7832
J. Phys. Chem. 1996, 100, 7832-7848
Femtosecond Real-Time Probing of Reactions. 18. Experimental and Theoretical Mapping of Trajectories and Potentials in the NaI Dissociation Reaction P. Cong,† G. Roberts,‡ J. L. Herek, A. Mohktari, and A. H. Zewail* Arthur Amos Noyes Laboratory of Chemical Physics,§ California Institute of Technology, Pasadena, California 91125 ReceiVed: NoVember 20, 1995; In Final Form: January 17, 1996X
We describe a method of tracking wave packet trajectories by probing both the temporal and spatial evolution of a coherent state prepared by a femtosecond laser pulse in real time. Application to the NaI dissociation reaction illustrates this technique, in which the direction of wave packet propagation within the quasi-bound A 0+ potential is observed and correlated to the physical processes of extension and contraction of the Na-I bond. From the combination of spatial and temporal information observed from experiments, self-consistent potential energy functions for the A 0+ state and the higher-lying state accessed by the probe laser pulse are constructed. The dynamics and the validity of the potentials so derived are tested via semiclassical and quantum-dynamical simulations of the wave packet motion, and comparisons with potentials obtained from other methods are drawn.
I. Introduction The seminal work by Professor Kinsey and co-workers1 on the evolution of wave packets in dissociation reactions has stimulated numerous theoretical and experimental studies. This paper focuses on the real-time observation of the evolution and directionality of wave packets in NaI photofragmentation. We are particularly pleased to present this work in honor of Jim Kinsey, who not only has made important experimental and theoretical contributions to the field but also has welcomed and encouraged new developments in the most scholarly and enthusiastic approach possible. Sometime ago, we reported in a letter2 a method for the studies of reaction trajectories by probing the direction of the motion of a wave packet. This method, which carefully tracks the trajectory in time (t) and space (R), was demonstrated for the dissociation reaction of NaI, for which the interaction of the covalent and ionic potentials along R defines the adiabatic coordinate of the reaction. In this paper, we present our full account of these initial studies which are relevant to general interest in the dynamics of wave packets.3 We describe in detail the experimental methodology and the new results on the NaI system. The clocking of the wave packet and the establishment of its directionality enable us to visualize the trajectory, establish the spatial resolution, and deduce the potential energy for the motion. The findings are supported by molecular dynamics and classical and quantal approaches are examined. Previous experiments2,4-10 have explored the real-time dynamics of the dissociation process that occurs when NaI is excited via dipole-allowed charge-transfer transitions11 from the ionic ground state to the lowest-lying Ω ) 0+ and 1 covalent excited states: ‡ + NaI(X 1Σ+ 0 ) f [Na‚‚‚I] *(Ω)0 ,1) f
Na(2S1/2) + I(2P3/2) (1) A femtosecond pump pulse (300 e λ1 e 390 nm) prepares a † Department of Chemistry, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. ‡ Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, England. § Contribution No. 9160. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(95)03403-4 CCC: $12.00
coherent superposition in the excited-state manifold which subsequently evolves as a wave packet that can be observed in real time2,4-8 according to the methodology of femtosecond transition-state spectroscopy (FTS).9 Oscillatory motion of the wave packet trapped within the adiabatic potential well of the quasi-bound A 0+ state (see Figure 1) combined with the finite escape probability to form Na + I atoms via nonadiabatic crossing to the X 1Σ+ 0 potential results in the real-time observation of a series of fluorescence peaks of decreasing amplitude that persists for about 10 ps. Likewise, detection of “free” Na atoms reveals a series of plateaus of increasing height as a function of time that corresponds to the formation of atomic products on each occasion that the wave packet encounters the intersection region between the diabatic potentials with its momentum vector directed toward increasing R.4-6,8 The contribution of Na arising from rapid dissociation via the Ω ) 1 channel has also been discerned in the time-dependent fluorescence signal.6 A particular feature of the earlier work was the careful determination of the oscillation frequency νosc of the wave packet Ψ2(R;t) ()〈R|Ψ2(t)〉) within the A 0+ adiabatic well as a function of the available energy Eav.4,6,8 This was achieved simply by measuring the period τosc ()1/νosc) between successive fluorescence maxima in the time-domain spectra recorded at different values of the pump laser wavelength.4,6 From the variation of νosc with Eav, Rose et al.4,6 determined an anharmonicity constant of ωexe ≈ 0.035 cm-1 for the adiabatic potential at Eav ≈ 5640-9200 cm-1. The value of the anharmonicity is in accord with the result obtained from analysis of the spectroscopic potential over the same energy region.12,13 The dependence upon Eav of the damping time τD of the realtime fluorescence signal was also characterized, revealing as expected a higher probability for transitions between adiabatic surfaces with increasing recoil velocity.4,6,8 Measurements of the adiabatic escape probability coupled with application of the Landau-Zener formalism for dynamical curve crossing lead to a value of V12(Rx) ) 415 cm-1 for the coupling matrix element between covalent and ionic Ω ) 0+ states, in close agreement with previous theoretical and experimental values.13-18 Of relevance to the experiments described in the current work is a point raised previously by Rose et al. concerning the appearance of the time-resolved fluorescence transient,6 namely, © 1996 American Chemical Society
NaI Dissociation Reaction
Figure 1. Low-lying potential energy curves relevant to FTS probing of NaI dissociation. Indicated by vertical arrows are the ultrashort pump and probe pulses. A thermal distribution of molecules is promoted by a pump pulse at wavelength λ1 from the ground-state potential E1(R) to the quasi-bound potential E2(R), which correlates adiabatically with Na+(1S0) + I-(1S0) ions. The wave packet formed on E2(R), comprising a distribution of eigenstates determined by the spectral profile of the pump laser, oscillates within the adiabatic well with diminishing amplitude as a fraction proceeds to Na(2S1/2) + I(2P3/2) products by nonadiabatic crossing to the E1(R) continuum states. A probe pulse at wavelength λ2 monitors the dynamics of the wave packet on E2(R) every time it enters the resonant absorption region by excitation to the higherlying potential E3(R), from which time-integrated Na(2PJf2S1/2) fluorescence is detected. In this diagram, λ1 ) 320 nm and λ2 ) 640 nm are shown for illustration.
that if absorption of the probe pulse by Ψ2(R;t) on the A 0+ potential takes place over a restricted range of coordinates centered about a particular R, then the second and subsequent real-time fluorescence peaks should comprise two closely separated maxima, corresponding to inward and outward bound propagation of the wave packet through the optically coupled region (OCR) defined by the probe pulse.19,20 An attempt was made by Rose et al. to observe such structure in the FTS transient by decreasing the pulse width of the probe laser; nevertheless, the anticipated splitting of the fluorescence peaks into a doublet line shape remained unobserved, though an increase in the width of the second peak relative to the first was noted.6 We return to this problem here, where we report results of experiments designed to probe the vibrations of [Na‚‚‚I]‡* complex in the A 0+ state with increased spatial resolution. A detailed study of the dynamics of Ψ2(R;t) constitutes the first objective of these investigations, with the use of improved temporal resolution offering a finer probe of the wave packet motion than has been available hitherto.4-7 The expected doublet structure of the fluorescence peaks in the FTS transients has now been revealed, allowing determination of the direction of wave packet motion and distinguishing the processes of extension and contraction of the [Na‚‚‚I]‡* bond. As in the initial report,2 in this paper, we examine the evolution of the wave packet at short times (e3 ps) rather than the non-BornOppenheimer dynamics of the excited-state electronic population per se; the former gives information on the shape of the potential energy curve while the latter reflects the partial Landau-Zener crossing at the conical intersection with the ground-state potential, as discussed above.
J. Phys. Chem., Vol. 100, No. 19, 1996 7833 In addition to a more detailed description of the motion of Ψ2(R;t) afforded by high-spatial-resolution FTS, a semiclassical analysis of the temporal positions of the transient line shapes recorded at various initial wave packet energies Eav also yields the form of the potential energy curves for the A 0+ state and the higher-lying state accessed by the probe laser. Although measurements of νosc as a function of Eav coupled with application of an RKR (or similar) procedure permit the separation ∆R(Eav) between outer and inner turning points of a potential curve to be obtained, it is necessary to know the average interfragment separation at different energies in order to fix the potential in configuration space; this is usually accomplished by measurement of the rotational constant B(Eav) of the molecule. These constants can, in principle, be obtained from measurements of rotational recurrences in the time domain, such as have been carried out for I2(B 3Π0+u),21,22 ICl(A 3Π1),23 and large aromatic molecules,24 for example. Here, the energy dependence of the interatomic separation is deduced from the vibrational period τosc together with clocking25 of the wave packet motion as Ψ2(R;t) propagates with time from the Franck-Condon region of the A 0+ state to longer distances. A set of timing experiments have been performed, analogous in principle to those carried out on the direct dissociation of ICN over a repulsive PES.26 During the course of this work, another method of clocking the reaction emerged, employing the resolution of the doublet structure within fluorescence peaks of the FTS transients. An account of the various tests that were carried out to verify the accuracy and robustness of the potentials so derived is also presented. The organization of the following sections of this paper is as follows. In section II, a brief description of the experimental arrangement adopted for high-spatial-resolution studies of NaI photodissociation is given. The observations of wave packet trajectories and the method of mapping the relevant potential energy curves are discussed in section III, where comparison is made with the potentials derived from frequency-resolved spectroscopy. Section IV provides an outline of the theoretical treatment employed to obtain a quantitatively accurate description of the observed time dependence with emphasis on the sensitivity of the calculated transients to the form of the potentials derived in section III. Brief concluding remarks are given in section V. II. Experimental Section The experimental methodology of FTS9,27 as applied to the study of alkali halide predissociation reactions has been presented in detail elsewhere.6 Therefore, only a brief outline of the procedure adopted in this work is given here, with attention focused on specific details pertinent to the current experiments. Femtosecond laser pulses (fwhm ∼ 100 fs) were generated in a colliding-pulse mode-locked ring dye laser (CPM) at a wavelength λ ≈ 620 nm. Pulse amplification was achieved using a four-stage pulsed dye laser amplifier pumped by a Q-switched frequency-doubled Nd:YAG laser operating at 20 Hz. Tunable probe pulses (λ2 ) 590-700 nm) were obtained by focusing part of the amplified CPM output into a D2O cell to generate a white-light continuum which was frequency selected by an interference filter. Tunable pump radiation (λ1 ) 320-365 nm) was generated by further amplifying a portion of the continuum with residual 532 nm light from the Nd:YAG laser followed by second-harmonic generation in a KD*P crystal; pump pulses at λ1 ) 391 nm were obtained by sumfrequency generation through mixing the amplified CPM light (λ ) 623 nm) with the Nd:YAG fundamental at λ ) 1064 nm.
7834 J. Phys. Chem., Vol. 100, No. 19, 1996
Cong et al. TABLE 1: Time τ0 for a Wave Packet Evolving on the Adiabatic Potential E2(R) To Reach the Center of the Optically Coupled Region Defined by Probe Laser Pulses of Varying Wavelength λ2 (Pump Wavelength λ1 ) 310 nm)
Figure 2. Schematic representation of the experimental configuration adopted for clocking experiments of NaI dissociation. The probe pulse is delayed in time relative to the pump pulse before the two are focused into the reaction cell containing NaI. The cross-correlation of the pump and probe pulses was measured at point A by deflecting the beams through a KD*P crystal and detecting the frequency difference signal as function of pump-probe time delay. By translating the cell out of the optical path and recording the cross-correlation at point B, the dispersion between the pulses due to the quartz window could be measured, corresponding to one half the difference between the two cross-correlation signals. A similar correction for dispersion from the KD*P crystal was also made. When these corrections were subtracted from the cross-correlation signal, time zero was obtained to within (20 fs.
