Faraday Discuss. Chem. SOC., 1983, 75, 23-43 Intramolecular Relaxation of Excited States of BY GBRALD DUJARDIN AND SYDNEY LEACH * Laboratoire de Photophysique Moleculaire du CNRS,t Bitiment 21 3, UniversitC de Paris-Sud, 91405 Orsay Cedex, France Received 27th January, 1983 Fluorescence quantum yields, qF(u’), and lifetimes, ~ ( v ’ ) , of selected vibrational levels of electronic excited states of C6F6’ have been measured by counting coincidences between threshold photoelectrons and ion-fluocescence photons (T-PEFCO technique). The derived non-radiative rates k,,(u’) for the B 2A2u state correspond to intramolecular electronic non-radiative transitions (ENRT) via coupling to high vibrational levels of the X *Elg ground state. Radiative rates kr and their variation with internal energy reflect ezectronic properties whereas the variations of knr also reveal vibrational dynamics of the ion.The knr(u’) values exhibit (i) a monotonic quasi-exponential increase for each of the two vibrational progres- sions 1” and 1”2l with increasing vibrational energy E, and (ii) mode-selective behaviour, knr(u’) being enhanced when mode 2 is excited. Model calculations of the vibrational part of knr(u’) were carried out on a non-communicating, harmonic-oscillator basis ; the results reproduce qualitatively the experimental findings. Good quantitative agreement is found between experimental and calculated values of knr(U’) for low values of E,. Progressive deviations which occur as E, incre_ases are interpreted as indicating the gradual onset of vibrational redistribution in the B state.This constitutes a new method for studying vibrational non-radiative transitions (VNRT). With increasing E,, the VNRT processes are considered to be analogous to the sparse-level (I; E, = 0-2500 cm-’), intermediate (11; E, = 2500-4500 cm-’) and statistical-limit (111; E, > 4500 cm-’) cases, respectively, of ENRT. Emission in coincidence with ex_cit_ation of the c 2B2u state is discussed and shown to involve coupling between and mixed B,X vibronic levels. 1. INTRODUCTION This study concerns two types of intramolecular relaxation processes of excited electronic states. These are (i) electronic non-radiative transitions (ENRT) in which coupling occurs between two or more electronic states (fig. 1) and (ii) vibrational non- radiative transitions (VNRT), involving intramolecular vibrational redistribution due to mode coupling within a single electronic state.ENRT can give rise to fluorescence quantum yields, qF, of the initially excited state, which are less than unity, especially in the case of large polyatomic species. VNRT can occur both in the optically excited state Is), which couples to the final states { I } , and in the latter. We will dis- tinguish these as VNRT(s) and VNRT(Z) processes. The species studied is a molecular ion, C6F6+ ; we have reviewed elsewhere the specific advantages of molecular ions for the study of intramolecular radiationless transitions.’,* We report experimental data on fluorescence quantum yields qF(j,u’) and life- * Also at Departement d’Astrophysique Fondamentale, Observatoire de Paris-Meudon, 921 90 t Laboratoire associe a I’UniversitC de Paris-Sud.Meudon, France.24 RELAXATION OF C6F,+ EXCITED STATES times z(j,u') of energy-selected levels of hexafluorobenzene cations, where j is the electronic and 0' the vibrational state initially excited. Radiative and non-radiative relaxation rates, k,(j,v') and k,,,(j,v'), respectively, are derived and are used to explore both ENRT and VNRT processes in C6F6'. Variations in the k, rate as a function of internal energy are shown to reflect electronic properties of the ion, whereas variations in k,, chiefly reveal its vibrational dynamics. A preliminary account of this work has been given el~ewhere.~ Two methods have been used to measure qF and z of energy-selected levels of 2 , v" X " \\\\ Fig.1. Vibronic coupling scheme of electronic non-radiative transitions in C6F6' as applied in model calculations (see text). C6F6+. Maier and Thommen4 have photoionized C6F6 with He(1) radiation and measured coincidences between energy-analysed photoelectrons (resolution E 100 meV) and fluorescence photons (PEFCO technique). The present work uses a different technique; level selection is by monochromatized V.U.V. radiation from a synchrotron-radiation source coupled with detection of photoelectrons having zero kinetic energy, and coincidence measurements are made between the threshold photoelectrons and ion-fluorescence photons (T-PEFC0).5 The energy resolution in our T-PEFCO study ((45 meV) makes it possible to study specific vibronic levels.Our time resolution enabled us to measure shorter lifetimes (> 1 ns) than in the PEFCO experiments, where the lower limit is of the order of 15 ns. This made it possible to determine directly decay rates of a greater number of levels in the T-PEFCO experi- ments. A further characteristic of the T-PEFCO method is that ion states can be formed not only by direct ionization, as in the PEFCO technique, but also via auto- ionization processes. As will be seen from our results on hexafluorobenzene, this makes accessible some vibronic states of the ion that are difficult or impossible to form by direct ionization alone, e.g. in Franck-Condon gap 2. EXPERIMENTAL The experimental technique for obtaining threshold photoelectron spectra (TPES) and for counting coincidences between threshold photoelectrons and the fluorescence photons emitted by the C6F6' ion excited into selected vibronic levels has been described in detaiL5 We recall here only the salient features of the experiment.G.DUJARDIN AND S. LEACH 25 The photon source is synchrotron radiation from the Orsay storage ring (ACO), dispersed by a 1 m normal-incidence grating monochromator (McPherson, 2400 lines/mm). This continuously tunable light source was operated with a band pass of 2 A. Photoelectrons produced by photoionization of ajet of C6F6 ( p zi 5 x Torr) * are accelerated towards a time-of-flight electron spectrometer by a 1.4 V cm-’ electric field and are detected by a microchannel-plate electron multiplier. The kinetic-energy spectrum of the photoelectrons is determined by measurement of the time interval between a synchrotron radiation pulse (1.2 ns f.w.h.m., 13.6 MHz) and detection of the photoelectrons.Threshold photoelectrons (TPE) can therefore be detected by gating the electron detector after each synchrotron radiation pulse with a time window centred on the specific arrival time of the zero-kinetic-energy photoelectrons. We recall that our electron energy resolution is better than 45 meV. The time-of-flight analysis requires a number of calibrations and corrections which are detailed elsewhere.’ Threshold photoelectron spectra are obtained by counting the TPE as a func- tion of the photon excitation energy. At certain excitaJion energies the C6F6’ ion emits fluorescence photons mainly correspond- ing to the B 2A2,-X 2Elg transition in the 400-600 nm spectral region.6 T-PEFCO experi- ments are done as follows.Coincidence detection and counting are carried out using a time-to-amplitude converter whose start input and stop input are, respectively, the TPE signal and the fluorescence photon signal. The latter was detected by an R943 Hamamatsu photo- multiplier with appropriate electronics. The coincidence signals are stored in a multichannel analyser. The time scale is calibrated by the method used by Dujardin et al.7 The lifetime z of the energy-selected state is determined by fitting the accumulated coincidence curve to an exponential-decay curve using a least-squares method. The total number of true coin- cidences, Nc, during the time of the coincidence count is obtained by time integration of the T-PEFCO curve after subtraction of the false coincidence background.During the same time interval we count N,, the total number of threshold photoelectron_s. The TPE can result from direct ionization producing the emitting level of CbF6+, e.g. B 2A2u(u), and also from autoionization processes which can form not only the fluorescing state of the ion but also nzn-radiative isoenergetic vibrational levels belonging to lower electronic states, notably the X 2E1g ground state.’ Methods for obtaining the branching ratios for these processes in CsF6+ are described in detail elsewhere.’ These methods enable us to determine N,(j,u’), the number of TPE corresponding to formation of the j,u’ state of the ion. With the T-PEFCO technique, autoionization processes can increase the relative number of ions formed at certain j,u’ levels with respect to the PEFCO technique, in which only direct ionization processes are measured.This is particularly true in the so-called Franck-Condon gaps ; the T-PEFCO technique therefore permits us to measure the fluorescence decay characteristics of j,u’ levels whose access is difficult with the PEFCO technique. The fluorescence quantum yield &,d) is then determined from the relation: VF(j,V’) = N c / ~ e ( j , u ’ l h v (1) where f h v is the fluorescence photon detection efficiency, whose method of determination is described elsewhere. l9’ 3. RESULTS The TPES of C6F6 has been measured over the 9-29 eV excitation energy range.’ In fig. 2 is reproduced only the 11.8-13.8 eV region of the TPES.The B” 2A2u state origin of the C6F6’ ion is at 12.595 eV; two vibrational progressions in this state are indicated; these involve the two totally symmetric vibrational modes vl(alg), which is a C-F stretching mode of frequency 1515 cm-l, and v2(alg), a ring vibration of frequency 525 cm-’. Peaks assigned to autoionizing levels of a Rydberg series converging to the 2; 2B2u state at 13.88 eV are also indicated. They have the same energies as known Rydberg bands in the V.U.V. absorption spectrum of C6F6 ;’ analysis * 1 Torr = 101 325/760 Pa.26 RELAXATION OF C6F6' EXCITED STATES r 1 - 0.5 - 112.738 13 I 14 1 12 1.R (19) 1.E (C) photon excitation energy/eV Fig. 2. Threshold photoelectron spectrum of C6F6 in the region of the state of the ion.Two vibrational progressions (1" and l"2l) are indicated, involving the two totally symmetric vibrational modes v1 and v2. Also shown is a Rydberg series (n is the principal quantum numier; quantum defect 6 = -0.02) with its vibrational components v and converging to the C 2Bzu state at 13.88 eV. The circled numbers correspond to excitation energies at which pF and r measurements were made. The zero signal at 12.39 eV corresponds to a check of detector signal background in the absence of photon excitation. of these TPES and V.U.V. absorption Rydberg features is discussed el~ewhere.~ These Rydberg peaks are also quasi-identical in energy with &state vibronic levels of C6F6+ ; threshold autoionization to form the ion levels is thus fa~oured.~ A remark on the energy resolution is in order.The symmetry of C6F6 is high (&) and is retained in the electronic states of concern to us in the ion. Furthermore, mode distortion or displacement is relatively small for all known vibrational modes in going from C,F6 to the Consequently, apart from sequence bands (whose maximum relative intensity we estimate as no greater than 30% of the B(0Qband in the PES or TPES), only the two totally symmetric modes are excited in the B 2A2u state. The energy resolution in our TPES and TPEFCO experiments, although not high in comparison with normal optical spectroscopy techniques, is quite sufficient to resolve the bands corresponding to these vibronic excitations. The 10 circled numbers in fig. 2 correspond to ten of the photon excitation energies Eexc at which T-PEFCO measurements were made.A measurement was also carried out at 13.88 eV, the band energy of the 2; 2B2u state origin. These excitation energies are given in column 1 of table 1. Columns 2 and 3 give the measured fluorescence quantum yields qF and lifetimes z. Columns 4 and 5 give, respectively, the radiative (k,) and non-radiative (k",) rates for the 8 and state levels, derived from relations (2) and (3): 2A2u state of the k r = PF/T (2) knr = (1 - M Z . (3)G. DUJARDIN AND S. LEACH 27 Table 1. Fluorescence quantum yields, qF, lifetimes, z, and radiative (k,) and non-radiative (knr) rates of selected vibronic levels of C6F6+. The band assignments are discussed in ref. (5). photon band energ y/eV VF zjns kr/106 s-' kn,/106 s-l Ev/cm-' assignment 12.59 12.65 12.77 12.84 12.92 12.98 13.04 13.22 13.40 13.68 13.88 1.07 f 0.10 0.81 f 0.12 0.97 & 0.10 0.75 f 0.15 0.81 f 0.12 0.81 f 0.12 0.54 f 0.08 0.45 f 0.07 0.25 f 0.04 0.07 f 0.01 0.008 f 0.001 49 f 2 41 f4 46 f 4 42 f 4 41 h4 41 & 4 43 f 4 29 f 2 17 f 2 8 f 1 1.7 f 1 20.4 f 3.0 19.8 f 5.3 21.1 f 4.3 17.9 f 5.7 19.8 & 5.3 0 to 0.6 3.7 4.6 & 3.1 2.4 0.6 & o,6 4.6 6.0 f 3.8 3.7 4.6 & 3.1 19.8 f 5.3 12.6 & 3.2 4.6 f 3.7 3.1 3.1 10.7 f 2,6 4.0 19.0 f 3.5 8.5 44.1 & 6.8 18 116 f 14 15.5 & 3.7 14.7 f 4.6 8.8 f 2.7 835 584 & 216 4.7 f 2.0 0 500 1500 2000 2650 3150 3650 5100 6550 8800 10 400 B: oo 2' 1' 1'2l valley; (1 122 ?) l2 1 221 1321 1421 1522 ? c' oo Column 6 gives the selected internal energy, E,, of the B and c states with respect to the Oo level of the B" state at 12.595 eV: the corresponding vibrational assignments are given in column 7.4. RADIATIVE RATE kr ( j , ~ ' ) In fig. 3 we compare our T-PEFCO k, results with those reported by Maier and Thommen from their PEFCO results;4 for clarity in the figure we have not reproduced the error limits of the Maier and Thommen values, which are of the order of *2 x lo6 s-l. There is excellent agreement between the two sets of values for common excit- ation energies. From table 1 and fig. 3 we see that the radiative decay rate is approxi- mately constant, with a value k, = (18 &- 3) x lo6 s-l for the nine &v' levels up to and including the photon excitation energy EeXc = 13.40 eV.It decreases to ca. 9 x lo6 s-l at Eex, = 13.68 eV in the vicinity of the c state and further drops to ca. 5 x lo6 s-l at the c-state origin. We note that the k, rate for the C(Oo) level is ca. 25% of that of the &OD) state. This result shows that corrections must be applied to the non-radiative rates for the c state region of C6F6+ (Eexc = 13.88-14.28 eV) determined by Maier and Thommen from the relation k,, = k,(l - v)F)/v)F with the assumption that k, is the same for the B and ~ t a t e s . ~ 5 . NON-RADIATIVE RATE knr(B,v') 5.1. PRINCIPAL OBSERVATIONS The experimental values of the non-radiative rate k,, in the h a t e region (table 1) are plotted in fig. 4 as a function of internal vibrational energy E,. This figure also28 l o 8 3 I \ 2 m 10'- lo6 RELAXATION OF CsF,j+ EXCITED STATES - 2' 1' 2' 1*2' 13z1 142' (15 22) t oo I 1 I I 1 @f wp .P p I! f u OP 1' l 2 I I 1 lo9 l o 8 - - ~ ~ 2 ~ 1 a 1 V 1* 2' l 3 2'r-b...1' 2' 1 o6 10' - 10000 E,/cm-' Fig. 4. Non-radiative decay rate k,, of the &' levels of C6F6'. Rectangles indicate the uncertainty limits of experimental measurements. T-PEFCO values are taken from table 1 and PEFCO values are taken from Maier and Th~mrnen.