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Comment on collision induced electronic relaxation
Volume 15, number 1
CHEMICAL PHYSICS LETTERS
COlMiMENT ON COLLISION
INDUCED
1.5
ELECTRONIC
July 1972
RELAXATION
William M. GELBART Deparrrnellt of Chemistry, UtriwrsiTyofCa~ifon?iu,
Berkelqv, Califinh
91720. US.4
and Danielle CAZES Laboratoire de Chinlie Thbon’que, Utiwrsite’ de Phs-Sud, RcLxiwd
in this note we comment on recent an alternative description
and sugest
steady
state
13 April
kineticannlyses
91-Orsa.v, Frculce
1972
of coLICon induced
intersystem
crossing
in gases
of the intia-and intermolecular dynamics.
1. Introduction In the last ten years there has been a lot of discussion in the literature regarding the nature of nonradiative electronic relaxation processes in polyatomic molecules [l] . Emphasis has centered on the “s@tistical” case in which the n-~olecule acts as its own heat bath in the radiationless conversion of electronic energy into vibrational degrees of freedom, In this limit the nonradiative behavior can be described in terms of a true intramolecular decay which competes with the usual radiative and photochemical chmnels. In a recent series of papers Jortner et al. [2-51 have considered the more problematic case in which the stringent conditions of the statistical limit are not satisfied and the spin-vibronic couplings do not give rise to a simple relaxation process in the isolated molecule. An extreme esample is that of diatomic systems in which spin-orbit and rotational-+lectronic interactions lead to static perturbations of excited states but not to new dynamical effects. It has been suggested that important cases intermediate between the above large and small molecule limits arise in: (i> aromatic hydrocarbons where, say, an excited singlet is perturbed by a nearly degenerate second triplet; and (ii) triatomics where the excited state is coupled through large Franck-Condon factors (arising from dramatic changes in nuclear geome-
tries) to n small number of vibronic levels beionging to a lower configuration. It was predicted that for molecules of this kind coltisions would be necessary to induce electronic relaxation between the coupled manifolds. Recent experiments on the pressure dependence of intersystem crossings have confirmed this behavior for several molecul% of type (i) and (ii): (i) Porter and co-workers [6] have measured the effect of added gas on the intensity of triplet-triplet absorption following flash photolysis of a series of aromatic hydrocarbons; (ii) similar experiments have been performed by Braun et al. [7] for the triatomic dindical methyl-ne. A possible mechanism for the observed pressure dependence of the nonradiative quantum yield has been discussed [6,9] * in terms of forward and backward intersystem crossing rates which are asserted [6,Y] to compete with fluorescence and/or collisional deactivation rates in establiskting the photostationary state: the overall molecular dynamics are analyzed by a conventional kinetic scheme involving first and second order rate constants. A correct description of the reacting molecule in the absence of collisions, however, precludes the possibility of a simpie internal relaxation process in the intermediate cases (i) and (ii). It is therefore not meaning* In ref. [8] the singbt-triplet mking is ascribed to ztn RXternal (collisional) heavy atom effect; see also ref. [Z 31.
Votume 15, nilmkr
1
ful to associate a time independent transition rate with these intramolecular mkings of singlet and triplet configurations, i.e., one cannot describe the change in excited state concentrations by the usual “master” equations involving microscopic transition probabilities. Correspondingly we feel that the steady state kinetic analyses do not allow for a proper interpretation of the relevant dynamics. In this note we suggest an alternative discussion which we believe provides a more correct description of coilisionaily induced nonradiative relaxation in molecules of types (i) and (ii). We shall consider intersystem crossing in methylene as a prototype for the general approach. The theoretical formalism which we use has been employed in other contexts, most recently by Jortner et al. to treat long-lived second singlets [4] and fluon?+ cence quenching [S] in type (i) intermediate molecules. We outline a simple model computation which includes bcth of the competing “mechankms” [8,9] considered in the literature and which relates the laboratory observables in collision induced chemical reluxation to well defmed molecular quantities. In this way we try to illustrate the conceptual failings of the conventional kinetic analyses of these reactions and to motivate the need for new theoretical caiculations and experimental data.
