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Fragmentation dynamics of Fe(CO)5 upon femtosecond excitation: a time-dependent statistical description
4 September 1998
Chemical Physics Letters 293 Ž1998. 485–490
Fragmentation dynamics of Fež CO/ 5 upon femtosecond excitation: a time-dependent statistical description O. Rubner, V. Engel Institut fur Am Hubland, 97074 Wurzburg, Germany ¨ Physikalische Chemie, UniÕersitat ¨ Wurzburg, ¨ ¨ Received 1 June 1998
Abstract We present a time-dependent statistical model for the multiple CO abstraction from iron pentacarbonyl which is initiated by femtosecond excitation. The theory is based on a classical RRKM-model. The short pulse excitation is treated within time-dependent perturbation theory and the fragmentation probability is calculated via the energy dependent density of states of the parent and fragment molecules. We consider sequential and concerted CO loss separately. Application to recent experiments ŽIhee et al., Chem. Phys. Lett. 281 Ž1997. 10. yields good agreement with the experimental results and points towards a sequential dissociation process. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The study of metal carbonyls is central to the understanding of basic photochemical processes in organometallic chemistry w1,2x. Whereas until recently the spectrocopic measurements were performed using cw-laser sources, newer studies have been performed on the femtosecond time-scale on gas phase systems w3–5x. The powerful methods used in the field of Femtochemistry w6–9x make it possible to attack dynamical questions which cannot be settled by other means. In particluar the photochemistry of FeŽCO.5 has been investigated using pumprprobe photoionization measurements w10,11x. The application of ultrafast electron diffraction ŽUED. technique as introduced by Zewail and co-workers w12,13x to iron pentacarbonyl gave insight into the details of the dissociation process and the structures of the fragments w14x. From a theoretical point of view the situation is rather unsatisfactory. Although improved quantum
chemical calculations on the FeŽCO.5 molecule have been reported w15–17x we are still far from a detailed understanding of the structure of excited electronic states. Furthermore, the absorption of several photons opens many fragmentation channels and without a knowledge of the potential energy surfaces and their non-adiabtic coupling elements the underlying dynamical processes cannot be described correctly. Concerning the fragmentation dynamics of FeŽCO.5 there has been an ongoing discussion if, after absorption of sufficient photon energy, the decarbonylation takes place sequentially or concerted. One of the major problems in the effort to answer this question is the timescale on which the dissociation process takes place, the latter beeing finished in less than a picosecond w10,11x. Even though most of the experiments have been carried out on a much longer timescale some evidence can be found concerning the time-evolution of the process. According to most of these studies a sequential loss is in better agreement with experimental data w18–20x though
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 8 8 - X
486
O. Rubner, V. Engel r Chemical Physics Letters 293 (1998) 485–490
some recent experiments are in favor of a partially concerted scheme w10x. Here we set up a statistical model to achieve a basic understanding of the fragmentation dynamics. Former approaches to the iron carbonyl photofragmentation have used statistical models w19,21x to describe cw-experiments and in particular the resulting fragment distributions. Here we present an extended model including a femtosecond excitation process thus providing a general and simple approach to complex dynamical processes initiated by short-pulse excitation. In Section 2 we briefly introduce the model. An application to the fragmentation process induced by a femtosecond pulse of 310 nm is presented in Section 3. The final section contains a short summary and outlook.
values were taken from Refs. w10,16,22,23x and the threshold energies for the production of FeŽCO. n were slightly modified taking experimental uncertainties into account. Femtosecond excitation prepares the parent molecule in an excited electronic state. The interaction of the parent molecule with an ultrashort laser pulse results in an energy Ž E5 . and time Ž t . dependent population. In our case the initial excitation step corresponds to a two-photon absorption. The population is calculated within second-order perturbation theory. Denoting the intermediate states of energy e m as < sm : the projection of the second-order state on the final state < E5 : of energy E5 is given by
c Ž E5 ,t . s ² E5 < c Ž2. Ž t . : ; Ý a Ž E5 ,m . b Ž m,i . m
= 2. Theory First we will discuss a model for a sequential fragmentation process of FeŽCO.5 initiated by femtosecond excitation. In this case a fragment channel FeŽCO. ny 1 q CO is populated via decay of the parent molecule FeŽCO. n . Multiphoton absorption by the fragments and ionization processes are not treated here. Fig. 1 shows the energy diagram of FeŽCO. n with the energies employed in our calculations. The
t
t2
Hy`d t Hy`rmdt 2
1
eyi E 5 Ž tyt 2 .
