J. Phys. Chem. 1996, 100, 7859-7866
7859
Spectroscopic Wave Function Imaging and Potential Inversion Moshe Shapiro Department of Chemical Physics, The Weizmann Institute, RehoVot 76100, Israel ReceiVed: May 11, 1995; In Final Form: June 19, 1995X
An accurate imaging technique of complex nonstationary wave functions and a method of inversion of excitedstate potentials is developed. The wave function imaging is accomplished by using both frequency-resolved and time-resolved fluorescence data, thereby reducing the problem to a solvable set of linear algebraic equations. The procedure is demonstrated for excited-state Na2 wave packets created by ultrashort pulse excitations. A method for evaluating the sign of the transition-dipole amplitudes from the spectral line strengths is developed. Using the transition-dipole amplitudes thus generated, plus the transition frequencies and the ground state potential, we show how to obtain excited-state potentials. Contrary to the RKR procedure, this inversion scheme is not limited to diatomic molecules and is capable of yielding potentials at energies above the 1 dissociation thresholds. Highly accurate inversions of the Na2(A1Σ+ u ) and Na2(B Πu) excited-state potentials are demonstrated.
I. Introduction The term “inversion”1 pertains to a procedure enabling one to derive phases or potentials from observed data while making no prior assumption of their form. Though complex entities, such as wave functions, are not directly measurable, a set of real observables may contain enough information to determine (up to an overall insignificant phase) both the magnitude and phase of such complex entities. That such phase inversion is possible has been demonstrated in the context of the scattering phase problem,1,2 where measurements of differential cross sections at all angles may, via unitarity relations, yield the (complex) scattering amplitudes. Spectroscopic inversions of polyatomic potentials have been achieved with great success in the harmonic or near-harmonic limits.3-6 Extensions of such inversion methods far from the harmonic limit are however impossible, although the use of algebraic techniques7 can substantially increase the range of the derived potential. One therefore resorts to fitting procedures which employ explicit parametrizations of the potentials.8-11 In these “forward” methods, a set of potential parameters is assumed and the spectrum is calculated and compared with experiment. The potential parameters are then iteratively modified until a satisfactory agreement with the experimental data is attained. Besides being very laborious, the main drawback of such forward procedures is in their dependence on the form of the analytic parametrization chosen. It is possible to have two different potentials derived from two different parametrizations which produce reasonably well the observed data. Thus, the development of spectroscopic inversion methods of polyatomic potential surfaces far from the harmonic limit is a highly desirable objective. Spectroscopic wave packet imaging is being made possible due to the development of ultrafast techniques for monitoring fluorescence and absorption. For example, using up-conversion techniques, Walmsley et al.12-14 were able to measure the temporal evolution of fluorescence over ultrashort (∼60 fs) time scales. The same authors then attempted to use “tomographic” techniques13 to convert their measurements to images of Wigner phase-space distributions. The tomographic analysis relies X
Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(95)01315-3 CCC: $12.00
heavily on assumptions about the harmonicity of the potentials and the existence of a one-to-one correspondence between each “fluorescence-frequency window” and an internuclear distance.14 Even with such restrictive assumptions, test runs of the tomographic method have failed to reproduce the source wave packets to sufficient accuracy. Because of the uncertainty principle, a one-to-one correspondence between emission frequencies and internuclear positions is exhibited only in the very short wavelength (WKB) regime. Under these circumstances one can indeed use timedependent techniques to image potentials.15 In the “large p” quantum regime, limitations on the energetic spread of the wave packet, either in the process of preparation or in the process of detection, result in its delocalization in coordinate space.16,17 Likewise, attempts to improve the coordinate definition are incompatible with the imposition of a frequency window. Inversion of anharmonic potentials in the frequency domain can be successfully performed by the use of the RKR procedure.18 The RKR method is however limited, owing to its use of only bound-state energy level information, to extracting potentials below the dissociation limit. In addition, the RKR procedure can only be applied to diatomic molecules (mainly because a good WKB theory for many-body systems does not exist) and is limited by the range of validity of the WKB approximation. Extensions of spectroscopic techniques to obtain