~ Data on levels involving mode 2 excitation are given by cross-hatched rectangles.G. DUJARDIN AND S. LEACH 29 l"2l progression as well as 0' and l", with a resolution better than 45 meV. The agreement is good between the PEFCO and T-PEFCO results for the common values of E,. Our two principal observations (table 1 and fig. 4) are as follows: (1) We note first of all that the 1" (n = 0-4) and l"2' (n = 0-4) progressions, taken separately, each exhibit a monotonic, quasi-exponential, increase in k,, as E, increases.(2) We see from table 1 and fig. 4 that the k,, rate depends not only on the vibrational energy of the optically excitedjp' state but also on the specific optical vibrational mode excited. This type of behaviour has previously been observed in neutral species, e.g. benzene.11*12 In the case of C6F6' it is clear from fig. 4 that the k,, rate is enhanced, with respect to the 1" progression, when the v2 mode is excited simultaneously with 1" in the 1"2l combinations. Most of the rest of this paper is devoted to interpretation of these two results and requires first of all the presentation of an adequate theoretical model of ekxctronic non- radiative transitions.We will use the simplified notation knr(u') = k,,(B,u'). 5.2. MODEL CALCULATIONS OF k,,(u') RATES : THEORY Dujardin et al.,7 using the photoion-fluorescence-photon coincidence technique,lp7 have previously shown that ENRT in Auorobenzene cations corresponds to the so- called large-molecule case of radiationless transitions.12 ENRT from vibronic &' levels was shown to be due to internal conversion to isoenergetic 2,u" vibronic levels of the electronic ground state. In a model appropriate to this case (see e.g. fig. l), each initial Born-Oppenheimer Is) state (i.e. each 8,u' level) is coupled by the intra- molecular non-adiabatic potential uSz to the set of I states (2,~" levels).We remark that for C6F6' other non-radiative transitions such as intersystem crossing to quartet states or predissociation are not expected to exist in the energy region studied here. In particular, competitive dissociation processes do not set in until ca. 15.3 eV.13 Furthermore the low pressure (ca. 5 x lov4 Torr) in our experiments makes the effect of collisions negligible in comparison with the electronic intramolecular pro- cesses, as has been verified by pressure-variation experiment^.^ The irreversible transition of a vibronic Is) state of energy E, to quasi-isoenergetic E states (energy El) occurs with a rate given by We note that in this formulation the non-radiative rate kSl is equivalent to the experi- mentally determined k,, rate, which is extracted from the fluorescence quantum yield q&) and lifetime z(s) of the corresponding Is) vibronic level by relation (3).We stress the fact that for open-shell species such as C6F6+, coupling with high vibrational levels of the ground state is the only non-radiative relaxation pathway from the first excited state of the same multiplicity as the ground state, whereas in closed-shell species (e.g. C6H6) one or more triplet levels lies below the s1 level and can give rise to several other relaxation channels competitive with S1-So coupling. We therefore consider that interpretation of non-radiative transitions in open-shell species such as C6F6+ can be less equivocal than for analogous studies in closed-shell species.l Theoretical studies predicting the monotonic increase of knr(u') as E, increases have previously been carried out by a number of authors.12 Such studies have shown that this behaviour is expected for moderate and large values of the electronic energy gap30 RELAXATION OF C,F, EXCITED STATES Eo (we recall that E, = Eo + &,).I4 In our case, the electronic energy gap is large (Eo = 21 601 cm-'> and our experimental findings on the trend of k,, as a function of E, for each of the two progressions 1" and 1*2' are thus qualitatively consistent with the general theoretical results.Our own theoretical work reported here concerns more specifically model cal- culations of relative k,, rates for each of the two progressions. Our aim is to confirm the monotonic trend for each progression and also, more importantly, to see whether we can reproduce the experimentalIy observed vibrational mode selective dependence of knr.The starting point of model calculations of the relative k,,(u') values is eqn (4). The key is in the assumption that the overall interaction energies usl can be factorized into single products of electronic and vibrational factors.12a*b The electronic factor, whose calculation would be compIicated, is taken to be constant for different vib- rational levels of the B state [the Condon approximation following from the Born- Oppenheimer (B.O.) approximation; problems arising in its application to radiation- less transitions are discussed by Freed and Lin "]. The electronic factor can there- fore be factorized out in the relative rate caIculations. The latter then reduces to a calculation of the appropriate Franck-Condon factors taking into account the energy conservation condition represented by the &function in eqn (4). The electronic-vibrational factorization approximation requires that the Born- Oppenheimer separation of the electronic and nuclear motions be valid.This is reasonably the case in describing the initial vibronic &' states in which onIy totally symmetric modes are excited. However, the orbitally degenerate ground state of C,F,+ is affected by dynamic Jahn-Teller distortion via four non-totally symmetric degenerate eZs modes,6*16 so that the €3.0. separation may no longer be valid for the final 2,u" states. Nevertheless, we consider, and assume in the folIowing, that it is reasonable to caIculate the k,, rates wchin the B.O.approximation, in view of the fact that selection rules restrict accepting X,d' levels to those that are totally symmetric. PROMOTING MODES AND ACCEPTING MODES In order to carry out the model calculations we need to know which are the promoting and accepting modes (as defined by Lin)." Fur this we shall make use of group-theoretical selection rules. A promoting mode K is considered to be any vibrational mode which induces extensive non-adiabatic coupling between initial S and final L electronic states via a the electronic matrix element (ySl -1 yL) l2 occurring in usi, where QK is the nuclear 8 Q K normal coordinate associated with mode K. From the form of this matrix element the selection rules given by Nitzan and Jortner l4 imply that the direct symmetry products Ts x rL x rK and Ti x rp x r, must contain the totally symmetric represent- ation of the C6F6+ (&) point group, where Ti and rp are the vibrational symmetries of the initial.and final states. Since rs = A2u, and r, = El,, it follows that r, = elu. Three vibrational modes of C6F, have el,, symmetry.6 it has been shown that the promoting mode K acts to reduce the effective energy gap ED, the reduc- tion being equal to the vibrational energy wK. It has been shown by many authors