J. Phys. Chem. 1993, 97, 7040-7047
7040
Three-Dimensional Reactive Quantum Mechanical Study for the H + Xz (X = H, D, T) Systems: Application of Negative Imaginary Arrangement Decoupling Potentialst I. Last, A. Baram, H. Szichman, and M. Baer’ Department of Physics and Applied Mathematics, Soreq NRC, Yavne 70600, Israel Received: February 12, 1993
-
In this work is presented a three-dimensional quantum mechanical study of three hydrogenic reactions, namely, X2 HX X (X = H , D, T) employing a newly developed approach based on negative imaginary arrangement decoupling potentials. Reactive vibrational-state selected cross sections are obtained from the accurate calculated J = 0 probabilities by employing a Jshifting procedure. The studied (total) energy interval is from threshold to 1.3 eV. Detailed comparisons with previous available quantum mechanical results and with quasi-classical trajectory results are presented.
H
+
+
I. Introduction The quantum mechanical (QM) treatment of chemical exchange processes is the subject of many studies. To a particular category belong those treatments which are called exact. Heretofore, there were several methods which yielded many interesting results (probabilities and cross sections).l-Il In the present work, we discuss one such method” whose characteristic feature is the application of negative imaginary potentials (NIPs). With the use of NIPS we achieve two aims: (a) we produce absorbing boundary conditions (ABCs) and (b) we smoothly decouple irrelevant arrangement channels (ACs). This method has been applied to series of three-atomllb@ and four-atom1Ifsystems in order to calculate exact reactive (exchange) state-to-state S matrix elements and probabilities. Recently, the method was exploited in several successful studies of the cumulative reaction probability functions.12 In the present work are considered the three isotopic reactions
H + H2(uj) H + D,(uj) H
+ T,(uj)
-
-
H,(u’j’)
+H
(1)
HD(v’j’)
+D
(11)
HT(v’j’)
+T
(111)
for which we calculated ( J = 0) state-to-state and total reactive probabilities. These reactions have always been of major interest, primarily due to the possibility of producing state-to-state reactive cross sections from first principles. The H H2 system was studied in detail by many groups,I-lO and the results of our study are shown here only for the sake of completeness. The H + D2 system was studied only sporadically,2~sa+’cJ3 and as for H T2, we were unable to find any published outcomes. Our study was carried out on the LSTH surface,l4 the surface for which the majority of the H + H2 calculations to date were performed. In order to compare the present results with the quasi-classical trajectory (QCT) and QM cross sections, the ( J = 0) probabilities were converted into cross sections by employing a reduced dimensionality approach.15 The paper is organized in the following way: in the next section are discussed some aspects related to the theory, results are presented in the third section, and conclusions are given in the fourth section.
system is converted into a (inelastic) single-arrangement system. In fact, two ACs are considered at a time1ld.c (those are the two for which the reactive S matrix elements have to be obtained): one, the PAC, serves as a sink AC, and the other, the LAC, serves as a source AC. As will be shown, the inelastic Schrodinger equation (SE), which contains the NIPs to create the ABCs, is solved in the sink AC, and an unperturbed elastic SE is solved in the source AC.
If H is the full Hamiltonian of the system, then the SE to be considered is
( E - H)QIA =0
where is the complete wave function and tA stands for a set of quantum numbers related to either an initial or a final state. The aim of the calculation is to obtain the state-to-state S matrix element, namely, S(t,--t,), which is defined aslldJ6
W,--t,) = (+fyJ*fA)
11. Theory
The uniqueness of our approach is in the application of NIPs. With the help of these potentials, a (reactive) multiarrangement t This work was supportedby theU.S.-Israel Binational Science Foundation, by the Israel Academy of Sciences and Humanities, and by ITAMP at the Harvard Smithsonian Center for Astrophysics, Cambridge, MA.