2. Method We write the total molecular hamiltonian in the form H= Ho t- V( Vis the spin-orbit interaction cperator) and restrict ourselves to two coupled electronic configurations, a singlet S and a lower triplet T - in the case of methylene S corresponds to the excited ‘Al state while T is the %, ground state *. Suppose that at time C= 0 the system is prepared in the lo\ve~t (V = 0) level of the singlet (this is a questionabie, but customary, assumption for the photolysis generation of methylene - more properly we shouid include in cur system th’: vibrationally excited ieveis OfS- see section 3, and also ref. [I I] for a recent review of “mechanism”). The pure spin vibronic states (H~+o~= enpn) are coarsely enough spaced in * A discussion of earlier electronic structure calculations aad m@lecuI,ustructure measurements is included jr! zef. [lo]. 38
15 July 1972
CHEMICAL PHYSICS LETTERS
this enera region so that the intramolecular coupling can tie approximately described by the one singlet level (gS, es) interacting with one (3.fold degenerate) triplet (@.(~=O,*l),e~). The initial state can then be written in terms OF the molecular eigenstates (N\k,, = E, ‘t,) as PP(t=O)f = Iq$ = 5
nT=l
+P,) .
(0
Allowing for radiation and collision effects, the time evolution of this compound state is most convenientIy described by 3 generalized Wigner-Weisskopf approximation [ 121
=25a:
exp
!I =
1
:-m .*-l
ir,)r] I@,,>,
(2)
where y,, is a sum of the radiative I?,, arid collisional A,, widrhs associated with !Vn (we neglect I-,, for methylene). In writing eq. (2) the damping matrix D, de-
fined by -(?i/i)d(ol(t),...,
a&N ldt = W,~~L..,~,(~)l 3
has been supposed to be diagonal in the (!Pk,} basis. The Nth molecular level width, i.e., the imaginary part can be interof DtV,3 4, = JQ;I~A, i-(1 -Ig12)AT, preted as a weighted average of the collision induced electronic quenching width As of the singlet and the vibrational deactivation width A, of the coupled triplet level. These simplifying ansatzes need to be investigated more rigorously and appropriately modified - the qproacll to the problem, however, remains the same regardless of these details. The probabilityPs(t) = I(qsPP(tj>i2 that the sysa rem Ml1 be in the singlet state at time f is
Now, writing 6 = cs - eT and v’ = (qSI Vfq+I) - aii other matrix elements of V=H,,vanishing for methylene in CzV symmetry 191 - the solution of the secular equation gives the following results for the weights of singlet in the molecular eigenstates (/v’/~ = 1~1~);
Volume 15, number 1
CMEXIIG%L PtiYSJCS
LETTERS
is July
1972
timates of the vibronic structure appropriate to the ground and low-lying electronic states of methyiene. = (*S/2) T [(S2/4) f z/vi2 ] =EI(2)-eT (3) can be rewritten BS
El (2) Eq.
p,(t)
=
c
n=1,2
I@;!4 exp(-M-”
A,, r) ;
l/2
.
3. DisWssion
(5)
here we have dropped the interference term since it is osciifatorywith period T= rifi[(6’/4)+21~1~] Uz < 10-l * set and can be assumed to give zero when averaged over experimental observation times. The rates Q-T z -( 1/P,$ dPJd t associated with the collision induced singlet-triplet reiaxatian are given below for two cases of interest:
kSdT =A-‘@,
i-A.-+ 9
S < Iul .
