J . Phys. Chem. 1989, 93, 5701-5717
proximated by the graphic method on the basis of eq 1 and 2. The examples discussed so far illustrate the following: (1) how variations in the magnetic field strength affect the general appearance of a C I D N P multiplet and (2) how changes in g-factor differences brought about by substituents affect the polarization of a multiplet. It remains to consider variations of Ag by exchanging one constituent of a radical pair. For this purpose, we have studied hydrogen (deuterium) abstraction reactions of photoexcited benzaldehyde with three additional substrates: cyclohexane-d12,methylene-d2 chloride, and chloroform-d.
w
PAIR D
I
&HI
/ PAIR E
The light-induced abstraction reactions from these substrates occur in competition with the self-reaction of benzaldehyde, generating radical pairs in which the benzoyl radical ( g = 2.0008) of pair C is replaced successively by cyclohexyl (g = 2.0026),35 dichloromethyl ( g = 2.0083),28 and trichloromethyl ( g = 2.0091).36 In each of the resulting radical pairs, the I3C and ‘H hyperfine coupling constants of the hydroxybenzyl radical determine the polarization observed for a coupling product which is, however, (35) Leung, P. M. K.; Hunt, J . W. J . Phys. Chem. 1967, 71, 3177. (36) Hudson, A,; Hussain, H. A. J . Chem. SOC.B 1969, 793.
5701
modified by the changing g-factor difference for the different counterradicals. The ratio of hyperfine couplings is small (lac/aHI = 1.5) whereas the g-factor difference varies from small (-0.0004 < Ag < -0.0005) for cyclohexyl to very large (+0.0060) for trichloromethyl. The radical pair with the small g-factor difference (pair D) gives rise to a nearly pure A / E multiplet effect a t 21 kG and to net emission, on which a strong A / E multiplet effect is superimposed, a t 58.8 kG (Figure 6). In contrast, the photoreactions with methylene-d2 chloride and chloroform-d give rise to radical pairs with substantially larger g-factor differences and, accordingly, to pure net effects. For these pairs, enhanced absorption is observed, in keeping with the sign change of the g-factor difference. This is illustrated in Figure 7 for the hydrogen abstraction from chloroform.
Conclusion The results presented in this paper lead to one overriding conclusion: the presence of strongly (hyperfine) coupled nuclei (most often I3C) in a radical pair profoundly influences the C I D N P effects of nuclei (most often ‘H) that are (J) coupled to the former in products resulting from that pair. As previously indicated, CIDNP effects observed in such cases can be interpreted only with caution. In particular, Kaptein’s sign rules can be applied only to the “major” effects, i.e., to the net effect, when both signals have the same direction, and to the multiplet phase, when the signals have opposite directions. Acknowledgment. W e a t Columbia acknowledge the support of this research by AFOSR, DOE, and N S F . Registry No. 4-Methoxydibenzyl ket~ne-~)CO, 121329-44-0; deoxy121329-45-1; benzoin-13C0, 104917-39-7; 4-~hlorodeoxybenzoin-’~CO, dibenzyl ketone-”CO, 68120-92-3; 4-chlorodibenzyl ketone-”CO, 121329-46-2; 1,3-diphenylacetone, 102-04-5; benzaldehyde, 100-52-7; cyclohexane-d,2, 1735-17-7; chloroform-d, 865-49-6; methylene-d2chloride, 1665-00-5; deuterium, 7782-39-0.
Determination of Excited-State Rotational Constants and Structures by Doppler-Free Picosecond Spectroscopyt J. Spencer Baskin and Ahmed H. Zewail*v* Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125 (Received: January 23, 1989)
Picosecond time-resolved fluorescence measurement of purely rotational coherence is developed as a Doppler-free technique for the determination of rotational constants of large molecules in their excited states. We present detailed analyses of purely rotational coherence measurements, supplying new information about the rotational constants and structures of the first excited electronic states of t-stilbene, four t-stilbene van der Waals complexes, and anthracene, including values for all three anthracene S, rotational constants. Evidence is considered in the case of stilbene for a transition dipole with a significant component perpendicular to the a inertial axis, and the consequences of such a dipole are explored by way of numerical simulations. Excited-state structures are proposed for stilbene and stilbene-rare-gas complexes and comparisons made with model calculations of the van der Waals potential. Application of the new spectroscopic technique to molecules of large asymmetry is demonstrated by the analysis of fluorene and fluorene-argon measurements, and the results are compared with data from previously published high-resolution frequency domain studies.
