Anomalous behavior of the anticrossing density as a function of excitation energy in the C2H2 molecule

Chemical Physics 152 ( 1991) 293-3 18 North-Holland

Anomalous behavior of the anticrossing density as a function of excitation energy in the C2H2 molecule P. Duprb, R. Jost Service Nationaides Champs Intenses, C.N.R.S., BP 166X. 38042 Grenoble Cedex, France

M. Lombardi Laboratoire de Spectromt%riePhysique, UniversitP Joseph Fourier BP 87, 38402 Saint Martin d’Heres Cedex, France

P.G. Green ‘, E. Abramson * and R.W. Field Massachusetts Institute of Technology, Department of Chemistry, Cambridge, MA 02139, USA Received 8 October 1990

We have recorded Zeeman anticrossing (ZAC) spectra of gas phase acetylene (HC=CH ) in the A ‘A. u; =0-3 (u; is the transbending normal mode of the trans-bent excited electronic state), J= K= I=0 levels. The energy range thus sampled was from 42200 to 45300 cm-’ above the rotationless zero-point level of the R ‘2: state. The magnetic field was scanned from 0 to 8 T and the ZAC spectra were recorded as decreases of intensity of the fluorescence excited by a pulsed, frequency doubled dye laser. The ZAC spectra were unassignably complex. We report here a surprisinglyrapid increase in the density of detectable anticrossings (AC) over a 3100 cm-’ energy interval (only 7% of the total excitation energy). In the u; =3 level the detected anticrossing density is IO2times larger than the maximum computed density of triplet T,-Ts vibrational levels and quite comparable to the computed density of 2 ‘Xl vibrational levels. Plausible mechanisms for the rapid increase in the ZAC density are discussed, especially one involving access to the highly excited vibrational levels of the & state via a cis-bent isomer of the triplet acetylene near the top of a cis-tram isomerization barrier on one triplet surface.

1. Introduction The spectroscopy and photophysics of the acetylene molecule, particularly in the gas phase, have been the focus of much research. Although the spectroscopy of the singlet 2, A and fi electronic states has begun to be well understood [ l-7 1, clear evidence of new types of intramolecular dynamics at high vibrational energy, where many spectroscopic quantum numbers and dynamical constants of motion vanish [ 8- 141, is opening a new field of spectroscopy. Moreover, very recent experiments have detected vibrational levels of the So vinylidene isomer [ l&l6 ]

’ Present address: Mail Stop 170-25, California Institute of Technology; Pasadena, CA 9 1125, USA. * Present address: University of Washington, Department of Chemistry, BH-10, Seattle, WA 98195, USA. 0301-0104/91/$03.50

and the triplet cis-bent acetylene [ 17,181. The metastability of acetylene triplet states seems to be experimentally proven by Hg photosensitized reactions [ 19,201 and by near-threshold electron impact investigations on molecular beams where direct singlet-triplet electronic transitions are allowed [ 2 l231. These observations seem to be in good agreement with quantum chemical calculations [ 24-29 1. Although it has been a subject of considerable recent controversy, the lowest energy dissociation limit of acetylene, D,(HCC-H), HCCH(R 5: u=O, J= 0)+HCC(%2Z+ v=O, N=O)+H( (Is)’ 2S) certainly lies below 46673 cm-’ where unmistakeable evidence from predissociation in the A ‘A, v; =4 level was obtained by Fujii et al. [ 3 1,321. A variety of other upper bound [ 17,18,30,34-361 and thermodynamic cycle [ 371 measurements and ab initio calculations [ 38,391 of Do( HCC-H) have been re-

0 1991 - Elsevier Science Publishers B.V. (North-Holland)

294

P. DuprPet al. /Level antlcromng m C,Hz

ported recently. With the exception of the probably erroneous low value proposed in refs. [ 17,18 1, there is a convergence of agreement on D,,(HCC-H) = 45900&300 cm-’ [37]. The value of D,,(HCC-H) is relevant to the present paper because the phenomenon discussed here and the dissociation limit occur at nearly the same energy. The density of levels coupled to specific rovibrational levels of the S, state has already been noted to be very large (many levels giving rise to quantum beats have been observed) especially in the 32 G subband [ 40,4 1 ] where in addition a Fermi perturbation has been detected [ 111. It has been shown, by using a magnetic field (up to 1 T), that triplet levels were involved in the perturbationoftheS,levels (K’=l,v;=l-3) [14,41] (by determining gyromagnetic g factors). Using an electric field [ 14,17,18 ] it has been recently shown (by Stark anticrossing spectroscopy) that some of the triplet levels which interact with S, have significant cis-bent or vinylidene character [ 14,18 1. We present here results obtained by recording the integrated fluorescence from 4 rotationless vibrational levels (trans-bending mode, v; = O-3, N’ = 0, K’ =O) of the S, level as a function of magnetic field (up to 8 T). As we will show, for v; = 0 the density of detected anticrossings agrees well with the triplet level density estimated according to the Marcus-Rice formula [42] or by a direct count of the levels at the given energy for a multidimensional harmonic potential surface. However, at only 1000 cm - ’ higher energy, the discrepancy (ratio between the observed and calculated level densities) is nearly one order of magnitude. At a total of 3 100 cm- ’ higher energy this discrepancy becomes larger than two orders of magnitude. It is important to distinguish between detectable and undetectable anticrossings. When the coupling matrix element between the laser populated bright level and the Zeeman tuned dark level is larger than a critical value, an anticrossing is detectable. Often, even when an anticrossing involves a coupling matrix element which is not required by symmetry to be zero, the matrix element is below the detection threshold and the anticrossing is undetectable. In this paper we discuss an abrupt change in the density of observed anticrossings. Our conclusion will be that this is due to an abrupt change in the average value of the SI -dark state matrix ele-

ment, not to an abrupt change in the density of dark levels allowed by symmetry to interact with the selected bright level. Up to now, there is no definitive explanation for these observations; we suggest several hypotheses in section 5. The coupling of the S, levels with S,, levels via T , or T2 levels is clearly implied because only the S,, state can supply the observed density of levels. We will show that the rapid increase in anticrossing density in the S, state cannot be explained by a change in the density or strength of So-T interactions alone. The key to the understanding of the S, anticrossing densities is a change in the S, -T coupling strength which occurs near the top of an isomerization barrier on T, or Tz.

2. The experimental set-up We excite various individual S, rovibrational levels (via S, +SO transitions) using a frequency doubled homemade dye laser [ 43 1. The pumping beam is produced by a frequency tripled and @switched Nd: YAG laser (Quantel, output energy = 40 mJ/ pulse). The dye laser has three stages of amplification and the dye used depends upon the vibrational band that we study (Coumarins: C440, LD466, C 102 or C47 ) . We frequency double the blue dye laser radiation with a /?-BaB204 crystal (Fujian Institute of Research, China) giving a UV beam whose typical energy is ~4009, spectral bandwidth is x4 GHz fwhm and divergence is less than 0.4 mrad. A tellurium oven is used to calibrate the blue wavelength [ 441. The experimental set-up is summarized on fig. 1. In order to produce an amplitude reference signal to improve the signal-to-noise ratio, the UV beam passes through a reference acetylene fluorescence cell before entering the magnet. This magnet is a Bitter coil (5 MW) whose magnetic field may be adjusted between 0 and 8 T. The spatial homogeneity is about 0.1 mT per cm3 along the center-line of the magnet bore (82 mm diameter). This homogeneity is obtained by superimposing on the main magnetic field trim fields produced by three small externally controlled homogenization coils. The short term temporal stability is roughly given by the stability of the power supply current (2.5 x 1O-5).

