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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-dependent theory 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 req~ency .~ 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 two- photon 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. 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) = p121xl ) , 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 wI is 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 time- dependent 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

( 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,

(4) Rousseau, D. L.; Williams, P. F. J . Chem. Phys. 1976, 64, 3519. 399.

0022-36S4/88/2092-3363$0~ .50/0

(5) (a) Tannor, D. J.; Heller, E. J. J. Chem. Phys. 1982, 77, 202. (b) Lee,

(6) Heller, E. J . Potential Energy Surfaces and Dynamics Calculations;

(7) Heller, E. J. Acc. Chem. Res. 1981, 14, 368.

S. Y.; Heller, E. J. J . Chem. Phys. 1979, 71, 4777.

Plenum: New York, 1981; p 103.

0 1988 American Chemical Society

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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 n ~ ~ ' ~ cast in a time-dependent for- mulation sheds some light on the connection between the experimental results and the dynamics. Here the Raman amplitude is

The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 Williams and Imre

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 off- resonance 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 con- stantly 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 (4j is time independent. Now define the Raman wave function )R,,,q,) 2 J"~e'"''-'lrld(t)) dt.lo This is the virtual state. 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

(9) Heller, E. J.; Sundberg, R. L.; Tannor, D. J. J . Phys. Chem. 1982,86,

(10) Sundberg, R. L.; Heller, E. J. Chem. Phys. Lett. 1982, 93, 586.

Spectroscopy of Molecule; Academic: New York, 1984.

1822.

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, + EJh, 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 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3365

0 0.0 1.0 2.0 3.0 4.0 5.0 DISPLACEMENT

Figure 1. Moving wave packet on the excited-state surface a t time r = 0 ( to ) and a t later times t l 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 Franck- Condon 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 final Raman vibration levels ( I (@i@( t ) ) I ) for u” = 0 (self-overlap) 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) Kru- penie, P. H. J . Phys. Chem. Ref: Dura 1972, 4, 423. (c) Julienne, P. S.; Krauss, M. J . Mol. Spectrosc. 1975, 56, 270.

(14) (a) Holm, 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 ’ l ’ 1 ’ l ’ l ~ l . i 2.5 5 7.5 IO 11.5 15 17.5 XI

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

l i I I I 1 I 1 I 1 I I I I

I I

I I 1 1

I I d.M 0.01 0.06 0.11

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

3366 The Journal of Physical Chemistry,

1

Vol. 92, No. 12, 1988 Williams and Imre

I ,

dispLocement

1

dispLacement

0 i a TIME(femto eecl

1 I I

displacement

1

dispLocement diapLocement

dispLocement dispLacement

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 eigen- function 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.

I - :

la1

[bl

(01

( d l

le1 I I

-0.04 0.01 0.m 0.11 PRWUENCY I au 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


la1

Ib)

I01

I I I I 0.00 0.01 0.01 0.1 0.04

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 The "virtual state" has been a common term which was in-

troduced 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 time- independent 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 a t every time step to account for the contribution of I+(t)) to the spectrum over all time. The term eiwf acts as a filter, and it projects out of Iq 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 the 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.

The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3367

I C I D

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))I2 with the intensity chosen such that each Raman wave function is normalized. For each excitation five Raman wave functions are shown. These correspond to 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)).

3368 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988

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. Fre- quencies used are resonant to u ’ = 1, 1 + 0.25wc, 1 + 0.5wc, 1 + 0.75w,, and u ’ = 2.

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 a t 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 con- tributions from numerous excited eigenstates. In fact, for r = 0.12 the absorption spectrum shows no band structure, while a t 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

Williams and Imre

(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).

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.

( 2 2 ) Sunberg, R. L.; Imre, D. G.; Hale, M . 0.; Kinsey, J. L.; Coalson, R. Chem. 1986, 90, 4997.

D. J . Phys. Chem. 1986, 90, 5001.


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 3251 of 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, a t 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 3251 Raman 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 1 l 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 1 1 b. 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,13c and second, the overlap of u’= 0 is rather ~ m a l 1 . l ~ ~ This spectrum demonstrates the effects described in Figure 8. The u’


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 1 la 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

1 ------*/ (a)

0.50 0.75 1.00 1.25 1.50 t.75 2.00 2.25 2.50 BOND LENGTH

tbl

o.m o h 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) Calcu- lated 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 a t 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-

Williams and Imre

(a)

tbl

Figure 13. Normalized (a) Raman wave function and (b) emission 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.

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 mono- tonically 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

Raman Spectroscopy: Time-Dependent Pictures The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3371

E in' h 3 h

(bl

(cl

(d)

(el

-10 5 0 5 10 15 a0 MOMENTUM I arb unite I

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) Aw = 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,

2 Re IR(w,q))I = ~ ~ e ' " I ' l + ( t ) ) dr

then for I4(0)) purely real as in our case

and

is the eigenstate a t 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 for Cl,,l4 Brz,15 CH,I,I6 and others, it is clear that what the laser prepares is the Raman wave function and not the scattering eigenstates.

Figure 15 shows the Raman wave functions from Figure 13 in momentum space. We have plotted IR(w,p))J2 vs p for different 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, a t 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 Williams and Imre

N - N-3 N- -1 4 h

-1 0 1 2 3 4 disphcement

-1 0 1 2 3 I displacement disolacement tal

N- N-

,J

O - , , . I . I .

0 25 50 75 lb lis 140 lk TIMEtfea eeol

- 1 0 I 2 3 4 dispLacement displacement displacement

N - N- m-

JL - 1 0 1 2 3 4

dispLacement displacement displacement

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

tbl

0

0.000 o.ms 0 . 0 9 ~ 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 (RJR( t ) ) . Note here that the first photon creates IR) on the excited state which is then propagated on the ground state.

N-

oL - 1 O I 2 3 r - 1 O I 2 3 4 - 1 0 1 2 3 4

diepLacement displacement displacement

N-

I,,,, - 1 O I 2 3 4

disDLacement - 1 o i a 3 r

disDt.ocement - I O 1 2 3 4

disdacement

N-

i d

-1 0 1 2 3 4 dispLacement dispiacement dispLacement

Figure 18. Similar to Figure 16 except that the laser is now on resonance (Au = 0).


c 1 I a


I i i

FREDUENCY ( ou 1 figure 19. Same as Figure 17 except that these results are for a resonant excitation (Am = 0).

I I 1 I 1 I i -0.S 0 0.5 1 1.9 a

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 ( -e -) of each wave function.

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 dis- cussions. However, the FC factor for u’= 4 is so small that after

I I 1 1 I I i

-1 0 1 a 3 4

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 real (-) and imaginary parts ( - e - ) of each wave function are also shown.

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

D ISPLACEHEN?

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*(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 O2 molecule. 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 O2 molecule. 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 a t 1.6 A, which represents a fairly large displacement of -0.4 from the ground state 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

(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,

(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:

issue.

93, 586.

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 dom- inated 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 O2 is excited by an excimer laser a t 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

New York, 1950. (11 ) Krupenie, P. H. J. Phys. Chem. ReJ Data 1972, 1, 423. (12) Shibuya, K.; Stuhl, F. J . Chem. Phys. 1982, 76, 1184.

0022-3654/88/2092-3374$01.50/0 Q 1988 American Chemical Society

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