J . Phys. Chem. 1985, 89, 10-13
10
I
iir
,
106
-05
1 , 0
\ , ! ,I
20
05 I O 15 A E (cV)
25
30
Figure 3. Calculated values of Wq and Was functions of the energy gap AE for some values of 6 in the charge recombination reaction. The curve of W for 0 = m was obtained by using the formula of Hopfield.'
confirmed that the results of our calculation for j3 = 0 . 1 4 . 3 can reproduce completely the experimental results2 of the k , vs. AG relation for the fluorescence quenching reaction by putting W = k13 in the reaction scheme of eq 17 and by using appropriate values* for the other rate constants, as shown in Figure 2B. A*
k12
k23
k2,
k32
+ B EA* ...B
A' ...B*
-
In contrast to a charge separation reaction, W for the charge recombination reaction calculated by our method shows a strong energy gap dependence as indicated in Figure 3, where W for /3 I0.1 is quite similar to Wq. With an increase of 6, the shape of Was a function of AE becomes much broader and for j3 = m, W(AE) is quite different from W,(AE). Since the peak position depends on the parameters used for the quantum modes of W(AE) (while this is not the case for the charge sepration reaction), we give here only a qualitative discussion. The results in Figure 3 show that, although the energy gap itself is affected very much by the solvation of ions with polar solvent, the solvent mode appears to play little role in the energy gap dependence, if we consider the large frequency change of the solvent mode in the charge recombination reaction. As is already discussed in the Introduction, the charge recombination reaction of photochemically produced ion pairs shows a strong dependence upon the energy gap in general,',) contrary to the photoinduced charge separation reaction in strongly polar solvents. Moreover, it should be noted here that the rate of the back electron transfer in the recombination reaction depends strongly also upon the In the nature of individual electron donor or acceptor framework of the present theory, the latter fact means that the rate of back electron transfer is determined by Wqeven in strongly polar solvents. This is just what our theory is predicting. Details of the derivations of above results as well as a more detailed discussion will be given in forthcoming paper^.^^^
km
(17)
(9) Kakitani, T.; Mataga, N., to be submitted for
publication.
Orientation Dependence of the H 4- D2 Reaction Cross Section: Steric Model vs. Trajectory Calculations N. C. Blais, Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
R. B. Bernstein,* Chemistry Department, University of California, Los Angeles, California 90024
and R. D. Levine Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel (Receiued: October 1 1 , 1984)
-
Quasiclassical trajectory calculations of the orientation dependence of the cross section for the H + D2 HD + D reaction have been carried out at collision energies of 0.55 and 1.30 eV. The results are well approximated by the angle-dependent barrier, line-of-centers steric model of Levine and Bernstein.
Introduction In a recent Letter, Levine and Bernsteinl (LB) presented an opacity analysis of the steric requirements of elementary reactions, based upon an angle-dependent line-of-centers model, which utilizes the full orientation dependence of the barrier height. The purpose of their study was to identify the primary factor governing the orientation dependence of the reaction probability. The model was applied to the reaction H + D2(vj=O) H D + D. It was used to calculate the orientation-averaged opacity function P(b) at collision energies of 0.55 and 1.30 eV, and the translational energy dependence of the reaction cross section o ( E ) , from threshold to 2.5 eV. Results agreed well with the quasiclassical
-
(1) R. D. Levine and R. B. Bernstein, Chem. Phys. Lett., 105, 467 (1984).
0022-3654/85/2089-0010$01.50/0
trajectory calculations of Blais and Truhlar2 (BT), which utilized the same ab initio calculated potential surface, LSTH (that of Siegbahn and Liu? as parametrized by Truhlar and Horowitz4). Here, we present results for the orientation dependence of the same reaction. It was pointed out by Pollak et aLS that the LB model, which uses the actual dependence of the barrier height Eo on the angle of approach, is a generalization of an earlier model of Smith,6 (2) N . C. Blais and D. G. Truhlar, Chem. Phys. Lett., 102, 120 (1983). (3) P. Siegbahn, and B. Liu, J . Chem. Phys., 68,2457 (1978). (4) D. G. Truhlar and C. J. Horowitz, J . Chem. Phys., 68,2466 (1978); 71, 1514 (1979).
