J . Phys. Chem. 1992, 96, 10608-10616
FEATURE ARTICLE Ionization, Charge Separation, Charge Recombination, and Electron Transfer in Large Systems E.W.Schlag* Institut fiir Physikalische und Theoretische Chemie. Technische Universitat Miinchen, W-8046 Garching, Germany
and R. D.Levine* The Fritz Haber Research Centerfor Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel, and Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90024- 1569 (Received: September 23, 1992; In Final Form: November 5. 1992)
Ionization of an isolated large molecule is discussed as a prototype of charge separation, in which the electron is removed in the absence of external environmental effects. The high density of electronic Rydberg states and the large distance Over which the electron is transported suggests a picture where the motion of the electron is taken to be incoherent with strong damping due to coupling to the rovibrational modes. The theory predicts a delay between activation and ionization, which has already been observed in both large molecules and in clusters. The relevant physical parameters include the size and constitution of the molecule or cluster, its initial temperature, and the wavelength of the photoexcitation. Particular attention is given to the competition of the delayed ionization with other processes such as dissociation. The incoherent limit discussed here, where the molecule acts as its own solvent, is qualitatively different from the well-known approach based on the work of Marcus and further developments where the motion of the electron is coherent. The implicationsof the incoherent model can be extended to a broad range of experiments involving charge separation or charge recombination and their competition.
1. Introduction
From molecular electronics’ to biophysics,2 electron-transfer processes) are of central interest. The recent literature on this topic is so rich and varied that for further background material we refer the reader to a review of reviews? The starting point for a theoretical analysis is the seminal work of Marcus5v6and the more recent related development^.^-'^ In these theories the solventI2has played a central role, and a variety of techniques,” including solvent friction have been invoked. However, for both activated and activationless processes, the relative importance of the intramolecular (i.e., those of the reactants) and of the solvent coordinates continues to be a subject of considerable interest, with particular reference to the role of temperature.I5J6 Recently, we have discuss6d” the ionization of large molecules. In that paper we coined the descriptive term that a large molecule can act as its own solvent. This enabled us to discuss the delay between the initial excitation of superexcited states capable of ionization and the actual departure of the electron. Of particular current interest‘8-2’is the production of very large biomolecular ions of molecular weight of many thousands. There are many ways of producing ions. However, for the purpose and illustration and clarity, we here confine our attention to photoinduced ionization. In this article we begin the discussion with the proposition that ionization of a large isolated molecule constitutes the simplest example of an electron-transfer process. This provides an ideal test case for which there are no environmental effects. We shall, however, argue that wen in the isolated molecule limit there are features that are quite reminiscent of a large body of important effects observed in electron transfer in the presence of an environment. Our purpose therefore is to extend the concept of electron transfer to a more general type of charge separation for which ionization is the archetype. Charge separation from Rydberg state atoms in condensed phasesz2provides a natural extension of our 0022-3654/92/2096-10608S03.00/0
point of view to systems where environmental effects are paramount. The photodetachment of an electron from an isolated large molecule (or a cluster2f27)is thus proposed as a first case for study. Systems of greater complexity such as charge separation of door charge recombination can then be exnor-acceptor pairsz2*28 amined within the same framework. It is our intention that such an extension will prove to be of particular interest whenever the physics of the problem admits of an incoherent description. Such will be the case when there are many coupled electronic states in which rovibronic effects are important. While we shall emphasize photoinduced prcwsses, the formalism we shall discuss is applicable to thermal processes and to the incorporation of the effects of an external environment. The traditional interpretation of electron-transfer processes follows upon the extensive foundations laid down by Marcus.s It is based on the coherent transfer between two displaced electronic states. (By coherent we mean that the electron is described throughout by a wave function so that definite phase relations exist between the two electronic states.) Depending on the strength of the coupling between these two states, an adiabatic or a nonadiabaticB approach may be more appropriate.’ The latter regime has been particularly useful in theories concerned with radiationless t r a ~ i t i o n s . * ~Typically, ~ however, the vibrational displacements of the two electronic states connected by a radiationless transition are small. In electron transfer these final distances can become much more substantial. The low orbital overlap35over such distances has often prompted the inclusion of one or more intermediate electronic states in order to facilitate the tran~fer.)~ This is particularly the case for electron transfer in biophysical systems2J7where the distances are large on the atomic scale. A widely accepted prediction of the coherent transfer between two displaced electronic states picture is the presence of an “inverted region” in which the electron-transferrate decream as the process becomes more exoergks This is attributed to the Q 1992 American Chemical Societv
Feature Article interplay between two parameters, A, the nuclear reorganization energy and the exoergicity AGO. The rate is an exponential and so is maximal at AGO = -A and function of (A decreases for more negative values of AGO. The inversion regime is well documented for charge recombination processes.38 There is, however, also an important class of systems in this model in which the distance d e p e n d e n ~ eof ~ ~A . and ~ ~ especially vibronic effectsls need to be invoked to explain the absence of such an inversion regime." Under such physical condition that it applies, the incoherent approach can be studied also for the very exoergic regime. The inversion regime is then found to be the exception rather than the rule. The reason is that in the exoergic regime, where the net reaction rate is high, the rate of the process is limited by diffusion control. (Of course, here the "diffusion" is intramolecular.) The competition between reaction and diffusion is discussed in detail in section 2c. In section 2 we discuss the physical view of delayed ionization. Somewhat more quantitative considerationsare given in section 3.