I. Introduction Recently, we have reported’-3 on the use of Doppler-free picosecond spectroscopy to determine excited-state rotational constants and molecular structures for large molecules in their excited states. This approach is based on the phenomenon of purely rotational coherence. T h e theoretical foundation of purely rotational coherence and observations of its effects in jet-cooled molecule^^-^ and in a room-temperature vaporS have been the Contribution No. 7747.
John Simon Guggenheim Foundation Fellow.
0022-3654/89/2093-5701$01.50/0
subjects of previously published works. A quantitative analysis of recurring transients in the polarization-analyzed fluore~cence’,~ (1) (a) Felker, P. M.; Baskin, J. S . ; Zewail, A. H . J . Phys. Chem. 1986, 90, 724. Baskin, J. S.; Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1986, 84, 4708. (b) Baskin J. S.; Semmes, D.; Zewail, A. H . J . Chem. Phys. 1986.85, 7488. Semmes, D. H.; Baskin, J. S.; Zewail, A. H. J . Am. Chem. SOC.1987, 109, 4104. (2) Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1987, 86, 2460. (3) Baskin, J. S . ; Felker, P. M.; Zewail, A. H . J . Chem. Phys. 1987, 86, 2483. (4) Scherer, N. F.; Khundkar, L.; Rose, T.; Zewail, A. H. J . Phys. Chem. 1987, 91, 6478.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
(or absorption4) limited to the case of a parallel dipole-induced transition in a near-symmetric top permitted determination of the sum of the B and C excited-state rotational constants of t-stilbene (and its van der Waals complexes) and indicated that additional information could be ~ b t a i n e d . ~ Rotational coherence “fingerprinting” of fluorescence has also been used as a simple method for the identification of fluorescing species.lb The goal of the present paper is to explore in greater detail the application of purely rotational coherence (PRC) spectroscopy as a practical technique for the measurement of excited-state structures and other properties of large molecules. Fundamental to this effort is an expansion of the scope of the analysis made to exploit all the information available in the P R C transient spectra. Included in this effort are quantitative investigations of the influence of asymmetry and of arbitrary transition dipole direction on P R C measurements. Picosecond polarization-analyzed fluorescence measurements on eight molecules will be presented, providing illustrative examples of various aspects of the analysis while permitting the derivation of new information about the excited states of six of the eight molecules. The results indicate that the usefulness of the technique is not severely limited by requirements of symmetry or dipole direction. This fact, in conjunction with the advantages inherent in the quantum beat nature of the phenomenon (sub-Doppler resolution6 and insensitivity to ground-state structure), helps establish P R C spectroscopy as a useful technique for the characterization of excited states of large molecules. Some general properties of P R C in symmetric and asymmetric tops are described in section I1 to provide a background for the later discussion. The experimental apparatus and procedures are described in section 111. Section IV provides details of the analysis applied to P R C measurements for the extraction of precise structural parameters. Experimental results, including P R C fluorescence measurements of t-stilbene, t-stilbene-rare-gas van der Waals complexes, fluorene, fluorene-argon, and anthracene, are presented and discussed in section V. 11. Properties of PRC A . P R C in Symmetric Top Molecules. The theoretical treatment of P R C has been detailed in ref 2. Only a brief description of a simple, physically intuitive picture of the phenomenon is given here. It is based on the fact that the properties (e& time and polarization dependence) of P R C fluorescence signals can be understood, both qualitatively and, to a great degree, quantitatively, as arising from the classical motion of each moleculefixed dipole as the molecules of the sample rotate freely in space.2 For example, Figure 1 illustrates the classical rotation of a rigid prolate symmetric top of moment of inertia I , about the symmetry axis and Ibabout any axis perpendicular to it. This can be described as consisting of two motions: (1) a precession or nutation of the symmetry axis about the total angular momentum vector J, a t angular frequency w 1 or frequency u l = w,/(277); ( 2 ) a rotation about the symmetry axis of the body at angular frequency w2. Figure I shows the position of the body a t two different times separated by half of the nutation period. The angle 0 between the symmetry axis and J remains constant so that the component of J along the symmetry axis (=K) has constant magnitude. The frequencies u I and v 2 2re given by’ 1 J h 4nB U , = -lJl/Ib = - - = 2BJ 277 27T h 1 v 2 = -lJl(cos 27r
i )( ,
K h 4aA
8)(
-
=
7-
47rB
-
2 ( A - B)K ( 2 . 2 )
( 5 ) Myers, A . B.; Hochstrasser, R. M. J . Chem. Phys. 1986, 85, 6301. (6) The resolution of PRC is sub-Doppler in the sense that the rotational level structure of the molecule’s excited state is accessible to study by the PRC
technique even when Doppler broadening in the sample is sufficient to obscure all rotational structure in the frequency domain spectrum of the studied t rami t ion. (7) Herzberg, G. Molecular Spectra ond Molecular Structure; Van Nostrand Reinhold: New York, 1945, Vol 11, p 23.
