1660
J. Phys. Chem. 1995,99, 1660-1665
How Large Are High Molecular Rydberg States? A Direct Experimental Test C. E. Alt, W. G. Scherzer, H. L. Selzle, and E. W. Schlag Institut f i r Physikalische und Theoretische Chemie, Technischen Universitat Munchen, Lichtenbergstrasse 4, 0-85747 Garching, Germany
L. Ya. Baranov and R. D. Levine" The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel Received: August 9, 1994; In Final Form: September 23, 1994@
Results of a three-laser experiment that probes the yield of charge transfer between high Rydberg states of C6H6 and C6D6+ are discussed in light of theoretical considerations. The primary observation is the rather low fraction of the long-living Rydberg states that do undergo such an electron transfer under realistic experimental conditions. A careful determination of the yield as a function of the energy of the Rydberg state and a determination of the absolute number of c a s + ions present in the focal volume enable the absolute cross section for this transfer to be determined as a function of the principal quantum number n of the Rydberg state. The experimental result is (7 = (26 f 6)n4 (au)2 vs the value of (7 = 20n4 (au)2 derived on the basis of physical considerations. This cross section is larger than the geometric size of the Rydberg orbit, and the reasons for this are discussed in detail. It is also argued that for high Rydberg states charge transfer is the dominant mechanism for the binary destruction of high Rydberg states. Despite the enormous cross section, the yield of the transfer can be made to be negligible under realistic experimental conditions, at typical ion densities.
1. Introduction Bill Chupka has recently's2 reiterated the importance of external perturbation^^-'^ for the understanding of the time evolution of high molecular Rydberg states.15-17 ZEKE (zero electron kinetic energy) spectroscopy9J9~20 of molecular cations makes use of the extremely long lifetimes of Rydberg states of the parent which are within a few wavenumbers of the ion state energy. The longevity of these states can be caused by other nearby ions andor fields, either intentionally applied or stray. Spoiling field^'^.'^ or other arrangements (such as a magnetic b ~ t t l e ~can ~ s be ~ ~introduced ) to separate the promptly produced charged species. Even so, it is certainly clear that the perturbation of the Rydberg states by external charges in their vicinity is always possible, unless precautions are taken. The question discussed in this paper is whether the perturbation due to these nearly charges is, in practice, unavoidable or whether, despite their unusually large size, Rydberg states can be de facto isolated so that their truly intramolecular evolution can be explored. It all comes down to the question, quite how large are the large Rydberg states? The promptly produced, direct, ions present in the excitation region may stabilize those Rydberg states on the red side of the ZEKE spectrum and quench the highermost Rydberg states which would otherwise be on the blue edge.8J0J1 Zhang et a1.* and Merkt and Zare" have emphasized that the concentration of ions can vary in a rather wide range (-1-lo6 ions/ mm3) from one ZEKE experiment to another. While the detailed picture of the role of these ions is yet to be fully validated, it is already clear that the dependence on the concentration of the ions may be complicated. This article presents experimental evidence that it is possible and realistic to achieve experimental conditions under which the extent of transfer of the Rydberg electron to neighboring ions is small. Analysis of the experiment enables us to @Abstract published in Advance ACS Abstracts, January 15, 1995.