+
2. Delayed Ionization
The electronic excitations and, at higher energies, ionization of a small isolated molecule are well studied and understood. In particular, these processes are prompt, and once the ionization energy is exceeded, the electron departs on a femtosecond time scale.41 There is a vast and important literature on the coupling to rovibrational states" and on the special effects ("above threshold e ~ c i t a t i o n " ~made ~ ) available by the presence of strong fields. Direct and prompt photoionization is however the rule for most systems examined so far which by nature typically represent classes of smaller molecules. Recently, it has been observed that as molecules become larger the ionization is no longer prompt. Instead, there is a measurable delay prior to the appearance of a free electron, the most prominent being C0 A similar delay has been observed in the ionization of ~ l u s t e r s . ~Just ~ - ~below ~ the threshold for ionization, the highest lying Rydberg states even for large molecules are foundMto be quite long-lived, unlike those at lower energies.47 We have suggested17 that as molecules become larger the ionization will no longer be prompt and that the delay will increase with the size of the molecule but can be made shorter by increasing the photon energy and/or the initial thermal energy of the molecule. In the simplest version of our model, the molecule acts like its own solvent,17so that photoionization of a large molecule is similar to a photoinduced charge separation in soluti0n.~9~~ The photon does provide the energy necessary to initiate electron evolution, but the actual departure of the electron, and hence ionization, is delayed, the delay being longer for larger molecules. For molecules of extreme size, the delay is so long that other processes (bond cleavage, in particular) dissipate the energy and preclude ionization. This then predicts that extremely large molecules cannot be ionized by processes (photons at energiesjust above the first ionization potential, electron impact at conventional energies) which do not provide a large energy excess. This is familiar to practitioners of mass spectrometry, who have to resort to special techniques to observe ions in the kilodalton mass range. In our point of view, the molecule acts as a solvent for the motion of its (highly excited) electron. We must emphasize that by this we do not necessarily mean that the electron is localized. We only mean that the very high density of Rydberg states just below ionization and the exceedingly high density of vibrational states lead us to discuss an incoherent equation of motion for the probability density of the electron. In particular, the large size of the molecule means that the density of vibrational states is an exponentially increasing function of the excess energy in the nuclear motion. This is the primary driving force for detaining the electron. The energy necessary to escape the long-range attraction to the positive core that is left behind is being efficiently drained into vibrational excitation. The total energy is sufficient for ionization, but it is not all available to the electronic degree of freedom. Ultimately, if the molecule has not dissociated in the meantime, energy will flow back and the electron will depart.