Baskin and Zewail
/
t2
Figure 1. Classical motion of a rigid symmetric top. The top is shown a t two times separated by half a nutation period. The total angular momentum vector J and its component K along the symmetry axis a r e indicated. w , and w2 a r e the angular frequencies of nutation and of rotation about the symmetry axis, respectively.
where J and K are the magnitudes of J and K, respectively, in units of A and the definitions of the rotational constants B = h/(4xIb) and A = h/(47rZO)have been used. (For an oblate top, v 2 simply changes to 2 ( B - C)K.) Thus, it is seen that a dipole that is parallel to the symmetry axis partakes only of the nutational motion, sweeping out a cone in space a t the frequency vl. It can be shown that, as a result, the radiation from this dipole in any given direction is modulated only by the nutation frequency 2BJ and its first harmonic. By contrast, when the dipole is perpendicular to the symmetry axis, it undergoes both a nutational and rotational motion so frequencies based on both v , and v 2 appear. For u2, only the first harmonic, i.e., 4 ( A - B ) K , is relevant since a rotation about the symmetry axis of 180’ returns the dipole to its original direction. Both the time dependence and polarization dependence of the rotational coherence of individual molecules can be derived from this model. When the above behavior of individual excited-state molecules, which are partially aligned by the excitation pulse a t t = 0, is coupled with the postulate that J and K may assume only integral values, the following quantitative picture of the macroscopic behavior of P R C is obtained: (1) For parallel transitions (dipole parallel to the symmetry axis), a total spatial realignment or rephasing of dipoles occurs a t the fundamental nutation frequency 2B‘since all molecules nutate a t some multiple of that frequency. ( 2 ) For perpendicular transitions, rephasings occur a t 4(A’-B’), which is twice the fundamental rotation frequency, as well as the fundamental nutation frequency 28: but these rephasings are only partial unless the two frequencies are commensurable. Recurrences in fluorescence or absorption intensities associated with these two patterns of dipole rephasings constitute the principle observable consequences of P R C in symmetric tops. B. PRC in .4symmetric Top Molecules. For the classical asymmetric top, no principal axis maintains a fixed angle 0 with J, leading to a more complicated motion than that described above. For a given J , the type of behavior will depend on the total rotational energy. At low and high energies, the motion will approximate that of a symmetric top but with the effective symmetry axis changing from the c-axis a t low energy to the a-axis at high energy. For a range of intermediate energies determined
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5703
Excited-State Rotational Constants and Structures by the degree of asymmetry, the motion is very irregular. This leads to a loss of commensurability of the nutation and rotation frequencies associated with different initial states, SO that the regularly spaced sharp recurrences seen in the symmetric limits are expected to broaden (initially) and diminish as the asymmetry increases. The same result derives quantum mechanically from the irregular spacing of the asymmetric top rotational energies. Since asymmetry shifts