experiments.
We now present discussion for two such experimental systems.
In the chemical timing experiments by Paramenter et al.,1L~’2
high
quencher gas pressure was used to vary the fluorescence lifetime
in p-difluorobenzene (pDFB). Here the laser pumps the 3251of
SI state. This state is coupled to a high density of neighboring
states. For zero added quencher gas the fluorescence lifetime is
long (5 ns). The zero-order pumped state (Le., the initially pumped
state) relaxes (via IVR) to the neighboring states which dominate
the emission spectrum. As a result the observed emission shows
no structure. However, at high pressures of quencher gas only
short-time dynamics is observed. The fluorescence lifetime is short
(10 ps for 24.6 kTorr of added), and the zero-order pumped state
dominates the emission spectrum. Consequently, a structured
emission spectrum is observed. The dynamics in the SI state of
pDFB is on two time scales. The first involves very fast dynamics
corresponding to the time evolution along the zero-order state to
produce the 3251Raman wave function which is analogous to the
1-D bound system discussed above. On a second, slower time scale,
this Raman wave function continues to evolve toward a single
eigenstate (for a very narrow laser); this second process is normally
treated with an IVR formalism. The experiment manages to
shorten the lifetime to the point where the slower time evolution
is insignificant, producing a Raman wave function which at zero
order can be assigned as 3251.
To further demonstrate some of these Raman phenomena, we
will look at a direct application to the 0, molecule. Figure 1l a
shows an experimental resonance Raman spectrum for 0, reproduced from ref 13a. The laser in this case was tuned onto
resonance with 0’ = 4 of the B3Z,-state. The authors have been
able to reproduce most of the spectrum by a simple FC calculation
shown in Figure 11b. However, note the calculated spectrum is
missing the lower vibrational levels which appear in the experimental spectrum. Two factors contribute to the relatively large
intensities in u” = 1, 2 , and 3 in this case. First, due to predissociation, the lifetime of u ’ = 4 was estimated at approximately
1 ps,13cand second, the overlap of u’= 0 is rather ~ m a l 1 . lThis
~~
spectrum demonstrates the effects described in Figure 8. The u’
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3369
FRWUENCY [au)
om
o h
O h
FREOUENCY ou 1
Figure 11. (a) Experimental resonance Raman spectrum of 0,excited
-
by ArF laser. The laser is on resonance with L” = 4 of the B3Z[ state
of 0,;(b) Calculated Franck-Condon factors for B3Z,,7v’=4)
X3Z;(v’9. These results were reproduced from the data presented in ref
13a. In part a each peak in the spectrum is fitted with a Gaussian
envelope.
= 4 component of the t = 0 wave packet is extremely small (low
FC). Therefore, it takes a long time for the Raman wave function
to produce that state; in other words, the memory of the initial
state lingers for a long time. A picosecond after the excitation
the spectrum still exhibits the effect of the short-time dynamics.
We have simulated the spectrum in Figure 1l a by applying the
same procedure discussed in this section. For these calculations
the RKR potential curves for the ground and B states of 0, from
ref 13b were used. At time t = 0, the system consists of the
displaced ground-state wave function on the excited B state
surface. Since the ground-state wave function is not an eigenstate
of the B-state Hamiltonian, it becomes an evolving wave packet
on the B potential energy surface. The evolving wavepacket is
then half Fourier transformed a t the energy of u’ = 4 to create
the Raman wave function shown in Figure 12a. Both the amplitude and the real part of the Raman wave function are shown
in Figure 12a. Inspection of Figure 12a shows that the Raman
wave function is made up of two distinct components: the u ’ =
4 eigenstate (real part) and an imaginary component in the local
FC region. It is this imaginary component, which is the memory
of the initial f = 0 wavepacket, that is responsible for the intensity
into u ” = 1, 2, and 3 as shown in Figure 12b. Since RKR potentials were used, we found it most convenient to calculate the
Raman spectrum in Figure 12b using eq 4. The application of
eq 4 in obtaining Raman spectra will be further discussed in section
IV. It is clear from these calculations that the real part of the
Raman wave function (which is the u ‘ = 4 eigenstate) is responsible
for intensities into high vibrational levels (u” > 3), that is, the
fluorescence part of the spectrum, while the imaginary component
is responsible for the Raman lines, that is, intensity into u “ = 1,
2, and 3. A more in-depth study of the O2 molecule is discussed
in the paper following this
One very important note to
make here is that both the Raman and fluorescence lines originate
from the same single process.