0022-36541992099- 1660$09.00/0
determine the dkpendence of the yield of the transfer on the concentration of the ions and on the excitation energy. For this purpose we provide a comparison between experimental data and a simple theoretical model for charge transfer from a high Rydberg state. The transfer cross section that is obtained by the analysis of the experimental results is rather large and will lead to the conclusion that, as seen by an ion, a high Rydberg state has a size which is roughly 4 times the area of the orbit. One can therefore, quite easily, ensure extensive perturbations of the Rydberg state by surrounding charges and that, at even low concentration of ions, the perturbation will not be linear in the concentration of ions. The present results suggest that the opposite limit, of isolated systems, can also be experimentally realized and is realized under typical conditions. We shall present both a direct experimental test and simple physical considerations in support of this conclusion. Beyond the perturbation by the promptly produced ions there are other external influences on the Rydberg states. One that is often noted' is an external dc field, whether of a stray nature or a field that is intentionally imposed. Ditto for a magnetic field.13 One can also consider the combined influence of the ions and the external fields. Here we do not address the possibility of synergetic effects between the different perturbations. Nor does this article discuss the very interesting question of what is the mechanism responsible for the rather long lifetimes of molecular high Rydberg states.16,21The demonstrations8-10 that the contribution of this channel can be enhanced by perturbations by the promptly produced ions do not appear to us sufficient to conclude that long-living states cannot be observed in the absence of such perturbations. In this article we only demonstrate that long lifetimes can be monitored under such conditions and that the role of the perturbations resulting in charge transfer can be quantitatively estimated and is small. In particular, we shall conclude that the perturbation by a low density of surrounding ions is, to leading order, equivalent to imposing an external weak dc field, of the order of 1 V/cm or
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99,No. 6,1995 1661
High Molecular Rydberg States less. One can therefore mimic or enhance the main role of the promptly produced ions by the intentional application of an external dc field. Elsewhere, we shall report on the role of such a field in determining the intensity of the long-living Rydberg states and on their lifetime. The three-laser experiment to be discussed uses two independent lasers to simultaneously excite C6H6 or C6D6 molecules to their respective first excited (SI, 6l) electronic state. The third laser then scans the energy region of the ionization potential. In this paper we specifically consider the region where the third laser pumps just below the ionization threshold of c& so that, as further discussed below, the third laser pumps Rydberg states of C6H6 and direct ions of C6D6. The question of interest is whether the Rydberg electron of C6H6 can transfer to the nearby C&6 ions, thereby creating "false" Rydberg states of C6D6. If produced, these false states will be detected in the ZEKE mass-resolved spectrum as C6D6 ions. A wider frequency spectrum of ZEKE states produced in the three-laser experiment has been reported before, for both the mixture of isotopomers22 and C a 6 on1y.l6 The isotopic shift of the ionization potential of benzene is such that those C6D6 molecules which absorb in this range are 100 cm-' or more above their ionization threshold and are promptly ionized. These, and any other, ions are removed by the dc spoiling field, and under the present experimental conditions this requires, on the average, about 3 p s . During this time, the high Rydberg electron can transfer from C6H6 to the C6D6 ions. The neutral Rydberg states of C6D6, produced by this charge transfer, will be detected as ZEKE ions by the delayed field ionization. Also detected as ZEKE ions will be any surviving Rydberg states of C6H6. The experimental finding22is that, for a 1:2 mixture of the two isotopomers, the ZEKE ions mass spectrum contains a very small fraction of C6D6 ions. The yield of the C6D6 ZEKE ions depends on the frequency of the laser used to pump the Rydberg states of C6H6. However, this dependence is clearly (Figure 1) different from that of the C6H6 ZEKE spectrum. The peaks of the two spectra do occur at the same frequency, and at the peak the C6D6 signal is about 2.5% O f the C6H6 ZEKE ion spectrum. The technical purpose of this paper is to examine this result in detail. Electron transfer from high Rydberg states has been observed, with large cross sections, for both atoms and molecule^.^^'^^^^^^ The point in the present experiment is that it examines the importance of this process for the conditions of interest in ZEKE spectroscopy. The quantitative estimate of the charge transfer cross section that we will present suggests that the size of a Rydberg state as seen by an ion is at most 8 times the maximal size32 of the orbit. This size scales with the fourth power of the principal quantum number n. The charge transfer will therefore primarily affect the highermost Rydberg states, and this is clearly seen in the detailed results to be presented. The experiment characterizes only the charge transfer process, and there may be other processes which are not detectable by observing the ZEKE signal of C6D6. The physical model, as discussed below, does however suggest how to estimate the size with respect to all changes of state. In particular, it suggests how experiments in the presence of an externally imposed dc field can mimic the role of ionic perturbers. Section 2 provides a brief outline of the experiment and the preliminary analysis of the data, leading to an effective transfer cross section u = (26 f 6)n4 (au)z. Section 3 provides a physical model which suggests that this value is reasonable and that the overall size of the C& Rydberg state, as seen by charge transfer to an ion, is about u = 20n4 (au)z. These are quite
1 0.8 0.6 0.4 0.2 0
-0.2''" ' " ' ' ' ' " ' 35880 35900 35920 35940 35960 "
"
"
frequency /cm-' Figure 1. ZEKE spectrum of ion yield vs the frequency of the third laser, in a three-laser experiment described in the text and in ref 22.
