David Perry
email: dperry@uakron.edu

Anthony Miller

Badr Amyay

André Fayt

Michel Herman
email: mherman@ulb.ac.be

Richard Zare

Prof M Herman

Vibration-rotation alchemy in acetylene (12C2H2), at low vibrational excitation: From high resolution spectroscopy to fast intramolecular dynamics

Keywords: IVR in acetylene, Rotational effects in IVR, Overtone spectroscopy in acetylene, Overtone intensity

The link between energy-resolved spectra and time-resolved dynamics is explored quantitatively for acetylene (

with up to 8,600 cm -1 of vibrational energy. This comparison is based on the extensive and reliable knowledge of the vibration-rotation energy levels and on the model Hamiltonian used to fit them to high precision (B.

Introduction

In molecular spectroscopy, when the density of vibrational states is large, a given vibrational bright state that carries oscillator strength from the vibrational ground state may interact with many dark vibrational states. Each dark state then borrows some oscillator strength from the bright state greatly increasing the number of observed features in a high resolution spectrum 1,2 . The set of molecular eigenstates that results from the interactions with a single rotational level of the bright state is often referred to as a feature state. Stimulated emission pumping (SEP) and infrared techniques have been applied to obtain eigenstate-resolved spectra of a wide variety of molecules and many insights into the nature of intramolecular vibrational couplings have been obtained 3,4 .

With the assumption that only one of the zeroth-order vibrational basis states is responsible for all of the observed intensity in a given feature state, the spectra contain enough information to calculate the time dependence following a hypothetical coherent excitation of the whole clump by a short light pulse 4,5,6 . The initially prepared non-stationary state is the bright state, which subsequently dephases and the vibrational amplitude moves into other zeroth-order basis states. This dynamical process, termed intramolecular vibrational redistribution (IVR), may be characterized by plotting the calculated survival probability of the bright state as a function of time. The time constant for the initial decay of the bright state is τ IVR , and in some cases, the probability of the initial bright state may The analysis of high resolution vibration-rotation spectra demonstrated the clustering of vibrational states in some molecules, to form polyads or clusters, each assigned to a set of quantum numbers. This structure reliably emerges whenever the ratios of a molecule's vibrational frequencies are approximately the ratios of small integers. Anharmonic resonances connecting the close vibrational states determine the level pattern within each cluster. Each polyad may in fact extend over a broad spectral range. Although there may be significant overlap between polyads, the strongest energy and intensity perturbations strictly arise from interactions within a polyad, at least within excitation ranges as dealt with in the present investigation. Under these circumstances, the polyads are examples of feature states as discussed above. Molecules that exhibit such polyads, when well characterized by conventional spectroscopy, are ideal targets for unraveling the interactions within the feature states and for exploration of the related dynamics.

Vibrational polyads are known to shape the vibrational structure in a number of triatomic species such as CO 2 , H 2 O, N 2 O and OCS 22,23 . They are also very well characterized in a larger species such as ethyne (acetylene, C 2 H 2 ) 22,[START_REF] Herman | [END_REF] . The analysis of conventional and SEP spectra show that there are well-defined polyads at least up to 8,600 cm -1 in 12 C 2 H 2 and most probably up to higher energies 22 . The increasing density of vibrational sub-states with excitation, from less than 1 (/cm -1 ) at low energies to dozens around 1 µm and 100's in the visible range . As the total energy increases, these resonances result in new kinds of vibrational motion. 31,32,33,34 An analysis based on classical mechanics 31,34 shows that the bending normal modes bifurcate to give rise to new kinds of motions, such as the local bend, the counter rotator, the orthogonal bend, and precessional modes. Vibrational structure in acetylene is thus an ideal test case for exploration of the connection between polyad spectra and dynamics. 25,26,30,35,36,37 Following early reports of rotational involvement in IVR 38,39 , Lawrance and Knight 40 analyzed the possible roles of Coriolis and centrifugal couplings. These rotational couplings may affect the observed rate of IVR with dependences on the rotational quantum numbers J and K that are characteristic of the particular coupling mechanism , 19,40,41, ,42 , and they may serve to increase the density of coupled levels 19,38,42 . Since Coriolis coupling links the rotational states in steps ∆K = ±1, the time required to populate all of the accessible K-levels may be very much longer than the decay time τ IVR of the bright state 20 .

Recently, a detailed global model has been reported which simultaneously treats both the vibrational and rotational structure in ... [36] ... [27] ... [22] ... [28] ... [26] ... [40] ... [29] ... [START_REF] Herman | [END_REF] ... [30] ... [START_REF] Knight | Excited States[END_REF] ... [35] ... [START_REF] Mcilroy | [END_REF] ... [31] ... [38] ... [43] ... [32] ... [39] ... [44] ... [33] ... [45] ... [34] ... [46] ... [START_REF] Perevalov | The NATO Programme for Security through Science C: Environmental security[END_REF] ... [47] ... [ The reliability and global nature of this spectroscopic Hamiltonian may shed some new light on the vibrational and also rotational aspects of IVR.

The aim in this work is to explore quantitatively the relationship between the energy-resolved spectra and the time-resolved dynamics of acetylene ( 12C 2 H 2 ), g X 1 + Σ % by taking advantage of the detailed knowledge of the vibration-rotation energy levels in this species up to 8,600 cm -1

. Specifically, we will examine how the collisionless time-dependent dynamics are affected by (i) the choice of the bright vibrational state, (ii) the extent of rotational excitation, and (iii) the excitation of a thermal ensemble of rotational levels.

Section II of this paper reviews the means by which time-dependent dynamics are deduced from frequency-resolved spectroscopic data. After recalling significant features of the global model in section III, the role of basic intramolecular couplings is illustrated in a series of pedagogical case studies in section IV, both in energy and time domains. Then in section V, the dynamical behavior of thermal distributions of J-levels is examined for two case studies, before concluding in section VI. The usual unperturbed / perturbed or zeroth-order / eigenstate scheme for a 2 by 2 perturbation is recalled in Figure 1, with notations similar to Xu et al 44 . We use the convention that states with lower numbering (I/1) have lower energy than states with higher numbering (II/2). Such a scheme is used in high resolution, energy-resolved spectroscopy to explain intensity borrowing that results from anharmonic or Coriolis mixing of two zeroth-order rotation-vibration states. In this case, the mixing is often presented as

II. Spectroscopy and Dynamics

0 0 1 2 0 0 1 2 I II a b b a ψ ψ ψ ψ ψ ψ = - = + (1)
where the coefficients a and b are identical to those in This scheme is also used in time-resolved spectroscopy when coherently exciting the two eigenstates ψ I and ψ ΙΙ with a short laser pulse and following the evolution with time of the total, non stationary wavefunction,

The time-dependent probability of finding the system in the bright and dark zerothorder states is oscillatory (5) where δε = E II ---E I , and the oscillation period is τ rec = h/δε.

This approach can be extended to a more complicated set of interactions involving n basis states 5,6 . In this case, the basis functions j ψ 0 can be expressed as a linear combination of the n eigenstates k ψ :

n k j jk k c ψ ψ = = ∑ 0 1 . (6)
If we assume that only one of the basis states, say j = 1, carries oscillator strength from the ground state, then the relative intensities of the eigenstates are

and the time-dependent wavefunction following a coherent excitation is

. ( 8
)
At t = 0, the bright state is prepared, ( ) ψ

Ψ = 0 1 0
, but thereafter the time-dependent probability of the various basis states may be calculated as

. ( 9
)
When j=1, this yields P 1 (t), the survival probability of the bright state. This timedependence is in general more complicated than the simple oscillatory pattern given in Eq. ( 5) and involves movement of vibrational amplitude among the various coupled rotation-vibration states, that is, IVR.

There are a number of physical phenomena that are not included in the present treatment. In particular, we include no effects, such as collisions or infrared radiative lifetimes, that would impart a finite lifetime and a homogeneous line width to individual eigenstates. Hyperfine structure is not considered here. Below, we do include a thermal distribution of rotational levels as an approximate treatment of the inhomogeneous rotational structure, but do not consider Doppler broadening. The neglected effects would cause P 1 (t) to decay to zero or at least damp out its fluctuations. As calculated from the above equations, P 1 (t) will continue indefinitely to fluctuate around a finite average value. Three different measures are used in this paper to characterize the timedependent dynamics. The first is the dilution factor 49,50,51 φ d , which is determined by the way the bright state character is distributed among each of the eigenstates,

φ d = c 1k 4 k=1 n ∑ = I k 2 k=1 n ∑ I k k=1 n ∑       2 ( 10
)
and is consequently related to the spectral intensities through Eq. ( 7). The effective number of coupled states is 1 φ d . For an unperturbed state, φ d = 1, and if all n eigenstates appeared in the spectrum with equal intensity, we would have

φ d = 1 n.
The dilution factor can be obtained from the time-dependent dynamics as the timeaveraged probability of the bright state, P 1 (t), where the average extends out to long times,

φ d = lim t →∞ 1 t P 1 t ( ) 0 t ∫ dt . ( 11
)
The second measure, IVR lifetime 3 τ IVR , is usually defined as the time required for the bright state probability P 1 (t) to decay from its initial value, P 1 (0) = 1, to 1/e.

Since in general the form of P 1 (t) is non-exponential, the 1/e value is simply a convenient reference point to take as a measure of the rate of IVR. In the present work, where we are dealing with the very sparse limit, we define τ IVR as the time for ( ) 1-φ d ( ). 51 In the limit of many coupled levels ( φ d → 0), this reduces to the usual definition of τ IVR .

The final measure is the recurrence time τ rec , which is just the oscillation period (Eq. ( 5)) when there are only two interacting states. When there are more interacting levels, multiple recurrence times may be evident in a single P 1 (t) curve.