P. DuprPet al. /Level anticrossingin C,H,

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We use pure acetylene gas (99.5%) from a gas cylinder (Prodair) where acetylene is dissolved in acetone (without further purification) and we adjust the acetylene pressure inside the reference and signal cells over a typical range of O-500 mTorr. Light is collected from the reference cell through coloured glass filters (Schott: UG5 and/or WG280, depending on the wavelength of excitation), without use of a lens. From the cell inside of the magnet light propagates through two serial quartz light pipes (whose total length is two meters) to a shielded photomultiplier tube (PMT) (Hamamatsu R2059). We collect 50 times fewer photons from inside the magnet than from the reference cell. The electrical signals produced by the reference and signal PMTs are added after one of them is delayed by z 200 ns. The fluorescence decays can then be digitized by a single transient recorder (Lecroy TR88 18, time resolution 10 ns), one after the other in the same

trace. The data collection is controlled by a personal computer (PC) and a GA210 module (L.A.L. Orsay) connected to a CAMAC bus. The PC is also used to control the scan of the main and trim magnetic fields and to make real time calculations (normalization, integration and taking the ratio in order to minimize the shot-to-shot fluctuations) at the laser repetition rate of 20 Hz. While we scan the magnetic field we also record a field-calibration nuclear magnetic resonance (NMR ) signal from a homemade probe [ 45 1. The associated electronics enable us to generate markers (every time that the magnetic field induces resonances in the NMR probe at a frequency accurately known in advance) which we use after each scan is completed to calibrate the magnetic field strength throughout this scan. A typical complete scan between 0 and 8 T (65536 data points) takes roughly one hour. The results obtained at low magnetic field (fig. 9 )

296

P. DuprP et al. /Level anticrosmg

come from a slightly different experimental set-up than the one described in fig. 1 [ 8,141. The major difference between the high (Grenoble) and the low (MIT) field set-ups concerns the use of an electromagnet (instead of a Bitter Coil) to give better field resolution ( x0.1 mT). The transient recorder was a Tektronix model 79 12AD. It was used to record the fluorescence decay as well as quantum beats [40]. The excitation dye laser was a Lambda Physik FL2002 model pumped by a Molectron (MY 34-20) Nd : YAG laser.

3. Principles of the experimental method and analysis We have recorded undispersed fluorescence versus the magnetic field over a range of O-8 T (Zeeman anticrossing (ZAC) spectra) for four vibrational levels of the A ‘A, electronic state. For each vibrational level we have chosen the N= 0 (N is total molecular angular momentum excluding electronic and nuclear spins ) , K= 0 (Kis projection of N onto molecular axis of least inertia), I=0 (I is nuclear spin) rotational level by optical excitation (hot-band transitions v;, where v is the trans-bent CCH vibrational mode, vj in the excited state and vq in the ground state, rotational pPI (1) transition) in order to minimize the number of detectable anticrossings. The selection rules for the anticrossings are AN=O, + 1, (O-O), M,=O, ? 1, coming from AJ=O, (J=N+S), and are discussed in Appendix C. They have been previously used to successfully interpret the seemingly analogous ZACs in the glyoxal molecule [ 47-541. By exciting rotationless levels of the S, state and by assuming that the singlet-triplet couplings are only those which occur at zero magnetic field, we expect only two anticrossings to occur for each interacting vibrational triplet level in the probed range. These two anticrossings correspond to the only two triplet rotational levels N, = 1, K,= 0 or 1 that are allowed to perturb Sr, N, = KS= 0.Correlations must exist between two such ZACs, but their expected energy (or field) separation is too large ( = 13.8 T: corresponding to the expected energy splitting at zero magnetic field of A = 12.9 cm- I, where A is the rotational constant of the molecule corresponding to its lowest moment of inertia) to be observable [ 46,471. More-

WIC,H,

over, for these conditions, the triplet fine structure is reduced to a single level (MN= + 1, or - 1, depending on the value of Mst of the anticrossing triplet level, respectively - 1 or + 1) [ 48,491. Thus from the total number of ZACs we can deduce the density of interacting triplet vibrational levels from the anticrossing spectra. At this stage we make no assumption on the vibrational symmetry of these interacting triplet levels. We assume further that the molecular electronic spin is decoupled from the nuclear rotation (strong field limit or Paschen-Back effect) over most of the range of the scanned magnetic field. This is suggested by the results of our previous experiments on the glyoxal molecule [ 47-541. Comparing the glyoxal and acetylene molecular moments of inertia, only a factor of 10 larger for the spin-rotation coupling constant is conceivable, thus fixing a maximum value of 0.5 cm-’ ( = 0.5 T) for the coupling range. We assume that Paschen-Back decoupling is sufficiently complete that the Zeeman effect is purely linear. The energy of a given triplet level (relative to E= 0 for the assumed B-independent S, level) is

E=@f&n(B-B,)

.

(1)

where g is the gyromagnetic factor of the level under consideration, M, is the projection of the electron spin on the quantization axis (z), pB is the Bohr magneton, B is the intensity of the magnetic field (along the z-axis), and B. is the field where the triplet level crosses the singlet level. For a singlet level A4, = 0. For a pure triplet level S= 1, M,= 0, f 1, and the g factor is assumed to be equal to 2.0023. The ZAC spectrum is obtained by recording the time integrated laser induced fluorescence (LIF) as a function of magnetic field. A slight (magnetic field) dependent) correction on the amplitude of the LIF spectrum is made to take into account the effect of the magnetic field on the PMT gain (decreasing by a factor of 2 between 0 and 8 T). A ZAC appears as a dip in the LIF intensity due to the coupling of an S, level with a triplet Zeeman sublevel, M, = + 1 or M, = - 1, tuned by the magnetic field [ 55 ] (see Appendix B). For the simplest model where only two levels interact, the ZAC lineshape is Lorentzian; the fractional depth of the dip is roughly one half (assuming that the collisionally relaxed singlet and trip-

P. Dupd et al. /Level anticrossing in C,H,

let lifetimes are the same) and the fwhm is given by the approximate relationship

AB==z,

(2)

where V is the coupling matrix element (see Appendix B). The variation of the magnetic field across the sample (ABinh) determines the minimum value of the coupling matrix element K we can only detect anticrossings whose fwhm is larger than ABinh (4 V/g,u, > A&,). The experimentally determined value of AB,nh is 0.25 mT (due to spatial and temporal inhomogeneities). In addition, the noise in the ZAC spectra is usually reduced by smoothing which results in an equivalent apparatus function of 0.4 mT (V> 2.8 MHz if g=2). This limit could impose an experimental bias on the observed anticrossing densities, because the narrowest (i.e. smallest) ZAC’s are not detected. On the other hand, the maximum fwhm that we could measure (8 T) is not really a limitation because we have never observed a ZAC broader than 0.5 T.