l2 that k,, can be expressed as an exponentially decreasing function of the electronic energy gap E,,. From this energy-gap law we would expect that the greater is the reduction of the effective energy gap (i.e. the smaller is Eo - mK>, the higher is the non- radiative rate.In the absence of information as to how the electronic factor From an explicit calculationG. DUJARDIN AND S. LEACH 31 a a Q K (ty&--~tyL) varies with different el, modes, we thus consider the most effective promo& mode to be the el, vibrational mode with the highest frequency. In the following we shall assume that only this promoting mode is effective [mode v20 in ref. ( 6 ) ] ; its frequency is taken to be that of the same mode in neutral C6F6, (0.1~~ = 1530 cm-I), since its value for the C6F,+ cation is unknown. The propensity rule derived by Freed and Jortner l8 predicts that the transition probability is highest when one quantum of the promoting mode is found in the final vibronic state (cf. also Nielsen and Berry,19 who first propounded a similar vibrational hexis for the radiationless process of autoionization).We recall that only the totally symmetric v1 and v2 modes are optically excited, so that Ti = alg. The vibrational selection rule then gives us that in addition to the promoting mode, the final vibrational state contains only v, and v2 totally symmetric modes as accepting modes (or totally symmetric combinations involving non-totally symmetric modes, the latter giving rise to weaker higher-order couplings). In going from the initial S state to the final L state, vibrational modes could be distorted, i.e. change in frequency, and/or be displaced, i.e. change in equilibrium position. Since the C6F6+ cation retains its D6,, symmetry group in relaxing from the S electronic state to the final L state (as can be assumed from optical spectroscopy results 6*10p16) symmetry conditions restrict the possibility of displaced modes to the two totally symmetric modes.Furthermore, apart from the special case of the eZg Jahn-Teller modes in the ground state of C6F6+, mode-frequency changes appear to be relatively minor between the B and 8 states.6*10 We therefore considered all modes other than the totally symmetric modes 1 and 2 to be effectively constrained to being undistorted as well as undisplaced. In this case they would have the same quantum number in both S and L, and so we factorize them out of our relative k,, expression. THE NON-INTEGRAL OCCUPATION NUMBER (NION) METHOD The specific method we used for calculating relative k,,(v’) rates is derived from the non-integral occupation number (NION) method developed by Kiihn et a1.20 (see also Prais et a1.).21 It enables us to confront in practical fashion the delicate problem of ensuring that energy is conserved in the radiationless transition, i.e.to deal adequately with the &function in eqn (4). The NION method has been shown to be consistent with other methods used for relative k,, rate calculations, such as the “ saddle-point approximation ” 12b*20 but is of simpler application. It is well known that the evaluation of rate expressions can be achieved by the use of generating functions.18 The &function in eqn (4) can be represented by a Fourier integral. Evaluation of this integral by the saddle-point approximation requires the analytical evaluation of infinite sums.This is avoided in the NION version of the generating function method, in which the generalized density- of-states functions do not have to be factorized into products of matrix elements and the vibronic density of states. Furthermore, conservation of energy and evaluation of a sum in vibrational quantum numbers in the final state are facilitated by introduc- tion of a non-integral vibrational quantum number y. The NION method is presented in detail in ref. (20), where it is shown how one can obtained closed form expressions for the Franck-Condon factors involving y ; see also the discussion in ref. (12b). To calculate the k,,(u’) values we have to determine the vibrational dependence of ks+l from eqn (4). From our previous analysis it results that the accepting modes are the two optical modes vl and v2 of C6F6+.For an initial vibronic B,v’ state in-32 RELAXATION OF C6F6' EXCITED STATES cluding n, and 12, quanta of modes v, and v,, the vibrational part of the non-radiative rate is given by a sum of products of Franck-Condon factors: The sum is over all the integer values of Il from 0 to (Eo + n,co; + n20; - wK)/o; where the primed and double-primed co refer, respectively, to the initial B,u' and final 2,u" states. The non-integral quantum number y is defined by where AE(I,) = Eo + n,o; + n,oi - coK - I,co," (7) is the vibrational energy to be redistributed into mode 2 in the final state. We derived relation ( 5 ) for the case of two optical modes by extension 22 of eqn (2.8) of ref.(20) in which the NION method of calculation of non-radiative rates is expounded. We remark also that we hEve used the normal-mode description in characterizing the vibrational levels of the B (and 2) states. A local-mode description might be more appropriate, especially for the higher members of the 1" and 1"2l progressions. However, from the results of Kiihn et al.,O there is not expected to be any significant consequence in our NION calculations, which are carried out on a harmonic- oscillator basis, in the use of a normal- rather than a local-mode description. 5.3. MODEL CALCULATIONS OF k,, (U'): APPLICATION TO C6F6+ In calculating the non-radiative rate knr(nl,nz) by the NION technique, mode number 2 in relation (5) is conventionally assigned to the best accepting mode.20 In the well studied case of non-radiative transitions from the 2 1B2y state of neutral C6H6, the v, = 3073 cm'l C-H mode has generally been considered to be a far better acceptor than the other al, vibration, the C-C mode v2 = 992 crn-l;,O [however, recent calculations of Hornburger and Brand 23 for S,+So internal conversion show that the two modes are of comparable accepting quality and that other accepting modes can be important on a fully (vibrational) communicating model].From the symmetry selection rules we deduced that for non-radiative transitions from the B state of C6F6' the totally symmetric modes v, and v2 are the principal accepting modes, and we shall now discuss which is the best accepting one. The accepting character of a given mode is related to its degree of excitation in the final vibrational states (I).Many different isoenergetic ( I ) states are non- radiatively coupled to the initial Is) state, and consequently the relative non- radiative rate k,, is expressed in relation (5) by a sum of products of Franck- Condon factors. We denote k;,"" as the maximum term value of this sum and p a x as the final state associated with k;,"". We shall assume in the following that the accepting character of v1 and v2 is related to the vibrational composition of the Pax state. Table 2 shows in column 3 and 4 the numbers L1 and L2 of v1 and v2 vibrational quanta included in the I""" state for different J,u' optically excited levels. The corresponding vibrational energies El = Llcoz and E2 = L2co," are listed in columns 5 and 6.v1 will be the best accepting mode when El > E2 and v2 when E, > El. Table 2 shows that for the 1" progression, v2 is the best