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Quasiclassical trajectory study of the effect of rotation on reactivity for the reactions D + H2, H + HD and H + D2 on the LSTH and DMBE potential energy surfaces
Chemical Physics ELSEVIER
Chemical Physics 222 (1997) 29-41
Quasiclassical trajectory study of the effect of rotation on reactivity for the reactions D + H 2, H + HD and H + D 2 on the LSTH and D M B E potential energy surfaces Sipora Hochman-Kowal, Avigdor Persky * Department of Chemistry, Bar-llan Universi~, Ramat Gan 52900, Israel Received 3 April 1997
Abstract The influence of rotation on reactivity for the isotopomeric reactions D + H 2 (v = 0, j = 0-6), H + HD (v = 0, j = 0-6) and H + D 2 (v = 0, j = 0-6) on the LSTH and DMBE potential energy surfaces has been investigated by quasiclassical trajectory calculations. The results are discussed in terms of the differences in the orienting character of the potential energy surfaces and the differences in the moments of inertia of the reacting molecules. The most distinct differences between the two potential energy surfaces are observed for the H + HD and H + D 2 reactions at low collision energies. The feasibility of carrying out experimental studies based on these differences in order to discern between the two surfaces is discussed. © 1997 Elsevier Science B.V.
1. Introduction The H + H 2 reaction and its isotopomeric analogs have been the focus of extensive experimental and theoretical studies, in the area of molecular dynamics, in recent decades. At present, three accurate ab-initio potential energy surfaces (PESs) are available for this system. These are the L S T H [1-3], D M B E [4] and B K M P [5] PESs. Recently, Truhlar and co-workers [6] carried out calculations of accurate quantum mechanical (QM) rate constants for the D + H 2 reaction on these three PESs, over a wide temperature range. They found that the LSTH and D M B E PESs give rate constants which are close to each other (within less than 8% over the temperature range 2 0 0 - 9 0 0 K) and which are in excellent agree-
* Corresponding author. Fax: 972 3 5351250.
ment with experimental data (for the D M B E PES within 13% or better over the whole temperature range). Much worse agreement with experiment was obtained for the B K M P PES, especially at low temperatures (higher than experiment by a factor of 2.2 at 200 K and by 42% at 300 K). Very recently, Aoiz et al. [7] carried out extensive quasiclassical trajectory calculations (QCT) for the reaction D + H 2 (v = 0, 1; j = 0 - 7 ) on these three PESs. Special emphasis was put on the influence of rotation on reactivity. For all three potential energy surfaces, at low collision energies, rotational excitation (for j < 5) was found to cause a decline in reactivity, but at sufficiently high collision energies, rotation becomes beneficial for reaction (for discussions of such effects for the H + H 2 reaction and for other a t o m - m o l e c u l e reactions see, for example, the review by Sathyamurthy [8] and Refs. [9-14]). The
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1-01 0 4 ( 9 7 ) 0 0 1 8 0 - 8
30
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
negative effect at low collision energies was found to be more marked for the DMBE PES than for the two other surfaces. Aoiz et al. showed that the influence of rotation on reactivity can be related to the anisotropy of the potential on the way to the barrier. The three PESs have nearly the same collinear barrier (9.65, 9.80 and 9.54 kcal/mol for the DMBE, LSTH and BKMP PESs, respectively), but the change with attacking angle for the DMBE PES is more significant than for the two other PESs (Ref. [7], figs. 12, 13), and therefore the orienting forces (towards collinearity) are stronger for this surface. In the absence of rotation, these orientation forces cause the reactivity on the DMBE PES to be larger than the reactivity on the two other surfaces. However, with the increase in the rotational energy of the H 2 molecule, the effect of orientation is diminished and reactivity decreases. For a rotating H 2 molecule, the average potential for all attacking angles is larger for the DMBE PES than for the two other surfaces and therefore the cross sections for this surface becomes the lowest. At sufficiently high collision energies, the effect of orientation and the differences in the average potentials become less important and cross sections for the DMBE PES tend to be similar to those on the two other surfaces. Aoiz et al. discuss the more limited data available from QM studies concerning the effect of rotation on reactivity on these PESs (D + H 2 (v = 0, j = 0, 1) on the DMBE PES [15] and D + H 2 (v = 1, j = 0 - 3 ) on the LSTH PES [16]). They conclude that trends similar to those obtained from their QCT calculations can also be expected from QM calculations. Based on their results, they suggest that careful measurements of cross sections or rate constants at low temperatures for the D + H 2 reaction with rotationally excited H 2 molecules, can provide data which may be helpful to discern between the various potential energy surfaces. Thus, their calculations indicate that at 300 K the values of the ratio of rate constants k(D + para-H2)/k(D + normal-H 2) are 1.16 for the DMBE PES, 0.93 for the LSTH PES and about 1.00 for the BKMP PES, and the differences between them increase with decrease in temperature. In this publication we report results of our study of the effect of rotation on the reactivity for the
isotopomeric reactions H + HD --* H 2 + D and H + D 2 ---) HD + D on the DMBE and LSTH PESs. The results are compared with results for the reaction D + H 2 ~ HD + H, and conclusions are derived concerning the systematic change in the effect of rotation on reactivity for these three isotopomeric reactions. Conclusions are also derived concerning the feasibility of experimental studies which may be helpful to discern between the DMBE and LSTH potential energy surfaces.
2. Results and discussion Detailed QCT calculations were carried out for the reactions H + HD (v = 0, j = 0-6) ~ H 2 + D and H + D 2 ( v = 0 , j = 0 - 6 ) ~ H D + D on the DMBE and LSTH potential energy surfaces. For the purpose of comparison, similar calculations were also carried out for the reaction D + H 2 (v = 0, j = 0 - 6 ) ~ HD + H. The computational procedure employed was similar to the procedure employed by us in earlier QCT calculations for other atom-molecule reactions, such as the O + HC1 [10] and the F + H 2 [17] reactions. For each set of initial conditions (collision energy E~ and rotational state j), between 20,000 and 160,000 trajectories were calculated. Based on preliminary calculations, the maximum impact parameter, in each case, was chosen to be somewhat larger than the largest one leading to reaction (by about 8%). The initial distance between the atom and the center of mass of the diatomic molecule was 6 ,~ in all the calculations. In the calculations for j = 0, we employed a procedure which we have used in earlier studies [ 18] in order to significantly improve the statistics of the calculations. This procedure involves calculations over a limited range of the initial azimuthal orientation angle Or , the angle between the molecular axis of the reacting molecule and the initial direction of the relative velocity vector. Thus, instead of randomly selecting Or (according to cos 0r) from the full possible range (0-180°), as is usually done in trajectory calculations [19,20], this parameter was selected between 0 ° and 0r 0r,max was appropriately chosen to be somewhat larger than the highest orientation angle leading to reactive collisions for the specific collision energy of each set of calculations. The ....
.
0.170 0.235 0.390
1.545 1.695 1.745 standard
0.078 + 0.002 0.145 + 0.003
0.220 f 0.002 0.300*0.005
0.465 + 0.006
0.625 f 0.008 0.775 *0.010
0.885 + 0.012 1.120+0.015 1.310*0.020 1.445 f 0.020
1.550 + 0.020 1.705 + 0.020 1.765 + 0.020
7.50 8.00
8.50 9.00
10.00
11.00 12.00
13.00 15.00 17.00 19.00
21.00 23.00 25.00
a In order to avoid overcrowding
0.835 1.065 1.260 1.435
0.560 0.715
0.043 0.105
0.027
4.95 + 0.04(-2)
1.585 1.670 1.760
0.750 0.990 1.205 1.385
0.465 0.610
0.300
0.090 0.155
1.530 1.715 1.825
0.675 0.970 1.170 1.430
0.370 0.530
0.220
0.040 0.085
4.85(-4) 0.010
j=3
are included
2.50(-3) 0.035
1.20(-4)
j=2
deviations
3.80(-4) 2.00(-3) 0.010
7.25
f 0.07(-3) + 0.06(-3) *0.02(-2) *0.05(-2)
3.90 6.80 1.30 2.85
6.50 6.60 6.75 7.00
j=l
1.00*0.02(-3)
j=O"
oa (AZ 1 (DMBE)
6.35
E,, (kcal/mol)
1.660 1.830
0.960
0.490
0.205
0.020
j=5
only in this column.
1.610 1.700 1.845
0.645 0.925 1.195 1.390
0.330 0.495
0.195
0.040 0.075
9.10(-4) 9.45(-3)
j=4
The standard
1.745 1.930
1.030
0.565
0.245
0.030
j=6
energy surfaces
deviations
1.415 1.500 1.605
0.760 0.970 1.150 1.300
0.515 0.640
0.365
0.160 0.225
0.057 0.100
0.036
2.80(-3) 5.OOG3) 9.55(-3) 0.020
6.05(-4)
j=O
1.445 1.615 1.710
0.725 0.970 1.160 1.275
0.445 0.595
0.300
0.100 0.160
0.015 0.050
4.75(-3)
5.80(-4)
j=2
for the other columns
1.445 1.530 1.645
0.740 0.950 1.135 1.315
0.500 0.640
0.350
0.155 0.220
0.048 0.095
0.030
3.55(-4) 1.35(-3) 5.25(-3) 0.013
j=l
0s (Z.2, (LSTH)
Table I Cross sections for the reaction D + H, (I: = 0, j = O-6) + DH + H on the DMBE and LSTH potential
1.570 1.760 1.855
0.680 0.935 1.195 1.385
0.365 0.525
0.240
0.060 0.105
6.30(-3) 0.025
1.25(-3)
j=4
1.655 1.810
0.995
0.560
0.240
0.025
j=5
are of similar magnitude.