The spectrum of eigenstates resulting from the excitation of a single ground state rotational level is referred to as a ''feature state'' or ''clump'' and represents a kind of homogeneous broadening. Thus, the ensuing dynamics are also called ''homogeneous''. This will be contrasted below with an inhomogeneously broadened spectrum resulting from of the excitation of a thermal distribution of rotational levels and the ensuing inhomogeneous dynamics.

III. A global spectroscopic Hamiltonian

The conventional normal mode numbering for acetylene is used throughout this paper. Modes 1 to 5 correspond to the symmetric CH g σ + (ν C 2 H 2 up to 8,600 cm -1 . The data base includes rotational levels up to J = 68 for some sub-states as well as data from highly excited trans-bend states, from Field and coworkers 59 . All lines could be reproduced within their stated measurement accuracy (3σ), meaning typically better than 0.001 cm -1 , except for a limited number of them (8%) These problems are all local and attributed either to measurement problems, due to line overlap or lower line intensity, or to Coriolis perturbation, that cannot be fixed without a dedicated reanalysis of the literature spectra. We are confident that such small errors do not affect the trends emerging from this work. In all, 331 vibration-rotation parameters were determined from the fit.

The present investigation is meant to highlight the basic processes.

Therefore, the case studies to be presented here were selected to be well within the energy range (≤ 8,600 cm -1 ) of the data fit by the global model.

IV. Basic intramolecular couplings

The definition of the zeroth-order states is a critical issue. We include in this concept the states whose energy results from all of the diagonal matrix elements in the global model, in the symmetrised e/f basis set as defined by Fayt and coworkers

57

. Therefore, the implicit definition used here is that the zeroth order states are those states that have the diagonal energies of the global model as their eigenvalues. In a multidimensional problem, even as small as for acetylene, it is a challenge to write out an explicit form for such ''implicit'' basis functions. Indeed this challenge constitutes one of the conceptual limitations of effective

Hamiltonians derived from fitting spectroscopic data.

The implicit set of zeroth-order basis states incorporates all of the major terms defining the vibration-rotation energies in linear molecules, including the role of diagonal vibrational 60,61 and rotational l-type resonances, shaping the regularly evolving vibrational and rotational energy spacings with increasing v i and J excitation. This set allows convenient implementation of the major transition selection rules, including identification of bright state. The remaining terms in the Hamiltonian are off-diagonal and arise from anharmonic interaction, Coriolis coupling and rotational l-type resonance. Their introduction into the Hamiltonian matrix provides coupling between the basis states and hence defines eigenstates.

The role of each of these terms in coupling the basis states and in the timedependent dynamics is investigated in the present section. Only the role of the As pointed out before, zeroth-order sub-states are labeled (

1 2 3 4 5 4 5 v v v v v l l , ).
Eigenstates are labeled using Roman numerals, numbered from lowest energy to highest.

A. Rotational l-type resonance

The rotational l-type resonance connects sub-states related by ∆k = ±2, with matrix elements scaling as J(J+1) . The resultant mixing of the basis states is illustrated in Figure 2 for the {0,2,k e ,e,u} polyad. This polyad contains two zerothorder sub-states, (00011,1-1), Σ eu and (00011,11), ∆ eu . The first of these is the only one allowed by the conventional ∆k = 0, ±1 IR selection rules from the ground state (GS), thus leading to the (00011,1-1), Σ eu ← GS transition. It is therefore the bright sub-state for IR absorption from the GS.

The rotational l-type resonance introduces |∆k| = 2 interaction terms connecting the zeroth-order states, ( )

1/ 2 e eff e 4 5
1 00011, H 00011, q q J ( J 1 ) J ( J 1 ) 2 2

Σ ∆   = + + + -       . ( 12
)
We consider only the e-levels defining the two eigenstates I and II. Their energies were calculated from the global model. The reduced energies presented in Figure The separation between the two sets of eigenenergies is determined by the parameters (in cm -1 ) q 4 = -5.231 10 -3 and q 5 = -4.698 10 -3 43 and is J(J+1)-dependent in agreement with Eq. ( 12). The zeroth-order sub-states are also presented on multiplying the lower intensity lines by a factor 3. They were simulated for temperature conditions high enough (500 K) for high J-levels to be populated in the GS. As a consequence of the rotational l-type resonance, intensity borrowing occurs, transferring some of the intensity of the I ← GS band to the II ← GS band. The intensity transfer is greater at high J-values, as expected from the increasing strength of the rotational l-type resonance matrix elements (Eq. ( 12)). The overall intensity transfer remains modest.

ii) Survival probability: Figure 2d presents the survival probabilities in the zeroth-order bright sub-state calculated for three selected values of J (2, 30 and 100). The same color and marker style convention is used in Figures 2a and2d for the zeroth-order sub-states. In this 2-state case, the time-dependences follow Eq.

(5). At J = 2, the wave packet remains in the bright zeroth-order sub-state. As J increases, oscillation occurs, with the other zeroth-order sub-state participating.

The frequency and amplitude of the oscillation increase with J, again as expected from the rotational l-type resonance matrix elements (Eq. ( 12)). The period of the oscillation, τ rec = h/δε, defined in Eq. ( 5) thus decreases as the energy difference ( δε )

increases. The dilution factor (φ d ) evolves from the limit of weak interaction (φ d = 1) at J = 2 to approach the limit of strong interaction between the two basis states (φ d = 0.5) at J = 100.

B. Anharmonic resonances

The next case study is the {1,5,k e ,e,u} polyad, which contains three zerothorder sub-states, (00100,00), Σ eu , (01011,1-1), Σ eu and (01011,11), ∆ eu . The same rotational l-type resonance perturbation as just discussed is again active. In addition to rotational l-type resonance, the anharmonic 3/245 coupling 66 connects the two-zeroth-order sub-states of Σ eu symmetry, (00100,00) and (01011,1-1). The interaction matrix elements are

[ ] ± - + - - = - + ± m % m 1 2 5 5 4 4 1 1 1 8 1 2 3 4 5 1 2 3 4 5 3 / 245 2 3 4 4 5 5
, , , , ,

)

l l l l v v v v v H v v v v v K v v v l v l (13) with |K 3/245 | = 18.3 cm -1 43 .
This additional coupling is strong, with J-independent matrix elements; however, the mixing is slightly J-dependent. In a second-order perturbation scheme, the mixing coefficients are related to the energy difference between the two interacting levels (Fig. 1), which varies for each pair of J-levels according to the vibrational dependence of the rotational constants. A reduced energy graph is presented in Figure 3a. The zeroth-order sub-states are also presented and identified on Figure 3a. At very low J-values, (000100,00) Σ eu is slightly higher in energy than (01011,1-1), Σ eu .

In order to allow for a direct comparison with the above case study, we consider first the fate of the (01011,1-1), Σ eu zeroth-order sub-state, which is taken for the moment as the bright state, and therefore plotted as a solid red line in Figure 3a. With allowance for the increased vibrational energy, the behavior would be very similar to the one exemplified in Figure 2 for the (00011,1-1), Σ eu zerothorder sub-state since they both involve the ν 4 +ν 5 vibrational combination. The interplay is however now more complex thanks to the anharmonic resonance bringing in the third sub-state. Compared to situation in Figure 2a, the anharmonic coupling now leads to a significant repulsion between the two zerothorder sub-states close to 3290 cm -1 , while the rotational l-type resonance acts more strongly at higher J-values. The situation is further detailed in Figure 3b, which quantifies the amount of the zeroth-order sub-state in eigenstates I to III. The fraction of the zeroth-order bright state is nearly identical in I and II for the lowest J-values, then increases slightly for II up to J = 45, where it falls below I again. As J increases further, the zeroth-order bright wavefunction is shared mainly between I and III. This behavior looks thus different from the previous case study because at low J-values the anharmonic resonance dominates. At high J-values, it is more similar to the previous case because the J-dependent rotational l-type resonance becomes the dominant interaction.

In this 3-level system, the time-dependence determined by Eqs. ( 8) and ( 9) is more complicated than that given in Eq. ( 5). The time-dependent probabilities in sub-states (J = 2, top graph), while the third (01011,11) ∆ eu zeroth-order sub-state is more and more involved (J = 30, middle graph) and even dominates over (00100,00) at high J-values (J = 100, bottom graph). Thus the rotational l-type resonance takes over as J increases.

The set of graphs is repeated in Figure 4, but now selecting the bright substate for IR absorption from the GS, i.e. (00100,00), Σ eu . This sub-state involves only one vibrational quantum excitation, as compared to three for (01011,1-1), Σ eu .

Furthermore, it corresponds to a CH stretch, which is expected to result in a strong absorption band. Figure 4a is thus identical to Figure 3a, but for the selection of the zeroth-order bright sub-state. The related line color and marker style were adapted accordingly. Figure 4b presents the fraction of this new zeroth-order bright sub-state in the eigenstates. Figure 4c shows the resulting intensity borrowing, using the same convention for the ortho:para relative line intensities, as in Figure 2c. It occurs as in the previous case study (Fig. 2c), but is now distributed over three bands in agreement with the mixing scheme. In other words the zeroth-order bright character is equally shared by the I ← GS and II ← GS bands at low J, then I ← GS gains intensity up to J = 55, after which II ← GS dominates. The band III ← GS is weaker but receives about 2% of bright character for 35 < J < 45. The population factor, in Figure 4c, makes this band appear stronger between J = 20 and 30, at 500 K. The survival probability for the infrared-active bright state is presented in Figure 4d for three selected J-values (2, 30 and 100). As could be expected from the previous graphs in Figure 4, the wavepacket first oscillates between the bright substate and the anharmonically coupled state, (01011,1-1), Σ eu (top graph). However, now the third zeroth-order state, (01011,11), ∆ eu , is only indirectly coupled to (00100,00), Σ eu through the (00100,00) The dilution factor provides a convenient way of quantifying the extent of mixing of the different bright states in this 3-state system. First, we note that the extent of mixing may vary non-monotonically with J. For example, with the bright sub-state (01011,1-1), Σ eu (Fig. 3c interacting with (01011), with their various k-components. The Coriolis matrix elements are given by (14) with .

