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Lifetimes of vibrational levels in the B̃2A′ state of HCO
19 April 1996
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 252 (1996) 333-342
Lifetimes of vibrational levels in the B
state of HCO*
Joseph D. Tobiason 1, Eric A. Rohlfing Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551, USA
Received 20 November1995; in final form 8 February 1996
Abstract
We measure the fluorescence decays for low-lying rotational states in the first eight vibrational levels of HCO B 2.aL Some long-lived decays exhibit weak quantum beats that we attribute to a higher-order hyperfine perturbation in the 13 state. We estimate fluorescence quantum yields by combining measured fluorescence intensities and a calculated electronic transition moment. The estimated nonradiative decay rates increase with increasing vibrational excitation and exhibit mode specificity. We discuss possible vibronic coupling mechanisms, including spin-orbit coupling to quartet states and predissociation via the ~kLH(2/~') and ~2p~ states.
1. Introduction
The 132/~_~2g bands, or 'hydrocarbon flame bands', of the formyl radical, HCO, have been the subject of numerous experimental investigations over the years [1]. The motivation for much of this work is the importance of the formyl radical in both combustion [2] and atmospheric [3] chemistry, particularly as a source of atomic hydrogen through its unimolecular dissociation on the ground-state potential surface [4]. The f3-X system provides an effective means for accessing bound vibrational states and quasibound vibrational resonances over a wide range of energies in the ground state [5-7]. We have used this system in a recent study of the energies and
* This work is supported by the US Department of Energy, Office of Basic Energy Sciences, Chemical Sciences Division. l Present address: Center for Technology,Kaiser Aluminum & Chemical Corporation, 6177 Sunol Boulevard, Pleasanton, CA 94566, USA.
widths of HCO resonances using dispersed fluorescence (DF) and stimulated emission pumping (SEP) spectroscop_ies [8]. The B - X system also offers an effective means of probing concentrations of HCO in combustion environments. Several recent high-resolution laser studies have provided the spectroscopic characterization necessary for such laser-based diagnostics. These studies include experiments using laser-induced fluorescence (LIF) in cells by Sappey and Crosley [5] and Adamson et al. [6,7], LIF in free-jet expansions and in cells by Chen and co-workers [9-12], and resonant two-photon ionization (R2PI) in free-jet expansions by Cool and Song [13]. Besides providing spectroscopic data that characterizes the rovibrational structure of B-state HCO, these studies have probed the dynamics of the excited state. For example, the LIF intensities drop dramatically with increasing energy in the B state [5] whereas the R2PI intensities [13] do not. Also, lifetime broadening is observed in the R2PI spectrum at high energies. These observations have provided the first evidence
0009-2614/96/$12.00 © 1996 Elsevier Science B.V. All rights reserved PI1 S0009-2614(96)00163-7
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J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
for nonradiative processes that quench the fluorescence from the B state. There have been several direct measurements of fluorescence lifetimes of the rovibrational levels in the 13 state, including the initial LIF and quenching work by Crosley and co-workers [5,14] and the more recent series of studies by Chen and co-workers [10-12]. Shiu and Chen [10] measured the lifetimes of rotational levels with K ' a = 0 and K 'a = 1 in the first six vibrational states. Subsequently, Lee and Chen [11,12] have concentrated their efforts on the effects of rotational motion on the lifetime of the vibrationless level of the B state. They find a sharp decrease in lifetime with K'a, which they attribute to a-type Coriolis coupling of the 132,~ state to the ,A2II (2A~') state. In addition, they observe that the lifetime decreases slowly with increasing N' and ascribe this effect to much weaker Coriolis coupling mechanisms: b-type coupling to the ,~ state or c-type coupling to the X state or both. The low-lying doublet electronic states of HCO have been extensively studied theoretically and are known to be coupled by nonadiabatic effects [15-21]. The ,A and X states form a Renner-Teller pair that are degenerate at the linear geometry and their coupling is manifested in the well-documented predissociation of the ,A state [4]. There is also a conical intersection between the B and X states that lies well above the low C - H bond dissociation energy on the ground state [20] and thus offers a mechanism for predissociation of the I3 state. The 13 state also undergoes another avoided crossing with a 3s(2/~) Rydberg state [18]. This crossing, which affects the electronic character of the B state at large H - C - O angles, should not influence the predissociation since the dissociation asymptote of the Rydberg state lies far above the B-state minimum. Finally, the two lowest lying quartet states of HCO, 4A" and 4~, which calculations show lie slightly above the 13-state origin [21], may spin-orbit couple to the 13 state. Thus, intersystem crossing is another candidate for nonradiative decay out of the B state. The purpose of this Letter is to describe our lifetime measurements for the first eight vibrational levels of HCO 132A~, which began as a part of our fluorescence-depletion SEP studies of the ground state [8]. The emphasis in our jet experiments is on purely vibronic effects, which we accomplish by
examining low-lying rotational states where rotational coupling mechanisms are minimized. The three vibrational modes of HCO X can be readily described as a C - H stretch(v~), a C - O stretch (v~), and a bend (v~). In the B state, v~ remains a C - H stretch, however there is ample evidence [13,22] that the character of v~ is predominantly bend and that of v~ is mostly C - O stretch. In addition to providing lifetime data, we also observe weak quantum beats in the fluorescence decays taken upon excitation of a specific rotational transition terminating on the long-lived vibrational levels in the B state. We do not believe that these beats are indicative of widespread couplings of the 13 state to another electronic state, but rather that they are due to an additional hyperfine perturbation within the 13 state. Finally, we use our previous DF measurements of the relative intensities in the B - X system [8] and a calculated value of the electronic transition moment [21] to estimate the quantum yield of fluorescence, and hence the nonradiative rate, for each rovibrational level. The nonradiative rate increases rapidly with increasing vibrational energy in the 13 state and also exhibits mode specificity. We discuss possible mechanisms for nonradiative processes, including intersystem crossing to nearby quartet states and predissociation via the A and X states.
2. Experimental details and lifetime determinations We generate HCO by the excimer-laser photolysis (308 nm) of acetaldehyde (CH3CHO), which is seeded at a concentration of ~ 5% into helium at a total pressure of 1-2 atm and expanded via a piezoelectrically actuated pulsed valve through a 1-mm orifice. A 500-mm focal-length lens focuses the photolysis beam (pulse energy 40-100 mJ) on the expansion 2-5 mm downstream from the orifice. The excitation laser is a Nd : YAG-laser-pumped dye laser (Continuum NY81C/Lumonics HD-500) that produces pulses of 10 IxJ-1.5 mJ in energy in the 239-259 nm region with a duration of 6 - 7 ns and a bandwidth of 0.1-0.2 c m - 1 . The excitation spot size is 1-2 mm in diameter where it crosses the expansion 10-20 mm downstream from the valve orifice. The LIF excitation spectra of HCO obtained in this
J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996")333-342
region of the expansion are consistent with rotational temperatures of 5-15 K. We collect fluorescence decays upon excitation of qR0, qPo, rQ0 and rR 0 transitions (notation: of the first eight (cold) vibronic bands of the B - X system. Table 1 lists the excited-state vibrational levels (notation: (v'l, v~, v~)), the rotational levels accessed (notation: and J ' = N ' + 1/2), and the vibrational energy in excess of the B-state origin. A photomultiplier tube (PMT, Hamamatsu R955), which is mounted directly on a window above the chamber, detects the fluorescence at right angles to the jet and excitation beams. We use a single lens to provide a 1 : 1 image of the PMT photocathode at the interaction region. This arrangement provides a viewing region of at least 8 mm along the jet axis and insures that the observed fluorescence decays are not affected by molecular transit through the viewing region. A 300 nm long-wave-pass filter in front of the PMT reduces scatter from the photolysis and excitation pulses. The photolysis pulse is also delayed in time by ~ 6 p.s from the excitation pulse. A 1 GHz digital oscilloscope (LeCroy 7200 with 7242B plug-in) digitizes the fluorescence decay signals from each laser pulse, averages 200 such decays, and stores the signal-averaged transient. The oscilloscope is triggered off a signal from a photodiode that detects the 532-nm light from the N d : Y A G pump laser. We also collect an averaged background obtained under the same conditions but with the excitation laser detuned from resonance. Subtraction of the background signal yields a corrected fluorescence decay, examples of which are given in panel (a) of Figs. 1-3. We determine lifetimes from the observed fluorescence decays from fits to a model function that is a convolution of an effective excitation pulse with an exponential decay. For each set of experimental runs we collect an effective excitation pulse either by reflecting an extremely attenuated excitation beam onto the PMT or by detecting Rayleigh scattered light from ambient air. Both methods give identical results. The effective excitation pulse is a convolution of the actual laser pulse with the temporal response of the detection system (PMT and subsequent electronics). At the low PMT bias voltage used in these experiments (450-540 V) the effective excitation pulse shape (shown in Fig. l(b)) is signifi-
ar~AJr,~)
N~,or,
335
cantly wider ( = 12 ns full width) than the actual laser pulse and is slightly asymmetric with a longer trailing edge. We use a fast iterative convolution technique [23] to numerically convolute (using trapezoidal integration) the effective excitation pulse with an exponential decay and we fit the corrected fluorescence decay to this convolution using a standard Levenberg-Marquadt nonlinear least-squares fitting routine [24]. The convolution of the effective excitation pulse is quite capable at modeling our observed decays and allows us to determine lifetimes down to a lower limit of ~"> 1 ns. In addition, because of the longer trailing edge of the effective excitation pulse, the convolution is necessary even for long-lived states (~-> 50 ns). In these cases, a fit tO the falling part of the observed decay with a simple exponential overestimates the lifetime by as much as 15 ns.
