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Time-Dependent Center-of-Gravity Metric Determines Key Dynamical Features of Doorway-Mediated Intersystem Crossing Kyle L. Bittinger,* Wilton Virgo, and Robert W. Field Massachusetts Institute of Technology, Cambridge, Massachusetts 02142
ABSTRACT Traditional statistical models provide only a phenomenological description of intersystem crossing, the spin-orbit-induced mixing of optically bright and dark excited electronic states with different values of total electron spin, S. The statistical models do not identify the dominant energy flow pathways and the state-specific mechanisms responsible for the mixing of zero-order bright state character into dark states. A time-dependent center of gravity metric, described herein, is sensitive to deterministic patterns of bright state-dark state mixing that deviate from mechanism-free statistical pictures. The center of gravity metric may be applied to incoherently excited, time-gated, high-resolution spectra of small molecules in a narrow energy region surrounding each selectable transition into an optically bright rovibronic state. The metric reveals the characteristic behavior caused by the nonstochastic bias in the local distribution of fractional bright state character, as well as the magnitude of the matrix element between bright and dark states. We use a simple two-state model to illustrate the metric, and then apply the metric to several transitions in the S1-S0 spectrum of acetylene. The metric is robust with respect to measurement deficiencies common in laser-induced fluorescence spectroscopy, and may be applied to numerically characterize state-specific mechanisms of intersystem crossing beyond the scope of statistical Golden Rule models. SECTION Kinetics, Spectroscopy
ntersystem crossing (ISC) is traditionally understood in terms of a Golden Rule model, where mixing between one optically selected zero-order bright state into many zero-order dark states is characterized by the average squared matrix element and the average dark state vibrational density of states.1 According to Fermi's Golden Rule, the rate of ISC is
I
kISC ¼
2π ~ SO jTn æj2 FE jÆS1 jH p
ISC, where a single dark state of distinct, knowable characteristics connects the bright state to the remaining ensemble of dark states. The bright state and doorway state are treated as having definite zero-order vibration-rotation characters, whereas all of the other dark states are treated as members of a statistical ensemble. This amounts to partially pulling back the statistical curtain, to uncover and exploit userspecifiable mechanistic pathways. In acetylene, C2H2, the ISC dynamics of S1 conform to doorway-mediated models. Experiment and theory have demonstrated that a sparse manifold of low-lying vibrational levels from the T3 state (10 per cm-1).4-6 To date, many experiments have focused on S1 3ν3 K0 =1, where the F2 ( J 0 = N0 ) spin component of a T3 level is accidentally near-degenerate with its singlet counterpart at low J 0 .7-12 It is understandable that this level has been so well studied. When assigning triplet perturbations in small molecules, high-resolution spectroscopists can only unambiguously identify those dark states that
ð1Þ
where FE is the density of triplet spin-rovibronic states that are near-degenerate with the optically selected singlet bright state, and E is the energy above the zero point vibrational ~ SO level of the triplet state, Tn. The eq 1 matrix element of the H perturbation operator is a product of an S1-Tn electronic factor and a vibrational overlap integral between S1 and Tn vibrational wave functions. The rate of ISC depends on the average squared singlet-triplet coupling matrix element and the Tn vibrational level density. However, the ISC dynamics of real molecules often deviate from such a statistical picture when an interaction of explicit dynamical mechanism occurs among the dark states. Tiered coupling models and network systems have been proposed to reveal and describe mechanistic (nonstatistical) ISC dynamics.2,3 The simplest of these models is doorway-mediated
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Received Date: April 23, 2010 Accepted Date: June 22, 2010 Published on Web Date: June 30, 2010
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DOI: 10.1021/jz100526g |J. Phys. Chem. Lett. 2010, 1, 2144–2148
pubs.acs.org/JPCL
cooperate by appearing via rotationally assignable extra lines, which are both rare and biased exclusively toward F2-component ( J = N) perturbations. The reason for this experimental bias toward F2-component level crossings is the small change in the singlet-triplet energy denominator, E( J 0 ) = J0 ( J0 þ 1)(BS - BT), as J0 is increased. The energy denominator for F1 and F3 components ( J0 =N0 ( 1) is much more strongly J0 -dependent, which makes it prohibitively difficult to detect or rotationally assign any predominantly triplet extra lines at the level crossing. However, a T3 doorway level in acetylene can imprint its presence on the spectrum of the S1 levels with which it interacts, even if the doorway level is neither near-degenerate nor slowly tuning in energy relative to the bright state. When an S1 level mixes with an energetically distant T3 doorway spinvibronic level, the energy of the doorway level relative to the S1 level influences the pattern of local mixing between the S1 level and the near-degenerate T1,2 levels. This leads to a previously unexploited local energy dependence of the average fractional S1 character in nominal T1,2 eigenstates. In this communication, we describe a new metric, the time-dependent center of gravity in an incoherently excited, time-gated, laser-induced fluorescence (LIF) spectrum, which is designed to be sensitive to this characteristic energy dependence of bright state character among a set of bright~dark mixed eigenstates. The time evolution of the center of gravity of the multieigenstate rotational features