=W Ž t 2 . eyi e m Ž t 2yt 1 . W Ž t 1 . eyi E i t 1 ,
Ž 1.
where aŽ E5 ,m.,bŽ m,i . are transition dipole matrix elements involving the initial Ž< i :., intermediate Ž< sm :. and final Ž< E5 :. states. All these numbers were set to unity in our numerical calculation. The molecule-field interaction is given by W Ž t . s yf Ž t . eyi v t .
Ž 2.
f Ž t . describes the ŽGaussian. envelope of the laser pulse with peak frequency v . We note that in the above model the energy distribution of final states is small and centered around the energy which equals 2 v. As always when classical and quantum descriptions are mixed one is faced with inconsistencies. Therefore we use the population ² c Ž E5 ,t .< c Ž E5 ,t .: to enter into the statistical description of the dissociation processes. The population g 5 of FeŽCO.5 in general depends not only on the energy E5 but also on the energy E4 of the fragment FeŽCO.4 . We calculated g 5 at time t q D t from the population at time t as g 5 Ž E5 , E4 ,t q D t . s ² c Ž E5 ,t . < c Ž E5 ,t . : yg 5 Ž E5 , E4 ,t . Fig. 1. Energy diagram of FeŽCO.5 and its fragments. The energy which corresponds to the absorption of two 310 nm photon is indicated as a horizontal arrow.
=P4 Ž E5 , E4 . eyk Ž E5 , E4 .D t . Ž 3 . Here k Ž E5 , E4 . is the energy dependent rate constant
O. Rubner, V. Engel r Chemical Physics Letters 293 (1998) 485–490
487
for the unimolecular decay, as specified below. P4Ž E5 , E4 . is the probability for dissociation from an initial state with energy E5 into a fragment state of FeŽCO.4 with energy E4 . This probability is assumed to be proportional to the product of the density of states of FeŽCO.4 Ž r4 Ž E4 .. and CO Ž r CO Ž E5 y E4 ..:
CO fragment. Accordingly the energy dependent probability is
P4 Ž E5 , E4 . s r4 Ž E4 . d E4 r CO Ž E5 y E4 .
The population of FeŽCO. 2 in the present case is
=d Ž E5 y E4 . ,
P˜2 Ž E5 , E2 . s r 2 Ž E2 . d E2 r CO
g 2 Ž E5 , E2 ,t q D t .
where proper normalization is assumed. In the next step the population of FeŽCO.4 is calculated as
s g 2 Ž E5 , E2 ,t .
yk Ž E 5 , E 4 .D t
q d E5 g 5 Ž E5 , E4 ,t . P4 Ž E5 , E4 . e
yg 4 Ž E4 , E3 ,t . P3 Ž E4 , E3 . eyk Ž E4 , E 3 .D t .
Ž 5.
Here the quantities are defined analogous as in the first dissociation step. The second term in the above equation belongs to an increase of the population due to decay of FeŽCO.5 and the second term represents the dissociation process into FeŽCO. 3 q CO. The subsequent fragmentation steps are then treated in a similar manner. The energy dependent fragment distributions are defined as g n Ž En ,t . s d Eny1 g n Ž En , Eny1 ,t .
H
Ž 6.
and the total population in the different channels is given by g n Ž t . s d En g n Ž En ,t . .
H
/
Ž 7.
The above approach clearly assumes a sequential fragmentation process. In contrast we may assume a simultaneous dissociation of more than one CO group. Let us outline the case where initially three CO groups are dissociated leaving FeŽCO. 2 which then dissociates sequentially. Because of symmetry, the three carbonyls which are built first carry away the same amount of energy and have the same distribution of final ro-vibrational and translational states. This restricts the function r CO since now only a third of the energy E5 y E2 is available for each
Ž 8.
q d E5 g 5 Ž E5 , E2 ,t . P˜2 Ž E5 , E2 . eyk Ž E 5 , E 2 .D t
H
yg 2 Ž E2 , E1 ,t . P1 Ž E2 , E1 . eyk Ž E 2 , E1 .D t .
s g 4 Ž E4 , E3 ,t .
H
3
d Ž E5 y E2 . .
Ž 4.
g 4 Ž E4 , E3 ,t q D t .
ž
E5 y E2
Ž 9.