Like the rotational l-type resonance, the Coriolis couplings depend explicitly on rotation. Although the value of the C 2/444 parameter at 7.8 10 -3 cm -1 43 is similar to the size of the q 4 and q 5 parameters in the rotational l-type resonance, the rotational dependence is, however more modest, rather than ~J J + 1 ( ), resulting in much smaller coupling matrix elements, ~0.5 cm -1 at J = 100 as opposed to ~50 cm -1 . Therefore, the Coriolis couplings appear mainly as local avoided crossings, affecting only few J's. The avoided crossings can be easily identified in Figure 5a as the locations of abrupt changes in the slope of the eigenstate reduced-energy curves.

The eigenstates in Figure 5a Coriolis-coupled to (01011), do take part in the dynamical behavior at J = 92. At J = 92, the total probability of the Coriolis-coupled states reaches a maximum of ~10% at 8 ps and proceeds to oscillate weakly with a period of 16 ps. Although there are numerous Coriolis resonances (Fig. 5a), and such resonances do affect the dynamics at the localized J's where they occur (Fig. 5d), most J's remain more-or-less unaffected.

V. Thermally averaged dynamics

In the previous section, we have explored the basic mechanisms perturbing the zeroth-order vibration-rotation picture in acetylene in the lower energy region (0 ---8,600 cm -1 ) where its rotation-vibration spectroscopy is most complete. The same mechanisms rule the time-dependent dynamics resulting from coherent excitations.

Since all of the interactions that are significant in this energy region are known to good precision, the rate at which energy arrives in the receiving modes as well as the rate at which it leaves the bright state can be calculated reliably. That is, the complete vibrational dynamics are obtainable from the experimental Hamiltonian.

The detailed rotational analysis in this region allowed us to examine the rotational dependence of the dynamics. Since some of the important intramolecular interactions, such as the rotational l-type resonance and Coriolis interactions, ). This means that a coherent excitation with an infrared source, which produces u upper states from the g GS, would yield a preponderance of even J upper state molecules. Likewise a coherent Raman or SEP excitation would yield odd J's preferentially (if proceeding through R-or P-line excitation). This rule applies, in both cases, for e-symmetry levels.

A. Polyad {7,e,u}

The zeroth-order IR bright sub-state in the {7,e,u} polyad is (00120,00), Σ eu .

The complex set of global and local perturbations is evident from the irregular Jdependence of the reduced energies in Figure 7a. All of the interactions are well characterized spectroscopically 43 . The contribution of the bright zeroth-order substate to the selected 10 relevant eigenstates, labeled I to X, is shown in Figure 7b.

Since there are a total of 45 sub-states in the polyad, each of the parts of Figure 7 are simplified to show only the most significant quantities.

The dynamics is presented in Figure 7c. Thermal averaging obviously smoothes out the detailed interaction pattern. Nevertheless, one can identify the main trends of the behavior in terms of the basic intramolecular couplings investigated in the previous section. At low temperature, since only the very first Jlevels are populated, the dynamics is dominated by the 3/245 anharmonic resonance connecting the (00120,00) and (01031,1-1) sub-states. The probability oscillates between these two sub-states with a period of 1.7 ps. This is the pattern expected for a 2-state system (Eq. ( 5)) with little involvement from other states in the polyad.

At 270 and 500 K, higher J-levels are involved and the rotational l-type resonance plays an increasing role. The rotational l-type resonance-coupled sub-state (01031,3-1) reaches a maximum probability after about 3 ps, and oscillations ensue Felker and Zewail 68 noted the damping of oscillations by small frequency differences among the thermally-averaged rotational levels. In the present case, the dephasing of the rotational l-type resonance oscillations is rapid because of the explicit J-dependence of the coupling (Eq. ( 12)). The anharmonic resonance oscillations damp out much more slowly because the anharmonic interaction has only the incidental rotational dependence that results from the slight variation of the rotational constants with different kinds of vibrational excitation. Thus, the apparently irreversible decay of the oscillations in Figure 7c does not mean that any Although Coriolis coupling is involved at intermediate J-values in the complex interaction pattern shown in Figure 7a, it remains very local and therefore does not play a significant role in the averaged dynamics.

B. Polyad {7,e,g}

There is no IR allowed transition from the GS to any state in the {7,e,g} polyad, all of which have g-symmetry. The SEP bright sub-states are (01040,00), Σ eg and (01040,20), ∆ g . Richer rotationally-dependent dynamics can, however, be expected from the (00070,10), Π eg sub-state in this polyad. This sub-state is indeed mixed through the 44/55 anharmonic resonance to other sub-states such as (00052).

In addition, rotational l-type resonance generates strong couplings for this sub-state spanning k-values from 1 to 7, by steps of 2. Furthermore, it is coupled to the previously identified SEP bright sub-states thanks to 2/444 Coriolis resonance. We have therefore selected it as our hypothetical zeroth-order bright sub-state. The reduced energy graph is presented in Figure 8a, with only relevant zeroth-order and eigenstates presented. The corresponding sub-state fractionation and dynamics are presented in Figure 8b andc, respectively. The time-dependent dynamics in Figure 8c look quite different from the previous case (Fig. 7c). At low temperature, the behavior is dominated by the 44/55 anharmonic resonance. It is too far off resonance to generate much redistribution, but it does result in high-frequency (0.2 ps), small amplitude (12%) oscillations in the zeroth-order bright sub-state survival probability. These small oscillations persist at the higher temperatures and out to long times. At the higher temperatures, the J-dependent interactions get stronger and induce what appears to be a rapid irreversible decay of the bright-state survival probability, τ IVR = 1.3 and 0.9 ps for 270 and 500 K, respectively. The absence of substantial recurrences results from the rotationally-dependent intramolecular couplings highlighted above.

Both rotational l-type resonance, bringing in the other sub-states from (00070), and

Coriolis coupling, bringing in the (01040) sub-states play an active role in the energy redistribution. The rotational l-type resonance within (01040) further fractionates the redistribution.

Insight into the nature of the apparent irreversible decays in Figure 8c can be obtained by examining the extent of mixing of the zeroth-order bright sub-state as measured by the dilution factor. We find φ d = 0.84, 0.51, 0.43 for the temperatures 27, 270, and 500 K respectively, which indicates that the fractionation for this gsymmetry state is comparable to but slightly less than for the u-, IR-active bright state represented in Figure 7. Just as for the above case (Fig. 7c), each of the constituent rotational levels contributing to Figure 8c recurrences extending out to long times (similar to Figs. 2d and4d). The curves in Figure 8c are the thermal average of curves with differing oscillation frequencies. In Figure 7, the strongest mixing of the basis states is caused by the anharmonic resonance (large fast oscillations in Fig. 7c), which is only indirectly J-dependent.

However in Figure 8, this interaction results in less mixing (small fast oscillations in Fig. 8c), and the dominant mixing comes from the J-dependent rotational l-type resonance. This gives rise to a stronger J-dependence of the constituent oscillation frequencies in Figure 8c, and the thermal average of such a wide range of oscillation frequencies has the appearance of an irreversible decay. Once again, this is inhomogeneous dynamics resulting from the inhomogeneous rotational structure within the coherent excitation bandwidth.

VI. Conclusion

The connection between the energy-resolved spectra and time-resolved dynamics has been explored in acetylene ( 12C 2 H 2 ),

g X 1 + Σ %
in a way that complements previous literature results 25,26,30,35,36,37 . Here, the emphasis is on the impact of rotational excitation on the time-dependent dynamics, which has not been considered in detail in the previous acetylene literature. The present results derive from the extensive and reliable knowledge of the vibration-rotation energy levels Finally, the effects of inhomogeneous broadening within the excited spectrum can be simulated, and these effects may change the qualitative form of the timedependent survival probabilities.

Formatted: English U.S. , and eigenstates, ψ ΙΙ and ψ Ι . The red/grey color code is selected to highlight the fraction of each zeroth-order wavefunction in the eigen wavefunctions. (medium) and 100 (lower). At t = 0, the bright state (red) begins at P 1 (t) = 1.0. The color scheme distinguishes the zeroth-order sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d). (medium) and 100 (lower). At t = 0, the bright state (red) begins at P 1 (t) = 1.0. The color scheme distinguishes the zeroth-order sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (c).

Formatted
( ) 1 173000 1 1 .
. (medium) and 100 (lower). At t = 0, the bright state (red) begins at P 1 (t) = 1.0. The color scheme distinguishes the zeroth-order sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d). Deleted: Reduced energies correspond to ( )

1 175076 1 1 .
. (middle) and 500 (lower) K. The color scheme and line marker styles distinguish zeroth-order sub-states according to their order of appearance in the time evolution.

The color and marker styles for the zeroth-order sub-states are the same in parts (a)

and (c).

