J . Phys. Chem. 1986, 90, 715-717 ionsg vs. values for Z 2 / a are shown in Figure 1. Here, except for halogenide ions, effective radii of Laidler and Muirhead-Gould9 were adopted as values for a,while Goldschmidt’s radiiI0 were used for a values of the halogenide ions. The plot for the largest (9) Laidler, K.J.; Muirhead-Gould, J. S. Trans. Furaday SOC.1967, 63,
9 53 __“.
(10) Goldschmidt, V. M. Chem. Ber. 1927, 60, 1263.
715
value of Z 2 / adeviates considerably from the line b in Figure 1. This deviation seems to be due to the use of the rather small a value of 0.07 nm for A13+. Tables I and I1 and Figure 1 clearly show that the present modified equation (eq 5 ) better reproduces the observed solvation free energies than does the Born equation (eq 1). However, there still remains the ionic size problem to be solved in order to obtain very_ good with the observed _ agreement values by eq 5 .
Differential Cross Sections and Rainbows in He-LiH Collisions Aristophanes Metropoulos Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation,‘ Athens 11635, Greece (Received: November 14, 1985)
Total, inelastic, and state-testate differential cross sections are obtained for the He-LiH collision system by using quasiclassical trajectories. The observed inelastic scattering in a nearly forward direction is discussed. The “outside” branch of the rainbow curve is obtained and its implications are discussed.
Introduction The rotational energy transfer of the He-LiH system has been investigated experimentally and the rotationally inelastic integral cross sections have been 0btained.l Integral cross sections have also been computed theoretically by employing both the coupled states approximation2and the quasiclassical trajectories approximation3 on an accurate potential energy surface4 Here, we use quasiclassical trajectories to give a rough estimate of the total, inelastic, and state-testate differential cross sections of the system and to obtain a branch of the rainbow curve. Description of Calculations The calculations were done at the peak of the experimental collision energy distribution1(0.3057 eV). The initial rotational angular momentum quantum number, j , was fixed at the experimentally selected initial state’ (i = 1). The Silver potential energy surface (rigid rotor)4 was used as previously’ and a total of 3000 trajectories were run on a Perkin-Elmer 3240 computer (- 1 min per trajectory). In order to increase the frequency of inelastic encounters the maximum impact parameter was fixed at 4.7 8,as opposed to 5.5 8,in the previous calculation^.^ In those calculations, no rotational energy transfer was observed for impact parameters larger than 4.5 A. Also, a comparison of their integral cross sections with those of the present calculations showed no significant differences. Further details of the calculation are given elsewhere’ and will not be repeated here. The various differential cross sections are obtained with the expressionS K
da/dw = b,,,/N
sin (AO) Cb,/sin 0, r=1
(1)
where b,,, is the maximum impact parameter, b, is the impact parameter of the ith trajectory, N is the total number of trajectories run, K is the total number of trajectories included in the sum (see below), A0 is the width of the angular bins (here A0 = 2), and 0, is the angle of the center of each angular bin and represents the scattering angle for the trajectories within this bin. For the total differential cross section one sets K = N and all the trajectories are to be included in the sum. For the inelastic differential
TABLE I: Total and Inelastic Differential Cross Sections (in A’lsr) Bi, deg total % error inelastic % error 1 4437.91 8.71 1135.44 17.21 3 619.44 13.82 379.32 17.63 282.94 16.05 188.16 19.31 5 7 127.29 19.75 21.90 99.61 105.77 9 18.87 20.04 91.88 22.15 11 59.77 22.58 57.39 44.81 13 46.49 22.76 23.05 21.94 38.80 15 40.85 22.34 17 24.78 28.28 28.28 24.78 25.47 25.47 23.58 23.58 19 24.29 21 23.31 23.31 24.29 18.45 23 25.50 25.50 18.45 28.15 28.15 13.67 13.67 25 27 36.37 36.98 7.33 7.14 29 9.78 9.61 30.04 30.38 30.28 31 8.12 1.96 30.65 32.85 33 6.54 6.54 32.85 32.08 35 6.93 6.93 32.08 37 4.24 40.44 40.44 4.24 39 4.50 35.35 35.35 4.50 41 3.54 40.37 40.37 3.54 44.26 43 2.46 2.46 44.26 47.19 47.19 45 2.10 2.10 47 2.48 42.77 42.77 2.48 49 2.74 39.96 39.96 2.74 2.27 44.11 45.22 51 2.18 41.16 53 2.29 2.12 41.56 1.64 45.02 45.02 55 1.64 49.72 57 1.57 1.57 49.12 57.47 59 1.30 1.22 59.83 41.72 61 2.19 2.19 41.72 63 2.08 2.01 38.78 39.56 41.39 1.83 65 1. I 7 42.26 45.18 45.18 1.43 1.43 67 2.19 2.07 69 38.02 36.73 48.87 48.87 1.18 1.18 71
cross section only trajectories with final quantum number j ’ # j are to be included in the sum. For the state-to-state cross sections (1) P. J. Dagdigian
48 Vas. Constantinou Ave.
and B. E. Wilcomb, J . Chem. Phys., 72,6462 (1980). (2) E. F. Jendrek and M. H. Alexander, J. Chem. Phys., 72,6452 (1980). ( 3 ) A. Metropoulos and D. M. Silver, J. Chem. Phys., 81, 1682 (1984).