The subsequent steps are then treated sequentially as above. The densities of states of the molecules for a given amount of rovibrational energy EÕ r distributed over s vibrational and r rotational degrees of freedom are calculated using the classical Whitten– Rabinovitch-formula w24–26x. Finally, the rate constants k Ž En , Em . which enter in the above equations have to be approximated. Therefore we employ a simple model where we first compute the velocity Õn m of a leaving carbonyl group in the dissociation process FeŽCO. n ™ FeŽCO. m q Ž n y m. CO. Assuming that fragmentation occurs after CO passed a critical distance L the time it needs to reach this length can be calculated. The rate constant then simply is its reciprocal: Õn m k Ž En , Em . s . Ž 10 . L Taking the weighted average over all CO states respective to their occupation gives the average dissociation constant for the cleavage of the Fe–CO bond. For the somewhat arbitrary value of L we take ˚ Physically this means that after an elongation of 2 A. the bond that corresponds to a Fe–CO distance of around twice the equilibrium distance the bonding should be weak enough to admit bond-rupture in most cases. We note that a change of the critical distance L within reasonable limits does not result in different asymptotic fragment distributions, rather it results in a re-scaling of the time axis. Within the simple model described above we now may predict the possible outcome of a femtosecond
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excitation experiment and in more detail answer the question how the CO groups are built as a function of time. This will be illustrated with a numerical example in the next section.
3. Application In a recent experiment performed in the Zewail group, FeŽCO.5 was excited with a 300 fs pulse of 310 nm w14x. The fragments then were probed by diffraction of an electron beam. We applied our model to the above experiment in order to study details of the fragmentation dynamics and decide whether the fragment distribution is in accordance to either sequential or concerted dissociation. Fig. 2 shows the time-dependence of the fragment distributions assuming a sequential dissociation. The population of the parent molecule belongs to the one prepared by femtosecond excitation Žtwo photons of 310 nm. from the ground state. It can be taken from the figure that at intermediate times all FeŽCO. n Ž n s 1– 5. fragments exist. However, at around two picoseconds, the populations settle to constant values. As is found in the experiments, only three fragments containing iron are present: FeŽCO. 2 Ž1%., FeŽCO. Ž4%. and Fe Ž1%.. The fragment distribution is in agreement with the findings of Ieeh et al. w14x: FeŽCO. 2 Ž2% " 1%., FeŽCO. Ž5% " 2%., Fe Ž5% " 1%.. We note that the branching ratio is quite sensitive to changes in the threshold energies and we did not try
Fig. 2. Sequential dissociation of FeŽCO.5 : time-evolution of the populations g nŽ t . in different fragment channels, as indicated.
Fig. 3. Same as Fig. 2 but for concerted dissociation of FeŽCO.5 : three CO ligands dissociate in the first fragmentation step which is followed by sequential dissociation of the remaining carbonyls.
an adjustment in order to achieve best agreement between theory and experiment. Fig. 3 shows the time-dependence of the fragment populations for a concerted dissociation of three CO molecules in the first step and sequential dissociation of the remaining two carbonyl groups. Here of course FeŽCO.4 and FeŽCO. 3 do not exist. As before we find a population of the other product channels at intermediate times. Nevertheless the asymptotic distribution is essentially different from the one obtained for sequential dissociation. Almost exclusively atomic iron and only a little fraction of FeCO is built in the fragmentation process. It is instructive to monitor the time-evolution of the energy dependent fragment distribution. Fig. 4 Ža. shows g 2 Ž E2 ,t . for two times, as indicated. This energy distribution of FeŽCO. 2 is compared for the sequential and concerted fragmentation mechanism, as discussed above. The calculation for sequential dissociation shows that with every loss of a CO the mean energy in the fragment is reduced by about 1 eV so that after three CO molecules are built the maximum of g 2 Ž E2 ,t . is found somewhere between 5 and 6 eV. This behaviour can be traced back to the form of the probability PnŽ Enq1 , En . ŽEq. Ž4.. which contains the product of the density of states of FeŽCO. n and CO. Whereas the former increases fast with En approaching Enq1 , the latter decreases smoother so that the product shows a maximum at
O. Rubner, V. Engel r Chemical Physics Letters 293 (1998) 485–490
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this way the energy distributions clearly predict the outcome of the multi-channel dissociation process for the different scenarios of sequential and concerted fragmentation. We calculated fragment branching ratios for the other concerted processes and found none to be in accord with the experimental findings. As a conclusion we find that the fragment distribution of Ihee et al. is in very good agreement with a sequential loss of CO after femtosecond excitation whereas a concerted loss of two or more carbonyl groups should yield a considerably different distribution.