potentials at values above the dissociation thresholds can be made using predissociative19,20 and continuous21 spectra. Inversions of elastic scattering data also yield potential values above the dissociation limit,22-25 but the extraction of the complex amplitudes from the differential cross sections1 can be accomplished in practice for only atom-atom pairs.2 In the present paper we consider spectroscopic inversion of both wave functions and excited-state potentials. Recently we have developed an approach26 which is capable of imaging complex wave packets with high fidelity while making no assumption concerning the fluorescence-frequencysinternuclear distance correspondence, the localization of the wave packet, or the degree of harmonicity. The approach utilizes both the undispersed time-dependent fluorescence and the dispersed frequency-resolved spectrum. We also derived a quantum mechanical formula which explicitly expresses any excited-state © 1996 American Chemical Society
7860 J. Phys. Chem., Vol. 100, No. 19, 1996
Shapiro
potential in terms of the ground-state potential, the fluorescence transition frequencies, and the line intensities. The formula, which is not limited by the goodness of the WKB approximation, is valid for polyatomic molecules. In the present paper we demonstrate the accuracy of this wave function imaging method and our excited-state potential inversion scheme. In particular, we show that our potential inversion formula can generate potentials above (as well as below) the dissociation limit, while using realistic fluorescence data. The organization of the paper is as follows: In section II we outline the theory for the extraction of wave functions, transitiondipole phases, and excited-state potentials. In section III we demonstrate the fidelity of our procedure by simulating the timeresolved and frequency-resolved fluorescence of Na2 from both the A state and the B state and using this data to image excitedstate wave packets and invert the excited-state potentials. Finally, we discuss the use of our inversion technique to derive the phases of arbitrary optical fields. II. Theory A. Wave Function Imaging. Consider the fluorescence from a molecular wave packet excited by a short pulse of light. We assume that the molecule, initially in the ground vibration of the ground electronic state, χg, with energy Eg, is subjected to the action of a linearly-polarized pulse of the form
E B(t) ) E ˆ Re(t) exp(iωat)
(1)
where Re signifies the real part, E ˆ is the polarization direction, ωa is a “carrier” frequency, and (t) is a (complex) pulse envelope. We choose the pulse envelope to be
[
(t) ) a exp -
]
(t - t0 + 2iR)2 4R2
Einstein A-coefficient27
Af,s )
j(ω) ) π-1/2 Ra exp[-R2(ω - ωa)2 - (2R + it0)(ω - ωa)] (3) Choosing t0 of eq 2 such that at t ) 0 the pulse is practically over, the excited wave packet at t ) 0 may be written as
ωf,s3 3π0pc3
|df,s|2
df,s ≡ 〈χf|µ|φs〉
as )
2πi j(ωs,g)〈φs|µ|χg〉 p
γs ) γ ∑ Af,s
where ωs,g ) (Es - Eg)/p At subsequent times the coherently excited wave packet, moving as
Ψ(t) ) ∑ asφs exp(-iEst/p - γst/2)
where γ is a strength parameter. If the decay process is due to pure radiative decay, we take γ to be equal to 1. The (“nondispersed”) rate of fluorescence, F(t), from the entire wave packet is given28,29 as a coherent (double) sum of amplitudes from the excited states comprising the wave packet, summed over all the possible final states. We obtain that
F(t) ) ∑∑ f s,s′
ωf,s3 + ωf,s′3 3
df,sds′,fasas′* exp[-i(Es - Es′)t/p -
6π0pc
(γs + γs′)t/2] (10) where ωf,s ) (Es - Ef)/p. Notice that contrary to the expression for the emission from a single state, given in eq 7, due to the ωf,s3 photon phase-space factors and the s, s′ double summation, F(t) cannot be written as a square of an amplitude. With φs and Es assumed known, the imaging task is equivalent to deriving the complex as coefficients from the measured F(t) signal. Denoting products of the unknown coefficients as Xs,s′ ≡ asa*s′, we can rewrite eq 10 as
F(t) ) ∑ Xs,s′es,s′(t)
where γs are (radiative or nonradiative) decay rates, may spontaneously emit photons while undergoing transitions to any of the electronically-ground vibrational wave functions χf. The situation is depicted for the Na2 case in Figure 1. The rate of emission from a given φs component of the excited wave packet to a given ground-state χf is given as a single
(11)
s,s′
(6)
s
(9)
f
(4)
(5)
(8)
Since the decay rates γs may arise from radiative processes, we parametrize them as
s
where φs are electronically-excited vibrational wave functions with energies Es. Because the pulse is over at t ) 0, as, the “preparation coefficients”, are given as27,28
(7)
where df,s are transition-dipole matrix elements, given as
(2)
Hence, j(ω), the pulse’s Fourier transform, is a Gaussian with a linear chirp,
Ψ(t)0) ) ∑asφs
1 + 1 Figure 1. Illustration of the Na2(X1Σ+ g ), Na2(A Σu ), and Na2(B Πu) potentials and sample wave functions (χg and φs) belonging to these surfaces. Also marked are typical frequencies of excitation and fluorescence.