0022-3654/86/2090-0715$01.50/00 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986
716
Letters
TABLE 11: Selected State-to-State Differential Cross Sections (in AZ/sr)
e,, deg 1
3 5 7 9 11
13 15
ul,o
%error ul-,
322 84 13 8
33 40 80 88
-
-
-
-
% error
-
3302 240 95 28 14 -
23 30 46 51 -
-
-
-
11
u,-~ %
error ule3 % error u1.t4 %error ul-(
127 198 95 38 21 9 6 4
22 25 28 37 46 63 I0 82
86 82 58 23 30 13
I 3
59 36 34 46 35 50 63 83
15 19 17 16 14 5 6
80 55 51 47 46 70 63 64 I7 -
1
5 II 10
13
I 3 3 2 2
%error
u,-~
%error
196 88 53 51 42 55 16 71
2 5
196 88 74 15 69 51 63 66 61 77 89
6
5 5 7 5 4 3 2
I1 91 92
1
1
u,-,
% ’ error
3 4 2 3
113 88 -
88 57 75 57 63 81 89
5 3 4 4 2 1
I
I
i
I
I
i 1
L------i 1 3
13 19 25
31
37
43
ffl
49
55 61 57
73
I
I
I
I
-1.3
1.0
3.0
5.3
x
SiAT, ANGLE LEG1 Figure 1. Upper curve (O),total differential cross section; lower curve (O), inelastic differential cross section (both in A2/s).
only trajectories with the desiredj’are to be included in the sum. The output of the trajectory calculations for the differential cross sections was also used to obtain the rotational rainbow curve.
Results and Discussion The total and inelastic differential cross sections are shown in Figure 1 and they are tabulated in Table I up to 0, = 71’. The percent error for each point is given in Table I and it is based on a 90% confidence leveL3 The state-to-state differential cross sections are tabulated in Table I1 up to j ’ = 7. Because of the relatively low number of trajectories in each j’bin the accuracy of these cross sections is rather low and it becomes worse for j ’ > 7. In Table 11, blank spaces signify that no trajectories were found in the indicated region for the indicated j ’ (classically forbidden regions). Dashes indicate that the corresponding cross sections were too inaccurate (95% error or higher) to be of any value. However, a few highly inaccurate points have been included in Table I1 in order to mark the smallest 0, for each indicated transition. Beyond 0, = 2 5 O , all cross sections are highly inaccurate. It is evident from Figure 1 and Tables I and I1 that there exist a large number of inelastic encounters in the small scattering angle (4) D.M. Silver, J . Chem. Phys, 72,6445 (1980). (5) A. Metropoulos, J . Phys. Chem., 88, 1 (1984).
191
Figure 2. A two-dimensional section of the He-LiH potential energy (ref 4) corresponding to a collinear approach. The LiH diatom is shown parallel to the X axis and with its center of mass (A)at (0,O). The Li atotm (0)is on the left and the H atom (0)is on the right. TABLE 111: Smallest Scattering Angles (in deg) Found in Each j’ Bin
i‘
OR
i’
OR
0
0.334 0.007 0.126 0.768 2.777 5.923
6 7 8 9 10
5.204 6.184 8.442 11.139 13.530
1
2 3 4
5
j’ 11 12 13 14 15 16
OR
13.498 18.533 24.159 33.525 59.071 95.159
region ( 0 - 2 O ) with b’-jl I2. In this region only elastic collisions are expected, however. The existence of such encounters may be attributed to the anisotropy and asymmetry of the potentiaL4 The Li side of the molecule has a very steep repulsive potential followed by a weakly attractive region while on the H side the potential is strictly repulsive but less steep. This is shown in Figure 2, which is a two-dimensional section of the potential function containing the X (Li-H) and the Z (potential) axes (compare with Figure 4, ref 4). The potential value of 0.3057 eV, for instance, is attained eitheratX=-l.619A(-l.419AfromLi) o r a t X = 3.336A (- 1.941 A from H). Thus, it seems possible that a He-Li approach can lead to small amounts of energy transfer in a more or less forward scattering direction. On the other hand, virtually no elastic scattering is observed in the high scattering angle region,
J . Phys. Chem. 1986, 90, 717-718 LD
0
I
I
I
I
20
40
60
80
SCAT. ANGLE [DEE1 Figure 3. Rainbow curve at 0.3057 eV collision energy and at j = 1. which is what one expects. The small amount of elastic scattering observed in this region may be considered as due to collisions which initially raise j’to a high level while upon separation they lower it back to j.6 (6) A. Metropoulos, Chem. Phys. Lett., 89, 405 (1982); A. Metropoulos and D. M. Silver, Chem. Phys. Lett., 93, 247 (1982).