1.505 1.625 1.755
0.690 0.940 1.185 1.360
0.395 0.530
0.250
0.065 0.110
4.50(-3) 0.025
1.40(-3)
j=3
1.780 1.940
1.055
0.585
0.270
0.035
j=6
? 2
;= 2 2 % s
s z.
D 9 ii’ e
3 \
P
% 2 _E ?
8
9 0 x
j=l 2.35(-4) 1.40(-3) 7.90(-3) 0.020 0.052 0.094 0.135 0.175 0.215 0.255 0.350 0.440 0.480 0.5 15
j=O"
4.85 *0.40(-4) 1.85~0.10(-3) 8.65 + 0.30(-3) 2.00* 0.05(-2) 0.053 * 0.001 0.096 + 0.002 0.135 *0.004 0.175 * 0.003 0.210*0.005 0.255 +0.005 0.360 + 0.007 0.440*0.010 0.490+_0.010 0.505 *0.010
~a (A*‘, (DMBE)
l.lO(-4) 3.80(-3) 0.015 0.052 0.092 0.135 0.180 0.225 0.255 0.350 0.460 0.490 0.520
j=2 1.40(-4) 2.80(-3) 0.012 0.046 0.089 0.125 0.185 0.225 0.270 0.370 0.475 0.510 0.560
j=3
0.023 0.110 0.215 0.315 0.430 0.565 0.590 0.620
0.016 0.094 0.195 0.290 0.405 0.495 0.570
3.50(-3)
j=5
7.75(-4)
j=4
0.245
0.129
0.036
8.90(-3)
j=6 1.20(-4) 1.15(-3) 6.15(-3) 0.015 0.043 0.079 0.120 0.150 0.180 0.220 0.305 0.395 0.425 0.470
j=O 1.25(-4) 1.30(-3) 7.10(-3) 0.016 0.046 0.087 0.125 0.155 0.190 0.230 0.320 0.405 0.430 0.480
j=l
(TR (2',(LSTH)
8.65(-4) 6.55(-3) 0.017 0.050 0.086 0.130 0.160 0.210 0.245 0.335 0.430 0.470 0.500
j=2
8.65(-4) 5.90(-3) 0.016 0.05 1 0.094 0.140 0.175 0.220 0.260 0.350 0.445 0.490 0.520
j=3
0.575
0.280 0.380 0.485
0.190
0.100
0.019
2.05(-3)
j=4
0.315 0.415 0.525 0.570 0.605
0.215
0.117
0.031
6.30(-3)
j=5
’ In order to avoid overcrowding standard deviations are included only in this column. The standard deviations for the other columns are of similar magnitude.
6.75 7.00 7.50 8.00 9.00 10.00 11.00 12.00 13.00 14.00 17.00 21.00 23.00 25.00
E, (kcal/mol)
Table 2 Cross sections for the reaction H + HD ( u = 0, j = O-6) + H, + D on the DMBE and LSTH potential energy surfaces
0.250
0.143
0.044
0.015
j=6
S&
Y 2
& 2 8 2
2 “. c
0 F 3 6’ %
4
P
5 g .-‘ ?
!I
oa (AZ’, (DMBE)
0.500 0.650 0.755 0.850 0.915 1.005 standard
0.495 * 0.010 0.630*0.011 0.755 +0.015 0.830*0.015 0.920+0.015 0.995*0.015
15.00 17.00 19.00 21.00 23.00 25.00
a In order to avoid overcrowding
1.70(-3) 4.35(-3) 0.014 0.033 0.097 0.175 0.260 0.340 0.420
j=l
3.30+0.16(-4) 1.70+0.07(-3) 4.35+0.08(-3) 1.50*0.03(-2) 3.30& 0.08(-2) 0.094 f .O.OOl 0.170+0.004 0.250 f 0.005 0.345 * 0.007 0.410~0008
j=O"
7.50 7.75 8.00 8.50 9.00 10.00 11.00 12.00 13.00 14.00
E,, (kcal/mol) j=3
deviations
0.520 0.650 0.780 0.855 0.960 1.030
4.10(-3) 0.017 0.038 0.102 0.185 0.270 0.340 0.435
are included
0.505 0.675 0.780 0.895 0.985 1.050
3.30(-3) 0.017 0.038 0.106 0.185 0.275 0.360 0.435
1.70(-3) 7.10(-4)
j=2
0.117 0.315 0.510
0.109 0.295 0.490
0.975 1.065 1.125
0.740
3.35(-3)
j=6
2.4@-3)
j=5
0.435 0.580 0.685 0.775 0.855 0.910
9.90(-4) 3.25(-3) 0.014 0.031 0.084 0.150 0.230 0.300 0.370
j=O
0.455 0.560 0.665 0.790 0.840 0.940
l.lCt-3) 3.70(-3) 0.015 0.03 1 0.084 0.155 0.230 0.305 0.385
j=l
0.455 0.595 0.715 0.775 0.870 0.900
1.20(-3) 4.25(-3) 0.015 0.035 0.095 0.170 0.240 0.320 0.400
j=2
0.480 0.615 0.735 0.825 0.910 1.005
1.50(-3) 4.10(-3) 0.018 0.041 0.103 0.180 0.260 0.335 0.415
j=3
0.530 0.675 0.790 0.855 0.940 1.070
1.20(-3) 4.35(-3) 0.017 0.046 0.109 0.190 0.270 0.360 0.440
j=4
0.925 1.005 1.075
0.690
0.460
0.290
0.115
5.05(-3)
j=5
only in this column. The standard deviations for the other columns are of similar magnitude.
0.535 0.680 0.820 0.925 0.980 1.065
2.50(-4) 2.55(-3) 0.014 0.034 0.108 0.195 0.290 0.380 0.480
j=4
(~a (A* ) (LSTH)
Table 3 Cross section for the reaction H + D, (U = 0, j = O-6) + HD + D on the DMBE and LSTH potential energy surfaces
0.495
0.310
0.126
7.75(-3)
j=6
S. Hochman.Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
34
cross sections which were obtained from the calculations with the limited ranges of Or were multiplied by the factor (1 - cos 0r,max)/2 in order to derive the cross sections that would be obtained if Or had been selected from the full possible range. Cross sections as a function of collision energy and initial rotational state, for the D + H 2, H + HD
and H + D 2 reactions, are presented in Tables 1-3. Standard deviations are indicated only in one column of each table (for j = 0 on the DMBE PES), in order to avoid overcrowding of the Tables. These are typical standard deviations. The standard deviations which are not indicated in the other columns are of similar magnitude.
D+H 2
H+HD
2 .... i .... i .... i .... a) (
,,
"~-,l''"='" ~ ,,
....
I ....
H+D 2
I''"
, . . . . . . . .
, ....
I ....
, ....
(g)
LSTH
[
I
I~
2
,'"1 .... I .... i .....
~-
+
(b)
.... i .... r .......
(e)
~
i .... i .... i ....
th)
OMBE
,~
1S
f
0
' ....
2
''''1''''1
....
---0-- j=O ~ j=l
~"
~
10
15
I ....
[ ....
~ ~
5
I ....
,
j=3
20
25 5
10
15
20
25 5
10
15
20
25
Err ( k c a l / m o l ) Fig. 1. Ratios of cross sections trR(j)/OrR( j = 0) on the LSTH (top row) and DMBE (middle row) potential energy surfaces and ratios of cross sections trR(DMBE)/o'R(LSTH) for specific values of j (bottom row) as a function of collision energy for the reactions D + H 2 (left column), H + H D (middle column) and H + D 2 (right column). ( O ) j = 0 ; ( l ) j = 1; ( v ) j = 3 ; ( A ) j = 5 . In order to avoid overcrowding the results for j = 2, 4 and 6 and part of the points for the other j states were not included in the figure.
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
rows, the ratios OrR(j)/o'R(j = 0) for the LSTH and DMBE PESs, respectively, are presented. In the bottom row, the ratios O-R(DMBE)/o'R(LSTH) for various values of j are shown. In order to avoid overcrowding, the results for j = 2, 4 and 6 are omitted from the figure. The different effects of
Effects of rotation on reactivity, based on Tables 1-3, are presented in Figs. 1 and 2. In Fig. 1, ratios of cross sections, rather than absolute values, are presented as a function of collision energy, for the D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) reactions. In the two upper
D+H 2
H+HD
J
J
023 4
,,
,
,
4
5
,
,
0234 2 , ,
(a)
5 ,
,
H+D 2
J
3456
6 ,
iii
i
i
i
i
7
8
i
i
(i)
(e) Err=7
35
kcal/mol
below threshold 2
~ 0
~,~,
LSTH DMBE
i , I , I ,
16 1, ,
i
i
,
0 ii
i
i
i
i
i
i
0.9
it i i
I
i
L
i
I
i
I
i
(b) Err=8
kcal/mol
0.6l
.< 8
0
2
0.3
o v
L
0 o 4J 50 0 G)
, (c)
40
,
,
1
,
~1 i
L
i
,
i
I
,
i
I
I
,
I
03 0
,
I
:
"t/
:
32::::
: :
:
13"':::
i
Etr=10 kcal/mol
0
0.0
9f
9
20
80
,
I
i
I
,
I
i
,
L
I
24
Etr=l 2 kcal/mol 70
20
L
;/ i
i
(d)
I
i,
i
i
i
h
i
24
16
50
I
0
2
4
6
2
4
6
L
0
2
~
I
4
Erot(kcal/mol)
Fig. 2. Cross sections (in units of 0.01 ,~2) as a function of rotational energy Erot (bottom scale) or rotational state j (top scale) for specific collision energies for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) on the LSTH and DMBE potential energy surfaces. The collision energies are 7 kcal/mol (top row), 8 kcal/mol (second row), 10 kcal/mol (third row) and 12 kcal/mol (bottom row). ( 0 ) LSTH PES; (E]) DMBE PES.
