7396
J. Phys. Chem. 1993,97, 7396-7411
FEATURE ARTICLE Real-Time Dependence of Photodissociation and Continuum Raman Experiments Moshe Shapiro Department of Chemical Physics, The Weizmann Institute of Science, Rehouot, 76100 Israel Received: January 21, 1993; In Final Form: April 14, 1993
A theory of photodissociation and continuum Raman scattering with coherent laser pulses is developed wih special emphasis on the nature of the prepared state under various modes of excitation. Time-dependent phenomena are treated using stationary quantum mechanics, and connections to wavepacket methodologies are made. Some universal features of the real-time dynamics of pulsed excitations are discussed. Detuning of the pulse center with respect to a transition frequency is shown to give rise to a nonmonotonic transient whose shape is only a function of the detuning and the pulse characteristics. “Time” in the usual way of propagating photodissociating wavepackets (under G(t)-pulse excitation conditions) is shown to be but a convenient way of obtaining the frequency-resolved photofragmentation cross sections. Observations in “real time” are shown to be intimately linked to the pulse characteristics, hence the nonexistence of a unique decay rate. This point is illustrated in the case of the femtosecond transition state spectra of NaI. The formulation is extended to the treatment of continuum Raman experiments. A uniform theory of Raman scattering and resonance-fluorescence from an intermediate dissociative manifold for excitation with pulses is presented. It is shown that transient effects lead to the appearance of additional terms, not included in the Kramers-Heisenberg formula. The relative importance of “true” Raman scattering vs “resonance fluorescence” is shown to be strongly dependent on the pulse parameters and the spontaneous emission lifetimes. “True” Raman dominates during the rise of the pulse, whereas resonance-fluorescencesets in and dominates theobserved signal at longer times. Computations of the continuum excitation-emission spectra of dissociating CH31 and IBr are presented. The computations explain the observation of a “reflection-like” structure in the spectrum. This structure, which is not explainable by the usual C W Kramers Heisenberg expression, is due to the dominance of resonance fluorescence in these nanosecond pulsed experiments. Introduction With the advent of chemistry with short laser pulses,14 it is often stated that “it is now possible to probe photodissociation processes while the products fall apart”. The feeling is that pulses of 5C100-f~duration allow a direct dissociation process to be followed in “real time” and that “real time” experiments are superior to frequency-resolved experiments.5 However, it is not always clear what molecular quantity is being measured in “real-time” experiments. This is because in quantum mechanics a unique decay rate is difficult to define: It is not possible to disentanglethe preparation from the dissociation step.6 That this is the case when the bandwidth of the laser pulse is smaller than the photoabsorption spectrum has been well appreciated by most researcher^.^^^ What is perhaps less wellknown is the point, demonstrated below for the case of NaI, that the concept of a unique decay rate runs into difficulties even if the bandwidth of the laser is larger than the individual spectral features. A different question,which is partly computationally motivated, is the extent to which Stationary quantum methods are equivalent to wavepacket propagation techniques. There seems to be a consensus that both methods are equivalent as far as obtaining energy-resoluedquantities.Are the two methodologies equivalent as far as exploring time dependent phenomena? Clearly, if one solves the exact SchrMingerequation, both methods agree exactly. However sometimes the exact solutionof the SchrMingerequation is difficult, such as when spontaneous emission is involved. It is of interest to compare the two methodologies for this case, encountered for examplein the “continuum Raman” experiments. To answer some of these questions, we develop in this paper the time-dependent theory of photodissociation and continuum 0022-3654f 93/2091-1396%O4.O0f 0
Raman scattering using energy eigenstates. We start the discussion by giving an explicit expansion of the wavepacket created by a laser pulse during and after the excitation process. We study in detail the effect of the detuning of the pulse center with respect to the energy-levels positions and show that while the pulse is on, and depending on the degree of detuning, the buildup of the coefficients multiplying each energy eigenstates is a nonmonotonic function of the time. For weak fields, the temporal buildup of each continuum level is a universal phenomenon which has nothing to do with intramolecular dynamics. The buildup of the preparation coefficients, which occurs simultaneously with the evolution of the wavepacket as it is being formed, and the postpulse values of these coefficients are shown below to affect the transients observed when a secondary (probe) photon, such as in the pump-dump or spontaneous continuum Raman experiments, is detected. This is illustrated for the case of the predissociation of NaI with short laser pulses. We show that in principle for this case the CW measurementsgivesuperior information to the “femtosecond transition state” measurements, for which the disentanglement of the laser characteristics from the decay dynamics is virtually impossible. We then use the stationary eigenstates approach to develop a time dependent theory of spontaneous continuum Raman scattering of a pulse of light. We study both resonance and offresonance effects and show that the photoemission is composed of two transients, one following the pulse, which may be termed a “true” Raman process, and the other, decaying via the combination of the natural radiative decay and wavepacket dynamics, representing resonance fluorescence. We show that for a pulsed excitation source one needs to substantially modify the continuous-wave (CW) Kramers Heisen0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7397
Feature Article berg (KH) expression,**g(beyond just introducing an integral over the pulse frequencies). Our modified expression is capable of explaining the observationIOJ1 of structures in the excitationemission spectrum (the emission signal to a given final state as a function of the excitation frequency), of directly dissociating molecules. These structures are not obtainable by the usual CW KH expression. The paper is organized as follows: In section 1 we briefly outline some of the basic time-dependent perturbation theory used to describeexcitation processes with pulses. In section 2 we derive an analytic expression in first-order perturbation theory for the coherent excitation of a set of levels by a pulse. In section 3 weuse this expression to examine the photodissociationdynamics during and after the excitation pulse. In section 4 we present numerical studies of the photopredissociation of NaI with ultrashort pulses. In section 5 we show how the present approach can be applied to continuum Raman spectroscopy. In section 6 we present numerical results for continuum excitation-emission spectra of CH3I and IBr and compare them to experiments.