4. The level densities Our ZAC spectra are presented in figs. 2-5. The observed densities (number of ZACs per Tesla) are summarized in table 1. Fig. 6 shows a segment of fig. 4 on an expanded scale. Anticrossings are counted by visual inspection. Direct counting of ZACs is unambiguous when each ZAC is well isolated, as is the case for the 0” level. In the 3’ level, most of the anticrossings are still isolated, but in the 32 level there are numerous unresolved ZACs. In the 33 level nearly all the anticrossings are overlapping. Consequently, for the 33 levels we can only estimate a lower bound for the observed ZAC density, the actual ZAC density could easily be more than 10 times higher than the density estimated by visual count. A seemingly single broad anticrossing may be the superposition of many smaller ones. We did not record the ZAC spectrum in the 34 (J= K= I= 0 ) level because the fl G sub-band is overlapped by the much stronger 26 L$KYsub-band and

297

we were unable to excite selectively only the pP, ( 1) rotational transition. The key observation here is the surprisingly rapid evolution of the ZAC density (from 2.2 to N 500 per Tesla ) over a small range ( 3 100 cm - ’ ) of the total excitation energy ( x 7%). It is instructive to compare the observed ZAC density against predicted ZAC densities based on plausible models. We must first consider the selection rules for anticrossings so that we know the expected number of anticrossings per vibrational level of triplet (T,, T2, T,) and singlet (So =X ‘Cc ) electronic states. Our ability to select a single, non-degenerate initial level of A ‘Au, where all the quantum numbers (J, K, Z, N, S, fMJ, MS, MN) are equal to zero, vastly simplities the possibilities for detectable anticrossings (see previous section ) . We do not take into account the possible effect of the non-zero ABinhvalue in our experimental measures of anticrossings densities. The effect is certain to cause some narrow anticrossings to be undetectable. We could attempt to take these into account by increasing the observed ZAC densities by a multiplicative factor > 1. This would enlarge the previously described discrepancy. In table 1 we compare the observed and expected values of the anticrossing densities. The first hypothesis is that all observed anticrossings are due to singlet-triplet intersystem crossings and so we calculated the triplet state vibrational level densities. The lowest energy potential minima of the triplet states (T,, T2 and T,) are known theoretically [24,26-291 and these energies agree approximately with the few already published experimental values [ 15,19-23 1. The triplet states are predicted to have minima in cisand trans-bent and vinylidene configurations (see fig. 7 ). We have used (see table 1) three energy differences (gap between the selected excited singlet level and the lowest energy minima of the specified Ti and T2 triplet states). We completely ignore the restriction on the number of anticrossings due to vibrational symmetry conservation. In other words, if vibrational symmetry is conserved the computed density of states must be reduced by a factor of up to 4 to obtain the density of symmetry accessible triplet vibrational levels. As a comparison, we know that in the glyoxal molecule the u-g vibrational selection

P. DuprP et al. /Level antrcromng In C,H,

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P. Duprb et al. /Level antlcrossmg rn C2Hz

300

Table 1 Comparison between observed and calculated level densities Vtbrational level ‘)

Electronic level

Energies b, (cm-‘)

Calculated density ‘) (per cm-’ )

O0

T2 (tram) T, (trans) T, (cis)

8900 11700 14500

0.6 0.8 1.5 2.0 3.3 4.2

3’

Tz (trans) T, (trans) T, (cis)

9950 12700 15550

0.9 2.0 4.3

1.1 2.6 5.7

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11000 13800 16600

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12050 14850 17650

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Measured density ‘) (per Tesla) 2.2

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a) The vibrational levels are overtones of the u; mode (a,, CCH trans-bending, wsx 1048 cm-’ [ 1 ] ) in the St state. b, Energies of the specified S, level relative to the zero-point level of the assumed triplet state [28,29]. ‘) Calculated total vibrational density of states (not sorted according to symmetry species). Left column: semi-classical (Marcus-Rice) approximation [ 391, right column: direct level count. The triplet states normal modes frequencies being unknown, we have computed the triplet leveldensitiesusing thenormal mode frequenciesofthe S, state: 3041 cm-’ [4], 1387 cm-’ [4], 1048 cm-’ [4], 670cm-’ [70], 2985 cm-’ [70] and 803 cm-’ [70]. d, These anticrossing densities (states per Tesla) are obtained from the spectra shown in figs. 2, 3 and 4 and can be compared with the calculated level densities (states per cm-‘) by assuming g% 2 (1 Tz0.935 cm-‘). The values of the densities derived from anticrossing spectra (number of peaks per Tesla) have been divided by four: two to take into account the two possible values of MS,, + 1 or - 1 (triplet levels above or below the singlet level), and also two to take into account the two possible types of detected anticrossings: (N,=O, &= 0) - (N,= 1, K, =O or 1) . Note that we see only one of the two K, anticrossings for each vibrational level because they are separated by Am 13.8 T. The corresponding factor of two is nevertheless necessary because we see in the magnetic field scan vibrational levels from two disconnected pure vibrational energy windows corresponding to levels with K,=O and to those with K,= 1. The numbers in parentheses are the estimated uncertainties of the anticrossing densities; for vi = 3 anticrossings are so overlapped that counting of anticrossings gives only a very conservative lower limit of 40 states per Tesla. The other number in the table corresponds to a more realistic estimation which uses two independent methods based on quantum beat experiments on this vibrational level which enables one to resolve the multiple levels which appear as a single anticrossing peak [ 141. The number in ref. [ 141 has been divided by 6 to take into account the average number of So rotational levels to which the KS, = 1, IV,,= 1 rotational level excited in these experiments can be coupled through a triplet level.

rule is valid but the a-b vibrational selection rule is broken. We see that whichever triplet is selected and whatever model is used for the density-of-states calculation (semi-classical Marcus-Rice approximation [ 421 or direct count) approximate agreement between the calculated and observed densities is only found for the O”level. For the 3’ level the discrepancy is about a factor of 5. For the 32 level, the discrepancy is about one order of magnitude. For the 33 level, the discrepancy is not realistically measurable by this method but is certainly more than a factor of 102.

The validity of this comparison between observed and calculated densities of states is, of course, limited because it assumes that the Zeeman tuning rate of each triplet level (eq. ( 1) ) follows the same rule for each anticrossing. This means that we assume that the gyromagnetic g factor is constant and equal to 2. Under this condition 1 T corresponds to a level tuning of 0.935 cm-‘. To examine the possibility that the anticrossing densities are related to very highly excited vibrational levels of the So electronic ground state, we give in table 2 some calculated values of level density for

P. DuprPet al. /Level anticrossing in Cfl*

Vibrational

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301

214

MAGNETIC FIELD (Tesla) Fig. 6. Details of one anticrossing spectrum (vi = 2 level; see fig. 3) showing isolated and superimposed anticrossings; noise is about i% ( 10 averaged spectra).

the S, electronic state in the vicinity of the four S1 vibrational levels under discussion. Two level densities have been computed, the total, non symmetry sorted vibrational density for all I’s (1 is the vibrational angular momentum) of a 7 mode molecule (i.e. two doubly degenerate modes), and the symmetry sorted vibrational density including only o: and a: vibronic states. The first density is appropriate if 1 is not a good quantum number for S, levels at the energy of the S, levels, the second in the opposite case. Note that it has been shown experimentally by SEP spectroscopy that I remains a good quantum number in the So state up to 28000 cm-’ [ 9, IO]. The calculated densities of states depend only weakly on the counting method (algebraic of direct count ). Moreover, in each case, we assume harmonic potential wells; taking reasonable anharmonicities into account does not drastically change this behavior. As seen in table 1, the increase of the computed density of states is very smooth and slow. The next section will show how it may be possible to explain a sudden increase in the detected ZAC density.