(23) Williams, S. 0.;
Imre, D.G.J . Phys. Chem., following paper in this
issue.
3370 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
Williams and Imre
(a)
(a)
1
------*/
tbl
0.50 0.75
1.00
1.25
1.50
t.75
2.00
2.25
2.50
BOND LENGTH
Figure 13. Normalized (a) Raman wave function and (b) emission
tbl
o.m
o h
spectrum as a function of the incident laser frequency for a repulsive
excited state. From front to back the plots correspond to detuning by
A@ = 8500, 5500, 4000, 1550, and 0 cm-' from the center of the absorption band toward the red end.
0; 10
FREOUENCY(aul
Figure 12. Raman wave function for resonant excitation to D' = 4 in 0,.
Both the amplitude (-) and the real part (- - -) are shown. (b) Calculated Raman spectrum by using the Raman wave function produced in
part a. The results were calculated by using RKR potentials for both
the ground and the excited (B) state of 0,from ref 13b.
We find that in the bound case the spectrum is very sensitive
to the incident laser frequency. When the laser is tuned into or
near resonance with a vibrational level, the frequency of motion
of I$(t)) and the laser frequency are in resonance. I$(t)) survives
on the upper state for a long time, and it interferes with itself to
create the eigenstate (provided r is large) at the laser energy. As
we tune away, eiA"'oscillates at the wrong frequency and the
integral in eq 2 decays rapidly in time. Consequently, the spectrum
is determined by short-time dynamics, and thus, the laser frequency can be used as a tool to determine the time scale over which
the molecule is observed.
b. Repulsive Excited State. The system for this study is the
same as in section IIa. The calculation was carried out with the
same formalism described in section I. Here we examine the
frequency dependence of the Raman wave functions rather than
the time evolution and excitation profiles.
Figure 13 shows Raman wave functions for these model potentials obtained at five different excitation frequencies, the first
(from the back) being on resonance (Le., laser frequency chosen
to match the center of the absorption band) while the other Raman
wave functions correspond to detuning by 1550, 4000, 5500, and
8500 cm-' from the center of the band toward the red end. Figure
13b represents the Raman spectra for each of the Raman wave
functions. It is interesting to note the similarity between these
Raman wave functions and spectra and the short-time results for
the bound case.
For a repulsive potential I $ ( t ) ) accelerates away from the FC
region, developing overlap with higher vibrational states later in
time. The gradual decrease in spectral intensity with vibrational
quantum numbers is due to the fact that as I$(t)) accelerates it
spends less time in any given region of larger internuclear sepa-
Figure 14. (a) Continuum eigenstates and (b) emission spectrum for the
same laser excitations as in Figure 12.
ration. The wave packet also spreads, although this is a smaller
effect. Note how the Raman wave functions reflect that dynamics,
with most of the amplitude in the FC region, decreasing monotonically toward larger displacement. The on-resonance spectrum
corresponding to the longest observation time of the dynamics
shows, as one may expect for such a system, a long progression
with a simple intensity pattern. As we detune the laser, we limit
the time over which the system is observed. Thus we would predict
that the farther off resonance the laser is the fewer overtones will
appear in the spectrum.
It is important to note that even though the laser is a CW
single-frequency laser and the excitation frequency is chosen to
be on resonance with the dissociative excited eigenstate, the laser
does not prepare an eigenstate but rather a Raman wave function.