3. Results and discussion We present representative fluorescence decays and fits in Figs. 1-3. As noted above, the convolution of the effective excitation pulse with an exponential decay provides a model that fits our observed decays quite well. The only significant deviations occur for decays from short-lived states. In these cases there is sometimes a discrepancy between the leading edge of the convolution (which is dominated by the leading edge of the effective excitation pulse) and the observed decay and this gives rise to large residuals near zero time (see Fig. 1). We summarize the lifetimes obtained from many decay fits in Table 1. The errors listed in Table 1 are determined from fits to several different decays for each level (often reached by different rotational transitions) and from the effects of varying the time offset in fitting an individual decay. The purely statistical error from a single fit to an individual decay is usually smaller. When we excite the 101 state of the (0, 0, 0) and (0, 0, 1) levels via the qR0(0) transition we observe reproducible oscillations in the residuals from the decay fits. Figs. 2 and 3 display typical examples of these very weak oscillations, which have modulation depths of 1-2%, that we attribute to hyperfine quantum beats. Quantum beats occur when a single longitudinal mode of the laser excites a coherent superposition of molecular eigenstates that are comprised of
336
J.D. Tobiason, E,A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
0.3
t-
0.2
~
0.'1
(a) Fluorescence Signal and Fit
~
0.0
(b) Excitation Pulse
I
50
I
100
I
150
Time (ns)
O
0"2 f (C) Residuals t 0.0 --":" ....y"":"--"~.~---.--.-.-~-."-".....................................
v~ -0.2
Fig. 1. Fluorescence decay from the 101 (J' = 3/2) state of the (0, 0, 2) level of HCO [~2~ excited via the qR0(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~-= 17.5 +0.2 ns. (b) Excitation pulse used in the convolution. (c) Fit residuals (fit minus experimen0.
However, we observe the quantum beats only for the 10t rotational state and only upon excitation of the qR0(0) transition. Excitation of the 10t state via the qP0(2) transition does not produce observable quantum beats. Because of the extreme specificity with respect to rotational transition we do not believe that the observed quantum beats are indicative of a general coupling to other quartet or doublet electronic states, since such couplings should produce quantum beats over a broad range of rotational levels and transitions. A more plausible mechanism for producing the quantum beats involves a higher-order interaction among the hyperfine levels within the 10l rotational state. Fig. 4 displays the energy level schematic appropriate to the 101 rotational state in the (0, 0, 0) level of [l-state HCO. Spin-rotation interaction splits the 101 level into the J ' = 3 / 2 (F 2) and J ' = 1 / 2 (F 1) components, which are further split by the magnetic hypeffine interaction with the hydrogen nucleus ( I = 1 / 2 ) to produce two pairs of hypeffine doublets ( F ' = 0 and 1; F ' = 1 and 2). The energy ordering of
1.0
0.8
optically bright and dark basis (zero-order) states [25-30]. This coherent superposition is time-dependent (since it is not an eigenstate) and the fluorescence beat pattern and modulation depth depend on the number or eigenstates coherently excited and their fractional bright character. The clean, singlefrequency beats seen in Figs. 2 and 3 suggests the superposition of just one bright and one dark eigenstate. We fit the residuals to a exponentially damped sinusoidal function in order to extract the beat frequencies listed in Table 1. There are several possibilities by which darker basis states might couple to the bright [i-state levels to produce hyperfine quantum beats. One is spin-orbit coupling to one or both of the quartet states. However, the calculated origins of the 4A¢' and anl state lie well above the (0, 0, 0) level of the [l state [21], thus precluding spin-orbit coupling as the source of the quantum beats. Another origin for the quantum beats might be Coriolis coupling to the ,~ or X states.
c "~ 0.6
~2 0.4.
a) Fluorescence Signal
0.2 o.o I
200
of O
I
I
I
400 600 Time (ns)
800
s
0.0 -0.5
Fig. 2. Fluorescence decay from the lol (J' = 3/2) state of the (0, 0, 0) level of HCO B2A~excited via the qRo(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~"= 100.1 +0.2 ns. (b) Residuals from exponential fit and fit to a damped sinusoidal function that gives a beat frequency of 16.1 + 0.1 MHz.
J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
the hyperfine levels depends on the relative magnitude of the spin-rotation and hyperfine splittings. The spin-rotation splitting is very small for K'o -- 0 states and is equal to the constant, ~', which is independent of N' [31]. For ground-state HCO the spin-rotation splitting (/z" = - 9 4 MHz) is roughly equal to the hyperfine splitting [32], but neither the spin-rotation splitting nor the hyperfine splitting has been accurately measured for the K ' a -- 0 states of the (0, 0, 0) level of the I~ state. The most recent measurement by Lee and Chen [11] suggests /z'-- - 3 0 MHz, however the error bars for this value are large ( + 180 MHz) and encompass both positive and negative values of/z'. In Fig. 4 we assume that /z' is negative and that the hyperfine splitting is significantly
337
(roughly four times) larger than the spin-rotation splitting. In this case, the two lowest hyperfine components are both F' = 1. These two states of common N' and F' can be further coupled by magnetic hyperfine matrix elements that are off-diagonal in J'. This additional hyperfine interaction is known to be significant in ground-state HCO and has been observed as significant perturbations in the microwave spectrum [32]. We postulate that the quantum beats observed upon q R 0 ( 0 ) excitation originate from the coherent excitation of the eigenstates labeled l i) and I k) in Fig. 4. These are formed from the mixing of the F' = 1 basis states that originate from J' = 3 / 2 and J' = 1/2, which are labeled l a) and I b) in Fig. 4.
Table 1 HCO B-state lifetimes and estimated fluorescence quantum yields Vibrational level
Excess energy
(v'l,v'2,v'3)
(cm- 1)
(0, 0, 0)
(0, O, 1)
(0, 1, O)
(0, 0, 2)
(0, 1, 1)
(1, O, O)
0
1065
1380
2114
2430
2597
(0, 2, 0)
2773
(0, 0, 3)
3146
Rotational level
,
NK'K'c lol 202 11o 211 lol 202 11o 211 lol 202 11o 211 101 202 11o 211 lol 202 11o 211 lol 2o2 11o 211 lol 202 lol 2o2
j, 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 3/2 3/2,
5/2
5/2
5/2
5/2
5/2
5/2
5/2 5/2
Lifetime
Estimated b
qRo(0)
(ns) a
qBf
beat freq. (MHz) a
99 (3) 89 (4) 48 (2) 39 (2) 73 (3) 64 (3) 15 (2) 16 (2) 12 (2) 12 (2) 5.5 (10) 5.5 (10) 17 (2) 20 (2) 8 (1) 8 (1) 1.5 (10) 1.2 (8) <1 <1 15 (2) 14 (2) 4 (1) 4 (1) <1 <1 < 1 < 1
0.92 0.83 0.45 0.36 0.73 0.64 0.15 0.16 0.13 0.13 0.06 0.06 0.17 0.20 0.08 0.08 0.01 0.01 < 0.01 < 0.01 0.15 0.14 0.04 0.04 < 0.01 < 0.01 < 0.01 < 0.01
16.1(1)
a Errors in parentheses. b Quantum yield estimates based on calculated transition moment from Ref. [21] (see text).
14.9 (1)
J.D. Tobiason, E~A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
338 0.4
As illustrated in Fig. 4(a) by the shaded boxes, I a) is completely bright in absorption via the qR0(0) transition while I b) is completely dark since it is not connected to the ground state rotational level (00o, J" = 1 / 2 ) by a dipole-allowed transition. Thus, quantum beats occur because l i> is mostly bright and I k) is mostly dark. However, the depth of the fluorescence modulation also depends on the bright and dark character of the eigenstates with respect to emission (which is not shown explicitly in Fig. 4). For each vibrational band in the B - X system state I a) (with J ' = 3 / 2 ) can fluoresce via four rotational transitions, q R 0 ( 0 ) , qP0(2), P Q I ( 1 ) , and PPI(1). By
-
0.3 (a) Fluorescence Signal ¢-
~d
0.2 -
0.1
0.0
-
-
"~
I 1 O0
I
I
I
200
300
400
Time (ns) 10
-10
l-
: ;" ii :'-'.
(b) Residuals and Fit
j
Fig. 3. Fluorescence decay from the 101 ( J ' = 3 / 2 ) state of the (0, 0, 1) level of HCO B2,K excited via the qR0(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~" = 74.7+0.5 ns. (b) Residuals from exponential fit and fit to a damped sinusoidal function that gives a beat frequency of 14.9 + 0.2 MHz.
(a)
lol
?/
qR0(O)
la~ ~ i
. . . . . . . . . . . . . . . . . . . . . . . .
N'K; K;
J'
F'
Eigenstates
. . . . . . . . . . . . . . . . . . . . . . . .