4. Summary
Fig. 4. Energy distribution g 2 Ž E2 ,t . of sequential fragmentation from FeŽCO.5 fragmentation of three CO molecules Žb.. shown, as indicated.
FeŽCO. 2 produced by Ža. and by concerted Two selected times are
intermediate energies. When the distribution function reaches the threshold energy of the FeŽCO. fragment only a fraction of the molecules can dissociate. This results in the characteristic step in the distribution function at the threshold energy of ; 5 eV Žsee Fig. 4 Ža... In the case of a concerted dissociation ŽFig. 4 b. the loss in energy amounts to only about 0.8 eV in the first dissociation step. As a consequence the FeŽCO. fragmentation channel is open for all FeŽCO. 2 molecules and the de-population occurs. In
We have presented a simple statistical model to describe the fragmentation dynamics of metalcarbonyls upon femtosecond excitation. The calculation of fragment distributions needs the threshold energies for fragmentation and the densities of states for the parent and the various fragment molecules as input. The population in the excited state of the parent molecule is calculated via time-dependent perturbation theory and the unimolecular rate constants are estimated using simple kinematic arguments. Thus our model does not contain any adjustable parameter. Surprisingly good agreement is found with fragment distributions obtained in recent short-puls experiments on FeŽCO.5 . The calculation supports the view of a sequential dissociation of the CO groups from FeŽCO.5 for the case of a two photon excitation with a femtosecond pulse of 310 nm. We note that the simple approach presented here does not include any information on the actual interaction energy of the nuclei in the different fragmentation channels, i.e. on the potential energy surfaces and possible non-adiabatic couplings. Thus it is unlikely that the calculations are able to predict detailed data from sophisticated experiments. However, in cases where accurate calculations are not feasible our method might be able to provide valuable information on the laser excitation of complex systems and their subsequent fragmentation dynamics. The future will show where the limits of the model are and if possible improvements can extend the applica-
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bility to situations realized in various experiments. In particular we are interested in time-resolved femtosecond experiments and aim at an explanation of recent studies of the multiphoton ionization of FeŽCO.5 w27x.
Acknowledgements Financial support by the DFG within the SFB 347 ŽTP C-5. and by the Fonds der Chemischen Industrie are gratefully acknowledged. We thank L. Banares, T. Baumert, M. Bergt, B. Kiefer and G. Gerber for helpful discussions.
References w1x G.L. Geoffrey, M.S. Wrighton, Organometallic Photochemistry ŽAcademic Press, New York, 1979.. w2x Ch. Elschenbroich, A. Salzer, Organometallics ŽVCH, Weinheim, 1992.. w3x S.A. Angel, P.A. Hansen, E.J. Heilweil, J.C. Stephenson, in Ultrafast Phenomena VII eds. C. B Harris, E.P. Ippen, G.A. Mourou, A.H. Zewail, Springer Series in Chemical Physics, Volume 53 ŽSpringer, Berlin, 1990. p. 480. w4x S.K. Kim, S. Pedersen, A.H. Zewail, Chem. Phys. Lett. 233 Ž1995. 500. w5x T. Lian, W.E. Bromberg, M.C. Asplund, H. Yang, C.B. Harris, J. Phys. Chem. 100 Ž1996. 11994. w6x A.H. Zewail, Femtochemistry, Vols. 1, 2 ŽWorld Scientific, Singapore, 1994.. w7x J. Manz, L. Woste, eds., Femtosecond Chemistry ŽVCH, ¨ Weinheim, 1995.. w8x M. Chergui, ed., Femtochemistry ŽWorld Scientific, Singapore, 1996..