(2)
where is a solution of the v-unperturbed SE, represented in terms of the unperturbed Hamiltonian H,
( E - H,>+r,= 0
(3)
and V, is the u-interaction potential defined as
v, = H-H”
+
+
(1)
(4)
The asymptotic behavior of V, is assumed to be lim V,(R,, ...) = 0 R,+m
(5)
where R , is the u-translational coordinate. To solve eq 1, \k,, is assumed to be of the form
(6) = XI, + +IA where $,A,like $,, is a solution of the (A) unperturbed SE *IA
( E - H,)+fA =0
(7)
and xf, is consequently the solution of the full inhomogeneous SE:
( E - H ) X $ = V,+fA
(8)
Here, Vi is the perturbation potential defined as
v, = H - H ,
0022-365419312097-7040%04.00/0 0 1993 American Chemical Society
(9)
3-D Quantum Mechanical Study
0.5
1
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7041
H+H2
1
0.1
0.5
0.6
0.7
0.8
-
0.9 Et&W
1.0
1.1
0.01
1.2
Figure 1. Reactive (J = 0) transition probabilitiesas a function of total energy: (-) H + Hz(u=Oj=O) H2(u’=O;W’) + H (present), (- - -) H + Hz(u=Oj=O) -.,H2(u’=l,G”) + H (present), (0)ZM results (ref 7a), (A) Zhang et al. results (ref 6b).
0.03
1
I
I
I
I
lim Vk(Rk, ...) = 0
I
()’
I
and behaves asymptotically as
I
0.02 -
RA--
To solve eqs 3 and 7, the unperturbed potentials W, (a= A, v; defined as U - V, where U is the full potential) are chosen to be angular independent and separable in the two a-radial coordinates r, (the interatomic distance) and Ra.17 Thus
w,= u,(r,) + w,(R,);
1
I
0.01 -
a = A, v
(11) where ua(ra) is the asymptotic (diatomic) potential and w,(R,) is a distortion potential which increases as R, decreases.” In the present calculations, w,(R,) (a = A, Y) were chosen to be either
w,(R,) = U(R,,r,=r,,,y,=O);
CY
= A, v
I
(12a)
or
w,(R,) = L2 s -1+ l d cos y, U(Ra,ra=rac,ya);a = A, v
(12b)
In the first case, w,(R,) is the a-collinear potential as calculated at the asymptotic vibrational equilibrium distance role,and in the second case, it is the angular averaged potential again as calculated at r = rae. It is important to emphasize that these two potentials behave differently as a function of R,; the first increases much faster than the second, as R, decreases. In order to solve eq 8, we employ the NIPSwhich are defined adla
where x stands for r, or R,, Ax, is the length of the range for which the N I P is different from zero, and UI* is the maximal height of the NIP. These two parameters must be chosen in such a way that the NIPS will not affect the scattering process. Neuhauser and BaerlO* have established the conditions for that to happen. Accordingly, the following two inequalities must be fulfilled (in atomic units):
where N is the number of sectors, M(n) is the number of vibrotational states included in the expansion a t the nth sector,f,(R,) are Gaussian functions located at R,,, gm(ruy,~R,,) are the vibrotational basis functions, and a(nmlnxj3 are coefficients to be determined. The value of M(n) is determined by either the energy valueECutor the energy value Er. Both will be discussed in section 111.2. 111. Results
E,,’~2/Ax1(8~)’/2 I uIx I A ~ ~ ( 8 p ) ’ / ~ E , , 3 ’ ~(14) where Et, is the translational energy in the x direction and p is the reduced mass of the two colliding species. As for XI, it has to be large enough not to affect the wave function in the interaction region; consequently, the translational coordinate R,I has to be in the range where the translational dependence of the potential
This section is divided into three parts. In the first, the present results are compared with those from other approaches. In the second, a sensitivity study is carried out, and in the third, a few new results are shown and analyzed in various ways. We make two comments: (a) in all cases the three atoms are considered to be distinguishable and (b) all probabilities and
Last et al.