T h e q R 0 ( 0 ) transition displayed in Fig. 4(a) contains
only one spin doublet ( J ' = 3 / 2 ~ J " = 1 / 2 ) and two hyperfine components ( F ' = 2 ~ F " = 1 and F' = 1 ~ F" = 0). The F' = 2 *-- F " = 1 component is not affected by the additional hyperfine perturbation and will not display quantum beats. Only the F' = 1 * - F " = 0 component allows for the coherent excitation of l i) and I k). The overall modulation is thus reduced because the total fluorescence is the sum of the modulated F ' = 1 ~ F " = 0 and unmodulated F ' = 2 ~ F " = 1 hyperfine components. As Fig. 4 indicates the beat frequency should be slightly larger than the spin-rotation splitting of the 101 level. Our observed beat frequencies of 16.1 and 14.9 MHz for the (0, 0, 0) and (0, 0, 1) levels, respectively, are consistent with the limited knowledge of the spinrotation splitting for K ' a = 0 levels in the B state noted above. Finally, we do not observe quantum beats for the q R 0 ( 0 ) decays from the other vibrational levels because their lifetimes (7 < 20 ns) are all significantly shorter than the beat period.
0 (b)
II
j
~
'
la>
,
i,>J_
qP0(2)
Fig. 4. (a) Energy-level diagram showing the postulated origin of the hyperfine quantum beats observed upon qR0(0) excitation of the (0, 0, 0) and (0, 0, 1) levels (Figs. 3 and 4, respectively). White levels are bright in absorption; black levels are dark. The basis states l a) and I b) are coupled by higher-order hyperfine perturbation (off-diagonal in J ' ) to give the eigenstates Ii) and I k). Because I k) is mostly dark and Ii) is mostly bright the coherent excitation of this pair produces quantum beats with frequency, toik. (b) Same diagram as in (a) but for excitation via the qPo(2) transition. In this case, all the hyperfine levels, including I i) and I k), are equally bright in absorption and no quantum beats are observed (see Fig. 2).
J.D. Tobiason, E.A. Rohlfing/ Chemical Physics Letters 252 (1996) 333-342 contrast, state I b) (with J ' = 1 / 2 ) fluoresces by only three transitions, qP0(2), PQI(I), and PPI(1). Thus the contrast in 'brightness' between states l a) and I b) is not nearly as strong in emission as in absorption. The exact modulation depth in this case, while somewhat complicated [28], is weak, as we observe. (If only the fluorescence from a OR0(0) rotational line were observed then the modulation should be much larger.) The additional hyperfine interaction shown in Fig. 4 that produces the quantum beats for qR0(0) excitation occurs for all rotational states with N ' > 0. (It cannot occur for 000, which has only J ' = 1/2.) However, all other transitions terminating on states with N' > 0 contain both spin-rotation doublets and thus all four hyperfine components. Fig. 4(b) illustrates this situation for the qP0(2) transition, which accesses the same hypeffine manifold as the qR0(0) transition shown in Fig. 4(a). In this case (as illustrated by the clear boxes), both eigenstates ]i) and [k) are equally bright in absorption and no modulation in the fluorescence is possible. Thus, the higher-order hyperfine mechanism explains why no quantum beats are observed for any rotational transition except qRo(0). We have measured lifetimes for both K'a = 0 and K'~ = 1 rotational states (with N' = 0 and 1 respectively) for the first six vibrational levels in the excited state, and Table 1 lists these levels and our measured lifetimes. Our lifetimes are in reasonably good agreement with those from the jet experiments of Chen and co-workers [10-12]. The only serious discrepancies occur for some of the longer lived states, specifically the K'~ = 0 states of the (0, 0, 0) and (0, 0, 1) levels, for which our lifetimes are longer by 10-25%. The collision-free lifetime of r = 43 ns for the (0, 0, 0) level extracted from the quenching study of Meier et al. [14] upon excitation of the qR 1 bandhead is consistent with our values for the K'a = 1 states in (0, 0, 0). Their estimated lifetimes for the higher lying vibrational levels are in rough agreement with our measurements, although in these cases they do not report rotational transition(s) or K'a values. In all cases the lifetimes of the K'a = 1 states are significantly shorter than those of the K'~ = 0 states. Additionally, for each long-lived state (both K'~ = 0 and K'~ = 1 in (0, 0, 0) and K'a = 0 in (0, 0, 1)), the
339
lifetime for N ' = 1 is noticeably longer than that for N' = 2. Recently, Lee and Chen [11,12] have characterized the rotational dependence of ~" in the (0, 0, 0) level over a broad range of rotational states (N' -- 0 26 and K'a = 0-3). They attribute the rapid decrease in r with increasing K'a to an a-type Coriolis coupling (strength proportional to K'a2) to vibronic levels in the ,~2II (22V) state, which subsequently predissociates via the )(2A/ state. In addition, they observe a weak decrease in r with increasing N' which they associate with b- or c-type Coriolis coupling (strength proportional to N ' ( N ' + 1)) to vibronic levels in the A and X states, respectively. We observe a somewhat larger decrease in the lifetime between the 101 and 202 states of the (0, 0, 0) level than do Lee and Chen [11,12]. We estimate fluorescence quantum yields for the eight vibrational levels listed in Table 1 by calculating the radiative decay rates and comparing these to the measured total decay rates. The total decay rate of an individual rovibrational level in the B state may be written as 1 "N' ' r'o)= k T ( Vi'
r(u;;
N ' , K'~)
=A(v'i) + knr(V;; N', K'a),
(1)
where A(v~) is the radiative rate, knr(Vl; N', K'~) is the nonradiative rate, and v'i is shorthand notation for the three vibrational quantum numbers in the excited state. The radiative rate is proportional to the transition dipole, which ab initio calculations indicate to be a weak function of the nuclear coordinates over the displacements involved in the B - X system [18]. We thus take the transition dipole moment for the B - X transition to be independent of the nuclear coordinates and write the radiative rate as [33] 64,rr 4 A(v'i) = ~ l /~¢12S(v;) = 2.026 x 10-61 txol2S(v'i),
(2)
where /% is the purely electronic transition moment (in atomic units) and S(v I) is given by S( v;) = ~ , q ( v'i, v,'.')~,3(v;, v;'). It
vi
(3)
340
,I.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
In Eq. (3) the q(v;, v~') are the sum-normalized Franck-Condon factors from the vibrational level in the 13 state to all possible final levels in the ground state and ~,(vl, v'i') are the transition frequencies (in cm-1). We convert the previously measured relative fluorescence intensities for the (0, 0, 0), (0, 1, 0), and (1, 0, 0) bands in the B - X system [8] to Franck-Condon factors and use /ze = 0.42 au from the ab initio calculations of Manaa and Yarkony [21] to produce radiative rates. The results for S(v~) and the radiative lifetimes, ~'r = 1/A(v~), are: S(v;)= 2.60 × 1013 2.98 × 1013, and 2.78 × 1013 cm -3 and ~'r -- 108, 94, and 101 ns for the (0, 0, 0), (0, 1, 0), and (1, 0, 0) bands, respectively. The frequencyweighted sum over Franck-Condon factors in Eq. (3) has a pronounced effect on the calculated radiative lifetimes because the 13-X emission is dominated by bands that are strongly red-shifted from the excitation wavelength. For instance, if one neglects the red shift by assuming that all the emission is produced at the excitation wavelength, one obtains r r = 48 ns instead of ~'r = 108 ns for the (0, 0, 0) band of the B state [21]. Since we find that S(v I) is not a strong function of vibrational level in the excited state, we use the average of the three values quoted above, S(v I) = 2.79 × 1013 cm -3, to provide e s t i m a t e s o f a(v;), k,r(Vl; N', K' a) = kT(V~; N', K'a) -A(v;), and the fluorescence quantum yield, ~ f =A(v;)/kT(V~; N', K'a), for each of the rovibrational levels studied here. We list the quantum yields in Table 1. The quantum yields and nonradiative rates are only estimates since they rest upon the accuracy of the ab initio value for the electronic transition moment and its assumed independence from the nuclear coordinates. The quantum yield for the 101 state in the (0, 0, 0) level is slightly less than unity. Because the 101 level is weakly coupled via b- and/or c-type Coriolis effects to other electronic states, the estimated quantum yield for the 00o state in (0, 0, 0) is even closer to unity. We are now in a position to investigate the effects of vibrational energy and mode specific excitation on the nonradiative rates. In Fig. 5 we plot the measured total decay rate and estimated nonradiative rate for the 101 rotational states of the first eight vibrational levels of the B state as a function of excess energy. We take this rotational state to be indicative of
..." • i ...• (0,2,0) ,......'"(o,1,1) (0,0,3}
109
(0,1,0), " il~ ," (0,0,2) ~
10a
//I"(,,o,o)
rr 1o7`
"" ../
•
(o,o,o) I
1000
,
,
I
I
2000
3000
Excess Energy (cm") Fig. 5. Measured total decay rates ( © ) and estimated nonradicative decay rates ( O ) for the first eight vibrational levels of HCO ~ 2 ~ as a function of excess energy. All rates are for the 10] rotational state; labels denote the vibrational levels. The rates for the (0, 2, 0) and (0, 0, 3) levels are lower limits. The lines are exponential fits to the nonradiative rates (not including (0, 2, 0) and (0, 0, 3)) for levels with v~ = 0 (solid) and levels with v~ = 1 (dashed).
purely vibronic coupling since the 101 level can exhibit only very weak b- or c-type Coriolis coupling. (In the remainder of this discussion 'vibrational level' refers to the 101 state of that level.) As the rates in Fig. 5 and the quantum yields in Table 1 illustrate, the total decay rate for the (0, 0, 0) level is dominated by the radiative rate. The nonradiative rate increases (and q~f decreases) rapidly with excess energy so that the total decay rate is dominated by the nonradiative portion for all levels above the (0, 0, 1) level. In addition, the nonradiative rate is noticeably mode dependent. Levels with v~ = 1 exhibit larger rates and a more rapid increase with excess energy than levels with v~ = 0. We show simple exponential fits to the nonradiative rates for these two sets of levels in Fig. 5 to illustrate this point. Others have indirectly observed the rapid increase in the nonradiative rate (and decrease in q0f) with excess energy, particularly for levels with v~ ~< 1. The sharp decrease in band intensities in the LIF spectra of Sappey and Crosley [5] are indicative of the rapid decrease in the fluorescence quantum yield.