36
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
rotation on reactivity on the two PESs for the three isotopomeric reactions is clearly seen from Fig. 1. The left column, which represents our results for the D + H 2 reaction, clearly illustrates the effect of rotation reported earlier by Aoiz et al. [7]. The negative effect of rotation at low collision energies and the beneficial role at high collision energies is obvious. The negative effect at low collision energies is much more pronounced for the DMBE than for the LSTH PES. It can also be seen that the collision energy at which rotation starts to become helpful for reaction is significantly higher for the DMBE than for the LSTH PES. The middle colunm of Fig. 1 shows the results for the H + HD reaction. As can be seen, the negative effect of rotation is much weaker than for the D + H e reaction. However, it is significantly stronger for the DMBE than for the LSTH PES. Actually, for the LSTH PES, a decay of reactivity with j (up to j = 3) is observed only for collision energies very near threshold. For higher collision energies, or higher rotational states even near threshold, rotation promotes reaction. For the DMBE PES, rotation (up to j = 3) causes a decline in reactivity up to a collision energy of about 12 kcal/mol, as compared to 19 k c a l / m o l for the D + H 2 reaction. For j = 5 rotation is helpful for reaction even for very low collision energies. Fig. 1 does not include results for j = 4, but from Table 2 it can be seen that for j = 4 rotation causes a decline in reactivity only up to a collision energy of about 10 kcal/mol. The right column of Fig. 1 shows the results for the H + D 2 reaction. In this case, rotation promotes reaction on the LSTH PES even for low collision energies, and causes a relatively small decrease in reactivity for the DMBE PES only at very low collision energies (below 8.5 kcal/mol, as compared to 12 k c a l / m o l for H + HD and 19 k c a l / m o l for D + H2). At higher collision energies rotation promotes reaction. The bottom row of Fig. 1, which shows ratios of O'R(DMBE)/trR(LSTH) for different values of j, and the data in Tables 1-3, indicate that at low collision energies and low values of j ( j = 0 for D + H 2, j = 0 and 1 for H + H D and j = 0 - 2 for H + D 2) reactivity on the DMBE PES is higher than on the LSTH PES for all three isotopomeric reactions, but the differences tend to become smaller with the
increase in collision energy. For higher values of j and low collision energies, the reactivity on the DMBE PES becomes smaller than the reactivity on the LSTH PES, but with the increase in collision energy they tend at first to become nearly equal and then to become higher on the DMBE PES. Fig. 1 indicates an obvious trend in the effect of rotation on reactivity in the series of the isotopomeric reactions D + H 2, H + HD and H + D 2, for each of the two potential energy surfaces. This trend is probably related to the increase in the moment of inertia, and therefore also in the difficulty in orienting the molecules towards collinearity, in the order H 2 < HD < D 2. This means that the effect of orientation becomes less important in the order D + H 2 > H + HD > H + D 2. As a result, the negative effect of rotation on reactivity also becomes less significant in the same order, and the beneficial effect of rotation starts at lower collision energies and lower values of j, for each of the two potential energy surfaces. It is clear from Fig. 1 that for each of the three isotopomeric reactions the orienting effect on the DMBE is significantly stronger than on the LSTH PES, in accord with the results and conclusions of Aoiz et al. [7] with regard to the D + H 2 reaction. In accord with this conclusion, we found that the average impact parameter and collision angle leading to reaction at low collision energies, are larger for the DMBE PES than for the LSTH PES, but become nearly the same at higher collision energies. The average values of these parameters for low collision energies are larger for the D + H 2 reaction than for the H + HD and the H + D 2 reactions. As might be expected, the examination of individual trajectories for non-rotating reagents under identical initial conditions, showed that indeed the orienting effect becomes weaker in the order D + H 2 > H + HD > H + D 2 and is stronger for the DMBE than for the LSTH PES. The effect of rotation on reactivity on the LSTH and DMBE PESs for the three isotopomeric reactions is displayed in a different manner in Fig. 2. In this Figure, cross sections are presented as a function of rotational energy Ero t (bottom scale), or rotational state j (top scale), for four specific collision energies (7, 8, 10 and 12 kcal/mol). The left column represents the results for D + H 2, the middle column for H + HD and the fight column for H + D 2. It should
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
be noted that the Ero t scales are the same for the three reactions, but the j scales are different. The significant differences in behavior between the three isotopomers, which was shown in Fig. 1, is clearly seen also in this Figure. For the D + H a reaction, a qualitatively similar behavior for the two PESs is observed. For both of them, cross sections decrease initially with j and then recover (except for the lowest collision energy (7 kcal/mol) for which the cross sections do not seem to recover at least up to j = 5). For low values of j, the reactivity on the DMBE PES is higher than on the LSTH PES, but it becomes smaller at higher j levels, since the decrease of cross sections with j is faster on the DMBE PES than on the LSTH PES. At still higher j levels, the cross sections on the two PESs tend to be nearly equal. A different behavior is observed for the H + HD reaction. In this case, for low collision energies (up to 10 kcal/mol), different trends are observed for the two PESs. While for the LSTH PES cross sections tend to increase with j, for the DMBE PES they first decrease down to a minimum value and then increase strongly. This behavior is qualitatively similar to the behavior observed for the D + H 2 reaction, but the initial decrease is much weaker and the increase for higher rotational energies is much stronger for H + HD than for D + H 2. This behavior is consistent with the observations from Fig. 1 concerning the weaker negative effect at low collision energies and the stronger positive effect at high collision energies for H + HD than for D + H 2. For the highest collision energy in Fig. 2 (12 kcal/mol), reactivity increases with j for both surfaces, without any initial decline. For the H + D 2 reaction, reactivity increases with j for the LSTH PES, for all values of collision energy. For the DMBE PES, for a collision energy of 8 kcal/mol, which is only slightly above threshold, reactivity decreases initially with j and then increases. For higher collision energies, reactivity increases with j without any initial decrease. This behavior is consistent with the observation from Fig. 1 that only near threshold and only for the DMBE PES rotation causes a decline in reactivity. For all other conditions, rotation promotes reaction. Some limited calculations were also carried out for the isotopomeric reaction H + T 2 for a collision
37
energy of 8 kcal/mol, which is only slightly above threshold. In this case no decline in reactivity with rotation was observed even on the DMBE PES. Moreover, the cross section on this PES was found to increase strongly with j (3.0 X l0 -4, 4.5 X l0 -4, 6.9X 10 -4, 1 . 0 0 X 10 - 3 and 1.23X 10 - 3 ~2 for j = 0, 1, 2, 3 and 4, respectively). This behavior is consistent with the trends observed for the other isotopomeric reactions. Cross sections for thermal distributions of rotational states, rather than for specific states, are displayed in Fig. 3. In this Figure, cross sections are plotted as a function of the rotational temperature of the reacting molecules Trot, for specific collision energies (the same energies as in Fig. 2: 7, 8, 10 and 12 kcal/mol), for the reactions D + H 2, H + HD and H + D 2, on the LSTH and DMBE PESs. Again, pronounced differences in the behavior for the three isotopomeric reactions are observed, which reflect the differences noticed in Figs. 1 and 2 and the change in the distribution of rotational states with temperature. At the lowest temperatures, mainly the lowest rotational levels are populated. With the increase in temperature, higher levels become more populated and the lower levels become depleted and this can lead either to a decay in total reactivity, or to an increase in reactivity, depending on the rotational distribution, the isotopomeric reaction and the PES. For the D + H e reaction (left column), Fig. 3 shows similar trends for both PESs. For this reaction, the population of the j = 5 level is very small, and that of higher states is negligible, even for the highest temperature in Fig. 3. In this case, as might be expected from Fig. 2, cross sections decrease with the increase in rotational temperature for all values of collision energy for both surfaces, but the decrease is faster for the DMBE than for the LSTH PES. A different behavior is observed for the H + HD (middle column) and H + D 2 (right column) reactions. For H + HD, cross sections on the LSTH PES increase strongly with Trot for all collision energies, except for the lowest one (7 kcal/mol), where the cross section is almost unchanged up to about 300 K and then increases strongly with Trot. For the DMBE PES, for the three lowest collision energies (7, 8 and 10 kcal/mol), reactivity decreases initially with increase in Trot, up to about 500 K, and then increases
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
38
with further increase in temperature. For the highest collision energy (12 kcal/mol), reactivity increases with Trot , even at low rotational temperatures. This behavior reflects the negative role of rotation at low collision energies and low values of j and the favorable role at higher collision energies and higher values of j, as noticed in Figs. 1 and 2.