(7)
The overall rate PQ..~ (proportional to the concentration of added gas) defined here should not be confused with the ~tramolecular rate constants appearing in the steadyestate kinetic analyses of the pressure dependent methylene intersystem crossing. As indicated in section 1 we feel that these latter quantiries are not well-defined, e.g., they are neither calculable from known properties of the reacting system, nor accessible from independent measurements. k.++T on the other hand corresponds directly to an experimental observable, and has been expressed above in terns of the intramolecular quantities Y and 6 and the quenching cross sections A, and AT; E/A, and Pi/AT are the pure collision lifetimes which would be calculated for the deactivatjon by inert gas (Z%jof the v = 0 electronically excited singlet and the v f: 0 via brationally e,ucited triplet, respectively. As has in fact been estimated in a recent note by Chu snd Dahler 1131 where they take into account the coupling of the methylene spins and the electronic orbital momenta of X; their approximate results suggest that 4, is proportional to pressure and increases with the mass of the inert collision partners (external heavy atom effect). AT is in principle calculabfe from the ox;ensive body of theory on the vibrational relaxation of small molecules. v and 6, as witnessed by considerable disagreement in the literature, await more reliable es-
Ob-viously the model employed above negIects many of the complicating features which are present in the actual coilision induceh inrersystem crossing process. Also, because ofits means of generation and unusual chemicai reactivity, methyfene presents a particularly problematic case. In spite of and because of this faut the CH, radical has attracted considerable attention. in particular, the two-!evel system described in section 2 has been used as a common starting point in the recent discussions of this type of reaction: our intent has been to provide a new Yanguage“ for the relevant dynamics and to thereby correctly relate experimental observablzs to well defined molecuiar properties. If, for example, we take: Y = 10 cm-l and 6 = 100 cm-l - “corn~fomise”vahres [S-10,14] for the disputed properties of the coupled singlet and iripfet levels - then (6) can be used with the observed pressure and heavy atom dependence of /cyzT to ascertain refative valuesof AS and AT, e.g., a reliable estimate ofeither cross section would determine the other. Note that A,/?Z corresponds to a bimofecuIar (coilision.induced) intersystem crossing rate (showing an external heavy atom effect): the in~~~o~ec~iar spin orbit mixings are contained in the Ymatrix eiements. If we had inciuded thermally excited S IeveIs (psi) in the origina model, then we would simply have needed to solve a higher order secular equation for the moiecuiar eigenstates, and to add pure vibrational deactivation components (4sf;bib)J to the electronic quenching cross sections (A,,), etc. The trivial example presented in section :! was referred specifically to methylene because of the recent interest and controversy concerning this particular case. As emphasized earlier (see also refs. [Z-S]), however, the simple method involved - replacing the conventional kinetic schemes - is appropriate to many different examples of ‘“intermediate” type molecules. ln, say, the experiments of Porter et al. [6] the increase in nonradiative quantum yields of aromatic hydrocarbons with pressure of inert gas has been analyzed [t;] in terms of a modeI which is essentially the same as that suggested 191 for methylene: forward 34
Voiurne 15, number
1
(k:Sc) and backward stants are introduced
IS July 1972
CHEMICAL PHYSICS LETTERS
(kLsC) i~ziramoiecular rate conto describe the reversible mix-
ing of singlet and trip& configurations in the ‘isolated” molecules - these rates are then treated OR an equal footing with those corresponding to emission? collisionti deactivation and quenching, and photochemical decay channels. As argued above, however, :!cisc and !Gsc b are not meaningful quantities (as opposed fo. say, interaction energies and level spacings, wEi& are AVLQJSwell-defined and calculable) and therefore do not provide the most useful interpretation of the laboratory data on coflibion induced eiectronic relaxation. Instead, an approach of the sort outlined in section 2 would be required. The simple method discussed above should also be appropriate to pressure dependent isomerism (pktnsr sin&t-twisted triplet mixing) of substituted ethylenes, and to quenching of fluorescence and phosphorescence in triatornics like SO,, NO, t etc. We are presently considering these latter problems in a general program involving the role of collisions in efectrorlic relaxation in “intermediate” size molecules.
the Miller Institute for their generous financial SUPport at BerkeIey. References
111J.
Jortncr,
S.A. Rice and RX.
Photo&em.