J.D. Tobiason, E.4. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
In addition, the R2PI spectra of Cool and Song [13] show noticeable lifetime broadening (full widths ~ 0.6-1.2 cm -1) at high excess energies, beginning at the (0, 1, 2) and (1, 1, 0) levels at excess energies of 3464 and 3915 cm-1, respectively. If we extrapolate the simple exponential fit for the v~ = 1 levels (the dashed line in Fig. 5) we obtain lifetimes of 60 and 15 ps for (0, 1, 2) and (1, 1, 0), respectively. These lifetime are roughly the same order of magnitude as those inferred from the line broadening (T ~ 3 - 8 ps), from which we conclude that the nonradiative rate increases fairly monotonically (but not necessarily exponentially) with increasing excess energy in the 13 state. There are several possible nonradiative pathways available for the 13 state, including intersystem crossing to the quartet states and predissociation via the .~ or X states. The 4g, and 4/~ are calculated to lie at energies above the 13-state minimum of = 1500 and = 2400 cm -1, respectively [21]. The lack of any dramatic increase in the nonradiative rate at these excess energies and the relatively weak spin-orbit coupling of the quartets to the B state both indicate that intersystem crossing is not the dominant mechanism. We note that all mechanisms for predissociation that depend on mixing of the B 2/~ and ,~ 2/~, states involve rotation (i.e., Coriolis coupling) and cannot be invoked for the nearly rotationless HCO studied here. Thus, the remaining candidate for purely vibronic predissociation is the ground state. The B and )( states have a conical intersection that is calculated to lie ~ 10500 cm -1 b e l o w the f3-state minimum [20]. To get from the minimum to the crossing point involves substantial H-atom motion from the carbon to the oxygen side of CO [20], i.e., large changes in bond angle and C - H bond length. The height of the barrier between the B-state minimum and crossing point is not well known. However, it seems reasonable that vibrational excitation, particularly in the bending mode (v~), should facilitate the traversal of this barrier (or tunneling through it) and thus enhance the vibronic coupling between the B and X states. The combination of C - H stretch (v'1) with the bend should be especially effective at promoting predissociation; this is consistent with the line broadening observed in the R2PI spectrum for the (1, 1, 0) level [13]. Finally, the c-type Coriolis effect enhances coupling to the X state for nonzero
341
values of N' and this may be the primary reason for the decrease in lifetime with increasing N' observed by us (Table 1) and by Lee and Chen [11,12].
4. Summary We have measured fluorescence decays for low-lying rotational states ( N ' = 0,1 and K ' a = 0,1) in the first eight vibrational levels of HCO B 2g (see Figs. 1-3 for representative decays). Fits to these decays, which involve convoluting the effective excitation pulse with an exponential decay, yield the lifetimes we list in Table 1. In all cases the K ' a = 1 states exhibit shorter lifetimes than the K'a = 0 states. This observation is in agreement with the previous measurements of Chen and co-workers [10-12] who attribute this effect to a-type Coriolis coupling between vibronic levels in the ~2/~ and /~2I-I (2~,) states. Fluorescence decays obtained via excitation of the qR0(0) transition terminating on the long-lived (0, 0, 0) and (0, 0, 1) levels exhibit weak quantum beats that can be observed in the residuals from the exponential fits (panel b in Figs. 2 and 3). Because these beats are observed only for the qR0(0) transition, we believe that they arise from an additional hyperfine perturbation within the B state that is off-diagonal in J ' (see Fig. 4) and are not indicative of a general coupling to another dark electronic state. We use our previous measurements of the fluorescence band intensities (Franck-Condon factors) in the B - X system [8] and an ab initio value for the 13-X transition moment [21] to estimate the radiative and nonradiative contributions to the total decay rates (fluorescence quantum yields in Table 1). In order to assess the effects of purely vibronic coupling we examine the estimated nonradiative rates for the 101 rotational states (Fig. 5), for which rotational couplings are very weak. These rates increase monotonically with increasing vibrational energy in the B state; the rate of increase is in qualitative agreement with previous observations of LIF intensities [5] and line broadening in the R2PI spectrum [13]. Finally, the nonradiative rates for levels with v~ = 1 are larger and appear to increase more rapidly with vibrational energy than those for levels with /)2' = 0. These observations are consistent with a picture in which the dominant nonradiative decay
342
,I.D. Tobiason, E,A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
path for HCO radicals with little or no rotational energy in the 13 2p~ state is predissociation that proceeds via the conical intersection with the X 2~ state.
Acknowledgement We thank Ed Bochenski, Thomas Reichardt, and James Dunlop for helpful technical discussions and assistance in performing these experiments. We also thank I-Chia Chen for helpful discussions and for providing manuscripts prior to publication. References [1] M.E. Jacox, Vibrational and electronic energy levels of small polyatomic transient molecules, NIST Standard Reference Database 26, Ver. 4.0 (NIST, 1995), and references therein. [2] J.A. Barnard and J.N. Bradley, Flame and combustion, 2nd Ed. (Chapman and Hall, New York, 1985). [3] B.J. Finlayson-Pitts and J.N. Pitts Jr., Atmospheric chemistry: fundamentals and experimental techniques (Wiley, New York, 1986). [4] D.W. Neyer and P.L Houston, The HCO potential energy surface: probes using molecular scattering and photodissociation, in: The chemical dynamics and kinetics of small radicals, eds. K. Liu and A. Wagner, Advanced Series in Physical Chemistry (World Scientific, Singapore, 1995) and references therein. [5] A.D. Sappey and D.R. Crosely, J. Chem. Phys. 93 (1990) 7601. [6] G.W. Adamson, Z. Zhao and R.W. Field, J. Mol. Spectry. 160 (1993) 11. [7] G.W. Adamson, Ph.D. Thesis (MIT, Cambridge, 1994). [8] J.D. Tobiason, J.R. Dunlop and E.A. Rohlfing, J. Chem. Phys. 103 (1995) 1448. [9] Y.J. Shiu and I.-C. Chert, J. Mol. Spectry. 165 (1994) 457. [10] Y.J. Shiu, and I.-C. Chen, Chem. Phys. Letters 222 (1994) 245. [11] S.-H. Lee and I.-C. Chen, J. Chem. Phys. 103 (1995) 104.
[12] S.-H. Lee and I.-C. Chen, J. Chem. Phys. (1995), submitted. [13] T.A. Cool and X. Song, J. Chem. Phys. 96 (1992) 8675. [14] U.E. Meier, L.E. Hunziker and D.R. Crosley, J. Phys. Chem. 95 (1991) 5163. [15] J.M. Bowman, J.S. Bittman and L.B. Harding, J. Chem. Phys. 85 (1986) 911. [16] K. Tanaka and E.R. Davidson, J. Chem. Phys. 70 (1979) 2904. [17] P.J. Bruna, R.-J. Buenker and S.D. Peyerimhoff, J. Mol. Struct. 32 (1976) 217. [18] H. Lorenzen-Schmidt, M. Peri6 and S.D. Peyerimhoff, J. Chem. Phys. 98 (1993) 525. [19] H.-J. Werner, C. Bauer, P. Rosmus, H.-M. Keller, M. Stumpf and R. Schinke, J. Chem. Phys. 102 (1995) 3593. [20] M.R. Manaa and D.R. Yarkony, J. Chem. Phys. 99 (1993) 5251. [21] M.R. Manaa and D.R. Yarkony, J. Chem. Phys. 100 (1994) 473. [22] D.E. Milligan and M.E. Jacox, J. Chem. Phys. 41 (1964) 3032. [23] J.N. Demas, Excited state lifetime measurements (Academic Press, New York, 1983), p. 102-111. [24] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical recipes in C: the art of scientific computing (Cambridge Univ. Press, New York, 1988). [25] J. Velazquez, N. Hemmi and T.A. Cool, Chem. Phys. Letters 221 (1994) 39. [26] N. Hemmi and T.A. Cool, J. Chem. Phys. 102 (1995) 1867. [27] J. Chaiken, M. Gurnick and J.D. McDonald, J. Chem. Phys. 74 (1981) 106. [28] W.R. Lambert, P.M. Felker and A.H. Zewail, J. Chem. Phys. 81 (1984) 2217. [29] M. Dubs, J. Miihlbach, H. Bitto, P. Schmidt and J.R. Huber, J. Chem. Phys. 83 (1985) 3755. [30] P.R. Stannard and W.M. Gelbart, J. Phys. Chem. 85 (1981) 3592. [31] G. Herzberg, Molecular spectra and molecular structure III. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand, New York, 1966) p. 117. [32] G.A. Blake, K.V.L.N. Sastry and F.C. DeLucia, J. Chem. Phys. 80 (1984) 95. [33] H. Okabe, Photochemistry of small molecules (Wiley, New York, 1978) p. 39.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 252 (1996) 333-342
Lifetimes of vibrational levels in the B
state of HCO*
Joseph D. Tobiason 1, Eric A. Rohlfing Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551, USA
Received 20 November1995; in final form 8 February 1996
Abstract
We measure the fluorescence decays for low-lying rotational states in the first eight vibrational levels of HCO B 2.aL Some long-lived decays exhibit weak quantum beats that we attribute to a higher-order hyperfine perturbation in the 13 state. We estimate fluorescence quantum yields by combining measured fluorescence intensities and a calculated electronic transition moment. The estimated nonradiative decay rates increase with increasing vibrational excitation and exhibit mode specificity. We discuss possible vibronic coupling mechanisms, including spin-orbit coupling to quartet states and predissociation via the ~kLH(2/~') and ~2p~ states.