For the H + D 2 reaction, a different trend for the two PESs (increase for the LSTH PES and decrease for the DMBE PES) is observed only for a collision energy near threshold (8 kcal/mol). At higher collision energies, reactivity increases with Trot for both surfaces. This behavior reflects the negative effect observed from Fig. 2 only for the DMBE PES at low H+HD
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Fig. 3. Cross sections (in units of 0.01 /],2) for thermal distributions of rotational states and specific collision energies as a function of rotational temperature Trot, for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) on the LSTH and DMBE potential energy surfaces. The collision energies are 7 kcal/mol (top row), 8 kcal/mol (second row), 10 kcal/mol (third row) and 12 kcal/mol (bottom row). ( 0 ) LSTH PES; ( D ) DMBE PES.
S. Hochman-Kowal,A. Persky/ Chemical Physics 222 (1997) 29-41
cern between the two PESs. For example, experiments based on Fig. 3(f) or Fig. 3(j). Such experiments, which should involve measurements with molecular beams of cross sections as a function of the rotational temperature of the reacting molecule for specific values of the collision energy, seem to be feasible. However, they may be quite difficult be-
collision energies, and the favorable effect under all other conditions for this PES, and for all collision energies and rotational states for the LSTH PES. The different trends observed in Fig. 3 for the H + HD and H + D 2 reactions at low collision energies on the LSTH and DMBE PESs, point to the possibility of designing experiments in order to dis-
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39
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T (K) Fig. 4. Ratios of rate constants k(j)/k(j = 0) on the LSTH (top row) and DMBE (middle row) potential energy surfaces and ratios of rate constants k(DMBE)/k(LSTH) for specific values of j (bottom row) as a function of temperature for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column). ( O ) j = 0; (11) j = 1; (zx) j = 2; ( v ) j = 3.
40
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
cause of the very low cross sections at the collision energies for which significant differences between the two PESs are predicted. Of course, the experiments suggested here are based on the assumption that trends similar to those found from our QCT calculations are also expected from QM calculations, as concluded by Aoiz et al. [7] with respect to the D + H 2 reaction. The cross sections listed in Tables 1-3 were used to calculate rate constants for specific values of j. Ratios of rate constants k ( j ) / k ( j = 0), as a function of temperature, are presented in the top row (LSTH PES) and middle row (DMBE PES) of Fig. 4, for the reactions D + H2 (left column), H + HD (middle column) and H + D 2 (right column). In the lowest row of Fig. 4 ratios of k ( D M B E ) / k ( L S T H ) for specific values of j ( j = 0-3), for the three isotopomeric reactions, are presented. As might be expected from the effect of rotation on cross sections (Figs. 1-3), the strongest effect of rotation on rate constants is observed for the D + H 2 reaction, being more marked for the DMBE than for the LSTH PES. The largest negative effects are observed at low temperatures, for which the contribution of low collision energies is largest, and therefore rate constants are drastically reduced with increase in j for both PESs. With the increase in temperature, the contribution of higher collision energies becomes more important and the reduction in rate constants with j becomes milder. For the H + HD reaction, the decline of rate constants with j is still significant for the DMBE PES at low temperatures, though much milder than for the D + H 2 reaction, and the effect becomes much weaker at high temperatures. For the LSTH PES, rather slight changes with temperature are observed. The ratio k ( j = 1 ) / k ( j = 0) is slightly higher than unity at low temperatures and decreases very slowly with temperature, remaining somewhat higher than unity over the whole temperature range. The ratios k ( j = 2 ) / k ( j = O) and k ( j = 3 ) / k ( j = O) are very close to unity at low temperatures and increase very mildly with temperature. For the H + D 2 reaction on the LSTH PES, all the ratios are higher than unity over the whole temperature range, being highest for k ( j = 3 ) / k ( j = 0) and lowest for k ( j = 1 ) / k ( j = 0), and decrease with increase in temperature. The ratio k ( j = 1 ) / k ( j
= 0) is nearly unity above 500 K. For the DMBE PES, the ratios of rate constants are slightly lower than unity at low temperatures and they increase very mildly with temperature, becoming slightly higher than unity. Experimental studies designed in order to discern between the LSTH and DMBE PESs, on the basis of the differences in behavior observed from Fig. 4 for the H + HD and the H + D 2 reactions, do not seem to be feasible at present, because of the extreme difficulty in preparing and in carrying out kinetic experiments with rotationally state selected reagents, especially in the present case where the rate constants are very small.
3. Summary The influence of rotation on reactivity for the three isotopomeric reactions D + H 2 (v = 0, j = 0 6), H + H D ( v = 0 , j = 0 - 6 ) and H + D 2 ( v = 0 , j = 0-6), on the LSTH and DMBE potential energy surfaces, was investigated by quasiclassical trajectory calculations. The results of this study can be rationalized in terms of the differences in the orienting forces (towards collinearity) of the two PESs, and in the moments of inertia of the reacting molecules. As was discussed by Aoiz et al. [7] with respect to the D + H 2 reaction, the orienting forces are stronger for the DMBE PES than for the LSTH PES. In accordance with this conclusion, they found for the D + H e reaction, and we found also for the H + HD and H + D 2 reactions, that for non-rotating reagents the reactivity on the DMBE PES is larger than on the LSTH PES. The differences are most significant for low collision energies and they tend to become smaller with the increase in collision energy, as the orienting forces become less significant. With the increase in rotational energy cross sections tend to decrease at low collision energies and to increase at high collision energies. A clear and systematic trend in the influence of rotation on reactivity in the series of isotopomeric reactions D + H 2, H + H D and H + D 2 was observed. This trend can probably be related to the increase in the moment of inertia and, correspondingly, in the difficulty in orienting the reacting
s. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
molecules towards collinearity, in the order n 2 < HD < D 2. Accordingly, the orienting effect is most significant for D + H 2 and least significant for H + D 2, and the decline in reactivity with rotation is strongest for D + H 2 and weakest for H + D 2. For the DMBE PES, the decline in reactivity with rotation is observed up to a collision energy of about 19 k c a l / m o l for D + H2, up to about 12 k c a l / m o l for H + HD and only up to about 8.5 k c a l / m o l for H + D 2. Above these collision energies rotation becomes beneficial for reaction. For the LSTH PES, a decline in reactivity with rotation is observed up to a collision energy of about 13 k c a l / m o l for D + H 2 and only near threshold for H + HD. For H + D 2, rotation does not cause a decline in reactivity even near threshold. The limited calculations which were carried out for the H + T 2 reaction gave results which are consistent with the trends observed for the other isotopomeric reactions. In this case rotation caused a significant increase in reactivity even on the DMBE PES, even near threshold. The most distinct differences between the two PESs were observed for the H + HD and H + D 2 reactions at low collision energies. These differences can, in principle, serve as the basis for experimental studies in order to discern between the two PESs. Such experiments should involve measurements with molecular beams of cross sections as a function of rotational temperature for specific collision energies. For example, Fig. 3(f) predicts for the H + HD reaction at a collision energy of 8 k c a l / m o l , an increase in the cross section with Trot for the LSTH PES, and a decrease down to a m i n i m u m value around T~ot = 500 K and then an increase for the DMBE PES. Fig. 3(j) predicts for the H + D 2 reaction, also for a collision energy of 8 k c a l / m o l e , an increase with Trot for the LSTH PES and a decrease for the DMBE PES over the whole temperature range (up to 1000 K). Such experiments seem to be feasible, though they may be quite difficult because
41
of the very low cross sections under the conditions for which the most significant differences between the two PESs are observed.
Acknowledgements We are grateful to professor D.G. Truhlar for providing us with the computer programs for the LSTH and DMBE potential energy surfaces
References [1] B. Liu, J. Chem. Phys. 58 (1973) 1925. [2] P. Siegbahn,B. Liu, J. Chem. Phys. 68 (1978) 2457. [3] D.G. Truhlar,C.J. Horowitz, J. Chem. Phys. 68 (1978) 2466; 71 (1979) 1514 (E). [4] A.J.C. Varandas, F.B. Brown, C.A. Mead, D.G. Tmhlar, N.C. Blais, J. Chem. Phys. 86 (1987) 6258. [5] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 95 (1991) 4343. [6] S.L. Mielke, G.C. Lynch, D.G. Truhlar, D.W. Schwenke, J. Phys. Chem. 98 (1994) 8000. [7] F.J. Aoiz, L. Banares, T. Diez-Rojo, V.J. Herrero, V. Saez Rabanos, J. Phys. Chem. 100 (1996) 4071. [8] N. Sathyamurthy,Chem. Rev. 83 (1983) 601. [9] J.C. Polanyi,J.L. Schreiber, Faraday Discuss. Chem. Soc. 62 (1977) 267. [10] A. Persky, M. Broida, J. Chem. Phys. 81 (1984) 4352. [11] H. Loesch, Chem. Phys. 112 (1987) 85. [12] H.R. Mayne, S.K. Minick, J. Phys. Chem. 91 (1987) 1400. [13] A. Persky, H. Kornweitz,Chem. Phys. 130 (1989) 129. [14] H. Kornweitz, A. Persky, I. Schechter, R.D. Levine, Chem. Phys. Lett. 169 (1990) 489. [15] M. Zhao, D.G. Truhlar,D.W. Schwenke,D.J. Kouri,J. Phys. Chem. 94 (1990) 7074. [16] S.M. Auerbach, W.H. Miller, J. Chem. Phys. 100 (1994)
1103. [17] E. Rosenman,A. Persky, Chem. Phys. 195 (1995) 291. [18] A. Persky, H. Kornweitz,J. Phys. Chem. 91 (1987) 5496. [19] M. Karplus, R.N. Porter, R.D. Sharma, J. Chem. Phys. 43 (1965) 3259. [20] D.G. Truhlar,J.T. Muckerman,in: Atom-MoleculeCollision Theory, A Guide for the Experimentalist,ed. R.B. Bernstein (Plenum Press, New York, 1979) ch. 16.