Pulse widths of ∼100 fs ensure adequate time resolution of nuclear dynamics, yet are comfortably longer than the time scale 28 for dephasing of the Ω ) 0+,1 r X 1Σ+ 0 electronic transition 29,30 (given by the inverse of the absorption line width ). The time variable was introduced into the experiment by altering the optical path length of the probe beam relative to the pump by means of a high-precision Michelson interferometer. As shown in Figure 2, the two separated beams were collinearly recombined by a dichroic beam splitter and focused into a sealed quartz cell of length 8 cm containing approximately 100 mTorr of NaI when heated to about 650 °C.6 Laser-induced fluorescence (LIF) was collected perpendicular to the incident laser beams direction as a function of pump-probe delay time τD. The LIF was dispersed through a 0.34 m monochromator tuned to the unresolved D-line fluorescence of atomic Na at λ ) 589 nm (2PJ f 2S1/2) and detected photoelectrically. The zero of time, i.e., the delay time at which the pump and probe pulses are temporally synchronous, was determined via the cross-correlation signal (see Figure 2). The extent of dispersion in the cell windows was determined by comparing the cross-correlation signals measured with and without the vapor cell in the beam path, at point B in Figure 2. III. Results and Discussion A. Potential Energy Curves and Methodology. Figure 1 displays the adiabatic potential energy curves relevant to the study of NaI photodissociation via the A 0+ state, labeled in accordance with the standard convention adopted for the alkali halides. Adiabatic potentials E1(R) for the ground electronic + state (X 1Σ+ 0 ) and E2(R) for the bound excited state (A 0 ) formed by mixing of the diabatic potentials V1(R) and V2(R) were obtained from the standard expression:31
E1,2(R) ) 1/2{V2(R) + V1(R) ( [(V1(R) - V2(R))2 + 4V12(R)2]1/2} (2) For the diabatic ionic curve V2(R) and coupling matrix element V12(R), the parametrized forms put forward by Faist and Levine are employed.32 The repulsive covalent state V1(R) is represented as an exponential function of distance, characterized by the parameters derived in this work from an analysis of the
λ2/nm
τ0/fs
λ2/nm
τ0/fs
600 623 642 652
118 96 66 28
662 676 698
22 35 43
experimental clocking data. The potential energy curve for the lowest-lying Ω ) 1 covalent multiplet that correlates with Na(2S1/2) + I(2P3/2),29,33,34 together with the remaining Ω ) 2, 1, 0- levels,15,34 is not depicted. Similarly, of the 15 multiplets that connect asymptotically with Na(2PJ) + I(2P3/2) atoms, only a single potential curve denoted E3(R) is shown in Figure 1, the anharmonic form of which is taken from the parameters determined herein; E3(R) represents the highest-lying potential(s) accessed by the probe laser. Indicated by vertical transitions are the laser excitation processes that constitute the essence of the FTS technique. An weak transform-limited pump pulse at wavelength λ1 promotes an ensemble of ground-state molecules to the predissociative potential E2(R), where a nonstationary state |Ψ2(t)〉 is prepared that comprises a coherent superposition of quasi-bound rotation-vibration levels whose relative contributions are determined by the distribution of Franck-Condon factors within the transform-limited bandwidth of the laser pulse. The evolution of |Ψ2(t)〉 with time is probed by a second ultrashort laser pulse λ2, temporally delayed with respect to the first, by transferring population to the state E3(R) which gives rise to fluorescence. Absorption of probe light occurs only when Ψ2(R,t) samples those coordinates on E2(R) that lie within the narrow range of positions spanned by the OCR,19,20 the central location of which is defined by hc/λ2 ) E3(R) - E2(R), and spectral width is determined by the temporal duration of the probe laser. By selecting a range of off-resonance wavelengths λ2 to the red (or blue) of the sodium D-line transition, the oscillating wave packet can be monitored at different spatial coordinates on E2(R). The final wave packet formed on E3(R) is detected as a function of pump-probe time delay τD by emission of fluorescence via the optically allowed transition to E1(R). B. Observation of the Trajectories: Tracking the Wave Packet Motion. A series of experiments were carried out to determine the time τ0 between coherent preparation of the initial wave packet on the repulsive limb of E2(R) and its subsequent detection by the probing laser pulse, as revealed by an intensity maxima in the FTS transient. Such measurements require an exact determination of time zero, i.e., the delay time at which the pump and probe pulses are temporally synchronous.9,26,27 In this work, τ0 was obtained as the difference between the first peak of the transient LIF recorded at off-resonance probe wavelengths and the cross-correlation signal between the pump and probe beams, minus the slight dispersion between pump and probe pulses introduced by the quartz window of the vapor cell. The uncertainty associated with values of τ0 determined in this manner was estimated to be (20 fs. This method may be contrasted with previous studies of the real-time dissociation of ICN,9,26 in which the zero of time was ascertained from the REMPI signal of N,N-diethylaniline, which gives the integrated response function of pump and probe lasers.26,27 Figure 2 presents a schematic illustration of the experimental arrangement adopted to obtain such data. τ0 was determined systematically for probe wavelengths in the range λ2 ) 600-700 nm at a fixed pump wavelength of λ1 ) 310 nm, yielding the values listed in Table 1.
NaI Dissociation Reaction
J. Phys. Chem., Vol. 100, No. 19, 1996 7835 TABLE 2: Splitting Time τs (fs) between the Doublet Intensity Maxima of the Second and Subsequent Fluorescence Peaks of FTS Transients at Different Pump (λ1) and Probe (λ2) Wavelengths λ2/nm λ1/nm
600
623
642
662
310 320 330 340 350 355 360 365 390
305b 326 343 351 367 373 373 374 431
NAa 233 262 241 278 289 314 313 355
NA 150 180 229 242 NA 260 NA 269
NA NA NA 173 189 NA NA NA NA
a NA ) not available. b Note that 305/2 ) 152 fs, which compares with the 118 fs value in Table 1 within the accuracy of the latter measurement.
Figure 3. Experimental FTS transients obtained at different pump and probe laser wavelengths: (a) variation of λ1 at a fixed probe wavelength of λ2 ) 605 nm; (b) variation of λ2 at a fixed pump wavelength of λ1 ) 391 nm.
TABLE 3: Wave Packet Periods τosc of the Quasi-Bound Ψ2(R;t) on E2(R) as a Function of Pump Wavelength λ1a λ1/nm
Eav/cm-1
τosc/fs
λ1/nm
Eav/cm-1
τosc/fs
310 320 330 340
6852 5844 4897 4005
1281 1095 1009 991
350 360 390
3165 2371 235
959 936 915
aE av is the classical excess energy available for wave packet oscillation, calculated as the difference between the photolysis energy E1 ) hc/λ1 and NaI dissociation energy De.
Figure 4. Typical FTS data of LIF signal intensity (arbitrary units) versus pump-probe time delay τD (ps) showing the splitting time τs between observed intensity maxima (other than the initial peak) and the average oscillation period τosc of the wave packet trapped within the adiabatic well E2(R).
During the course of our investigations, a direct and rather simple method of measuring τ0 became apparent that made use of the observed splittings in the off-resonance transients. For these experiments, “high-resolution” transients were recorded by numerically averaging 20-100 individual data scans to facilitate a thorough examination of the transient line shapes. As shown in Figure 3a, for a fixed probe wavelength of λ2 ) 605 nm, the second and subsequent (later time) peaks of the FTS transients become increasingly broad as the energy Eav available to the oscillating wave packet is decreased, eventually splitting into two distinct maxima for pump wavelengths λ1 g 330 nm. Similar behavior is depicted in Figure 3b, which shows the effect of tuning the probe laser to shorter wavelengths approaching λ2 ) 589 nm at a constant photolysis wavelength of λ1 ) 391 nm. In order to analyze these measurements quantitatively, the observed transients were fitted to a sum of Gaussian line shapes by a nonlinear least-squares fitting routine. Figure 4 defines the splitting time τs as the difference between the two maxima of the second (or subsequent) fluorescence peaks of the FTS signal. In addition, the oscillation period τosc can also be determined from the experimental data, and this too is shown. Values of τs determined from FTS measurements at different pump and probe wavelengths are presented in Table 2, while the oscillation periods are given in Table 3. The simple connection between τ0 and τs is presented in section IV.
C. Mapping the Potential Energy Curves. The low-lying covalent potential energy surfaces curves of NaI have been the object of thorough investigations dating back to the 1920s.35 Of the five excited multiplet levels that correlate asymptotically with neutral Na(2S1/2) and I(2P3/2) atoms in the diabatic representation (distinguished by the quantum numbers Ω ) 2, 1, 1, 0+, and 0- according to Hund’s case c coupling scheme36), the Ω ) 0+ and one of the Ω ) 1 levels lie at lower energies and shorter internuclear distances than the remainder.34 The nondegenerate ground state X 1Σ+ 0 (for which Hund’s case a provides the most appropriate description of angular momentum coupling36) leads diabatically to Na+(1S0) + I-(1S0) ions located energetically at ∆E ≈ INa - AI ) 16 770 cm-1 above the neutral asymptote.37,38 Non-Born-Oppenheimer coupling between the diabatic potentials of the ionic ground state and excited covalent state of Ω ) 0+ symmetry gives rise to an electronically excited state labeled A 0+ that is strongly bound relative to the separated ions. In accordance with the Neumann-Wigner rule, the adiabatic potential for the A 0+ state exhibits an avoided crossing 39 In contrast, with the X 1Σ+ 0 curve centered at Rx ) 6.93 Å. the lowest-lying Ω ) 1 covalent level is essentially dissociative at all internuclear separations29,33,34 and is only weakly coupled to the ground state. 1. FTS-Based DeriVation of E2(R) and E3(R). In this section, the experimental data presented in Figure 3 are employed to obtain the NaI potentials probed by FTS, namely, those for the intermediate A 0+ state and fluorescing state correlating with Na(2PJ) + I(2P3/2) atoms. The methodology is based on classical trajectory calculations of the oscillatory motion over E2(R) and makes use of the classical description of real-time dissociation put forward by Bersohn and Zewail.19 Kosloff and Baer have provided a quantum-mechanical iterative procedure for reduction of the data obtained from realtime experiments in the impulsive limit to the form of the underlying potential function that is based on time reversal of the quantum equations of motion.40 FTS data have previously been employed to derive potential energy curves for bound electronic states such as I2(B3Π0+u)21,22 and ICl(A3Π1).23 In
7836 J. Phys. Chem., Vol. 100, No. 19, 1996
Cong et al.