717
We now turn to the classical rotational rainbow curve shown in Figure 3 and tabulated in Table 111. Each point in Figure 3 represents the smallest scattering angle found in each j’bin. The line through these points is a least-squares fit and separates the classically forbidden from the classically allowed region^.^ For heteronuclear diatoms there exist a second rainbow curve within the classically allowed region* but only the “outside” curve can be identified by the method used here. In light of the comparative study of Schinke et al.,9 Figure 3 can be interpreted as a manifestation of the dominance of the repulsive part of the potential in the collision process at 0.3057 eV. This of course has made possible the use of the coupled states approximationZand it is partly due to the short range of the attractive part of the potential. The minimum of the He-LiH potential is -15.24 meV and it occurs on the X Z plane at X = -2.3862 A. This value of the minimum is between the minima values for cases I1 and I11 of the model potential of Schinke et al. but its range is much shorter (see Figure 1, ref 9). This implies that in order to obtain the type of rainbow curve corresponding to collisions influenced by the attractive region (see Figure 14, ref 9) a collision energy much less than 100 meVg may be needed. At such low energies, it is uncertain whether the quasiclassical trajectories method would give reliable results around the region of the rainbow angles (strong interference effects), although they may give good cross section results provided the collision energy is sufficently above the excitation threshold (1.86 meV in our case). Registry No. He, 7440-59-7; LiH, 7580-67-8. (7) L. D. Thomas in “Potential Energy Surfaces and Dynamics
Calculations”,D. G. Truhlar, Ed., Plenum, New York, 1981; R. Schinke and J. M. Bowman in “Topics in Current Physics”,Vol. 33, J. M. Bowman, Ed., Springer-Verlag,New York, 1983. (8) H. J. Korsch and R. Schinke, J . Chem. Phys., 73, 1222 (1980). (9) R. Schinke, H. J. Korsch, and D. Poppe, J . Chem. Phys., 77, 6005
(1982).
1,P-Hydrogen Shift in Neutron-Irradiated Dichloroethanes K. Berei Central Research Institute f o r Physics of the Hungarian Academy of Sciences, Budapest, Hungary (Received: August 20, 1985; In Final Form: December 2, 1985)
Radical rearrangement involving a 1,2-hydrogen shift has been assumed as a means of interpreting the formation of isomeric labeled products in neutron-irradiated dichloroethanes.
Radical rearrangements involving 1,Zhydrogen transition are considered to be very However, Madden and Bernhard,3,4while studying radical reactions in X-irradiated methyl a-D-glucopyranoside (a-MeGlu) single crystals, observed the formation of a product via unimolecular radical conversion involving a 1,2-hydrogen shift: namely, the transition of a C6centered deprotonated primary hydroxy alkyl radical into a C5centered secondary oxyalkyl radical upon slowly warming the irradiated ‘a-MeGlu single crystals. Later, the same transition was reported5 for radicals produced by photolysis in neutral (1) Semenov, N. N. “On Some Problems of Chemical Kinetics and Reactivity”; Academy of Sciences Publishing House: Moscow, 1958; p 89. (2) Freidlina, R. Kh. Adv. Free-Radical Chem. 1965, I , 272. (3) Madden, K. P.; Bernhard, W. A. J . Chem. Phys. 1979, 70, 2431. (4) Madden, K. P.; Bernhard, W. A. J . Phys. Chem. 1980, 84, 1712.
aqueous solutions of a-MeGlu. The assumption has been made that the negative charge on the primary alcohol group attached to C6 may be a critical factor in promoting the hydrogen shift from C6 to C5.4 We have found an additional case where radical rearrangement involving a 1,Zhydrogen shift offers the most plausible explanation for the formation of some radioactive-labeled products observed in neutron-irradiated dichloroethanes. On studying the chemical interactions of recoil 3sCl atoms produced in the nuclear process 37Cl(n,y)38C1by irradiating crystalline 1,l-dichloroethane (1,l-DCE) at 77 K (for experimental details see ref 6 and 7) we observed a considerable yield of the K. P.; Fessenden, R. W. J . Am. Chem. SOC.1982, 104, 2578. (6) Berei, K.; VasPros, L.; Ache, H. J. J . Phys. Chem. 1980, 84, 1063. (7) Berei, K.; Vaslros, L.; Kiss, I., to be submitted for publication. ( 5 ) Madden,
0022-3654/86/2090-07 17$01.50/0 0 1986 American Chemical Society