Chemical Physics 222 (1997) 29-41
Quasiclassical trajectory study of the effect of rotation on reactivity for the reactions D + H 2, H + HD and H + D 2 on the LSTH and D M B E potential energy surfaces Sipora Hochman-Kowal, Avigdor Persky * Department of Chemistry, Bar-llan Universi~, Ramat Gan 52900, Israel Received 3 April 1997
Abstract The influence of rotation on reactivity for the isotopomeric reactions D + H 2 (v = 0, j = 0-6), H + HD (v = 0, j = 0-6) and H + D 2 (v = 0, j = 0-6) on the LSTH and DMBE potential energy surfaces has been investigated by quasiclassical trajectory calculations. The results are discussed in terms of the differences in the orienting character of the potential energy surfaces and the differences in the moments of inertia of the reacting molecules. The most distinct differences between the two potential energy surfaces are observed for the H + HD and H + D 2 reactions at low collision energies. The feasibility of carrying out experimental studies based on these differences in order to discern between the two surfaces is discussed. © 1997 Elsevier Science B.V.
1. Introduction The H + H 2 reaction and its isotopomeric analogs have been the focus of extensive experimental and theoretical studies, in the area of molecular dynamics, in recent decades. At present, three accurate ab-initio potential energy surfaces (PESs) are available for this system. These are the L S T H [1-3], D M B E [4] and B K M P [5] PESs. Recently, Truhlar and co-workers [6] carried out calculations of accurate quantum mechanical (QM) rate constants for the D + H 2 reaction on these three PESs, over a wide temperature range. They found that the LSTH and D M B E PESs give rate constants which are close to each other (within less than 8% over the temperature range 2 0 0 - 9 0 0 K) and which are in excellent agree-
* Corresponding author. Fax: 972 3 5351250.
ment with experimental data (for the D M B E PES within 13% or better over the whole temperature range). Much worse agreement with experiment was obtained for the B K M P PES, especially at low temperatures (higher than experiment by a factor of 2.2 at 200 K and by 42% at 300 K). Very recently, Aoiz et al. [7] carried out extensive quasiclassical trajectory calculations (QCT) for the reaction D + H 2 (v = 0, 1; j = 0 - 7 ) on these three PESs. Special emphasis was put on the influence of rotation on reactivity. For all three potential energy surfaces, at low collision energies, rotational excitation (for j < 5) was found to cause a decline in reactivity, but at sufficiently high collision energies, rotation becomes beneficial for reaction (for discussions of such effects for the H + H 2 reaction and for other a t o m - m o l e c u l e reactions see, for example, the review by Sathyamurthy [8] and Refs. [9-14]). The
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1-01 0 4 ( 9 7 ) 0 0 1 8 0 - 8
30
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
negative effect at low collision energies was found to be more marked for the DMBE PES than for the two other surfaces. Aoiz et al. showed that the influence of rotation on reactivity can be related to the anisotropy of the potential on the way to the barrier. The three PESs have nearly the same collinear barrier (9.65, 9.80 and 9.54 kcal/mol for the DMBE, LSTH and BKMP PESs, respectively), but the change with attacking angle for the DMBE PES is more significant than for the two other PESs (Ref. [7], figs. 12, 13), and therefore the orienting forces (towards collinearity) are stronger for this surface. In the absence of rotation, these orientation forces cause the reactivity on the DMBE PES to be larger than the reactivity on the two other surfaces. However, with the increase in the rotational energy of the H 2 molecule, the effect of orientation is diminished and reactivity decreases. For a rotating H 2 molecule, the average potential for all attacking angles is larger for the DMBE PES than for the two other surfaces and therefore the cross sections for this surface becomes the lowest. At sufficiently high collision energies, the effect of orientation and the differences in the average potentials become less important and cross sections for the DMBE PES tend to be similar to those on the two other surfaces. Aoiz et al. discuss the more limited data available from QM studies concerning the effect of rotation on reactivity on these PESs (D + H 2 (v = 0, j = 0, 1) on the DMBE PES [15] and D + H 2 (v = 1, j = 0 - 3 ) on the LSTH PES [16]). They conclude that trends similar to those obtained from their QCT calculations can also be expected from QM calculations. Based on their results, they suggest that careful measurements of cross sections or rate constants at low temperatures for the D + H 2 reaction with rotationally excited H 2 molecules, can provide data which may be helpful to discern between the various potential energy surfaces. Thus, their calculations indicate that at 300 K the values of the ratio of rate constants k(D + para-H2)/k(D + normal-H 2) are 1.16 for the DMBE PES, 0.93 for the LSTH PES and about 1.00 for the BKMP PES, and the differences between them increase with decrease in temperature. In this publication we report results of our study of the effect of rotation on the reactivity for the
isotopomeric reactions H + HD --* H 2 + D and H + D 2 ---) HD + D on the DMBE and LSTH PESs. The results are compared with results for the reaction D + H 2 ~ HD + H, and conclusions are derived concerning the systematic change in the effect of rotation on reactivity for these three isotopomeric reactions. Conclusions are also derived concerning the feasibility of experimental studies which may be helpful to discern between the DMBE and LSTH potential energy surfaces.
2. Results and discussion Detailed QCT calculations were carried out for the reactions H + HD (v = 0, j = 0-6) ~ H 2 + D and H + D 2 ( v = 0 , j = 0 - 6 ) ~ H D + D on the DMBE and LSTH potential energy surfaces. For the purpose of comparison, similar calculations were also carried out for the reaction D + H 2 (v = 0, j = 0 - 6 ) ~ HD + H. The computational procedure employed was similar to the procedure employed by us in earlier QCT calculations for other atom-molecule reactions, such as the O + HC1 [10] and the F + H 2 [17] reactions. For each set of initial conditions (collision energy E~ and rotational state j), between 20,000 and 160,000 trajectories were calculated. Based on preliminary calculations, the maximum impact parameter, in each case, was chosen to be somewhat larger than the largest one leading to reaction (by about 8%). The initial distance between the atom and the center of mass of the diatomic molecule was 6 ,~ in all the calculations. In the calculations for j = 0, we employed a procedure which we have used in earlier studies [ 18] in order to significantly improve the statistics of the calculations. This procedure involves calculations over a limited range of the initial azimuthal orientation angle Or , the angle between the molecular axis of the reacting molecule and the initial direction of the relative velocity vector. Thus, instead of randomly selecting Or (according to cos 0r) from the full possible range (0-180°), as is usually done in trajectory calculations [19,20], this parameter was selected between 0 ° and 0r 0r,max was appropriately chosen to be somewhat larger than the highest orientation angle leading to reactive collisions for the specific collision energy of each set of calculations. The ....
.
0.170 0.235 0.390
1.545 1.695 1.745 standard
0.078 + 0.002 0.145 + 0.003
0.220 f 0.002 0.300*0.005
0.465 + 0.006
0.625 f 0.008 0.775 *0.010
0.885 + 0.012 1.120+0.015 1.310*0.020 1.445 f 0.020
1.550 + 0.020 1.705 + 0.020 1.765 + 0.020
7.50 8.00
8.50 9.00
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11.00 12.00
13.00 15.00 17.00 19.00
21.00 23.00 25.00
a In order to avoid overcrowding
0.835 1.065 1.260 1.435
0.560 0.715
0.043 0.105
0.027
4.95 + 0.04(-2)
1.585 1.670 1.760
0.750 0.990 1.205 1.385
0.465 0.610
0.300
0.090 0.155
1.530 1.715 1.825
0.675 0.970 1.170 1.430
0.370 0.530
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0.040 0.085
4.85(-4) 0.010
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2.50(-3) 0.035
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3.80(-4) 2.00(-3) 0.010
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The standard
1.745 1.930
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0.565
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energy surfaces
deviations
1.415 1.500 1.605
0.760 0.970 1.150 1.300
0.515 0.640
0.365
0.160 0.225
0.057 0.100
0.036
2.80(-3) 5.OOG3) 9.55(-3) 0.020
6.05(-4)
j=O
1.445 1.615 1.710
0.725 0.970 1.160 1.275
0.445 0.595
0.300
0.100 0.160
0.015 0.050
4.75(-3)
5.80(-4)
j=2
for the other columns
1.445 1.530 1.645
0.740 0.950 1.135 1.315
0.500 0.640
0.350
0.155 0.220
0.048 0.095
0.030
3.55(-4) 1.35(-3) 5.25(-3) 0.013
j=l
0s (Z.2, (LSTH)
Table I Cross sections for the reaction D + H, (I: = 0, j = O-6) + DH + H on the DMBE and LSTH potential
1.570 1.760 1.855
0.680 0.935 1.195 1.385
0.365 0.525
0.240
0.060 0.105
6.30(-3) 0.025
1.25(-3)
j=4
1.655 1.810
0.995
0.560
0.240
0.025
j=5
are of similar magnitude.