these systems, the dependence of the oscillation frequency νosc on the wavelength λ1 permits the separation ∆R(Eav) between the outer and inner turning points of the potential to be determined as a function of Eav, and the dependence of the rotational constant B(Eav) (≈1/2τrot) on λ1 enables the average interatomic separation 〈R〉(Eav) at different energies within the well to be elucidated, thereby locating the excited-state potential in configuration space.21-23 For NaI(A 0+), although the variation of νosc with Eav can be readily obtained by FTS, the short dissociation lifetimes of the majority of vibration-rotation levels compared to an estimated τrot ≈ 569-1054 ps (ref 41) preclude the determination of B(Eav) from measurements of rotational recurrences. Instead, the energy dependence of 〈R〉(Eav) is derived from the vibrational periods τosc and clocking times τ0 for wave packet motion within the adiabatic well at different pump wavelengths. In the following sections, the inner repulsive limb of E2(R) is obtained first by optimizing the parameters of an assumed functional form for the short-range potential, and then the longrange Coulombic interaction is recovered from a numerical analysis of the energy dependence of τosc. The complete process can be thought of as the separate determination of covalent and ionic diabatic potentials V1(R) and V2(R), which are then combined to generate the complete adiabatic potential well. From the shape of E2(R) so derived and measured values of τ0(Eav) at different probe wavelengths, a potential for the higherlying electronic state(s) accessed by the probe laser pulse is recovered. a. RepulsiVe Branch of the Quasi-Bound Potential E2(R). The repulsive branch of E2(R) has been studied experimentally 12,13,30,42,43 by frequency-resolved A 0+ r X 1Σ+ 0 spectroscopy, 14,15 atomic scattering, and photofragment spectroscopy.34,44-46 A number of workers have presented numerical47,48 and analytical32,49,50 forms for the diabatic potential V1(R). Here, a parametrized expression for the potential describing the shortrange covalent interaction in the A 0+ state is adopted. A particularly convenient choice is the exponential form
V1(R) ) V0 exp[-(R(R - R0)] + De′′
(3)
proposed by van Veen et al.45 for the lowest-lying Ω ) 0+ and 1 covalent potentials in the Franck-Condon region. Although it fails to take account of long-range van der Waals attraction, the main broad features of the absorption spectrum of NaI can be adequately recovered by this simple expression. The four parameters that characterize V1(R) are determined by applying the classical equations of motion for the time evolution of a single heavy-particle trajectory over an exponentially repulsive potential. For a particle whose initial coordinate is R1 at time t ) 0, the time taken to reach the position R2 is given by
t ) ∫R
R2 1
1
x2/µ[Eav - V1(R)]
dR )
(
)
1 + x1 - exp[-R(R2 - R1)] 1 ln (4) RV∞ 1 - 1 - exp[-R(R - R )] x 2 1 is the terminal for V1(R) given by eq 3, where V∞ ) recoil velocity of the particle in the center-of-mass frame. In the limit that R(R2 - R1) . 1, the last expression can be simplified to (2Eav/µ)1/2
t)
r2 - r1 1 + (ln 4) V∞ RV∞
(5)
This equation clearly delineates the effect of the exponential force field on the classical motion:8 the first term is simply the time taken for a particle of velocity V∞ to travel the distance R2 - R1, and the second represents the influence of the potential via the reciprocal length parameter R. For a hypothetical pump laser pulse that is instantaneously short but nevertheless has a definite frequency, the incipient (infinitely narrow) wave packet Ψ2(R;t)0) is prepared at a precisely defined coordinate R1(λ1) on V1(R) that depends upon the laser wavelength. For the purpose of determining the potential, it is assumed that R1 corresponds to the inner turning point on V1(R) at an energy Eav ) hc/λ1-De′′ above V1(R)∞) (the A 0+ state minimum); that is to say, Ψ2(R;t)0) is initially located on the repulsive branch of V1(R) irrespective of λ1, from where it begins to oscillate within the adiabatic potential well.51 The time t given by eq 5 represents the time taken for λ2(R;t) to propagate from R1(λ1) to the OCR centered at R2(λ2); it can therefore be identified as τ0 and related to the splitting time τs via eq 10 (Vide infra). Clearly, t depends on R1(λ1), R2(λ2), and the steepness of the potential V1(R) between these two points, governed by the magnitude of R: if R2(λ2 is kept constant and R1(λ1) varied over a range of values, then it is possible to sample different positions on the covalent branch and gain specific information about its shape by measuring the time required to reach the OCR as a function of λ1. The overall goal is therefore to determine that set of parameters V0, R, R0, and De which yields the closest agreement between values of t calculated from eq 5 for known values of R1(λ1) and R2(λ2) and the experimental splitting times listed in Table 2. The procedure becomes particularly straightforward when V0, R0, and De′′ are independently assigned fixed values, leaving only the inverse length parameter R to float as a variable parameter. Our justification for fixing V0, R0, and De′′ is that these quantities can be determined accurately by time-integrated spectroscopies12,13,45 whereas R cannot: V0 and R0, for example, are determined largely by the location of the absorption maximum;29 R, on the other hand, which reflects the slope of V1(R), is derived from the width of the absorption profile and is a much less accurately known parameter due to hot-band contributions in the absorption spectrum and other inhomogeneous broadenings. Results from ultraviolet laser absorption spectroscopy and photofragment spectroscopy give V0 ) 6557 52 cm-1,12,13,45 while R0 is taken to be Re′′(X 1Σ+ 0 ) ) 2.71 Å. The dissociation energy depends upon the choice of groundstate potential: the value De′′ ) 25 406 cm-1 selected here represents the average of estimates from photofragment spectroscopy44,45 coupled with the modified Rittner form of V2(R) suggested by Faist and Levine.32 Knowing V0, R0, and De′′, it is a simple matter to determine Eav and R1 for any given λ1, and the task of obtaining the repulsive potential reduces to determining the value of R which is consistent with the measured times τ0 required to reach different positions R2(λ2) located on V1(R). If a classical trajectory is allowed to evolve for a time t ) τ0 according to eq 5, it will reach a final position designated R2′(λ1;R) given by
R2′(λ1;R) ) R1(λ1) + V∞t - (ln 4)/R
(6)
If R is correctly chosen, then for a given probe wavelength λ2 the deviation between calculated values of R2′(λ1;R) should be very small for different starting coordinates R1(λ1), because each R2′(λ1;R) corresponds to the same central position R2(λ1) of the OCR defined by the spectral bandwidth of the probe laser. In this way, an optimal value of R ) 2.5 ( 0.5 Å-1 is determined by solving eq 6 for R2′(λ1;R) for λ2 ) 605 nm and the different
NaI Dissociation Reaction
J. Phys. Chem., Vol. 100, No. 19, 1996 7837
values of λ1 in the range 310-391 nm listed in Table 2. This result is smaller than the value R ) 4.088 Å-1 derived from photofragmentation spectroscopy46 but is consistent with other literature values.14,15,32 To examine the consequence of varying the analytical form of V1(R), we have investigated the effect of including a physically reasonable attractive component to the covalent potential: when a -C6/R6 term is added to the purely repulsive form of eq 3 to give a shallow well 258 cm-1 deep located at R ≈ 5.0 Å,32 a slight increase in τosc of less than 4% at λ1 ) 320 and 360 nm (Eav ≈ 5840 and 2370 cm-1) was noted, though it was not possible to discern any significant differences in τs. The uncertainty of (0.5 Å-1 ascribed to R reflects the slightly lower estimate obtained using the RKR ground-state potential determined from absorption spectroscopy,13 rather than the Rittner model, and the effect of incorporating an attractive term in the analytical form for V1(R). The result R ) 2.5 Å-1 obtained for λ2 ) 605 nm also gives excellent overall agreement with the values of τs recorded at λ2 ) 620 and 640 nm. Using eq 6 in conjunction with the splitting times listed in Table 2, it is possible to deduce R2(605 nm) ) 6.5 Å, R2(620 nm) ) 5.6 Å, and R2(640 nm) ) 5.0 Å for these three probe wavelengths. These data are of crucial importance in determining the shape of E3(R) (Vide infra). b. The Quasi-Bound Potential E2(R). Having obtained the repulsive branch of E2(R), the oscillation periods τosc of Ψ2(R;t) can be used to determine the separation ∆R(E2) between inner and outer turning points of the adiabatic well at different vibrational energies. To this end, we invoke the classical procedure presented by Bernstein and Zewail for inversion of data obtained by FTS.22 For a known ground-state potential V2(R)32 and the exponentially repulsive form of V1(R) derived above, the corresponding pairs of values of Eav and τosc listed in Table 3 can be fitted to a third-order polynomial of the form
∆R(E2) ) 11.154 × 10-5a0V1/2[1 + (a1b1V + a2b2V2 + a3b3V3)/a0] (7) where a0 ) 909 fs, a1 ) -1.1225 × 10-3, a2 ) -2.2019 × 10-6, a3 ) 1.2802 × 10-9, b1 ) 0.6667, b2 ) 0.5333, and b3 ) 0.4571 for V expressed in cm-1. From values of R1(E2) obtained from the known exponential form of V1(R) and R2(E2) given by
R2(E2) ) ∆R(E2) + R1(E2)
(8)
it is straightforward to construct a complete adiabatic potential curve for the A 0+ state: the result is displayed in Figure 5. c. The Fluorescent Potential E3(R). Given the form of the adiabatic potential E2(R) derived above, the times τ0 for Ψ2(R;t) to reach R2(λ2) may be further employed to derive the form of the potential E3(R) for the fluorescing state(s) accessed by the probe laser. For a fixed pump energy, values of τ0 effectively constitute a direct measurement of the difference potential ∆E32(t) ) E3(t) - E2(t) expressed as a function of time, the conversion of which to ∆E32(R) is readily facilitated by means of the trajectory R(t). Since E2(R) is known, values of ∆E32(R) therefore yield a set of points on E3(R) at interfragment separations corresponding to the times τ0. From the values of τ0 listed in Table 1 for λ1 ) 310 nm, we may determine E3(R) at seven different coordinates; R(t) is computed by the numerical integration method described above. Further information on the shape of E3(R) is provided by the values of τs recorded at λ2 ) 605 nm (16 529 cm-1), 620 nm (16 000 cm-1) and 640 nm (15 625 cm-1), which give ∆E32(R) directly at
Figure 5. Adiabatic potential curve for the A 0+ state of NaI determined by FTS.