Figure 14 shows the continuum eigenstates corresponding to the
energy of the laser excitations shown in Figure 13. These eigenfunctions were obtained by using the same trajectory as the
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3371
Raman Spectroscopy: Time-Dependent Pictures
(bl
Ein'
h
h
(cl
3
(d)
(el
-10
5
0
5
MOMENTUM I arb
10
15
a0
uniteI
Figure 15. Raman wave functions in momentum space: (a) IR(w,p)) for
resonant excitation; (b) IR(w,p)) for Aw = 1550; (c) IR(w,p)) for Aw =
4000 cm-I; (d) Au = 5500 cm-I; and (e) A w = 8500 cm-I.
one used to generate the Raman wave functions; in fact, in this
case (since I+o)is real) the eigenfunctions are just the real part
of the Raman wave function. Since
IR(w,q)) = L m e i u W t ) dt,
)
then for I4(0)) purely real as in our case
2 Re IR(w,q))I = ~ ~ e ' " I ' l + ( dr
t))
and
is the eigenstate at energy wI.
Emission spectra corresponding to these scattering states are
shown on the lower trace. Note the different behavior exhibited
by the Raman wave function (Figure 13) and the scattering states
(Figure 14) as the energy changes. As we detune the laser, the
Raman wave functions move toward shorter displacements and
the spectral intensity moves with them to lower vibrational levels.
On the other hand, as the energy of the scattering states decrease,
they move toward larger displacements and their corresponding
spectral intensities shift with them to higher vibrational levels.
It is important to reemphasize that the simple absorption of a
photon prepares the Raman wave function, while the scattering
states can only be produced in an atom-atom collision experiment.
The differences between the Raman wave function and the
scattering or eigenstates will be further discussed in section V.
When comparing these emission spectra to experimental results
CH,I,I6 and others, it is clear that what the
for Cl,,l4 Brz,15
laser prepares is the Raman wave function and not the scattering
eigenstates.
Figure 15 shows the Raman wave functions from Figure 13
vs p for different
in momentum space. We have plotted IR(w,p))J2
excitation frequencies. Figure 15 illustrates a very interesting
property of the Raman wave function. Despite being time independent, the average momentum; ( p ) = (R(w,q)lPIR(w,q))is
not zero; in fact, the results show that ( p ) is a function of the
incident frequency: the average momentum increases as we get
closer to resonance. In effect we have the capability of tuning
the momentum of the Raman wave function by simply tuning the
laser frequency.
IV. Total Raman Spectrum
The full Raman spectrum is given bylo eq 4:
I(w,)
a
JI_=e"'(R(w,q)lR(w,q,r)
) dt
where P = w1 - o,+ E J h and ws is the frequency of the scattered
light. It is important to note that the above equation is equivalent
to the expression for the absorption spectrum (eq 5), with IR(w,q))
and P substituted for I+(O)) and w , respectively. The incident
photon places the ground-state wave function on the excited state,
where I+(i)) evolves in time producing the Raman wave function,
IR(w,q)). Emission of a photon returns IR(w,q)) to the ground
state. The Raman wave function is not an eigenstate of the
ground-state Hamiltonian, H,,, so it must evolve in time on the
ground-state surface according to IR(w,q,t)) = eJHgJ/*lR(w,q)).
The total emission spectrum is then given by the full Fourier
transform of (R(w,q)lR(w,q,t)).One of the advantages of this
formalism is that there is no need to know any of the eigenstates
of the ground or excited surface. The only eigenstate that we need
to calculate is u" = 0, our initial wave function. In most cases,
it can be well approximated by the harmonic ground state.
This approach to the spectrum presents the Raman process from
yet another perspective. The Raman spectrum/experiment can
be viewed as a study of the dynamics on the ground state. In order
to conduct such a study, we need to produce a displaced wave
packet on the ground electronic surface and obtain its spectrum;
the displacement can be in p or q space. This can be easily
achieved in a two-photon experiment. The first photon prepares
the displaced wave packet (Raman wave function) which is no
longer an eigenstate of the ground surface. The second photon
transfers it back to the ground state. The dynamics of IR(w,q))
on the ground surface is reflected in the Raman spectrum. The
displacement on the ground state, as we illustrated in the previous
section, is determined by the excited potential and the laser frequency.