Deleted: Reduced energies correspond to ( )

! " "# ! " " ! ! " # $ # $ % & ! "# ! # " ! # ' ! ( ( ! # "# ! " # $ % & " # ! " ! $ $$ $ $$ % & & % ( ( ( ( ( ( ' )
' * ) , selecting the {0,2,ke,e,u} polyad with the bright state taken to be the (00011,1-1), Σeu zeroth-order sub-state:

0 0 0 ( $$ " $$ ! " ( 0 " 0 2 0 1 0 " " +, * ' ( $ " $ ! # ( 0 1 " 2 " +, ! $ " # " ! # % & $ # $$ $ " " " ' ) -+, ( . / 1 2 2 0 2 12 4! ! " +, ' +, ) # $ % & 1 0 2 ( ) / 2 % ! " ' +, ) +, +, % & % # '& #( )(" & ! " ' +, * +, +, % & 1 0 2 ( ) / 2 . / # $ % &
(a) Reduced energy graph as a function of J(J+1). The eigenstates (black) are labeled I and II according to the ordering of their energies. The zeroth-order sub-states are colored and identified on the left using (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the bright state in the composition of the energy eigenstates plotted as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Calculated relative intensities of the R-branch lines in the I ← GS and II ← GS IR absorption bands. The simulated intensities are given in terms of the upper J quantum number (J') for a temperature of 500 K. The 3:1 intensity alternation due to ortho and para acetylene has been cancelled by multiplying the lower intensity lines by a factor 3;

(d) Calculated time-dependent probability of the basis states for J = 2 (upper), 30 (medium) and 100 (lower). At t = 0, the bright state (red) begins at P1(t) = 1.0. The color scheme distinguishes the zerothorder sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d). , selecting the {1,5,ke,e,u} polyad with the bright state taken to be the (01011,1-1), Σeu zeroth-order substate:

(a) Reduced energy graph of the sub-states in the polyad as a function of J(J+1). The eigenstates (black) are labeled I to III according to their energies. The zeroth-order sub-states shown in color and identified on the left as (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the bright state in the composition of the eigenstates of the polyad, plotted as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and are internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Calculated time-dependent probability of the basis states for J = 2 (upper), 30 (medium) and 100 (lower). At t = 0, the bright state (red) begins at P1(t) = 1.0. The color scheme distinguishes the zeroth-order sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (c). , selecting the {1,5,ke,e,u} polyad with the bright state taken to be the (00100,00), Σeu zeroth-order sub-state:

(a) Reduced energy graph of the sub-states in the polyad as a function of J(J+1). The eigenstates (black) are labeled I to III according to the ordering of the energy of the corresponding eigenvalues. The zeroth-order sub-states are colored and identified on the left as (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the bright state in the composition of the eigenstates of the polyad, plotted as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Calculated relative intensities of the R-branch lines in the I, II and III ←GS IR absorption bands. The simulated intensities are given in terms of the upper J quantum number (J') for a temperature of 500 K. The 3:1 intensity alternation due to ortho and para acetylene has been cancelled by multiplying the lower intensity lines by a factor 3;

(d) Calculated time-dependent probability of the basis states for J = 2 (upper), 30 (medium) and 100 (lower). At t = 0, the bright state (red) begins at P1(t) = 1.0. The color scheme distinguishes the zerothorder sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d). , selecting the {5,e,u} polyad with the bright state taken to be the (00100,00), Σeu zeroth-order sub-state:

(a) Reduced energy graph of selected sub-states in the polyad as a function of J(J+1). The eigenstates are identified by the marker style (black) and are labeled I to VII according to their energies. The three most relevant zeroth-order sub-states are colored and identified on the left as (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the eigenvector in the composition of the eigenstates in the polyad, as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Resulting intensity borrowing on related R-branch lines simulated at a temperature of 2,000 K, in function of the upper J quantum number (J').

The two major contributions to the intensity are summed up, defining the grey bar;

(d) Calculated time-dependent probability of the basis states for the Coriolis perturbed J = 92 level (lower), compared J = 88 level (upper), which is not Coriolis-perturbed. Portions of the graphs shown on an expanded time and probability scales. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d). Population distribution in the rotational levels of the GS in 12 C2H2, , at 27, 270 and 500 K. The ortho and para distributions are color coded. , with the zeroth-order bright sub-state taken to be (00120,00), Σeu: (a) Reduced energy graph of selected sub-states as function of J(J+1). The eigenstates are identified by the marker style (black) and are labeled I to X according to their energies. The four most relevant zeroth-order sub-states are colored and identified on the left as (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the zeroth-order bright sub-state eigenvector in the composition of the eigenstates in the polyad, as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and are internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Calculated time-dependent probability of the basis states for a thermal averaged population in the upper zerothorder bright sub-state, for T = 27 (upper), 270 (middle) and 500 (lower) K. The color code and line marker styles distinguish zeroth-order sub-states according to their successive rank in the time evolution. The color and marker styles for the zeroth-order sub-states are identical in parts (a) and (c). time / ps 0 0 0 7 0, 1 0eg 0 0 0 5 2, 1 0eg 0 1 0 4 0, 0 0eg 0 1 0 4 0, 2 0eg 0 0 0 7 0, 3 0eg 0 0 0 7 0, 5 0eg other The {7,e,g} polyad in 12 C2H2, , with the zeroth-order bright sub-state taken to be (00070,10), Πeg: (a) Reduced energy graph of selected sub-states in function of J(J+1). The eigenstates are identified by the marker style (black) and are labeled I to VIII according to their energies. The five most relevant zeroth-order sub-states are colored and identified on the left as (v1v2v3v4v5,l4l5eu);

(b) Squared coefficients of the (00070,10), Πeg, zeroth-order sub-state eigenvector in the composition of the other eigenstates in the polyad, as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and are internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);

(c) Calculated time-dependent probability of the basis states for a thermal averaged population in the upper zerothorder bright sub-state, for T = 27 (upper), 270 (middle) and 500 (lower) K. The color scheme and line marker styles distinguish zeroth-order sub-states according to their order of appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (c).

  region up to the threshold for isomerization to vinylidene has yielded a wealth of spectroscopic and dynamical insights.Low-resolution dispersed fluorescence spectra from the 1 u A A % state, reaching low-J, low-k states in the 10,000 to 15,000 cm -1 range, have allowed extensive analysis of the various vibrational resonances 26,27,28,29,30

12 C 2 H 2 ,

 1222 up to 8,600 cm -1 43 .

  Furthermore, it allows for the first time a quantitative exploration of the role of rotational degrees of freedom in acetylene IVR.Deleted: to link quantitatively the energy-and time-resolved approaches, including the rotational dependence,

Figure 1 . 2 I II ∝ b 2 ( 3 ) 2 = 1 .Figure 1

 1223211 Figure 1 by the red fractions in ψ I and ψ ΙΙ .

Formatted

51 Deleted: 51 Deleted: 10 Deleted: 3 Deleted: 12 P 1 (

 5151103121 At long times, Deleted: of the bright state Deleted: is the dilution factor 49,50,The effective number of coupled states is 1 φ d . For an unperturbed state, φ d = 1, and if all n eigenstates appeared in the spectrum with equal intensity, we would have φ d = 1 n. where only a few levels are strongly coupled and the long time limit P 1 t ( ) → φ d may be comparable to -or possibly even greater than -1/e . Therefore for the present purposes Deleted: τ IVR to take the finite value of φ d into account 51 ; t) to decay to φ d + 1 e

57 ,

 57 and then . include all of the diagonal matrix elements in the global model Deleted: TIn the present global model, the diagonalse elements, thus including the role of diagonal vibrational60,61 and rotational l-type resonances Deleted: , Deleted: shape the regularly evolving vibrational and rotational energy spacings with increasing v i and J excitation. each coupling case is discussed in the text, but all terms are included in the calculations.

Formatted

  For completeness, one should add that off-diagonal vibrational l-type resonance terms also contribute to eigenstates. They are included in the Hamiltonian, but their role is not considered further in the discussion below. by subtraction of the rotational energy contribution determined using a selected set of B and D parameters, identical for both sub-states.

Figure 2a ,

 2a Figure2a, and identified on the left using the (v 1 v 2 v 3 v 4 v 5 ,l 4 l 5 eu) label. The Jdependence of the rotational l-type resonance is obvious. The bright sub-state corresponds to the solid line, in red. The same presentation will occur in all similar graphs.

Formatted

Figure 3c reflect the

  Figure 3c reflect the fractionation discussed above. The probability first oscillates between the anharmonically coupled, (01011,1-1) and (00100,00) Σ eu zeroth-order

  Figure 2d, but the interactions involved are different.

25 anharmonic 2 /

 252 Figure 5b. As in Figure 5a, the abrupt changes in marker style along the smooth curves in Figure 5b indicate the locations of the Coriolis resonances.

  with rotational angular momentum, there are significant changes in the pattern of the dynamics with rotational excitation as has been demonstrated above. In the present section, we shall examine the results of exciting a thermal distribution of rotational levels. In this case, a blend of the basic mechanisms just highlighted is involved. The temperature regulates the rotational contribution in the dynamics. The population distributions in the GS are presented for the three selected temperatures, 27, 270 and 500 K, in Figure6. The behavior is also expected to depend strongly on the polyad and on zeroth-order bright substate. Polyads {7,e,u} and {7,e,g} were selected as working examples, both near 4500 cm -1 and therefore in the spectroscopically well characterized energy region. All of the interaction mechanisms contained in the global Hamiltonian are included in the calculations. For the two case studies, the GS rotational distribution is simply projected onto the e-levels of the upper zeroth-order bright sub-state. We did not include the detailed rotational line strengths or the particular rotational selection rules appropriate to a given spectroscopic technique. As a result, rotational coherences67 have been suppressed. However, we did include the appropriate nuclear spin weights appropriate: 1:3 for even:odd J for sub-states in {7,e,g} and 3:1 in {7,e,u} (see Figure27in Herman

24

 24

29 with a 7

 297 Figure 7c results from the dephasing of the oscillations of different J-levels that have different oscillation frequencies. Indeed, in their early ultrafast experiments,

  decays irreversibly, but it is an artifact of the inhomogeneous rotational structure within the coherent excitation bandwidth.