7042 The Journal of Physical Chemistry, Vol. 97,No. 27, 1993
both cases, for a large range of energies. One of the most crucial tests for any method has recently become the reproduction of the detailed vib-rotational reactive transition probabilities for the ortho-para processes:
0.5
H+D*
H
0.5
0.6
0.7
0.9
0.8
1.0
-+
+
1.1
H + D,(u=Oj=OlJ=O)
Figure 3. Reactive ( J = 0) transition probabilities as a function of total HD(d=O,U’) D (present), (- -) energy: (-) H D&=Oj=O) H D&=Oj=O) HD (u’=l,u’) D (present), (0)WL results (ref
+
2b).
-
+
-
+ H,(u=Oj=01J-O)
-
+H
H,(u’=O,l;&IJ=O)
+ Tz( dJ=O)
-
(Ib)
+
HD(v’=O,I;E~J=O) D (IIa)
H + T,(ujlJ=O)
-
-
HT(v’jlJ=O)
+T
(IIIa)
Table I lists 34 different vib-rotational state-to-state transitions for the energy Ebt= 0.7 eV, as calculated by varying ninedifferent characteristic parameters. These parameters are presented in Table I1 and are explained in the following: (1) The first column gives the number of Gaussians, which is identical to the number of sectors employed in the calculations. Most of the calculations were carried out with 25 Gaussians, but we give a few examples calculated with smaller numbers of Gaussians. (2) The second
(Ia’)
as a function of the total energy. The results are compared with those due to Zhang and Miller (ZM)’a and with those of Zhang et a1.6b It can be seen that a very encouraging fit is obtained in
TABLE I: Parametric Study for the System H
+H
H,(u’=lj=1,3)
The results are shown in Figure 3, where they are compared with those of Webster and Light (WL).2b In general, a reasonably good fit is obtained except along the central part of the energy interval, where the WL (0 -0) transition probabilitiesare 5-10s larger than ours. 111.2. Sensitivity Study. To carry out the sensitivity study, we chose the third reaction on our list:
cross sections are multiplied by 2 to account for the two ends of the reagent molecule. III.1. Comparison witb Other Calculations. The use of the NIPS that create ABCs helps to avoid some complications which are encountered in the other approaches. By comparing our results with results due to other methods, we hope to show that employing the NIPS yields results of equal quality as do the more established methods. In Figure 1 are shown the reactive probabilities for the two vibrational state-to-state transition probabilities
H
-
These are shown in Figure 2, where they are compared with those due to ZM7Cand those of Launay and LeDourneuf (LL).4 It can be seen that the present results deviate somewhat from those of ZM along the high-energy range but fit reasonably well with those of LL. In most cases, the deviations are less than 5%. A third comparison is carried out for the two vibrational stateto-state H Dz systems:
Etot(eV)
+
+ H,(u=Oj=O)
HT( v!/lJ=O) + T As Performed for I&
= 0.7 eV’