1. Introduction
The 132/~_~2g bands, or 'hydrocarbon flame bands', of the formyl radical, HCO, have been the subject of numerous experimental investigations over the years [1]. The motivation for much of this work is the importance of the formyl radical in both combustion [2] and atmospheric [3] chemistry, particularly as a source of atomic hydrogen through its unimolecular dissociation on the ground-state potential surface [4]. The f3-X system provides an effective means for accessing bound vibrational states and quasibound vibrational resonances over a wide range of energies in the ground state [5-7]. We have used this system in a recent study of the energies and
* This work is supported by the US Department of Energy, Office of Basic Energy Sciences, Chemical Sciences Division. l Present address: Center for Technology,Kaiser Aluminum & Chemical Corporation, 6177 Sunol Boulevard, Pleasanton, CA 94566, USA.
widths of HCO resonances using dispersed fluorescence (DF) and stimulated emission pumping (SEP) spectroscop_ies [8]. The B - X system also offers an effective means of probing concentrations of HCO in combustion environments. Several recent high-resolution laser studies have provided the spectroscopic characterization necessary for such laser-based diagnostics. These studies include experiments using laser-induced fluorescence (LIF) in cells by Sappey and Crosley [5] and Adamson et al. [6,7], LIF in free-jet expansions and in cells by Chen and co-workers [9-12], and resonant two-photon ionization (R2PI) in free-jet expansions by Cool and Song [13]. Besides providing spectroscopic data that characterizes the rovibrational structure of B-state HCO, these studies have probed the dynamics of the excited state. For example, the LIF intensities drop dramatically with increasing energy in the B state [5] whereas the R2PI intensities [13] do not. Also, lifetime broadening is observed in the R2PI spectrum at high energies. These observations have provided the first evidence
0009-2614/96/$12.00 © 1996 Elsevier Science B.V. All rights reserved PI1 S0009-2614(96)00163-7
334
J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
for nonradiative processes that quench the fluorescence from the B state. There have been several direct measurements of fluorescence lifetimes of the rovibrational levels in the 13 state, including the initial LIF and quenching work by Crosley and co-workers [5,14] and the more recent series of studies by Chen and co-workers [10-12]. Shiu and Chen [10] measured the lifetimes of rotational levels with K ' a = 0 and K 'a = 1 in the first six vibrational states. Subsequently, Lee and Chen [11,12] have concentrated their efforts on the effects of rotational motion on the lifetime of the vibrationless level of the B state. They find a sharp decrease in lifetime with K'a, which they attribute to a-type Coriolis coupling of the 132,~ state to the ,A2II (2A~') state. In addition, they observe that the lifetime decreases slowly with increasing N' and ascribe this effect to much weaker Coriolis coupling mechanisms: b-type coupling to the ,~ state or c-type coupling to the X state or both. The low-lying doublet electronic states of HCO have been extensively studied theoretically and are known to be coupled by nonadiabatic effects [15-21]. The ,A and X states form a Renner-Teller pair that are degenerate at the linear geometry and their coupling is manifested in the well-documented predissociation of the ,A state [4]. There is also a conical intersection between the B and X states that lies well above the low C - H bond dissociation energy on the ground state [20] and thus offers a mechanism for predissociation of the I3 state. The 13 state also undergoes another avoided crossing with a 3s(2/~) Rydberg state [18]. This crossing, which affects the electronic character of the B state at large H - C - O angles, should not influence the predissociation since the dissociation asymptote of the Rydberg state lies far above the B-state minimum. Finally, the two lowest lying quartet states of HCO, 4A" and 4~, which calculations show lie slightly above the 13-state origin [21], may spin-orbit couple to the 13 state. Thus, intersystem crossing is another candidate for nonradiative decay out of the B state. The purpose of this Letter is to describe our lifetime measurements for the first eight vibrational levels of HCO 132A~, which began as a part of our fluorescence-depletion SEP studies of the ground state [8]. The emphasis in our jet experiments is on purely vibronic effects, which we accomplish by
examining low-lying rotational states where rotational coupling mechanisms are minimized. The three vibrational modes of HCO X can be readily described as a C - H stretch(v~), a C - O stretch (v~), and a bend (v~). In the B state, v~ remains a C - H stretch, however there is ample evidence [13,22] that the character of v~ is predominantly bend and that of v~ is mostly C - O stretch. In addition to providing lifetime data, we also observe weak quantum beats in the fluorescence decays taken upon excitation of a specific rotational transition terminating on the long-lived vibrational levels in the B state. We do not believe that these beats are indicative of widespread couplings of the 13 state to another electronic state, but rather that they are due to an additional hyperfine perturbation within the 13 state. Finally, we use our previous DF measurements of the relative intensities in the B - X system [8] and a calculated value of the electronic transition moment [21] to estimate the quantum yield of fluorescence, and hence the nonradiative rate, for each rovibrational level. The nonradiative rate increases rapidly with increasing vibrational energy in the 13 state and also exhibits mode specificity. We discuss possible mechanisms for nonradiative processes, including intersystem crossing to nearby quartet states and predissociation via the A and X states.
2. Experimental details and lifetime determinations We generate HCO by the excimer-laser photolysis (308 nm) of acetaldehyde (CH3CHO), which is seeded at a concentration of ~ 5% into helium at a total pressure of 1-2 atm and expanded via a piezoelectrically actuated pulsed valve through a 1-mm orifice. A 500-mm focal-length lens focuses the photolysis beam (pulse energy 40-100 mJ) on the expansion 2-5 mm downstream from the orifice. The excitation laser is a Nd : YAG-laser-pumped dye laser (Continuum NY81C/Lumonics HD-500) that produces pulses of 10 IxJ-1.5 mJ in energy in the 239-259 nm region with a duration of 6 - 7 ns and a bandwidth of 0.1-0.2 c m - 1 . The excitation spot size is 1-2 mm in diameter where it crosses the expansion 10-20 mm downstream from the valve orifice. The LIF excitation spectra of HCO obtained in this
J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996")333-342
region of the expansion are consistent with rotational temperatures of 5-15 K. We collect fluorescence decays upon excitation of qR0, qPo, rQ0 and rR 0 transitions (notation: of the first eight (cold) vibronic bands of the B - X system. Table 1 lists the excited-state vibrational levels (notation: (v'l, v~, v~)), the rotational levels accessed (notation: and J ' = N ' + 1/2), and the vibrational energy in excess of the B-state origin. A photomultiplier tube (PMT, Hamamatsu R955), which is mounted directly on a window above the chamber, detects the fluorescence at right angles to the jet and excitation beams. We use a single lens to provide a 1 : 1 image of the PMT photocathode at the interaction region. This arrangement provides a viewing region of at least 8 mm along the jet axis and insures that the observed fluorescence decays are not affected by molecular transit through the viewing region. A 300 nm long-wave-pass filter in front of the PMT reduces scatter from the photolysis and excitation pulses. The photolysis pulse is also delayed in time by ~ 6 p.s from the excitation pulse. A 1 GHz digital oscilloscope (LeCroy 7200 with 7242B plug-in) digitizes the fluorescence decay signals from each laser pulse, averages 200 such decays, and stores the signal-averaged transient. The oscilloscope is triggered off a signal from a photodiode that detects the 532-nm light from the N d : Y A G pump laser. We also collect an averaged background obtained under the same conditions but with the excitation laser detuned from resonance. Subtraction of the background signal yields a corrected fluorescence decay, examples of which are given in panel (a) of Figs. 1-3. We determine lifetimes from the observed fluorescence decays from fits to a model function that is a convolution of an effective excitation pulse with an exponential decay. For each set of experimental runs we collect an effective excitation pulse either by reflecting an extremely attenuated excitation beam onto the PMT or by detecting Rayleigh scattered light from ambient air. Both methods give identical results. The effective excitation pulse is a convolution of the actual laser pulse with the temporal response of the detection system (PMT and subsequent electronics). At the low PMT bias voltage used in these experiments (450-540 V) the effective excitation pulse shape (shown in Fig. l(b)) is signifi-
ar~AJr,~)
N~,or,
335
cantly wider ( = 12 ns full width) than the actual laser pulse and is slightly asymmetric with a longer trailing edge. We use a fast iterative convolution technique [23] to numerically convolute (using trapezoidal integration) the effective excitation pulse with an exponential decay and we fit the corrected fluorescence decay to this convolution using a standard Levenberg-Marquadt nonlinear least-squares fitting routine [24]. The convolution of the effective excitation pulse is quite capable at modeling our observed decays and allows us to determine lifetimes down to a lower limit of ~"> 1 ns. In addition, because of the longer trailing edge of the effective excitation pulse, the convolution is necessary even for long-lived states (~-> 50 ns). In these cases, a fit tO the falling part of the observed decay with a simple exponential overestimates the lifetime by as much as 15 ns.