rotational levels away from the regular spacing seen in the symmetric top, determination of the expected P R C recurrence periods is no longer trivial. In fact, strictly periodic recurrences are not expected since no single frequency will be shared by all molecules of the sample. However, simulations have shown that, even for fairly large asymmetries, the equal spacing of clearly defined nutation-like recurrences is retained to good approximation. In contrast, distinct, periodic rotation recurrences are seen only for very small asymmetries. We will refer to the (quasi-) periods associated with these two types of recurrences as T~ and 72, respectively. To derive approximate expressions for the excited-state rotational constants in terms of 7 , and T ~ one , must start with the rotational energies. For low values of J , it is possible to represent the energies of an asymmetric rigid rotor in power series expansions in J ( J 1) and the asymmetry parameter b = 1/2(C- B ) / [ A 1 / 2 ( B+ C ) ] ,which goes to zero in the prolate symmetric limit.* (An equivalent expansion in b* = 1 / 2 ( A- B ) / [ C+ B)] is useful when the molecule is oblate. The discussion below will be based on the prolate case.) Energy levels are each designated by a value of K from 0 to J with two levels for each nonzero K . From these expansions, the following conclusions may be drawn: (1) For small b, the energy expressions are approximated by those of a symmetric top with B replaced by B s ‘12(B+ C ) . A first estimate of rotational constants is thus provided by the modified symmetric top relations
+
B’+ C’
N
l/T1
(2.3)
and 4 [ A ’ - J/z(B’+ cq]
N
1/72
(2.4)
(2) Leading corrections to the symmetric top like expressions are of second order in b for all levels, except for a first-order splitting of the K = 1 levels. Since a pure level splitting results in quantum beat frequencies symmetrically displaced from the unperturbed value, it will affect recurrence amplitudes but have little effect on recurrence periods in thermally averaged P R C signals. Therefore, the b2 terms are the first terms capable of causing deviations from the expressions given above. The derivation of an approximate correction to relation 2.3 based on expansion terms up to b2 is given in Appendix A. This equation takes the following form
B’+
c’=
1/71
- U[A’- 1/(2T1)]b2
(2.5)
where a is a molecule- and temperature-dependent factor. P R C simulations indicate that this expression accounts well for the dependence of the asymmetry correction on b, while the empirically determined values of a are within the range 1.4-4.7 for all rotational constants and temperatures considered. These results suggest that, given an experimentally determined recurrence period and approximate values of A’and b, one may obtain a first estimate of the asymmetry correction to B’ C’by setting a = 3 and applying eq 2.5. When the correction is small, this estimate will be sufficiently accurate for most purposes. When the correction is larger, or higher accuracy is desired, simulations specific to the system of interest may be required.
+
111. Experimental Section
The experimental apparatus and procedures used in our laboratory for PRC fluorescence measurements have been described previou~ly.~ Briefly, we employ mode-locked and synchronously pumped dye laser excitation of jet-cooled molecules and time(8) Polo, S. R. Con. J . Phys. 1957, 35, 880.