Formatted

  global spectroscopic Hamiltonian based on the block diagonalisation of vibrational polyads. Three kinds of intramolecular interactions, anharmonic, rotational l-and Coriolis resonances, perturb the zeroth-order vibration-rotation picture. The roles of each were quantitatively investigated for selected case studies well within the range of the fitted spectroscopic data (0 -8,600 cm -1 ). The fractionation of each bright sub-state into the various eigenstates of its polyad was demonstrated to affect both the intensities in the spectra and the IVR survival probabilities. The collisionless dynamics resulting from the excitation of thermal distributions of rotational levels was investigated for the ν 3 +2ν 4 and 7ν 4 bright states. The resulting inhomgeneous dynamics showed damped oscillatory survival probabilities for ν 3 +2ν 4 and apparent irreversible decay for 7ν 4 . It will be interesting to build on the present results to investigate the more extensive IVR that occurs at higher excitation energies.Although actual time-dependent experiments haven't yet been performed on this system, the detailed spectroscopic Hamiltonian allows the results of a wide variety of virtual time-dependent experiments to be calculated. This kind of simulation enlightens key issues in time-dependent experiments. First, the identity of the vibrational bright state is critical in determining the time-dependent dynamics.Different bright states in the same polyad or same energy region produce very different results, whereas similar bright states tend to produce similar results over a range of energies. Second, in the case of overlapping polyads, virtual The results quantitatively assess the roles of anharmonic, rotational l-and Coriolis resonances in the processes. IVR from the ν 3 +2ν 4 and 7ν 4 states was also investigated for Deleted: at T = 27, 270 and 500 K. performed on individual polyads separately, as opposed to the superposition of polyads that one might get in a practical experimental excitation.

Figure 1 :Formatted

 1 Figure 1: Basic scheme for IVR and intensity borrowing issues defining zeroth-

Figure 2 : 12 C 2

 2122 Figure 2: Illustration of the effects of the rotational l-type resonance in 12 C 2 H 2 ,

Figure 3 : 12 C 2 41 -

 312241 Figure 3: Illustration of the effects of the rotational l-type resonance combined with

Figure 4 : 12 C

 412 Figure 4: Illustration of the effects of the rotational l-type resonance combined to

42 -

 42 (c) Calculated relative intensities of the R-branch lines in the I, II and III ← GS IR absorption bands. The simulated intensities are given in terms of the upper J quantum number (J') for a temperature of 500 K. The 3:1 intensity alternation due to ortho and para acetylene has been cancelled by multiplying the lower intensity lines by a factor 3; -(d) Calculated time-dependent probability of the basis states for J = 2 (upper), 30

Figure 5 : 43 -

 543 Figure 5: Illustration of the effects of the 2/444 Coriolis resonance combined to

Figure 6 : 12 C 2 44 Figure 7 : 12 C 2

 6122447122 Figure 6: Population distribution in the rotational levels of the GS in 12 C 2 H 2 , 1 g X + Σ % ,

45 Figure 8 : 12 C 2

 458122 Figure 8: The {7,e,g} polyad in 12 C 2 H 2 , 1 g X + Σ %, with the zeroth-order bright sub-state

B. Amyay 2 *Figure 1 :

 21 Figure 1: Basic scheme for IVR and intensity borrowing issues defining zeroth-order states, ψ 1

!

  Fig .1

Figure 2 :

 2 Figure 2:Illustration of the effects of the rotational l-type resonance in12 C2H2, , selecting the {0,2,ke,e,u} polyad with the bright state taken to be the (00011,1-1), Σeu zeroth-order sub-state:(a) Reduced energy graph as a function of J(J+1). The eigenstates (black) are labeled I and II according to the ordering of their energies. The zeroth-order sub-states are colored and identified on the left using (v1v2v3v4v5,l4l5eu);(b) Squared coefficients of the bright state in the composition of the energy eigenstates plotted as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);(c) Calculated relative intensities of the R-branch lines in the I ← GS and II ← GS IR absorption bands. The simulated intensities are given in terms of the upper J quantum number (J') for a temperature of 500 K. The 3:1 intensity alternation due to ortho and para acetylene has been cancelled by multiplying the lower intensity lines by a factor 3;(d) Calculated time-dependent probability of the basis states for J = 2 (upper), 30 (medium) and 100 (lower). At t = 0, the bright state (red) begins at P1(t) = 1.0. The color scheme distinguishes the zerothorder sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (d).

  Figure 3a

Figure 3 :

 3 Figure 3: Illustration of the effects of the rotational l-type resonance combined with the 3/245 anharmonic resonance in 12 C2H2,, selecting the {1,5,ke,e,u} polyad with the bright state taken to be the (01011,1-1), Σeu zeroth-order substate:(a) Reduced energy graph of the sub-states in the polyad as a function of J(J+1). The eigenstates (black) are labeled I to III according to their energies. The zeroth-order sub-states shown in color and identified on the left as (v1v2v3v4v5,l4l5eu);(b) Squared coefficients of the bright state in the composition of the eigenstates of the polyad, plotted as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and are internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);(c) Calculated time-dependent probability of the basis states for J = 2 (upper), 30 (medium) and 100 (lower). At t = 0, the bright state (red) begins at P1(t) = 1.0. The color scheme distinguishes the zeroth-order sub-states according to order of their appearance in the time evolution. The color and marker styles for the zeroth-order sub-states are the same in parts (a) and (c).

Figure 4 :

 4 Figure 4a

  Figure 5a

Figure 5 :

 5 Figure 5d

  Figure 6:Population distribution in the rotational levels of the GS in12 C2H2, , at 27, 270 and 500 K. The ortho and para distributions are color coded.

  Fig6Page 59 of 60

Figure 7 :

 7 Figure 7:The {7,e,u} polyad in12 C2H2, , with the zeroth-order bright sub-state taken to be (00120,00), Σeu: (a) Reduced energy graph of selected sub-states as function of J(J+1). The eigenstates are identified by the marker style (black) and are labeled I to X according to their energies. The four most relevant zeroth-order sub-states are colored and identified on the left as (v1v2v3v4v5,l4l5eu);(b) Squared coefficients of the zeroth-order bright sub-state eigenvector in the composition of the eigenstates in the polyad, as a function of J. The numbering and line marker styles in parts (a) and (b) are related to the eigenstate label and are internally consistent. The zeroth-order bright sub-state is indicated as (v1v2v3v4v5,l4l5 e/f g/u);(c) Calculated time-dependent probability of the basis states for a thermal averaged population in the upper zerothorder bright sub-state, for T = 27 (upper), 270 (middle) and 500 (lower) K. The color code and line marker styles distinguish zeroth-order sub-states according to their successive rank in the time evolution. The color and marker styles for the zeroth-order sub-states are identical in parts (a) and (c).

  Figure 8a

Figure 8 :

 8 Figure 8c

: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Formatted: Superscript Deleted: 52 Deleted: 43 Deleted: 53 Deleted: 24 Deleted: pseudo Deleted: 54 Deleted: 55 Deleted:

	of quantum numbers. Sub-state indicates a l i -component of a state notated as either
	1 2 3 4 5 4 5 v v v v v l l ,	or	5 1 2 3 4 5 4 l l v v v v v . The symmetry labels include	52 and u/g. Level refers
	to a specific J-value of a state or sub-state.
	Over the last years, a global vibration-rotation model of acetylene (	12 C 2 H 2 ) in
	the ground electronic state has emerged from a series of investigations taking into
	account the complete set of experimental data published in the literature, now up to F 8,600 cm -1 43 . This model is based on the block diagonalization of the vibration-o r rotation Hamiltonian into polyads 24,53 .
	P Two vibrational quantum numbers can be defined, N r = 5v 1 +3v 2 +5v 3 +v 4 +v 5 and
	N s = v 1 +v 2 +v 3 Rotational l-type resonance terms connect sub-states with either even (e) or odd (o) 54,55,56 . Symmetrization factors the Hamiltonian into e and f blocks 57 . e e r values of k. Polyads thus include sub-states with either even or odd k values. That	56
	1 ) and CC g σ + (ν ), and the trans-g 3 π (ν 4 ) and cis-u 2 ) π is Σ (k=0), ∆ (k=2), Γ (k=4), etc. are in a separate polyad from Π (k=1), Φ (k=3). etc. stretches, the antisymmetric CH stretch u σ + (ν R anharmonic resonances, namely 11/33, 14/35, 15/34, 1/244, 1/255, 3/245 and 44/55, w The sub-states within a polyad are connected by a restricted set of e i rotation matrices are diagonal in J. A complete polyad label is {N s ,N r ,k e/o ,e/f,u/g,J}. v e Vibrational l-type resonance terms are also included. The resulting vibration-	Deleted: The e and f partition is applied to build the Hamiltonian in the symmetrised basis
	(ν usual vibrational angular momentum quantum numbers, l 4 and l 5 , with k = l 4 + l 5 . 5 ) doubly degenerate bends, respectively. The bends are characterized by the In this paper, state refers to a vibrational state characterized by the ( 1 2 3 4 5 y l v v v v v ) set all determined from the spectral analysis 24,53 . The reader is referred to Table I in O n Fayt et al 58 for the definition of the matrix elements and parameters in the model.
					13

Formatted

Deleted: 57 Deleted: 24 Deleted: 53 Deleted: As in the

			The Coriolis interaction, of the 2/444 type in particular, was recently
	introduced in the model by Fayt and coworkers	43 . The resulting couplings are weak
	but significant. Since they introduce matrix elements with ∆k = ±1, the even and
	odd k-blocks are now coupled. The N s quantum number is no longer well-defined
	and the resulting polyads are larger. They are labelled {N r ,e/f,u/g}.
			This global model was applied to all 15,562 published vibration-rotation lines
	in	12	o F
			r
			P
			e e r
			R e v i e w	introductory
				O n l y	section, the ij/mn labelling refers to an interaction coupling vibrational states with i j v v ∆ = ∆ = ±1 and m v v ∆ = ∆ = m1 . n
				14

Formatted: Superscript Formatted: Superscript Deleted: 43 Deleted: different Deleted: parities Deleted: mixed Deleted: now Deleted: and Deleted:

	F o
	r
		59
	P e e r	Deleted: Except for a limited number (8%) of local ized errors in the fitproblems, Deleted: a Deleted: but Deleted: . Deleted: However, w
	R e v i e w	Deleted: do not believe that Deleted: considered in Deleted: The results
		nevertheless involve energy
	O n l y	predictions from the model which may fail to exactly match local, J-specific features such as avoided crossings, at the spectroscopic accuracy. Such problems are however not believed to affect the trends emerging from the present investigation.
	15

  , φ d varies from 0.49 to 0.48 to 0.89 as J increases. The large φ d at J = 100 reflects the decoupling of this bright state at high J. The practical way to modify the nature of the bright state is to exploit different spectroscopic techniques, such as using IR absorption or stimulated emission pumping (SEP). Thus when different spectroscopic techniques are used to prepare states in the same energy range of the same molecule, and even in the same polyad, one can expect to see different dilution factors and rather different time-dependent behavior. As a practical matter in acetylene, the g/u selection rules prevent the IR and SEP techniques from reaching the same polyad, so a 3-photon variant of SEP 37 would be needed to reach the bright sub-state (01011,1-1), Σ eu .