The frst two, low-intensity, fixed-frequency, lasers pump the SI intermediate state of c6& and c&. The third laser is scanned in the frequency range just below the ionization threshold of C a 6 and hence is above the ionization threshold of C a s . The initially produced c6D6 ions are removed by a spoiling field, and no such ions are detectable in the ZEKE spectrum if the laser which pumps the SIstate of C6H6 is turned off. Each point on this plot is a separate experiment at a given frequency of the third laser. Hence, such c& ions which are detected are due to charge transfer from a C& Rydberg state pumped at that 6 ion. The neutral c& molecule frequency to an initially produced w is then not carried away by the (delayed by 50 ns) spoiling field but is detected as an ion by the delayed (by 12 ps) field ionization. Note that the intensity of the c@6ions has been multiplied by a factor of 40. Otherwise it will hardly be seen. The factor S,used in the text to estimate the transfer cross section, is the ratio of the c6D6 to c,& ZEKE ions. Note that S is a rapidly increasing function of the excitation energy of the c6H6 Rydberg state. large cross sections and imply that, contrary to what one might think, a high radius Rydberg orbit, as seen by an ion, is not "empty" on the inside. On the other hand, these cross sections are not so large that one simply cannot design an experiment in which Rydberg states unperturbed by charge transfer to their surroundings are produced. The general question of an interaction of a Rydberg state with an ion is also discussed in section 3. The question is not a simple one because of the very high degeneracy of states of given principal quantum number n, a degeneracy which is lifted at long distances by (small) Starklike splitting in the field of the ion.
2. Experimental Section The three-laser scheme for the simultaneous pumping of the ions and the Rydberg states has been previouslyz2 described. In essence, two low (10 pJ/pulse)-intensity, frequency-doubled, synchronized dye lasers at fixed frequencies (of 38 607 and 38 787 cm-') are used to excite the S1 6l intermediate state of benzene and perdeuterated benzene. A third, counterpropagating, frequency-doubled, dye laser, at a power of 80 pJ/pulse, is optically delayed by 3 ns. The frequency of this third laser is scanned in the range of the ionization potential of benzene. We shall be particularly concerned with the frequency range of about 35 800-35 950 cm-', which is above the ionization threshold Of C6D6 but below that Of C6H6. In this range, the third laser accesses the Rydberg states of C a 6 which are just below the lowest ionization threshold, and it also ionizes the C6D6 molecules. (That ionization of c&, does take place can be checked by turning the spoiling field off.) Those Rydberg states that survive for many microseconds are detected downstream. No Rydberg states of c&6 are detectable when the laser which pumps the C6H6 intermediate state is switched off. The experimental arrangement of delayed field ionization with mass
Alt et al.
1662 J. Phys. Chem., Vol. 99, No. 6, 1995 0
Separation of Ions and Rydbergs Due to Spoiling Field
-2 4
E
c
-4
>
W
-6 /+
*'++ -8 -10 -0.5
-b
V.
Jet
Figure 2. Schematic illustration of the focal volume. The three lasers, LI, L2, and L3, define the axis of the cylinder. The initially produced Rydberg states of C6H6 and the C6D6 ions drift in opposite direction, perpendicular to the axis of the cylinder. E is the spoiling field which removes the promptly produced ions, and v is the velocity of the jet.