distances 6.5, 5.6, and 5.0 Å according to the analysis of the trajectories. Since E2(R) is known and the Na 2PJ-2S1/2 separation gives the asymptotic energy spacing, this generates an additional four points on the attractive branch of the higherlying potential. The total of 11 discrete values of E3(R) can then be fitted to an appropriate analytical expression, such as the Morse function
E3(R) ) De′′′[1 - exp(-a(R - Re′′′))]2 - De′′′
(9)
which yields the results De′′′ ) 1863 ( 210 cm-1, Re′′′ ) 4.0 ( 0.1 Å, and a ) 0.83 ( 0.07 Å-1, where the uncertainties represent 1σ statistical fitting errors. Because the coordinate positions on E3(R) corresponding to τ0 depend on the shape of the intermediate potential via the conversion of ∆E32(t) to ∆E32(R), it is of interest to examine the sensitivity of E3(R) to the functional form of E2(R). When, as before, a dispersion contribution is added to the repulsive interaction to give a potential minimum 258 cm-1 deep at R ) 5.0 Å on E2(R), the resulting Morse curve for E3(R) is 280 cm-1 deeper but has the same equilibrium location and curvature as those given above within the uncertainty limits. 2. Comparison with Other Studies. a. The A 0+ Potential. The first absorption band of NaI arising from the Ω ) 0+, 1 r X 1Σ+ 0 transitions exhibits a complex fragmentary band structure that extends across the range λ ) 295-390 nm.29 Under high resolution, the spectrum is seen to consist of alternating discrete and diffuse regions due to rapid electronic predissociation of the A 0+ state.12,13,42,53,54 The diffuse segments arise from transitions to short-lived levels of the A 0+ state that are strongly coupled to the dissociation continuum of the X 1Σ+ 0 state.12,13,42,43,54 The discrete spectrum, in contrast, is associated with the sharpest rotational energy levels and has been assigned and interpreted by Tiemann and co-workers12-14 using an early version of Child’s semiclassical curve-crossing theory for predissociation line widths55 in conjunction with the assumption of intermediate coupling between rotation-vibration levels + constructed from diabatic (X 1Σ+ 0 ) and adiabatic (A 0 ) poten12,13 The resulting line assignments were than fitted to a tials. Dunham series expansion from which RKR turning point data + for both the X 1Σ+ 0 and A 0 states were computed for energies up to 34 450 cm-1 above the ground-state minimum (about 9040 cm-1 above the atomic asymptote). These spectroscopic data12,13 have been subject to reanalysis by Wang et al.,47 who cast potential functions for the same states in analytical form
7838 J. Phys. Chem., Vol. 100, No. 19, 1996 using a numerical optimization procedure, and by Chapman and Child,48 who derived hybrid-RKR potentials for use in conjunction with a semiclassical calculation of the wave packet recurrence observed9 at long times. An important limitation associated with conventional laser absorption spectroscopy of NaI12,13 is that the interesting region near the A 0!+ state minimum, where non-Born-Oppenheimer perturbations of the vibration-rotation manifold are greatest, cannot be accessed directly due to poor Franck-Condon overlap with the ground-state population. In an attempt to overcome this problem, Bluhm and Tiemann have applied the technique of ultraviolet-optical double-resonance spectroscopy to excite individual levels in the range 1600-2700 cm-1 above the neutral asymptote,43 with a discrete vibration-rotation level of the higher-lying C 0+ electronic state constituting the intermediate level. Unfortunately, the RKR potential curve derived from a combination of these data43 and the laser absorption measurements previously published by the same group12,13 is somewhat different than the earlier result obtained by ultraviolet laser spectroscopy alone.13 An early attempt to construct the form of the adiabatic potential was also made by Oppenheimer and Berry,30 who employed the results of their matrix-isolation absorption experiments in conjunction with the low-resolution gas-phase spectrum recorded by Davidovits and Brodhead29 for the repulsive branch. Approximate potential energy curves for the A 0+ state derived from the fluorescence excitation and emission spectra of supersonically cooled NaI in the region 318 e λ e 393 nm have been determined by Ragone et al.42 Spectroscopic studies of NaI carried out before 1979 have been discussed by Berry.35 Relative differential cross sections for Na + I f Na+ + Ichemiionization obtained from atomic beam scattering experiments have permitted determination of the matrix element V12 for coupling between the adiabatic curves at the avoided crossing14,15 and a parametrized expression for the covalent potential that includes a C6/R6 attractive term.15 Though as pointed out by Anderson et al.,34 such a potential represents the weighted average of the five multiplet levels that correlate with neutral Na + I atoms, since all available covalent states contribute to the measured differential reaction cross section in molecular beam scattering experiments of this type.14,15 Photofragment spectroscopy has been applied to the dissociation of NaI at various photolysis wavelengths (λ ≈ 266,44 300337,45 347.1,34 and 355 nm46) in studies designed to characterize the symmetries, energetic ordering, and transition probabilities to the Ω ) 0+ and 1 levels of the first excited-state manifold from measurements of the angular anisotropy of photofragment recoil. This technique permits straightforward determination of the degree of parallel and perpendicular character of the optically allowed transition connecting the ground and two excited states. By application of the Condon reflection method to determine the fraction of parallel and perpendicular transitions contributing to the total absorption cross section, van Veen et al. have utilized such photofragment data to characterize the repulsive limbs of the Ω ) 0+ and 1 covalent potentials in the Franck-Condon region in terms of simple exponential expressions.45 At the present time, however, it appears that a degree of ambiguity remains concerning the exact ordering of the Ω ) 0+ and 1 levels, the variation of their relative positions with internuclear distance, and the absolute value of the transition probabilities.46 Ab initio CI calculations have yielded potential energy curves for the ground and first excited states over a wide range of internuclear distances together with the off-diagonal matrix coupling elements between them in the region of the avoided
Cong et al.
Figure 6. Comparison potential energy curves for NaI(A 0+) determined by time-independent and time-integrated spectroscopies: (s) FTS (this work); (‚‚‚) Condon reflection analysis of photofragment spectra (ref 45); (- ‚ -) RKR analysis of ultraviolet absorption spectrum (ref 13); (- - -) numerical optimization of ultraviolet absorption spectrum (ref 47); (---) RKR analysis of UV-ODR spectrum (ref 43).
crossing at Rx.16,17 Spectroscopic constants (ωe, ke, Be, and Re) for the X 1Σ+ 0 state derived from the calculated potential by Sakai et al.17 were found to be in close accord with experimental measurements; for the A 0+ state, the corresponding parameters agree qualitatively with the earlier spectroscopic investigation of Tiemann and co-workers12 but, interestingly enough, not so well with the later experimental values.13 The splitting ∆E12(Rx) between ab initio adiabatic curves at Rx16,17 is in good agreement with the result obtained from a semiempirical calculation by Grice and Herschbach18 and the value derived from chemiionization measurements.14,15 Parametrized expressions for the ground- and excited-state diabatic potential curves have been given by numerous authors and have been widely used in a variety of time-dependent theoretical calculations; typically, these stem from the Rittner and Lennard-Jones potentials describing the ionic and covalent interactions, with a modification to the ionic curve first proposed by Faist and Levine32 that results in more desirable repulsive characteristics at short internuclear distances. The many analytical expressions for the shape of the ionic ground-state potential employed before 1972 have been reviewed by Patel and Gohel,49 while a more recent discussion by Kumar and Shanker is slanted toward the calculation of electronic polarizabilities and spectroscopic constants.50 As shown in Figure 6, the overall agreement between the various potential functions is quite good, except that the repulsive wall of E2(R) derived from UV-ODR spectroscopy43 is less steep and shifted to longer intermediate separations (giving rise to a sharper minimum) than those determined by FTS, photofragmentation recoil distributions,45 and a numerical analysis47 of earlier laser absorption data.13 A rigorous test of the adiabatic potential derived by FTS will be to calculate the frequency-resolved spectrum of rotation-vibration transitions using the result for E2(R) given above and to compare this with the experimental spectrum reported by Tiemann and coworkers.12,13 The important point, however, is the general agreement confirming the validity of the approach and the accuracy of the clocking. b. The Higher-Lying Potential. Polanyi and co-workers33 were the first to make a positive identification of a bound molecular state correlating with electronically excited-state
NaI Dissociation Reaction
Figure 7. Comparison of the potential energy curve E3(R) at energies E ≈ 40-44 cm-1 determined by FTS with the potentials for the NaI(C 0+) derived from time-integrated spectroscopy: (s) FTS (this work); (- ‚ -) far-wing emission (ref 33); (---) laser excitation spectroscopy (ref 50).
Na(2PJ) and ground-state I(2P3/2) atoms. They established the nature of the state from a spectral analysis of the ultraviolet and visible emission extending from λ ≈ 237 to 695 nm following broad-band excitation of jet-cooled NaI at λ ) 248.5 nm. The intense fluorescence monitored by these authors was attributed to bound f bound and bound f free transitions of NaI, which swamped the weak free f bound emission from the [Na‚‚‚1]‡* transition states of reaction 1 observed by the same group in separate experiments at shorter photolysis wavelengths.33,56,57 Bluhm et al. have subsequently established the depth of the potential minimum of a state labeled C 0+ leading to the same covalent asymptote by laser excitation spectroscopy across the wavelength range 239 e λ e 256 nm.43,58 Prior to these two studies, the existence of a state with a shallow potential energy well in the same energy range had only been tentatively inferred from absorption cross section measurements29 and discussed in relation to an early fluorescence quenching study.59 A repulsive form for a molecular state derived from Na(2PJ) + I(2P3/2) atoms has also been postulated by Herm and co-workers60 and by Barker and Weston61 in order to fit fluorescence efficiency spectra. Polanyi and co-workers33 have recorded the emission spectrum of NaI during dissociation following broad-band excitation to a high-lying state, believed to be C 0+.58 By analyzing the portion of the emission originating from bound states on the upper potential,33 they derived parameters for a Morse potential with a dissociation energy De′′′ ) 530 cm-1. A laser excitation investigation of the C 0+ potential in the frequency domain has recently been reported by Tiemann and co-workers.58 From an analysis of some 27 vibronic transitions connecting + the X 1Σ+ 0 state with levels V′ ) 0-15 of the C 0 state, an RKR potential was constructed with a well depth of De′′′ ) 1422.6 ( 8.6 cm-1 and an equilibrium bond distance of Re′′′ ) 3.306 73 ( 0.000 54 Å.58 Figure 7 offers a diagrammatic comparison of the potential energy curve for the fluorescing state determined by FTS with those for the C 0+ state derived from spectroscopic studies in the frequency domain.33,58 The value of De′′′ ) 1863 cm-1 reported here quantitatively accounts for the splitting times obtained at different probe wavelengths as a function of photolysis energy. Owing to the close proximity of the highest-lying potential probed by FTS to those in the same energy range derived from time-integrated spectroscopy, it is tempting to assume that the
J. Phys. Chem., Vol. 100, No. 19, 1996 7839 Morse curve for E3(R) represents the potential for the C 0+ state. There are, however, a number of caveats to bear in mind: first, that of the 15 molecular levels that correlate with Na(2PJ) + I(2P3/2) atoms in Hund’s case c coupling scheme, five with Ω ) 1 symmetry and three with Ω ) 0+ can be accessed optically from the lower A 0+ electronic state via dipole-allowed transitions. At excitation energies of (40-42) × 103 cm-1 above the X 1Σ+ 0 minimum, these multiplets are likely to be very closely spaced, with avoided crossings between levels of the same symmetry type: it is not unreasonable to suspect that more than one such level may lie within the bandwidth of the ultrashort probe pulse (∆ν2 ≈ 60 cm-1), especially since both spin-orbit levels of the 2PJ term of Na (∆E ) 17 cm-1) are excited at λ∞2 ) 589 nm via the on-resonance transition. Furthermore, owing to the considerable variation in electronic charge distribution in NaI(A 0+) at different distances within the adiabatic well, the electronic transition moment µ32 ) 〈3|µ|2〉 connecting E3(R) and E2(R) may vary significantly with internuclear distance, especially on either side of the avoided crossing between E2(R) and E1(R). Consequently, transitions to different upper multiplet levels may be favored at different probe wavelengths corresponding to [Na(2PJ)‚‚‚I]‡* r [Na‚‚‚I]‡* excitation at R < Rx and [Na(2PJ)‚‚‚I]‡* r [Na+‚‚‚I-]‡* charge transfer at R g Rx. Here, the use of a polarized probe laser beam would be of invaluable assistance in verifying the symmetry of the upper state accessed at different interfragment separations; in the absence of theoretical calculations of the highlying states, it is difficult to predict a priori whether parallel or perpendicular femtosecond excitation will predominate as the NaI selection rule ∆Ω ) 0, (1 becomes ∆J ) 0, (1 for the free Na atom. Hence, it may be concluded that the possibility exists that more than one electronic state is populated by the ultrashort probe laser and/or that different electronic states are reached at different probe wavelengths. Accordingly, the curve E3(R) shown in Figure 7 may represent the composite force field of two or more electronic states, rather than the potential of a single adiabatic state. IV. Theory: Classical, Semiclassical, and Quantal Treatments Before proceeding to a more formal consideration of the temporal behavior of NaI dissociation in terms of the quantum dynamics of wave packet motion in the A 0+ state, we first present an intuitive semiclassical interpretation of the ultrafast vibrational dynamics. Some 35 years ago, Berry62 provided a lucid qualitative picture of the vibrational motion of NaI and KI in their lowest-lying electronically excited states, supported by rigorous quantum-mechanical calculations of the coupling between ionic and covalent potentials. This adiabatic description was enunciated in terms of the degree of ionic/covalent character of the oscillating wave function at different positions within the potential well E2(R) at either side of the diabatic crossing point Rx.62 Where appropriate, the account of the timedependent dynamics given below is related to the nature of the electronic wave function at different distances along the vibrational coordinate, following Berry.62 It turns out that a simple classical description of the nuclear motion provides an insightful physical picture by which the experimental results depicted in Figure 3 can be readily understood. A. Semiclassical Description of Wave Packet Motion. It is assumed that absorption of pump laser light always promotes the system from the ground state via a resonant vertical transition to the inner turning point Rin of the adiabatic potential E2(R), as illustrated in Figure 1; i.e., there is no opportunity for tunneling of Ψ2(R;t) through the classically forbidden region
7840 J. Phys. Chem., Vol. 100, No. 19, 1996
Cong et al.