where
es,s′(t) ≡ fs,s′ exp[-i(Es - Es′)t/p - (γs + γs′)t/2] (12) fs,s′ ≡ ∑ df,sds′,f f
ωf,s3 + ωf,s′3 6π0pc3
(13)
Wave Function Imaging and Potential Inversion
J. Phys. Chem., Vol. 100, No. 19, 1996 7861
Equation 11 can be written in matrix form by treating s, s′ as a single index k. We obtain that
F ) e‚X
(14)
where Ft ≡ F(t), et,k ≡ es,s′(t), and Xk ≡ Xs,s′. By strobing the time intervals such that their number equals the number of k values, we can try to invert the e matrix of eq 14 to obtain for the unknown X vector
X ) e-1‚F
(15)
However, it follows from eq 12 that the e matrix cannot be inverted, as it contains a number of columns, explicitly all the s ) s′ columns, composed of a single number. This is due to the fact that for s ) s′ the Es - Es′ terms vanish, leaving the γs decay rates as the only source of time dependence. Since for spontaneous radiative decay (and many other processes), the decay times, 1/γs, are orders of magnitude longer than the duration of the subpicosecond measurement, the es,s(t) matrix elements are essentially time-independent and virtually identical to one another. As a result, the e matrix, which becomes nearly singular, cannot be inverted. We can solve this problem by recognizing that we can factor F(t) by defining Foff(t) as
Foff(t) ) F(t) - Fdiag(t)
that in order to affect the imaging we would need to know the excited-state potential. It is of interest to see if it is possible to derive the di,s amplitudes from experimental data, thereby obviating the need for prior knowledge of the excited-state potential. In ref 26 we showed that this indeed can be done and that it is possible to extract the dipole transition amplitudes di,s from their squares |di,s|2 which in turn are (see, e.g., eq 7) directly related to the measurable line strengths. This procedure, which, as described in section II.C, also yields the necessary input to the potential inversion, is briefly described here. For bound-bound transitions the phase problem is fortunately simpler than in the scattering case,1,2 because both χi and φs may be chosen real. Hence, the phase problem reduces to finding the sign of the di,s matrix elements. We start the derivation by looking at sums of line strengths. Summing over excited quantum numbers, we obtain that
Ji ≡ ∑ di,s2 ) ∑〈χi|µ|φs〉〈φs|µ|χi〉 ) 〈χi|µPµ|χi〉 (20) s
s
where P ≡ ∑s|φs〉〈φs|. The more states summed over, the more local does µPµ become, and the more accurately can one represent it as a Taylor series,
µPµ ) ∑µl(R - Req)l/l!
(16)
(21)
l
where
Fdiag(t) ) ∑∑ f
s
ωf,s3 3
3π0pc
|df,s|2|as|2 exp(-γst) ) Af,s|as|2 exp(-γst) ∑ f,s
(17)
We see that Foff(t) contains only the s * s′ terms in eq 10. Defining the e matrix of eq 12 to include k indices that only correspond to s * s′, allows us to solve eq 14 as
X ) e-1‚Foff
(18)
What remains therefore is to find a way of deriving Foff(t) from the observed signal F(t). This can be done by recording, in addition to the time-dependent nondispersed fluorescence, the dispersed signal. The dispersed signal is composed of a series of lines at different ωf,s frequencies, each line strength yielding a single Af,s|as|2 exp(-γst) term of eq 17. Since the Einstein A-coefficients are known, or can be measured separately, the strength of each line yields directly |as|2, while the sum of all the line strengths is exactly Fdiag(t)0). With the diagonal term known, the off diagonal component is obtained from eq 16 and the X vector determined from eq 18. The phases of the as coefficients are determined from X and the |as|2 coefficient (obtained from the dispersed fluorescence signal), in the following way: We can choose one phase factor at will, say by deciding that some am coefficient be real and positive. In that case we can write am ) Xm,m1/2 and determine the phase of the other as coefficients from the Xs,m column, since
as ) Xs,m/am
where R - Req is the internuclear displacements about some equilibrium Req, and µl are (as yet) undetermined coefficients. When P ) I, (i.e., we sum over all the excited states), µPµ ) µ2, which is strictly local, and the Taylor series expansion can be made as accurate as desired. In most cases, µPµ is nonlocal and the Taylor series only serves as a reasonable local approximation to it. Fortunately, as shown below, that is all we really require. Using eq 21 we see that eq 20 can be written as
(19)
B. Transition-Dipole Phase Inversion. The imaging method developed above might be assumed to require prior knowledge of φssthe excited state eigenfunctionssbecause these functions enter the transition-dipole matrix elements di,s used in the definition of the e matrix of eq 12. It seems therefore
Ji ) ∑µlRl,i
(22)
Rl,i ≡ 〈χi|(R - Req)l|χi〉/l!