(5) E. Pollak, M. Baer, N . Abu-Salbi, and D. J. Kouri, J . Chem. Phys., submitted for publication.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985 11
Letters
,
TABLE I: Traiectorv Parameters and Results initial state (D2) u=O.i=O
u=O.i=4
ET = 0.55 eV no. reacted bmm
282 0.95 0.27 f 0.02 134
A
a. A2
e, deg
0
SWITCHEDa
ET = no. reacted bmw
336 0.95 0.32 f 0.02 128 316
A
a, A2
e, deg SWITCHED‘
1.3 eV
I
1101
SI8
1.16 1.04 f 0.03 114
1.29 1.11 f 0.04 113 260
6
DNumberof trajectories that exchanged the nearest H-D atom pair before reaching RSHELL.
termed “the modified simple collision the or^".^ The Smith theory assumes a linear dependence of the barrier height upon the cosine of the “angle of attack”, say y, which is strictly valid only in the limit cos y G 1. The LB equation for P(b) in this limiting case (Le., ref 1, eq 13), is identical with Smith’s P(b) (ref 7, eq 15). (Other, recent related work includes that of Agmon*, of Walker and Hayes,9 and of Pollak et al.lO) A unique feature of the LB model is its ability to predict directly the dependence of the reaction cross section upon the angle of attack. From eq 2 and 3 of ref 1 ~R(ET)
=
x ~ l d ( c o 7)s aR(y)
We define the relevant partial cross section da/d cos y such that
=
1, 1
d(cos y) do/d
COS
y
(2)
v.0,
-
y = j/za~(r)
(3)
COS
=
‘/z rd2[1 -Eo
ET]
=4
I .5
Y
1.0 -
V -0 \
b
Using eq 8 of ref 1, we obtain the explicit result da/d
j
OQ
0
COS
I
N
u)
da/d
05
Figure 1. Orientation dependence of reaction cross section. Plot of da/d(cos y) vs. cos y for HD products from trajectories and from the LB model. The solid curves are calculated (via the LB model; eq 4) with the parameter d = 2.25ao. The trajectories are represented by points with error bars indicating a statistical error of one standard deviation. The D2 initial state for the trajectories was u = 0, j = 0. Solid points, obtained with the trajectory geometry at RSHELL = 2.Sa0: ‘ Ia t 1.3 eV and 0 at 0.55 eV. Open points from the initial geometry of the trajectory: V at 1.3 eV and 0 at 0.55 eV.
x Thus
0
cos y
(1)
-1
-0.5
-I
U
(4)
for ET 3 Eo(y),zero for ET < E,(y). Consistent with LB, E o ( y ) is the a b initio derived barrier height for given orientation y. In what follows, eq 4 will be tested by comparison with quasiclassical trajectory calculations for the partial cross section do/d cos y. The LB model refers only to the orientation of the reagents at a critical configuration. To compare the predictions of the model with experiments using oriented reagent molecules,” it is necessary that the experiments be carried out under such conditions that the rotation of the molecule is slow compared with the transit time. Some of the computations to be presented below explore the validity of this assumption. 11. Calculational Procedures
The quasiclassical trajectories were computed with the same methods2 and with the same potential energy surface (the LSTH p ~ t e n t i a l )as ~ .for ~ the BT investigation, with the following special (6) I. W. M. Smith, “Kinetics and Dynamics of Elementary Gas Reactions”, Butterworths, London, 1980. (7) I. W. M. Smith, J . Chem. Ed., 59, 9 (1982). (8) N. Agmon, Chem. Phys., 61, 189 (1981). (9) R. B. Walker and E. F. Hayes, J . Phys. Chem., 87, 1255 (1983). (10) J. Jellinek and E. Pollak, J. Chem. Phys., 78, 3014 (1983). E. Pollak, in “Theory of Chemical Reaction Dynamics”, M. Baer, Ed., CRC Press, New York, 1984. E. Pollak and R. E. Wyatt, Chem. Phys. Lett., to be published. ( 1 1) P. R. Brooks, Science, 192, 1 1 (1976); S. Stolte, Ber. Bumenges. Phys. Chem., 86, 413 (1982); S . Stolte, K. K. Chakravorty, R. B. Bernstein, and D. H. Parker, Chem. Phys., 71,353 (1982); D. H. Parker, K. K. Chakravorty, and R. B. Bernstein, Chem. Phys. Lett., 86, 113 (1982). In the analysis of the experimental data, the “angle of attack” is considered to be the angle between the electric dipole moment of the molecule fi (or the long axis of the molecule) and the initial relative velocity vector v,. This is closely related to, but not identical with, the present y .