where wm,n I
b,(t
=-a) =
t+(t)
(1.1)
where z is the direction of propagation of the light, w is the mode frequency, k is the mode’s wave-vector (k = w / c ) , and t is the retarded time: t = t - z/c (1.2) The above formula allows each mode-amplitude e(w) to be a complex number, e(w) = Ie(w)lexp[i4(w)], where 4(w) are frequency-dependentphase factors. Obviously, only the relative magnitudes of these phases are of importance. It follows from the reality of e(t) that e(+) = e*(w), Le., $(-w) = +(a) and I4-w)l = l4w)l. We denote the radiation-free molecular Hamiltonian as HO and its (discrete or continuous) set of energy eigenvalues and eigenfunctions as and $:
e
In the presence of the field, the temporal evolution is governed by the full Hamiltonian H , defined as
H ( t ) = H,
+ V(t)
(1.4) where V(t)-the light-matter interaction-is given, in the dipole approximation, as V(r,t) = -p(r)*e(t) (1.5) where ~1 is the electric dipole operator. We can solve the time-dependent SchriMinger equation i h d 9 / d t = H ( t ) 9 = [Ifo+ V(f)l9
(1.6) by expanding the full time-dependent wave function9 ( t ) in terms of $: 9(t)=
zb,(t)
+exp(-iet/h I
-y,t/2)
(1.7)
n
where in the present treatment yn are a set of phenomenological spontaneous emission rates. Using the orthonormalityof the basis functions and substituting eq 1.7 into eq (1.6), we obtain a set of O.D.E.for b,(t): db,,,/dt = ( l / i h ) x b n ( t ) exp(iw,,,t)(+O,IV(t)l$:) n
(1.8)
fork # s
(1.10)
and that
c,(t)
p
(l/ih)(+:lV(t)l+j)
exp(iwijt)
(1.11)
is a weak perturbation, i.e., the set of Rabi frequencies is small such that
> l/r
(2.15)
This gives us the desired criterion as to how long it takes the resonance condition (eq 1.17) to be established. Contrary to the CW domain, in which the goodness of eq 1.17 depends only on the passage of a large number of optical cycles, it follows from eq 2.15 that in the pulsed case the relevant parameter is the pulse duration l/r. This quantity can, in principle, be shorter than a single optical cycle. As expected, the CRW coefficients do not survive the pulse and therefore form pure transients. In contrast, the RW coefficients are not pure transients as some portion of ci(t) survives the pulse, depending as shown below, on the detuning. To see this explicitly, we have computed
(2.5)
c:(t)
c;(t) = -(2r/r)[i(wmJ
ci(t)
Shapiro
(2.14)
and Y = t - a2ym,. w[z] is the complex error function (see ref 13, eqs 7.1.3 and 7.1.8). Similar formulas for real frequencies were derived for the Gaussian case by Rhodes,14 and by Taylor and Brumer.15 It follows from eq 2.12 that
foryms = O,usingeqs2.12-2.14. Theresultsfor(c’,,,(t)l,Rp’,,,(t), and I,,,c’,,,(t) for a Gaussian pulse whose intensity bandwidth (4 = 2(ln 2)1/*/a) is 120cm-1, for different A,,=w,,- wo detunings, are presented in Figure 1. It is immediately evident from Figure l a that while at the end of the pulse the amplitude for populating a state with a given wmJ transition frequency is proportional to Ic(w,)l, the path leading to this value is remarkably different for different values of A,,. For wmjnear thelinecenter, c’,,,(t) rises smoothly to its asymptotic value. However at off-center energies c’,,,(t) does not rise so monotonically: At early times all the em's respond to the field in almost the same manner since the system has insufficient “information”to determine the true spectral composition of the pulse. It therefore ‘thinks” that it is exposed to a much broader band of frequencies, hence a more slowly varying e(A). Only at later times does the system “realize” its “mistakenand corrects for it by depleting the off-center c’,,,(t)’s. In the extreme case of detuning, i.e., when e(w,,) = 0, although the probability of observing the level after the pulse is essentially zero, it can be easily verified from eq 2.12 and from Figure 1 that c’,,,(t) during the pulse is not necessarily zero. This means that the level in question gets populated and depopulated during the pulse. In the usual jargon what we have described here is a virtual state. The above description gives a physically meaningful content to the concept of a virtual state, which is often treated as a pure mathematical construct. The role of the phase is also interesting: The phase of b,(t) at the end of the pulse is guaranteed by eq 1.17 to be that of ;(w,,,,,), which means, using eqs 2.4 and 2.15, that c’,,,(t) is real at the end of the pulse. As Figure lb,c shows c’,,,(t) is real at all times for zero detuning (A,, = 0). At finite detunings, c’,,,(t) is complex during the pulse. In the extreme wings of the pulse, the imaginary part of c’,,,(t) essentially follows the pulse shape, reaching a maximum at t = 0, while the real part exhibits a (temporal) “dispersion-like” curve. At intermediate detuning from the pulse center the behavior is more complicated: The imaginary part changes sign several times whereas the real part starts varying more monotonically. It is also of interest to look at dlc’,,,(t)lZ/dt, which is proportional to the rate of populating the mth level. If the pulse is incoherent, most textbooks16 derive the existence of a constant rate at long times by first calculating the rate for a single mode and then averaging over some frequency spread. Since this procedure presupposes that there are no multimode coherent effects, it obviously cannot be adopted in the present case. With coherent pulses we must first calculate the ‘action of the whole pulse and then compute the rate. The results of the rates derived in this way are given in Figure 2. As expected for a Gaussian pulse, the rate reaches a steady state at an isolated point only. This steady-state point, as evident in Figure 2, depends on the detuning: at near-center frequencies
Feature Article
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7399
Pulse p r e p a r a t i o n coefficients
1
-1
I
I
I
- 0 . 3 - 0 . 2 - 0 . 1
I
0 . 0
I
0.1
9.2
3.3
I 3.4
tbsec)
-0.4
-0.2
0.2
0.0
0. 6
3 . 4
t(psec)
8 1
Figure 2. Excitation rata (dlc’,,,(f)lz/dt)for a 120-cm-’-wide pulse at different detunings from the pulse center: (-) & = 0; (0)A, = 24 cm-1; (X) A, = 48 cm-I; (H) A, = 72 cm-1; (+) 4= 96 cm-I; ( 0 )A, = 120
cm-1. E=O. E=24.
eq 1.17 into eq 1.7, is given for an absorption process (E, > E8) by
E=48.
E=72. E=O8.
E=lZO.