1.505 1.625 1.755
0.690 0.940 1.185 1.360
0.395 0.530
0.250
0.065 0.110
4.50(-3) 0.025
1.40(-3)
j=3
1.780 1.940
1.055
0.585
0.270
0.035
j=6
? 2
;= 2 2 % s
s z.
D 9 ii’ e
3 \
P
% 2 _E ?
8
9 0 x
j=l 2.35(-4) 1.40(-3) 7.90(-3) 0.020 0.052 0.094 0.135 0.175 0.215 0.255 0.350 0.440 0.480 0.5 15
j=O"
4.85 *0.40(-4) 1.85~0.10(-3) 8.65 + 0.30(-3) 2.00* 0.05(-2) 0.053 * 0.001 0.096 + 0.002 0.135 *0.004 0.175 * 0.003 0.210*0.005 0.255 +0.005 0.360 + 0.007 0.440*0.010 0.490+_0.010 0.505 *0.010
~a (A*‘, (DMBE)
l.lO(-4) 3.80(-3) 0.015 0.052 0.092 0.135 0.180 0.225 0.255 0.350 0.460 0.490 0.520
j=2 1.40(-4) 2.80(-3) 0.012 0.046 0.089 0.125 0.185 0.225 0.270 0.370 0.475 0.510 0.560
j=3
0.023 0.110 0.215 0.315 0.430 0.565 0.590 0.620
0.016 0.094 0.195 0.290 0.405 0.495 0.570
3.50(-3)
j=5
7.75(-4)
j=4
0.245
0.129
0.036
8.90(-3)
j=6 1.20(-4) 1.15(-3) 6.15(-3) 0.015 0.043 0.079 0.120 0.150 0.180 0.220 0.305 0.395 0.425 0.470
j=O 1.25(-4) 1.30(-3) 7.10(-3) 0.016 0.046 0.087 0.125 0.155 0.190 0.230 0.320 0.405 0.430 0.480
j=l
(TR (2',(LSTH)
8.65(-4) 6.55(-3) 0.017 0.050 0.086 0.130 0.160 0.210 0.245 0.335 0.430 0.470 0.500
j=2
8.65(-4) 5.90(-3) 0.016 0.05 1 0.094 0.140 0.175 0.220 0.260 0.350 0.445 0.490 0.520
j=3
0.575
0.280 0.380 0.485
0.190
0.100
0.019
2.05(-3)
j=4
0.315 0.415 0.525 0.570 0.605
0.215
0.117
0.031
6.30(-3)
j=5
’ In order to avoid overcrowding standard deviations are included only in this column. The standard deviations for the other columns are of similar magnitude.
6.75 7.00 7.50 8.00 9.00 10.00 11.00 12.00 13.00 14.00 17.00 21.00 23.00 25.00
E, (kcal/mol)
Table 2 Cross sections for the reaction H + HD ( u = 0, j = O-6) + H, + D on the DMBE and LSTH potential energy surfaces
0.250
0.143
0.044
0.015
j=6
S&
Y 2
& 2 8 2
2 “. c
0 F 3 6’ %
4
P
5 g .-‘ ?
!I
oa (AZ’, (DMBE)
0.500 0.650 0.755 0.850 0.915 1.005 standard
0.495 * 0.010 0.630*0.011 0.755 +0.015 0.830*0.015 0.920+0.015 0.995*0.015
15.00 17.00 19.00 21.00 23.00 25.00
a In order to avoid overcrowding
1.70(-3) 4.35(-3) 0.014 0.033 0.097 0.175 0.260 0.340 0.420
j=l
3.30+0.16(-4) 1.70+0.07(-3) 4.35+0.08(-3) 1.50*0.03(-2) 3.30& 0.08(-2) 0.094 f .O.OOl 0.170+0.004 0.250 f 0.005 0.345 * 0.007 0.410~0008
j=O"
7.50 7.75 8.00 8.50 9.00 10.00 11.00 12.00 13.00 14.00
E,, (kcal/mol) j=3
deviations
0.520 0.650 0.780 0.855 0.960 1.030
4.10(-3) 0.017 0.038 0.102 0.185 0.270 0.340 0.435
are included
0.505 0.675 0.780 0.895 0.985 1.050
3.30(-3) 0.017 0.038 0.106 0.185 0.275 0.360 0.435
1.70(-3) 7.10(-4)
j=2
0.117 0.315 0.510
0.109 0.295 0.490
0.975 1.065 1.125
0.740
3.35(-3)
j=6
2.4@-3)
j=5
0.435 0.580 0.685 0.775 0.855 0.910
9.90(-4) 3.25(-3) 0.014 0.031 0.084 0.150 0.230 0.300 0.370
j=O
0.455 0.560 0.665 0.790 0.840 0.940
l.lCt-3) 3.70(-3) 0.015 0.03 1 0.084 0.155 0.230 0.305 0.385
j=l
0.455 0.595 0.715 0.775 0.870 0.900
1.20(-3) 4.25(-3) 0.015 0.035 0.095 0.170 0.240 0.320 0.400
j=2
0.480 0.615 0.735 0.825 0.910 1.005
1.50(-3) 4.10(-3) 0.018 0.041 0.103 0.180 0.260 0.335 0.415
j=3
0.530 0.675 0.790 0.855 0.940 1.070
1.20(-3) 4.35(-3) 0.017 0.046 0.109 0.190 0.270 0.360 0.440
j=4
0.925 1.005 1.075
0.690
0.460
0.290
0.115
5.05(-3)
j=5
only in this column. The standard deviations for the other columns are of similar magnitude.
0.535 0.680 0.820 0.925 0.980 1.065
2.50(-4) 2.55(-3) 0.014 0.034 0.108 0.195 0.290 0.380 0.480
j=4
(~a (A* ) (LSTH)
Table 3 Cross section for the reaction H + D, (U = 0, j = O-6) + HD + D on the DMBE and LSTH potential energy surfaces
0.495
0.310
0.126
7.75(-3)
j=6
S. Hochman.Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
34
cross sections which were obtained from the calculations with the limited ranges of Or were multiplied by the factor (1 - cos 0r,max)/2 in order to derive the cross sections that would be obtained if Or had been selected from the full possible range. Cross sections as a function of collision energy and initial rotational state, for the D + H 2, H + HD
and H + D 2 reactions, are presented in Tables 1-3. Standard deviations are indicated only in one column of each table (for j = 0 on the DMBE PES), in order to avoid overcrowding of the Tables. These are typical standard deviations. The standard deviations which are not indicated in the other columns are of similar magnitude.
D+H 2
H+HD
2 .... i .... i .... i .... a) (
,,
"~-,l''"='" ~ ,,
....
I ....
H+D 2
I''"
, . . . . . . . .
, ....
I ....
, ....
(g)
LSTH
[
I
I~
2
,'"1 .... I .... i .....
~-
+
(b)
.... i .... r .......
(e)
~
i .... i .... i ....
th)
OMBE
,~
1S
f
0
' ....
2
''''1''''1
....
---0-- j=O ~ j=l
~"
~
10
15
I ....
[ ....
~ ~
5
I ....
,
j=3
20
25 5
10
15
20
25 5
10
15
20
25
Err ( k c a l / m o l ) Fig. 1. Ratios of cross sections trR(j)/OrR( j = 0) on the LSTH (top row) and DMBE (middle row) potential energy surfaces and ratios of cross sections trR(DMBE)/o'R(LSTH) for specific values of j (bottom row) as a function of collision energy for the reactions D + H 2 (left column), H + H D (middle column) and H + D 2 (right column). ( O ) j = 0 ; ( l ) j = 1; ( v ) j = 3 ; ( A ) j = 5 . In order to avoid overcrowding the results for j = 2, 4 and 6 and part of the points for the other j states were not included in the figure.
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
rows, the ratios OrR(j)/o'R(j = 0) for the LSTH and DMBE PESs, respectively, are presented. In the bottom row, the ratios O-R(DMBE)/o'R(LSTH) for various values of j are shown. In order to avoid overcrowding, the results for j = 2, 4 and 6 are omitted from the figure. The different effects of
Effects of rotation on reactivity, based on Tables 1-3, are presented in Figs. 1 and 2. In Fig. 1, ratios of cross sections, rather than absolute values, are presented as a function of collision energy, for the D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) reactions. In the two upper
D+H 2
H+HD
J
J
023 4
,,
,
,
4
5
,
,
0234 2 , ,
(a)
5 ,
,
H+D 2
J
3456
6 ,
iii
i
i
i
i
7
8
i
i
(i)
(e) Err=7
35
kcal/mol
below threshold 2
~ 0
~,~,
LSTH DMBE
i , I , I ,
16 1, ,
i
i
,
0 ii
i
i
i
i
i
i
0.9
it i i
I
i
L
i
I
i
I
i
(b) Err=8
kcal/mol
0.6l
.< 8
0
2
0.3
o v
L
0 o 4J 50 0 G)
, (c)
40
,
,
1
,
~1 i
L
i
,
i
I
,
i
I
I
,
I
03 0
,
I
:
"t/
:
32::::
: :
:
13"':::
i
Etr=10 kcal/mol
0
0.0
9f
9
20
80
,
I
i
I
,
I
i
,
L
I
24
Etr=l 2 kcal/mol 70
20
L
;/ i
i
(d)
I
i,
i
i
i
h
i
24
16
50
I
0
2
4
6
2
4
6
L
0
2
~
I
4
Erot(kcal/mol)
Fig. 2. Cross sections (in units of 0.01 ,~2) as a function of rotational energy Erot (bottom scale) or rotational state j (top scale) for specific collision energies for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) on the LSTH and DMBE potential energy surfaces. The collision energies are 7 kcal/mol (top row), 8 kcal/mol (second row), 10 kcal/mol (third row) and 12 kcal/mol (bottom row). ( 0 ) LSTH PES; (E]) DMBE PES.