inside the repulsive potential wall. As pointed out by Berry,62 such a transition results in a rearrangement of the electronic charge distribution along the Na-I axis from the halide to the metal ion to give a bond that is predominantly covalent in nature. The wave packet Ψ2(R;t) formed by coherent excitation begins to propagate from the Franck-Condon region toward larger distances, subsequently reaching the center of the OCR projected onto E2(R) by the probe pulse at a time τ0 later. At the distance R2 (where R2 < Rx < Rout), the probe laser is assumed to be in resonance with the energy separation between E2(R) and the higher-lying potential E3(R), resulting in the appearance of the first fluorescence peak in the FTS transient. This peak also includes a component arising from prompt dissociation6 over the repulsive potential of the Ω ) 1 covalent state, which is also populated by a dipole-allowed transition from the ground state. As time proceeds, Ψ2(R;t) continues to evolve over E2(R) toward longer interatomic separations. Approximately 11%4-6 of the original electronic population in the A 0+ state escapes to form Na + I at the avoided crossing point Rx as Ψ2(R;t) samples the interaction region between E1(R) and E2(R). At distances beyond Rx, the wave function takes on a more ionic character as electrostatic forces build, causing a retardation of the nuclear motion and a contraction in size of Ψ2(R;t). At the outer turning point Rout, Ψ2(R;t) is reflected back to shorter separations from the attractive branch of the potential and becomes more covalent again at R < Rx.63 En route to the inner turning point, the wave packet reenters the OCR, giving rise to the leading (earlier) maximum of the doublet structure that follows the initial peak. The later-time maximum of the doublet arises from propagation toward longer distances as it rebounds from the inner repulsive wall of E2(R) and undertakes a second round trip within the adiabatic well. The wave function continues to oscillate between Rin and Rout until depletion of the electronic population initially promoted to E2(R) is complete as a result of repeated passage across the conical intersection with E1(R). If the excess energy Eav is large, then the group velocity of the wave packet is sufficiently great so as to render the spacing between the two maxima of the doublet unresolvable with probe pulse widths on the order of 100 fs. At longer pump wavelengths, where the group velocity is lower, the peak-to-peak spacing is large enough to result in splitting of the second and subsequent fluorescence maxima. Inspection of Figure 3a shows that the splitting time τs increases with increasing values of λ1; this result indicates that the covalent region of E2(R) is highly repulsive at short distances, such that τs is dominated by velocity effects rather than the round-trip distance traveled by Ψ2(R;t) within a single oscillation period. The observed decrease in the oscillation period τosc upon lowering Eav, also discernible in Figure 3a, simply reflects the shorter distance between classical turning points as the minimum of the potential well is approached. By the same reasoning, tuning the probe laser to shorter wavelengths is equivalent to shifting the OCR to larger internuclear separations, thereby creating sufficient spatial (and hence temporal) spacing between the center of the OCR and Rin to distinguish the direction of the wave packet propagation. Hence, it follows that the splitting time τs between the doublet maxima constitutes a direct measurement of the round-trip time for Ψ2(R;t) to propagate from the OCR on E2(R) to the inner turning point and return to the OCR again, i.e.
τs ) 2τ0
(10)
B. Quantum-Classical Correspondence. In light of the above discussion, it might reasonably be inquired if a classical
treatment can quantitatiVely describe the early-time motion in NaI(A 0+), even for wave packets of nonnegligible width, in addition to providing a simple qualitative picture of the ultrafast dynamics. The question is of worthy significance because the method of inverting FTS data to E2(R) rests upon a classical trajectory analysis of the vibrational evolution. According to Ehrenfest’s principle, a classical approach will furnish a quantitatively accurate description of the nuclear motion during the half-collision only if the following two conditions are satisfied:64 (1) the mean values of physical quantities such as average position and momentum follow closely the classical laws of motion, and (2) the size of the wave pocket representing the nonstationary state is small compared to the characteristic dimensions of the problem and remains so throughout the time interval of the evolution (e3 ps in the present case). The force acting on the wave function Ψ2(R;t) temporally located at an average coordinate 〈R〉(t) on E2(R) is given by
(
µ
∫dR Ψ2*(R;t)[dE2[R(t)]/dR(t)]Ψ2(R;t) d〈R〉(t) 2 )dt ∫dR Ψ2*(R;t) Ψ2(R;t) (11)
)
where the range of integration extends either side of 〈R〉(t) to distances where Ψ2(R;t) ≈ 0, and the normalization term ∫dR Ψ2*(R;t) Ψ2(R;t) is introduced to take care of the timedependent decay of the bound population in the A 0+ state by curve crossing to X 1Σ+ 0 . Equation 11 states that the temporal evolution of Ψ2(R;t) over E2(R) will obey the classical laws of motion when the spatial distribution of |Ψ2(R,t)|2 about 〈R〉(t) is sufficiently narrow that the right-hand side can be replaced by dE2[R(t)]/dR(t), yielding Newton’s second law. It follows from this that the spread of Ψ2(R;t) located at 〈R〉(t) must be smaller than the characteristic length L[R(t)] ) |E2[R(t)]/ {dE2[R(t)]/dR(t)}| over which E2(R) changes at that distance.64,65 A convenient measure of the size of the wave function is the fwhm ∆R(Ψ2), so that we may express this requirement as
∆R(Ψ2) , L[R(t)]
(12)
When this condition is met, the mean values of the position and momentum of Ψ2(R;t) during the course of the dissociation will closely follow the classical prediction.66 To test the criterion expressed by eq 12 during the earlytime vibrations of NaI(A 0+), both the quantum and classical evolutions have been calculated for an excitation wavelength of λ1 ) 350 nm (Eav ∼ 3165 cm-1).2,7 Using model diabatic potentials to construct E2(R),32,45 it was found2 that the classical trajectory tracks the wave packet motion in the A 0+ state almost exactly, except in the region close to Rin where it penetrates to shorter distances by some 0.2 Å. The effect is illustrated in Figure 8, which shows the temporal evolution of the mean position 〈R〉(t) of Ψ2(R;t) and the classical trajectory R(t) on E2(R) at times up to 1.2 ps. 〈R〉(t) is obtained from the standard expression
〈R〉(t) )
∫dR Ψ2*(R;t) RΨ2(R;t) ∫dR Ψ2*(R;t) Ψ2(R;t)
(13)
and R(t) from
R(t) ) R0 + ∫0 2/µxEav - E2[R(t′)] dt′ t
(14)
with the initial conditions p(t)0) ) -µ(dR/dt) ) 0 and R0 (≡r(t)0)) ) Rin as before. That the classical trajectory samples
NaI Dissociation Reaction
J. Phys. Chem., Vol. 100, No. 19, 1996 7841
Figure 8. (a) Schematic representation of the wave packet trajectory within the adiabatic potential well. (b) Comparison of the expectation value 〈R〉(t) of the quantum state |Ψ2(t)〉 with the classical trajectory R(t) on E2(R) during the first oscillation period. E2(R) is constructed from the modified Rittner potential for V2(R)32 and the exponential form for V1(R) derived from photofragment spectroscopy.45 The initial coordinate r(t)0) ≡ Rin ) 2.8 Å of the classical particle corresponds to a total energy of Eav ) 3165 cm-1 for product recoil; in the quantum picture, this is equivalent to preparation of |Ψ2(t)〉 by a pump pulse of nominal wavelength λ1 ) 350 nm, in this case with a Gaussian temporal envelope of fwhm ∆t1 ) 100 fs centered at t01 ) 0 fs.
slightly shorter bond lengths than Ψ2(R;t) at Rin simply reflects the fact that the center of a wave packet of finite dimensions turns around before it reaches the classical turning point.67,68 Although the classical trajectory undertakes a longer journey within the adiabatic well, it also requires a shorter time to reverse direction than does the wave packet,68 such that agreement between values of their mean positions is excellent at all other times up to 1.2 ps. The wave packet calculations below, like those of Engel and Metiu,65 show that, for NaI excited by a laser pulse of 85 fs duration, Ψ2(R;t) can extend over an appreciable range of coordinate space within the A 0+ potential well, even during the first oscillation period: at Eav ≈ 5850 cm-1 for example, 〈R〉(t) ) 7.7 and 5.0 Å and ∆R(Ψ2) ) 2.1 and 2.9 Å after t ) 200 and 1000 fs, respectively, compared to the width ∆R(E2(R)) ) Rout-Rin ≈ 9 Å of the potential well at this energy. It has already been noted that, during the first few oscillations, the Coulombic force field tends to counteract the dephasing effect of potential anharmonicity whenever Ψ2(R;t) reaches the ionic region of E2(R);65 at these distances L[R(t)] ≈ R(t) > Rx, and eq 12 is clearly satisfied. However, in the region of E2(R) well between the classical turning points (and where Ψ2(R;t) is broadest), dE2[R(t)]/dR(t) ≈ 0 such that L[R(t)] ≈ ∞ and eq 12 is still satisfied. It is only when Ψ2(R;t) impinges on the repulsive wall of the covalent potential that eq 12 is violated, since ∆R(Ψ2) is large close to Rin, and for the modified LennardJones potential32 at R g 2 Å, L[R(t)] ≈ 0.435 Å is comparatively small. During the first oscillation period, Ψ2(R;t) thus spends most of its time in those regions of E2(R) where L[R(t)] . ∆R(Ψ2). This pattern of wave function “breathing ” characterized by a frequency 2νosc continues throughout the first 10 or so vibrations until destructive interference between the phases of the long-lived rotation-vibration levels present within Ψ2(R;t) curtails the coherent oscillatory behavior. The calculations of Engel and Metiu indicate that, for times up to t e 2 ps (approximately two complete vibrations), Ψ2(R;t) is broad only at those intermolecular distances where the potential energy is essentially constant and narrow only where the gradient of the potential is large.32 For wave packets excited by laser pulses with widths 50 e ∆t1 e 100 fs, eq 12 is satisfied for t e 10 ps.