-2
I
~
-0.4
I
-0.2
I
0.0
0.2
I
I
0.4
0. 6
tbsec)
Thus, after the pulse, the absorption of a photon has created a wavepacket in which the coefficients of preparation are proportional to the field amplitude at the omsfrequency. During the excitation pulse the above picture must be corrected via the use of eqs 2.3-2.4. We obtain that
In the next section we extend these results to the case of excitation in the continuous spectrum. -2
1
I
I
- 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1
I
I
0.0
’
I
I
I
0.1
0.2
0.3
3. Photodissociation Dynamics during and after the Pulse
t(psec)
Figure 1. Time evolution of the c’,,,(t) coefficients at different detunings from the center of the pulse for a Gaussian pulse with fwhm of 120 cm-l: (a) Ic’m(t)l; (b) Rlc’m(t);(c) Lc’dt).
the rate is always positive, at off-center frequencies the rate actually becomes negative towards the later part of the pulse. This corresponds to an actual depletion of the excited levels, which, for extreme detunings approach the status of virtual state. In Figures 1 and 2 we only plotted the (RW) cG(t) coefficients. It can be shown on the basis of eq 2.13 that the behavior of the (CRW) c;(t) coefficients resembles that of the highly detuned cG(t) coefficients, save for the fact that the CRW coefficients rigorously vanish in the long time limit. Thus the CRW coefficients make a noticeable contribution only at short times. This justifies the usual practice of neglecting the counter rotating contribution whenever detuning with respect to some molecular levels is small. The behavior noted above for c’,,,(t) and d(c,12/dt is essentially independent of any particular molecular system save for the location of the molecular energy levels. The attributes of any particular system enter mainly through the the pms matrix elements. At the end of the pulse, * ( t ) , obtained by substituting
Photodissociation arises when the states accessed after absorption of a photon are in the continuous spectrum. Equation 1.7 can be easily extended to that regime by using scattering states as our basis functions. Allowing for possible degeneracy of the scattering states, we denote these states by their energy E and a set of extra quantum numbers n. Degeneracy alwaysoccurs in a (polyatomic) molecule because it can breakapart to fragments with different internal states. Using the energetically accessible internal states of the fragments (also called open channels) to distinguish between the degenerate continuum states, we denote the continuum eigenstates of HOas I&&??) : lim,+,(E - iq - Ho)l$;(E)) = 0 (3.1) Each I+;@)) state correlates in the long-time limit to a welldefined internal state In) and translational state Ik.) (where k. is the asymptotic wave vector of the freely moving fragments): lim?-&;(E)
) = lk.)In )
(3.2)
We can now modify eq 2.18 to include continuum states by replacing the m summation by integration over the energy plus
Shapiro
7400 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
pulse as
summation over the open channels. We obtain
s,(t)= $: exp(-i@/h)
+ (i/h) XJ d E [c:(t)
Z(wE,,,)+
n
where WE,,, = (E - e ) / h - i y ~ / 2and we have dropped, for simplicity, the n subscript in Y E and assumed that ys = 0, which is rigorously true if $: is the ground state. In analogy to eq 2.17 we can write \E,(t) after the pulse as
*,(t) = $: exp(-iet/h) (Z*i/h)CJdE n
(3.11)
It is now possible to use the spectral resolution of the evolution operator in thesubspacespannedby thecontinuumwave functions:
($;(a4$:)$;(a x
C J d E exP(-iw4l$;(E))
i(oE,,,)
The boundary conditions implied by +;(E) allow for an immediate evaluation of P,(E)-the photodissociation probability. Defining the photodissociation probability-amplitude, A,@), as the amplitude of observing a Ik’,)lm) free state in the long-time limit Am@) = lim,-- exp(iEt/h)(k,,,mJq,(t))
($;(E)( (3.12)
n
to obtain that
\k’$(t)= ( i / h ) J ’ dt’e(t‘) X -0
exp[-(iHi,/h
+y / W -
t?lPl$:(t?)
(3.13)
where we have used the identity, &t’) = exp(- i,?$’/h)&O). It is instructive to interpret this result by looking at the ultrashort laser pulse limit:
(3.5)
we have that
c(t)
eoS(t)
(3.14)
which is equivalent to choosing a completely white pulse:
P ~ ( E=) Pm(E)12
exp(-iEt/h - yEt/2) (3.7) Hence from eq 3.5
Am(E) = (2*i/ h)~(uE+) ($~(E)IPI$:)
ex~(qEt/2)
(3.8)
where we have used the orthonormality of the (k,)ln) functions. a,(E), the photodissociation cross section, is defined as the photon energy absorbed due to a transition to afinal fragment state, divided by the incident intensity of light per unit energy. The energy absorbed is Pn(E)hvE, (v 27ro), and the incident intensity per unit energy, Z(E), is Ie(o~,,,)]~c/h. We thus have from eqs 3.5 and 3.6, for negligible Y E , that ~ n ( ~ ) h v E s / l ( ~()8 * 3 ~ E s / ~ ) I ( $ ~ ( ~ ) I ~ I + ~ )(3.9) 12
This formula forms the basis for many of the computations of detailed photodissociation cross sections and the angular distribution of photofragments reported in the literature.17-26 We next turn our attention to investigating the real-time dependence of the photodissociation process under various excitation conditions. We first connect with wavepacket methodologies by writing the preparation coefficients of eq 2.4 as the finite time Fourier transform of the pulse
J-Ldt’e(t’)
e(o)
(3.6)
Letting t m, we obtain from eqs 3.2 and 3.4 that the excited part of the wavepacket is
an(E)
t
exp(-iHct/ h) =
+ exp(-iEt/h - yEt/2) (3.4)
-
[ e 91
exp - i-+-
exp(io,,t?