state initial
final
I
I1
I11
IV
v
VI
VI1
VI11
IX
x
XI
XI1
XI11
XIV
xv
00
00
1.79 3.46 2.55 1.11 0.285 0.029 1.52 2.92 2.08 0.842 0.198 0.018 1.12 2.09 1.37 0.477 0.091 0.721 1.27 0.731 0.201 0.027 0.081 0.141 0.079 0.021 0.066 0.1 1 1 0.058 0.014 0.042 0.067 0.030 0.005
1.77 3.42 2.53 1.11 0.291 0.031 1.51 2.90 2.06 0.840 0.202 0.019 1.12 2.09 1.37 0.477 0.093 0.720 1.27 0.731 0.201 0.027 0.082 0.142 0.080 0.022 0.066 0.112 0.058 0.014 0.042 0.067 0.030 0.005
1.84 3.56 2.64 1.14 0.280 0.026 1.56 2.99 2.14 0.864 0.196 0.017 1.15 2.15 1.42 0.488 0.090 0.720 1.29 0.743 0.204 0.027 0.081 0.141 0.080 0.021 0.065 0.110 0.057 0.013 0.041 0.066 0.030 0.005
1.75 3.13 2.34 1.17 0.318 0.025 1.52 2.70 1.95 0.917 0.231 0.017 1.24 2.21 1.49 0.588 0.116 0.833 1.49 0.899 0.270 0.033 0.081 0.142 0.084 0.026 0.062 0.106 0.060 0.016 0.037 0.062 0.031 0.006
1.80 3.48 2.57 1.12 0.286 0.029 1.53 2.93 2.09 0.848 0.199 0.018 1.12 2.10 1.38 0.479 0.091 0.720 1.27 0.729 0.200 0.027 0.086 0.150 0.086 0.024 0.069 0.117 0.062 0.015 0.044 0.070 0.032 0.006
1.75 3.43 2.57 1.13 0.290 0.032 1.49 2.90 2.09 0.854 0.201 0.020 1.10 2.08 1.39 0.484 0.091 0.709 1.27 0.741 0.204 0.026 0.080 0.141 0.081 0.022 0.065 0.111 0.059 0.014 0.041 0.067 0.031 0.005
1.78 3.40 2.49 1 .08 0.274 0.026 1.52 2.89 2.02 0.819 0.191 0.016 1.11 2.06 1.33 0.464 0.089 0.716 1.25 0.708 0.196 0.028 0.081 0.139 0.077 0.021 0.065 0.109 0.056 0.013 0.042 0.066 0.029 0.005
1.75 3.34 2.49 1.07 0.249 0.023 1.49 2.83 2.02 0.807 0.176 0.014 1.10 2.03 1.33 0.456 0.083 0.708 1.24 0.703 0.192 0.029 0.080 0.138 0.077 0.020 0.066 0.108 0.056 0.013 0.042 0.065 0.029 0.005
1.78 3.44 2.53 1.10 0.282 0.028 1.52 2.90 2.06 0.835 0.196 0.018 1.11 2.08 1.36 0.473 0.090 0.716 1.27 0.726 0.200 0.027 0.080 0.140 0.079 0.021 0.065 0.110 0.057 0.013 0.042 0.066 0.029 0.005
1.78 3.44 2.53 1.11 0.281 0.028 1.52 2.90 2.06 0.836 0.196 0.018 1.11 2.08 1.36 0.474 0.090 0.716 1.27 0.726 0.200 0.027 0.080 0.140 0.079 0.021 0.065 0.110 0.057 0.013 0.042 0.066 0.030 0.005
1.79 3.46 2.55 1.11 0.285 0.029 1.52 2.92 2.07 0.841 0.198 0.018 1.12 2.09 1.37 0.476 0.091 0.721 1.27 0.731 0.200 0.027 0.081 0.141 0.079 0.021 0.066 0.11 1 0.056 0.014 0.042 0.067 0.030 0.005
1.80 3.46 2.54 1.11 0.283 0.028 1.52 2.90 2.06 0.834 0.197 0.018 1.12 2.09 1.37 0.474 0.090 0.723 1.28 0.731 0.200 0.027 0.091 0.157 0.086 0.023 0.100 0.180 0.088 0.018 0.043 0.069 0.032 0.006
1.79 3.46 2.56 1.12 0.286 0.029 1.52 2.92 2.08 0.845 0.199 0.018 1.12 2.10 1.38 0.479 0.092 0.722 1.28 0.734 0.202 0.027 0.080 0.142 0.080 0.022 0.066 0.11 1 0.058 0.014 0.042 0.067 0.030 0.005
1.79 3.49 2.61
3.31
01 02 03 04 05 01
02
00 01 02 03 04 05 00 01 02 03 04
03
00
10
01 02 03 04 00 01 02 03
11
00
12
01 02 03 00 01 02 03
The meaning of the various columns is given in Table 11. The numbers are percentages.