selection has also been previously d e ~ c r i b e d . ~The ~ , ~central ~.~~ point for the interpretation is that each frequency of the third laser corresponds to a separate experiment. In such an experiment, Rydberg states of C6H6 and ions of C6D6 are simultaneously produced. The ions are removed by a delayed (50 ns) spoiling field of 1 V/cm. Removal of the ions requires about 3 ps. The neutrals drift in a field of 1 V/cm for 12 ps and are then ionized by a high-voltage (450 V/cm) pulse. The ions are detected in a reflectron time-of-flight mass spectrometer. The mass spectrum vs frequency plot determined in this way is shown in Figure 1. The main points to notice are that the yield of C6D6 ions is low (in the figure it is shown multiplied by 40) and that to the red of the peak the frequency dependences of the yield of the C6H6 and C6D6 ions are different. The experimental setup is such that any C6D6 ions which are detected by the delayed ionization are a result of charge transfer from the Rydberg state of C6H6 pumped at a specific frequency. Examination of Figure 1 immediately verifies that the transfer cross section must be a steep function of the binding energy of the Rydberg state of C6H6: The number of detected C6D6 ions is far higher the nearer is the C6H6 Rydberg state to its ionization threshold. The experiment cannot verify that the Rydberg state of C6D6 as produced in the charge transfer has the same binding energy as that of the Rydberg state of C6H6. It does, however, ensure that any C6D6 ions observed at a particular frequency are due to a Rydberg state of C6H6 pumped at the same frequency. It is this which enables us to estimate the cross section as a function of the excitation energy of the Rydberg state. The measured ratio, S, of C6DdC6H6 ZEKE ion intensities is converted to an effective cross section o as follows. The principle of the computation is that an effective cross section is the area of the Rydberg state as seen by the ion which accepts the electron. It is illustrated by the geometrical considerations shown in Figure 2. The volume V of the focal region excited by the three lasers is that of a cylinder of a diameter D = 0.5 mm and a length L = 3.5 mm, whose axis is perpendicular to the direction of the molecular beam. The number, N , of direct ions produced, per shot, in this region is determined from the ratio of the area of the C6D6 ZEKE peak compared to a single ion event in the mass spectrum. The number, N , of direct ions was then found to be 5 times this value when the spoiling field is turned off. It is found to be N = 3 x lo3. The effective cross section for charge transfer is computed on the basis of
0
0.5
1
1.s
r/R Figure 3. Potential (solid line) seen by a Rydberg electron, initially located in the well on the left, due a single charge at a distance R away. (The Coulomb well of the isolated center on the left is plotted as a dotted line.) Shown as a dashed horizontal line is the energy of the electron which can just classically make it to the other side. Electrons at higher energies can cross the barrier. Electrons at lower energies have to tunnel through. Because the potential about the bamer is very shallow, such tunneling occurs with a very low probability. From this diagram one can readily determine the largest distance R at which an electron with an initial value of n can transfer. As discussed in the text, the shift in the unperturbed potential and the result that R is twice the barrier location combine to yield that R = 8n2.
the geometrical considerations shown in Figure 2. The mean free path A of the ion through the focal volum is A = 40/3x. (The ion moves in a direction perpendicular to the long axis of the cylinder.) The number density of Rydberg states is NRIV. (This number is unknown but will cancel out of the expression for the cross section.) The number, Nl, of ions produced by charge transfer is NI = ( N R / V ) N A ~where , 0 is the effective cross section. (Appendix A provides a more detailed derivation.) The fraction of Rydberg states detected after charge transfer, S, is thus S = N A d V , independent of the number, NR, of the Rydberg states (as long as that number is low so that only binary perturbations need to be considered). This leads to o = 1.08 (or -3.86 x lolls (au)2). x 10"s Since the fractional yield, S, of C6D6 ZEKE ions increases rapidly with the excitation frequency, the transfer cross section is much higher for the higher Rydberg states. It is reasonable to expect this cross section to scale up with the size of the Rydberg orbit, and in the next section we discuss a simple model which suggests that the scaling is with n4 or as E-2 where E is the binding energy of the initially accessed Rydberg state of C6H6. The model will be shown to agree with the data not only regarding the scaling but also in providing an absolute magnitude.