7 (1969) 139;
Hochstiasser, &inn.
G.W. Robinson, J. Chem. Phys. 47 (X967) 1967, and rcferenats therein; K.F. Freed, Topics Current Chem., to be published. r2: K.F. Freed and J. Jortner, J. Chem. Phys. 50 (1969) 19x6. 131 Xi. Bison and 3. Jortner, J. Chem. Phys. 50 (1969)
3284. f4l P. Wannier, P.M. Rentzepis and J. Jortner, Chem. Phys. Lcttcrs LO(1971) !02,193. [51 A. Nitzan, J. Jortner, J. Kommandeur and E. Drent, Chem. Phys. Letters 9 (1971) 773.
t61 C.W. Ashpoia, S.J. Formosinho
and G. Porter, Proc.
Roy. Soc.A323 (1971) II. f7f W. Braun, AN. Bass and ?+I.Pilling, J. Chem. Phys. 52 (1970) 5131.
[81 T.W. E&r and R.W. Carr Jr., 3. Chem. Phys. 53 (1970) 2258.
Acknowledgements
[9J T.Y. Chang and H. Basch, Chem. Phys. Letters 5 (1970) 147. fIOl P.J. Hay, W.J. Hunt and WA. Goddard III, Chem. Phys. Letters 13 (1972) 30. IfI1 W-L. Hase, R.f. ~~~~ps and J.W. Simons, Chem. Phys.
It is Gpleasure to thank Lionel Salem for his hospitality at the Laboratoire de Chirnie Th6orique, where this work was started. W.M.G. acknowledges the National Science Foundation for a NATO Postdoctoral Feilowship allowing him to visit Orsay, and
1121 M. Bison, J. Jortner and Y. Dothan, Xfol. Phys. 17 (1969) 109. 1131 hf.Y. Chu and J.S. Dahler, Chem. Phys. Letters 8 (1971) 369. 1141 R.F.W. Bader and J.I. Generosa, Can. J. Chem. 43 (1965) 1631.
Letters
40.
I2 (1971) 161.
CHEMICAL PHYSICS LETTERS
COlMiMENT ON COLLISION
INDUCED
1.5
ELECTRONIC
July 1972
RELAXATION
William M. GELBART Deparrrnellt of Chemistry, UtriwrsiTyofCa~ifon?iu,
Berkelqv, Califinh
91720. US.4
and Danielle CAZES Laboratoire de Chinlie Thbon’que, Utiwrsite’ de Phs-Sud, RcLxiwd
in this note we comment on recent an alternative description
and sugest
steady
state
13 April
kineticannlyses
91-Orsa.v, Frculce
1972
of coLICon induced
intersystem
crossing
in gases
of the intia-and intermolecular dynamics.
1. Introduction In the last ten years there has been a lot of discussion in the literature regarding the nature of nonradiative electronic relaxation processes in polyatomic molecules [l] . Emphasis has centered on the “s@tistical” case in which the n-~olecule acts as its own heat bath in the radiationless conversion of electronic energy into vibrational degrees of freedom, In this limit the nonradiative behavior can be described in terms of a true intramolecular decay which competes with the usual radiative and photochemical chmnels. In a recent series of papers Jortner et al. [2-51 have considered the more problematic case in which the stringent conditions of the statistical limit are not satisfied and the spin-vibronic couplings do not give rise to a simple relaxation process in the isolated molecule. An extreme esample is that of diatomic systems in which spin-orbit and rotational-+lectronic interactions lead to static perturbations of excited states but not to new dynamical effects. It has been suggested that important cases intermediate between the above large and small molecule limits arise in: (i> aromatic hydrocarbons where, say, an excited singlet is perturbed by a nearly degenerate second triplet; and (ii) triatomics where the excited state is coupled through large Franck-Condon factors (arising from dramatic changes in nuclear geome-
tries) to n small number of vibronic levels beionging to a lower configuration. It was predicted that for molecules of this kind coltisions would be necessary to induce electronic relaxation between the coupled manifolds. Recent experiments on the pressure dependence of intersystem crossings have confirmed this behavior for several molecul% of type (i) and (ii): (i) Porter and co-workers [6] have measured the effect of added gas on the intensity of triplet-triplet absorption following flash photolysis of a series of aromatic hydrocarbons; (ii) similar experiments have been performed by Braun et al. [7] for the triatomic dindical methyl-ne. A possible mechanism for the observed pressure dependence of the nonradiative quantum yield has been discussed [6,9] * in terms of forward and backward intersystem crossing rates which are asserted [6,Y] to compete with fluorescence and/or collisional deactivation rates in establiskting the photostationary state: the overall molecular dynamics are analyzed by a conventional kinetic scheme involving first and second order rate constants. A correct description of the reacting molecule in the absence of collisions, however, precludes the possibility of a simpie internal relaxation process in the intermediate cases (i) and (ii). It is therefore not meaning* In ref. [8] the singbt-triplet mking is ascribed to ztn RXternal (collisional) heavy atom effect; see also ref. [Z 31.