(23)
l
where
The Rl,i matrix can be calculated using our (assumed) prior knowledge of the χi vibrational states belonging to the groundstate potential. It follows from eqs 22 and 23 and the µl coefficients can be obtained from the measurable Ji as
µl ) ∑Rl,i-1Ji
(24)
i
Once µl are known we can use them in the off-diagonal analogue of eq 20
Ji,j ≡ ∑di,sdj,s ) 〈χi|µPµ|χj〉 ) ∑µl〈χi|(R - Req)l|χj〉/l! s
l
(25) Since 〈χi|(R - Req)l|χj〉 can be calculated, knowledge of µl via eq 24 allows us to calculate Ji,j. By writing di,s ) |di,s|ni,s, where ni,s ()(1) is the desired sign of di,s, we can use the knowledge of Ji,s to write an equation for ni,s,
Ji,j ) ∑|di,s||dj,s|ni,snj,s s
(26)
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The solution of this equation is accomplished by a direct search for the set of ni,s ()(1) numbers (note that |di,s| are known) that minimize
M ) |Ji,j - ∑|di,s||dj,s|ni,snj,s|
where µ j is some average electronic transition dipole. Hence, by substituting eq 32 and eq 31 in eq 30, we have that
Vex(R) )
(27)
s
The search starts by covering all 2N+1 possible combinations (where N is the number of s states included in eq 27 of ni,snj,s numbers for one of the i,j pairs. Once these numbers are found, the search is repeated (over all 2N combinations) with the same i number and different j numbers. Our experience is that the solution that minimizes M indeed yields the correct ni,s signs, subject to the fact that, due to our freedom to choose the overall sign of χi and φs, N + N′ ni,s numbers can be arbitrarily assigned the value of +1. The minimum is much better defined the more s states are included, since as N increases, both the locality of µPµ and the Taylor series expansion improve. On the other hand, the number of possible combinations that need be searched through increases very rapidly with N. With the knowledge of the phase of di,s the wave packet imaging described in the previous section can be accomplished with no prior knowledge of the excited potential. All that is needed is the ground-state potential (from which χi are calculated), the transition frequencies, ωi,f, and the line strengths, |di,f|2. C. Inversion of Excited-State Potentials. We now show how the transition-dipole amplitudes derived in section II.B, and the transition frequencies can be used to derive the excitedstate potentials. We assume that we know the ground-state potential Vgr(R), where R ≡ (R1, R2, ..., RN) is a collection of internuclear distances specifying the shape of a polyatomic molecule. Our aim is to obtain the excited-state potential Vex(R). In the inversion we make use of χi, the eigenstates of the (known) ground-state Hamiltonian, satisfying the Schro¨dinger equation,
[Ei - K(R) - Vgr(R)]χi(R) ) 0
(28)
where K(R) is the nuclear kinetic energy operator and Ei is the bound state energy. In contrast, φs of eq 8 are eigenstates of the unknown excited-state Hamiltonian,
[Es - K(R) - Vex(R)]φs(R) ) 0
(29)
with eigenvalues Es. Using eq 29, we formally express the excited-state potential as
Vex(R) )
1 [E - K(R)]φs(R) φs(R) s
(30)
and expand the φs(R) wave functions in terms of the known ground-state wave functions,
φs(R) ) ∑χi(R)〈χi|φs〉
(31)
i
By employing the Franck-Condon approximation,3 we can write the 〈χi|φs〉 overlap integrals in terms of the transitiondipole matrix elements,
di,s 1 〈χi|φs〉 ) 〈χi|µ|φs〉 ) µ j µ j
(32)
∑idi,s[Es - K(R)]χi(R) ∑iχi(R)di,s
(33)
Thus, the exact value of µ j need not be known. We next estimate the action of K(R) on χi(R) via eq 28, from which it follows that
K(R) χi(R) ) [Ei - Vgr(R)]χi(R)
(34)
and we obtain (from eq 33) that
Vex(R) )
∑idi,s[pωi,s + Vgr(R)]χi(R) ∑iχi(R)di,s
(35)
Equation 35 is already a closed form solution of the excitedstate potential. In practice, however, since the basis of χi states is always incomplete, eq 35 cannot be used as is, because whenever the denominator is zero, or nearly zero, large errors occur. (Only with a complete basis set can the numerator vanish simultaneously with the denominator.) We therefore need to modify eq 35 to a more robust form, preferably one in which the denominator never vanishes. A more robust form is attained by using the redundancy of eq 35, namely, that an identical equation can be written for any excited φs state. We exploit this fact by multiplying eq 35 by |∑iχi(R)di,s|2 followed by a summation over s. We obtain that
∑s |∑i χi(R)di,s|2Vex(R) ) ∑s ∑i di,s[pωi,s + Vgr(R)]χi(R){∑j χj*(R)dj,s* }
(36)
and hence that
Vex(R) )
* p ∑s∑i,jdi,sdj,s ωi,sχi(R)χj*(R)
∑s|∑iχi(R)di,s|2
+ Vgr(R) (37)
Since the denominator in eq 37 is a sum of squares, it can be zero if all the terms vanish simultaneously. This can only occur in the extreme nonclassical regions where all of the φs functions are practically zero. In other regions, since it is not possible for all of the φs(R) functions to have a node at the same R values, the sum of squares never vanishes. Because of this eq 37 can be readily used in practice. Our procedure is expected to be accurate only for R values in which eq 31 is valid, i.e., when the φs excited states can indeed be expanded in terms of the ground wave functions χi. It is naturally subject to the goodness of the Franck-Condon approximation. In spite of these limitations, eq 37 has an immense appeal because it is parameter-free and it does not necessitate assignments of levels. Another advantage is that we only need to solve the (polyatomic) Schro¨dinger equation for the ground state (eq 28). Contrary to brute-force fitting procedures where one iteratively searches for the potential, a procedure involving repeating the solution of the nuclear Schro¨dinger equation many times, here we need to solve the (ground-state) nuclear Schro¨dinger equation only once.