3. Model In this section we provide a simple physical picture, outlined in Figure 3, for the estimate of the charge transfer cross section. The picture is atomic-like and suggests that the cross section is numerically larger than the area of the Rydberg orbit but scales with n as the area does. It will be shown that there are two factors which contribute to the cross section being larger than the area of the orbit. The physical argument is quite simple, but in view of the large value of the cross section, we also seek to make the result plausible by providing a link to the quantum theory of resonant charge transfer.2s The experiment only measures the transfer of charge, but the high Rydberg state can be perturbed in other ways. One of these is the change in the orbital angular momentum of the Rydberg electron by an external field, as emphasized by Bill Chupka.' The physical picture will suggest that the role of ionic
J. Phys. Chem., Vol. 99, No. 6,1995 1663
High Molecular Rydberg States perturbers in inducing such processes can be mimicked by an experiment using an extemal dc field of a magnitude which is just sufficient to reach the classical threshold for ionization of the Rydberg state of interest. (Note thatz6the fraction of states which will ionize at threshold is significantly below unity.) We will report on such experiments elsewhere. The purpose of the model is to estimate the largest distance at which a high Rydberg electron will transfer, between two centers, of unit charge each. It will be simpler to discuss a modified question; that is, what is the lowest energy at which a high Rydberg electron will transfer between two such centers which are at a distance R apart? Figure 3 shows the potential seen by the electron. It is a sum of two Coulombic terms centered at a distance R from one another. Such a potential will have a maximum at the midpoint between the two centers of charge. For the problem of interest R is quite large so that the maximum is rather shallow. Even a fraction of a wavenumber lower in energy, the width of the potential barrier is rather large. The tunneling probability of an electron through such a barrier is therefore quite low.z6 Hence, the lowest energy of an electron which can transfer is when its binding energy just matches the value of the potential at its maximum in the midpoint between the two charges. At the midpoint, the potential is twice as deep as it would be if the perturbing charge was not there.33 For an isolated Rydberg state, the classical turning point ro (in atomic units) is related to the binding energy by -1/2n2) = -l/ro. But, ( 1 ) due to the presence of the ion, the potential is deeper. At the midpoint it is twice as binding. Hence, the tuming point of an electron that can just reach midway between the two centers of charge is at r = 4n2. ( 2 ) The ion is at R = 2r or the maximal distance at which a singly charged ion at a distance R apart can induce a transfer of high n Rydberg electron is R = 8nzau. This construction is shown in Figure 3 . The reason why the linear “size” scale of the Rydberg state is 4 times the linear size scale of an isolated orbit is due to two reasons: It is sufficient if the ion is within twice the distance of the turning point of the electron, for the transfer to be possible (point 2 above). In the language of transition state theory, the “point of no retum” is midway between the two centers of charge. Just as in transition state theory, there is a transmission coefficient. The electron will cross to the right only half the time. The other consideration is the Coulomb potential of the ion which adds to the Coulomb well of the isolated Rydberg state (Figure 3) and thereby enables the electron to sample a larger region. This aspect is similar to the lowering of the potential caused by an external dc field. The simple physical model discussed above is oversimplified in two ways. The first is that accompanying the lowering of the potential seen by the Rydberg electron, its unperturbed energy will also be lowered. This is familiar from the conventional Stark effect where it is known that degenerate states of the same n fan out into a Stark manifold. Some states will be lowered in energy while others may have their energy go up. (This depends on the orientation of the field-induced electric dipole of the Rydberg electron in the field of the ion.) In the Appendix we show that if we discuss the first state of a given n that can cross the barrier, the result R = 8n2 changes to R = 6n2. The other manifestation of the Stark manifold is that not all electrons of a given unperturbed energy can cross the same barrier height. This is familiar from studies of ionization of Rydberg states in an extemal field,z6 where the yield of ionization vs the external field rises from threshold in an S-shaped functional form. This further reduces the cross section by a factor k that is derived in the Appendix. At the high n’s
...’...
I .I.
1
-0.2 -40
-30
-20
-10
,
)
0
E - 74556.5 cm-1 Figure 4. Same C a s ZEKE spectrum shown in Figure 1: dashed line, including now the fit, solid line, to an n4 scaling law of the transfer cross section. The plot is vs the binding energy of the Rydberg electron. The fit shown uses u = 27.7n4 (au)*to compute the ratio S;cf. section 2. The value of n is related to the frequency of the laser using energy - 74 556.5 cm-I = -Rh2 (= binding energy), where R is the Rydberg
constant. in question, the value k = 0.36 is realistic. Both these corrections show that the model considerations provide an upper bound to the cross section. Figure 4 shows a fit of the ZEKE spectrum of Figure 1 , using the result S = NLu/V; cf. section 2. The transfer cross section is written as u = a0 n4, and the scale uois fitted to the data by fitting S’(ca6 intensity) to the c& intensity at each frequency. The result is 00 = 26 & 6 (au)2 vs the value of 00 = 3 2 ~ ( 6 / 8 ) ~ k = 20 (au)2. The value = 32n is the size expected on the basis of the simple physical considerations above, and (0.75)% is the correction due to the Stark-like splitting of the unperturbed states of given n. The error bound34 quoted in the fitted value of 00 reflects the several sources of uncertainty in the fit. Primarily, these are (i) the number N of ions in the focal volume is somewhat uncertain, (ii) the conversion of the frequency scale to an n scale, and (iii) the unavoidable experimental noise