This result is a direct consequence of the unusual shape of the adiabatic potential, with its dominant Coulombic interaction near the outer turning point, square-well characteristics in the middle region, and an abrupt, almost vertical repulsive potential at short distances. It may be concluded that the classical approximation provides an adequate quantitative description of the transition-state dynamics within the first 3 ps following excitation by an ultrashort light pulse. Indeed, several classical analyses of the temporal evolution of NaI dissociation over times up to 10 ps2,6,69 have succeeded in modeling the vibrational dynamics monitored by FTS with quantitative accuracy. Finally in this section, we consider briefly the dephasing of Ψ2(R;t) as a function of time,7,8 a quantum phenomenon that cannot be accounted for by classical mechanics. Cong et al. have previously shown that the short-time decay of wave packet oscillations within the potential well exhibits a strong nonMarkovian time dependence, resulting from both vibrational dephasing of Ψ2(R;t) and the nonadiabatic electronic population dynamics.7 Initial loss of vibrational coherence was found to occur over a time scale of about 10 ps following ultrafast excitation7,8 and arises from dispersion of Ψ2(R;t) due to the local anharmonicity of the E2(R) potential.70 It was further shown that the time dependence of wave packet spreading within the adiabatic well is identical to the short-time population buildup of product Na atoms and takes place over a longer period than vibrational dephasing.7,8 The spreading thus reflects the dynamics of the adiabatic population arising from nonBorn-Oppenheimer coupling between E2(R) and E1(R). An analogy with the characteristic times T2 for dephasing and T1 for level depletion from the quantum theory of relaxation is self-evident in this context. Each rotation-vibration level excited by the pump laser has a finite lifetime τdiss(V′,J′), ranging from below 1 ps to more than 500 ps, that is determined by the strength of non-BornOppenheimer coupling between V1(R) and V2(R) and the degree of resonance between covalent and ionic levels. After a time τD ≈ 8-10 ps, dephasing of the coherent vibrations of Ψ2(R;t) appears to be complete. However, although a substantial fraction of the excited-state population created on E2(R) decays out of rotation-vibration levels with τdiss(V′,J′) < 40 ps, longer-
7842 J. Phys. Chem., Vol. 100, No. 19, 1996 lived levels supported by the adiabatic potential dephase and rephase over a longer time scale and give rise to a recurrence in the regular oscillatory pattern of the fluorescence signal some 15-20 ps later, depending upon Eav.7,8 Both a semiclassical calculation of the predissociation line widths48 and an exact quantum dynamical treatment71 show that the recurrence signal is determined by constructive interference between groups of long-lived eigenstates contained within the initial superposition state, which brings about a resumption of coherent vibrational motion at τD ≈ 25-30 ps. The exact structure and rephasing time of the wave packet recurrence have been found to depend critically upon the anharmonicity of E2(R) and its coupling to the ground-state potential, as well as the wavelength and temporal duration of the pump pulse.48,71 C. Quantum Dynamical Treatment of Wave Packet Motion. The simulation of FTS transients by an exact quantum dynamics treatment provides a rigorous examination of the procedure applied to elucidate E2(R) and E3(R) by FTS within the limitations of the theory. In this section, we first summarize the principal findings of previous wave packet calculations of the predissociation of NaI,65,72-76 emphasizing the potential curves used and the degree of agreement with FTS results.2,4-8 We then present our own theoretical calculations describing the quantal evolution over early times and highlight the quantitative agreement between the computed wave packet evolution and the forms for E2(R) and E3(R) derived by FTS. 1. PreVious Quantum Dynamical Studies. Recent years have witnessed a large number of quantum dynamical investigations of photodissociative and reactive scattering processes that aim to solve the time-dependent Schro¨dinger equation for suitably chosen initial wave packets (see, e.g., ref 77). The appeal of the time-dependent approach lies primarily in the physical insight provided by a wave packet picture of the nuclear dynamics, which often corresponds to intuitive ideas based on classical mechanics. In addition, its relative computational simplicity offers advantages in comparison to coupled-channel calculations of collisional phenomena which entail determination of fully converged numerical solutions to the time-independent Schro¨dinger equation. The ease of implementation of timedependent methods stems from the fact that they are initial value problems and are thus able to treat the nuclear dynamics of complex rearrangement reactions, as well as allowing a facile description of continuum states. In contrast, the coupled-channel method has mainly been applied to the H + H2 reaction and its isotopic analogues, though other small systems have also been investigated.78 For problems in which the total angular momentum is nonzero, or that evolve over multiple potential energy surfaces, the necessary computational effort rapidly becomes prohibitive due to the large number of rotational and vibrational channels required at each energy. For this reason, approximate approaches to the full coupled-channel method have been developed, such as the infinite-order sudden approximation79 and the T-matrix method.80 Heller was the first to appreciate fully the insight provided by a time-dependent description of molecular spectroscopy and dynamics and pioneered the development and application of semiclassical methods for propagation of Gaussian wave packets over excited-state potential surfaces.81 In quantum-mechanical propagation schemes, the state vector |Ψ(t)〉 ahd Hamiltonian operation H|Ψ(t)〉 are represented in a common discretized Hilbert space. The major breakthrough in the application of such methods was the introduction of the Fourier or pseudospectral technique by Kosloff and Kosloff in 1983,82 in which the potential energy operation V|Ψ(t)〉 is calculated locally in a grid representation of coordinate space and the kinetic energy
Cong et al. operation (p2/2µ)|Ψ(t)〉 is calculated locally in momentum space. The approach is both accurate and computationally efficient, due to the use of the fast Fourier transform (FFT) algorithm for transformation from the discrete coordinate representation to the discrete momentum representation, and Vice Versa. The recent realization83 of time-dependent calculations in the interaction representation (IR) has enabled the major kinematic obstacles associated with integration of the time-dependent Schro¨dinger equation to be overcome: because of the localized nature of the IR wave function, a much smaller number of grid points on the potential energy surface of interest are required, and because the wave function is essentially independent of energy and varies only slowly with time, a larger temporal increment may be applied in the numerical integration procedure. This formulation of the time-dependent approach should permit treatment of larger scale problems than have been attempted hitherto over longer time periods and over a wide range of initial energies.83 Several quantum dynamics calculations of the short-time behavior of NaI predissociation up to about 10 ps have been presented recently. Engel et al. were the first to report a wave packet calculation of the dynamics, using first-order perturbation theory to determine the time-dependent population created on E2(R) by the pump laser.65,72 In this work,65 the exponential form for V1(R) due to van Veen et al.45 and the Rittner model for V2(R) proposed by Faist and Levine32 were invoked to describe the diabatic covalent and ionic potentials, respectively. It was assumed65,72 that the off-resonance LIF signal recorded in FTS experiments corresponded directly to the population of [Na‚‚‚I]‡* at R < Rx, while the population on E1(R) at R2 g Rx resulting from adiabatic curve crossing was considered to contribute solely to on-resonance detection of free Na atoms. The time-dependent populations obtained by these workers compare well with the experimental results, though some differences were noted: inaccuracies in the computed product population of Na atoms at times less than 50 fs and in the relative intensities of the first peaks of the off-resonance spectrum were postulated to arise from neglect of the dissociative Ω ) 1 excited state; the broader peak widths of the experimental signals compared to the calculated time-resolved spectra were attributed to factors such as the omission of rotational degrees of freedom and the assumption of zero probepulse absorption at distances beyond Rx.65 A careful examination of the influence on the calculated populations of the pump laser pulse width and shape, the initial vibrational level of the ground electronic state, the magnitude of the coupling matrix element V12(R), and the temperature dependence of the transient behavior was also presented.65 Calculations of nonadiabatic transitions between V1(R) and V2(R) revealed, not surprisingly, that the semiclassical Landau-Zener treatment failed to predict the correct escape probability with increasing number of passes through the avoided crossing due to neglect of dephasing of the vibrating wave packet, especially at higher recoil velocities. A similar effect was found for short pump pulse widths on the order of 10 fs, since a spatially narrow wave packet spends less time in the crossing region.65 In an extension of their previous work, Engel and Metiu subsequently treated the two-photon excitation of NaI in the time domain by means of second-order perturbation theory.72,73 To calculate the wave function on the higher-lying potential E3(R) as a function of pump-probe delay time, these workers chose the Morse form for E3(R) determined by Bower et al.33 by time-integrated emission spectroscopy, though it was realized that this potential may not be that reached by the probe pulse in FTS experiments.72 Again, reasonably good agreement
NaI Dissociation Reaction between calculated and experimental time-resolved spectra was obtained, though some discrepancies between calculated and observed peak widths were noted.72,73 Perhaps the most striking feature of the real-time spectra calculated by Engel and Metiu using second-order perturbation theory was the existence of doublet maxima in the fluorescence peaks, including the first, at excitation wavelengths of λ1 ) 300, 310, and 328 nm.72,73 At lower photolysis energies, the calculated doublet structure became increasingly pronounced and was present in a greater number of individual fluorescence peaks stretching to longer times (τD < 6 ps). For the particular value of λ1 ) 310 nm, the doublets were observed to disappear at probe wavelengths further removed to the red from onresonance detection of Na atoms at λ2 ) 589 nm. The number of fluorescence peaks in a given transient exhibiting doublet maxima was found to increase for longer pulse widths of the pump laser, but not for the probe.72,73 It was argued by Engel and Metiu that the doublet maxima appeared only for those delay times at which the center of the wave packet created by the pump pulse was located at the inner turning point Rin on the neutral part of E1(R). The explanation put forward by these authors invoked the concept of a wave train:72,73 at the covalent turning point Rin the wave train created by the pump laser was considered to consist of one portion with its wave vector directed toward the Franck-Condon region and another component, recently reflected from the repulsive wall, in transit to longer distances. Absorption of a probe photon by both inward and outward traveling wave trains then admitted the possibility of a reduction in absorption probability, and hence the existence of a minimum in the anticipated fluorescence peak, due to destructive interference between the two wave trains. Choi and Light have also presented quantum dynamical calculations of NaI predissociation in real time.74 These authors employed the Gauss-Chebyshev discrete variable representation to obtain solutions to the time-dependent Schro¨dinger equation as the amplitudes of the time-evolving wave function |Ψ1(t)〉 on V1(R) and V2(R) at a defined set of coordinate points. Reasonable agreement was found between the calculated absorption spectrum,74 obtained as the discrete FFT of the autocorrelation function 〈Ψ1(t)0)|Ψ1(t)〉, and the experimental spectra determined by ultraviolet laser excitation spectroscopy.12,13 The nature of the potential energy surfaces proposed by earlier workers12,13,32,45 was examined in detail, resulting in modification of the diabatic potentials V1(R) and V2(R).74 An analysis was also carried out of the diabatic curve crossing probability for different values of the coupling matrix element V12(R): for a pump laser pulse at λ1 ) 320 nm that was assumed to be a δ-function of time, a crossing probability of 0.094 was obtained with V12(Rx) ) 350 cm-1, in good agreement with the experimental result PLZ ) 0.11 and value of V12(Rx) derived by Rose et al.4,6 from a Landau-Zener analysis. A novel description of the curve-crossing dynamics through the vicinity of the avoided crossing has been given by Kono and Fujimura.75,84 These workers took into account motion within the ionic well of E1(R) of the wave packet |Ψ1(t)〉 that is formed on each occasion that the wave packet |Ψ2(t)〉 initially prepared on E2(R) samples the conical intersection during the inward-bound phase of a given oscillation cycle. The quantumdynamical picture presented by these workers predicts that formation of Na atoms during the outward-bound phase of the second and subsequent oscillation cycles comprises two distinct product “waves”:84 the first due to adiabatic passage of Ψ2(R;t) over E1(R) and the second resulting from the diabatic behavior of Ψ2(R;t) as it traverses the avoided crossing between E2(R) and E1(R). From the phase difference between the two
J. Phys. Chem., Vol. 100, No. 19, 1996 7843 dissociation waves at the crossing point Rx, estimated using the JWKB approximation, the dependence of dissociation probability on the energy and pulse width of the pump laser was rationalized in terms of constructive and destructive interference between the two waves leading to Na + I products.84 These same authors have proposed a method of tracking the wave packet evolution over E2(R) induced by a femtosecond pulse by monitoring the instantaneous emission intensity from E3(R) excited by a monochromatic continuous wave probe beam.75 Lin et al. have applied generalized linear response theory to compute transition-state spectra for ultrafast processes in the gas and condensed phases, including an analysis of NaI predissociation.10,76 In comparison to calculations of the realtime behavior based on perturbation approaches that invoke pure states, this method offers the advantage that both the electronic population dynamics and coherence are readily obtained from the appropriate density matrix for the molecule subjected to the electric field of the pump laser. Starting from the quantum Liouville equation for the appropriate density matrix in the IR, these authors10,76 have derived an expression for the time- and frequency-dependent linear susceptibility of the nonstationary system that contains two terms: one due to the time dependence of the electronic coherence induced by the pump laser and a band shape function determined by the characteristics of the probing pulse. Application of the formalism to reaction 1 results in a temporal variation of the susceptibility that closely mimics the real-time signal monitored by FTS.10,76 2. Quantum-Dynamical WaVe Packet Calculations of FTS Transients. To simulate FTS probing of the oscillatory time dependence of the state vector |Ψ2(t)〉 subject to the force field of E2(R), we have adopted a time-dependent perturbation treatment to calculate the interaction between the two ultrashort laser pulses and the NaI molecule. The calculations reported here are focused on the resonant photofragment trapping within the A 0+ state adiabatic well, in particular the direction of motion of |Ψ2(t)〉 at early times. The time evolution of |Ψ2(t)〉 is monitored by detection of fluorescence from E3(R); the total fluorescence is proportional to the probability density 〈Ψ3(t)|Ψ3(t)〉 for finding the system on E3(R) at a time t immediately following the probe pulse, as the fluorescence detection in timeintegrated, intensity modulations depends only upon the pumpprobe time delay τD rather than any coherence effects on E3(R). The computed transients consist of plots of 〈Ψ3(t)|Ψ3(t)〉 versus τD. The absolute magnitude of the LIF intensities depends on a number of factors, namely, the topologies and relative locations of the three potentials in configuration space (reflected by different Franck-Condon factors), the transition dipole moments that optically couple the different electronic states, the probability for undergoing an avoided crossing from E2(R) to E1(R), and the laser line shapes. The accurate computation of fluorescence intensities is not of primary concern here; rather, our goal is to calculate the periodic modulations in the observed FTS spectra which arise solely from the oscillatory motion of |Ψ2(t)〉. a. Methodology. In this study, two types of calculations have been performed, depending upon the pump wavelength. At λ1 ) 320 nm, Franck-Condon arguments together with a Boltzmann distribution of the vibrational population in the X 1Σ+ 0 state indicate that the lowest vibrational level carries most intensity for absorption of a pump photon. As the case of temporally overlapping pump and probe pulses needs to be treated at short times, the coupling of the electric field to the molecule is calculated up to second order. For weak transformlimited laser pulses, the state |Ψ3(t)〉 created on the final (fluorescent) potential E3(R) is given by the well-known
7844 J. Phys. Chem., Vol. 100, No. 19, 1996
Cong et al.