0.5 -
-I
-05
0
0.5
1.0
cosy Figure 2. Influence of D2 rotation on orientation dependence. Same plot as in Figure 1, except that the initial state of the D2 for the trajectories was u = 0, j = 4. Only the geometry of RSHELL is shown. The calculated curves from the LB model, not dependent on initial state, are the same as those in Figure 1.
feature: when the approaching H atom first reached a distance of 2.5~20(denoted RSHELL) from the nearest D atom of the diatomic, the trajectory coordinates were recorded. The RSHELL was chosen to be comparable with the parameter d used in ref 1. The angle y (see Figure 1 of ref 1) was calculated from the trajectory coordinates at RSHELL and also at the start of the trajectory, a distance of 11Sao between the H atom and the center of mass of the diatomic. Because these trajectories were intended to test the LB model, only one initial state at a time was used for the D2. Two separate cases were examined, u = 0, j = 0 and c = 0, j = 4 . I 2 3 l 3 For each of these, two collision energies (relative translational energies), (12) The present notation is a departure from that of BT.2 Here, it was desired that the initial rotational momentum J , be zero for j = 0, while also being 4.5 for j = 4. This is best represented by J , = (ju+ l ) t ~ ) ’ / The ~ . BT work used the Langer replacement J , = (j + 1/2)h. For vibration, however, the BT representation was still used here, so that the zero-point vibrational energy was included at u = O.I3 (13) N. C. Blais and D. G . Truhlar, J . Chem. Phys., 65, 5335 (1976).
12 The Journal of Physical Chemistry, Vol. 89, No. I, 1985
ET, were used, 0.55 and 1.3 eV. The choice of j = 0 was made to test the trajectories against the model under conditions most favorable for comparison with an oriented-molecule experiment. That of j = 4 was made to test the extent over which the model could be applied: j = 4 having an energy well above the mean initial rotational energy of the reacted trajectories in the BT study. All other variables were chosen by unweighted Monte Carlo procedures. A total of 13 500 trajectories were run, of which 2237 reacted. Quantities of importance of this work are listed in Table I. The statistical errors in all of the figures and in the tabulated quantites represent one standard deviation because of the Monte Carlo procedures, Le., the fractional deviation is l/N'lz. A fixed time step of 5 X s was used, after testing to ensure that the final state state properties were independent of further reductions in this parameter. As part of the analysis, the reacted trajectories were sorted into bins of equal width in the variable cos y. This was done both at the initial geometry and at RSHELL. When the entire set of trajectories for each case was used, the bins were of width 0.1; the resulting differential cross sections, do/d(cos y), are plotted in Figures 1 and 2. A record was kept of whether an H atom following its trajectory reacted with the D atom closest to the H atom at RSHELL or whether subsequent exchange of D-atom reaction partners occurred. N o trajectory was found to undergo such an exchange for any of the initial conditions. Also examined was the question of whether the closest H-D partners at the start of the trajectory were the same as those at RSHELL. Those that switched partners in going from the start of the trajectory to reach RSHELL (and react) are designated as SWITCHED in Table
I.
Letters ,
0.61
r
1
t
0,4
I
0,z
9
1.2
,
I
I
"
"
l
"
"
1
b
Q
b) l30eV.
fi = 1 1 4 O 1
0.8 -
#L
-
A : O
o
D
$1
A
04 0
.