in counting the very few C6D6 ions which are produced at the lower frequencies and the confidence limits of the fit. In view of all of this, we regard the agreement as very satisfactory. The fit does not, of course, validate the model. All it does is to not rule it out. The model discussed above assumes that any electron which energetically can transfer will do so with a probability of 0.5. This conclusion may be subject to the limitation that the time needed by the high n electron to reach and/or to cross the midway barrier can be very long, by which time the ion will have departed. We have therefore sought to estimate the rate of charge transfer using the (microcanonical) transition state analogy mentioned above. The result is that at the low drift velocities of the ions (ca. 10-100 m / s ) the cross section is nearly at its maximal value as computed above. It will, however, decrease as the velocity is increased. The experimental results reported in this paper are for a density of ions, 4 x 103/mm3,which is below higher concentrations (say, 106/mm3)which have been discussed in experiments where there is a strong role of the environment. The model we propose indeed suggests that, at higher concentrations of the ions, their role will be much more dominant and that, at such concentrations, a binary picture involving only one ion perturbing a given Rydberg molecule will break down. The physical considerations which gave rise to the n4 scaling law of the transfer cross section can also be discussed on the basis of the quantum mechanical theory of resonant charge transfer. At a given impact parameter, the transfer probability
1664 J. Phys. Chem., Vol. 99, No. 6, 1995
Alt et al.
is sin2 (Ad), where Ad is the phase shift difference for the gerade and ungerade potentials of the two centers. While the phase shifts can be computed using the H2+ potentials, this is not really needed because for lower values of the impact parameters the phase difference will be large and rapidly varying with impact parameter. For the purpose of integrating over the impact parameter the transfer probability can be replaced by l/2 (hence the factor of l/2 already mentioned earlier). According to the Massey-Mohr appr~ximation,~’ all that one really needs is that value of the impact parameter, say b*, at which Ad decreases for the last time through d 4 . At higher impact parameters one reaches the classically forbidden regime where the transfer occurs by tunneling. The transfer cross section is ( ~ r / 2 ) b The *~. physical picture discussed earlier can therefore be regarded as a simple attempt to delineate the boundary between the classically allowed and forbidden regions for the electron transfer. So far we have discussed the transfer cross section as this is the measure which is determined by the present experimental results. For the general discussion of ZEKE spectroscopy, one would also like to know the cross section for any possible change of the initially accessed Rydberg state. Since it is not simple to measure it directly, we make the following two points. First, note that by our physical picture the size of the Rydberg state is determined by the location of the maximum in the potential seen by the Rydberg electron in the presence of the ionic perturber. But up to the maximum, this potential is not all that different from the potential due to the presence of an external dc field. The largest distance at which an electron will transfer was earlier determined by considering the lowest value of n for which the classical electron can reach out all the way to the maximum in the potential. At the maximum, the potential is twice as deep as it would be in the absence of the perturber. An external dc field also lowers the potential. The classical threshold for ionization is when the extemal field has lowered the unperturbed potential by just the binding energy. The classical threshold for ionization due to a dc field is thus the analog of the classical threshold for charge transfer. An ion which is closer in is the equivalent of a stronger field. The second point regarding the total cross section is that one can try to estimate it by considerations similar to those invoked to characterize the transfer cross section. The “conservation” of the total cross section28provides the criterion that the total cross section would be equal in magnitude to the elastic cross section if all inelastic and/or transfer channels are turned off. The high degeneracy of the unperturbed Rydberg state leads one to expect that at very long range the ion-molecule potential will vary in the manner expected for a Stark shift, viz. -An2/ R2, where the coefficient A depends on the quantum numbers of the Rydberg state. Explicit results for the A coefficient are known29for the interaction with a Rydberg electron in a purely hydrogenic orbital, Le., neglecting the quantum defect. It is simple to get the A coefficients, and also the higher-order terms in the 1/R expansion, by the expressing the perturbation due to the ion at R on the electron at r, using the well-known identity30
initial Rydberg state, the perturbation (3.1) can induce I and ml transitions, with obvious selection rules, but on average it vanishes and the leading adiabatic long-range perturbation is the familiar R-4 polarization potential. For an R-4 long-range potential, the computation of the barrier in the effective potential (which includes the centrifugal term) is standard.31 Using n6 au3 as the averaged polarizability leads to a cross section for binary events of3l UT = Jtn3E~-1/2 where ET is the ion molecule relative translational energy. (In practical units, UT (au2) = 5.16 x 104n3/v( d s ) , where v is the velocity.) To compare with the charge transfer cross section a, it is convenient to measure the energy in units of the binding energy E of the Rydberg electron, viz. E (au) = 1/2n2, so that UT = 0 . 2 a ( E / E ~ ) ’In~ ~other . words, as long as the relative energy of the ion and the Rydberg state is smaller than the energy required to detach the Rydberg electron, the binary cross section (averaged over the manifold of degenerate Rydberg states of given n) is somewhat larger than the charge transfer cross section. For high Rydberg states, the charge transfer cross section dominates. This is to be expected since the very weak binding of the high Rydberg electron implies that it is practically a ZEKE transfer.