1
correlated single photon counting detection to time resolve fluorescence from isolated molecules. Response times of 60 ps and under are regularly attainable from a microchannel plate photomultiplier. The excitation bandwidth is either 5 or 2 cm-l, depending on the tuning element employed. The sample compounds were obtained from commercial sources (Aldrich, MCB, Pfaltz and Bauer) and used without further purification. In the interest of obtaining the lowest possible rotational temperature of the sample, neon at a pressure of approximately 6 atm was used as the backing gas in the molecular beam expansions for most experiments. Argon a t low concentrations was mixed with the neon to product t-stilbene-argon van der Waals complexes, while helium complexes of t-stilbene were formed with backing pressures of pure helium up to 45 atm. One aspect of these measurements that was not fully discussed in ref 3 was the question of accuracy. Since the accuracy of quoted rotational constants depends fundamentally on an accurate determination of the time scale of our experiments, we describe here the procedure used for the time calibration. A primary calibration standard of high precision is provided by the mode-locked laser system used in our experiments. The repetition rate of our argon ion laser is locked to twice the frequency of the acoustic wave in the mode-locker crystal. The driving signal is derived from a stabilized frequency source that also governs the dye laser cavity dumper. The agreement of the displayed mode-locker frequency with the measured dye laser repetition rate to better than 1 part in lo5 serves to confirm the accuracy of this value. From it, the average argon ion pulse separation of about 12 ns is determined with the same high degree of accuracy. When the full time range of the time-to-amplitude convertor (TAC) is greater than this 12-11s separation, two or more pulses may be detected simultaneously and recorded on the multichannel analyzer (MCA). .”Under the assumption of strict periodicity of the laser pulses and both stability and linearity of the TAC-MCA combination, uncertainty in the measured channel separation of the pulses becomes the limiting uncertainty in the time base calibration. To perform a calibration, scattered laser light was collected and recorded in a calibration file showing two or more well-defined laser pulses. When the recorded pulses are regular in shape, their separation may be measured by determining the shift of data points required to bring one pulse into congruence with another. In this case, a clear degradation of the overlap of the pulses occurs for changes of a fraction of a channel in the shift. An upper limit of a channel half-width uncertainty was assumed in the typical measurement of an 800-channel separation equaling a relative uncertainty of less than 0.07%. Both the short- and long-term stability of the time base have been tested by frequent calibration. The results have shown that variations in the calibration over spans of days or even weeks often did not exceed the uncertainties of a single measurement. Periods of fluctuation and abrupt changes on the order of 1% have been observed, however. Therefore, for measurements requiring the highest degree of accuracy, the time base was calibrated both before and after the experiment. If either the TAC or M C A has a nonlinear response, a calibration of the system establishes only an average time scale, while the true local time scale will vary from position to position. A standard test of linearity consists of measuring the arrival times of randomly distributed pulses. Since these events occur with equal probability in each time division of equal length, the number of counts accumulated in each channel after a sufficient collection time, and in the absence of pulse pileup effects, indicates the duration to which that channel corresponds. Preliminary tests of this type have been carried out and the small influence of pulse pileup accounted for. These indicate that the effect of nonlinearity on our measurements may be as large as 0.1% in the case that the measurement is made over a small segment of the calibration range. The effect is substantially reduced when the measurement spans the full range of the calibration. To push the accuracy of this technique beyond the present level, a fuller characterization of this possible source of error is needed. Its bearing on the measurements presented here will be considered where appropriate.
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The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
I
I
I
I
I
I
I
I
Time (ns) Figure 2. Example of a parallel fluorescence measurement used in the determination of the PRC recurrence period of I-stilbene. The data are shown as measured and after suppression of the fluorescence decay. (Deviation from exponential decay at long times, due presumably to ion feedback in the microchannel plate detector, have also been compensated for in the flattened data.)
When the time range is shorter than the 1241s pulse separation, an indirect method of calibration is required. The most convenient is the use of a delay cable that has itself been calibrated against the laser repetition rate. When the increased uncertainty inherent in such a secondary method is acceptable, the simplicity of this procedure makes it useful for longer time scales as well. I n summary, the calibration procedure described above appears to provide an accuracy that is limited principally by the number of channels on the MCA and the width of the laser pulses, provided that the influence of nonlinearities is given proper consideration.