	Page 23 of 60	Molecular Physics Molecular Physics	Page 24 of 60
	Deleted: effects Deleted: change Deleted: use Deleted: { Hoshina, K.; Iwasaki, A.; Yamanouchi, K.; Jacobson, M. P.; Field, R. W. J. Chem. (Fig. 4d)Formatted: Font: Century F o r P e e r F o r Phys. 2001, 114, 7424-7442.} P Schoolbook Formatted: Font: (Default) Helvetica, Complex Script Font: Helvetica e e r
		R e v i e w R e v i e w	Formatted: Don't adjust right indent when grid is defined, Space After: 0 pt, No widow/orphan control, Don't adjust space between Latin and Asian text, Don't adjust space between Asian text and numbers, Tabs: Not at 0 pt + 18 pt + 134.7 pt Deleted: sub-states with different Deleted: Deleted: -parity Deleted: , previously
		O n O n
		l y l y
		23 24

), φ d varies from 0.49 to 0.40 to 0.48 for J = 2, 30 and 100, respectively. The extent of mixing also depends on which basis state is identified as the bright state. For the infrared-active bright state, (00100,00), Σ eu URL: http://mc.manuscriptcentral.com/tandf/tmph

C. Coriolis resonances

As pointed out above, Coriolis resonances act as symmetry-breaking interactions. They mix even and odd k blocks otherwise belonging to different polyads. As an example, we will use the same infrared-active zeroth-order bright sub-state (00100,00), Σ eu as in the previous example. With the Coriolis interaction included, the polyad size is now increased to 17, from the 3 in Figures

3 and 4

. A reduced energy graph of a selected, relevant subset of these 17 sub-states is represented in Figure 5a. Coriolis resonances are induced in the present case by the 2/444 coupling scheme, bringing in the polyad states such as (00041), URL: http://mc.manuscriptcentral.com/tandf/tmph

  will exhibit substantial

	Molecular Physics	Page 32 of 60
	F o
	r
	P	Deleted: is observed for the first time in the selected case studies. It
	e e r	Deleted: entirely from the basic Deleted: of rotational origin Deleted: Rotational
	R e v i e w	Deleted:
	O n	Deleted: like
	l y
	32

URL: http://mc.manuscriptcentral.com/tandf/tmph

:

  French Belgium

	Page 35 of 60	Molecular Physics
	F o F o F o F o r r r r		Deleted: the various comments Deleted: which
	P P P P e e r e e r e e r e e r	Deleted: helped improving the pedagogical Deleted: in the
		R e v i e w R e v i e w R e v i e w R e v i e w
		O n O n O n O n
		l y l y l y l y
		36 37 38

URL: http://mc.manuscriptcentral.com/tandf/tmph

2010 11:37:00 AM Superscript Page 4: [5] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [6] Deleted Mike 1/25/2010 11:20:00 AM 9 Page 4: [6] Deleted Mike 1/20/2010 5:28:00 PM In

  general, the notation ij/mn indicates that quanta vibrational modes i and j involved are exchanged for quanta of modes m and n.

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URL: http://mc.manuscriptcentral.com/tandf/tmph URL: http://mc.manuscriptcentral.com/tandf/tmph

2010 11:37:00 AM Superscript Page 4: [13] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [14] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [15] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [16] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [17] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [18] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 4: [19] Formatted Mike 1/25/2010 11:37:00 AM Superscript Page 48 of 60 URL: http://mc.manuscriptcentral.com/tandf/tmph

			Molecular Physics
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		Molecular Physics
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  as already partly investigated in the literature, see e.g

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	Molecular Physics
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		Molecular Physics
	Vibration-rotation alchemy in acetylene ( 12 C2H2),	at low vibrational excitation:
	From high resolution spectroscopy to fast intramolecular dynamics
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Acknowledgements

The referee is thanked for a detailed commentary that has led to a clearer, more insightful paper. This work was supported, in Brussels, by the Fonds National de la Recherche Scientifique (F.R.S.-FNRS, contracts FRFC and IISN) and the « Action de Recherches Concertées de la Communauté française de Belgique ». The work was supported by the Division of Chemical Sciences, Offices of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy under Grant No. DE-FG02-90ER14151. Support of this work does not constitute endorsement by the DOE of views expressed in this paper. B.A. thanks FRIA for a research grant.

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Richard Zare

T Peter Rakitzis

Maurice H M Janssen

Photofragment angular momentum distributions from oriented and aligned polyatomic molecules: beyond the axial recoil limit

Keywords: photodissociation, polarization, photofragment alignment

Photofragment angular momentum distributions from oriented polyatomic molecules: beyond the axial recoil limit

Introduction

For several decades, information about the photodissociation process in the molecular frame has been obtained from the measurement of laboratory-frame vector properties. The most well-known example has been the measurement of the angular distribution of photofragments from isotropic parent molecules, with respect to the photolysis laser polarization, introduced by Zare and Herschbach [START_REF] Zare | Proc. IEEE[END_REF]: 2 I( ) 1 P (cos )

ε ε θ = + β θ , (1)
where ε θ is the angle between the recoil direction v and the polarization of the photodissociating light, and P 2 is the second-order Legendre polynomial (note that this equation is valid only for one-photon photodissociation in the dipole approximation).

The parameter β ranges from -1 to +2. For prompt photodissociation in the axial recoil limit, a pure parallel transition produces photofragments with β = +2, and a pure perpendicular transition produced photofragments with β = -1. If the photodissociation dynamics of a molecule is known to be prompt and the recoil is axial, then the measurement of the spatial anisotropy β of the photofragments can be used to infer the relative contributions of parallel and perpendicular transitions to the photodissociation process. However, in the general case where axial recoil approximation does not hold, further information is required to unravel the dissociation mechanism.

In recent years, the measurement of photofragment angular momentum polarization has been used as an extremely sensitive probe of photodissociation dynamics, particularly as a unique probe of the interference effects arising from the coherent excitation of multiple dissociative states with different symmetry. In a seminal Rakitzis and Janssen 3 paper, Siebbeles et al. [START_REF] Siebbeles | [END_REF] gave the full quantum mechanical description of photofragment polarization from the prompt dissociation of molecules in the axial recoil limit. In this treatment the photofragment polarization is decomposed into coherent and incoherent contributions from transitions to multiple dissociative states. This formalism, and an equivalent molecular-frame formalism by Rakitzis and Zare [3], was used to successfully describe the photofragment polarization measured from the photodissociation of a number of diatomic molecules (well described by the axial recoil approximation) [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. More recently, photofragment polarization has been measured from the photodissociation of polyatomic molecules, for which the axial recoil approximation breaks down [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. The polarization measurements can be fit with the same polarization parameter basis functions (as they form a complete basis), however the interpretation of the values of the polarization parameters is no longer straightforward or well-understood.

Recently, Vasyutinskii and coworkers have expanded the Siebbeles formalism beyond the axial recoil approximation, by including the effects of parent-molecule rotation, and by producing general expressions valid for polyatomic systems independent of reaction mechanism [34,35,36,37,38]. However, the theoretical methods for extracting polarization from polyatomic molecule photodissociation [39], where the transitions to the dissociative states are neither pure parallel nor pure perpendicular transitions.

Direct observation of non-axial recoil dynamics has been achieved by Janssen and coworkers, by observing the angular distribution of photofragments from the photodissociation of hexapole-oriented parent OCS molecules [40,41]. The angular distributions of the photofragments were fit using a theoretical treatment for non-axial recoil dynamics from a single dissociative surface [42,43], and the directional non-axial recoil angle α were determined for each rotational state J of the CO(v=0,J)

photofragments [41].

Pipes et al. [44], and Underwood and Powis [45] extended the Siebbeles treatment to the photodissociation of polarized diatomic molecules in the axial recoil limit.

However, a treatment of the photodissociation of polarized polyatomic molecules, either within or without the axial recoil limit, is lacking.

The aim of this paper is to provide a framework for the description of photofragment polarization for the photodissociation of polarized polyatomic molecules.

We combine the molecular-frame k q a polarization-parameter formalism [3] with the semi-classical non-axial recoil dynamics treatment for dissociation of oriented molecules [46,47]. The aim is to allow the experimental decoupling of the measurement of the photofragment polarization and the non-axial recoil dynamics (deflection angles of the photofragment recoil). This will provide a general approach to describe the photofragment polarization from oriented molecules. We emphasize that we decouple the geometrical factors that arise from the fact that the recoil direction v results at angles of the mechanism that produced these angles, and thus represent purely geometrical factors (see Figure 1). The general polarization parameters k q a (i) which we use to phenomenologically describe the photofragment polarization (depending on the photodissociation mechanism) may also depend on the angles α,χ, and ϕ µd .

Here, excitation to only a single excited state is discussed. The goal of future research will be to generalize this approach to describe the coherent excitation of multiple states. The immediate goal of this work is to provide an approach for the fitting of existing S( 1 D) photofragment polarization data from the photodissociation of oriented state-selected OCS molecules [48,49], while taking into account the effects of the orientation of the parent molecules. We show that the experimental sensitivity to the k q a (p) parameters varies as a function of the degree of the orientation and alignment of the parent molecules, and we quantify this variation in sensitivity. Furthermore, as discussed below, the orientation and alignment of the parent molecules allow the measurement of the photofragment polarization to be sensitive to additional k q a (p) parameters (such as those with q>2), which normally would not be the case for the onephoton photodissociation of an isotropic sample of parent molecules.