(3.10)
Using eq 3.10 in eq 3.3 and ignoring the energy dependence of Y E , we can write the excited portion of the wavepacket during the
= eo/2*
(3.15)
Substitutingeq3.14intoeq 3.13 (oreq 3.15ineq3.4) weobtain that
Equation 3.16 has been often used to interpret photodissociation*’ in the following way: At t = 0 the light pulse creates a wavepacket given by which subsequently evolves under the action of exp(-iHcf/h) while decaying due to spontaneous emission (which is usually neglected). Strictly speaking, as noted several times in the past,22,2*in order for eq 3.16 to hold, WE,,,) must vary more slowly with energy than any other variable in eq 3.4. In particular, it should vary more slowly than the energy (or frequency) dependence of the photodissociation amplitudes, (&(E)I&:). An easy way of deciding whether this is the case is tocompare the bandwidthof the laser with the frequency width of the absorption spectrum of the molecule. Typically for direct dissociations, the absorption spectrum extends over 4000 cm-’. To have a broader bandwidth than this, e ( t ) must, by eq 1.1, be as short as =2 fs. Since most pulses used in real photodissociation experiments are much longer than that, eq 3.16 is not a true description of present day experimentsof direct photodissociation processes. The form of C(W) is however, as shown by Heller,27 immaterial for the computations of the cross section. As can be seen by eq 3.9, E(W) cancels out in the cross-section expression, hence the t limit can be carried out with the particular choice of C ( W ) of eq 3.15. After propagating *’At) of eq 3.16 to t = m, the photofragmentation amplitude {+&Z)l&:) is obtained by equating eqs 3.5 and 3.8, and the cross section obtained from eq 3.9. Thus, wave-packet propagation with ultrashort pulses of the type giving rise to eq 3.16 find their greatest utility in calculating frequency-resolved quantities. Coming back to eq 3.13, we see that the correct real-time dynamics can be interpretedZG3O as a superposition of wavelets created by a sequence of 6 pulses. Equation 3.13 is a result of integrating all the wavelets resulting from a e(t) = Jdt’ e(t’). 6(t - t 9 pulse, each evolving according to eq 3.16 with a different starting time t’,
):$IP
-
0)
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The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7401
An alternative way of propagating the real time wavepackets is to explicitly include the external field in the Hamiltonian.31-35 Such computations are however much more time consuming, because they involve both the ground and excited electronic states and must follow faithfully the oscillation of the field (at optical-I0l5 s-’-frequencies). This can be amended in the RW approximation by subtracting a “center frequency” from the oscillating field by dressing the ground potential surface with the center-frequency photon. The method has an advantage in that so far it is the only effective way of dealing with strong pulses.33334 If the field is not too strong (Le., perturbation theory can be used) the route described by eqs 3.3, 3.4, in which one first calculates the frequency-resolved amplitudes by some timeindependent method17-26 and then folds them with the field profile, is much faster. In most direct photodissociationsthe opposite limit to eq 3.16 is realized: The laser’s bandwidth is usually much narrower than that of the absorption spectrum. Under these circumstances we can replace in eq 3.4 the narrow range of energies accessed by the laser by a single continuum energy level Eo, and write W,(t) as
0.20
0.15
-0.05
-0.10 -0.15
7
1
I 5
10
15
20
25
30
R ( i n a.u.) Figure 3. NaI diabatic potential curves.
transform of the pulse-modulated absorption spectrum: (3.22) We show below that the autocorrelation function “remembers” the shape of the pulse that created it at all times. This point is illustrated below for the NaI case.
We recognize the term in the curly bracket as E+(t), the positively rotating-wave part of the laser pulse of eq 1.1. Hence we see that under these circumstances the wave packet created by the laser simply follows the time dependence of that quadrature of the pulse. Classically, this corresponds to a situation in which the dissociation is “faster” than the laser: Every photon absorbed leads to instantaneous dissociation, and the only time evolution that can be observed is due to the laser itself. The case of NaI, studied in the next section, is intermediate between the two limits discussed above. It is of interest to look at the wave packet created by the laser while the pulse is on. The complete answer is given in eq 3.3, but it contains the spatial dependence of WS(r,t),which is not always desirable. We can integrate out the spatial dependence by considering the autocorrelation function, defined as
It follows from eq 3.3 that F,(t,to) can be calculated as
where we assumed for simplicity that E(W) is real which allows lumping together the RW and CRW terms in one preparation coefficient CE(t) = ci(t)
+ c&)
If both t and to occur after the pulse (Le., t, obtain, using eq 3.4, that
(3.20) to
>> l/r) we
F,(t, to) = - ( 2 r / W 2 C J d E le(wE,)(+~(E)lP111~)12 x n
We see that after the pulse the autocorrelation function, which no longer depends on z, is essentially the (complex) Fourier
4. Application: Photopredissociation of NaI with Ultrashort Pulses
To illustrate the nature of the photodissociationautocorrelation functions we present here a study of the photopredissociation of NaI.l2.42,43 This system has been extensively studied experimentally, using a “pulseprobe” technique (called FTS, femtosecond transition-state spectroscopy)with short (-100 fs) pulses, by Zewail et a1.36~3~In addition, a number of theoretical studies3228-78 aimed at explainingthe NaI FTS experiments, have been presented. We shall not attempt here to exactly reproduce Zewail’s experiments, which involve an additional “probe” laser pulse used to promote the system to a higher state correlating with the fluorescent Na(2P) atomic state. Computations pertaining to the entire pump + probe experiment are reported in a forthcoming publication.45 Our purpose here is to use the NaI system as an example of how attributes of a short pulse affect the long-time dynamics. We shall therefore concentrate on the study of Fs(t, r01. The potential surfaces we use for the NaI system, given in the diabatic representation, are illustrated in Figure 3. In the FTS experimentNaI wasexcited at -32 000-33 300 cm-1. Therefore, initially,the system is deposited almost exclusively on the repulsive branch of both the excited 0 = O+ and fl = 1 statesM Following Engel and Meti@ we neglect the rapidly dissociating 0 = 1state and concentrate on the O+ and the ground states. In the Franck-Condon region, the O+ state is dominantly covalent and repulsive. At larger internuclear separations its potential is affected by the attractive branch of the ionic structure. As a result, the O+ Born-Oppenheimer state is binding, with a short-range covalent repulsion and a long-range ionic attraction. Due to nonadiabatic coupling, and the proximity of the ground state, a slow leakage (resulting in dissociation) of molecules from the O+ state to the ground state occurs. In Figure 3 we show the original ionic and covalent (diabatic) potentials, as well as the next excited state, correlating to zP Na atoms. Using available spectroscopicinformation and following Berry47 and Grice and Hershbach,48 we parametrize the NaI ionic, covalent and ionic-covalent matrix elements (in au) as follows:
7402 The Journal of Physical Chemistry, Vol. 97, No. 29, I993
V.JR) = 0.02572514 exp[-2.51(R - 5.124)]
V.,,,(R) = 0.00077458/(1
+ exp[S(R - 24.5)])
Shapiro
(4.1)
(4.3) x
59
where
600
-c
4
Vfitber(R) = 58.63 1 exp(-l,3867R) - 1/ R 24.771/R4 0.07629 (4.4)
400
+
V,,
20 0
= 0.1931556(exp[-0.3562(R- 5.124)] - 1j2-
0 32600 3 2 7 0 0 32800 32900 33000 3 3 1 0 3 3 3 2 0 0 3330C
0.1931556 (4.5)
&,(R) = 2/(1 + exp[0.25621R- 5.12411)
(4.6)
The ionic state is parametrized as a linear combination of a Morse potential (at near-equilibrium separations) and a Rittner potential49 (at other separations). The linear combination is controlled by&&?) which is used to smoothly switch between the two forms. In this way it is possible to fit well the nearequilibriumseparations,which affect the vibrational frequencies at low energies, while at the same time to maintain the correct long-range Coulombic attraction (as well as the short-range repulsion) of the Rittner potential. In Figure 4 we present exact quantum computations,using the artificialchannel meth0d,~7Pof the photodissociation cross section (a.(E) of eq 3.9) for the above set of potentials. The range of frequencies chosen is that spanned by some of the pulses used in the FTS experiment^.^^ The displayed high-resolution spectrum reveals a very interesting sequence of narrow Fano-type lines, slowly changing to a series of broader features. The complexity of the spectrum is due to interferences induced by the curve crossing processes. To obtain the actual wave packet created by the laser (eq 3.3) or its autocorrelation function (eqs 3.18-3.21), weneed to "fold" the laser amplitude c ( w ~ with ~ ) the above spectrum. We first look at the autocorrelation function for an 80-fs pulse at short times, while the pulse is on. In Figure 5 we present the correlation function between \k',(t) during the pulse and @',(to) after the pulse. We choose to to be a "post-pulse" reference time of 0.5 ps. Because we have generated the wavepacket as a sum of energy eigenstates, changing t in F,(t,to) is done analytically in a trivial manner, (see eqs 3.19 and 3.21). Figure 5 should be compared with the c,'s of Figure 1, which lack the ($;(E)lpI$:) molecular attributes. In contrast to Figure 16, c, both real and imaginary parts of F,(t,to) are of comparable quantity. However IF,(t,to)l is very small due to the poor overlap of W,(t) at short times with itself at to = 0.5 ps. This is due to the fact that the wave packet moves very quickly away from the Franck-Condon region during this time span. To negate the effect of this fast motion, we can propagate the post-pulse wave packet back to to = 0 under the action of the radiation-free Hamiltonian. When this is done, the autocorrelation function, shown in Figure 6,displays a more meaningful overlap. Contrary to Figure 5 , we see a full fast rise of F,(t,to) during the pulse. This is followed by a somewhat slower fall off, as the wavepacket moves away from the Franck-Condon region. In contrast to Figure 5 , in this case F,(t,to) is mostly real. The initial falloff of the autocorrelation function, although slower than its buildup during the pulse, is still characterized by the same time scales of the laser pulse. It is of interest to see the extent to which the post-pulse wavepacket carries with it the "memory" of the pulse that created it. In Figure 7 we study the behavior of F,(t,0.05 ps) with a 120-cm-'-wide excitation pulse, centered at 32 700 cm-l, for t up to 6 ps. The initial rise and fall of F,(t,0.05 ps), is followed by a sequence of structuredrecurrences, spaced -1.3 ps apart. The train of recurrences decays, due to
frequency (cm-')
Figure4. PhotdiationcrosssectionofNaIinthe32 6 W 3 3 200-cm-1
range.
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
time (psec)
Figure 5. F,(t,to = 0.5 ps) NaI autocorrelation function for a 120-cm-l-wide Gaussian pulse centered at 32 900 cm-I.
-0.02
~
-0.4
I
-0.2
I
0.0
I
I
0.2
0.4
I
0.6
I 0.8
t i m e (psec)
Figure6. Same as in Figure 5 with the referencestate, *',(to), propagated back to to = 0.
the dissociation, with half-life (2112) of =8 ps. The spacings and decay rate are in reasonable agreement with the FTS experiments.36 Thewidth and structureof each recurrencepeakaredetermined by a combination of the laser pulse shape and the shapes of the individual resonances lines (shown in Figure 4) accessed by the laser pulse. In contrast, the temporal spacings between successive recurrences are thought to be a measure of "pure" molecular attributes: Zewail et interpreted them to bedue to reflections from the potential walls and their spacings to be given by the classical travel timebetweenthe two turning points of the adiabatic O+ potential. In Figure 7b we show the real and imaginary parts of F,(t,O.OOS ps). Inspectionreveals that the phase jumps by =+r/2 from one recurrence to the next. This is in quite good agreement with the
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 1403
Feature Article
6=120cm-' center=33000cm-'
1
012
-
,025
1
-
1
n
n
,008
Y 4
3.
h
in
x
v
3. -
. 004
. 000 -
I
-1
1
0
1
2
3
4
5
6
7
I
I
I
I
I
I
I
1
0
1
2
3
4
5
6
7
Figure 7. (a) lF8(t,to = 0.05 p)(NaI autocorrelation function for a 120-cm-1-wide Gaussian pulse centered at 32 700 cm-l. (b) The same as in (a) showing the real and imaginary parts of F,(t,to). 6 =240cm-' c e n t e r =32700cm-' .030
7
-1
0
1
2
0
1
2
3
4
5
6
7
t(psec)
t(p==c)
3
4
5
6
7
t(psec)
Figure 8. The same as in Figure 7a for a 240-cm-'-wide pulse.
WKB connection formulas, which spell out the phase changes suffered by a wave packet reflected from a turning point. Wenow wish toseewhether thedecay ofthe trainofrecurrences is determined solely by the molecular attributes. To this end, we compute F,(t,O.OS ps) for a shorter pulse, whose width is 240 cm-1, centered at the same (32 700 cm-1) frequency. The results are shown in Figure 8. While the fact that the shape of each recurrence has changed and its duration has shrunk is in accordance with the above expectations, the shortening of the decay time of the whole recurrencepattern to =6 ps is surprising. How is this possible? A glance at Figure 4 reveals that a 240-cm-'-wide pulse centered at 32 700 cm-1 encompasses more of the broad resonancesthan the 120-cm-'-wide pulse. This results in a faster decay by some of the resonances.
Figure 9. (a) Same as in Figure 7a for a pulse centered at 33 OOO cm-l. (b) Same as in Figure 7b for a pulse centered at 33 000 cm-l.