IV. Data Analysis The nature of purely rotational coherence lends itself to two semi-independent types of analysis. The first is the determination of recurrence periods, and the second is the fitting of data to numerical simulations. For the first, the optimal time range for data collection includes many recurrence periods. For the second, resolution of the details of recurrence shapes and amplitudes is of primary importance and a shorter time scale is usually preferable. A full description of each of these methods of analysis follows. With the established time-scale calibration, rotational recurrence periods and associated error bars are determined in the following manner. Recurrence positions in a measured decay are located by inspection, and uncertainties corresponding roughly to 90% confidence intervals are estimated for each one. If the fluorescence lifetime is short, it is first removed to prevent biasing of the positions. This may be accomplished by multiplying the data by erIr, where T is the measured lifetime. A weighted least-squares fit to the estimated positions yields the recurrence period. The quoted uncertainties are found conservatively by considering the maximum and minimum periods consistent with the error bars on the extreme recurrences. For the values reported here, these conservative error estimates have been reduced only when warranted by the good agreement between a series of independent measurements. Figure 2 presents an example of a rotational coherence measurement used to determine the recurrence period of t-stilbene. The decaying curve is the measured fluorescence signal with the polarization analyzer set to select the fluorescence component with polarization vector parallel to that of the excitation pulse (henceforth referred to as parallel fluorescence). Also shown at
Baskin and Zewail the top of the figure is the same data after processing to remove the fluorescence decay. In this, the regularly spaced in-phase recurrences may be located and their positions accurately measured. On the basis of a single measurement of this type, the PRC recurrence period of t-stilbene can be determined with an uncertainty of 5 4 ps. As seen in section 11, precise values for the rotational constants of asymmetric tops are not given directly by P R C recurrence periods as in the symmetric top case. Even the approximate expressions derived there do not permit the determination of independent B’ and C’values without some additional constraint (such as the assumption of planarity). T o extract as much potential information as possible from the experimental data will therefore require comparison with numerical simulations designed to accurately reproduce the conditions of the measurement. The fundamental component of these simulations is the expression for the time evolution of polarization analyzed and thermally averaged fluorescence derived in ref 2, section V. The pertinent equations, with correction of some minor errors appearing in the original printing, and a brief description are given in Appendix B. There are three important assumptions involved in each P R C simulation in this paper, and these are stated here together for completeness: ( 1) The initial population distribution is assumed Boltzmann. (2) A rigid-rotor rotational Hamiltonian is assumed. (3) The possibility of axis switching in excitation is n e g l e ~ t e d . ~ It should be kept in mind that any of these assumptions may lead to inaccuracies since they need not be justified in all cases. Further limitations are placed on the accuracy of the calculations by the finite temporal and frequency resolution, frequency bandwidth, and range of ground-state rotational levels included. However, checks can and have been made to ensure that the choice of these parameters has no significant influence on the results. Typical values used in the present work are 4 ps and 1 M H z for temporal and frequency resolutions and 50 G H z for the frequency bandwidth. An energy cutoff was set for the inclusion of ground-state rotational levels. This was preferable to a simple limit on Jo since the overlap of J manifolds is very pronounced in many of the molecules studied. It was found that simulated intensities reached their asymptotic values when ground-state levels with J o up to 25-35 (depending on temperature and molecule) were included in the summation. Similarly, the contributions of minor isotopic species due to the natural abundance of I3C have been considered and will be discussed in relation to the t-stilbene results. These contributions can be important but were found negligible for the purposes of the present work. The calculation of P R C intensities requires, in principle, the specification of eleven parameters: the rotational temperature, six rotational constants (A”, B”, C”, A’, B’, C?, and the directions of the absorption and emission dipoles ( O , , @,, 82, @2). The ground-state structure exerts an influence on P R C in two ways: the ground-state energies determine the initial state distribution, and the eigenstates govern the transition strengths to each upper state. However, it has been found that results of simulations are quite insensitive to the exact values of the ground-state constants, as might be expected since the beat frequencies responsible for the appearance of recurrences depend only on the excited-state energies. Thus. it is sufficient in practice to specify a single set of rotational constants. In addition, in the cases considered here, a single electronic transition is involved in both absorption and emission, implying a single transition moment. This reduces the effective number of simulation parameters to six. The computer fitting routines that have been employed in the data analysis for this paper treat only four of these six as adjustable variables, leaving the transition moment fixed. To perform the least-squares fitting, calculated intensities were convoluted with an instrument response function and compared to the experimental data. The response was obtained by measuring ( 9 ) Herzberg, G . Molecular Spectra and Molecular Structure; Van Nostrand Reinhold. New York, 1966, Val 111, p 208.