in the LIF spectrum reveals the energy location of the doorway state and the magnitude of the bright state~doorway state interaction matrix element. The metric takes as input the incoherently excited, timegated (time after the excitation pulse), high-resolution, fluorescence excitation spectrum of a cluster of bright~dark mixed eigenstates. Furthermore, it requires that all bright state character in a spectral feature originate from a single zeroorder state. To define the metric, we consider a spectrum that is incoherently excited (i.e., the spectrum consists of spectrally unresolved transitions into eigenstates with energy separations larger than h/2πτlaser) by a high-resolution pulsed (∼1 GHz fwhm, τlaser∼6 ns) laser. A single bright state is admixed into an ensemble of near-degenerate dark states via an energetically distant (unidentified) doorway state. Laserpopulated eigenstates sampled in the spectrum have a wide range of fluorescence lifetimes, which are determined by their fractional bright state character. The relative fluorescence intensities of the members of the incoherently excited ensemble of mostly dark eigenstates change as a function of time delay after excitation, causing fine details of the spectrum to evolve. The time-dependent center of gravity is defined as the intensity-weighted average energy, Z E IðE, tÞ dE ð2Þ Eave ðtÞ ¼
Figure 1. The fluorescence intensity of mixed eigenstates at various time delays after the laser excitation pulse, plotted as a function of bright state character. At t = 0, eigenstates with the largest bright state character have the strongest fluorescence intensity. After a delay of 5 times the zero-order bright state lifetime, τs, eigenstates with fractional bright state character of 40% appear with the largest relative intensity. When the delay time is increased to 15τs, eigenstates with only 15% fractional bright state character appear with greatest intensity.
fluorescence intensity is Is ðtÞ ¼
ð3Þ
R normalized such that ¥ 0 Is dt = 1. When the bright state is mixed into an ensemble of dark states, the resultant eigenstates, {|mæ}, each have a fractional bright state character of |Æs|mæ|2 = am2. If the lifetime of a pure dark state is much longer than τs, the lifetime of a mixed eigenstate is τs/am2, and its time-dependent fluorescence intensity is ! 4 2 am am t ð4Þ Im ðtÞ ¼ exp τs τs R 2 The integrated fluorescence intensity is ¥ 0 Im(t) dt = am , relative to the unit intensity for a pure bright state, under the assumption that the fluorescence signal from a mixed eigenstate is detected with the same efficiency as that of a pure bright state. This reflects a factor of am2 smaller probability for excitation of a mixed eigenstate. Figure 1 shows the dependence of fluorescence intensity on the fractional bright state character at several values of time delay. When t = 0, the fluorescence intensity is monotonically increasing with respect to fractional bright state character. In contrast, at a delay of t=5τs, the fluorescence intensity described by eq 4 is “tuned” to states with a fractional bright state character of 40%. Figure 1 also shows the dependence of fluorescence intensity on fractional bright state character for t=15τs. At this time delay, the fluorescence intensity is at a maximum for mixed eigenstates with fractional bright state character of approximately 15%. At intermediate delay times, molecules in eigenstates containing too much fractional bright state character are discriminated against, because they have already fluoresced with
where the unit-normalized intensity distribution, I(E,t), is measured within a time window t to t þ dt after excitation. To illustrate how the center of gravity evolves with time delay, we investigate the relative fluorescence intensities from individual eigenstates as the time delay is increased. For a pure bright state, |sæ, with radiative lifetime, τs, the time-dependent
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1 1 exp τs τs
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DOI: 10.1021/jz100526g |J. Phys. Chem. Lett. 2010, 1, 2144–2148
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high probability. Traditional methods of lifetime-separated spectroscopy use precisely this principle to investigate spectra of predominantly dark eigenstates.13 Conversely, molecules in eigenstates with too little fractional bright state character are also discriminated against at intermediate delay times, because they have a low probability of fluorescing in the time window under consideration. A simple model may be used to illustrate how bright state~dark state mixing influences the behavior of the center of gravity as a function of delay time. To construct this model, we momentarily set aside the energetically distant doorway state and consider a simple case of mixing between a bright state, |sæ, and a single dark state from the near-degenerate manifold, |tæ. The two mixed states |1æ and |2æ are j1æ ¼ ð1 - a2 Þ1=2 jsæ þ ajtæ j2æ ¼ ajsæ þ ð1 - a2 Þ1=2 jtæ
ð5Þ
where a is the mixing coefficient between |sæ and |tæ, 0 e a2 e 0.5. The fractional bright state character of |2æ (the nominal dark state) is a2, and that of |1æ (the nominal bright state) is (1 - a2). The ratio of the fluorescence intensities, I1(t)/I2(t), has the following time dependence: !2 ! I1 ðtÞ 1 - a2 ð1 - 2a2 Þt ð6Þ ¼ exp I2 ðtÞ τs a2
Figure 2. Time development of the intensity ratio and center of gravity for a model system containing two basis states. At small values of the mixing fraction, 0.001e a2 e 0.2, the relative intensity changes rapidly from E1 to E2 at a time delay between 5τs and 15τs, leading to a rapid shift in the intensity-weighted center of gravity. When the mixing fraction is nearly 0.5, the eigenstates have similar intensities at t = 0, and the center of gravity at t = 0 lies midway between E1 and E2. The magnitude of the change in center of gravity is decreased, and the center of gravity changes slowly with time delay.