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10609 Other things being equal, the higher the total energy, the faster will the energy be relocalized in the electronic degree of freedom. At a given energy, the more vibrational degrees of freedom are available to couple into the e x a s energy, the longer will ionization be delayed. Overall, we view the ionization as reflecting the competition between the "recombination" of the geminate electron-core pair and the charge separation at large distances, a strongly mass dependent process. This competition is also basic to all other electron-transfer processes. A similar picture will apply to photodetachment of large negative The one technical difference is the potential that attracts the electron to the core. For a negative ion, the "core" is neutral so that the long-range potential is due to polarization with an R4asymptote. The lifetime to ionization (reflecting, in our model, the incoherent migration of the electron enroute to its final departure) should be a sensitive probe of the structureof the molecular system. In a series of similar large molecules (e.g., peptides''), the prime variation would be with size but even could go beyond size effects. One pwible study is to compare the ionization, in the gas phase, of proteins in their native or denatured state. This would allow a comparison of two rather different geometries at the same We believe that another important example is the prompt emission of photoelectrons from metals. Small metallic clusters exhibitz6 substantial delays in photoionization, as we would expect from the model. Yet, the conduction band in the metal with the long coherence length of electron motion and the slow dissipation by the coupling to the vibration can result in prompt emission. The smaller metallic clusters do not seem to exhibit the coherent electronic motion characteristic of a bulk metal. This suggests that the size dependence of the delayed ionization can provide a very interesting diagnostic for the metallic character. Such a signature for a nonmetal to metal transition with increasing size of the cluster is different from other tests hitherto proposed. It further emphasizes that the delayed ionization is not simply a function of size but does depend in an essential way on the constitution of the system. Electron-hole recombinations in semic o n d u c t o r ~and ~ ~in amorphous solidsssare examples of proasses in the bulk and there are available examples of the importance of constitution and of structure. One experimental way to approach the prompt ionization would be the use of ultrashort pulses. The electron would then depart prior to the onset of coupling to the dissipative quasicontinuum of vibronic states. In this fashion one would ensure a direct ionization. The onset of dissipation is not only a function of the constitution but also of the energy. It would be of particular interest to examine the role of electron-attracting group, but even before that it would be necessary to demonstrate that fast pumping can compete with dissipation. A variety of estimates (including thermalization in waters6lifetimes of lower lying Rydberg states , ~ ~the ) and of vibrationally-excited-coreRydberg s t a t e ~ ~all~ put fast, prompt processes at the subpicosecond scale, possibly as low as 100 fss9-60 so that femtosecond pumping will be required. This effect has, since writing this paper, been confirmedtosfor 700-fs excitation of Cw An important parameter that governs the ultimate fate of the electron is the mean vibrational energy per 111ode.l~ It is this energy, which one can characterize as a vibrational temperature, which determines the rate of the thermal fluctuations that can provide the energy for the final ejection of the electron. This energy is made up by any initial thermal energy of the large molecule or of the luster^^.",^^ plus the energy transferred to the vibration during the fast initial thermalization of the highly excited electron. This latter contribution increases with the excitation wavelength. In both molecules and clusters, there is a competition between ionization and dissociation, both of which require localization of the internal energy which is otherwise randomly distributed. The electron is not however necessarily thermalized down to the same temperature as that of the vibration, unlike the case of thermionic emission from bulk metals.61 In the full development of the model it is not necessary to even assume that the electron reaches a (quasi)stationary distribution. All that we
10610 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
shall assume is that the motion of the electron is strongly damped. Even that assumption is necessary only in order to have a simple, analytically tractable theory. The required modifications if the electron is not strongly damped will be spelled out. The two competing fates of the Rydberg electron can also be used to discuss the spectroscopic observation of Rydberg prog r e s s i o n ~ . ~ ~ * ~The ~ - ~strong * * damping of the Rydberg electron implies that the line is not sharp but diffuse. When the line width begins to exceed the spacings in the progression, the spectral features merge and ultimately result in a featureless background. If the ionic core of the Rydberg state is vibrationally hot, ionization will become much more efficient and could compete with the intramolecular damping. One could then detect the Rydberg progression via the formation of ions. In a two-photon resonance pumping of the Rydberg state, one can establish the required conditions by