w9x V. Sundstrom, ¨ ed., Femtochemistry and Femtobiology ŽWorld Scientific, Singapore, 1997.. w10x L. Banares, T. Baumert, M. Bergt, B. Kiefer, G. Gerber, ˜ Chem. Phys. Lett. 267 Ž1997. 141. w11x L. Banares, T. Baumert, M. Bergt, B. Kiefer, G. Gerber, J. ˜ Chem. Phys. 108 Ž1998. 5799. w12x J.C. Williamson, J. Cao, H. Ihee, H. Frey, A.H. Zewail, Nature 386 Ž1997. 159. w13x M. Dantus, S.B. Kim, J.C. Williamson, A.H. Zewail, J. Phys. Chem. 98 Ž1994. 2782. w14x H. Ihee, J. Cao, A.H. Zewail, Chem. Phys. Lett. 281 Ž1997. 10. w15x L.A. Barnes, M. Rosi, C.W. Bauchschlicher, J. Chem. Phys. 94 Ž1991. 2031. w16x A. Marquez, C. Daniel, J.F. Sanz, J. Phys. Chem. 96 Ž1996. 121. w17x O. Rubner, V. Engel, C. Daniel, Chem. Phys. Lett. in preparation. w18x J.T. Yardley, B. Gitlin, G. Nathanson, A.M. Rosan, J. Chem. Phys. 74 Ž1981. 370. w19x B.K. Venkataraman, G. Bandukwalla, Z. Zhang, M. Vernon, J. Chem. Phys. 90 Ž1989. 5510. w20x K. Norwood, A. Ali, G.D. Flesch, C.Y. Ng, J.Am. Chem. Soc. 112 Ž1990. 7502. w21x I.M. Waller, J.W. Hepburn, J. Chem. Phys. 88 Ž1988. 6658. w22x P.C. Engelking, W.C. Lineberger, J.Am. Chem. Soc. 101 Ž1979. 5569. w23x K.E. Lewis, D.M. Golden, G.P. Smith, J.Am. Chem. Soc. 106 Ž1984. 3905. w24x G.Z. Whitten, B.S. Rabinovitch, J. Chem. Phys. 38 Ž1963. 2466. w25x G.Z. Whitten, B.S. Rabinovitch, J. Chem. Phys. 41 Ž1964. 1883. w26x K.A. Holbrook, M.J. Pilling, S.H. Robertson, Unimolecular Reactions, 2nd ed. ŽWiley, New York, 1996.. w27x O. Rubner, M. Bergt, B. Kiefer, G. Gerber, V. Engel, in preparation.
Chemical Physics Letters 293 Ž1998. 485–490
Fragmentation dynamics of Fež CO/ 5 upon femtosecond excitation: a time-dependent statistical description O. Rubner, V. Engel Institut fur Am Hubland, 97074 Wurzburg, Germany ¨ Physikalische Chemie, UniÕersitat ¨ Wurzburg, ¨ ¨ Received 1 June 1998
Abstract We present a time-dependent statistical model for the multiple CO abstraction from iron pentacarbonyl which is initiated by femtosecond excitation. The theory is based on a classical RRKM-model. The short pulse excitation is treated within time-dependent perturbation theory and the fragmentation probability is calculated via the energy dependent density of states of the parent and fragment molecules. We consider sequential and concerted CO loss separately. Application to recent experiments ŽIhee et al., Chem. Phys. Lett. 281 Ž1997. 10. yields good agreement with the experimental results and points towards a sequential dissociation process. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The study of metal carbonyls is central to the understanding of basic photochemical processes in organometallic chemistry w1,2x. Whereas until recently the spectrocopic measurements were performed using cw-laser sources, newer studies have been performed on the femtosecond time-scale on gas phase systems w3–5x. The powerful methods used in the field of Femtochemistry w6–9x make it possible to attack dynamical questions which cannot be settled by other means. In particluar the photochemistry of FeŽCO.5 has been investigated using pumprprobe photoionization measurements w10,11x. The application of ultrafast electron diffraction ŽUED. technique as introduced by Zewail and co-workers w12,13x to iron pentacarbonyl gave insight into the details of the dissociation process and the structures of the fragments w14x. From a theoretical point of view the situation is rather unsatisfactory. Although improved quantum
chemical calculations on the FeŽCO.5 molecule have been reported w15–17x we are still far from a detailed understanding of the structure of excited electronic states. Furthermore, the absorption of several photons opens many fragmentation channels and without a knowledge of the potential energy surfaces and their non-adiabtic coupling elements the underlying dynamical processes cannot be described correctly. Concerning the fragmentation dynamics of FeŽCO.5 there has been an ongoing discussion if, after absorption of sufficient photon energy, the decarbonylation takes place sequentially or concerted. One of the major problems in the effort to answer this question is the timescale on which the dissociation process takes place, the latter beeing finished in less than a picosecond w10,11x. Even though most of the experiments have been carried out on a much longer timescale some evidence can be found concerning the time-evolution of the process. According to most of these studies a sequential loss is in better agreement with experimental data w18–20x though
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 8 8 - X
486
O. Rubner, V. Engel r Chemical Physics Letters 293 (1998) 485–490
some recent experiments are in favor of a partially concerted scheme w10x. Here we set up a statistical model to achieve a basic understanding of the fragmentation dynamics. Former approaches to the iron carbonyl photofragmentation have used statistical models w19,21x to describe cw-experiments and in particular the resulting fragment distributions. Here we present an extended model including a femtosecond excitation process thus providing a general and simple approach to complex dynamical processes initiated by short-pulse excitation. In Section 2 we briefly introduce the model. An application to the fragmentation process induced by a femtosecond pulse of 310 nm is presented in Section 3. The final section contains a short summary and outlook.