5. Discussion For each of the following mechanisms, we must consider both the variation of the density of states versus energy and the distribution law for the coupling strengths. We will assume that the distribution law for the coupling strength is approximately a single parameter (&) law, whose fractional form is identical for all vibrational levels, but whose parameter TV,;,the average coupling strength, varies with vibrational quantum number vs. Such a universal law has been found previously in glyoxal [ 53 1. We will not assume that in acetylene this law has the same functional form as in glyoxal, only that the form of this distribution law does not vary’ greatly with vi. The value of pti5 has two distinct influences on the detectable anticrossings. As r;iy;increases: (i ) Broader anticro~ings will appear in the spectra, corresponding to the maximum values of coupling strength contained in the distribution law. (ii) A larger number of narrower anticrossings, corresponding to the small value tail of the distribution law, will become detectable because they are

P. DuprP et al /Level antlcrossing rn C2H2

302

Table 2 Evolution of the vibrational level densittes of the So state at the energies of the 4 vibrational states (O’, 3’. 3*, 33) of the S, electromc state S, mode

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4

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q 30 000 -

Energy (cm- ’ )

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33

42200

43250

44300

45300

3800 4275 4550

4250 4950 5275

4800 5675 6075

5550 6350 6950

62 70 85

67 80 95

72 90 105

82 100 120

‘G

Fig. 7. Summary of energies on S,, T,( i= l-3) and So surfaces in the acetylene molecule. We have represented the approximate energies of the different potential surfaces minima according to refs. [ l&24,27-29,57,69]. The various isomers are shown for each of the five lowest electronic surfaces considered. The horizontal long-dash lines indicate the approximate energies of the barrier maxima for cis-trans isomerixation and the short-dash line indicates the tram-bent acetylene++vinylidene isomerixation barrier.

broader than the inhomogeneities.

3’

a) Total non-sorted vibrational densities for all I’s (vibrational angular momentum) of a 7 mode (i.e. two doubly degenerate modes) molecule. b, Symmetry sorted vibrational densities including only vibronic o: and x: states. In both cases the upper value is due to a dtrectcount calculation of the energy levels with the harmonic normal mode frequencies taken from ref. [ 3 1,the middle value is computed by replacing the two normal modes CH symmetric and anttsymmetric stretches by two local modes described by Morse oscillators with dissociation bmit 45900 cm-‘, the lower value IS computed with anharmonic constants taken from ref. [ 7 I].

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5.1. The coupling between singlet and triplet states The physical basis for the unexpected behavior of the observed anticrossing density can only be understood once the dark levels that anticross with the selected S, level are characterized. We will suggest several plausible hypotheses. First, notice that the entire preceding analysis is based on rotational selection rules. Such rules, strictly valid at zero magnetic field, have been shown in

glyoxal to be only approximately valid in the presence of non-zero magnetic field, enabling for example detection of nominally forbidden m= + 2 anticrossings. Calculations of singlet-triplet interaction matrix elements are summarized in Appendix C, in a more general formalism than was used previously [ 481. The conclusion is that N-mixing effects could at most produce a doubling (or may be a tripling if one considers higher orders of perturbation in the spin-rotation tensor) in the detectable ZAC density. As, except for the O” level, the observed ZAC density is compatible only with the S,, highly vibrationally excited level density, So - S, ZACs must occur. Since the So level density does not increases drastically in the probed region, according to the discussion of section 4 we must search for a mechanism for a very rapid increase of the S-T coupling strength. If such a mechanism can be found, the obvious explanation would be the following: as the excitation energy increases each triplet level is increasingly mixed (at zero field) with more and more SOlevels, so that true $, molecular eigenstates (zero-order So levels

P. Duprt!et al. /Level anttcrossing in C2H2

303

contaminated by zero-order T, triplet level (i = 1, 2 or 3 ) ) would have wavefunctions

%&, being the spin-orbit perturbing term in the Hamiltonian and (Ya normalizing factor. Such a contaminated level would tune in a magnetic field with a gyromagnetic factor gQ

gQ=


(4)

(g=2.0023) and this spin-orbit induced Zeeman tuning of so levels would result in many so-S, anticrossings. We must now address the question of the detectability of these so N S, anticrossings. Before going any further in the analysis, it is essential to realize that such a two-level model cannot explain the observed high density of ZACs: an n-level (plus one triplet level) problem cannot be understood as a set of n two-level problems. Indeed if one computes the molecular eigenstates resulting from a dense set of magnetic field-invariant, nominally So levels anticrossed by a magnetic field-tuned triplet level, the resultant levels will appear as sketched in fig. 8. The energy of every so molecular eigenstate is confined between the energies of the adjacent zeroorder So basis states. This is true regardless of the strengths of the matrix elements (T, 1&so ISo ) . If the S, _ T, coupling strength for the V; = 1,2 and 3 levels remained identical to that observed in the O” level, namely a few mT (maximum anticrossing linewidth 3.9 mT hwhm), or a few tens of MHz, which is comparable to the So average level spacing ( x 10 MHz), a negligible number of S, -3, anticrossings would be detectable over the O-8 T range. To further clarify this point, it is certainly true that many nominally So levels (i.e. 5,) are contaminated by one triplet T, level spanning an energy range of the order of the (T, I&, 1So ) matrix element as described by eq. ( 3 ), but it is certainly false that levels tune according to the perturbation eq. (4) to any significant extent. This equation is only valid for displacements of levels by much less than the spacing between two consecutive zero-order So levels. The solution to this problem is to consider what happens if the S, -T, coupling is increased. The re-

Fig. 8. Energies of a set of SOlevels coupled to S, and T, levels as a function of magnetic field (arbitrary units) in two opposite coupling strength limiting cases: (a) V,a: V, (f&-T couplinga:S, -T coupling). This gives rise to a number of anticrossmgs which is on the order of the number of S,, levels (see Appendix A): ZAC density z S, vibrational level density. (b) V, Q: VO.The only effect of the SO-T coupling, V,, is to slightly shift the So levels (the triplet contaminated $, levels are constrained to remain within an energy interval defined by the adjacent zero-order SOlevels). The existence of a T level in the vicinity of an St level induces only one anticrossing. ZAC density =T, level density. The eigenenergies are shown as heavy solid lines. The triplet state (zero-order) is shown as a long-dashed line. The So states (zero-order) are shown as light solid lines. The singlet S, state (zero-order) is shown as a short-dashed line.

sults are sketched in fig. 8. When, as in fig. 8a, the S, m T, coupling matrix element is large, the result is a complex anticrossing consisting of a broad fluorescence dip. Every time one of the levels of this mixed S,-T, pair (s, and T,) crosses a nominally So level (so), there is a narrow So- (S, -Ti) anticrossing. Notice that, as in the discussion above, triplet con-