These findingsare in fact in agreement with more recent results of the Zewail group3' in which shorter decay times were observed when shorter (e.g., 40 fs) pulses were used. Thus, for NaI, subpicosecond laser pulses give us information only about an average decay time of many individual resonances. The shorter the pulse, the more averaged is this information. We therefore turn back to Figure 4, as the one closer to the detailed dynamics, and make the immediate prediction that a shift of the pulse to higher frequencies, while maintaining its width at 120 cm-l, would further decrease the recurrence-train decay time. In Figure 9a we display IF,(t,0.05ps)]obtained with a 120-cm-'-wide pulse centered at 33 000 cm-1. This is thecenter of the wide resonances of Figure 4 and indeed, the recurrences decay time has shrunk to -1.5 ps. In this range, as revealed by Figure9b, thesimple+u/2phasejumpsdonot occur. Thesecond recurrence has roughly the same phase as the first recurrence. Only the third and fourth recurrences are seen to suffer the +u/2 phase jumps. Some of the frequencies studied by Zewail et al. (e.g., 32 900 cm-1) are in an in-between region. In this region we expect to see an effect due to two essentially different lifetimes, those of the narrow 32700-cm-l resonances and those of the broad 33 000-cm-1 resonances. This is in fact the case, as can be seen from Figure loa, where we plot IF,(t,0.05 ps)J at 32 900 cm-l. The decay time is intermediate (tip 2.5 ps). The FTS spectra of Figure 10a does not reveal however that the decay pattern is influenced by two types of resonances. The general shape of its envelope looks very similar to the 32 700- or 33 000-cm-1 cases. Only the phase plot of Figure lob, where the real and imaginary parts of F,(t,0.05 ps) are displayed, hints at the different nature of this wave packet: Each successive recurrence is seen to suffer a larger than usual phase jump (approaching =3u/4). The existence of more than one time scale was in fact measured experimentally by Cong et al.3' There they found that due to the
Shapiro
1404 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
because it ignores the fact that the excitation is usually done with pulses, and that allowance must be made to the existence of resonancefluorescence which, contrary to the Raman process, can take place even in the absence of the pulse.
6 = 1 ~ 0 c m - 'c e n t e r = 32900cm-'
h
,025
In what followswe use the approach developed in the previous Sections, based on stationary quantum mechanics, to investigate the transient emissions. The analysis presented below reveals what percentage of the observed signal is comprised of a "true" Raman process, arising from "virtual" states, and what percentage is made up of "resonance fluorescence" process, which is usually ascribed to emission from "real" (continuum) levels. Contrary to "ordinary" (bound state) Raman scattering, in which this issue was settled some time ago?8 in continuum Raman scattering this question is a source of some confusion, as evidenced by the titles of the experimental papers dealing with the subject.
,020
. 01 5 ,010
,005
,000
I
-
l
1
l
0
I
1
I
2
I
3
I
A
5
,
I
6
7
It is evident from the preceeding sectionsthat according to our formulation there is no distinction between "real" and "virtual" levels, all levels are real. The difference between them is a quantitative one, it is simply a function of their detuning with respect to the pulse center. Levels which are way off-resonance with respect to the pulse center simply exist only when the pulse does. If the pulse frequencies overlap the continuum spectrum, then, depending on the detuning, the ;(wE,,)cE(t) preparation coefficients tell us what fraction of the state exists both during and after the pulse. Obviously, the fact that fragments are detected at the wavelengths used for the Raman experiment tells us that continuum states are being populated even after the pulse is over. One cannot, however, determine from the such photodissociation experiments what is the importance of the "true" Raman process vs resonance fluorescence.
t(psec)
Figure 10. (a) Same as in Figure 7a for a pulse centered at 32 900 cm-'. (b) Same as in Figure 7b for a pulse centered at 32 900 cm-I.
coexistence of broad and narrow resonances the whole recurrence pattern does not die after 8 ps, in perfect accord with our findings. Rather the recurrence pattern is "revived" after about 30 ps. The time scale of the revival has been explained by Chapman and Childu to be due to the existence of two subgroupsof resonances characterized by interresonancespacings differing by 1.13 cm-l. The very short pulses used by Cong et al.37 were postulated to excite the two groups of resonances. The same point, originally made in refs 12, 42, and 43,is shown in Figure 8.
5. Resonsnce Raman and Resonance Fluorescence via Dissociative States The so-called "continuum Raman" experimentslOJl,S55involve exciting a moleculeto a dissociative electronic state and monitoring the ("secondary") emission emanating from the molecule as it falls apart. The theoretical framework so far u ~ e d is5 based ~ ~ on a straightforward generalization of the Kramers-Heisenberg (KH) expression819to the continuum case. According to the KH expression, which is derived in the CW case using secondsrder perturbationtheory, the observed emission rate is proportional to
where, as in sections 2 and 3, w,, is the laser frequency, wi,, is the transition frequency between the final and initial states (the Stokes shift), $: and $p are the initial and final states, and c(f is the component of the transition dipole in the direction of the field of the scattered light. Application the KH formula for structureless continuum ("direct dissociation") usually results in a structureless excitationemission spectrum, irrespective of the initial or final quantum number.66172 As shown below the above formula is deficient
Contrary to ordinary (bound-bound)Raman scattering, where because of relaxationsone needs to use the (optical) von Neumann equations,70171the use of nanosecond or shorter pulses for directly dissociating molecules in the gas phase excludes all other relaxation process, save for spontaneousemission. This means that we can extend the formulation presented above by including explicitly spontaneous emission processes to treating continuum Raman scattering within the framework of the SchrMinger equation. The SchrMinger equation approach was used in the context of the continuum Raman pr0blem,61.63.66.6~by running wave packets aimed at getting the CW KH cross sections. Contrary to eq 3.16, which is a valid computational tool for obtaining the cross sections in straightforward photodissociation, the situation in continuum Raman scattering is different. We show below (following ref 72) that the CW KH expression is inapplicable for dealing with pulses. Among other things, it predicts that the Raman excitation spectrum of directly dissociating molecules is completely structureless, whereas such structures have in fact been measured for C H 3 P and IBr.11
5.1. SpontaneousEmission from Dissociative States. We now wish to explicitly treat the process of spontaneous emission from
a dissociating wavepacket. This we do by generalizing the expansion of the wave function to include both zero-photon and one-photon states. The field cannot be treated classically, as done above, and we find it useful to define two types of radiation states: (1) (a) Ial, u2, ..., a N , 0, ..., 0) which is composed of a multimode coherent state, la1 > (a2> la^), representing the incoming pulse, and zero photons in all other modes.