4 o
o.-.-a,
r 4
,
,
1
I
I
,
I
,
A more detailed binning of the trajectories was also carried out. Here, trajectories that reacted with a final scattering angle 8 greater than the mean angle 8 are binned separately from those with 8 less than 8 (the bin size is 0.2 in cos y). The values of the mean scattering angle are tabulated in Table I. The model calculations of do/d cos y involved simply evaluating eq 4, using the E o ( y ) from Figure 1 of ref 1. The choice of the range parameter d was different from that adopted in ref 1, however. Here, a 10% smaller value of d, namely, 2.25ao, was chosen; it yielded better agreement in the integral reaction cross section uR (at both values of ET) between model and trajectory calculations. 111. Results
Figure 1 shows the dependence of the reaction cross section upon the cosine of the angle of orientation, i.e., the partial cross section du/cos y. Errors bars denote statistical uncertainties, as in BT.z Two different choices of y were used: the initial value at the start of the trajectory, and the value at RSHELL. The solid curves are predictions of the LB model. Table I lists the integral cross sections for comparison. Figure 2 shows the influence of initial rotational angular momentum, j = 4, compared with the results for j = 0. Table I also lists the uR values from these calculations. The small effect of j is partly due to the additional energy (E,,, = 0.075 eV), but mainly due to r e ~ r i e n t a t i o n . ' ~ J ~ It is of interest to explore the correlation of du/cos y with the angle of scattering 8. However, the large number of trajectories required to yield statistically meaningful results precluded such an evaluation. The present study considered only two "bins" in scattering angle, one for backscattering ( H D scattered back relative to the initial velocity of the H atom, Le., e > the other for forward scattering (0 d The results (for u = j = 0) are shown in Figure 3, plotted for y(initia1) and for y evaluated at RSHELL. The results are very similar for y(RSHELL) but show a very strong deviation for ?(initial). The origin of this difference is discussed in Section IV. The important conclusion from this
e),
e).
~
~~~~
(14) H. R. Mayne and J. P. Toennies, J . Chem. Phys., 74, 1017 (1981); 75, 1794 (1981). (15) N. Sathyamurthy, Chem. Rev., 83, 601 (1983).
(16) G. H. Kwei and D. R. Herschbach, J. Phys. Chem., 83, 1550 (1979).
J . Phys. Chem. 1985, 89, 13-17 (at R = 11.54) orientation of the diatomic (with respect to the initial relative velocity v,), and the orientation at RSHELL (which determines the outcome, reaction vs. nonreaction). To obtain cos y a 1 (near-collinear) a t RSHELL (and thus high reaction probability) for small b, one requires the initial orientation to be 'parallel", Le., Dzllvl while for the large b, an off-axis, "more perpendicular" orientation is required. From Figure 3, it is seen that at 1.3 eV the "preferred" orientation for the largeb collisions (where du/d cos y is a maximum). On (i.e. 0 C 8) is cos y the other hand, to obtain cos y i= 0 a t RSHELL (and thus low reaction probability) a t small 6, one requires a perpendicular orientation (cos y i= 0), while for large b, a more parallel orientation (cos y = 1) is required. Figure 3 shows that, for cos y a 1, da/d cos y for large-b collisions (i.e., 0 < 8) is small, while that for small b (0 > 8) is large. Thus, the qualitative explanation of the discrepancy (Le., between the low-angle and high-angle scattering points) when interrogating orientations at R = 1 1.5ao is essentially purely geometrical in nature. For the 1.3-eV case, Figure 3b, the differential cross section at RSHELL is largely independent of the scattering angle 0, indicating that, for any orientation, high- and low-impact parameters contribute equally. (This is also true of the overall reaction, because it is found that b is almost identically equal to 1 Iz~max.) For 0.55 eV, Figure 3a, an angle dependence to the differential cross section (Le., da/d cos y) persists even at RSHELL. Much of the difference between the 1.3- and the 0.55-eV cases can be understood in terms of the LB model. In order to react at the angle y, the model requires the energy along the line-of-centers ET (1 - b2/&) to exceed the barrier height E,,(?) a t that orientation. When ET is as high as 1.3 eV, even large impact parameters can lead to reaction at, say, cos y 0.7. This is no longer the case at 0.55 eV. This effect should also be present at 1.3 eV, but it would be apparent only a t much smaller values of cos y (where the large relative statistical variations may mask the effect).