To illustrate this point of view, we chose a Morse ground state
and a repulsive excited state of the form we used in section 111.
We first need to obtain the initial wave function u" = 0. This
was accomplished numerically. u" = 0 is then placed on the
repulsive excited state, and its dynamics are obtained as before,
by using the grid method. The Raman wave function was calculated at two frequencies, at 4000 cm-l off the center of the
absorption band and on the center of the absorption band. Since
a large component of IR(w,q))is the initial wave function u" =
0, we first subtract it out. This is equivalent to removing the
Rayleigh line. That produces a new wave function IR'(w,q)) where
IR'(w4)) = lR(w,q)) - (R(w,q)l+(O))I+(O))
which represents only the displaced parts of our initial wave
function. This is done mainly in an effort to show the part of
IR(w,q)) contributing to the dynamics on the ground state. Figure
16 shows the dynamics of IR'(w,q)) on the ground-state potential
for an off-resonance excitation (Aw = 4000 cm-I). The correlation
function (R'(o,q)lR'(w,q,r))for this case is shown in Figure 17a,
and the Raman spectrum is shown in Figure 17b. The overall
decay in the correlation function is due to two factors: an artificial
lifetime which we have introduced and the anharmonicity of the
potential. Since the laser excitation is off-resonance, only
short-time dynamics of the initial wavepacket contributes to the
Raman wave function. Therefore, IR(wJ)) is not very different
from the initial ground-state wave function and has a very small
average momentum on the ground-state surface. That is, the
off-resonance excitation produces a Raman wave function that
is only slightly displaced in both q and p space from the equilibrium
position in the ground state. As a result, the initial decay of the
correlation function is rather slow, producing few overtones in the
Raman spectrum (narrow spectrum).
Figure 18 shows the dynamics of IR'(o,q))for an on-resonance
excitation (Aw = 0 ) . We expect the Raman wave function to span
larger displacements and momenta. Note the complex dynamics
as part of the wave packet moves toward larger displacement due
to net initial momenta, while other parts respond to the force that
acts on the wave packet to move it toward smaller displacement.
Overall, the dynamics in this case exhibit larger amplitude motion
as is expected. The correlation function shown in Figure 19a shows
a rapid initial decay which results in a broad spectrum, Le., long
progression.
3312
The Journal of Physical Chemistry, Vol. 92, No. 12, I988
N-
4
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-1
N-
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disolacement
Williams and Imre
0
1
2
3
4
-1
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1
2
3
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displacement
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2
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-
displacement
tbl
Figure 16. Trajectory of IR)on the ground state for At 5 fs (Le., each
frame represents 5 fs on the ground state) for off-resonant excitation (Au
= 4000 cm-I). In order to show only the part of the Raman wave
function contributing to the dynamics on the ground state, the initial
wave packet I&) (or 0’’ = 0 in this case) is subtracted from IR).
In polyatomic molecules we can use these ideas to study the
dynamics of highly excited local modes-for example, in molecules
where the excited state produces a force predominantly along a
single local mode as in CHJ. The excited state produces a Raman
wave function which is displaced almost exclusively along the C-I
stretch. The Raman spectrum shows that, for vibrational levels
as high as 15 quanta in the C-I stretch, the energy which is initially
deposited in that mode remains in that mode.I6 At higher levels,
we find mixing or energy flow into the C-H umbrella mode. A
similar study on CD31 produces virtually the same initial wave
packet on the ground state, since the dynamics on the excited state
is very similar. Yet the Raman spectrum21*22
shows strong mixing
between C-I stretch and the C-D umbrella mode at very low
energies, due to a 1:2 resonance between the two modes.