pumping via a vibrationally excited intermediate state. The available evidence58is that Rydberg progressions not detectable by pumping via a cold intermediate state are observable if the intermediate state is hot. Neat molecular clusters are reported65to ionize with a cross section which exceeds the value for a single molecule. This is attributed to the weak vibronic coupling which allows mutual diffusion leading to fusion of the localized excitons.65 The weak coupling is also demonstrated by the ability of a “restricted” RRKM approach, where only the van der Waals modes participate in the energy exchange, to account for the lifetimes of the cluster ions as a function of Here, the construction of the cluster and not merely its size is central to delineating the relevant time scales. A very recent experiment6’ provides a direct probe of both the coupling of the high-lying Rydberg states to the nuclear niotion and of the energy dependence of the thermally assisted ionization. These were zero electron kinetic energy (ZEKE) experiments in which there was a delay between photoinitiation and the application of a drawout pulse which causes a field ionization of any surviving Rydberg states. The delay in the application of the field is used to measure the lifetime of the Rydberg states. We expect two opposing effects. On the one hand, the higher Rydberg states of a molecule become increasingly longer lived. In benzene typical spectroscopic Rydberg states have lifetimes 100 fs, whereas the last 20 cm-l before ionization display lifetimes 1 ps. On the other hand, our model predicts that the time to ionization becomes shorter as the energy of the molecule is increased. The rate of disappearance of Rydberg states (prior to the application of the drawout field) is the sum of these two rates. Hence, as we increase the energy, the lifetimes of the Rydberg states will increase until the ionization step takes over, leading to a reduction in lifetimes. This produces the otherwiseunexpected result of a maximum in a plot of overall time to ionization vs energy. In a classic set of experiments-mainly involving phenanthrene-this cusp of a maximum lifetime in the energy plot has been observed.67
-
N
3. Model The density of bound rovibrational states of a large molecule increases exponentially with energy. At the total energy sufficient to ionize an electron, the molecule can also dissociate (with a low but finite rate according to the RRKM theory). Hence, there is a continuum of rovibrational states coupled to the quasicontinuum of bound rovibrational states. A state of the molecule with an energy below or above the ionization limit where most of this energy is in one electronic degree of freedom is degenerate with a veritable continuum of states where more of the energy is in rovibrational motion. This density of states p is so high that even weak coupling V between electronic and nuclear degrees of freedom suffices to damp the motion of the electron. The criterion Vp > 1 made familiar by the theory of radiationless transition^^,^^ is well satisfied, nor is the coupling necessarily weak. The high density and hence low spacings of pure Rydberg states near the ionization limit mean that the breakdown of the adiabatic approximations will be severe. In physical terms, while the electron is indeed light, when its binding energy is low its orbital frequency is quite comparable to vibrational (or even rotational) frequencies.
Schlag and Levine Distribution of Vibrations in Angiotensine
0
1000
2006
Frequency ( cm
YO00
4000
-’ )
Figure 1. Distribution of vibrational frequencies of Angiotensine. Shown is a histogramic representation with bins of 25-cm-l width.
The effective coupling of the Rydberg electron to the rotational degrees of freedom is evident via the distribution of rotational states of the ions as determined by ZEKE ~pectroscopy.~~ It is however the vibrations that allow a rapid climbing down of the Rydberg ladder. In a large molecule there are many vibrations of low frequency (Figure l), so that there is a wide distribution of both principal and rotational quantum numbers of the electron that is rapidly established. The conclusion that in a large molecule a highly excited electronic state will be strongly damped can also be cast in a spectroscopiclanguage.30 Consider first the “true” eigenstates. Due to the degeneracy and the coupling, each eigenstate is a superposition of very many zero-order states corresponding to the multitude of different partitionings of the total energy among the electronic and nuclear degrees of freedom. The state which is optically a d has much or all of the energy in one electronic degree of freedom. It is not an eigenstate but is a nonstationary state which is a superposition of very many eigenstates. As the initial state evolves, the different eigenstates will dephase. The sheer number of eigenstates (Le., the high value of p) ensures that rapidly the initially accessed state becomes, for all practical purposes, an incoherent mixture. After a short time (in the sub-picosecond range, as discussed in the Introduction) the electron is executing an incoherent damped motion. It can continue to lose energy, leading ultimately to the dissociation of the molecule into neutral fragments, or it can acquire rovibrational energy and, ultimately, escape. It is this diffusive motion that we wish to discuss further. Before that we examine the potential for the motion of the electron. a. The Potential. The couplidg of the slow-moving excited electron to the molecule will modify the Coulomb attraction expected between