304

P. DuprP et al. /Level antrcrossrng rn C,H,

taminated S, levels can tune only between the energies of two consecutive zero-order So levels. The key difference between the correct multi-level interaction picture and the naive two-level picture is the fact that the S, level, which carries the oscillator strength, indirectly contaminates (through the T, level) So levels over an energy range of the order of the S, -T, coupling. Our conclusion is that So NT, couplings can cause the density of detected anticrossings to increase through induced 3, -s,, anticrossings only as the S, - T, coupling strength increases. There may or may not be a simultaneous increase of S,-T, couplings, but the increase of ZAC density implies nothing about the S,-T, coupling strength. However, the S,-T, coupling strength could be deduced from an evaluation of the widths (and of the gyromagnetic g factor values) of the anticrossings, but a discrimination between S, -T, “direct” ZACs and “extra” (or induced) S, - S, ZACs is not straightforward and this is a prohibitive obstacle to such an analysis. It seems reasonable to make the assumption that V,, (the So - T matrix element ) is smaller than V, (the S, - T matrix element ) because the vibrational overlap should be much larger for an energy separation (between the potential energy minima) of the order of 15000 cm-’ than for a splitting of = 40000 cm-‘. the formalism for three-states We present (S, -T - S,) anticrossings in Appendix A. The next logical question is: are the broad S, - T, anticrossings required by the preceding analysis indeed present? They are, and in fact we had noticed them at our first glance at our first ZAC spectra in v; = 3, well before we became aware of their central importance in explaining the variation of anticrossing density versus energy. For example, we were able to resolve optically a very broad anticrossing near 7.16 T [ 561. The directly measured matrix element and gyromagnetic factor of this anticrossing is compatible with a pure triplet state. A further confirmation of this explanation is given in fig. 9. Induced S, -3, anticrossings must have a local gyromagnetic g, factor (measured by quantum beat spectroscopy) which is proportional to the local dE/dB derivative of the mixed (S, -T,) level. Examples of such narrow anticrossings, which lie near the top of a broad anticrossing have been analyzed this way, giving matrix elements ranging between 0.5 and 3 MHz, and gyromagnetic

g, factors ranging between 0.3 and 1. The analysis of the complex anticrossing shown in fig. 9 is consistent with So anticrossings induced by a main Si -T, anticrossing with matrix element 103 + 20 MHz. However, in considering the complex anticrossings observed in the v; = 2 and especially the V; = 3 level more carefully, one must become uncomfortable with this explanation which implies a level clumping phenomenon. Our ZAC spectra show very broad anticrossings, and small anticrossings on the top of them, but also many small anticrossings which do not fall obviously on top of broad ones. In Appendix A, we show, by a direct calculation of a three-level (So, S,, T) anticrossing that an induced S,, - (Sr -T) anticrossing of significant amplitude can occur at a surprisingly large energy offset (see Appendix A and fig. A4) from the broad S, -T inducing anticrossing and therefore that the common origin of the induced and inducing anticrossings would not be experimentally obvious. Accompanying the increase of the anticrossing density versus energy of excitation, the ZAC spectra also show the expected significant broadening of the multiple anticrossings. Note that the nature of the clump formation we describe here is different from that observed in SEP spectra of acetylene at = 28000 cm-’ of vibrational [9,10]. In the SEP spectra the clumps correspond to a set of neighboring lines with large intensities. In the anticrossing spectra, they correspond to many narrow So - (S, -T) anticrossings induced by a large S, -T anticrossing. The experiment described previously in fig. 9 was devised to prove that the clumps in our ZAC spectra correspond to this indirect process, not to “accidental” clumping of narrow anticrossings similar to the type observed in SEP spectra. 5.2. An isomerization barrier on a triplet surface as

the reason for the increase in SO- 3, coupling A mechanism is required which explains the sudden increase in the coupling matrix element ( V0 and V, ) as v\ levels progressively higher above the vibrationless level of the S, electronic state are sampled. In this discussion direct S,, - S, couplings (internal conversion) are excluded as a plausible mechanism because there is no mechanism that could explain a sudden increase of such a coupling in the probed energy region. Since the So vibrational levels are ob-

P. Duprh et al. /Level anticrossing in Cf12

0 0.555

I ““I” 0.560

-

“I” 0.565

305

0.570

r ,” 0.575

n ’ 0.580

MAGNETIC FIELD (Tesla)

Fig. 9. Example of small anticrossings (T-S,) induced by a large (S, -T) anticrossing. The rotationless v; = 3 vibrational level is excited: (a) shows a detailed segment of fig. 5, on (b) we display the frequencies resulting from the analysis of quantum beats in the time-resolved fluorescence. The solid line is a plot of the results of the best least squares fit of the quantum beats to the equation: Y= ,/4V2+ [g&B-&)]*. The deduced g, values are plotted in (c) and they are interpreted as the local g* values of a large S, -T anticrossing (the small anticrossings permit a large anticrossing to be locally probed). The value of the St -T coupling strength is 103 + 20 MHz. The solid lines display the evolution of the best fitted g, values assuming g= 2. One curve is from the derivative (ti/dE) of E_ and the other one from the derivative of E, (see eq. (A2a) ).

served to be fully mixed (i.e. “quantum chaotic”) at energies above 27900 cm- ’ [ 8 1, it seems implausible that some large fraction of So phase space could be

inaccessible to low vibrational levels of S, yet suddenly become accessible to higher vibrational levels of S,. Moreover u-g interactions are electronically

306

P. Duprb et al /Level anticrossmg m C,H,

forbidden, thus S, -So interactions could at best be vibronically allowed. We suggest three plausible mechanisms for the increase of the singlet-triplet coupling strength: (i ) An abrupt increase in the S, -T coupling strength could be induced by an S, - T3 curve crossing which is predicted by an ab initio [ 24,27 ] calculation to occur above the S, O” level (see fig. 7). (ii)theS,-T, (or&-T,)andS,-T, (orSo-T2) interaction strengths increase abruptly near an isomerization barrier between cis- and trans-bent T, isomers. (iii) The effects of the lowest energy HCCH-+HCC+H dissociation limit on the So total density of states and the So-T, interaction strength. These mechanisms are plausible because in case (i) ab initio calculations predict the existence of S, - T3 surface crossings in the probed energy range; in case (ii) the top of an isomerization barrier on T, (or more probably T,) [ 24,271 between trans-bent acetylene and either cis-bent acetylene or vinylidene also lies in the sampled energy region; and in case (iii) the lowest HCC-H dissociation limit certainly lies in the probed energy range, even though the precise value of Do( HCC-H) is controversial [ 15,17,18,30391. Each of the three above mechanisms required that, above the V; - 1 level, the observed increase in ZAC density is due to new and stronger s, -so interactions induced via triplet levels.

5.2.1. SI - T, surface crossing Demoulin’s and Lishka’s calculations [ 24,27 ] predict an S, -TJ (trans-bent acetylene) surface crossing near V; = 3. This prediction should be reexamined by more accurate ab initio calculations. This surface crossing mechanism could also be tested by careful modelling of the V; and HCCH, DCCD isotopic dependence of the anticrossings density and coupling matrix elements. The idea is that, at energies near the S, -T, surface crossing, the S, -TJ interaction matrix elements will abruptly increase by several orders of magnitude. However, since the vibrational density of states on T3 in this 42200-45300 cm-’ energy region is very small, the increase of induced triplet character in nominal So (for V; 2 1) levels would necessarily be produced by a very small