1.15 0.300 0.028 1.52 2.94 2.12 0.869 0.208 0.017 1.12 2.11 1.40 0.495 0.097 0.725 1.29 0.746 0.211 0.029 0.081 0.143 0.081 0.022 0.066 0.112 0.059 0.014 0.042 0.068 0.031 0.005
1.85 2.38 1.14 0.335 0.038 1.57 2.82 1.96 0.869 0.231 0.023 1.14 2.05 1.34 0.497 0.104 0.703 1.26 0.139 0.208 0.028 0.080 0.140 0.080 0.023 0.064 0.109 0.060 0.015 0.039 0.065 0.032 0.006
3-DQuantum Mechanical Study
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 1043 H+H2 v.1
0.60
0.50
0.70
1.10
0.90
0.40
(e) 0.30 -
0.20
-
0.10
-
-0.50
I
0 -
0.50
0.70
1.10
0.90
0.90
0.70
0.30
(9
H+T2 v=1
0.50
-
0.70 Et01(eV)
0.20
-
0.10
-
0 0.90
0.50
0.70
0.90
Etot (ev)
-
Figure 4. Reactive (J = 0) state-selected probabilities as a function of total energy. (a) Results for H + Hz(u=Oj) H~(Zu’j3+ H. (b) Results + HZ(u=lj) Hz(Zu’J? + H. Parts (b) and (e) are for H + D2 and (c) and ( f ) are for H + Tz,respectively. Results for (-) j = 0, (- -) j I 1, (-,-) j I 2, (...) j = 3. for H
column represents the total number of trans-vib-rotational states employed in the expansion of the wave function. The number stands for the sum of the even and the odd (rotational) states. In the calculations, due to the fact that Tz is a homonuclear molecule, the even and the odd states are decoupled, and therefore only about half of the states are included in a single calculation. (3) The third and the fourth columns represent different values of Emt and El employed in the calculations. The difference between the two is as follows. We distinguish between two regions along the R interval: region I, defined by Ri IR IRI (along which the translational NIP is zero), and region 11, defined by RI S R 5 RI + ARI (along which the NIP is different from zero). Here, Ri is the smallest R value. Along region I, the number of states in each sector is controlled by Emt;along region 11, it is controlled by El. In other words, the first region includes all vib-rotational eigenfunctions with vib-rotational eigenvalues smaller than Emtand the second those with eigenvalues smaller than EI. Eat is usually much larger than the total energy Etot, whereas EI may be equal to or even smaller than Etot. (4) The next four columns relate to the parameters of the NIPS (see eq 13). (a) ARI stands for the width of the translational NIP which is located along the range R1 IR 5 Rf, where Rr = RI + ARI.
-
In all cases, Rf was taken as Rf = 3.4 A. (b) Ar1 stands for the width of the vibrational NIP which is located along the range r~I r I rf, where rf = r~ Ar1. In all calculations, the value of rf was taken as rf = 3.2 A. (c) URI and U,I are the heights of the translational and vibrational NIP, respectively (see eq 13). (5) The last column refers to the translational distortion potential used in the calculation. We distinguish between the collinear distortion potential and the average distortion potential (see eqs 12). Table I is arranged in such a way that the results given in column I (which were calculated for the parameters given in the first row of Table 11) are assumed to be the ‘best” results. All other results should be compared with those (the probabilities that are closest to those given in column I are considered to be the more accurate ones). From inspection of the results presented in Table I, the following conclusions can be drawn: (a) The state-to-state reactive probabilities are surprisingly insensitive to the NIP parameters. The variations of uxl’s (x = r, R) in the range 0.3-0.7 eV hardly affect the probabilities, and even making them as small as 0.2 eV causes some changes. The same applies to variations in the ranges of the NIPS (columns
+
Last et al.