values were taken from Refs. w10,16,22,23x and the threshold energies for the production of FeŽCO. n were slightly modified taking experimental uncertainties into account. Femtosecond excitation prepares the parent molecule in an excited electronic state. The interaction of the parent molecule with an ultrashort laser pulse results in an energy Ž E5 . and time Ž t . dependent population. In our case the initial excitation step corresponds to a two-photon absorption. The population is calculated within second-order perturbation theory. Denoting the intermediate states of energy e m as < sm : the projection of the second-order state on the final state < E5 : of energy E5 is given by
c Ž E5 ,t . s ² E5 < c Ž2. Ž t . : ; Ý a Ž E5 ,m . b Ž m,i . m
= 2. Theory First we will discuss a model for a sequential fragmentation process of FeŽCO.5 initiated by femtosecond excitation. In this case a fragment channel FeŽCO. ny 1 q CO is populated via decay of the parent molecule FeŽCO. n . Multiphoton absorption by the fragments and ionization processes are not treated here. Fig. 1 shows the energy diagram of FeŽCO. n with the energies employed in our calculations. The
t
t2
Hy`d t Hy`rmdt 2
1
eyi E 5 Ž tyt 2 .
=W Ž t 2 . eyi e m Ž t 2yt 1 . W Ž t 1 . eyi E i t 1 ,
Ž 1.
where aŽ E5 ,m.,bŽ m,i . are transition dipole matrix elements involving the initial Ž< i :., intermediate Ž< sm :. and final Ž< E5 :. states. All these numbers were set to unity in our numerical calculation. The molecule-field interaction is given by W Ž t . s yf Ž t . eyi v t .
Ž 2.
f Ž t . describes the ŽGaussian. envelope of the laser pulse with peak frequency v . We note that in the above model the energy distribution of final states is small and centered around the energy which equals 2 v. As always when classical and quantum descriptions are mixed one is faced with inconsistencies. Therefore we use the population ² c Ž E5 ,t .< c Ž E5 ,t .: to enter into the statistical description of the dissociation processes. The population g 5 of FeŽCO.5 in general depends not only on the energy E5 but also on the energy E4 of the fragment FeŽCO.4 . We calculated g 5 at time t q D t from the population at time t as g 5 Ž E5 , E4 ,t q D t . s ² c Ž E5 ,t . < c Ž E5 ,t . : yg 5 Ž E5 , E4 ,t . Fig. 1. Energy diagram of FeŽCO.5 and its fragments. The energy which corresponds to the absorption of two 310 nm photon is indicated as a horizontal arrow.
=P4 Ž E5 , E4 . eyk Ž E5 , E4 .D t . Ž 3 . Here k Ž E5 , E4 . is the energy dependent rate constant
O. Rubner, V. Engel r Chemical Physics Letters 293 (1998) 485–490
487
for the unimolecular decay, as specified below. P4Ž E5 , E4 . is the probability for dissociation from an initial state with energy E5 into a fragment state of FeŽCO.4 with energy E4 . This probability is assumed to be proportional to the product of the density of states of FeŽCO.4 Ž r4 Ž E4 .. and CO Ž r CO Ž E5 y E4 ..:
CO fragment. Accordingly the energy dependent probability is
P4 Ž E5 , E4 . s r4 Ž E4 . d E4 r CO Ž E5 y E4 .