V. Note on Raman vs Fluorescence
Up to this point, we have not made any distinctions between
Raman scattering and fluorescence. Some of the spectra we
produced in sections I1 and I11 have the appearance of typical
FC fluorescence spectra while others look like typical Raman
spectra. It is important to remember that all these spectra have
been generated by using one formula and we do not find it necessary to switch from one formula to another to generate these
two types of spectra. Our results, particularly for the bound case,
demonstrate that the transition from Raman to fluorescence is
a smooth one and can be achieved either by tuning the laser onto
resonance with a long-lived eigenstate or by changing the lifetime.
Since tuning on and off resonance amounts to changing the time
scale over which the experiment is conducted, we conclude that
what would typically be associated with a normal Raman spectrum
corresponds to a very short time scale (see ref 17).
Figure 20 shows three Raman wave functions for the two bound
potentials (section 11) at different frequencies. The lifetime for
the trajectory is about 60 vibrational periods. The same trajectory
was used to generate all three wave functions. Also shown are
the real and imaginary parts of IR(w,q)). The lower trace corresponds to an off-resonance excitation. The other two are on
resonance with u’= 0 and 1. We note that the phase of the Raman
wave function strongly depends on the laser frequency. For the
normal Raman case (off resonance), the Raman wave function
0
o.ms
0.09~
0.m
o m
FRWUENCY tau 1
Figure 17. (a) Correlation function (RIR(t))for off-resonant excitation
(Au = 4000 cm-I) and (b) total emission spectrum calculated by taking
the full Fourier transform of ( R J R ( t ) )Note
.
here that the first photon
creates IR)on the excited state which is then propagated on the ground
0.000
state.
N-
oL
-
1
O
I
2
3
r
-
displacement
1
O
I
2
3
4
-
1
displacement
0
1
2
3
4
diepLacement
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-
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i
a
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disDt.ocement
2
3
4
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-
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1
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2
3
4
disDLacement
N-
i
d
-1
dispLacement
dispiacement
0
1
2
3
4
dispLacement
Figure 18. Similar to Figure 16 except that the laser is now on resonance
(Au = 0).
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3373
Raman Spectroscopy: Time-Dependent Pictures
I
I
I
-1
i
1
0
i
1
1
I
I
a
3
i
4
D ISPLACEHEN?
Figure 21. Raman wave function for the repulsive excited state (from
section IIIb) for laser excitation (a) on resonance, Aw = 0; (b) off resonance, A o = 4000 cm-’; and (c) off resonance, Aw = 8500 cm-’. The
of each wave function are also
real (-) and imaginary parts
shown.
1I
(-e-)
c
a
FREDUENCY( ou 1
figure 19. Same as Figure 17 except that these results are for a resonant
excitation (Am = 0).
1 ps there is still an appreciable imaginary component in JR(w,q));
hence, the emission spectrum contains both Raman and
fluorescence lines.
Figure 21 shows a similar study for the repulsive potential. The
results show similar effects. As the laser is detuned from resonance, the Raman wave function becomes mostly imaginary. In
this case, however, when we are on resonance, the Raman wave
function is composed of approximately 50% real and 50% imaginary, with the two being out of phase such that IIR(w,q))J2
has
no nodes. It is not possible to create an eigenstate on a repulsive
potential in a photodissociation experiment. The eigenstate is
composed of two waves, one moving to the right and the other
moving to the left, to create an interference pattern that results
in the formation of nodes. In the photodissociation experiment
only one of the two waves is present. This argument becomes
apparent when we compare the expressions for the eigenstates to
those for the Raman wave functions. For the Raman wave
function
and for the eigenstate
The only difference is in the limit of integration. The expression
for i*(u,q)) contains in it the dynamics of I+(t)) for times -m
to m and thus both waves. The Raman wave function, on the other
hand, is obtained from the dynamics from t = 0 to m, which
includes only the wave moving to the right.