Votume 15, nilmkr
1
ful to associate a time independent transition rate with these intramolecular mkings of singlet and triplet configurations, i.e., one cannot describe the change in excited state concentrations by the usual “master” equations involving microscopic transition probabilities. Correspondingly we feel that the steady state kinetic analyses do not allow for a proper interpretation of the relevant dynamics. In this note we suggest an alternative discussion which we believe provides a more correct description of coilisionaily induced nonradiative relaxation in molecules of types (i) and (ii). We shall consider intersystem crossing in methylene as a prototype for the general approach. The theoretical formalism which we use has been employed in other contexts, most recently by Jortner et al. to treat long-lived second singlets [4] and fluon?+ cence quenching [S] in type (i) intermediate molecules. We outline a simple model computation which includes bcth of the competing “mechankms” [8,9] considered in the literature and which relates the laboratory observables in collision induced chemical reluxation to well defmed molecular quantities. In this way we try to illustrate the conceptual failings of the conventional kinetic analyses of these reactions and to motivate the need for new theoretical caiculations and experimental data.
2. Method We write the total molecular hamiltonian in the form H= Ho t- V( Vis the spin-orbit interaction cperator) and restrict ourselves to two coupled electronic configurations, a singlet S and a lower triplet T - in the case of methylene S corresponds to the excited ‘Al state while T is the %, ground state *. Suppose that at time C= 0 the system is prepared in the lo\ve~t (V = 0) level of the singlet (this is a questionabie, but customary, assumption for the photolysis generation of methylene - more properly we shouid include in cur system th’: vibrationally excited ieveis OfS- see section 3, and also ref. [I I] for a recent review of “mechanism”). The pure spin vibronic states (H~+o~= enpn) are coarsely enough spaced in * A discussion of earlier electronic structure calculations aad m@lecuI,ustructure measurements is included jr! zef. [lo]. 38
15 July 1972
CHEMICAL PHYSICS LETTERS
this enera region so that the intramolecular coupling can tie approximately described by the one singlet level (gS, es) interacting with one (3.fold degenerate) triplet (@.(~=O,*l),e~). The initial state can then be written in terms OF the molecular eigenstates (N\k,, = E, ‘t,) as PP(t=O)f = Iq$ = 5
nT=l
+P,) .