accepting mode for n = 0,1,2 but that v1 is a better accepting mode for n = 3 and 4; for the 1"2' progression v2 is a better acceptor than v1 for n = 0,1,2,3 and v1 better than v2 when n = 4.G. DUJARDIN AND S. LEACH 33 Table 2. Numbers L1 and L2 of v1 and v2 vibrational quanta included in the Zmnx state (_see text) and corresponding vibrational energies El and E2 for different optically excited B,v’ levels of C,5F,5+ optically excited B,v’ level EJcm - L1 L1 = El = L~o’; E2 = L20i v‘ = 1”12”2 /em- /cm- O0 2l 1’ 1 l2l l2 1 221 13 1321 14 1421 0 525 1515 2040 3030 3555 4545 5070 6060 6585 4 4 6 5 7 7 9 8 10 10 24.6 25.6 21.9 25.6 21.9 22.9 19.3 22.9 19.3 20.2 6 060 6 060 9 090 7 575 10 605 10 605 13 635 12 120 15 150 15 150 13 874 13 440 11 497 14 438 11 497 12 916 10 132 12 916 10 132 11 393 Whether vL or v2 is a better accepting mode therefore depends on the initial B,u’ level, so that no intrinsic relative accepting character can be given overall for vl and v2.We remark that for the 2 lBz,-8 ‘Al, internal conversion process in neutral benzene it has also been shown that the relative accepting qualities of particular modes can vary with Ev.23 Although the relative importance of v1 and v2 as accepting mode changes with excess vibrational energy E, in the state, we have uniformly taken the C-C mode v2 as the best accepting mode in the calculations reported in table 3.This does not affect the relative values of kn,(v’) since the latter are diminished by a constant factor of 10 when mode numbered 2 in relation (5) is switched from the C-C mode to the C-F mode. This result is interesting and requires further study. Table 3. Experimental and calculated relative non-radiative rates k,,(c’) of selected 1 state vibronic levels of CsF6+. Data used to determine relative experimental values of kn,(V’) are taken from table 1. The calculated knr(v’) values are normalized to the knr(v2) experimental value. optically relative k,,(o’) excited level experimental calculated 0 = oo v2 = 2‘ V 1 = 1‘ v1 + v2 = 1121 2v1= l2 2Vl + v2 = 1221 3Vl ---r 13 3V1+ v2 = 1321 4Vl= 14 4Vl+ v2 = 1421 <0.13 0.03 1 1 0.13 0.17 1.3 5 1 0.67 2.3 16.7 1.7 4.1 45 5 9.6 105 - - NION calculation parameter values (see text) : electronic energy gap, Eo = 21 495 cm-I ; accep- = 525 cm-’; 0: = 564 cm-’; promotor tor mode, w: = o: = 1515 cm-’; best acceptor mode, mode, W K = 1530 cm-l; reduced displacements, x1 = 0.0027; x2 = 0.284 (see Appendix 2).34 RELAXATION OF CsFs -i- EXCITED STATES The NION calculation k,,(u') values presented in coIumn 3 of table 3 were obtained from relation ( 5 ) by calculating Franck-Condon factors for dispIaced and distorted harmonic oscilIators.The harmonic F.C.-cakulation method of Katriel 24 was generalized to the non-integral quantum-number case (Appendix I). The molecular parameters used are shown in table 3. The energy gap Eo and vibrational frequencies of the [Note that the Eo value used here was taken from our TPES data.It differs by ca. 100 cm-l from the optically determined valuem6 This difference is insignificant in the present context of the NION calculations of k,,(v') rates.] Since the w; value is uncertain we took w; = w;. The reduced dispIacements state are taken from data presented in ref. (5). miw; 2fi xi (i = 1,2) = - (Q; - Q ; ) 2 of the normal coordinate Qi were determined 22 from the photoelectron spectrum of Duke et aLZ5 using the Franck-Condon-analysis method given in Appendix 2. We remark that mode 1 corresponds to a displaced oscillator, whereas the mode-2 oscillator is both displaced and distorted. Some comments on the sensitivity of the non-radiative rate calculations to mole- cular parameters are in order.Calculated k,, values have been shown 20-21*26 to 0 5000 EJcm - ' 10000 Fig. 5. Compari2on of experimental and calculated non-radiative relaxation rate k,, of selected levels B,d of C6F6+. VaIues from tables 1 and 3 and ref. (4). Calculated values were normalised to the experimental value knr(21). 0, Experimental ; 0, calculated. depend strongly on the vibrational frequencies in initial and final states and on vib- rational anharmonicities when the electronic energy gap is large. Further calculations in the case of C,F6+ require a better knowledge of the vibrational frequency of the v1 (C-F) mode and of the anharmmicity parameters involving the v1 and v2 modes, which are unknown at present. The present results are very satisfying in that they reproduce the two principal experimental observations on k,, behaviour.The calculated non-radiative rates given in table 3 and in fig. 5 have been normalized to the k,,,(2') experimental value.G. DUJARDIN AND S. LEACH 35 The results show a monotonic increase of k,, with E, for each vibrational progression 1" and 1"2' taken separately, in accordance with experiment. The calculations re- produce the mode-selective behaviour of k,, in predicting that for " equal excitation energy," the k,, (l"2') values should be greater than the k,,(l") values. For the 1" progression the quantitative agreement between theory and experiment is good, but for the l"2' progression we observe an increasing deviation of the experi- mental values from the calculated ones. This important result will be further dis- cussed.We also note that with k,,(2') normalized to the experimental value our calculations lead to a value of k,,(Oo) = 1.4 x lo5 s-' for the B 2A2u origin level, consistent with our experimental results and with those of Maier and Th~mmen.~ Before ending this section we briefly discuss, and establish as non-operative in our case, another possible source of apparent mode-selective behaviour of k,,. This relates to the case where a promoting mode is excited via sequences that overlap with the band b corresponding to excitation of one quantum of the optical mode. A theoretical treatment by Nitzan and Jortner l4 has shown that the k,,(b) rate could then be enhanced by a factor of 3. This cannot be the origin of the marked increase of the k,,(v2) rate in our case, because the frequencies of the three (possible promoting) el, modes in C6F6, and their expected changes in going to the B state of CsF6+, are unlikely to create a sequence band in the ~ ( v J region under our experimental con- ditions.We note also that the experimental " enhancement factor " is rather greater than 3 (see table 3). 