The population of FeŽCO. 2 in the present case is
=d Ž E5 y E4 . ,
P˜2 Ž E5 , E2 . s r 2 Ž E2 . d E2 r CO
g 2 Ž E5 , E2 ,t q D t .
where proper normalization is assumed. In the next step the population of FeŽCO.4 is calculated as
s g 2 Ž E5 , E2 ,t .
yk Ž E 5 , E 4 .D t
q d E5 g 5 Ž E5 , E4 ,t . P4 Ž E5 , E4 . e
yg 4 Ž E4 , E3 ,t . P3 Ž E4 , E3 . eyk Ž E4 , E 3 .D t .
Ž 5.
Here the quantities are defined analogous as in the first dissociation step. The second term in the above equation belongs to an increase of the population due to decay of FeŽCO.5 and the second term represents the dissociation process into FeŽCO. 3 q CO. The subsequent fragmentation steps are then treated in a similar manner. The energy dependent fragment distributions are defined as g n Ž En ,t . s d Eny1 g n Ž En , Eny1 ,t .
H
Ž 6.
and the total population in the different channels is given by g n Ž t . s d En g n Ž En ,t . .
H
/
Ž 7.
The above approach clearly assumes a sequential fragmentation process. In contrast we may assume a simultaneous dissociation of more than one CO group. Let us outline the case where initially three CO groups are dissociated leaving FeŽCO. 2 which then dissociates sequentially. Because of symmetry, the three carbonyls which are built first carry away the same amount of energy and have the same distribution of final ro-vibrational and translational states. This restricts the function r CO since now only a third of the energy E5 y E2 is available for each
Ž 8.
q d E5 g 5 Ž E5 , E2 ,t . P˜2 Ž E5 , E2 . eyk Ž E 5 , E 2 .D t
H
yg 2 Ž E2 , E1 ,t . P1 Ž E2 , E1 . eyk Ž E 2 , E1 .D t .
s g 4 Ž E4 , E3 ,t .
H
3
d Ž E5 y E2 . .
Ž 4.
g 4 Ž E4 , E3 ,t q D t .
ž
E5 y E2
Ž 9.
The subsequent steps are then treated sequentially as above. The densities of states of the molecules for a given amount of rovibrational energy EÕ r distributed over s vibrational and r rotational degrees of freedom are calculated using the classical Whitten– Rabinovitch-formula w24–26x. Finally, the rate constants k Ž En , Em . which enter in the above equations have to be approximated. Therefore we employ a simple model where we first compute the velocity Õn m of a leaving carbonyl group in the dissociation process FeŽCO. n ™ FeŽCO. m q Ž n y m. CO. Assuming that fragmentation occurs after CO passed a critical distance L the time it needs to reach this length can be calculated. The rate constant then simply is its reciprocal: Õn m k Ž En , Em . s . Ž 10 . L Taking the weighted average over all CO states respective to their occupation gives the average dissociation constant for the cleavage of the Fe–CO bond. For the somewhat arbitrary value of L we take ˚ Physically this means that after an elongation of 2 A. the bond that corresponds to a Fe–CO distance of around twice the equilibrium distance the bonding should be weak enough to admit bond-rupture in most cases. We note that a change of the critical distance L within reasonable limits does not result in different asymptotic fragment distributions, rather it results in a re-scaling of the time axis. Within the simple model described above we now may predict the possible outcome of a femtosecond
488
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excitation experiment and in more detail answer the question how the CO groups are built as a function of time. This will be illustrated with a numerical example in the next section.
3. Application In a recent experiment performed in the Zewail group, FeŽCO.5 was excited with a 300 fs pulse of 310 nm w14x. The fragments then were probed by diffraction of an electron beam. We applied our model to the above experiment in order to study details of the fragmentation dynamics and decide whether the fragment distribution is in accordance to either sequential or concerted dissociation. Fig. 2 shows the time-dependence of the fragment distributions assuming a sequential dissociation. The population of the parent molecule belongs to the one prepared by femtosecond excitation Žtwo photons of 310 nm. from the ground state. It can be taken from the figure that at intermediate times all FeŽCO. n Ž n s 1– 5. fragments exist. However, at around two picoseconds, the populations settle to constant values. As is found in the experiments, only three fragments containing iron are present: FeŽCO. 2 Ž1%., FeŽCO. Ž4%. and Fe Ž1%.. The fragment distribution is in agreement with the findings of Ieeh et al. w14x: FeŽCO. 2 Ž2% " 1%., FeŽCO. Ž5% " 2%., Fe Ž5% " 1%.. We note that the branching ratio is quite sensitive to changes in the threshold energies and we did not try
Fig. 2. Sequential dissociation of FeŽCO.5 : time-evolution of the populations g nŽ t . in different fragment channels, as indicated.