At t = 0, the intensity ratio is determined by the prefactor [(1 - a2)/a2]2. In the limit of long time delay, I1(t)/I2(t) always approaches zero, because the lifetime of |2æ is longer than that of |1æ, by definition. When the mixing fraction is large, close to its limiting value of 0.5, the prefactor [(1 - a2)/a2]2 causes the intensities I1 and I2 to be of comparable magnitude at t = 0. The intensity-weighted center of gravity of the model system may be written as a sum of two terms, Eave ðtÞ ¼ E1 I1 ðtÞ þ E2 I2 ðtÞ
overall magnitude of indirect, doorway-mediated coupling is represented by the mixing fraction, a2, and the energy dependence of the mixing is represented by the Es - Et energy difference. The time-dependent center of gravity metric is applied to individually resolved rovibrational transitions from the ~ 1Σþ ~ 1Au r X A g spectrum of acetylene. Each acetylene transition is a cluster of approximately 5 unresolved eigenstates, arising from a single S1 bright state and several near-degenerate T1,2 dark states. Three transitions are selected to illustrate the weak, medium, and strong mixing regimes. The approximate degree of mixing may be deduced from the average eigenstate lifetime and the energy width of the cluster. The Q(2) transition of the 2v2 þ v3 K0 =1 sub-band, which contains only one vibrational quantum of ν3, provides an example of the weak mixing case. The R(1) transition of 3v3 K0 =2 is an example of strong mixing, and the Q(2) transition of v2 þ 2v3 K0 =1 is an intermediate case example. Figure 3 shows the spectra surrounding the three selected transitions. The LIF intensity is integrated in two time windows, showing the difference between early (0.27 - 1.35 μs) and delayed spectra (1.89 - 4.05 μs). The redshifts of line centers in the delayed LIF spectra arise from changes in relative intensity among partially resolved eigenstates. The presence of near-degenerate T3 perturbers is ruled out in these spectra, because we observe no telltale, systematic level splittings (“extra lines”) that would indicate a near-degenerate triplet F2 spin component, as in 3ν3 K0 =1.12 A short segment of spectrum surrounding the transition, approximately the energy width of the observed clusters of nominal T1,2 mixed eigenstates, is used as input for the center of gravity metric.
ð7Þ
where E1 and E2 are the energies of the |1æ and |2æ eigenstates, respectively. The intensity-weighted center of gravity is plotted as a function of time delay in Figure 2. The behavior of the center of gravity follows that of the intensity ratio, which is also shown. At small values of the mixing fraction, 0.001 e a2 e 0.2, the intensity-weighted center of gravity evolves rapidly from E1 to E2 at a time delay between 5τs and 15τs. When the mixing fraction is nearly 0.5, the initial center of gravity at t = 0 lies midway between E1 and E2, because |1æ and |2æ have similar intensities at t = 0. In this case of near50:50 mixing, the center of gravity changes slowly with time. The intensity-weighted center of gravity still approaches E2 in the limit of long time delay, because the lifetime of |2æ is always longer than that of |1æ. The total magnitude of the change in center of gravity is reduced, due to the greater relative intensity of |2æ at t = 0. For a doorway-mediated system containing many dark states, the behavior of the center of gravity is more complex, but the two-state model captures the essential features. The presence of an energetically distant doorway state gives rise to an energy dependence of the mixing between the bright state (represented by |sæ) and the ensemble of near-degenerate dark states (represented by |tæ). In the simplified model, the
r 2010 American Chemical Society
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DOI: 10.1021/jz100526g |J. Phys. Chem. Lett. 2010, 1, 2144–2148
pubs.acs.org/JPCL
0 ~ 1Au r X ~ 1Σþ Figure 3. LIF spectra of three rotational transitions in the A g spectrum of acetylene, C2H2: the Q(2) line of 2v2 þ v3 K = 1 (plot a, blue), the R(1) line of 3v3 K0 = 2 (plot b, green), and the Q(2) line of v2 þ2v3 K0 =1 (plot c, red). The transitions consist of a single S1 state and approximately 5 T1,2 dark states, which are mixed by an energetically distant (unidentified) T3 doorway level. The LIF intensity is integrated in two time windows, showing the difference between early (0.27 - 1.35 μs, solid lines) and delayed spectra (1.89 - 4.05 μs, dashed lines). Red shifted line centers in the delayed LIF spectra arise from changes in relative intensity among partially resolved eigenstates.