In the bound case when we are on resonance for long enough
times
I
I
-0.S
1
0
I
1
0.5
1
I
1.9
ai
OISPLACEHENT
Figure 20. Raman wave functions for the bound excited state (from
section IIIa) for laser excitation (a) resonant to u / = 1, (b) resonant to
v / = 0, and (c) lOho below the bottom of the well. Also shown are the
real (-) and imaginary parts
-) of each wave function.
(-e
is purely imaginary (true for I+(O)) purely real), while as we tune
onto resonance the spectra turn to fluorescence and the Raman
wave function after 60 vibrational periods becomes almost purely
real. At that point, they are almost indistinguishable from the
true eigenstates u’ = 0 and 1. The only small, but apparent
imaginary piece is in the FC region. That part of the Raman wave
function becomes negligible as the time gets longer. The application to O2 presented earlier is also consistent with these discussions. However, the FC factor for u’= 4 is so small that after
I*(w,q))
= IR(w,q))
since the dynamics of I+(t)) are such that it revisits the same
regions many times.
VI. Conclusion
We have examined the two-photon spectroscopy of a simple
two-state, one-dimensional system within the FC approximation.
We chose to cast the two-photon process in a picture that makes
an intuitive connection between the spectrum and the dynamics
that produces it. As experimentalists, we find it comforting to
know what we produce when we shine the laser onto a sample,
whether we tune the laser onto resonance or off-resonance. The
time-independent treatment of Raman spectroscopy described the
state that the first photon creates as a “virtual” state. The
time-dependent treatment removes the mystery associated with
3374
J . Phys. Chem. 1988, 92, 3374-3379
this state; the Raman wave function is the state prepared by the
laser. We have provided a pictorial overview of this state and we
are now in a position to explore its unique properties.
In our picture, a single trajectory provides all the necessary
information. The dynamics of I $ ( t ) ) contains the absorption
spectrum, the Raman wave function prepared by the laser at any
frequency, and even the eigenstates of the excited-state surface.
It also makes it clear that, by choosing the laser frequency, we
essentially have the ability to choose which part of the dynamics
we will study. The connection between time and detuning means
that by detuning the excitation source we limit the time over which
the dynamics contribute to the spectrum. The further off resonance we tune the laser, the more we limit the time.
Another way to put it is that the off-resonance spectra provide
information about the local FC region, whereas on-resonance
spectra provide information about the equilibrium position in the
case of a bound excited state or the large internuclear separations
for a repulsive potential.
We find that it is not necessary to make a distinction between
Raman spectroscopy and fluorescence. One formula, and
moreover the same trajectory, will produce both. We conclude
that these are two extremes of one physical process. As the
incident laser frequency is tuned from off to on resonance, the
spectrum continuously go from Raman to fluorescence (if the
excited-state lifetime is long with respect to a vibrational period).
The dissociative case presents an interesting halfway point where,
no matter how narrow the bandwidth of the laser, it is impossible
to create an eigenstate by simple photon absorption. We find it
interesting that in these two extremes the states that the laser
prepares exhibit very clear differences. For normal off-resonance
Raman, the Raman wave function is purely imaginary, whereas
for what would normally be called fluorescence, the Raman wave
function is purely real. That is why we chose to designate the
repulsive case as the halfway point since here the Raman wave
function is complex with equal amounts of real and imaginary
components.
In our treatment, we have excluded, for the sake of simplicity,
many important effects, such as the nonconstancy of the transition
moment with intermolecular separation and the influence of more
than one excited state on the process. These will be dealt with
in future work.
Acknowledgment. This work has been supported by NSF Grant
CHE-8507168 and by the donors of the Petroleum Research Fund,
administered by the American Chemical Society. We thank
Professor Judy Ozment and the Eric Heller group for helpful
suggestions and David Tannor for introducing us to the grid
method used in these calculations.
Time-Dependent Study of the Absorption and Emission Spectra of 0,
Stewart 0. Williams and Dan G. Imre*
Department of Chemistry, University of Washington, Seattle, Washington 981 95 (Received: October 19, 1987)
We use a time-dependent theory to investigate the absorption spectrum to the B3Z; state, as well as the emission spectrum
corresponding to laser excitation to u’= 4 of the B32; state of the O2molecule. We present detailed discussion of the relationship
between the dynamics and the resulting spectra and correlate our results with experimental data.