7044 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE II: Parametric Study for the System H Tz( &=O) HT( V ~ I J = O+) T’
+
-+
Eat EI MI b URI Or1 (ev) (ev) (4 (A) (eV) (ev) w(R) I 25 3409 1.8 0.7 1.0 1.0 0.5 0.5 d I1 ae a a a a a 0.7 0.7 a I11 a a a a a a 0.3 0.3 a IV a a a a a a 0.2 0.2 a V a 3180 a a 1.2 a a a a VI a a a a a 1 . 2 a a a VI1 a a a a a 0.8a a a VI11 a a a a a 0 . 6 a a a IX a 3021 1.6 a a a a a a X a 2511 1.4 a a a a a a XI a 3362 a 0.6 a a a a a XI1 a 3305 a 0.5 a a a a a XI11 22 2978 a a a a a a a XIV 19 2546 a a a a a a a XV a a a a a a a a f a The parameters are those that were varied during the sensitivity study. NumberofGaussians (sectors) employed in thecalcu1ation.eTotal number of states employed in expanding the x function (=zNn.lM(n)). See eq 15. Average distortion potential. a designates the same value as in the first row. /Collinear distortion potential.
Nb
0.04
Ntot
t
.A
H+Dp
TABLE IIk Estimated Values of the Translational Coordinate (R0)at the (Collinear) Transition State and the CorresDondinn Calculated Rotational Constant ( B ) system Ro (A) B (ev) H+H2 1.3 1.8(-3)” H + D2 1.4 1.2(-3) H + T2 1.6 9.7(4) a Numbers in parentheses stand for powers of ten. ~~~~~
~
~
applied even to cases where &I was reduced from 1 to 0.6 A (the range of h r ~is taken to be 0.6 A I&I I1.2 A). (b) From the next four columns (IX-XII), we learn how the probabilities withstand convergence tests. In columns IX and X are listed results calculated for different Eat values (3 1.4, 1.6 eV) and in columns XI and XI1 for different E1 values (=0.5,0.6 eV). These variations affect the total number of states included in the expansion. In one case, this number decreased from 3409 to 251 1, and still minor changes were observed. The results are more sensitive to the value of E,. We found that making E1 equal to 0.5 eV (which is a value much smaller than the highest open state in the asymptotic region) affects only transitions from u = 1 but has no effect on transitions from u = 0 (as was to be expected). (c) In the next two columns are listed the results as calculated for different numbers of Gaussians (or sectors). The ones in column XI11 were calculated for 22 Gaussians. They are, for all practical purposes, identical to those in column I. The values in column XIV were calculated with 19 Gaussians, and only minor changes are observed. (d) We found that the results were somewhat sensitive to the choice of the (elastic) asymptotic distortion potential w(R).The calculations were carried out employing an average (asymptotic) potential (see eq 12b). In column XV are presented results with a collinear distortion potential (see eq 12a), and they should be compared with those in column I. Only negligible variations are observed. III.3. ReactivehobabilitiesandCrossSections. 111.3.1. Statetestate and State-Selected Probabilities. Reactive state-selected probabilities for the processes
H + H,(o=O, 1$=O. 1,2,31J=O)
-
+
H2(~u’j1J=O) H (Ib)
H
+ D2(u=0,1$=0,1,2,3~J=0)
-*
H D ( ~ U ’ ~ ~ J+ - 0D) (IIb)
H
+ T2(u=0,1;i=0,1,2,3)J=0)
--+
+
HT(~u’~~J=O T ) (IIIb)
I’ Figure 5. Reactive ( J = 0) vib-rotational state-to-state probabilities as
-
a function of fina! rotational quantum number (final vibrational state u‘ = 0). (a) Results for H + H&j) H2(u’=OJ’) + H. The calculations were done for Etot= 1.3 eV. (b) Results for H + Dz(uj) HD(u’=Oj’) + D. The calculations were done for Emt= 0.9 eV. (e) Results for H + T2(uj) HT(u’=OJ’) + T. The calculationswere done for Etot= 0.7 eV. Results for (--) ( u = 0,j = 0), (u = 0,j = l), (- - -) ( u = 0, j = 2), (- -) (u = 0,j = 3), (-) ( u = 1, j = 0).