number of energetically remote Tj vibrational levels. The number of very broad anticrossings on the v; = 3 level is probably not compatible with such a model. 5.2.2. SI - T, and/or SO- T, coupling increase near the top of an isomerlzation barrier on T, (i= 1 or 2) The S, -T, and So-T, coupling mechanisms are distinct. We will first discuss in section 5.2.2.1 the plausible role of the isomerization barrier on the S, -T, coupling and in section 5.2.2.2 on the So-T, coupling. 5.2.2. I. S, - T, coupling increases near the top of an isomerization barrier on T,. We expect that, by increasing the excitation energy in the S, electronic state, the corresponding vibrational levels (OO, - 33) will have increasingly larger spin-orbit (or spin-orbit - spin-rotation: see Appendix C) matrix elements with isoenergetic levels of the T, surface belonging to the cis-bent acetylene (or vinylidene) isomer than with the trans-bent acetylene isomer [ 24,27,28,57,58]. The vibrational overlap effects between S, and the T, (or T2) cis-bent isomer should be greater than the ones between S, and T, (or T2) trans-bent isomer. We make this claim because the geometries of the S, and T, trans-bent isomers are very similar [ 28 1, so that the overlap integral between near degenerate S, and T, trans vibrational levels is greatly reduced by the near orthogonality of the vibrational wavefunctions. On the contrary, near the top of an isomerization barrier on T, there would be large overlap between near linear turning point region of the vibrational wavefunctions of the sufficiently v; excited trans-bent S, and highly vb excited cis-bent T, isomers (fig. 10). Both wavefunctions have relatively large amplitudes in lobes which are located near the classical turning points. The overlap between these turning point lobes would increase exponentially with energy near the top of a barrier (due to exponential fall-off of the vibrational wavefunctions beyond the classical turning point). Note that the v; = 3 level in S, is far below the cis-trans isomerization barrier on S,; we are proposing the occurrence of an isomerization barrier on T,. This barrier has the effect of trapping a large and nodeless amplitude in the T, wavefunction in a region of configuration space which overlaps well with the near-linear region of the v; excited S, wavefunction. Note also that the orthogonal-

P. DuprPet al. /Level anticrossingin C2H2

307

5.2.2.2. T,- SOcoupling does not increase abruptly near the top of an isomerization barrier on Tl. Highly excited vibrational levels of the So state must have larger spin-orbit (or spin-rotation) matrix elements with isoenergetic levels of the T, (or T,) surface belonging to the cis-bent acetylene (or vinylidene) isomer than the trans-bent acetylene isomer because the So u T, trans-bent spin-orbit interaction is electronically g/u forbidden while the So NT, cis-bent spinorbit interaction is electronically A, _ B2 or Al -AZ [ 28,571 allowed:
I

4

TRANS

>

Cl.5

Fig. 10. Principle of our preferred mechanism (section 5.2.2). Heavy lines sketch (very roughly) potential wells in S, (dashed line) and Ti (z= 1 or 2 ) (solid line) electronic states as a function of trans (left) and cis (right) bending angle. Light lines sketch some vibrational wavefunctions of S, (dashed lines) and T, (solid lines) electronic states. Overlap integral between S, trans and T, trans (not represented to avoid confusion) isoenergetic vibrational wavefunctions is greatly reduced by near orthogonality of different wavefunction in similar potential wells. Overlap integrals between S, trans and low vibrational T, cis levels are small

because they are located in different parts of configuration space. There is to the contrary a good overlap integral between near linear configuration of v; excited trans S, levels and vb excited, near barrier cis T, levels. ity of vibrational

S, wavefunctions to all other vibrational wavefunctions in the same potential well, including cis-trans mixed wavefunctions near and above the S, cis-trans isomerization barrier, does not extend to orthogonality between all S, and all T, vibrational wavefunctions because, even if the geometries of the S, and T, potential surfaces are very similar near their equilibrium position, the energy differences between cis and trans minima are rather different in S, and T, surfaces. At energies above the top of the barrier the vibrational overlap would decrease rapidly. An exponential increase of the vibrational overlap would be diagnostic of near-barrier tunneling. The observed increase of the linewidths of the ZACs, especially for the V; = 3 level, also agrees well with this model.

) .

Furthermore, the same argument we have used to exclude direct internal conversion as a plausible mechanism at the beginning of section 5.2 applies also here, namely that, since the So vibrational levels are fully mixed at energies above 27900 cm-’ [ 81, it seems implausible that some large fraction of S,, phase space could be inaccessible to T, levels isoenergetic with low vibrational levels of S,, yet become suddenly accessible to higher vibrational levels of T,. But although the So N T, (cis) interaction is strong at all energies, the resultant cis-triplet contaminated 5, levels have negligibly weak interactions with zeroorder S, levels as well as with trans-triplet contaminated 3, eigenstates. At the energy of the S, 0’ level, the only ZAC-detectable triplet levels are those of the purely trans-bent isomer. The cis-trans isomer barrier on Ti is too high for appreciable overlap between the exponential tails of isoenergetic S, (trans) and T, (cis) wavefunctions. Therefore, all T,-mediated 3, N so anticrossings are undetectably weak. However, at higher energy close to the energy of the T, cis-trans isomerization barrier maximum, the T, mediated s, N “soanticrossings become detectable because of the barrier-induced trapping localization of the T, wavefunctions in the near-linear configuration. This is the mechanism discussed above in section 5.2.2.1. The exponential increase in the density of triplet mediated 3, -so anticrossings cannot be related to an abrupt increase in the So N T, coupling strength. The decisive factor is the barrier related change in the S, _ T, coupling strength.

308

P. DuprP et al. /Level antlcrowng tn C,H,

The existence of a large S’ NT, anticrossing ( %3 GHz matrix element ) [ 53 ] (as observed in the V; = 3 level at 7.16 T) is only compatible with an increase of the S, NT coupling strength. This is why we suggest that the mechanism described in section 5.2.2.1 is the most probable explanation of our results. This hypothesis is strongly supported by the values of the electric dipole moment observed for several ?‘, N 5, levels by Stark quantum beat spectroscopy [ 14,18 1. These Stark measurements indicate that, in the energy region of the v; = 2 level and above, some rovibrational levels have important cis-bent character. However, there is no way that the present ZAC and ZQB (Zeeman quantum beats) experiments can identify which one of the three triplet states (T,, Tz or T,) is involved. However this S’ -T,(barrier) -So hypothesis must be confirmed by theoretical calculations and additional experiments, notably to determine the values of the gyromagnetic g factors for a larger number of anticrossings and by examination of the behavior of higher energy levels in magnetic and electric fields. 5.2.2.3 Dissociation of the SOstate. Although controversial [ 15,17,18,30-39 1, the dissociation limit of HCCH, DO( HCC-H), is probably 45900 + 300 cm-’ which lies just above the Y; = 3 level of S’. The R’z: state dissociates into HCC (z *C+ ) + H( ( 1s) ’ 2S) fragments, probably without any barrier along the dissociation coordinate [ 72 1. Since the rapid increase in the density of detectable anticrossings in S, requires the involvement of high vibrational levels of So, a rapid near-dissociation increase in the density of S,, vibrational levels might explain the rapid increase in ZAC density at an energy just below D,,( HCC-H ). What happens to the So vibrational density of states near the HCC-H dissociation limit? The frequencies of one local C-H stretching and one doubly-degenerate local CCH bending mode decrease rapidly. It is plausible that the “soft” near-dissociation C-H stretching will have a very small effect on the vibrational density of states because the mass of the proton is too small to produce a sufficiently small splitting between the two final C-H stretching levels: this is supported by direct count density-of-states calculations we have done using Morse potentials to model C-H stretches (table 2 ). However, taking into ac-

count a doubly degenerate CCH low frequency bending vibration, it is possible to generate a high density of states near the dissociation limit. Although the combination of large amplitude C-H stretch and CCH bend vibrations could possibly generate a rapid increase in the S,, vibrational density of states near the HCC-H dissociation limit, these states would exhibit a rapid decrease in their overlap with far-fromdissociation S, and T, vibrational levels. This would cause the triplet mediated s, N 3, coupling strength to decrease rapidly, and the density of detectable anticrossings would decrease rather than increase. Therefore, we reject this mechanism.