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5705
Excited-State Rotational Constants and Structures
V. Results and Discussion In each of the four subdivisions of this section, we present analyses of PRC measurements on a different molecule (or related group of molecules). Special features of each case permit us to highlight different aspects of the P R C analysis in each discussion. In figures accompanying these discussions, the characteristic decay of fluorescence intensities has been suppressed (as described in connection with Figure 2) in order to facilitate the visual comparison of those features relevant to PRC. A . t-Stilbene. Extensive measurements of the polarizationanalyzed fluorescence of 1-stilbene (CI4Hl2)have been described Our purpose here is to use the information contained previ~usly.~ in these measurements to improve our knowledge of the t-stilbene S, electronic state to the fullest extent possible. Specifically, we
(1) account for the influence of asymmetry and possible nonlinearity of the measurement-system time scale in deriving a value of B’ + C’ and (2) fit measurements to P R C simulations in an effort to establish values for all three excited-state rotational constants. Since the results of these fits are found to be inconsistent with any reasonable t-stilbene structure when an a-axispolarized transition is assumed, the consequence of varying the direction of the transition moment in the simulations is investigated. Our attention will be restricted to the SI So Ooo transition exclusively (Acx = 3101.4 A). The t-stilbene recurrence period of 1948.4 ps reported in ref 3 was derived from a number of independent measurements taken over various portions of the calibration range. A consistent trend can be seen in these measured periods, ranging from 1946.5 to 1950.8 ps, which agrees well with other indications of a slight systematic nonlinearity in the measurement time base (see section 111). To eliminate any possible bias from this effect, we use only measurements over the full calibration range to derive a revised value of 1947.7 A 2.5 ps. The quoted uncertainty is justified on the basis of the small variation among many measurements, including those over only part of the calibration range. The apparent correlation between measurement range and measured period suggests that this uncertainty is an overestimate, but this correlation has not been convincingly established. We emphasize that the period and uncertainty given above are not based on any assumptions about the system linearity. To derive an accurate value of B’ + C’ from this period, the effect of asymmetry must be considered. This was not taken into account in earlier treatment^.^ As the following discussion will indicate, 2.6 GHz and -0.0054 may be taken as reasonable values for A’and b. On the basis of these values, both application of eq 2.5 (with a = 3) and analysis of P R C simulations show that the asymmetry correction combined with the slightly changed value of 71 results in a B’ + C’ that differs negligibly from the value of 0.5 132 G H z previously reported. The additional uncertainty in B’ + C’ arising from the uncertainty in this correction is no more than 0.0001 GHz, giving a final revised value of B’ C’ = 0.5132 A 0.0008 G H z . This value applies to ‘*CI4Hl2and is unaffected by the presence of I3C isotopes (see text below). Preliminary values of the individual rotational constants were also deduced in ref 3 by visually comparing the widths and amplitudes of measured recurrences to those produced by simulations in which the dipole was held fixed along the a inertial axis. Simulation parameters reported to produce the best agreement with experimental data obtained by excitation of the Ooo transition were T = 2 K, A ’ = 2.676 GHz, B’= 0.273 GHz, and C‘= 0.240 GHz. The larger than expected separation of the B’and C’values was needed to approximate the small amplitudes and large widths of the measured recurrences. Note that A’was left fixed for these simulations at a value calculated from an experimental groundstate structure,” while relation 4.1 actually requires that A’be less than 2 G H z to be consistent with the proposed B’and C’. Fits of data to PRC simulations assuming an a-axis dipole have now been carried out and are in basic accord with the earlier conclusions; Le., a rather large separation of B’and C’is needed to reproduce the measurements.,For example, we show in Figure 3 a measurement of the first oud-of-phase and first in-phase recurrence in the parallel component of stilbene fluorescence and the corresponding best-fit, planar molecule, a-axis-polarized PRC simulation. The weighted residual is shown above the data; the reduced x2 value of 1.04 indicates the generally satisfactory quality of the fit. The resulting parameters are T = 2.8 K, A ’ = 2.182 GHz, B’= 0.2716 GHz, and C’= 0.2416 GHz. (The short time scale of this experiment was not calibrated directly but determined by setting B’+ C’ = 0.51 32 GHz.) When the constants were not constrained by planarity, a slightly better fit was found by further separation of B’and C’to 0.2730 and 0.2402 G H z without reducing A’. This latter is again a physically unrealizable set of constants, as seen from relation 4.1.
(10) Meyer, P. L. Introductory Probability and Statistical Applications; Addison-Wesley: Reading, MA, 1970; p 139.