Fig. 3. Same as Fig. 2 but for concerted dissociation of FeŽCO.5 : three CO ligands dissociate in the first fragmentation step which is followed by sequential dissociation of the remaining carbonyls.
an adjustment in order to achieve best agreement between theory and experiment. Fig. 3 shows the time-dependence of the fragment populations for a concerted dissociation of three CO molecules in the first step and sequential dissociation of the remaining two carbonyl groups. Here of course FeŽCO.4 and FeŽCO. 3 do not exist. As before we find a population of the other product channels at intermediate times. Nevertheless the asymptotic distribution is essentially different from the one obtained for sequential dissociation. Almost exclusively atomic iron and only a little fraction of FeCO is built in the fragmentation process. It is instructive to monitor the time-evolution of the energy dependent fragment distribution. Fig. 4 Ža. shows g 2 Ž E2 ,t . for two times, as indicated. This energy distribution of FeŽCO. 2 is compared for the sequential and concerted fragmentation mechanism, as discussed above. The calculation for sequential dissociation shows that with every loss of a CO the mean energy in the fragment is reduced by about 1 eV so that after three CO molecules are built the maximum of g 2 Ž E2 ,t . is found somewhere between 5 and 6 eV. This behaviour can be traced back to the form of the probability PnŽ Enq1 , En . ŽEq. Ž4.. which contains the product of the density of states of FeŽCO. n and CO. Whereas the former increases fast with En approaching Enq1 , the latter decreases smoother so that the product shows a maximum at
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this way the energy distributions clearly predict the outcome of the multi-channel dissociation process for the different scenarios of sequential and concerted fragmentation. We calculated fragment branching ratios for the other concerted processes and found none to be in accord with the experimental findings. As a conclusion we find that the fragment distribution of Ihee et al. is in very good agreement with a sequential loss of CO after femtosecond excitation whereas a concerted loss of two or more carbonyl groups should yield a considerably different distribution.
4. Summary
Fig. 4. Energy distribution g 2 Ž E2 ,t . of sequential fragmentation from FeŽCO.5 fragmentation of three CO molecules Žb.. shown, as indicated.
FeŽCO. 2 produced by Ža. and by concerted Two selected times are
intermediate energies. When the distribution function reaches the threshold energy of the FeŽCO. fragment only a fraction of the molecules can dissociate. This results in the characteristic step in the distribution function at the threshold energy of ; 5 eV Žsee Fig. 4 Ža... In the case of a concerted dissociation ŽFig. 4 b. the loss in energy amounts to only about 0.8 eV in the first dissociation step. As a consequence the FeŽCO. fragmentation channel is open for all FeŽCO. 2 molecules and the de-population occurs. In
We have presented a simple statistical model to describe the fragmentation dynamics of metalcarbonyls upon femtosecond excitation. The calculation of fragment distributions needs the threshold energies for fragmentation and the densities of states for the parent and the various fragment molecules as input. The population in the excited state of the parent molecule is calculated via time-dependent perturbation theory and the unimolecular rate constants are estimated using simple kinematic arguments. Thus our model does not contain any adjustable parameter. Surprisingly good agreement is found with fragment distributions obtained in recent short-puls experiments on FeŽCO.5 . The calculation supports the view of a sequential dissociation of the CO groups from FeŽCO.5 for the case of a two photon excitation with a femtosecond pulse of 310 nm. We note that the simple approach presented here does not include any information on the actual interaction energy of the nuclei in the different fragmentation channels, i.e. on the potential energy surfaces and possible non-adiabatic couplings. Thus it is unlikely that the calculations are able to predict detailed data from sophisticated experiments. However, in cases where accurate calculations are not feasible our method might be able to provide valuable information on the laser excitation of complex systems and their subsequent fragmentation dynamics. The future will show where the limits of the model are and if possible improvements can extend the applica-
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bility to situations realized in various experiments. In particular we are interested in time-resolved femtosecond experiments and aim at an explanation of recent studies of the multiphoton ionization of FeŽCO.5 w27x.
Acknowledgements Financial support by the DFG within the SFB 347 ŽTP C-5. and by the Fonds der Chemischen Industrie are gratefully acknowledged. We thank L. Banares, T. Baumert, M. Bergt, B. Kiefer and G. Gerber for helpful discussions.
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