6. Conclusion The main result of this work is the unexpected behavior of the Zeeman anticrossing spectra above 43000 cm-’ in the acetylene molecule. The anticrossing density in the spectrum of the A ‘A, (S, ) O” level is in good agreement with the expected density of triplet levels. At only 1000 cm-’ higher energy the observed ZAC density is near order of magnitude larger than the expected triplet level density. The size of this discrepancy continues to increase rapidly at higher energy. We have discussed three mechanisms to explain this behavior: ( i ) An S l - T, surface crossing. (ii) A cis-trans isomerization barrier maximum on either the T’ or T2 surface, located at an energy near the S’ v; = 3 level. (iii) The lowest energy dissociation limit located just above the Sl V; = 3 level. We prefer mechanism (ii). The strong coupling between T, (cis) and So, which should exist at all energies above the T, 0’ level and varies slowly with energy, yields triplet-contaminated nominal So levels: 3,. In contrast, Sl can only interact strongly with T, at energies near the top of an isomerization barrier on T, to yield triplet contaminated S’ levels: s’. Direct S, u So interaction is electronically forbidden, but the Sl and So levels contaminated by levels of the same T, surface, s’ and so, can interact. This triplet-mediated indirect s, _ so interaction reaches a maximum strength at energies near the top of the isomerization barrier. A prediction, that may be of use in testing hypothesis (ii), is that the ZAC density will

P. hprk et al. /Lk-vei anticrossing in C,H,

decrease rapidly at energies higher than the top of the isomerization barrier. The present results for acetylene contrast greatly with previous studies of the glyoxal and propynal molecules where the calculated triplet vibrational densities of states [ 53 ] were in good agreement with anticrossing densities.

309

Acknowledgements We are grateful to NATO for supporting this work (NATO grant 693/84). The MIT portion of this research was supported by grants from the US Department of Energy [ DE-FG02-87ER1367 11.

Appendix A A three-level model Our favored hypothesis to explain the anomalous observed anticrossing densities involves highly excited vibrational levels of the So electronic state. We construct a 3 x 3 matrix to examine this model. We assume that I’,, the singlet S, -triplet T, matrix element, is signiticantly larger than V,, the singlet S,,-triplet T, matrix element. The vibrational overlap integrals between S and T levels are expected to be much larger when A&., the energy separation between the S and T potential energy minima is smaller (i.e. = 10000 cm- ’ as compared to = 40000 cm-’ ). Moreover, in agreement with Appendix B, we will assume that the depopulation lifetimes (including collisional effects) are comparable and will thus be taken as equal to 1/r. In the ( ISo), IS, >, 1T) ) basis set, the interactions are represented by E,-$ifir 0 (

0 - 4irir

vo

V,

E, - hifir 1

Vo

V,

,

(Al)

where the Zeeman-tunable energy of IT), Et, is defined by (B2). Note that the interaction matrix does not include direct S, -So couplings (i.e. direct internal conversion is neglected). By restriction to the sub-space ( 1S, ), IT) ), the eigenenergies are obtained by diagonalization [ 601 ,

E-+ =fE,+fdm

Wa)

where the eigenvectors are I+>=~IS,>+BIT>

I->=-Pls,>+alT>

(A2b.l)

7

>

(A2b.2)

with tan 0= -2V,lE,,

(A2c. 1)

a=cos je,

(A2c.2)

/3=sin jf?.

(A2c.3)

Inthenewbasis()So>,

l+),l-)thematrix(Al)becomes (A3)

P.Duprket al. /Level anticrossing in C,H,

310

Restricting the diagonalization now to the subspace ( ISO), I + ) ) (because we assumed that /~VOc IEO Ek 1) the new eigenenergies (E,> 0) are E+ =j(E,+E+)+j,/(E+

-E0)‘+4B2V;,

(Ada)

and the new eigenvectors are

I+‘>=Isb>=a’ISo)+8’aIS,>+P’BIT>

>

(A4b.l)

I-‘)=IS;)=-~‘IS,)+a!‘aISI)+cr’/3~T),

(A4b.2)

IT’)=I-)=-BIS,>+alT>,

(A4b.3)

where tan 19’= - 2WO E,-E,

’

(A4c.l)

d=cosfey

(A4c.2)

/?‘=sin 40’ .

(A4c.3)

Fig. Al describes the level evolution as E, is tuned by scanning the magnetic field in such conditions. Transition probability

The transition probability of observing fluorescence from the excited state I@(t =O) ) = IS, ) to the ground state lg) (the state pig) has a projection onto the 1S,) state but not onto IT) or ISO) ) is

Fig. Al. A three-level avoided crossing in the case where the S, (horizontal dashed line) -T (diagonal dashed line) coupling matrix element ( V,,) is smaller than the S, - T coupling matrix element ( V, ). We have plotted the normalized eigenenergies (E_/2V,, E’+/2V,, E’_/2V,) as a function of the reduced triplet energies (E,/2V,) (see Appendix A). In this particular case: V,/V,~0.015, Eo/ V,~0.4,where E,, is the splitting between So and S,.

P. DuprP et al. /Level anticrossing in C2H2

311

(A5) where ,u is the transition dipole moment (b, = (g 1p 1S, ) ). From eq. (A4) we obtain

(‘46) Introducing eq. (A6 ) into eq. (A5 ) gives ~(t)=~(~~)2e-‘“A’+(~ucu’)2e-i”B’+~2e-i”C’~2~,

.

(A7)

By introducing the values of the mixing coefftcients as functions of 8 and 19’we find g(t)=l-4

sin*B-f cos4(O/2) sin*@++

c0d(e/2)

sin*8’ coso,,t+$

sin*OC(r)

(Aga)

,

with C(t)=cos*(8’/2)

cosw,ct+sin*(8’/2)

cosc0,&,

(Agb)

and fir&U =E;

-EL,

fio,=E’_

-E_

,

fiWAC=E;

-E_

.

(A8c)

Taking into account our assumption that VO<< Vi the term C(t) is approximated cos* [ (E, -E_ which we obtain b(t)=l-sin”Bc0s2(E+~E~

t)-cos4(6/2)

sin*@ cos*(tw.d)

.

) t/n],

from

(A9)

It is now easy to integrate eq. (A9 ) to obtain the total fluorescence m I=

s

P(t)

e-“dt.

(Alo)

0

The result is

(All) This is the sum of two inverted Lorentzian curves. The first one is the usual result of a two-level avoided crossing centered at B=Bo (i.e. E,=O) and whose amplitude is one half if the linewidth of the levels (fir) is negligible compared to the coupling term Vi. The second term is an additional anticrossing centered at B= B1 (i.e. E+ = Eo) with

(Al2) The amplitude and fwhm of this second anticrossing are greatly dependent on the zero magnetic field separation of the So level from the center of the main S, _ T, anticrossing (see figs. A2, A3 and A4). In this model the apparent gyromagnetic factor is deduced from the derivative dE+ /dB (E+ detined in eq. (A2a) ), which varies from zero (far from the center of the main anticrossing) to 2 (nearly pure triplet state): see fig. 9c.

P. Duprk et al. /Level antlcrossmg In C2H2

312 0.5

0.4

+

0.3

.z

I

la i!