5.4 VIBRATIONAL REDISTRIBUTION We have seen (fig. 5) that the model calculations reproduce qualitatively the two principal results, i.e. the observed increase of k,, with E, for each vibrational pro- gression, and the mode-selective behaviour of k,,. However, quantitative agreement falls off as E, increases, especially for the l"2' progression. In fig. 5 the dashed and full lines join, respectively, the successive calculated and experimental points up to the last calculated level, 14F.The dotted line links the experimental values for E, >, E(14Y). The experimental k,, values oscillate with smaller and smaller amplitude as E, increases and show little variation from an expo- nential increase above E, = 4500 cm-'. We note in particular that there is no marked discontinuity at the onset of the state. The damping of the k,, value oscillations is in marked contrast to the calculated values. The calculations on relative k,,(&u') rates were carried out using the model scheme of fig. 1, on a harmonic-oscillator basis. This implies the absence of vibra- tional redistribution in the optically excited state. The calculated values thus provide a non-communicating oscillator limit, from which the progressively increasing devi- ations of experimental results can be used to argue that vibrational redistribution does indeed occur with increasing importance as E, grows.The model represented by fig. 1 is therefore insufficient. A more appropriate model scheme is illustrated in fig. 6. Optically excited vibrational states {s} in the excited electronic state S ( d ) can undergo coupling via matrix element D , , ~ to other quasi-isoenergetic vibrational levels in the {s'} manifold of the S state. These can be considered to form nuclear molecular eigenstates {.?I in the excited electronic state, which can couple in turn to isoenergetic highly excited vibrational levels {I} of the L(=8) ground state. However, it is more useful to continue our discussion in the zero-order basis of ($1, {s'] and ( I ) ; coupling will be discussed in terms of the matrix elements vSl and36 RELAXATION OF CsFs' EXCITED STATES In the present scheme we have no need to consider vibrational redistribution in the {I} states.The effect of VNRT ( I ) would be to add to the level width of the {I) states. The density of the { I } states isoenergetic with a particular s level is very high (ca. 10' per cm-l according to a Haarhoff-type calculation'), so that the effective quasi- continuum of ( I } states would be little affected by the introduction of VNRT(I). We remark that information on the VNRT(I) process can be obtained in studies of the so- called intermediate case of ENRT. An example we have studied is the chloro- acetylene cation.27 The operators in the various matrix elements will have the following physical significance : (i) vss': vibrational anharmonicity and Coriolis (rotational-electronic) S Fig. 6.Vibronic coupling scheme including electronic non-radiative transitions (ENRT) (coupling matrix elements uSz and uStZ) and the vibrational non-radiative transition [VNRT(s)] (coupling matrix element t],,*). coupling; we may use the term Fermi resonance as a generic name for any interaction which takes place between close-lying zero-order vibrational levels and which is caused by terms in the vibrational potential-energy expression cubic or higher in the vibrational coordinates. We remark that our concern is with anharmonicity in the optically excited S state and not, as in previous studies, 12*20*21 with the effects of an- harmonicity in the L state acceptor modes.(ii) vsl and vsnl : nuclear kinetic-energy operators resulting from the incomplete Hamiltonian which is implied in deriving the zero-order states. No spin-orbit operator contributions are necessary in our case of open-shell molecular ions, where the lowest accessible quartet state is above the first, and possibly the second, excited state. vsl and v s r l correspond to coupling with the quasi-continuum of ( t } states and thus lead to irreversible transitions whose rates are, respectively, ksl and ksfl. Let us consider successively I, I1 and 111, three different regions of E, corresponding to increasing level density p(E,v'). The boundaries between these regions are not rigid. Our experimental results (e.g.fig. 5) lead us to propose as approximate ranges in the B state of C6F6' : E,(I) = o to ca. 2500 cm-', E,(II) = ca. 2500 to ca. 4500 cm-', E,(III) > 4500 cm-l. From the viewpoint of VNRT(s), these regions corres-G. DUJARDIN AND S. LEACH 37 pond, respectively, to the sparse level (I), intermediate case (11) and statistical limit (111) analogues of ENRT.28-31. At low-level densities p(8,v') (region I), and in the absence of Fermi resonance and neglecting Coriolis coupling [both of which can lead to mixing of zero-order (de- perturbed) levels within the B" vibrational manifold], we expect the optically excited &I' levels to be isolated in the B" state and to couple individually to g,;v". Thus mode; selective behaviour of k,,, reflecting mode specific coupling matrix elements between B and 2, is reasonable to observe at small Eu.At higher internal energies we enter region IT. As E, increases, anharmonicity factors will become more important and p(B,u') will also increase; more Fermi reson- ance and Coriolis coupling possibilities will result. Greater zero-order level mixing and dynamic vibrational redistribution within the 8 vibrational manifold will gradually set in. This can also be envisaged in terms of a gradual increase, with Eo, of the phase- space occupancy.32 The marked decrease of observed k,,(1"21) with increasing n as compared with the calculated values is in contrast to the behaviour of the 1" levels, and indicates that the presence of one quantum of mode 2 facilitates vibrational redistribution towards levels that have poorer Franck-Condon factors in the ENRT process.At very high E, (region 111) we approach the situation of a quasi-continuous set of 8,v' levels which would correspond to the fully communicating " statistical limit " case of complete vibrational redistribution. The existence of such a continuum can be predicted for energies E, such that the mean energy interval p-l between levels in the (s} vibrational manifold is smaller than the average width Ts of these levels. The lower limit of I?, (in cm-l) is (27~c.r)"~, where z is the lifetime observed in fluorescence and reflects only those processes which depopulate the fluorescing electronic state S. Vibrational redistribution certainly adds a further contribution to r,.We can thus estimate an-upper limit EI,, beyond which B,v' levels form a quasi-continuum. Cal- culating p(B,v') by the Haarhoff approximation l and using the lifetimes @,d) from table 1, we find that p-' = (27~.cz)-~ at energy E,,, = 6900 cm-l. This calculated upper limit, neglecting vibrational redistribution, is compatible with the 4500 cm-l value estimated for the onset energy of region 111, as evidenced by the quasi-collinearity of the k,,(l") and k,,(1n21) values as a function of E, above 4500 cm-I (fig. 5). Relaxation processes mentioned above have their analogues in models of uni- molecular dissociation (UMD) reactions.33 In our case the ground-state vibrational quasi-continuum (I> plays the role of the dissociation continuum in unimolecular dissociation.The operator responsible for the s-+Z and s'-Z transitions will have the same nature in ENRT and in electronic predissociation (neglecting spin changes), whereas it will be of the VNRT(s) type in vibrational predissociation. In another area where similar problems arise, that of multiphoton dissociation (MPD),34 the initial optical excited vibrational level has access to a denser set of vibrational levels (analogous to the isoenergetic s-Z and d-+Z processes) by the non-isoenergetic mechanism of photon absorption. In both UMD and MPD, vibrational redistri- bution is essential in order to achieve the adequate nuclear configurations for tran- sition to the dissociation continuum; for example, in UMD the R.R.K.M. model supposes the existence of very rapid VNRT prior to dissociation, so that the whole phase space corresponding to all vibrational (and rotational) substates is equally accessible, subject only to conservation of energy and of angular-momentum res- traints.Recent