(1 1) Traetteberg, M.; Frantsen, E. B.; Mijlhoff, F. C.; Hoekstra, A. J. Mol. Struct. 1975, 26, 57.
laser light scattered off the nozzle through the normal detection system immediately before or after the experiment. The comparison of simulation and experimental data entailed the introduction of two additional fitting parameters: a scaling factor to adjust the simulation amplitude and a time shift to synchronize the temporal evolution of the two files. The experimental data could be fit either in the form of parallel fluorescence or of polarization anisotropy (R(t)). Methods employed to find the best-fit parameters included a simple grid search of likely regions of the parameter space and optimization routines either without constraint or constrained by the condition of planarity. In choosing initial values for the rotational constants, two generally applicable relations were useful. The first is A-’ 2
p - E’
(4.1)
which holds (to very good approximation) for any set of rotational constants. (Relation 4.1 is not rigorous for effective rotational constants; Le., A-I may be slightly smaller than C1- B1as a result of averaging over the vibrational state.) The second depends on the measured nutation recurrence period: B’+ C’= 1/7] if the molecule is prolate or A ’ + B’= l / r l if the molecule is oblate. The rotational constants of the ground state could either be held fixed at predetermined values or be equated to the excited-state values. A x 2 test was used to judge the goodness of fit. Poisson statistics for our single photon counting data are assumed for this purpose. In the case of R(t), approximate expressionsI0 are used to characterize the distribution that is not Poisson. When the parallel fluorescence was fitted, the decay lifetime was eliminated from the fitting process by multiplying both the data and the measured response by e‘/‘, with appropriate adjustment of weights used in the x2 calculation. For many of the most time-consuming calculations of asymmetric top PRC intensities, a near-symmetric treatment was used. This treatment is based on the fact that the asymmetric top eigenfunctions IJTM) must transform smoothly into symmetric top eigenfunctions as B and C (or A and B) converge. Thus, only coefficients with a single value of IKI in the expansion J
IJTM) =
2 ~(JTK)(JKM)
K=-J
can have appreciable amplitude for small asymmetries. In the near-symmetric treatment, all other coefficients are assumed negligible in the evaluation of eq B2. This is equivalent to applying the approximate selection rules AKsp = 0, f l , where Ksymstands for Kpror Kob,the K designation of the closest symmetric top limit. In this approximation, the only consequence of asymmetry is the shift in rotational energies that causes changes in P R C beat frequencies. This treatment was used regularly for fitting and off-axis dipole calculations and was found to give satisfactory results even for moderately large degrees of asymmetry while providing a substantial savings in computer time. In some cases, the influence of noise on our measurements has been tested by the addition of random noise to simulated data. Poisson statistics are assumed for this purpose.
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The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 I
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Figure 4. Structure of t-stilbene in SI based on calculations of ref 13. See text for details. The in-plane inertial axes are shown. I
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0.5
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1.o 1.5 Time (ns)
u 2.0
2.
Figure 3. Fit of the experimentally measured first out-of-phaseand first in-phase recurrences of t-stilbene PRC with a n a-axis transition dipole (0 = OO). An instrument response of 52-ps fwhm was measured with the data and used in the fit. The rotational constants found are given in the text and are not compatible with any reasonable stilbene structure. The weighted residual is plotted above the data. The fit of Figure 3, as for others in this paper, was carried out with the constants for a single isotopic species, in this case l2CI4Hl2. Since the seven distinct species of 12C13‘3CH12 comprise 13.3% of the total sample and have recurrence periods ranging up to -22 ps longer than that of the principal isotope, their effect on the observed signal must be considered. (Multiply substituted isotopes make up only I % of the total and can be ignored.) At long times, accumulated phase differences will disperse the recurrences of the minor isotopes and the observed recurrences should have amplitudes approaching -86% of those of a single species. Simulations confirm that this limit is reached after five or six periods but also show that the isotope effect on the fluorescence intensity over the time range shown in Figure 3 is negligible. This justifies the fitting procedure used. The shifts in recurrence peaks ( € 2 ps at all times) induced by the inclusion of minor isotope fluorescence in the simulations result in an inconsequential change (