-
-
(e-)
V-VIII). Most of the differences did not exceed 2% or 3%, and only on rare occasions was the difference larger than 5%. This
as a function of total energy are shown in Figure 4. It is well noticed that in general the most effective mode of motion for promoting the reaction is the translational mode and that the less effective is the rotational mode. Another observation concerns the mass of the diatomics; it can be seen that the heavier the mass the smaller are the reactive probabilities. However, this mass dependence seems to diminish as the initial rotational state becomes larger. In other words, the screening effect caused by the rotational motion seems to be more efficient the lighter the mass of the target molecule. Other interesting findings concern the oscillatory nature of the probability function. As is seen for all threecases, theoscillations gradually disappear as the initial rotational state increases. They would be expected to gradually disappear as the mass of the target molecule became heavier, but this does not seem to be the case; the oscillations are smaller in Dz than in Hz but then become larger again in Tz. More detailed information is given in Figure 5 . This figure presents vib-rotational state-to-state transition probabilities as a
3-DQuantum Mechanical Study
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7045
t
,(, ,e* ,.e'
0
I 0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
1.0
h o t (W
Figure 6. Reactive state-selected integral cross sections as a function of total energy. See Figure 4 for notations. 1.60
1.20
0.80
0.40
0.50
0.70
0.90
r rj-
-
n 0.50
1.10
0.70
1.10
0.90
1
1.20
5. a
s
p
0.80
a
-8
,$ Y
0.40
K
a 0.50
0.50
-
0.70
1.10
0.90
0.70 Eta (eV)
0.50
0.70
0.90
1.10
0.90
Figure 7. Reactive state-selected integral cross sections as a function of total energy, a comparison between QM and QCT results. (a) Results for H H&=OJ=O) Hz H. Parts (b) and (c) are results for Dfand Tz, respectively. (-) QM JSP results (present), (0)QCT results (ref 20a),
+
+
result (ref 21), ( 0 ) QM RIOSA result (ref 22), (A) QM result (ref 23), (A) QM result (ref Sc), (m) QM CS result (ref 5a). function of the final rotational state (but v' = 0), calculated for and those for H Tz for Eta = 0.7 eV. It is noticed that shifting different initial vib-rotational states. All calculations were done more and more energy into internal degrees of freedom strongly for a fixed energy value Ew: the calculations for H + H2 were affects the final rotational state distribution. However, in the done for E,,, = 1.3 eV, the ones for H + D2 for E,,, = 0.9 eV, case of H + Tz this shift only affects the absolute values of the (0)QCT
+
Last et al.