4. Concluding Remarks The transfer of a high Rydberg electron to an ion to produce a false Rydberg state is probably the most effective rearrangement process possible in the system. The cross section for this charge transfer is enormous, but the yield is small and can be made to be negligible under realistic experimental conditions for ZEKE experiments.
Acknowledgment. This work was supported by the Stiftung Volkswagenwerk and by the German-Israel Foundation (GIF) for Basic Research. Appendix A. Effective Cross Section for Charge Transfer
This appendix provides a detailed derivation of the expression a = 64~n(0.75)~k n4 for the effective charge transfer cross section. K is the transmission coefficient at the barrier for the transfer. As discussed in the text, both intuition and the quantum mechanical theory of resonant charge transfer support the value K = 0.5. The factor (6/8)*k accounts for the (Stark) shift of the energy of the unperturbed Rydberg states due to the presence of the field of the ion. The numerical value of k depends on what assumptions are made as to how the perturbation is applied. If it is adiabatic, k = 1. At high n’s it is more realistic to make a diabatic correlation.26 Below we shall obtain, for the diabatic limit, k = 0.36 or (T = 20n4, which is the result quoted in the text. The cross section is computed, as usual, by considering first the subset of events where the impact parameter is in the range b to b db. In terms of the number, WR,of Rydberg states that undergo charge transfer and the number, N , of ions
+
dNR/N = (NR/VAd o = (NR/V)AKfLzbdb (3.1) where 8 is the angle between the two vectors and the P’s are the Legendre polynomials. The coefficients are the matrix elements of the numerators in (3.1) in the hydrogenic zerothorder states.35 It follows directly from (3.1) that on averaging the perturbation, over the entire manifold or unperturbed Rydberg states of given n, the result vanishes. For a particular
(A.l)
A, as defined in the text, is the mean path of the ion, V is the interaction volume, andfis fraction of states whose unperturbed energy, in au, is -1/2n2, that ionize in the given dc electrical field, F, of the ion. This fraction has been previously26 computed as a function of the effective field strength, 4; 4 = n4F. The function f = A@)is thus known. It has a so-called S-shape with a value of unity at high field strengths, and graphs of it are shown in ref 26.
High Molecular Rydberg States The computations in ref 26 are for a uniform dc field, which is not quite the case here. Since the role of the field is to lower the barrier, we take the value of the equivalent uniform field to be such that it provides the same barrier height for ionization as that due to the ion. In ref 26 we show that at the barrier the potential energy, in au, is lowered by uniform dc field F by -2dF. (Note that the potential of the uniform field is defined so that it vanishes at the center of the Rydberg state system of coordinates.) In the present problem, when the distance of the ion and Rydberg state is b, the potential in the middle is r-* (b - r)-l b-l, where the last term is introduced to ensure that the potential of the dc field vanishes at r = 0. The maximal value of the potential, which is the classical barrier to charge separation, is -3/b. Equating the two barrier heights, F = 9/4b2 or 4 = 9n4/4b2, which specifies how f depends on the impact parameter b. With this transformation between the reduced field 4 and the impact parameter eq A.l can be integrated over b:
+
where in the integral we changed variables from b to 4, and the actual lower limit of the integration is determined by the vanishing o f f already at a finite value of the field. The integration is carried out numerically and, for f computed in the diabatic case, leads to 0 = (9/4)mcn4(5.7474) s 20n4 or k = 0.36. A secondary result of this appendix is the magnitude of an extemal dc field which, as judged by the facility for charge separation, is equivalent to the field of the ion. For an ion at a distance b, F = 9/4b2 or 4 = n4F = 9n414b2.