36
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
rotation on reactivity on the two PESs for the three isotopomeric reactions is clearly seen from Fig. 1. The left column, which represents our results for the D + H 2 reaction, clearly illustrates the effect of rotation reported earlier by Aoiz et al. [7]. The negative effect of rotation at low collision energies and the beneficial role at high collision energies is obvious. The negative effect at low collision energies is much more pronounced for the DMBE than for the LSTH PES. It can also be seen that the collision energy at which rotation starts to become helpful for reaction is significantly higher for the DMBE than for the LSTH PES. The middle colunm of Fig. 1 shows the results for the H + HD reaction. As can be seen, the negative effect of rotation is much weaker than for the D + H e reaction. However, it is significantly stronger for the DMBE than for the LSTH PES. Actually, for the LSTH PES, a decay of reactivity with j (up to j = 3) is observed only for collision energies very near threshold. For higher collision energies, or higher rotational states even near threshold, rotation promotes reaction. For the DMBE PES, rotation (up to j = 3) causes a decline in reactivity up to a collision energy of about 12 kcal/mol, as compared to 19 k c a l / m o l for the D + H 2 reaction. For j = 5 rotation is helpful for reaction even for very low collision energies. Fig. 1 does not include results for j = 4, but from Table 2 it can be seen that for j = 4 rotation causes a decline in reactivity only up to a collision energy of about 10 kcal/mol. The right column of Fig. 1 shows the results for the H + D 2 reaction. In this case, rotation promotes reaction on the LSTH PES even for low collision energies, and causes a relatively small decrease in reactivity for the DMBE PES only at very low collision energies (below 8.5 kcal/mol, as compared to 12 k c a l / m o l for H + HD and 19 k c a l / m o l for D + H2). At higher collision energies rotation promotes reaction. The bottom row of Fig. 1, which shows ratios of O'R(DMBE)/trR(LSTH) for different values of j, and the data in Tables 1-3, indicate that at low collision energies and low values of j ( j = 0 for D + H 2, j = 0 and 1 for H + H D and j = 0 - 2 for H + D 2) reactivity on the DMBE PES is higher than on the LSTH PES for all three isotopomeric reactions, but the differences tend to become smaller with the
increase in collision energy. For higher values of j and low collision energies, the reactivity on the DMBE PES becomes smaller than the reactivity on the LSTH PES, but with the increase in collision energy they tend at first to become nearly equal and then to become higher on the DMBE PES. Fig. 1 indicates an obvious trend in the effect of rotation on reactivity in the series of the isotopomeric reactions D + H 2, H + HD and H + D 2, for each of the two potential energy surfaces. This trend is probably related to the increase in the moment of inertia, and therefore also in the difficulty in orienting the molecules towards collinearity, in the order H 2 < HD < D 2. This means that the effect of orientation becomes less important in the order D + H 2 > H + HD > H + D 2. As a result, the negative effect of rotation on reactivity also becomes less significant in the same order, and the beneficial effect of rotation starts at lower collision energies and lower values of j, for each of the two potential energy surfaces. It is clear from Fig. 1 that for each of the three isotopomeric reactions the orienting effect on the DMBE is significantly stronger than on the LSTH PES, in accord with the results and conclusions of Aoiz et al. [7] with regard to the D + H 2 reaction. In accord with this conclusion, we found that the average impact parameter and collision angle leading to reaction at low collision energies, are larger for the DMBE PES than for the LSTH PES, but become nearly the same at higher collision energies. The average values of these parameters for low collision energies are larger for the D + H 2 reaction than for the H + HD and the H + D 2 reactions. As might be expected, the examination of individual trajectories for non-rotating reagents under identical initial conditions, showed that indeed the orienting effect becomes weaker in the order D + H 2 > H + HD > H + D 2 and is stronger for the DMBE than for the LSTH PES. The effect of rotation on reactivity on the LSTH and DMBE PESs for the three isotopomeric reactions is displayed in a different manner in Fig. 2. In this Figure, cross sections are presented as a function of rotational energy Ero t (bottom scale), or rotational state j (top scale), for four specific collision energies (7, 8, 10 and 12 kcal/mol). The left column represents the results for D + H 2, the middle column for H + HD and the fight column for H + D 2. It should
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
be noted that the Ero t scales are the same for the three reactions, but the j scales are different. The significant differences in behavior between the three isotopomers, which was shown in Fig. 1, is clearly seen also in this Figure. For the D + H a reaction, a qualitatively similar behavior for the two PESs is observed. For both of them, cross sections decrease initially with j and then recover (except for the lowest collision energy (7 kcal/mol) for which the cross sections do not seem to recover at least up to j = 5). For low values of j, the reactivity on the DMBE PES is higher than on the LSTH PES, but it becomes smaller at higher j levels, since the decrease of cross sections with j is faster on the DMBE PES than on the LSTH PES. At still higher j levels, the cross sections on the two PESs tend to be nearly equal. A different behavior is observed for the H + HD reaction. In this case, for low collision energies (up to 10 kcal/mol), different trends are observed for the two PESs. While for the LSTH PES cross sections tend to increase with j, for the DMBE PES they first decrease down to a minimum value and then increase strongly. This behavior is qualitatively similar to the behavior observed for the D + H 2 reaction, but the initial decrease is much weaker and the increase for higher rotational energies is much stronger for H + HD than for D + H 2. This behavior is consistent with the observations from Fig. 1 concerning the weaker negative effect at low collision energies and the stronger positive effect at high collision energies for H + HD than for D + H 2. For the highest collision energy in Fig. 2 (12 kcal/mol), reactivity increases with j for both surfaces, without any initial decline. For the H + D 2 reaction, reactivity increases with j for the LSTH PES, for all values of collision energy. For the DMBE PES, for a collision energy of 8 kcal/mol, which is only slightly above threshold, reactivity decreases initially with j and then increases. For higher collision energies, reactivity increases with j without any initial decrease. This behavior is consistent with the observation from Fig. 1 that only near threshold and only for the DMBE PES rotation causes a decline in reactivity. For all other conditions, rotation promotes reaction. Some limited calculations were also carried out for the isotopomeric reaction H + T 2 for a collision
37
energy of 8 kcal/mol, which is only slightly above threshold. In this case no decline in reactivity with rotation was observed even on the DMBE PES. Moreover, the cross section on this PES was found to increase strongly with j (3.0 X l0 -4, 4.5 X l0 -4, 6.9X 10 -4, 1 . 0 0 X 10 - 3 and 1.23X 10 - 3 ~2 for j = 0, 1, 2, 3 and 4, respectively). This behavior is consistent with the trends observed for the other isotopomeric reactions. Cross sections for thermal distributions of rotational states, rather than for specific states, are displayed in Fig. 3. In this Figure, cross sections are plotted as a function of the rotational temperature of the reacting molecules Trot, for specific collision energies (the same energies as in Fig. 2: 7, 8, 10 and 12 kcal/mol), for the reactions D + H 2, H + HD and H + D 2, on the LSTH and DMBE PESs. Again, pronounced differences in the behavior for the three isotopomeric reactions are observed, which reflect the differences noticed in Figs. 1 and 2 and the change in the distribution of rotational states with temperature. At the lowest temperatures, mainly the lowest rotational levels are populated. With the increase in temperature, higher levels become more populated and the lower levels become depleted and this can lead either to a decay in total reactivity, or to an increase in reactivity, depending on the rotational distribution, the isotopomeric reaction and the PES. For the D + H e reaction (left column), Fig. 3 shows similar trends for both PESs. For this reaction, the population of the j = 5 level is very small, and that of higher states is negligible, even for the highest temperature in Fig. 3. In this case, as might be expected from Fig. 2, cross sections decrease with the increase in rotational temperature for all values of collision energy for both surfaces, but the decrease is faster for the DMBE than for the LSTH PES. A different behavior is observed for the H + HD (middle column) and H + D 2 (right column) reactions. For H + HD, cross sections on the LSTH PES increase strongly with Trot for all collision energies, except for the lowest one (7 kcal/mol), where the cross section is almost unchanged up to about 300 K and then increases strongly with Trot. For the DMBE PES, for the three lowest collision energies (7, 8 and 10 kcal/mol), reactivity decreases initially with increase in Trot, up to about 500 K, and then increases
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
38
with further increase in temperature. For the highest collision energy (12 kcal/mol), reactivity increases with Trot , even at low rotational temperatures. This behavior reflects the negative role of rotation at low collision energies and low values of j and the favorable role at higher collision energies and higher values of j, as noticed in Figs. 1 and 2.