perturbation theory expression85
|Ψ3(t)〉 ) (-i/h)2∫-∞dt2∫-∞dt1 T3(t-t2) U32(t2) T2(t2-t1) × t
t2
U21(t1) exp(-iωV′′t1)|φV′′〉|1〉 (15)
where the electric dipole approximation is assumed for interaction of the molecule with the two radiation fields. The initial state |Ψ1(t)〉 ) exp(-iωV′′t1)|φV′′〉|1〉 is the Gaussian form of the V′′ ) 0 level in the ground electronic state |1〉 with a nuclear eigenfunction |φV′′〉 and associated phase factor exp(-iωV′′t1). The time-dependent perturbations due to the laser pulses are represented by the Uij(tk) given by
Uij(tk) ) µij(R)‚Ek(tk) gk(tk)
2
2
∑ |n〉[Kδnm + Vnm] < m|
(17)
n)1 m)1
where K is the KE operator, V11 and V22 are the potential curves denoted V1(R) and V2(R) in Figure 1, and V12 ) V21 are the nonadiabatic coupling matrix elements. At λ1 ) 320 nm, the pump pulse prepares a wave packet |Ψ2(t)〉 that is initially localized at the classical turning point of E2(R), with no tunneling through classically forbidden regions of configuration space. The higher-lying potential E3(R) is populated by the probe pulse perturbation U32(t2) which acts at the time t2 and the state vector |Ψ3(t)〉 propagated by T3(t-t2) between the times t2 and t. Ek(tk) is the electric field strength of the kth laser pulse of frequency ωk which we write as
Ek(tk) ) E0k exp(iωktk)
(18)
the temporal line shape gk(tk) of which is taken to be the Gaussian form
gk(tk) ) exp[-(tk - t0k )2/(σk)2]
|Ψ3(t)〉 ) (i/h)∫0 dt2 T3(t-t2) U32(t2)|Ψ2(t2)〉 t
(16)
where k labels the laser pulse that optically couples electronic states |i〉 and |j〉 and µij(R) denotes the relevant transition dipole moment operator. Thus, the effect of the pump pulse is given by U21(t1), which promotes the ground-state vibrational eigenfunction to the covalent part of E2(R) at time t1. The resulting superposition state |Ψ2(t)〉 is then propagated over E2(R) for a period t2 - t1 by the propagator T2(t2-t1) ) exp[-iH2(t2-t1)]. The Hamiltonian H2 that enters the calculation via T2 governs the motion of |Ψ2(t)〉 and is given by
H2 ) ∑
excitation, with V′′ ) 4 and 5 providing about 80% of the transition strength.86 In this case, the ground-state eigenfunction should ideally be replaced by the appropriate density matrix representing the vibrational population distribution at T ) 923 K, or alternatively, the solutions to eq 15 for a number of different starting levels should be Boltzmann averaged to give the final thermalized result. We have found, however, that it is possible to mimic the effect of excitation from a number of low-lying vibrational levels at λ1 ) 360 nm by adopting a firstorder perturbation treatment of the action of the probe laser pulse on a Gaussian intermediate state |Ψ2(t)〉. The final state |Ψ3(t)〉 is calculated using the expression85
(19)
with σ1 ) σ2 ) 50 fs for both pump (k ) 1) and probe (k ) 2) pulses, giving a fwhm ∆t1 ) ∆t2 ≈ 85 fs. The pump-probe time delay τD is defined in the usual manner as the temporal difference between the intensity maxima of the two Gaussian pulses: τD ) t20 - t10. The durations of both pulses are curtailed at (2σk so that the operators Uij(tk) act on the appropriate wave functions for a total period of 4σk, resulting in a concomitant reduction in the time available for wave function propagation. For a given value of τD, the complete sequence of wave packet promotion and propagation steps starts at a time t ) -2σ1 (σ2), depending upon whether the pump (probe) precedes the other, and terminates at t ) τD + 2σ2, corresponding to the end of the probe pulse. At λ1 ) 360 nm, it is necessary to consider absorption of pump photons from a thermal distribution of vibrational population in the ground electronic state: in this case vibrational levels V′′ ) 0-9 all contribute to the initial A 0+ r X 1Σ+ 0
(20)
where t ) 0 is equivalent to τD ) 100 fs. At this wavelength, the intermediate state |Ψ2(t)〉 is prepared within the classically forbidden region of E2(R). An important prerequisite for implementing eqs 15 and 20 is knowledge of the coordinate dependence of the dipole transition moments µ21(R) and µ32(R). For absorption of pump-pulse radiation, we invoke the Condon approximation to describe the (non)variation of µ21(R) with internuclear distance; i.e., µ21(R) ) µ0|1〉〈2|. Semiempirical calculations of the ground and lowest-lying excited states of the fluorides and chlorides of Li, Na, and K have been carried out by Zeiri and Balint-Kurti.87 From the potentials and wave functions obtained, transition dipole moments connecting the ground state and covalent levels correlating with Na(2S1/2) + X(2P3/2) were derived for different bond lengths. For NaCl for instance, Figure 4 of ref 87 predicts that the Ω ) 0+ r X 1Σ+ 0 transition dipole passes through a maximum of approximately 2.5 D at about twice Re′′ before declining with increasing ionic character of the excited-state wave function to a minimum of about 1 D just short of the diabatic crossing point. Although analogous calculations for the alkali iodides do not appear to exist at the present time, the variation of µ21(R) for NaCl would seem to cast doubt on the validity of the Condon approximation for pump-pulse absorption as an extended function of wavelength by NaI. Calculations of the probability density ∫Ψ2*(r;t) Ψ2(r;t) dr on E2(R) at distances Rin < R < Rx show, however, that the time-dependence of the oscillating population is similar for that different starting vibrational levels V′′ ) 0, 1, and 2 of the X 1Σ+ 0 state: an increase in λosc of about 10% is observed when one and two quanta of vibrational energy are added to the ground-state zeropoint energy.65 This indicates that |Ψ2(t)〉 is insensitive to the detailed shape of the initial wave function or density matrix excited by the pump laser, which in turn implies that the inclusion of a coordinate-independent dipole transition moment function is at least unlikely to lead to a time dependence that is qualitatively incorrect, given the small range of distances (2.52.8 Å) sampled by the lowest-energy vibrational eigenfunctions. For probing of transition states, it is postulated that µ32(R) for the [Na(2PJ)‚‚‚1]‡* r [Na‚‚‚I]‡* + hν2 transition is also constant at internuclear separations Rin < R < Rx in the absence of any pertinent (theoretical or experimental) information. Following Engel and Metiu,65,72,73 it is assumed that only the neutral transition-state species [Na‚‚‚I]‡* are excited to the higher-lying state |3〉 by the probe pulse; i.e., µ32(R) for the optically induced charge transfer transition [Na(2PJ)‚‚‚I]‡* r [Na‚‚‚I]‡* + hν2 at R > Rx is set to zero. It was noted by Engel and Metiu73 that when such a mechanism is included in wave packet simulations of FTS spectra, the computed off-resonance peaks appear broader, though their temporal positions remain the same.
NaI Dissociation Reaction
Figure 9. Quantum dynamics simulations of short-time FTS transients at λ1 ) 320 nm showing the variation of 〈Ψ3(t)|Ψ3(t)〉 with time delay τD. Probe wavelengths are (a) 600 and (b) 640 nm. Indicated on each diagram are the oscillation periods τosc, the splitting times τs between doublet maxima and the time τ0 ) τs/2 of the center of the first LIF peak measured from τD ) 0. Both transform-limited laser pulses have Gaussian profiles with fwhm ∆t1 ) ∆t2 ) 85 fs, and values of 〈Ψ3(t)|Ψ3(t)〉 are normalized with respect to the peak maximum amplitude in each transient. The time step for wavelength propagation is ∆tpr ) 1 fs.