0.2

0.1

0.0

0

Fig. A2. Amplitude of an induced anticrossing as a function of V, /E. (ratio of the main S , -T coupling matrix element to the Z&-S, splitting) for 5 values of the ratio, I’, =W/2Vo, where fir is the common linewidth of all 3 levels.

Fig. A3. Reduced linewidth of an induced anticrossing (reduction factor: g&4Vo) plotted as a function of V,/E, (ratio of the main S, -T coupling matrix element to the &,-S, splitting) for 5 values of the ratio, I’, =W/2 V,, where W is the common linewidth of all 3 levels.

P. Lluprb et al. /L.evel anticrossing in Cfi2

V./V,

=

E./V,

0.3-I . . . , -6

r

.

I

.

.

.

-2

-4

,

0

.

.

,

2

.

0.015 =

.

.

313

0.4

,

4

.

.

1

8

Fig. A4. Fluorescence of a three-level anticrossing for 5 different values of Eh the splitting S&, as a function of E,/2 V,. The ratio Vo/ V, has been fixed to 0.015 and the value of the ratio W/ 2 V. has been furedto 0.1. me middle curve shows the anticrossing resulting from the avoided crossing of fig. A 1.

Appendix B

Choice of identical decay rates for S,, z., and SO With the goal simplifying the equations, we examine the case of a 2 x 2 matrix including the coupling V, the lifetimes of the two unperturbed levels I s) and I t ) 1/I” and 1/r,, and the energy of one level (I&) being tunable by the Zeeman effect. The interaction matrix is

C

ifir,/2 v*

v E, - i fir,/2

4 =g~B~,(~-~o)

1’

-

@I)

WI

The fluorescence signal for such a model can be deduced to be, from ref. [55a] :

Wa) where

P. Duprb et al. /Level antlcrossrng WIC,H,

314

2+v2 (rs+rd2 rsr, .

Wb)

Defining I,,,, = Z(E, = 0)) the amplitude of the anticrossing at its center, is

Wa) where

Wb)

V=

2vpir,.

(B4c)

We display the y-dependence of I,,, in fig. Bl for five different values of v (from 25 to 0.5). Two main conclusions can be drawn from fig. B 1: (i) If V/W, < 0.25, the amplitude of the anticrossing will be too small to be distinguished from noise. (ii) If V/fir,> 0.25, we obtain a detectability condition for anticrossings: Z’,/r, > 0.2. Physically, if Z’,is too small compared to r,, the population oscillates back and forth between the I s) and 1t) states, but eventually decays only through the 1s) channel, which is the only open decay channel. In addition, we have never observed an individual anticrossing for which the amplitude is larger than 0.5, except for the very large one in which the Zeeman splitting is greater than the laser spectral linewidth (the anticrossing at 7.16 T on the v; = 3 vibrational level in fig. 5). Thus the ratio r,/r, cannot be significantly larger than unity. If r,/r, =S1 were not always true, some of the largest isolated anticrossings would have an amplitude close to unity. The deepest and widest anticrossings in the v’3 = 3 level are analysed as a superimposition of many anticrossings, which explains why these anticrossings can have amplitudes larger than 0.5.

Fig. Bl. Evolution of the amplitude of a two-level anticrossing in the CXLW of S, +-SC,optical excitation as a function of the linewidth of the triplet level (f,) divided by the S, level linewidth (r,). Calculations are illustrated for 5 different values of the ratio v= 2 V/W% (see Appendix B).

P. DuprP et al. /Level anticrossing in Cf12

315

The rough equality of the decay lifetimes (at our normal operating pressure of 100 mTorr), regardless of the nature of the interacting levels is corroborated by the measured lifetimes in this molecule [ 18,30,56] and in the glyoxal molecule [ 46 1. We never observe lifetime variations larger than a factor of 1.4. For our experimental conditions intermolecular collisions dominate the relaxation processes, thus equalizing the deexcitation rates 161 I.

Appendix C Selection rules

We will assume that isolated acetylene molecule follows the Hund’s case (b ) formalism [ 62,63 ] (J=N+S) . Further we assume that the spin (S) and the rotation angular momenta (IV) are decoupled. Thus the effective Hamiltonian is described by X= G c,T’(N)@T’(S)

,

where T( IV) and T(S) are tensors acting in the respective sub-spaces 1NM,)

(Cl)

and l S MS).

First-order spin-orbit coupling

Singlet and triplet states can only be coupled by an antisymmetric tensor of the first order [ 48,49,59,64-671. This coupling could come from the spin-orbit Hamiltonian: &&=aLZTOY

(C2a)

with

(C2b)

9ip=CI -1, .Y=s,

-s2

(C2c)

.

The total angular momentum (L = I, + 1,) is quenched [ 64,65 ] but 9 is not. Furthermore, as electronic (9) and nuclear rotation (N) angular momenta have the same rotational symmetry, matrix elements of 9’89 between two lNM,SM,) states are proportional to those of A%9 between the same states, according to the Wigner-Eckart theorem [ 68 1. Thus the following selection rules apply: AN=o, fl AM,=o,

(OcH.0)) z!I1.

Second-order coupling

We now consider a second-order effective Hamiltonian, &&,_sR,arising from a cross term between spinrotation or spin-spin (operating within a triplet state) and spin-orbit (operating between singlet and triplet states) terms. The matrix elements of &, are defined by the second-order perturbation theory,

~MT,)9 &o-SIC=
(C3)

where (C4)

P. DuprP et al. /Level antlcrowng

316

in C,H,

According to eq. (C 1) and the above section &o = c, S@Nfl”T’ ,

(CSa)

xsR = C CL S~ST~@N~~NITK . K

t-b)

Usingeq. (C87) of ref. [68] to evaluate eq. (C5b), introducingeq. ref. [ 681, we obtain &,_SR =

c Cl (Ns IIN~~‘T’(~N,~)(O~~~~~l>

NY

I3

_;NS

is,,

;zt,

7-c

IIN”N’TWt > ( 1IIt~TKII1) ( _ZN,

(4,

lc MS, --MS,.

(C5b) into (C4), and using eq. (C84) of

>

(-

)N”-NS+MS

MN~

NOWby using (C34) of ref. [ 681 we obtain %O-SR

=

,c

t

Cl (N,

II

N”vt’T’IIN1~ ) (Oll9II 1) C c:(N,,

Ic

IIN1’NtTKIIN> ( 1II “VI

1)

.

(C7)

We deduce from eq. (C7 ) the selection rules M=sk,

(C8a)

m,,GK,

(C8b)

II-K1
(Cf3c)

For example, if we consider a first-order tensor for &R (K= 1), k= 2 is allowed so that non-zero AN= 2 matrix elements result. However, if we consider the case of zero magnetic field where E,, is independent of Mst , the term 1/ (Et-E,, ) must be extracted from the sum over MS and, according to the orthogonality relation of 3J symbols (eq. (C Isa) of ref. [ 681)) which implies the selection rule k= 1, eq. (C7 ) becomes &CJ-SR

=

,c

t

CI

x (_)kW+t_


(OllYlll

) C c:( Nt, IIN”N’TKllN,)( 1II“TKll I) K Nt MN,

(C9)

In this B= 0 limit eq. (C8 ) shows that the selection rules are restricted to AN= 0, + 1, the same as for simple spin-orbit coupling, as expected from the independent proof given in the Appendix of ref. [ 521.

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