For the H + D 2 reaction, a different trend for the two PESs (increase for the LSTH PES and decrease for the DMBE PES) is observed only for a collision energy near threshold (8 kcal/mol). At higher collision energies, reactivity increases with Trot for both surfaces. This behavior reflects the negative effect observed from Fig. 2 only for the DMBE PES at low H+HD
D+H 2 '
'
I
'
'
I
'
'
'
I
I
'
'
I
H+D 2 '
'
I
I
~
I
r
I
'
'
I
(i) (a)
0.3
0.2
ot
threshold
below
0.1 ,
0 12
,
,
i
'
'
t
'
'
2.5
i
I
,
,
I
,
,
I
, l ~ , l , , i
r---t
1
8
2.0
v
O -,-I -IJ rJ (9
4
40
o3 O
35
,
,
,
'
-
i
,
,
I
,
,
r
,
'
'
I
'
'
I
,
1.5
,
,
I
,
i
I
I
'
'
I
'
'
I
,
L
I
I
,
I
,
,
I
'
'
I
'
'
1
'
'
I
,
,
I
,
~
I
~
,
]
'
I
'
'
,
'
'
I
'
0.5
Y
c)
I
'
,
(fl
o
i
,
I
,
,
I
0.4
,
,
I
0.3
10
(c)
L) 8
30
,
'
'
i
'
'
I
'
'
I
'
,
t
,
J
I
~
i
i
20
8
30
'
(h) 70
) Etr=12
(I)
kcal/mol-
60 ,
300
600
900
0
=
I
300 T r o
L
,
I
600 t
,
I
I
900
0
300
600
900
(K)
Fig. 3. Cross sections (in units of 0.01 /],2) for thermal distributions of rotational states and specific collision energies as a function of rotational temperature Trot, for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column) on the LSTH and DMBE potential energy surfaces. The collision energies are 7 kcal/mol (top row), 8 kcal/mol (second row), 10 kcal/mol (third row) and 12 kcal/mol (bottom row). ( 0 ) LSTH PES; ( D ) DMBE PES.
S. Hochman-Kowal,A. Persky/ Chemical Physics 222 (1997) 29-41
cern between the two PESs. For example, experiments based on Fig. 3(f) or Fig. 3(j). Such experiments, which should involve measurements with molecular beams of cross sections as a function of the rotational temperature of the reacting molecule for specific values of the collision energy, seem to be feasible. However, they may be quite difficult be-
collision energies, and the favorable effect under all other conditions for this PES, and for all collision energies and rotational states for the LSTH PES. The different trends observed in Fig. 3 for the H + HD and H + D 2 reactions at low collision energies on the LSTH and DMBE PESs, point to the possibility of designing experiments in order to dis-
D+H 2
H+HD
1.5
39
H+D 2
' (d)' I ' ' i ' ' i 'l
'
~ I ' ' ' i ' _~ I
(a) LSTH 1.0
0.5
G li
"r-~ v
, l I , J l l , I
0.0 ,
"r--I
i
'
'
I
'
'
I
' I ' ' I ' ' I '
'
' ' 1 ' ' 1 ' ' 1
(h)
(e)
(b) DMBE
1.0
0.5
I,,,I,,I
LI,,I,,I
0.0 1.5
I ' ' 1 ' ' 1
E~ 1.0 ,-.M
m
0.5
0.0
i
0
L
30O
i
i
i
600
~
,
[
900
I
I
1
,
300
,
I
600
,
,
L
900
I
0
300
i
i
I
600
i
i
h
900
T (K) Fig. 4. Ratios of rate constants k(j)/k(j = 0) on the LSTH (top row) and DMBE (middle row) potential energy surfaces and ratios of rate constants k(DMBE)/k(LSTH) for specific values of j (bottom row) as a function of temperature for the reactions D + H 2 (left column), H + HD (middle column) and H + D 2 (right column). ( O ) j = 0; (11) j = 1; (zx) j = 2; ( v ) j = 3.
40
S. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
cause of the very low cross sections at the collision energies for which significant differences between the two PESs are predicted. Of course, the experiments suggested here are based on the assumption that trends similar to those found from our QCT calculations are also expected from QM calculations, as concluded by Aoiz et al. [7] with respect to the D + H 2 reaction. The cross sections listed in Tables 1-3 were used to calculate rate constants for specific values of j. Ratios of rate constants k ( j ) / k ( j = 0), as a function of temperature, are presented in the top row (LSTH PES) and middle row (DMBE PES) of Fig. 4, for the reactions D + H2 (left column), H + HD (middle column) and H + D 2 (right column). In the lowest row of Fig. 4 ratios of k ( D M B E ) / k ( L S T H ) for specific values of j ( j = 0-3), for the three isotopomeric reactions, are presented. As might be expected from the effect of rotation on cross sections (Figs. 1-3), the strongest effect of rotation on rate constants is observed for the D + H 2 reaction, being more marked for the DMBE than for the LSTH PES. The largest negative effects are observed at low temperatures, for which the contribution of low collision energies is largest, and therefore rate constants are drastically reduced with increase in j for both PESs. With the increase in temperature, the contribution of higher collision energies becomes more important and the reduction in rate constants with j becomes milder. For the H + HD reaction, the decline of rate constants with j is still significant for the DMBE PES at low temperatures, though much milder than for the D + H 2 reaction, and the effect becomes much weaker at high temperatures. For the LSTH PES, rather slight changes with temperature are observed. The ratio k ( j = 1 ) / k ( j = 0) is slightly higher than unity at low temperatures and decreases very slowly with temperature, remaining somewhat higher than unity over the whole temperature range. The ratios k ( j = 2 ) / k ( j = O) and k ( j = 3 ) / k ( j = O) are very close to unity at low temperatures and increase very mildly with temperature. For the H + D 2 reaction on the LSTH PES, all the ratios are higher than unity over the whole temperature range, being highest for k ( j = 3 ) / k ( j = 0) and lowest for k ( j = 1 ) / k ( j = 0), and decrease with increase in temperature. The ratio k ( j = 1 ) / k ( j
= 0) is nearly unity above 500 K. For the DMBE PES, the ratios of rate constants are slightly lower than unity at low temperatures and they increase very mildly with temperature, becoming slightly higher than unity. Experimental studies designed in order to discern between the LSTH and DMBE PESs, on the basis of the differences in behavior observed from Fig. 4 for the H + HD and the H + D 2 reactions, do not seem to be feasible at present, because of the extreme difficulty in preparing and in carrying out kinetic experiments with rotationally state selected reagents, especially in the present case where the rate constants are very small.
3. Summary The influence of rotation on reactivity for the three isotopomeric reactions D + H 2 (v = 0, j = 0 6), H + H D ( v = 0 , j = 0 - 6 ) and H + D 2 ( v = 0 , j = 0-6), on the LSTH and DMBE potential energy surfaces, was investigated by quasiclassical trajectory calculations. The results of this study can be rationalized in terms of the differences in the orienting forces (towards collinearity) of the two PESs, and in the moments of inertia of the reacting molecules. As was discussed by Aoiz et al. [7] with respect to the D + H 2 reaction, the orienting forces are stronger for the DMBE PES than for the LSTH PES. In accordance with this conclusion, they found for the D + H e reaction, and we found also for the H + HD and H + D 2 reactions, that for non-rotating reagents the reactivity on the DMBE PES is larger than on the LSTH PES. The differences are most significant for low collision energies and they tend to become smaller with the increase in collision energy, as the orienting forces become less significant. With the increase in rotational energy cross sections tend to decrease at low collision energies and to increase at high collision energies. A clear and systematic trend in the influence of rotation on reactivity in the series of isotopomeric reactions D + H 2, H + H D and H + D 2 was observed. This trend can probably be related to the increase in the moment of inertia and, correspondingly, in the difficulty in orienting the reacting
s. Hochman-Kowal, A. Persky / Chemical Physics 222 (1997) 29-41
molecules towards collinearity, in the order n 2 < HD < D 2. Accordingly, the orienting effect is most significant for D + H 2 and least significant for H + D 2, and the decline in reactivity with rotation is strongest for D + H 2 and weakest for H + D 2. For the DMBE PES, the decline in reactivity with rotation is observed up to a collision energy of about 19 k c a l / m o l for D + H2, up to about 12 k c a l / m o l for H + HD and only up to about 8.5 k c a l / m o l for H + D 2. Above these collision energies rotation becomes beneficial for reaction. For the LSTH PES, a decline in reactivity with rotation is observed up to a collision energy of about 13 k c a l / m o l for D + H 2 and only near threshold for H + HD. For H + D 2, rotation does not cause a decline in reactivity even near threshold. The limited calculations which were carried out for the H + T 2 reaction gave results which are consistent with the trends observed for the other isotopomeric reactions. In this case rotation caused a significant increase in reactivity even on the DMBE PES, even near threshold. The most distinct differences between the two PESs were observed for the H + HD and H + D 2 reactions at low collision energies. These differences can, in principle, serve as the basis for experimental studies in order to discern between the two PESs. Such experiments should involve measurements with molecular beams of cross sections as a function of rotational temperature for specific collision energies. For example, Fig. 3(f) predicts for the H + HD reaction at a collision energy of 8 k c a l / m o l , an increase in the cross section with Trot for the LSTH PES, and a decrease down to a m i n i m u m value around T~ot = 500 K and then an increase for the DMBE PES. Fig. 3(j) predicts for the H + D 2 reaction, also for a collision energy of 8 k c a l / m o l e , an increase with Trot for the LSTH PES and a decrease for the DMBE PES over the whole temperature range (up to 1000 K). Such experiments seem to be feasible, though they may be quite difficult because
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of the very low cross sections under the conditions for which the most significant differences between the two PESs are observed.
Acknowledgements We are grateful to professor D.G. Truhlar for providing us with the computer programs for the LSTH and DMBE potential energy surfaces
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