Wave Function Imaging and Potential Inversion
J. Phys. Chem., Vol. 100, No. 19, 1996 7863
Figure 2. Time-dependent Na2 fluorescence signal, F(t), following excitation of the χg ground state by a 1200 cm-1 wide pulse: (, 90 points strobed for the inversion procedure. (a, top) X r A case with fast (nonradiative) decay (γ ) 103). The pulse, whose center frequency is 15 500 cm-1, simultaneously excites the φ4 to φ13 Na2(A1Σ+ u) vibrational states. (b, bottom) X r B case with slow (radiative) decay (γ ) 1). The pulse, whose center frequency is 22 400 cm-1, simultaneously excites the φ6 to φ15 Na2(B1Πu) vibrational states.
We note that eq 37 has a semiclassical analogue, obtained by assuming that for every line frequency ωs0,i there is a distance R ) R0 such that * p ∑ di,sdj,s ωi,sχi(R0) χj*(R0) ≈ i,j
* χi(R0) χj*(R0) (38) pωs0,i ∑ di,sdj,s i,j
from which we obtain that
Vex(R0) - Vgr(R0) ) pωs0,i
(39)
Equation 39 equates each transition energy with the excitedto-ground difference potential at some distance. Such correspondence between distances and frequencies is consistent with the semiclassical stationary-phase approximation,30 according to which the nuclear kinetic energy is conserved during the transition and the frequency of the emitted photon becomes simply the (instantaneous) difference between the potential energies. Even if eq 39 were accurate (which is usually not the case), it cannot be used as is for inversion purposes because it does not tell us which of all the measured transition frequencies is to be associated with a particular internuclear
Figure 3. Dispersed fluorescence spectrum following excitation by a 1200 cm-1 wide pulse. (a, top) Emission from the s ) 4-13 levels of 1 + the A(1Σ+ u ) state to the first 35 vibrational levels of the X( Σg ) state after excitation by a pulse centered at 15 500 cm-1. (b, bottom) Emission from the s ) 6-15 levels of the B(1Πu) potential to the first 35 vibrational levels of the X(1Σ+ g ) state after excitation by a pulse centered at 22 400 cm-1.
configuration R0. In contrast, the quantum mechanical expression of eq 37 does not require such an association. III. Results A. Wave Packet Imaging. We first test the imaging procedure developed in section II.A. In order to do this we simulate the fluorescence emitted by an excited wave packet, generate from the fluorescence an image of the wave packet, and compare the image to the source. We consider the case, depicted in Figure 1, of Na2 in the (χg) ground Na2(X1Σ+ g) vibrational state, excited by a pulse [of the form given in eqs 2 and 3] to a superposition of (φs) vibrational states belonging to 1 the Na2(A1Σ+ u ) or Na2(B Πu) electronic-states. In order to make this a realistic simulation, we have used the ab-initio Na2 curves of Meyer et al.31 to calculate the df,s dipole matrix elements for both the X r A and X r B transitions. Given these matrix elements, the preparation coefficients, as, are obtained from eq 5 and the time-dependent fluorescence signal, F(t), is from eq 10. Two cases are studied: that, shown in Figure 2a, of fast decay (γ of eq 9 ) 1000) from the A state, and that, shown in Figure 2b, of slow decay (γ ) 1) from the B state. In both cases the pulse bandwidth is taken to be 1200 cm-1, i.e., a pulse of ∼8 fs in duration. Depending on its center frequency (15 500 cm-1 for the A state and 22 400 cm-1 for the B state), the pulse simultaneously excites 10 φs vibrational states in either electronic