3. Results and discussion We present representative fluorescence decays and fits in Figs. 1-3. As noted above, the convolution of the effective excitation pulse with an exponential decay provides a model that fits our observed decays quite well. The only significant deviations occur for decays from short-lived states. In these cases there is sometimes a discrepancy between the leading edge of the convolution (which is dominated by the leading edge of the effective excitation pulse) and the observed decay and this gives rise to large residuals near zero time (see Fig. 1). We summarize the lifetimes obtained from many decay fits in Table 1. The errors listed in Table 1 are determined from fits to several different decays for each level (often reached by different rotational transitions) and from the effects of varying the time offset in fitting an individual decay. The purely statistical error from a single fit to an individual decay is usually smaller. When we excite the 101 state of the (0, 0, 0) and (0, 0, 1) levels via the qR0(0) transition we observe reproducible oscillations in the residuals from the decay fits. Figs. 2 and 3 display typical examples of these very weak oscillations, which have modulation depths of 1-2%, that we attribute to hyperfine quantum beats. Quantum beats occur when a single longitudinal mode of the laser excites a coherent superposition of molecular eigenstates that are comprised of
336
J.D. Tobiason, E,A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
0.3
t-
0.2
~
0.'1
(a) Fluorescence Signal and Fit
~
0.0
(b) Excitation Pulse
I
50
I
100
I
150
Time (ns)
O
0"2 f (C) Residuals t 0.0 --":" ....y"":"--"~.~---.--.-.-~-."-".....................................
v~ -0.2
Fig. 1. Fluorescence decay from the 101 (J' = 3/2) state of the (0, 0, 2) level of HCO [~2~ excited via the qR0(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~-= 17.5 +0.2 ns. (b) Excitation pulse used in the convolution. (c) Fit residuals (fit minus experimen0.
However, we observe the quantum beats only for the 10t rotational state and only upon excitation of the qR0(0) transition. Excitation of the 10t state via the qP0(2) transition does not produce observable quantum beats. Because of the extreme specificity with respect to rotational transition we do not believe that the observed quantum beats are indicative of a general coupling to other quartet or doublet electronic states, since such couplings should produce quantum beats over a broad range of rotational levels and transitions. A more plausible mechanism for producing the quantum beats involves a higher-order interaction among the hyperfine levels within the 10l rotational state. Fig. 4 displays the energy level schematic appropriate to the 101 rotational state in the (0, 0, 0) level of [l-state HCO. Spin-rotation interaction splits the 101 level into the J ' = 3 / 2 (F 2) and J ' = 1 / 2 (F 1) components, which are further split by the magnetic hypeffine interaction with the hydrogen nucleus ( I = 1 / 2 ) to produce two pairs of hypeffine doublets ( F ' = 0 and 1; F ' = 1 and 2). The energy ordering of
1.0
0.8
optically bright and dark basis (zero-order) states [25-30]. This coherent superposition is time-dependent (since it is not an eigenstate) and the fluorescence beat pattern and modulation depth depend on the number or eigenstates coherently excited and their fractional bright character. The clean, singlefrequency beats seen in Figs. 2 and 3 suggests the superposition of just one bright and one dark eigenstate. We fit the residuals to a exponentially damped sinusoidal function in order to extract the beat frequencies listed in Table 1. There are several possibilities by which darker basis states might couple to the bright [i-state levels to produce hyperfine quantum beats. One is spin-orbit coupling to one or both of the quartet states. However, the calculated origins of the 4A¢' and anl state lie well above the (0, 0, 0) level of the [l state [21], thus precluding spin-orbit coupling as the source of the quantum beats. Another origin for the quantum beats might be Coriolis coupling to the ,~ or X states.
c "~ 0.6
~2 0.4.
a) Fluorescence Signal
0.2 o.o I
200
of O
I
I
I
400 600 Time (ns)
800
s
0.0 -0.5
Fig. 2. Fluorescence decay from the lol (J' = 3/2) state of the (0, 0, 0) level of HCO B2A~excited via the qRo(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~"= 100.1 +0.2 ns. (b) Residuals from exponential fit and fit to a damped sinusoidal function that gives a beat frequency of 16.1 + 0.1 MHz.
J.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
the hyperfine levels depends on the relative magnitude of the spin-rotation and hyperfine splittings. The spin-rotation splitting is very small for K'o -- 0 states and is equal to the constant, ~', which is independent of N' [31]. For ground-state HCO the spin-rotation splitting (/z" = - 9 4 MHz) is roughly equal to the hyperfine splitting [32], but neither the spin-rotation splitting nor the hyperfine splitting has been accurately measured for the K ' a -- 0 states of the (0, 0, 0) level of the I~ state. The most recent measurement by Lee and Chen [11] suggests /z'-- - 3 0 MHz, however the error bars for this value are large ( + 180 MHz) and encompass both positive and negative values of/z'. In Fig. 4 we assume that /z' is negative and that the hyperfine splitting is significantly
337
(roughly four times) larger than the spin-rotation splitting. In this case, the two lowest hyperfine components are both F' = 1. These two states of common N' and F' can be further coupled by magnetic hyperfine matrix elements that are off-diagonal in J'. This additional hyperfine interaction is known to be significant in ground-state HCO and has been observed as significant perturbations in the microwave spectrum [32]. We postulate that the quantum beats observed upon q R 0 ( 0 ) excitation originate from the coherent excitation of the eigenstates labeled l i) and I k) in Fig. 4. These are formed from the mixing of the F' = 1 basis states that originate from J' = 3 / 2 and J' = 1/2, which are labeled l a) and I b) in Fig. 4.
Table 1 HCO B-state lifetimes and estimated fluorescence quantum yields Vibrational level
Excess energy
(v'l,v'2,v'3)
(cm- 1)
(0, 0, 0)
(0, O, 1)
(0, 1, O)
(0, 0, 2)
(0, 1, 1)
(1, O, O)
0
1065
1380
2114
2430
2597
(0, 2, 0)
2773
(0, 0, 3)
3146
Rotational level
,
NK'K'c lol 202 11o 211 lol 202 11o 211 lol 202 11o 211 101 202 11o 211 lol 202 11o 211 lol 2o2 11o 211 lol 202 lol 2o2
j, 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 1/2 3/2 3/2 3/2, 3/2 3/2,
5/2
5/2
5/2
5/2
5/2
5/2
5/2 5/2
Lifetime
Estimated b
qRo(0)
(ns) a
qBf
beat freq. (MHz) a
99 (3) 89 (4) 48 (2) 39 (2) 73 (3) 64 (3) 15 (2) 16 (2) 12 (2) 12 (2) 5.5 (10) 5.5 (10) 17 (2) 20 (2) 8 (1) 8 (1) 1.5 (10) 1.2 (8) <1 <1 15 (2) 14 (2) 4 (1) 4 (1) <1 <1 < 1 < 1
0.92 0.83 0.45 0.36 0.73 0.64 0.15 0.16 0.13 0.13 0.06 0.06 0.17 0.20 0.08 0.08 0.01 0.01 < 0.01 < 0.01 0.15 0.14 0.04 0.04 < 0.01 < 0.01 < 0.01 < 0.01
16.1(1)
a Errors in parentheses. b Quantum yield estimates based on calculated transition moment from Ref. [21] (see text).
14.9 (1)
J.D. Tobiason, E~A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
338 0.4
As illustrated in Fig. 4(a) by the shaded boxes, I a) is completely bright in absorption via the qR0(0) transition while I b) is completely dark since it is not connected to the ground state rotational level (00o, J" = 1 / 2 ) by a dipole-allowed transition. Thus, quantum beats occur because l i> is mostly bright and I k) is mostly dark. However, the depth of the fluorescence modulation also depends on the bright and dark character of the eigenstates with respect to emission (which is not shown explicitly in Fig. 4). For each vibrational band in the B - X system state I a) (with J ' = 3 / 2 ) can fluoresce via four rotational transitions, q R 0 ( 0 ) , qP0(2), P Q I ( 1 ) , and PPI(1). By
-
0.3 (a) Fluorescence Signal ¢-
~d
0.2 -
0.1
0.0
-
-
"~
I 1 O0
I
I
I
200
300
400
Time (ns) 10
-10
l-
: ;" ii :'-'.
(b) Residuals and Fit
j
Fig. 3. Fluorescence decay from the 101 ( J ' = 3 / 2 ) state of the (0, 0, 1) level of HCO B2,K excited via the qR0(0) transition. (a) Fluorescence signal and exponential fit using the fast iterative convolution. The fit yielded a lifetime of ~" = 74.7+0.5 ns. (b) Residuals from exponential fit and fit to a damped sinusoidal function that gives a beat frequency of 14.9 + 0.2 MHz.
(a)
lol
?/
qR0(O)
la~ ~ i
. . . . . . . . . . . . . . . . . . . . . . . .
N'K; K;
J'
F'
Eigenstates
. . . . . . . . . . . . . . . . . . . . . . . .