I. Introduction
In recent papers,’-2 we used a time-dependent theory to take
an in-depth look at the Raman process. In this paper, we will
use this the0ry~9~
to investigate various spectroscopic results for
the O2molecule. We will concentrate on the B3Z; excited state
of 02,which is the state responsible for the Schumann-Runge
(S-R) bands.
Because of the extensive work done on the S-R band^,^,^ this
band system and indeed the B state are well characterized. The
minimum in the B-state potential is at 1.6 A, which represents
from the ground state
a fairly large displacement of -0.4
of 02.Approximately 21 bound states have been identified on
the B excited
in fact, it has a shallow dissociation energy
of 8121 cm-’ (0.037 au). Because of the large displacement of
the minimum in the B-state potential from that of the ground-state
potential, and because of the shallowness of the B-state potential
(1) Williams, S. 0.;Imre, D. G. J . Phys. Chem., preceding paper in this
issue.
(2) Williams, S. 0.;
Imre, D. G., manuscript in preparation.
(3) Tannor, D. J.; Heller, E. J. J . Chem. Phys. 1982, 77, 202.
(4) Heller, E. J.; Sunberg, R. L.; Tannor, D. J. Chem. Phys. Lett. 1982,
93, 586.
(5) Bethke, G. W. J. Chem. Phys. 1959, 31, 669.
(6) Ackerman, M.; Biaume, F.J. Mol. Spectrosc. 1970, 35, 73.
(7) Goldstein, R.; Mastrup, F. N. J. Opt. SOC.Am. 1966, 56, 765.
(8) Bhartendu; Currie, B. W. Can. J. Phys. 1963, 41, 1929.
(9) Julienne, P. S.; Krauss, M. J . Mol. Spectrosc. 1975, 56, 270.
(10) Herzberg, G. Spectra ofDiatomic Molecules, 2nd ed.;Van Nostrand:
New York, 1950.
( 1 1 ) Krupenie, P. H. J. Phys. Chem. ReJ Data 1972, 1 , 423.
0022-3654/88/2092-3374$01.50/0
well, most of the Franck-Condon (FC) envelope is above the
dissociation limit for the B state as reflected by FC factors and
absorption s p e c t r ~ m . ~ ~ * * ’ ~
Both ab initio9 and detailed high-resolution experimental
studies6 have revealed that there are variations in the line widths
for different vibrational levels of the B state. This effect has been
shown to be due to predissociation. The predissociation is dominated by the
(at least for v’ = 4),9 which crosses the B3Z,state at 1.875 A, which is close to the turning point for u’ = 4.
As a result, the lifetime for v’ = 4 is 1 ps. Because of this, we
will ihtroduce a phenomenological lifetime r in our calculations.
In general, one would expect a fluorescence spectrum from a
resonant excitation to a discrete state. However, when O2is excited
by an excimer laser at 193 nm, which is resonant with a few
rotational levels of u’ = 4 of the B state, the dispersed emission
spectrum obtained by us and by Shibuya and Stuhl12 shows that
the spectrum cannot be unambiguously characterized as either
fluorescence or Raman and in fact appears to be a hybrid. It was
this result that first prompted us to undertake this study.
The results we presented in ref 1 and 2 showed that indeed
hybrid spectra could be expected in a variety of situations. The
O2 spectrum provided an example where the two “parts” of the
spectrum are of the same order of magnitude and neither can be
ignored. Thus this spectrum could serve as a test of our approach.
As we mentioned above, the absorption spectrum shows a few
discrete vibrational levels but most of the FC envelope is in the
continuum. The absorption spectrum in itself presents a challenge
-
(12) Shibuya, K.; Stuhl, F. J . Chem. Phys. 1982, 76, 1184.
Q 1988 American Chemical Society