Theory

The calculation presented here of the photofragment angular momentum distribution in the molecular frame from the photodissociation of oriented parent molecules combines the similar methods used for the calculation of photofragment angular momentum distribution of from a sample of isotropic parent molecules [3,39],

and the unpolarized photofragment angular distributions from oriented parent molecules Rakitzis and Janssen 6 [42,43]. We will consider here the excitation to only a single excited state of the parent molecule. The incoherent and coherent excitation of multiple states is discussed briefly in the Discussion section. given by the distribution D(cosδ), where d⋅ ⋅ ⋅ ⋅O=cosδ [51,52]:

The molecular frame

∑ = + = N k k k P c D 1 ) (cos 2 1 ) (cos δ δ , ( 2
)
where the values of the c k describe the degree of orientation (for example, for a distribution D(cosδ) proportional to cos 2 δ, c 2 =1 and all other c k are zero). The probability of absorption of a photodissociating photon is given by

2 μ ε μ ε μ ε μ ε ⋅ ⋅ ⋅
⋅ , which can be expressed in terms of the second Legendre polynomial as:

Deleted: ¶ F o r P e e r R e v i e w O n l y [ ] ) (cos 2 1 3 1 2 2 µε θ P + = ⋅ ε ε ε ε μ μ μ μ , (3)
where θ µε is the angle between µ µ µ µ and the photolysis polarization direction ε ε ε ε.

Finally, the detection probability of the photofragments in the molecular frame, as a function of the polarization parameters k q a (i) [3,39,53,[START_REF] Zare | Angular Momentum, Understanding Spatial Aspects in Chemistry and Physics[END_REF], is given by I( k q a ):

2 1 (a ) 1 a (i)C ( , ) J k k k k q k q q dp k q k I s p ϕ = =- = + ∑ ∑ , (4)
where k q C ( , ) θ ϕ is a reduced spherical harmonic [52], p is the angle between the probe polarization direction P and v, ϕ dp is the azimuthal angle between P and d about v, given by ϕ dp =ξ+ϕ Oε -ϕ Op (see Fig. 1), and the index i in k q a (i) refers to a particular dissociative excited state, so that the k q a (i) describe the photofragment angular momentum polarization of the photofragments produced by excitation to state i.

The molecular frame angular momentum angular distribution is given by the product of three quantities, the parent orientation distribution D(cosδ), the photodissociation probability

2 μ ε μ ε μ ε μ ε ⋅ ⋅ ⋅
⋅ , and the angular momentum detection probability

I[ k q a (i) ],
which is then integrated over the angle ξ to integrate all the molecular geometries that can yield photofragments along v: where the numerical factor of 6 is included for the normalization of subsequent equations.

2 2 0 1 6 (cos ) [a (i)] 2 π k q I D I d δ ξ π = ⋅ ∫ μ ε μ ε μ ε μ ε , ( 5
Substituting Eqs. (1-3) into Eq. ( 4), we obtain:

2 2 2 1 1 0 1 1 2 (cos ) 1 2 (cos ) 1 a (i)C ( ) 2 dp π N J k iq k k k k k q q k k q k I P c P s p e d ϕ µε θ δ ξ π = = =-     = + × + × +               ∑ ∑ ∑ ∫ , (6)
where P 2 (cosθ µε ) and P k (cosδ) can be expressed in terms of the molecular frame angles χ, γ, ξ, ϕ µd and α, δ, ξ, ϕ Οε respectively, using the spherical harmonic addition theorem, and ϕ dp = ξ + ϕ Οε -ϕ Οp :

( )

2 2 2 2 2 2 2 0 0 1 1 2 2 0 2 2 2 2 2 2 0 0 1 1 2 2 1 2 ( ) 1 1 1 2 ( ) ( ) 4 ( ) ( ) cos( ) 4 ( ) ( ) cos 2( ) 2 1 2 ( ) ( ) 4 ( ) ( ) cos( ) 4 ( ) ( ) cos 2( ) 1 a (i)C ( ) O Op π d d N k O O k J k iq k k k q q k q k I C C C C C C c C C C C C C s p e ε µ µ ε ε ξ ϕ ϕ χ γ χ γ ξ ϕ χ γ ξ ϕ π α δ α δ ξ ϕ α δ ξ ϕ = + - = =-   = + + + + +     × + + + + +      × +  ∫ ∑ ∑ ∑ dξ       (7)
Evaluation of the integral in Eq. ( 7) yields the photofragment angular momentum distribution in the molecular frame. We evaluate Eq. ( 7) for fitting experimental results

for the S( 1 D 2 ) photofragment from the photodissociation of OCS oriented in the (JlM=111) or (JM=10) states. For both of these states, the degree of OCS orientation is described by c k parameters c 1 and c 2 only, whereas all other c k vanish. The total angular momentum J of the S( 1 D 2 ) atoms is J=2, so that the k q a multipole moment expansion describing the angular momentum distribution is terminated at 2J (i.e. k=4).

Furthermore, for linearly polarized probe light, the detection step is only sensitive to k q a with even k; therefore the second summation in Eq. ( 7) includes only k=2 and k=4 terms. In addition, Eq. ( 7) can be expressed as purely real by combining the k q a terms with +q and -q and expressed in terms of Re[ k q a ] and Im[ k q a ], using k q k q q a ( 1) a * -=and C ( ) ( 1) C ( )

k q k q q p p - = - [49,50]: ( ) dp k q dp k q k q iq k q k q iq k q k q q q p p p dp dp ϕ ϕ ϕ ϕ sin ] a Im[ cos ] a Re[ ) ( C 2 e ) ( C a e ) ( C a + = + - - - (8)
Using the constraints described above, Eq. ( 7) is evaluated and presented below:

2 2 4 4 2 2 k k 2 0 0 4 0 0 0 0 k 0 0 k 2 2 k k 2 2 k k k 1 1 1 1 d O k 2 2 2 2 d O k k 2 2 2 1 1 4 I 1 s a (i)C (p) s a (i)C (p) 1 2C ( )C ( ) 1 2 c C ( )C ( ) 8c C ( )C ( )C ( )C ( ) cos( ) 8c C ( )C ( )C ( )C ( ) cos 2( ) s Re[a (i)]C (p) s Re µ ε µ ε         = + + + χ γ + α δ              + χ γ α δ ϕ -ϕ + χ γ α δ ϕ -ϕ    + + ∑ ∑ ∑ 4 4 2 2 k k 1 1 0 0 k 1 1 Op k 2 2 k k 2 2 k k 1 1 p d k 0 0 k 2 2 1 1 d O p k k 2 2 k k k 1 1 2 2 d [a (i)]C (p) 1 2C ( )C ( ) 4c C ( )C ( ) cos( ) 4C ( )C ( ) cos( ) 1 2 c C ( )C ( ) 8c C ( )C ( )C ( )C ( ) cos(2 ) 8c C ( )C ( )C ( )C ( ) cos( ε µ µ ε ε µ       + χ γ α δ ϕ          + χ γ ϕ -ϕ + α δ + χ γ α δ ϕ -ϕ -ϕ     + χ γ α δ ϕ ∑ ∑ ∑ O Op k 2 2 4 4 2 2 k k 2 2 2 4 2 2 0 0 k 2 2 Op k 2 2 k k 2 2 k k 2 2 p d k 0 0 k 1 1 1 1 d p O k ) s Re[a (i)]C (p) s Re[a (i)]C (p) 1 2C ( )C ( ) 4c C ( )C ( ) cos 2 4C ( )C ( ) cos 2( ) 1 2 c C ( )C ( ) 8c C ( )C ( )C ( )C ( ) cos( ε ε µ µ ε   -ϕ -ϕ          + + + χ γ α δ ϕ          + χ γ ϕ -ϕ + α δ + χ γ α δ ϕ + ϕ -ϕ     ∑ ∑ ∑ p k 4 4 2 2 k k 4 3 3 k 2 2 1 1 d Op p k 2 2 k k k 1 1 2 2 d Op p k 4 4 2 2 k k 4 4 4 k 2 2 2 2 d Op p k ) s Re[a (i)]C (p) 8c C ( )C ( )C ( )C ( ) cos( 2 ) 8c C ( )C ( )C ( )C ( ) cos(2 2 ) s Re[a (i)]C (p) 8c C ( )C ( )C ( )C ( ) cos 2( ) µ ε µ ε µ ε        + χ γ α δ ϕ + ϕ + ϕ      + χ γ α δ ϕ + ϕ + ϕ      + χ γ α δ ϕ + ϕ + ϕ  ∑ ∑ ∑ ∑        (9)
Notice, for convenience, that the angle ϕ pε = ϕ Οp -ϕ Οε is used for the azimuthal angle of P and ε ε ε ε about v. Inspecting Equation ( 9), we can compare the molecular-frame detection probability for photofragments from the one-photon photodissociation of oriented parent molecules and for isotropic parent molecules. In the latter case, it is well-known that (for the one-photon photodissociation in the dipole approximation) experimental signals are sensitive to the k q a (i) with |q| ≤ 2. In Eq. ( 9) we see terms for oriented and aligned parent molecules (which are proportional to the degree of parent orientation c k ) which have q = 3 and q = 4.