T h e q R 0 ( 0 ) transition displayed in Fig. 4(a) contains
only one spin doublet ( J ' = 3 / 2 ~ J " = 1 / 2 ) and two hyperfine components ( F ' = 2 ~ F " = 1 and F' = 1 ~ F" = 0). The F' = 2 *-- F " = 1 component is not affected by the additional hyperfine perturbation and will not display quantum beats. Only the F' = 1 * - F " = 0 component allows for the coherent excitation of l i) and I k). The overall modulation is thus reduced because the total fluorescence is the sum of the modulated F ' = 1 ~ F " = 0 and unmodulated F ' = 2 ~ F " = 1 hyperfine components. As Fig. 4 indicates the beat frequency should be slightly larger than the spin-rotation splitting of the 101 level. Our observed beat frequencies of 16.1 and 14.9 MHz for the (0, 0, 0) and (0, 0, 1) levels, respectively, are consistent with the limited knowledge of the spinrotation splitting for K ' a = 0 levels in the B state noted above. Finally, we do not observe quantum beats for the q R 0 ( 0 ) decays from the other vibrational levels because their lifetimes (7 < 20 ns) are all significantly shorter than the beat period.
0 (b)
II
j
~
'
la>
,
i,>J_
qP0(2)
Fig. 4. (a) Energy-level diagram showing the postulated origin of the hyperfine quantum beats observed upon qR0(0) excitation of the (0, 0, 0) and (0, 0, 1) levels (Figs. 3 and 4, respectively). White levels are bright in absorption; black levels are dark. The basis states l a) and I b) are coupled by higher-order hyperfine perturbation (off-diagonal in J ' ) to give the eigenstates Ii) and I k). Because I k) is mostly dark and Ii) is mostly bright the coherent excitation of this pair produces quantum beats with frequency, toik. (b) Same diagram as in (a) but for excitation via the qPo(2) transition. In this case, all the hyperfine levels, including I i) and I k), are equally bright in absorption and no quantum beats are observed (see Fig. 2).
J.D. Tobiason, E.A. Rohlfing/ Chemical Physics Letters 252 (1996) 333-342 contrast, state I b) (with J ' = 1 / 2 ) fluoresces by only three transitions, qP0(2), PQI(I), and PPI(1). Thus the contrast in 'brightness' between states l a) and I b) is not nearly as strong in emission as in absorption. The exact modulation depth in this case, while somewhat complicated [28], is weak, as we observe. (If only the fluorescence from a OR0(0) rotational line were observed then the modulation should be much larger.) The additional hyperfine interaction shown in Fig. 4 that produces the quantum beats for qR0(0) excitation occurs for all rotational states with N ' > 0. (It cannot occur for 000, which has only J ' = 1/2.) However, all other transitions terminating on states with N' > 0 contain both spin-rotation doublets and thus all four hyperfine components. Fig. 4(b) illustrates this situation for the qP0(2) transition, which accesses the same hypeffine manifold as the qR0(0) transition shown in Fig. 4(a). In this case (as illustrated by the clear boxes), both eigenstates ]i) and [k) are equally bright in absorption and no modulation in the fluorescence is possible. Thus, the higher-order hyperfine mechanism explains why no quantum beats are observed for any rotational transition except qRo(0). We have measured lifetimes for both K'a = 0 and K'~ = 1 rotational states (with N' = 0 and 1 respectively) for the first six vibrational levels in the excited state, and Table 1 lists these levels and our measured lifetimes. Our lifetimes are in reasonably good agreement with those from the jet experiments of Chen and co-workers [10-12]. The only serious discrepancies occur for some of the longer lived states, specifically the K'~ = 0 states of the (0, 0, 0) and (0, 0, 1) levels, for which our lifetimes are longer by 10-25%. The collision-free lifetime of r = 43 ns for the (0, 0, 0) level extracted from the quenching study of Meier et al. [14] upon excitation of the qR 1 bandhead is consistent with our values for the K'a = 1 states in (0, 0, 0). Their estimated lifetimes for the higher lying vibrational levels are in rough agreement with our measurements, although in these cases they do not report rotational transition(s) or K'a values. In all cases the lifetimes of the K'a = 1 states are significantly shorter than those of the K'~ = 0 states. Additionally, for each long-lived state (both K'~ = 0 and K'~ = 1 in (0, 0, 0) and K'a = 0 in (0, 0, 1)), the
339
lifetime for N ' = 1 is noticeably longer than that for N' = 2. Recently, Lee and Chen [11,12] have characterized the rotational dependence of ~" in the (0, 0, 0) level over a broad range of rotational states (N' -- 0 26 and K'a = 0-3). They attribute the rapid decrease in r with increasing K'a to an a-type Coriolis coupling (strength proportional to K'a2) to vibronic levels in the ,~2II (22V) state, which subsequently predissociates via the )(2A/ state. In addition, they observe a weak decrease in r with increasing N' which they associate with b- or c-type Coriolis coupling (strength proportional to N ' ( N ' + 1)) to vibronic levels in the A and X states, respectively. We observe a somewhat larger decrease in the lifetime between the 101 and 202 states of the (0, 0, 0) level than do Lee and Chen [11,12]. We estimate fluorescence quantum yields for the eight vibrational levels listed in Table 1 by calculating the radiative decay rates and comparing these to the measured total decay rates. The total decay rate of an individual rovibrational level in the B state may be written as 1 "N' ' r'o)= k T ( Vi'
r(u;;
N ' , K'~)
=A(v'i) + knr(V;; N', K'a),
(1)
where A(v~) is the radiative rate, knr(Vl; N', K'~) is the nonradiative rate, and v'i is shorthand notation for the three vibrational quantum numbers in the excited state. The radiative rate is proportional to the transition dipole, which ab initio calculations indicate to be a weak function of the nuclear coordinates over the displacements involved in the B - X system [18]. We thus take the transition dipole moment for the B - X transition to be independent of the nuclear coordinates and write the radiative rate as [33] 64,rr 4 A(v'i) = ~ l /~¢12S(v;) = 2.026 x 10-61 txol2S(v'i),
(2)
where /% is the purely electronic transition moment (in atomic units) and S(v I) is given by S( v;) = ~ , q ( v'i, v,'.')~,3(v;, v;'). It
vi
(3)
340
,I.D. Tobiason, E.A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
In Eq. (3) the q(v;, v~') are the sum-normalized Franck-Condon factors from the vibrational level in the 13 state to all possible final levels in the ground state and ~,(vl, v'i') are the transition frequencies (in cm-1). We convert the previously measured relative fluorescence intensities for the (0, 0, 0), (0, 1, 0), and (1, 0, 0) bands in the B - X system [8] to Franck-Condon factors and use /ze = 0.42 au from the ab initio calculations of Manaa and Yarkony [21] to produce radiative rates. The results for S(v~) and the radiative lifetimes, ~'r = 1/A(v~), are: S(v;)= 2.60 × 1013 2.98 × 1013, and 2.78 × 1013 cm -3 and ~'r -- 108, 94, and 101 ns for the (0, 0, 0), (0, 1, 0), and (1, 0, 0) bands, respectively. The frequencyweighted sum over Franck-Condon factors in Eq. (3) has a pronounced effect on the calculated radiative lifetimes because the 13-X emission is dominated by bands that are strongly red-shifted from the excitation wavelength. For instance, if one neglects the red shift by assuming that all the emission is produced at the excitation wavelength, one obtains r r = 48 ns instead of ~'r = 108 ns for the (0, 0, 0) band of the B state [21]. Since we find that S(v I) is not a strong function of vibrational level in the excited state, we use the average of the three values quoted above, S(v I) = 2.79 × 1013 cm -3, to provide e s t i m a t e s o f a(v;), k,r(Vl; N', K' a) = kT(V~; N', K'a) -A(v;), and the fluorescence quantum yield, ~ f =A(v;)/kT(V~; N', K'a), for each of the rovibrational levels studied here. We list the quantum yields in Table 1. The quantum yields and nonradiative rates are only estimates since they rest upon the accuracy of the ab initio value for the electronic transition moment and its assumed independence from the nuclear coordinates. The quantum yield for the 101 state in the (0, 0, 0) level is slightly less than unity. Because the 101 level is weakly coupled via b- and/or c-type Coriolis effects to other electronic states, the estimated quantum yield for the 00o state in (0, 0, 0) is even closer to unity. We are now in a position to investigate the effects of vibrational energy and mode specific excitation on the nonradiative rates. In Fig. 5 we plot the measured total decay rate and estimated nonradiative rate for the 101 rotational states of the first eight vibrational levels of the B state as a function of excess energy. We take this rotational state to be indicative of
..." • i ...• (0,2,0) ,......'"(o,1,1) (0,0,3}
109
(0,1,0), " il~ ," (0,0,2) ~
10a
//I"(,,o,o)
rr 1o7`
"" ../
•
(o,o,o) I
1000
,
,
I
I
2000
3000
Excess Energy (cm") Fig. 5. Measured total decay rates ( © ) and estimated nonradicative decay rates ( O ) for the first eight vibrational levels of HCO ~ 2 ~ as a function of excess energy. All rates are for the 10] rotational state; labels denote the vibrational levels. The rates for the (0, 2, 0) and (0, 0, 3) levels are lower limits. The lines are exponential fits to the nonradiative rates (not including (0, 2, 0) and (0, 0, 3)) for levels with v~ = 0 (solid) and levels with v~ = 1 (dashed).
purely vibronic coupling since the 101 level can exhibit only very weak b- or c-type Coriolis coupling. (In the remainder of this discussion 'vibrational level' refers to the 101 state of that level.) As the rates in Fig. 5 and the quantum yields in Table 1 illustrate, the total decay rate for the (0, 0, 0) level is dominated by the radiative rate. The nonradiative rate increases (and q~f decreases) rapidly with excess energy so that the total decay rate is dominated by the nonradiative portion for all levels above the (0, 0, 1) level. In addition, the nonradiative rate is noticeably mode dependent. Levels with v~ = 1 exhibit larger rates and a more rapid increase with excess energy than levels with v~ = 0. We show simple exponential fits to the nonradiative rates for these two sets of levels in Fig. 5 to illustrate this point. Others have indirectly observed the rapid increase in the nonradiative rate (and decrease in q0f) with excess energy, particularly for levels with v~ ~< 1. The sharp decrease in band intensities in the LIF spectra of Sappey and Crosley [5] are indicative of the rapid decrease in the fluorescence quantum yield.