7864 J. Phys. Chem., Vol. 100, No. 19, 1996
Shapiro
TABLE 1: Inverted Preparation Coefficients for the Na2(A1Σu+) State as coefficient a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
source real 0.433 38 -0.913 05 1.493 6 -1.967 5 2.132 2 -1.925 1 1.456 9 -0.923 80 0.485 75 -0.205 32
image
+i imaginary -i +i -i +i +i -i +i -i +i -i
0.526 64 0.707 93 0.699 94 0.436 68 0.0 0.424 40 0.672 69 0.697 50 0.563 61 0.375 83
real 0.426 48 -0.913 79 1.479 1 -1.967 5 2.132 2 -1.925 9 1.456 4 -0.923 80 0.4857 7 -0.229 30
+i imaginary -i +i -i +i +i -i +i -i +i -i
0.532 25 0.706 98 0.730 11 0.436 64 0.0 0.420 82 0.673 72 0.697 50 0.563 59 0.361 70
state. We computed the fluorescence due to transitions to the first 35 vibrational states in the ground state. As clearly seen in Figure 2, the time-dependent fluorescence F(t) is made up of a complicated beat pattern, with almost complete recurrences in the absence of a nonradiative decay (Figure 2b). Also shown in Figure 2 are the time points actually sampled in the solution of eq 18. As explained in section II.A, in order to affect the wave packet imaging we need, in addition to the time-dependent fluorescence, to compute Fdiag(t), derived from the dispersed fluorescence. The dispersed fluorescence, consisting of a series of lines, is shown in Figure 3a for the A state and in Figure 3b for the B state. Fdiag(t) is obtained as the sum over the strengths of these lines (see eq 17). Given Foff(t) from eq 16, we can now use eq 12 and the line strengths of the dispersed spectrum to compute
TABLE 2: Inverted Preparation Coefficients for the Na2(B1Πu) State source
as coefficient
real
a6 a7 a8 a9 a10 a11 a12 a13 a14 a15
0.025 36 -0.076 64 0.184 02 -0.363 03 0.599 82 -0.840 80 1.011 0 -1.045 6 0.925 62 -0.695 98
image
+i imaginary -i +i -i +i +i -i +i -i +i -i
0.029 67 0.057 35 0.083 21 0.077 56 0.0 0.176 37 0.439 09 0.728 94 0.960 59 1.071 9
real 0.025 36 -0.076 64 0.184 02 -0.363 03 0.599 82 -0.840 80 1.011 0 -1.045 6 0.925 62 -0.695 98
+i imaginary -i +i -i +i +i -i +i -i +i -i
0.029 67 0.057 351 0.083 21 0.077 565 0.0 0.176 37 0.439 09 0.728 94 0.960 59 1.071 9
the e matrix and, using eqs 18 and 19, to invert for the preparation coefficients, as. The derived (“image”) and the true (“source”) preparation coefficients are given in Table 1 for the A state and in Table 2 for the B state. In the A state the derived coefficients reproduce the source with an accuracy of 2-3 significant figures for both the real and imaginary parts. In the B state the agreement is even better; the derived coefficients agree with the source to at least 5 significant figures. The errors in the A state derived coefficients mainly arise from clumping of some lines around certain values of Es - Es′, leading to similar time dependencies of some of the e columns. This clumping, which results from the nearly harmonic form of the A state at the levels of excitation considered, though insufficient to render the e matrix singular, makes the inversion
Figure 4. Real and imaginary parts of the source Ψ(t) and its image at different times. Source real part (s; source imaginary part (- -); images ((). (a, top left) A state wave packet at t ) 0. The left hand ordinate scale is of the real part; the right hand ordinate scale is of the imaginary part. (b, top right) A state wave packet at t ) 1.1 ps. (c, bottom left) B state wave packet at t ) 0. (d, bottom right) B state wave packet at t ) 1.7 ps.