approaches to these problems equate the onset of rapid intramolecular vibrational redistribution with the onset of dynamical but no quantitative correlations have been reported for specific examples.38 RELAXATION OF C6F,5+ EXCITED STATES Rett~chnik,~~ Tramer and Voltz,12c Parmenter 36 and Smalley 37 have reviewed results on VNRT in excited electronic states of large neutral molecules. Three different techniques which have given information on VNRT in neutral species 36 are (i) a comparison of level widths determined by high-resolution absorption spectroscopy and from the fluorescence decay constant; (ii) a study of the fluores- cence spectrum as a function of E,, in single vibrational-level excitation experiments, where the development of a background and the broadening of spectral features are considered as indicators of the onset of VNRT processes; (iii) studies based on the time-dependent development of background emission from the excited species.For single-ring aromatics without side chains, VNRT sets in at E,, z 2000 cm-1.36 For example, in the case of the S1 state of p-difluorobenzene, Covaleskie et al.38 have inferred that no significant VNRT occurs below E,, = 1600 cm-l, but that a high VNRT rate, ca. 10l1 s-', exists at E,, > 2400 cm-'; Halberstadt and Tramer 39 also showed that at E, = 2190 cm-I the VNRT rate is greater than 3 x lo9 s-l.In our work, evidence for vibrational redistribution is obtained by a new method, (inherently also based on time-dependent phenomena since radiative and non-radiative rates act as internal clocks and probes), in which the criterion of VNRT is deviation of experimentally determined radiationless rates from values calculated with a model where vibrational redistribution is excluded. This has led us to evaluate the onset of significant VNRT in the B 'Azu state of C6F6+ to occur at E, z 2500 cm-', which is of the same order of magnitude as the vibrational energy for onset of VNRT in the S1 The model illustrated in fig. 6 suggests that VNRT in C6F6+ could also be followed by time- and energy-dependent spectral analysis of the fluorescence.Emission from unrelaxed (insignificant VNRT) s levels should have a different spectral distribution than the emission from relaxed levels. Such observations require a different experi- mental approach than the T-PEFCO method, where the number of fluorescence photons produced is at present too small for studies of dispersed spectra. state Of C6H6 40 and P - C ~ H ~ F ~ . ~ ~ ' ~ ~ 6. EMISSION IN COINCIDENCE WITH Zf-STATE EXCITATION Emission resulting from excitation in the c-state region of C6F6+ was first observed by Ames et aL41 in PEFCO experiments and was later confirmed by Maier and Thom- men, who showed that the emission in coincidence with excitation of the c,O* level was in the 400-650 nm spectral region.4 We will now discuss what is the nature of the emission connected with the c state of C6F6+.Three possible origins of this emission _have previously been proposed :' (i) the Zf+8 radiative transition, (ii) rapid c-+B cascade process followed by emission &8, (iii) c,B vibronic mixing and subsequent emission to the 8 state. Our T-PEFCO results throw new light on this problem and we will therefore examine the three possible mechanisms in more detail than in our earlier discu~sion.~ (i) The c 2B2,,-2 'El, transition is electric dipole forbidden, but could be vibronically induced. For a Herzberg-Teller-induced transition, the c+8 vibronically induced origin band should occur in the 320 nm_region, but because of possible relative dis- placement and distortion of the c and X potential hypersurfaces, the c-8 Franck- Condon emission might occur to high vibrational levels of the 8 state in the same 400- 600 nm region as that of the normal Franck-Condon 'Azu-+Z '3, emission. The closest allowed state to the c 'BZu state is indeed the lower-lying B 'Azu state whose electronic origin is some 10000 cm-l away.However, the D,, species C6F6' is vibrationally deficient in that it possesses no vibration of appropriate symmetry capable of inducing Herzberg-Teller coupling between the c 2B2u and 2A2uG. DUJARDIN AND S. LEACH 39 The next-closest-allowed state is the P2E1, state, which lies ca. 20 000 cm-' above the Herzberg-Teller coupling to this state would involve excitation of odd quacta of e29 vibrations. The situation is similar to that of the well known A 'BZu- 2 lA,, forbidden transition in neutral where eZg vibrations are Herzkerg- Teller active and the perturbing allowed state is ca.16 500 cm-l above the A 'Bzu state. If we assume that a Herzberg-Teller effect of similar magnitude occurred for the c 2B2u-f 2Elg transition in C6Fs+, the oscillator strength would be fz The corresponding radiative decay rate would then be of the order k , ( C ) z 7 x lo5 s-l (calculated from the relation k, = QF/z = v2fl.5, using the appropriate unitsP6 where v is the mean transition frequency). Our experimentally determined value for c,Oo (table 1) is several times greater than this value. We therefore do not consider direct &f transition, albeit Herzberg-Teller-induced, as having a dominant contri- bution to the emission in coincidence with c state excitation. 2Bzu+B 2Azu optical transition would have an origin band in the 970 nm region, beyond the range of our photomultiplier, Furthermore, since this is a_forbidden dipole transition, whose frequency (ca.10 000 crn-l) is relatively low, a c-+B radiative cascade would consequently be a very slow process (analysis of Herzberg-Teller possibilities leads us to expect an even smaller k, rate than that derived for c-2 above), and we will therefore neglect this relaxation pathway. (iii) Let us now examine the possibility of a B-state contribution to the emission. Vibronic coupling between the c and 8 states should be a weak process since the promoting mode would have to have bl, symmetry (see the theoretical discussion in section 5.2), and this is not a proper vibrational mode for a DCh species.42 We note that non-radiative relaxation of the state can also occur by direct coupling, via e,, vibrations as promoting modes, to high vibrational levels of the 2 state.For the state to be involved in the radiative relaxation process occurring subsequent to c- state excitation the molecular (ion) states must have three components, namely c,R,v' and 2 , v " zero-order states. In an imperfect sense, but consistent with symmetry properties, Cne can consider that c-state intramolecular relaxation occurs by inter- action with B-state levels already contaminated by mixing with high vibrational levels of the 8 2Els ground state. This discussion leads us to propose that the radiative transitions in the 400-600 nrn spectral region resulting from (?-state excitation arise from molecular states derived from vibronic coupling between c, B and 3 states.