(0
Allowing for radiation and collision effects, the time evolution of this compound state is most convenientIy described by 3 generalized Wigner-Weisskopf approximation [ 121
=25a:
exp
!I =
1
:-m .*-l
ir,)r] I@,,>,
(2)
where y,, is a sum of the radiative I?,, arid collisional A,, widrhs associated with !Vn (we neglect I-,, for methylene). In writing eq. (2) the damping matrix D, de-
fined by -(?i/i)d(ol(t),...,
a&N ldt = W,~~L..,~,(~)l 3
has been supposed to be diagonal in the (!Pk,} basis. The Nth molecular level width, i.e., the imaginary part can be interof DtV,3 4, = JQ;I~A, i-(1 -Ig12)AT, preted as a weighted average of the collision induced electronic quenching width As of the singlet and the vibrational deactivation width A, of the coupled triplet level. These simplifying ansatzes need to be investigated more rigorously and appropriately modified - the qproacll to the problem, however, remains the same regardless of these details. The probabilityPs(t) = I(qsPP(tj>i2 that the sysa rem Ml1 be in the singlet state at time f is
Now, writing 6 = cs - eT and v’ = (qSI Vfq+I) - aii other matrix elements of V=H,,vanishing for methylene in CzV symmetry 191 - the solution of the secular equation gives the following results for the weights of singlet in the molecular eigenstates (/v’/~ = 1~1~);
Volume 15, number 1
CMEXIIG%L PtiYSJCS
LETTERS
is July
1972
timates of the vibronic structure appropriate to the ground and low-lying electronic states of methyiene. = (*S/2) T [(S2/4) f z/vi2 ] =EI(2)-eT (3) can be rewritten BS
El (2) Eq.
p,(t)
=
c
n=1,2
I@;!4 exp(-M-”
A,, r) ;
l/2
.
3. DisWssion
(5)
here we have dropped the interference term since it is osciifatorywith period T= rifi[(6’/4)+21~1~] Uz < 10-l * set and can be assumed to give zero when averaged over experimental observation times. The rates Q-T z -( 1/P,$ dPJd t associated with the collision induced singlet-triplet reiaxatian are given below for two cases of interest:
kSdT =A-‘@,
i-A.-+ 9
S < Iul .
(7)
The overall rate PQ..~ (proportional to the concentration of added gas) defined here should not be confused with the ~tramolecular rate constants appearing in the steadyestate kinetic analyses of the pressure dependent methylene intersystem crossing. As indicated in section 1 we feel that these latter quantiries are not well-defined, e.g., they are neither calculable from known properties of the reacting system, nor accessible from independent measurements. k.++T on the other hand corresponds directly to an experimental observable, and has been expressed above in terns of the intramolecular quantities Y and 6 and the quenching cross sections A, and AT; E/A, and Pi/AT are the pure collision lifetimes which would be calculated for the deactivatjon by inert gas (Z%jof the v = 0 electronically excited singlet and the v f: 0 via brationally e,ucited triplet, respectively. As has in fact been estimated in a recent note by Chu snd Dahler 1131 where they take into account the coupling of the methylene spins and the electronic orbital momenta of X; their approximate results suggest that 4, is proportional to pressure and increases with the mass of the inert collision partners (external heavy atom effect). AT is in principle calculabfe from the ox;ensive body of theory on the vibrational relaxation of small molecules. v and 6, as witnessed by considerable disagreement in the literature, await more reliable es-
Ob-viously the model employed above negIects many of the complicating features which are present in the actual coilision induceh inrersystem crossing process. Also, because ofits means of generation and unusual chemicai reactivity, methyfene presents a particularly problematic case. In spite of and because of this faut the CH, radical has attracted considerable attention. in particular, the two-!evel system described in section 2 has been used as a common starting point in the recent discussions of this type of reaction: our intent has been to provide a new Yanguage“ for the relevant dynamics and to thereby correctly relate experimental observablzs to well defined molecuiar properties. If, for example, we take: Y = 10 cm-l and 6 = 100 cm-l - “corn~fomise”vahres [S-10,14] for the disputed properties of the coupled singlet and iripfet levels - then (6) can be used with the observed pressure and heavy atom dependence of /cyzT to ascertain refative valuesof AS and AT, e.g., a reliable estimate ofeither cross section would determine the other. Note that A,/?Z corresponds to a bimofecuIar (coilision.induced) intersystem crossing rate (showing an external heavy atom effect): the in~~~o~ec~iar spin orbit mixings are contained in the Ymatrix eiements. If we had inciuded thermally excited S IeveIs (psi) in the origina model, then we would simply have needed to solve a higher order secular equation for the moiecuiar eigenstates, and to add pure vibrational deactivation components (4sf;bib)J to the electronic quenching cross sections (A,,), etc. The trivial example presented in section :! was referred specifically to methylene because of the recent interest and controversy concerning this particular case. As emphasized earlier (see also refs. [Z-S]), however, the simple method involved - replacing the conventional kinetic schemes - is appropriate to many different examples of ‘“intermediate” type molecules. ln, say, the experiments of Porter et al. [6] the increase in nonradiative quantum yields of aromatic hydrocarbons with pressure of inert gas has been analyzed [t;] in terms of a modeI which is essentially the same as that suggested 191 for methylene: forward 34
Voiurne 15, number
1
(k:Sc) and backward stants are introduced
IS July 1972
CHEMICAL PHYSICS LETTERS
(kLsC) i~ziramoiecular rate conto describe the reversible mix-
ing of singlet and trip& configurations in the ‘isolated” molecules - these rates are then treated OR an equal footing with those corresponding to emission? collisionti deactivation and quenching, and photochemical decay channels. As argued above, however, :!cisc and !Gsc b are not meaningful quantities (as opposed fo. say, interaction energies and level spacings, wEi& are AVLQJSwell-defined and calculable) and therefore do not provide the most useful interpretation of the laboratory data on coflibion induced eiectronic relaxation. Instead, an approach of the sort outlined in section 2 would be required. The simple method discussed above should also be appropriate to pressure dependent isomerism (pktnsr sin&t-twisted triplet mixing) of substituted ethylenes, and to quenching of fluorescence and phosphorescence in triatornics like SO,, NO, t etc. We are presently considering these latter problems in a general program involving the role of collisions in efectrorlic relaxation in “intermediate” size molecules.
the Miller Institute for their generous financial SUPport at BerkeIey. References
111J.
Jortncr,
S.A. Rice and RX.
Photo&em.
7 (1969) 139;
Hochstiasser, &inn.
G.W. Robinson, J. Chem. Phys. 47 (X967) 1967, and rcferenats therein; K.F. Freed, Topics Current Chem., to be published. r2: K.F. Freed and J. Jortner, J. Chem. Phys. 50 (1969) 19x6. 131 Xi. Bison and 3. Jortner, J. Chem. Phys. 50 (1969)
3284. f4l P. Wannier, P.M. Rentzepis and J. Jortner, Chem. Phys. Lcttcrs LO(1971) !02,193. [51 A. Nitzan, J. Jortner, J. Kommandeur and E. Drent, Chem. Phys. Letters 9 (1971) 773.
t61 C.W. Ashpoia, S.J. Formosinho
and G. Porter, Proc.
Roy. Soc.A323 (1971) II. f7f W. Braun, AN. Bass and ?+I.Pilling, J. Chem. Phys. 52 (1970) 5131.
[81 T.W. E&r and R.W. Carr Jr., 3. Chem. Phys. 53 (1970) 2258.
Acknowledgements
[9J T.Y. Chang and H. Basch, Chem. Phys. Letters 5 (1970) 147. fIOl P.J. Hay, W.J. Hunt and WA. Goddard III, Chem. Phys. Letters 13 (1972) 30. IfI1 W-L. Hase, R.f. ~~~~ps and J.W. Simons, Chem. Phys.
It is Gpleasure to thank Lionel Salem for his hospitality at the Laboratoire de Chirnie Th6orique, where this work was started. W.M.G. acknowledges the National Science Foundation for a NATO Postdoctoral Feilowship allowing him to visit Orsay, and
1121 M. Bison, J. Jortner and Y. Dothan, Xfol. Phys. 17 (1969) 109. 1131 hf.Y. Chu and J.S. Dahler, Chem. Phys. Letters 8 (1971) 369. 1141 R.F.W. Bader and J.I. Generosa, Can. J. Chem. 43 (1965) 1631.
Letters
40.
I2 (1971) 161.