an electron and a point positive core. Almost all realistic mechanisms lead to a screening of the l/r potential, i.e., to the form69 V(r) = -(e2/tr) exp(-rr) (1) Here t is an effective static dielectric constant and K-I is the screening length. The different screening mechanisms provide for different estimates of K . It is also possible to reexpress the screening in terms of a dynamic dielectric constant c ( q ) , i.e., a function of the momentum q of the electron. Then V(q) = V,(q)/r(q)where V ( ) is the Coulomb potential in the momentum representation (4;ep/q2). The screened Coulomb potential, eq 1, corresponds to the general expectation69that for low q ((4)= 1 + K 2 / q 2 . Computationsof K for nearly free electrons, including the concept of pseudopotential, are discussed in refs 69-7 1. An important point is that the concept must not be used once K-I becomes of the order of the size of the system. Purely electronic effects give rise to screening which is independent of the vibrational energy of the molecule. Two models which explicitly incorporate this point are the linear coupling of the electron to the motion of a harmonic lattice (the polaron mode172-74)and the Debye-Hiickel screening the or^.^^^^^ The
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10611
Feature Article derivation of the screening by Mayer and M a ~ e shows r ~ ~ explicitly how the “bare” Coulomb potential becomes screened by the presence of other charges in the system. Common to all derivations of the DebyeHOckel screening potential is the introduction of the temperature T via the Boltzmann factor exp(-U(r)/kT) as the probability density for the relative separation r in the presence I of the potential V(r). This dependence ties directly to the T dependence of K~ K~
= ( 4 r e 2 / c k r ) l ( p ( r )-
dr
(2)
where the integration is over the volume of the system and p(r) is the local charge density (whose average value is p ) in units of the electronic charge e. The forefactor in (2) can be rewritten in terms of the O n ~ a g e rradius ~ ~ r, r, = e2/ckT = 1.67 X 105/cT
A
(3)
which is the length at which the Coulomb energy between two charges equals kT. The numerical value quoted in eq 3 is to emphasize that r, is quite large on the molecular scale. Unless the variance of the charge density is low, the screening length I ’ will thus be high. In practice, we shall find it sufficient to use the Kr < 1 limit of the screened potential V(r) = -($/cr) (e2/e)K (1’)
mass Figure 2. Scaling of the yield of one-photon ionization of amino acids and peptides at hv = 10.5 eV vs molecular weight (normalized to one for tryptophan, molecular weight 204). Dots: experimental.” Curve: fit to exp(-rtr,) = exp(-(M/Mo)’/*). Adapted from ref 17.
+
1
It is important to note that the screening of the electron-core interaction by the lattice vibrations7)gives rise to ‘0
Here, as usual, e, is the high-frequency dielectric constant. The temperature dependence of K computed using the electron-vibration coupling will be discussed in detail e l s e ~ h e r e .Here ~ ~ we I limit can be obtained for an electron just mention that the K~ a T interacting with a thermal distribution of rovibrational modes. b. The Ionization Length. The simplest approximation for the dynamics of the highly excited electron is that it is very effectively dissipated by transitions into lower lying electronic states. Most photoexcited molecules will thus eventually dissociate into neutral fragments as a result of the conversion of electronic into vibrational energy. For large molecules, the density of vibrational levels is so overwhelmingly larger than the density of Rydberg states (apart from just below the ionization threshold) that this provides a reasonable first approximation. In section 3c, this approximation will be avoided. The motion of the electron is thus diffusive, under the potential V(r),the rate of dissipation depends on the distance of the electron from the core. We mimic this dependence by the simplest assumption that at a distance r = u the electron is removed at unit efficiency. One can do better and account for the distance dependence of the di~sipation?~ but in the limit where dissipation is assumed to be the dominant process this refinement is not warranted. There are other limits, particularly relevant to electron transfer, where the distance dependenece of the %apture” is important. The physical model we have described allows us to take over the well-developed mathematical theory of diffusion-controlled reactions in We emphasize that it is the mathematical derivation rather than the physical context that we take over. In section 3c we discuss a new and more general approachE7where the result of this section appears as a special case. The simple model is then that the electron, once damped, has one of two possible fates. Either it ionizes by diffusing to very far away from the core or the diffusive motion brings it to r = u where it is captured with a high efficiency at a rate k,. The fraction of electrons that ionize is then given Here lo is the reaction radius given byE4,s6,E7 lo-’= l m r - 2exp(V(r)/kT) dr
(5)
\I
twophoton
L 500
lo00 1500 mass
2000
2500
Figure 3. Scaling of the yield of two-photon ionization of amino acids and peptides at a total photon energy of 10.5 eV vs molecular weight. Other details as in Figure 2.
and D is the diffusion constant of the electron. For the special case of a weened, attractive Coulomb potential, eql ( 1 ”’) V(r) = -(r,/Or) exp(-Kf) one obtains, in the typical limit Kr
u F= fc-’(l + Kf,) The result lo = lo(K=O)(l - Kf,),
Kfc