(This refines our original con- cl~sion,~ later adopted by Maier and Th~mmen,~ that the emission in coincidence with the (? state involves a very rapid c+8 irreversible non-radiative transition, and amounts to a &z emission from high vibrational levels of the state to high vib- rational levels of the lower 2 state.) The kr and k,, results reported in table 1 add further support for our interpretation of the process of emission in coincidence with the c state. We discuss first the radiative rate k,. It is approximately constant for most vibrational levels of the state, but decreases for energies close to and at the origin of f-state excitation.The k, values in the c-state region cannot therefore be the radiative rates of pure high vibrational levels of the 8 state. We have found that the same situation holds for the analogous case af 1,3,5-C6F,H,+, where the relevant experimental evidence is even more extensive in that not only the 0' level can be measured for this ion in the state but also several higher vibronic le~eXs.~'-~ The lessened k, rate for the c-state region confirms that in this energy region the (ion) molecular states are mixed. The situ- ation is similar to that found in small molecules exhibiting the " lifetime-lengthening " Douglas effect, except that in the mixed states, here, in contrast to the reported small- molecule cases,44 it is the (low) oscillator strength of the optically excited state (c) 2B2u (ii) The40 RELAXATION OF C6F6 -i- EXCITED STATES that is apparently reinforced by dilution with a perturbing state (B).Consequently, again in contrast to the original Douglas-effect cases (NO2, SO2, CS2),44 it is a final " acceptor "-state component whose radiative rate kr is " reduced " by this dilution. Fig. 4 and 5 show that there is no significant discontinuity in the k,, value on passage to the c,Oo level, where the excess vibrational energy in the fi state is 10.400 cm-l. This can be taken as evidence that very efficient intramolecular vibrational redistri- bution takes place in the vibronic levels to which the c,OO state is coupled, similar to the explanation of the continuity in k,, found for P-naphthol and P-naphthylamine on crossing the threshold to the S3 To sum up, the spectral range of the emission in coincidence with the c state and the radiative and non-radiative rates associated with this emission decay indicate that high vibrational levels of the B state are involved along with the 2; state.The reduced apparent radiative rate k, implies that the molecular state, resulting from excitation at the 2;-state energy, is compounded of zero-order, emitting, B vibronic states plus some other non- (or weaker) emitting states that " dilute " the oscillator strength. From our previous discussion we propose that these contributing states include both the state and high vibrational levels of the 8 state, and that efficient vibrational redistribution occurs in the molecular ion states to which c,O0 is coupled.If a quartet state exists that is both isoenergetic and accessible to the 2B2u it might also provide a component for the molecular state. We thank the LURE synchrotron radiation facility for technical support concern- ing the experimental work reported here. APPENDIX 1 : CALCULATION OF FRANCK-CONDON FACTORS WITH NON-INTEGRAL QUANTUM NUMBERS We calculated the Franck-Condon factors with non-integral quantum numbers by generalising the Katriel method 24 available for the usual case of integral quantum numbers. Let us consider a harmonic vibrational mode which is both displaced and distorted. The frequencies in the initial and final electronic states are, respectively, o' and o" and x is the reduced displacement of the normal coordinate between the two electronic states (see Appendix 2).The Franck-Condon factor for this vibrational mode with n and I quanta, respectively, in the initial and final states is l(n\Z)12. For non-integral values of I, the overlap integral (nll) was calculated from the following expression: with B = o'/c"'. ( I - 2iz -- i3) are positive or zero. function of I for both integral and non-integral values of 1. The SU-M is over all positive integral values of il, iz and 4 such that (n - 2il - 4) and We remark that from this expression (nll) is a continuous APPENDIX 2 : DETERMINATION OF THE REDUCED DISPLACEMENT x OF A NORMAL COORDINATE Q BETWEEN TWO ELECTRONIC STATES OF A POLYATOMIC CATION Heilbronner et al.46 proposed a semi-quantitative method for estimating the displacement of a normal coordinate Q between the electronic state v' of a cation M+ and the groundG.DUJARDIN AND S . LEACH 41 state X of the corresponding neutral molecule M (fig. 7). We used the following similar but simpler method to estimate this displacement, and to obtain the reduced displacement x of the normal coordinate Q between the electronic states v/’ and v/” of the ion. The relative intensity Zzo/Iyo of two vibrational components z and y of the t,v’ electronic Fig. 7. Schematic potential-energy curves showing the displacement of normal coordinate Q of cation M+ in state v/’ relative to the corresponding neutral molecule M in its ground state. band in the photoelectron spectrum is given in the Born-Oppenheimer approximation by the ratio of Franck-Condon factors: 46 ]q’(u’)) and lp(O)), respectively, represent the M+(t,v’) vibrational state d and the M(X) ground state II = 0.For harmonic displaced and distorted vibrational modes, the expression (Al) with z = 1 and y = 0 gives: !!!! = ( -) 4o’ (g (Q’ - Q o ) 2 ) zoo (1 + PI2 Q’ and Qo are the Q values in the M+( v/’) and M(X) states at equilibrium. rn is the reduced mass of the vibrational mode. co’ and oo are the angular frequencies of this mode in the M+(y/’) state and in the neutral ground-state, respectively (/3 = o’/wo). The reduc_ed displacements x1 and x2 of the two totally symmetric modes of C6F6+ between the X and B states of the cation were derived as follows. For each vibrational mode we first determined 2/(rn/2h)JQJ - QO), using expression (A2), from the corresponding vibrational progression in the B band of the C6F6 He(1) photoelectron ~pectrum.~’ The sign of (Q’ - Qo) was obtained according to the following rule: (Q’ - Qo) is positive when o’ < oo and negative when or > coo.The same method was used20 deteLmine -v‘(rn/2h) (Q” - Qo) (the prime and double prime refer, respectively, to the B and X states of the cation). From these two quantities we then calculated the reduced displacement x = (rno’/2h) (Q’ - Q”)’. The values x1 and x2 (corresponding to the two totally symmetric vibrational modes) obtained by this method are given in table 3. As an illustration of the method we give in table 4 the results of calculations of x and AQ = Q” - Qo for ionization of H2 and 02. The experimental PES band intensities Zlo/loo were taken from Gardner and Samson 47 and the vibrational frequencies from Huber and H e r ~ b e r g .~ ~ The values of AQpes derived from photoelectron spectra are in good agreement with the change of internuclear distance AQop, determined by optical spectroscopy.42 RELAXATION OF C6F6' EXCITED STATES Table 4. Ionization of H2 and 02. Reduced displacement x and change in internuclear distance derived from photoelectron (AQpes) and optical (AQopt) spectral data. ionization process ~ 1 0 / ~ 0 0 B = woo X A QpeslA A Qop t I A ~~ ~ Hz(X 'Z;) + Hz (X'Z,') 2.06 0.53 1.21 0.26 0.31 0,+(X211g)+ 02(X 'Xi) 2.32 1.21 2.80 -0.1 1 -0.09 S. Leach, G. Dujardin and G. Taieb, J. Chim. Phys., 1980,77, 705. Findley and R. Huebner (D. Reidel, Dordrecht, 1983).G. Dujardin and S. Leach, J. Chem. Phys., in press. J. P. Maier and F. Thommen, Chem. Phys., 1981,57, 319. G. Dujardin, S. Leach, 0. Dutuit, T. Govers and P. M. Guyon, J. Chem. Phys., in press. C. 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