change in center of gravity is larger than that of 2v2þv3 K0 =1, because the more strongly interacting T3 doorway state induces a more strongly varying energy dependence of fractional S1 character in the local ensemble of T1,2 dark states. However, the mixing is not so large as to degrade the resolution in relative eigenstate intensity as the delay time is increased. This phenomenon causes the overall magnitude of the change in the center of gravity for v2 þ 2v3 K0 = 1 to be larger than that of 3v3 K0 =2. The useful upper limit of time delay is determined by the field-of-view of the fluorescence detection optics in the experimental apparatus. In this study, the field-of-view of the fluorescence detection optics is about 5 mm. The maximum viewing time is therefore about 5 μs ≈ 18τs for molecules traveling in the molecular beam with average velocity ≈ 103 m/s. At a time delay longer than 18τs, no molecules remain to be seen within the fluorescence field of view. The two-state model system indicates that a mixing fraction of only 0.001 is sufficient to cause an observable change in the center of gravity within the maximum viewing time. The center of gravity metric is robust with respect to common sources of measurement deficiencies (resolution, dynamic range, and pulse-to-pulse laser intensity fluctuations) in fluorescence excitation spectra. To analyze the local cluster of lines surrounding an S1 r S0 rovibronic transition, relative intensities are compared over a very small energy range, typically less than 1 cm-1. This has the effect of minimizing relative intensity errors, which arise in LIF experiments mostly from slowly drifting laser intensity, spatial mode, and baseline effects. Unlike traditional measurements of fluorescence lifetime, the center of gravity metric is not biased by molecules leaving the LIF detection area. We have introduced a new metric, the time-dependent center of gravity, which may be applied to incoherently excited, time-gated, high-resolution spectra of small molecules in the energy region surrounding a bright state transition. The metric exhibits characteristic behavior that reveals the magnitude of the bright~dark matrix element and the energy of the doorway state relative to the bright state. Additionally, the metric is sensitive to bias in the local distribution of fractional bright state character, which is indicative of mechanistic interactions beyond the scope of Golden Rule models.
Figure 4. Dependence of the intensity-weighted center of gravity ~ 1Au r X ~ 1Σþ on delay time for three rotational transitions in the A g spectrum of acetylene. The Q(2) transition for 2v2 þ v3 K0 = 1 (trace a, blue) provides an example of a weakly mixed level, where a small shift in the center of gravity occurs rapidly near t = 8τs. The R(1) transition of 3v3 K0 = 2 (trace b, green) is an example of a strongly mixed level, where the center of gravity changes slowly and at a nearly constant rate as the time delay is increased. The Q(2) transition of v2 þ 2v3 K0 = 1 (trace c, red) is an example of the intermediate case.
Figure 4 shows the time-dependent center of gravity as a function of delay time, in units of τs. For S1 acetylene, the zeroorder bright state lifetime, τs, is largely independent of vibrational and rotational levels within S1, and is determined to be approximately 270 ns.14,15 The v3-dependent trend of the measured, in contrast with the zero-order lifetime toward larger τ is due to the dilution of S1 character into the Tn manifold.16 In the weakly mixed 2v2 þ v3 K0 =1 Q(2) transition, a small shift in the center of gravity occurs rapidly near t=8τs. In the strongly mixed R(1) transition of 3v3 K0 =2, the center of gravity changes slowly and with a nearly constant rate as the time delay is increased. The Q(2) transition of v2 þ 2v3 K0 =1 illustrates the intermediate case. The oscillations observed in this trace are residual effects of quantum beats, which occur between eigenstates of strongly mixed character with energy spacing of