...
(2) (a! + ik) lal, a2,..., aN, 0,...,0,l k , 0,..., o), representing the pulse plus one (spontaneously emitted) photon in the kth mode.
Using these radiation states, we expand the system's wave
The Journal of Physical Chemistry, Vol. 97,No. 29, I993 7405
Feature Article function q ( t ) as
positive times are given as
e(t)= I&)(b,(t)$gexp(-igt/h) + J d E bng(t) $;(E) ex~(-iEt/ h - ~ E t / 2 ) } + 0
El&+ i,)b:(t)$y exp(-i@t/h)
(5.1)
k,i
This expansion implicitly assumes that the absorptionof one pulse photon does not affect the pulse, which is justified for 1. >> 1. We now add to V,(t), the interaction of the molecule with the pulse (eq 1.5), written in the quantum form as
Vl(t) = --I&)p-€(t)(&l
(5.2)
the spontaneous emission term
- wk
+ wk - w,)t
- I't/2] 27r(wE,, - w, + iI'/2)i(wi,, + wk - W, + i r / 2 ) Z, exp[i(w,,
(5.8)
and negative times bTk(t) = l / h J d E a:,,@) X
Z, exp[i(wi,, + wk - w,)t
+ I't/2]
+
(wES- W, - iI'/2)i(wi9 wk - W, - i r / 2 ) For the CRW terms we obtain, at positive times In the above, u is the cavity volume, t o is the permittivity of free space, and pf is the component of the electric dipole operator in thedirection of the scattered light field. With the useof coherent states, the expectation value of the pulse's electric-field, r(t), assumes the classical form of eq 1.1. In the continuum Raman experiment only a small fraction of the excited molecules manage to fluoresce (cf. ref 51). Under these conditions, we can use the solution of section 3 to solve the augmented time-dependentSchradinger equation, in which V(t) = Vl(r) Vz(t). Substituting eq 2.3 into eq 5.1 and assuming that e(w) is real so that we can group together ci(t) as in eq 3.20, we obtain
+
+ wk + w,)t - 17/21 2 7 r ( ~+~ w, , ~- iI'/2)i(wi,, + wk + W, - iI'/2) z, exp[i(w,,
(5.10) and at negative times
(wE,,
To determine the b:(t) coefficients, we denote as
exp(ik.r)c($ylrfl$~(E)) ($&?~IPJ$~)( 5 . 5 ) 0
and substitute eq 5.4 into eq 1.6. We obtain,
ih dbfldt = JdE af,,(E) Z(wE,,) cE(t)X exp[-i(wE,i - Wk)tl (5*6) where wE,f = (E - @ ) / h - i 7 ~ / 2 . Using the definition of ci(t) (eq 2.4) and performing the time integration we have
(e
b:k(r)
+ b?(t)
(5.7)
where wi,, 3 - l$')/h. This result was originally derived in ref 72. A similar result has been also obtained in ref 73. We first illustrate the outcome of light scattering with Lorentzian pulses (eq 2.5), postponing the use of Gaussian pulses to the calculation of the emission rates. For a Lorentzian pulse the required bFk(t) coefficients are easily obtained by contour integration of eq 5.7. We obtain that the RW coefficients at
+ w, + ir/2)i(wi,, + wk + w, + i ~ / 2 ) J '
t < O (5.11) The first two terms in eq 5.8 (or eq 5.10) are written in a form that suggests their independenceon our choice of the pulse shape. We can do the same thing for the third term by noting that for a Lorentzian frequency profile the numerator of the third terms of eqs 5.8 and 5.9 is t+(t) exp[i(wt, + w&] and that of eqs 5.10 and 5.1 1 is t ( t ) exp[i(wi,, + w&], where c+(t) are defined in eq 1.1. The first term of eq 5.8 is easily recognizable as representing resonancefluorescence: It exists only if t(w) is nonnegligible in the range spanned by the continuum of the WE,, resonance frequencies. For CW excitation mode, the resonance fluorescence term simply decays exponentially with Y E , the radiative decay rates, (since WE,^ = (E - E i ) / h - i 7 ~ / 2 ) .In the pulsed mode, for I' >> Y E ,the resonance fluorescence term in fact, as shown below, decays at a much faster rate. The third term of eqs 5.8 and 5.9 is the Raman transient, which rises and falls with the pulse. The second term represents the long time limit of the sum of the resonance fluorescence and Raman terms. When the excitation is on resonance with a real continuum the contribution of the CRW terms is negligible compared with the RW terms. Therefore we shall henceforth concentrate on the RW terms. We now proceed to obtain expressions for the main experimental observables which are the transient and timeaveraged emission rates. Before doing that we briefly discuss the reduction of our formulae to the CW case. 5.2. 'I = 0 (CW) Case. The CW case is usually obtained by assuming a pure cosine-wave form for the electric field, which is equivalent to setting
(5.12) le(w)l= (e,/2) [ 6 ( ~ wa) + 6(w + wJ1 in eq 1.1 and then computingthe rate. Alternatively,we can first compute the rate and then let I' (the pulse bandwidth of eqs 2.5
7406 The Journal of Physical Chemistry, Vol. 97, No.29, 1993
-
and 2.9) 0. This is the more corrcct procedure because the r = 0 case is never attained in any real situation.
As shown below, both procedures ultimately yield the same results for the dominant CW RW term. However, it can be shown that the two procedures yield different results for the CW CRW terms. The second procedure also sheds light on the interplay between resonance fluorescence and true Raman in the CW limit. This will be further developed in the next subsection. For now, we follow the first procedure for the RW term, i.e., we assume eq 5.12. Substituting eq 5.12 into eq 5.7 and remembering that when e(w) has a singularity the second term in eq 2.1 is not zero, we obtain
Shapiro Using the identity
s a
IJI;(E))
(E - E,)/h - iy/2 - w,
+ E,/h)t]
exp[i(o,
=iKdt X
exp(-yt/2) exp(-iH,t/h)
(5.20)
we obtain that the CW Raman amplitude is given as
= i($;lPJ=dt exp[i(w,
& .),
+ E,/h)tI x
e x p ( q t l 2 ) exp(-iHct/ h )P;I&
(5.2 1)
where we assumed that the spontaneous decay rate independent of the energy (TE = 7 ) .