Wave Function Imaging and Potential Inversion somewhat nonunique. The B state, which is less harmonic and where higher vibrational states are considered, does not, as shown in Figure 3b, exhibit the same degree of clumping. The as coefficient inversion is therefore more stable in that case. Our basic task is however to image Ψ(t), involving sums of products of the as coefficients with individual eigenstates. It turns out that the errors in imaging Ψ(t) are much smaller than the errors in deriving the individual coefficients. This is demonstrated by a series of snapshots, shown in Figure 4a-d, where the real and imaginary parts of the Ψ(t) source are contrasted with their images. As shown in Figure 4a,b, even for the A state the imaged wave functions are practically indistinguishable from the true wave functions for all of the times considered. Obviously, the procedure is sensitive, as it should be, to the shape of the overall Ψ(t) and less so to the individual values of the expansion coefficients. The same fidelity is obtained even when the wave packet becomes quite distorted and delocalized, a case demonstrated (see Figure 4d) for the B state at 1.7 ps after the excitation pulse. B. Inversion of Excited-State Potentials of Na2. We examine our potential inversion procedure for the Na2(A1Σ+ u) and Na2(B1Πu) excited-state potentials, using the simulated Na2 frequency-resolved fluorescence of Figure 3a,b. These simulations are based on the ab-initio potentials and the transition dipoles produced by Meyer et al.31 The Franck-Condon approximation is used throughout. With the frequencies and line strengths of Figure 3a, and the di,s matrix elements obtained by the method of section II.B, we invert the A(1Σ+ u ) potential using eq 37. The result is shown in Figure 5a, where it is also compared with the ab-initio source potential. We see that over a substantial range of R values the inverted potential is essentially indistinguishable from the “true” potential. Outside the shown R values large errors occur due to the vanishing of the φs functions, whose highest member used is also depicted in Figure 5a. 1 + The inversion of the A(1Σ+ u ) potential from the X( Σg ) potential is a very stringent test of the method because the two potential minima are substantially shifted from one another. As a result, a large χi basis is needed to expand φs where the χi functions tend to zero or where the φs functions are vanishingly small. We have repeated the inversion for the B(1Πu) potential, for which the shift in the equilibrium position of the excited relative to the ground state is smaller than that of the A(1Σ+ u) case. Using the fluorescence spectrum shown in Figure 3b, we have extracted the B state potential. The results are shown in Figure 5b. Figure 5b exhibits an almost perfect agreement between the inverted potential and the true potential over the 4.6 < R < 12 au range. This much extended range of R values for which the inverted potential coincides with the true potential is a result of the smaller shift in the equilibrium separation of the B(1Πu) potential relative to the ground potential (see Figure 1). Outside this range, the excited-state vibrational wave functions considered (φ6 - φ15) are zero and no inversion can be performed. Notice that even with the use of these wave functions, which are of limited spread in R, the repulsive part of the potential is obtained at energies well above the dissociation limit. IV. Discussion In this paper we have shown how to image complex wave packets using the total undispersed time-dependent fluorescence and the frequency-resolved fluorescence. We have also shown how to express the excited-state potential in terms of the groundstate potential, using the frequency-resolved emission spectrum.
J. Phys. Chem., Vol. 100, No. 19, 1996 7865
Figure 5. Inversion of the A and B excited-state potentials: the “true” potential (obtained from ref 31) (() and the inverted potential (s) (obtained from eq 37). Also show, (- -) is the highest φs wave function used in the inversion. The left hand ordinate scale is for the potentials; and the right hand ordinate scale is for the wave function. (a, top) Inversion of the Na2(A1Σ+ u ) potential. (b, bottom) Inversion of the Na2(B1Πu) potential.