13
Figure 1 shows that this problem disappears when averaging over all impact parameters (to obtain the orientation dependence of the total reaction cross section du/cos y). It is also to be recalled that the applicability of the LB model rests upon two additional physical assumptions: one, that the collisions are fast with respect to rotation of the reagents and, two, that trajectories that have crossed the barrier will proceed to products and will not "recross". For most diatomics, and for reactions with a barrier, both assumptions are likely to be well satisfied. Dz is an almost extreme case of a fast rotation, yet even at j = 4 the classical rotation period is much longer than the transit time (defined with relation to RSHELL). As seen in Figure 2, the model still works. It was pointed out by Smith7 that, at collision energies well above the barrier, the assumed "no-return" character of the trajectories will become valid (see, e.g., ref 17), and at higher energies the model should begin to break down. Very recently, a trajectory study on a modified LSTH surface was carried out.'* The results are consistent with the present model provided that a somewhat smaller value of d is used. The magnitude of d is thus somewhat sensitive to the shape of the "conen of acceptance. Acknowledgment. The work a t Los Alamos National Laboratory (by N.C.B.) was performed under the auspices of the US. Department of Energy; support from the US. National Science Foundation, Grant CHE83-16204 (R.B.B.), is also gratefully acknowledged. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fur die Forschung, mbH, Munich, FRG. Registry No. H, 12385-13-6; Dl, 7782-39-0. (17) R. A. LaBudde, P. J. Kuntz, R. B. Bernstein, and R. D. Levine, J . Chem. Phys., 59, 6286 (1973). (18) N . C. Blais, D. G . Truhlar, and B. C. Garrett, submitted for publication.
Supercharged Cations of Benzene J. R. Appling, G. W. Burdick, M. J. Hayward, L. E. Abbey, and T. F. Moran* School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received: October 22, 1984)
Stable, supercharged benzene ions produced in electron impact ionization have been observed by techniques of collisional mass spectrometry in which single electron transfer reactions of incident C&"+ ions at kiloelectronvolt energies give rise to stable CsHs("')+product ions. Semiempirical SCF-MOcalculations employed for the computation of C6H6"+ structures indicate that geometry optimized minimum energy C6H6"+ cyclic ions exhibit a progression of elongation and expansion as n increases. Structural and energetic data have been obtained for the n = 1 to 6 charge states of benzene.
Introduction Spectroscopic studies have provided a wealth of information' on the energy levels of multiply charged states of atoms. Formation of multiply charged atomic ions is a commonplace occurrence in high-energy electron impact ionization of atomic Although multiply charged states of molecules can be produced in electron impact ionization, their presence can be difficult to monitor by conventional mass spectrometric methods. Supercharged molecular ions, M"+, of even mass (M) and charge (ne) appear a t the same mass-to-charge ratio as { M - X ( M / n)j(** ions, where X = 1,2, ..., (n - 1). Singly charged fragment (1 1 C. E. Moore, Nat. Bur. Stand. Circ., No.467 (1 949). (2) R. A. Falk, G. Stefani, R. Camilloni, G. H. Dum, R. A. Phaneuf, D. C. Gregory, and D. H. Crandall, Phys. Reo. A , 28, 91 (1983). (3) D. C. Gregory, P. F. Dittner, and D. H. Crandall, Phys. Reu. A , 27, 724 (1983).
0022-3654/85/2089-0013$0 1.50/0
( M / n ) + ions are often intense and in many cases obscure any multiply charged M"+component a t the same mass-to-charge ratio. In the case of doubly charged ions, a collisional mass spectral technique has been developed to overcome this interference problem. This method4 involves fully accelerated, doubly charged Mz+ ions which undergo single electron transfer reactions to form fast M+ products. These M+ product ions pcwess the same velocity as the incident doubly charged ions and are the only species that can be transmitted by an analyzing electrostatic sector maintained at twice the normal operating voltage. Subsequent momentum analysis serves to conclusively identify these M+ products arising from Mz+ electron transfer reactions. The product ion mass distributions so obtained in the case of doubly charged ions have (4) D. L. Kemp and R. G. Cooks in "Collision Spectroscopy",R. G. Cooks, Ed., Plenum Press, New York, 1978, p 257.
0 1985 American Chemical Society