Rotational degrees of freedom are not included in our calculations. The initial rotational energy of NaI provides a centrifugal component VJ(R) ) J(J + 1)p2/2µR2 to the predissociative potential which permits Ψ2(R;t) to spend a proportionally longer time at larger internuclear separations, thereby increasing the vibrational period. Semiclassical calculations65 of the predissociation dynamics show that when a rotational energy of rot ≈ kT ) 642 cm-1 is added to the initial zeropoint energy, the oscillation period of Ψ2(R;t) on E2(R) increases by about 11% relative to the rotationless reaction across the range of photolysis energies investigated by FTS. For accurate computations of LIF intensities it would be necessary to adopt an analogous procedure to that for vibrationally hot molecules, repeating the wave packet propagations for all rotational levels of the X 1Σ+ 0 state contained within the initial Boltzmann ensemble and then averaging the resulting fluorescence signals.73,88 Since these calculations are focused on the motion of Ψ2(R;t) within the A 0+ adiabatic well, the off-diagonal matrix elements V12 ) V21 that are responsible for diabatic leakage to form Na + I atoms are set to zero, though nonzero values could be included to estimate the population decay due to repeated nonadiabatic transitions. Finally, the rapid fragmentation of NaI over the Ω ) 1 covalent potential is not included in the calculations. Although inadvisable for accurate determination of the magnitude of the LIF intensity as a function of time, these constraints do not in any way detract from the ability of quantum dynamics calculations to yield quantitative information regarding the time dependence of the ultrafast motions. The state vectors |Ψi(t)〉 are propagated over the potentials Ei(R) (i ) 1, 2, 3) on a one-dimensional discretized grid of 1024 points. Several choices exist for advancing the solution to the
J. Phys. Chem., Vol. 100, No. 19, 1996 7845
Figure 10. Quantum dynamics simulation of short-time FTS transients at λ1 ) 360 nm. Probe wavelengths are (a) 600 and (b) 640 nm. All other information is identical to that give in the caption to Figure 9.
time-dependent Schro¨dinger equation with time, the relative computational efficiencies and accuracies of which have been carefully examined by several workers.89 In this work we used the split operator method due to Feit and Fleck for wave packet propagation,90 which takes advantage of the ease of treating operators in their diagonal representations. Aliasing and edge effects of the discretized wave functions were avoided by sampling above the Nyquist limit in momentum and coordinate space. The time dependence of 〈Ψ3(t)|Ψ3(t)〉 is calculated with high temporal resolution by employing a very fine time increment (∆tpr ∼ 1 fs) in the discretized versions of eq 15 or 20 to propagate each time-dependent wave function. Because ∆tpr is so small in comparison to τosc, wave packet propagations were carried out over the comparatively small range of delay times from τD ) -0.1 to +3 ps. In order to check for fully converged solutions, calculations were also performed with half the number of discretizations and time step. b. Results. Wave packet simulations of the FTS experiments were carried out for two probe wavelengths of λ2 ) 600 and 640 nm at each pump wavelength λ1 ) 320 and 360 nm. As discussed above, the longer probe wavelength monitors the dynamics over a range of distances (determined by the spectral bandwidth) centered about R ) 5.0 Å on the broad potential minimum of E2(R), while the shorter wavelength corresponds to probing the passage of the wave packet at R ≈ 6.5 Å, closer to the outer turning point. Figure 9 displays simulated FTS transients for a pump wavelength of λ1 ) 320 nm (Eav ∼ 5840 cm-1). Both spectra exhibit the anticipated series of fluorescence maxima separated by a constant period of τosc ) 1096 fs, identical to the experimentally measured value. Only at λ2 ) 600 nm, however, is the splitting of the second and third peaks of the calculated transients resolvable into a doublet structure. The temporal separation of the two doublets centered at τD ) 1096 and 2192 fs in Figure 9a is τs ) 308 fs in both cases, in reasonable agreement with the experimental result listed in Table 2.
7846 J. Phys. Chem., Vol. 100, No. 19, 1996
Cong et al.
Figure 11. Snapshots of the time-dependent wave packet evolution over the adiabatic potential well for laser excitation at λ1 ) 320 nm. The initial wave function was calculated by a first-order perturbation treatment, given a laser pulse width (fwhm) of 85 fs.
Although no doublet structure can be distinguished at λ2 ) 640 nm (and likewise in the experiment), τs can be determined from eq 10 as twice the time taken for the calculated fluorescence intensity to reach an initial maximum amplitude. The maximum of the first peak shown in Figure 3b occurs after some 77 fs, and so the underlying splitting time is τs ) 154 fs, in accord with the measured value. In agreement with the experimental findings of Figure 3, these result indicate that |Ψ(t)〉 takes longer to travel from the OCR defined by the probe pulse to the inner turning point of E2(R) and then return to the OCR for the more energetic probe pulse; i.e., for shorter wavelengths the OCR is located at larger internuclear distances along the vibrational coordinate. When the energy available for wave packet vibration is lowered to Eav ≈ 2370 cm-1 at λ1 ) 360 nm, the calculated value of τosc decreases to 870 fs, as shown in the plots of 〈Ψ3(t)|Ψ3(t)〉 versus τD presented in Figure 10. At this energy, the splitting of intensity modulations into doublets is resolvable at both probe wavelengths, again in agreement with experimental results. Values of τs are calculated to be 379 and 225 fs for λ2 ) 600 and 640 nm, respectively. While the calculated values of τs for this pump energy are in good agreement with the experimental results, the calculated value of τosc is some 7% lower than the FTS value. This indicates that, at energies near the bottom of the well, the Coulombic limb of E2(R) obtained by fitting the experimental oscillation periods to the thirdorder polynomial given in eq 8 is in need of slight readjustment, whereas the simple form of the repulsive covalent potential of eq 3 with parameters given in section III is essentially correct. Overall comparison of the theoretical values of τs and τosc indicated in Figures 9 and 10 with the experimental results listed in Tables 2 and 3 reveals the similarity between calculated and experimental transients, insofar as temporal aspects are concerned. (Inaccuracies in intensities resulting from the calculated population of [Na‚‚‚1]‡* have already been discussed.) When different forms of the potentials E1(R), E2(R), and E3(R) obtained from frequency-resolved spectroscopy are substituted for those
determined by FTS, the measured splitting times cannot be reproduced by the wave packet calculations to the same level of quantitative accuracy as given above. To demonstrate the robustness of the potentials derived in section III, we have carried out calculations using a Morse curve postulated for E3(R).33 In this case, it was not possible to reproduce satisfactorily the experimental splitting data to quantitative accuracy: values of τs ) 202 and 96 fs were calculated for λ2 ) 600 and 640 nm, respectively, at λ1 ) 320 nm, and τs ) 225 and 102 fs for the same probe wavelengths at λ1 ) 360 nm. The periods τosc remain unchanged from before, of course, since the oscillation frequency depends only upon the λ1 shape of E2(R). When a small attractive component was added to the repulsive form of V1(R) given in eq 3, a slight increase in values of τosc was found of the same magnitude (3-5%) as that obtained in the classical trajectory analysis. In general, the good agreement between FTS experiments and wave packet calculations lends considerable support to the classical inversion procedure for determination of the low-lying potentials of NaI by FTS or, viewed from the opposite perspective, the ability of time-dependent quantum dynamics to describe the early-time vibrational motion in NaI dissociation with quantitative accuracy. V. Conclusions In this contribution, we have described the methodology of femtosecond high-spatial resolution of wave packet motion with application to the NaI dissociation reaction. By probing both the temporal and spatial evolution of the system, we can determine the direction of the oscillating wave packet, visualize its trajectory, and deduce characteristics of the potential energy curves involved, both the reactive surface and the final template state. Complementary theoretical calculations have been carried out which permit the short-time evolution to be understood in terms of the time-dependent wave packet prepared by the ultrashort laser pulse (see snapshots in Figure 11). The calculated wave
NaI Dissociation Reaction packet motion on the reactive potential offers quantitative support to the form of the quasi-bound and high-lying potentials derived from FTS. The sensitivity of the calculated transients to the postulated forms for the potential functions has been investigated. Recent comparisons by Tiemann’s group support the forms of the potentials discussed here.92 The recent elegant work by the Leone group,91 defining the initial rotational state, would be a natural extension to the case reported here. Both the experimental and theoretical findings establish the degree of spatial locality and directionality of the motion. In addition, they provide the nature of the wave packet reflecting the basic features, classical and quantal, common to all molecular systems prepared on the femtosecond time scale. Acknowledgment. This work was supported by the Air Force Office of Scientific Research. References and Notes (1) Imre, D.; Kinsey, J. L.; Sinha, A.; Krenos, J. J. Phys. Chem. 1984, 88, 3956. Johnson, B. R.; Kinsey, J. L. In Femtosecond Chemistry; Manz, J., Wo¨ste, L., Eds.; VCH: New York, 1994; Vol. 1, p 353. (2) Mokhtari, A.; Cong, P.; Herek, J. L.; Zewail, A. H. Nature 1990, 348, 225. (3) Manz, J., Wo¨ste, L., Eds. Femtosecond Chemistry; VCH: New York, 1994; Vols. I and II. (4) Rose, T. S.; Rosker, M. J.; Zewail, A. H. J. Chem. Phys. 1988, 88, 6672. Rosker, M. J.; Rose, T. S.; Zewail, A. H. Chem. Phys. Lett. 1988, 146, 175. (5) Zewail, A. H. J. Chem. Soc., Faraday Trans. 1989, 85, 1221. (6) Rose, T. S.; Rosker, M. J.; Zewail, A. H. J. Chem. Phys. 1989, 91, 7415. (7) Cong, P.; Mokhtari, A.; Zewail, A. H. Chem. Phys. Lett. 1990, 172, 109. (8) Zewail, A. H. Faraday Trans. Chem. Soc. 1991, 91, 207. (9) Zewail, A. H. FemtochemistrysUltrafast Dynamics of the Chemical Bond; World Scientific: Singapore, 1994; Vols. I and II. Zewail, A. H. In Femtosecond Chemistry; Manz, J., Wo¨ste, L., Eds.; VCH: New York, 1994; Vol. 1, p 15. (10) Lin, S. H.; Fain, B.; Hamer, N. In AdVances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; Wiley: New York, 1990; Vol. 74, pp 123267. (11) Zare, R. N.; Herschbach, D. R. J. Mol. Spectrosc. 1965, 15, 462. (12) Schaefer, S. H.; Bender, D.; Tiemann, E. Chem. Phys. Lett. 1982, 92, 273. (13) Schaefer, S. H.; Bender, D.; Tiemann, E. Chem. Phys. 1984, 89, 65. (14) Moutinho, A. M.; Aten, J. A.; Los, J. Physica 1971, 53, 471. (15) Delvigne, G. A. L.; Los, J. Physica 1973, 67, 166. (16) Yamashita, K.; Morokuma, K. Faraday Discuss. Chem. Soc. 1991, 91, 47. (17) Sakai, Y.; Miyoshi, E.; Ano, T. Can. J. Chem. 1992, 70, 309. (18) Grice, R.; Herschbach, D. R. Mol. Phys. 1974, 27, 159. (19) Bersohn, R.; Zewail, A. H. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 373. (20) Bernstein, R. B.; Zewail, A. H. J. Chem. Phys. 1989, 90, 829. (21) Gruebele, M.; Roberts, G.; Dantus, M.; Bowman, R. M.; Zewail, A. H. Chem. Phys. Lett. 1990, 166, 459. (22) Bernstein, R. B.; Zewail, A. H. Chem. Phys. Lett. 1990, 170, 321. (23) Janssen, M. H. M.; Bowman, R. M.; Zewail, A. H. Chem. Phys. Lett. 1990, 172, 99. (24) Baskin, J. S.; Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1986, 84, 4708. Felker, P. M.; Baskin, J. S.; Zewail, A. H. J. Phys. Chem. 1986, 90, 724. Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1987, 86, 2460. Baskin, J. S.; Zewail, A. H. J. Phys. Chem. 1989, 93, 5701. (25) The term “clocking” refers to timing a reaction with respect to a known zero of time. (26) Dantus, M.; Rosker, M. J.; Zewail, A. H. J. Chem. Phys. 1988, 89, 6128. Rosker, M. J.; Dantus, M.; Zewail, A. H. Science 1988, 241, 1200. Scherer, N. F.; Knee, J. L.; Smith, D. D.; Zewail, A. H. J. Phys. Chem. 1985, 89, 5141. (27) Rosker, M. J.; Dantus, M.; Zewail, A. H. J. Chem. Phys. 1988, 89, 6113. (28) Fried, L. E.; Mukamel, S. J. Chem. Phys. 1990, 93, 3063. Beswick, J. A.; Jortner, J. Chem. Phys. Lett. 1990, 168, 246. (29) Davidovits, P.; Brodhead, D. C. J. Chem. Phys. 1967, 46, 2968. (30) Oppenheimer, M.; Berry, R. S. J. Chem. Phys. 1971, 54, 5058. (31) Bernstein, R. B. Chemical Dynamics Via Molecular Beam and Laser Techniques; Clarendon Press: Oxford, 1982; Chapter 8, p 169. (32) Faist, M. B.; Levine, R. D. J. Chem. Phys. 1976, 64, 2953.
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