We have demonstrated that, using this method, highly accurate 1 potentials can be obtained for the Na2(A1Σ+ u ) and Na2(B Πu) states. The main limitations of the potential inversion part have to do with the goodness of the Franck-Condon approximation and the limited spread in coordinate space of the excited vibrational states. Because of the use of many lines, we expect that errors introduced by the Franck-Condon approximation will tend to cancel each other, but this point must be investigated in detail. The limited spread in R space of the vibrational wave functions can be overcome by going to higher vibrations and by considering emission from continuum states. The extension of eq 37 to continuum-to-bound emissions is currently being studied, as is the possibility of using this method for curve crossing situations. Besides the intrinsic interest in imaging of potentials and wave functions, our technique has an immediate application in optics: It follows directly from eq 5 that the phase of a given as is proportional (up to the sign of 〈φs|µ|χg〉 which can be determined from the line strengths, as explained in section II.B) to the phase of j(ωs,g), the field at the ωs,g transition frequency. Thus, knowledge of the as coefficients yields j(ωs,g) at a sufficient number of frequencies to determine (by simple interpolation) the phases of all the frequencies that make up the pulse. The problem of phase determination of optical fields is receiving much attention recently.32 In available techniques the phase is extracted by measuring a third order correlation function
7866 J. Phys. Chem., Vol. 100, No. 19, 1996 of the field, followed by a nontrivial deconvolution procedure.32 The present method is much more direct as it uses the linear relation (which always exists for weak enough fields) between the molecular preparation coefficients and the field amplitudes. Acknowledgment. This work was supported by the Israel Academy Fund for Basic Research. References and Notes (1) Newton, R. G. Scattering Theory of WaVes and Particles; McGraw-Hill: New York, 1966; Chapter 20. (2) Gerber, R. B.; Shapiro, M. Chem. Phys. 1976, 13, 227. (3) Herzberg, G. Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules; Van Nostrand Reinhold: New York, 1950. Herzberg, G. Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules; Van Nostrand Reinhold: New York, 1966. (4) Dunham, J. L. Phys. ReV. 1932, 41, 721. (5) Ogilvie, J. F.; Tipping, R. H. Int. ReV. Phys. Chem. 1983, 3, 3. (6) Herman, R.; Wallis, R. F. J. Chem. Phys. 1955, 23, 637. (7) Iachello, F.; Levine, R. D. Algebraic Theory of Molecules; Oxford University Press: Oxford, U.K., 1995; and references cited therein. (8) Sorbie, K. S.; Murrell, J. N. Mol. Phys. 1976, 31, 305. (9) Law, M. M., Hutson, J. M., Ernesti, A., Eds. Fitting Molecular Potential Energy Surfaces; CCCP6: Daresbury U.K., 1994. (10) Watson, J. K. G. Chem. Phys. 1995, 190, 291. (11) Ho, T.-S.; Rabitz, H. J. Phys. Chem. 1993, 97, 13447. (12) Dunn, T. J.; Sweetser, J.; Walmsley, I. A.; Radzewicz, C. Phys. ReV. Lett. 1993, 70, 3383. (13) Dunn, T. J.; Walmsley, I. A.; Mukamel, S. Phys. ReV. Lett. 1995, 74, 884. (14) Kowalczyk, P.; Radzewicz, C.; Mostowski, J.; Walmsley, I. A. Phys. ReV. A 1990, 42, 5622. (15) Zewail, A. H. Faraday Discuss. Chem. Soc. 1991, 91, 207. Gruebele, M.; Roberts, G.; Dantus, M.; Bowman, R. M.; Zewail, A. H. Chem. Phys. Lett. 1990, 166, 459. Bernstein, R. B.; Zewail, A. H. Chem. Phys. Lett. 1990, 170, 321. Janssen, M. H. M.; Bowman, R. M.; Zewail,
Shapiro A. H. Chem. Phys. Lett. 1990, 172, 99. Gruebele, M.; Zewail, A. H. J. Chem. Phys. 1993, 98, 883. (16) Krause, J. L.; Shapiro, M.; Bersohn, R. J. Chem. Phys. 1991, 94, 5499. Shapiro, M. Faraday Discuss. Chem. Soc. 1991, 91, 352. (17) Shapiro, M. J. Phys. Chem. 1993, 97, 7396. (18) Rydberg, J. R. Z. Phys. 1931, 73, 376. Klein, O. Z. Phys. 1932, 76, 226. Rees, A. L. G. Proc. Phys. Soc. 1946, 59, 998. (19) Child, M. S.; Essen, H.; LeRoy, R. J. J. Chem. Phys. 1983, 78, 6732. LeRoy, R. J.; Keough, W. J.; Child, M. S. J. Chem. Phys. 1988, 89, 4564. (20) Shapiro, M.; Reisler, H. J. Chem. Phys. 1995, 103, 4150. (21) Blum, H.; Lindner, J.; Tiemann, E. J. Chem. Phys. 1990, 93, 4556. Schaefer, S. H.; Bender, D.; Tiemann, E. Chem. Phys. 1984, 89, 65; Chem. Phys. Lett. 1982, 92, 273. Johnson, B. R.; Kinsey, J. L. J. Chem. Phys. 1993, 99, 7267; In Femtosecond Chemistry; Manz, J., Wu¨ste, L., Eds.; VCH: Weinheim, Germany, 1995; p 353. (22) Firsov, O. B. Zh. Exp. Theor. Phys. 1953, 24, 279. (23) Vollmer, G. Z. Phys. 1969, 226, 423. (24) Buck, U. AdV. Chem. Phys. 1975, 30, 313. (25) Shapiro, M.; Gerber, R. B. Chem. Phys. 1976, 13, 235. Gerber, R. B.; Shapiro, M.; Buck, U.; Schleusener, J. Phys. ReV. Lett. 1978, 41, 236. (26) Shapiro, M. J. Chem. Phys. 1995, 103, 1748. (27) Loudon, R. The Quantum Theory of Light, 2nd ed.; Clarendon Press: Oxford, U.K., 1983. (28) See the derivation leading to eq 5.26 of ref 17. (29) Haroche, S. In High Resolution Laser Spectroscopy; Shimoda, K., Ed.; Springer-Verlag: Berlin, 1976; p 253. In eq 7.8 of that paper an approximation is made in that the different ωs,f3 photon phase-space factors are replaced by their average. This approximation is not made in eq 10 or in ref 17. (30) Child, M. S. Semiclassical Mechanics with Molecular Applications; Clarendon Press: Oxford, U.K., 1991; pp 108-112. (31) Schmidt, I. Ph.D. Thesis, Kaiserslautern University, Germany, 1987. (32) Chilla, J. L.; Martinez, O. E. IEEE J. Quantum Electron. QE 1991, 27, 1228. Trebino, R.; Kane, D. J. J. Opt. Soc. Am. A 1993, 10, 1101.
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