J.D. Tobiason, E.4. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
In addition, the R2PI spectra of Cool and Song [13] show noticeable lifetime broadening (full widths ~ 0.6-1.2 cm -1) at high excess energies, beginning at the (0, 1, 2) and (1, 1, 0) levels at excess energies of 3464 and 3915 cm-1, respectively. If we extrapolate the simple exponential fit for the v~ = 1 levels (the dashed line in Fig. 5) we obtain lifetimes of 60 and 15 ps for (0, 1, 2) and (1, 1, 0), respectively. These lifetime are roughly the same order of magnitude as those inferred from the line broadening (T ~ 3 - 8 ps), from which we conclude that the nonradiative rate increases fairly monotonically (but not necessarily exponentially) with increasing excess energy in the 13 state. There are several possible nonradiative pathways available for the 13 state, including intersystem crossing to the quartet states and predissociation via the .~ or X states. The 4g, and 4/~ are calculated to lie at energies above the 13-state minimum of = 1500 and = 2400 cm -1, respectively [21]. The lack of any dramatic increase in the nonradiative rate at these excess energies and the relatively weak spin-orbit coupling of the quartets to the B state both indicate that intersystem crossing is not the dominant mechanism. We note that all mechanisms for predissociation that depend on mixing of the B 2/~ and ,~ 2/~, states involve rotation (i.e., Coriolis coupling) and cannot be invoked for the nearly rotationless HCO studied here. Thus, the remaining candidate for purely vibronic predissociation is the ground state. The B and )( states have a conical intersection that is calculated to lie ~ 10500 cm -1 b e l o w the f3-state minimum [20]. To get from the minimum to the crossing point involves substantial H-atom motion from the carbon to the oxygen side of CO [20], i.e., large changes in bond angle and C - H bond length. The height of the barrier between the B-state minimum and crossing point is not well known. However, it seems reasonable that vibrational excitation, particularly in the bending mode (v~), should facilitate the traversal of this barrier (or tunneling through it) and thus enhance the vibronic coupling between the B and X states. The combination of C - H stretch (v'1) with the bend should be especially effective at promoting predissociation; this is consistent with the line broadening observed in the R2PI spectrum for the (1, 1, 0) level [13]. Finally, the c-type Coriolis effect enhances coupling to the X state for nonzero
341
values of N' and this may be the primary reason for the decrease in lifetime with increasing N' observed by us (Table 1) and by Lee and Chen [11,12].
4. Summary We have measured fluorescence decays for low-lying rotational states ( N ' = 0,1 and K ' a = 0,1) in the first eight vibrational levels of HCO B 2g (see Figs. 1-3 for representative decays). Fits to these decays, which involve convoluting the effective excitation pulse with an exponential decay, yield the lifetimes we list in Table 1. In all cases the K ' a = 1 states exhibit shorter lifetimes than the K'a = 0 states. This observation is in agreement with the previous measurements of Chen and co-workers [10-12] who attribute this effect to a-type Coriolis coupling between vibronic levels in the ~2/~ and /~2I-I (2~,) states. Fluorescence decays obtained via excitation of the qR0(0) transition terminating on the long-lived (0, 0, 0) and (0, 0, 1) levels exhibit weak quantum beats that can be observed in the residuals from the exponential fits (panel b in Figs. 2 and 3). Because these beats are observed only for the qR0(0) transition, we believe that they arise from an additional hyperfine perturbation within the B state that is off-diagonal in J ' (see Fig. 4) and are not indicative of a general coupling to another dark electronic state. We use our previous measurements of the fluorescence band intensities (Franck-Condon factors) in the B - X system [8] and an ab initio value for the 13-X transition moment [21] to estimate the radiative and nonradiative contributions to the total decay rates (fluorescence quantum yields in Table 1). In order to assess the effects of purely vibronic coupling we examine the estimated nonradiative rates for the 101 rotational states (Fig. 5), for which rotational couplings are very weak. These rates increase monotonically with increasing vibrational energy in the B state; the rate of increase is in qualitative agreement with previous observations of LIF intensities [5] and line broadening in the R2PI spectrum [13]. Finally, the nonradiative rates for levels with v~ = 1 are larger and appear to increase more rapidly with vibrational energy than those for levels with /)2' = 0. These observations are consistent with a picture in which the dominant nonradiative decay
342
,I.D. Tobiason, E,A. Rohlfing / Chemical Physics Letters 252 (1996) 333-342
path for HCO radicals with little or no rotational energy in the 13 2p~ state is predissociation that proceeds via the conical intersection with the X 2~ state.
Acknowledgement We thank Ed Bochenski, Thomas Reichardt, and James Dunlop for helpful technical discussions and assistance in performing these experiments. We also thank I-Chia Chen for helpful discussions and for providing manuscripts prior to publication. References [1] M.E. Jacox, Vibrational and electronic energy levels of small polyatomic transient molecules, NIST Standard Reference Database 26, Ver. 4.0 (NIST, 1995), and references therein. [2] J.A. Barnard and J.N. Bradley, Flame and combustion, 2nd Ed. (Chapman and Hall, New York, 1985). [3] B.J. Finlayson-Pitts and J.N. Pitts Jr., Atmospheric chemistry: fundamentals and experimental techniques (Wiley, New York, 1986). [4] D.W. Neyer and P.L Houston, The HCO potential energy surface: probes using molecular scattering and photodissociation, in: The chemical dynamics and kinetics of small radicals, eds. K. Liu and A. Wagner, Advanced Series in Physical Chemistry (World Scientific, Singapore, 1995) and references therein. [5] A.D. Sappey and D.R. Crosely, J. Chem. Phys. 93 (1990) 7601. [6] G.W. Adamson, Z. Zhao and R.W. Field, J. Mol. Spectry. 160 (1993) 11. [7] G.W. Adamson, Ph.D. Thesis (MIT, Cambridge, 1994). [8] J.D. Tobiason, J.R. Dunlop and E.A. Rohlfing, J. Chem. Phys. 103 (1995) 1448. [9] Y.J. Shiu and I.-C. Chert, J. Mol. Spectry. 165 (1994) 457. [10] Y.J. Shiu, and I.-C. Chen, Chem. Phys. Letters 222 (1994) 245. [11] S.-H. Lee and I.-C. Chen, J. Chem. Phys. 103 (1995) 104.
[12] S.-H. Lee and I.-C. Chen, J. Chem. Phys. (1995), submitted. [13] T.A. Cool and X. Song, J. Chem. Phys. 96 (1992) 8675. [14] U.E. Meier, L.E. Hunziker and D.R. Crosley, J. Phys. Chem. 95 (1991) 5163. [15] J.M. Bowman, J.S. Bittman and L.B. Harding, J. Chem. Phys. 85 (1986) 911. [16] K. Tanaka and E.R. Davidson, J. Chem. Phys. 70 (1979) 2904. [17] P.J. Bruna, R.-J. Buenker and S.D. Peyerimhoff, J. Mol. Struct. 32 (1976) 217. [18] H. Lorenzen-Schmidt, M. Peri6 and S.D. Peyerimhoff, J. Chem. Phys. 98 (1993) 525. [19] H.-J. Werner, C. Bauer, P. Rosmus, H.-M. Keller, M. Stumpf and R. Schinke, J. Chem. Phys. 102 (1995) 3593. [20] M.R. Manaa and D.R. Yarkony, J. Chem. Phys. 99 (1993) 5251. [21] M.R. Manaa and D.R. Yarkony, J. Chem. Phys. 100 (1994) 473. [22] D.E. Milligan and M.E. Jacox, J. Chem. Phys. 41 (1964) 3032. [23] J.N. Demas, Excited state lifetime measurements (Academic Press, New York, 1983), p. 102-111. [24] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical recipes in C: the art of scientific computing (Cambridge Univ. Press, New York, 1988). [25] J. Velazquez, N. Hemmi and T.A. Cool, Chem. Phys. Letters 221 (1994) 39. [26] N. Hemmi and T.A. Cool, J. Chem. Phys. 102 (1995) 1867. [27] J. Chaiken, M. Gurnick and J.D. McDonald, J. Chem. Phys. 74 (1981) 106. [28] W.R. Lambert, P.M. Felker and A.H. Zewail, J. Chem. Phys. 81 (1984) 2217. [29] M. Dubs, J. Miihlbach, H. Bitto, P. Schmidt and J.R. Huber, J. Chem. Phys. 83 (1985) 3755. [30] P.R. Stannard and W.M. Gelbart, J. Phys. Chem. 85 (1981) 3592. [31] G. Herzberg, Molecular spectra and molecular structure III. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand, New York, 1966) p. 117. [32] G.A. Blake, K.V.L.N. Sastry and F.C. DeLucia, J. Chem. Phys. 80 (1984) 95. [33] H. Okabe, Photochemistry of small molecules (Wiley, New York, 1978) p. 39.