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J. Phys. Chem., Vol. 99, No. 6, 1995 1665 (12) Pratt, S. T. J . Chem. Phys. 1993, 98, 9241. (13) Nemeth, G. I.; Selzle, H. L.; Schlag, E. W. Chem. Phys. Lett. 1993, 215, 151. (14) Merkt, F.; Mackenzie, S. R.; Rendall, R. J.; Softley, T. P. J. Chem. Phys. 1993, 99, 8430. (15) Bahatt, D.; Even, U.; Levine, R. D. J . Chem. Phys. 1993,98, 1744. (16) Scherzer, W. G.; Selzle, H. L.; Schlag, E. W.; Levine, R. D. Phys. Rev. Lett. 1994, 72, 1435. (17) Rabani, E.; Baranov, L. Ya.; Levine, R. D.; Even, U. Chem. Phys. Lett. 1994, 221, 473. (18) Even, U.; Ben-Nun, M.; Levine, R. D. Chem. Phys. Lett. 1993,99, 5800. (19) Muller-Dethlefs, K.; Schlag, E. W. Annu. Rev. Phys. Chem. 1991, 42, 109. (20) Schlag, E. W.; Peatman, W. B.; Muller-Dethlefs, K. J . Electron Spectrosc. 1993, 66, 139. (21) Reiser, G.; Habenicht, W.; Muller-Dethlefs, K.; Schlag, E. W. Chem. Phys. Lett. 1988, 152, 119. 1221 Alt. C.: Scherzer. W. G.: Selzle. H. L.: Schlag. -. E. W. Chem. Phvs. ~ e t t i994,224, . 366. (23) Kellert, F. G.; Smith, K. A.; Rundel, R. D.; Dunning, F. B.; Stebbings, R. F. J . Chem. Phys. 1980, 72, 3179. (24) (a) Hansen, S. B.; Ehrenreich, T.; Horsdal-Pedersen, E.; MacAdam, K. B.; Dube, L. H. Phys. Rev. Lett. 1993, 71, 1522. (b) Wang, J.; Olson, R. E. Ibid. 1994, 72, 332. (25) Mott, N. F.; Massey, H. S. W. The Theory of Atomic Collisions; Clarendon Press: Oxford, 1965. (26) Baranov, L. Ya.; Kris, R.; Levine, R. D.; Even, U. J . Chem. Phys. 1994, 100, 186. (27) Massey, H. S. W.; Mohr, C. B. 0. Proc. R. Soc. Landon 1934, A144, 188; 1933, A141, 434. (28) Levine, R. D. J . Chem. Phys. 1972, 57, 1015. (29) Krogdahl, M. K. Astrophys. J. 1944, 109, 333. (30) Zare, R. N. Angular Momentum; Wiley: New York, 1988. (31) Levine, R. D.; Bemstein, R. B. Molecular Reaction Dynamics and Chemical Reactivity; Oxford University Press: New York, 1987. (32) We will explain the “about” 8 below. The maximal distance of the electron from the core is, in atomic units, 2122. The maximal size is therefore 4m4. (33) The above can be done better by recognizing that what we are discussing is binding in a high Rydberg state of H2+. At the high n’s of present interest, the simple picture suffices. (34) The sources of error noted above preclude a meaningful simultaneous fitting of both the size scale and the n dependence of the transfer cross section. In particular, we cannot strictly rule out a dependence on n3, Such a dependence is expected if the charge does not necessarily align the axis of the orbit along the lines of centers. In this case the orbit is elliptical, of area equal to x times the lengths of the two axes, one of which scales as n2 and the other as nl. Since the orbital angular momentum 1 is bounded by n, an n3 power law requires a larger scale factor 00. The fit of au= functional form to the data yields 00 =z 2000 (au)2. It is harder to justify theoretically such a high scale factor, but due to the noise in the determination of the C6D6 ZEKE ions at the lower n values, one cannot rule out an n3 power law in terms of a fit to the data. We further discuss the n3 power law at the end of this section. (35) For example, the A coefficients as defined above are 3(nl - n2)/2n where nl and n2 are the familiar parabolic quantum numbers. JP942095V