When χ=0 and α=0, then we have axial recoil dynamics, and the state i has been accessed via a pure parallel transition; in this case, we expect the photofragment polarization to be described by the k 0 a (||) parameters only [3]. As expected, we see that the terms with q > 0 vanish, and that the molecular frame expression becomes equal to the product of the parent bond polarization multiplied by the detection expression for photofragments from a parallel transition only. Similarly, when χ=90° and α=0, then again we have axial recoil dynamics, but now the state i has been accessed via a perpendicular transition, and we expect the photofragment polarization to be described by the k 0 a ( ) ⊥ and k 2 a ( ) ⊥ parameters only [3]. As expected, we see that the terms with q=1 and q > 2 vanish, and that the molecular frame expression becomes equal to the product of the parent bond polarization multiplied by the detection expression for photofragments from a perpendicular transition only, with, however one exception: the q = 2 parameters are reduced by a factor of is not fixed in the molecular frame (whereas for a diatomic µ µ µ µ ⊥ is always in the plane defined by v and the photolysis polarization ε ε ε ε).

Below, we evaluate, quantitatively, the sensitivity to all the k q a (i) in the laboratory frame.

The laboratory frame

Figure 2 shows the laboratory-frame coordinates of the recoil velocity v, with respect to ε ε ε ε, P, and O. The Z axis is parallel to the detection axis (e.g. the time-of-flight axis of a mass spectrometer), and the Z-X plane is defined by O. The expressions for the polar angles in the molecular frame, γ, δ, and p, in terms of the laboratory-frame angles are given by: O co cos cos sin sin cos( ),

ε γ = Ω Γ + Ω Γ Θ -Φ s (10a) co cos cos sin sin cos , δ = Ω ∆ + Ω ∆ Θ s (10b)
OP cos p cos cos P sin sin P cos( ),

= Ω + Ω Θ -Φ (10c)
The expressions for trigonometric functions of the molecular-frame azimuthal angles, Oε ϕ , Op ϕ , and pε ϕ , in terms of the laboratory-frame angles (calculated using Eqs. (22- 24) from reference [3]) are given by:

= Ω Γ ∆ + Γ ∆ Φ -Ω Ω ∆ Γ Φ -Θ + Γ ∆ Θ - Ω Γ ∆ Φ -Θ Θ O O O O ε ε ε ε ϕ γ δ (
= Ω Γ ∆ Φ -Ω ∆ Γ Φ -Θ + Γ ∆ Θ O O O c ε ε ε ϕ γ δ (11b)
= Ω Γ + Γ Φ -Φ -Ω Ω Γ Φ -Θ + Γ Φ -Θ - Ω Γ Φ -Θ Φ -Θ p Op
ϕ γ = Ω Γ Φ -Φ -Ω Γ Φ -Θ + Γ Θ -Φ (11f)
Note that Φ pε = Φ Οp -Φ Οε . Inserting Eqs. ( 10) and ( 11) into ( 9

Notice that the expansion is terminated at n=8; the degrees of spatial anisotropy contributed by the orientation, photodissociation, and angular momentum detection steps are 2 for the orientation (as we considered orientation parameters c 1 and c 2 only), 2 for the photolysis step (for a one-photon photodissociation), and 4 for the angular momentum detection step (given by 2J, where J=2 for S( 1 D 2 ) atoms). Note that a similar expansion of an isotropic sample of parent molecules would be terminated at n=6 [START_REF] Rakitzis | [END_REF].

The coefficients b n are expressed in terms of a constant k n , and the 8 possible k q a polarization parameters with k = 2 (for which there are k+1 = 3 parameters) and k = 4

(for which there are k+1 = 5 parameters) as follows:

2 0 ( ) Re[ (i)], k k n n k q q even k q b k s w n a = = + ∑ ∑ (13)
where the ( )

k q
w n quantify the experimental sensitivity to each particular k q a parameter.

We consider first the experimental geometry with O perpendicular to the imaging plane (∆=0), which has been used in several of the authors experimental studies [26,38,39,46,47]. The sensitivity factors ( )

k q
w n are calculated for three degrees of orientation, the (JlM=111) state with c 1 = 3/4 and c 2 = 1/4, the (JM=10) state with c 1 = 0 and c 2 = 1, and an isotropic sample of parent molecules with c 1 = 0 and c 2 = 0; in addition the molecular frame angles are chosen to be χ = π/6, α = π/4, and ϕ µd = 0. The ( )

k q
w n factors are calculated for the geometry A (∆=0, Γ = π/2, and P = π/2) are presented in Table I this geometry, we note that there is no experimental sensitivity to the parent orientation (c 1 ). We see that the sensitivity factors ( ) k q w n do not change too strongly (about 10%) between those from an isotropic sample or from a modestly aligned sample of OCS molecules in the (JlM=111) state (with the exception of the q=2 parameters), whereas the deviations become more significant (about 30% or larger) for strongly aligned OCS molecules in the (JM=10) state.

Next, we consider the geometry B (∆=π/2, Γ = π/2, and P = π/2). When the molecules are isotropic (c 1 =c 2 =0), then the results are the same as for isotropic molecules in geometry A (as the only difference is the geometry of the orientation field O, which is not yet being utilized). We consider somewhat different dissociation dynamics, by choosing the molecular frame angles to be χ = π/6, α = π/4, |ϕ µd | = π/2. As OCS molecules have planar symmetry, the distribution of the angle φ µd is an even function, therefore we can set <sinqφ µd > = 0. The sensitivity factors ( )

k q
w n are calculated and shown in Table II. Notice that the sensitivity to the q=1 parameters, for the choice of this special case of |ϕ µd | = π/2, now vanishes (as their sensitivity is proportional to cosφ µd ); also the sensitivity to the q=2 parameters changes sign (as their sensitivity is proportional to cos2ϕ µd ). However, when the molecules are aligned (c 2 =1), we see that the sensitivity to the q=1 parameters, from zero, now becomes large, due to the anisotropy of the aligned parent molecules. This is an example of how polarizing parent molecules allows sensitivity to molecular-frame dynamical information, which was not available in experiments with isotropic samples of molecules. Rakitzis and Alexander [39], in the general case of polyatomic photodissociation,

show that both single surface (classical) and multiple-surface interference (quantum mechanical) contributions to the q=1 parameters are allowed. However, they also show that, for isotropic parent molecules, these two contributions cannot be distinguished from the photofragment angular distributions alone, at a single photolysis wavelength. Future work will involve showing, using the methods described in this paper, how to distinguish the single and multiple-surface contributions to the photofragment angular momentum distributions, using polarized parent molecules. Finally, these methods may be useful in combination with emerging techniques for orienting polyatomic molecules [56,57]. w n of the k q a (i) parameters calculated for the geometry (∆ = 0, Γ = π/2, and P = π/2), and for parent molecular alignment of an isotropic sample (c 2 = 0), the (JlM=111) state (c 2 = 1/4), and the (JM=10) state (c 2 = 1, i.e. maximal alignment). For the chosen laboratory slice geometry there is no sensitivity to c 1 (see text). The dynamical angles describing the mutual orientation of permanent dipole moment, transition dipole moment and recoil velocity are chosen to be χ = π/6, α = π/4, and ϕ µd = 0.

  dynamical information from experiments on polyatomic molecule photodissociation have not yet been developed; therefore it is not yet clear that all the dynamical information (such as non-axial recoil deflection angles, in addition to transition amplitudes and phases) can indeed be obtained sensitively, and this will become more clear as these methods are developed. Very recently, Rakitzis and Alexander have extended the k q a (p) polarization parameter formalism to allow for the description of photofragment

(α, 0 )

 0 to the dipole moment d and (χ,ϕ µd ) to the transition dipole moment µ

Formatted

Figure 1

 1 Figure1shows the molecular-frame coordinates of the polarization directions of the photolysis laser ε ε ε ε, the probe laser P, the Orientation field O, and the dynamically

  ), we obtain the full threedimensional (3-D) detection probability. Methods for collapsing this expression to explain signals from 1-D or 2-D experiments are given in Sections 3.1 and 3.2 of reference[40]. Here, we consider slice imaging detection[40], for which we set Ω=π/2, and then the photofragment angular distribution is a function of the angle Θ only. The resulting laboratory-frame slice-imaging angular distribution, I s (Θ), can be expressed as an expansion of Legendre polynomials, for experimental geometries where the angles ∆ and Γ are either 0 or π/2 (for other angles, the expansion requires more terms which have different symmetry from the P n (cosΘ):

  parameters calculated for the geometry (∆ = π/2, Γ = π/2, and P = π/2), and for parent molecular alignment of an isotropic sample (c 2 = 0) and the (JM=10) state (c 2 = 1). The dynamical angles describing the mutual orientation of permanent dipole moment, transition dipole moment and recoil velocity are chosen to be χ = π/6, α = π/4, and |ϕ µd | = π/2.

Fig. 1 :

 1 Fig. 1: Polar coordinates of the dynamical vectors (v, µ µ µ µ, and d), the photolysis (ε ε ε ε) and probe (P) polarization directions, and the orientation field (O), in the molecular frame.

Fig. 2 :

 2 Fig. 2: Polar coordinates of the dynamical vectors (v, µ µ µ µ, and d), the photolysis (ε ε ε ε) and

  Figure 2:

  11a)

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	sin	{ os sin sin sin
		sin [sin cos sin(	) sin cos sin ]} / (sin sin ),
			Deleted: ¶
		F o
		r
		P
		e e r
			R e v i e w
			O n
			l y

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	Molecular Physics
	Rakitzis and Janssen
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	15

Table I :

 I The sensitivity coefficients ( )

	k
	q

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Acknowledgements

This research has been financially supported by the councils for Chemical Sciences and Physical Sciences of the Dutch Organization for Scientific Research (NWO-CW, NWO-FOM). TPR thanks the EU for support to access the facilities of the Laser Centre Vrije Universiteit for which funding was received from the EC's Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 228334, and for support through the European Research Council grant "TRICEPS"(GA No. 207542). The authors would like to acknowledge further support through the EU-Marie Curie ITN Programme ICONIC (PITN-GA-2009-238671). Finally, the authors thank Dick Zare for his pioneering achievements and inspirational vision in the field of reaction dynamics; moreover we are grateful both for his mentoring and friendship.