YE
is
The CW Raman amplitude can also be written as
b:(t) = Z J d E a:,(E) exp[i(wi, 1imT--
X
+ wk - w,)t] - exp[-i(wi, + u k - w,)T] (WE,
= ($fbl$R(hU, +
e))
(5.22)
where $R is the so-called “Raman wave function”:Z*.64.65
+ O k - wu)
- w.)
(5.13) The total rate of emission is given as the sum over all the field modes of the individual emission rates:
where the UWk2/?r2c3
.ij(w,)
,dlbf(t)12 (~/?r’c~)$dw kWk -(5.14) dt factor comes from the angular integration.
Substituting eq 5.13 we obtain
+R(E) E
i K d t exp(iEt/h) exp(-yt/2) X exp(-iHCt/h)p&
(5.23)
The Raman wave function is obtained by taking the half-Fourier transform of a wave packet created as at t = 0 and propagated to time t while decaying via spontaneous emission. Physically speaking, t is not a “real” time, because in the CW case the wave packet is not created suddenly at t = 0. Rather t should be thought of as a useful device for getting the CW result. We now proceed to analyze the real temporal evolution of the emission. 5.3. Transient Emission Rates. The transient rate of emission to a specific kth mode, given as dlb:IZ/dt, is most conveniently computed as 2R,by db:/dt. Multiplying eq 5.6 by eq 5.7 we obtain
dlbY
Using the identity 1imT+- sin[(w,,
A*,,(E) X
dt = 2Re( ”)sdE h3e#
+ O k - o,)(t + n1-wi,s
+ wk-
Ou
r6(wfJ
+ wk-w,)
(5.16)
we obtain that the total rate of populating the ith level in the CW case is given as Using the equality 1 ( w * E ~- w)(w
where A,,(E)
c$;lPl$;(a)
=
clt;
(5.18)
This is essentially the Kramers-Heisenberg expression with continuum intermediate states, in which (as explained above) we have for simplicity ignored the CRW term. It is of interest to make connections at this stage with wavepacket methodologies. Following Heller et al.28.31.61,68 we can useeq 5.17 toimplicitlydefineatJ(ua),theCW Ramanamplitude:
and the definition of the CE(t) coefficients (eq 2.4), we obtain
wk)
z(oE?+)c*~,+y(~) CE(t) exp[-i(wE,i - wk)tl -
p*(w*E,) e(wE’,s)C*E(t) c L y ( t ) exp[-i(oEt,i - W*E,t)tl1 (5*25) The rate of emission to all the field modes, defined in eq 5.14, involves multiplying dlb:(t>12/dr by (uwk2/?r2c3)and integrating over Wk. When this is done, it can be shown22 using contour integration, that the second term in the curly brackets of q 5.25
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 I401
Feature Article vanishes. The remaining first term yields dP,,( t ) /d t = (8 / 7rc3h cO)ReJdE (0*E,,) 3A*(,E .) :*(O*E,)
C*E(f)
~Xp(iw*,,,t)JdE’A,,(.E~
X
-*
WE,,) CE,(?) x
exp(-iw,,,,t)
111
(5.26)
a
d m
Usually the (W*E,,)3 factor varies much more slowly than all other variables and its imaginary part is relatively small. We therefore replace JdEJdE’(wE,r)3 ... by JdE (o~,,)3/2JdE’(@EJ,,)3/2 and write eq 5.26 as
sx
...
dP,,(t)/dt
.-
d
-E 42
1
= (8/rc3h3c0)lJdE (WE,$)3’2A,,(E)X z ( u E ~ )C E ( t )
exp(-iwE,,t)
0
1’
(5.28)
where rn(E)is a set of turning points on the dissociative channel potentials:
E
(5.29)
and is some average transition dipole moment. With this approximation, the A,,(E) molecular factor of eq 5.18 assumes the form
If there is only one open channel A,,(E) is expected to have in energy space the nodal structure of the X $: product in
$7
3303
4000
( *27)
+
Vn,n[rn(E)I
2000
frequency (cm-’)
We first use eq 5.27 to examinethe transientand time-averaged emission rates with pulsed excitation sources. It follows from the definition of A&?) (eq 5.18) and from eq 5.27 that the rate of emission is simply the square of a sum of contributions from each of the $;(E) eigenstate components of the wave packet of eq 5.1. Each transition amplitude, given as [ 2 ( W E , l ) / ( C h ) ] ’ / ’ / ( € 0 7 r ) ’ ($Apl$;(E)), is multiplied by the preparation coefficient ( $;(E)lp;l$,)a(wE,)cE(t). The emission changes with time due to the natural evolution and radiative decay of the wave packet (governed by the exp(-iwE,,t) factor) and, while the excitation pulse is on, the time dependence of the CE(t) preparation coefficients. Notice that eq 5.27 involves an incoherent sum of probabilities to the individual radiation modes (as befits a spontaneous emission process) and a coherent sum over the scattering eigenstates (as befits a time evolving wavepacket). The CE(t) preparation coefficients determine the character of the emission process. Figure 1 shows that at early times ;(WE,) cE(t) extends over the whole continuous spectrum. In fact at these times the preparation coefficients are independent of €(wE,,) as they tail off as WE, - a,). This form is obtained from the asymptotic properties of w[z] of eq 2.12.” Thus, at sufficiently early times, eq 5.27 becomes essentially identical to the KH expression (eq 5.17), and the process may be termed “true Raman”. As time progresses, a transition occurs: C(WE,) cE(t) narrows down until it assumes the shape of the pulse itself. This is a result of the fact that after the pulse is over only the fl(t‘) term in the curly brackets of eq 2.12 remains. If the laser bandwidth is much narrower than the absorption spectrum, it follows from eq 5.27 that the emission rate now becomes essentially proportional to A,,Z(E, hw,). As a result of this transition, eq 5.27 describes resonance fluorescence from an evolving wave packet. To follow the emission process in greater detail, we now use eq 5.27 to compute the time-dependent excitation-emissionspectra for several pulse configurations. Postponing the more realistic treatment of the molecular aspects to section 6, we look at a diatomic molecule whose ground state is described by a harmonic potential for which the bound-free transition-dipole matrix elements follow the “reflection” a p p r o x i m a t i ~ n ~ ~ . ~ ~
(+~(.E)IP~I+!)= P+