7046 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
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probabilities, in the case of H H2 it also affects the shape of the entire distribution. An intermediate situation is encountered for H + D2. III.3.2. State-Selected Cross Sections. To calculate cross sections (CS) from the ( J = 0) probability functions,weemployed the J-shifting procedure (JSP) as suggested previo~s1y.l~ Having obtained the reactive state-testate ( J = 0) probabilities, we formedthe corresponding (J # 0) probabilities by employing the following JSP:I9
P(r,t)@3,,) = P(t,t)J=OJEtr-BJ(J+l))
(16)
where t x and t , stand for sets of quantum numbers in the X and v ACs, respectively, and B is the (translation) rotational constant: 15.19
B = ft2/2pRt
(17)
Here p is the translational reduced mass and Ro is the translational coordinate of the transition state. Once B is known, eq 16 can be used to calculate the corresponding CSs:I9
a(txf)Ettr) =2J(z--
7r
+ l)P(t,t)J=OIE,,-BJ(J+l)) ( 1 8 )
k%h To employ eqs 16 and 18, one needs to know the values of B or, better, of Ro. These were estimated based on the collinear configuration and are presented in Table I11 (reagents' Ro values were used). The state-selected CSs u(u=O,l;i=0,1,2,3) as a function of total energy for the three systems are shown in Figure 6. As can be seen, most of the oscillatory characteristics disappeared except in the case of H + H2 and ( D = 0, j = 0), for which some of the oscillations withstood the summation over J. One of the reasons to extend the (J = 0) probabilities to CSs is the ability to compare QM CSs with quasi-classicaltrajectory (QCT) CSs. A comparison of the three systems is shown in Figure 7 (the QCT results are due to Mayne2O). All the results are for (u = 0, j = 0). It is noticed that whereas the fit along the low-energy region is reasonably good the two kinds of results deviate considerably as the energy increases. Whenever available, we also show other QM cross sections. Such are the full three-dimensional calculations at Et,, = 0.65 eV for the H H222 system and at Etot= 0.74 eV for the H D2 system9c as well as the two approximate calculations for the H + D2 system, namely, the coupled state treatment at Etot= 0.74 eVS* and the reactive infinite order sudden treatment at Etot = 1.03 eV.23 Whereas a good fit between the JSP calculation and the accurate QM calculation is obtained for the H + H2 system, it seems that for the H D2 system the JSP results increase faster with energy as compared to the QM ones. This is best noticed at the (low) energy value (Etot= 0.74 eV), for which we have two QM results. Some support to this behavior can be found in a recent similar study for the D + H2 system.lgc There also the JSP cross section increased faster than both the QCT and the QM cross sections.
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IV. Conclusions In this article is presented a 3-D QM study of three hydrogenic reactive systems (reactions 1-111). The method we used is based on a solution to the time-independent Schrodinger equation employingnegative imaginary potentialswhich produce absorbing boundary conditions. The study was divided into two parts: 1. The numerical aspect of the study was devoted to convincing the reader that the method is reliable and stable and that it can produce the required QM results. The reliability was demonstrated by reproducing a great number of 3-D vib-rotational stateselected as well as state-to-state reactive probabilities for H H2 and H D2. The stability feature was demonstrated in a
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detailed study of the H T2 system. For that system, we varied nine different characteristicparameters (particularlythose related to the NIPS) and showed that, for all practical purposes, the results remain unchanged. Among other things, we showed that the method possesses the convergence feature; namely, when the size of the basis set is increased, the results converge to definite values. It is important to mention that this study complements a 3-D study of the D H2 systemllC and several collinear studies.l1b,c 2. The scientific part was devoted to some features which characterize these three reactions. In particular, we showed first that the ( J = 0) probabilities and subsequently the JSP cross sections were strongly dependent on the initial rotational state. This is in contrast to what is known from the QCT situations,20 where only a minor dependence was observed. The other important finding of our study is that in the higher energy region (- 1 eV) the JSP CSs were much larger (-30%) than the QCT ones (this applies for all three reactions). These large deviations could be artifacts of the JSP model, and it undoubtedly is important to determine the actual situation. If these deviations are close to real, then QCT results, whether in the low- or in the high-energy region, should be considered with care.
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Acknowledgment. M. Baer thanks Prof. A. Dalgarno for his kind hospitality during a two-month stay at the Institute for Theoretical Atomic and Molecular Physics at the HarvardShithsonian Center for Astrophysics. He also thanks Roi Baer for carrying out the calculation of the JSP cross sections. References and Notes (1) Kupperman, A.; H i p , P. G. J . Chem. Phys. 1986.84,5962. (2) (a) Webster, F.;Light, J. C. J . Chem. Phys. 1986,85,4744;(b) 1989, 90,300. (3) (a) Parker, G. A.; Pack, R. T.; Archer, B. J.; Walker, R. B. Chem. Phys. Lett. 1987,137,564. (b) Pack, R.T.; Parker, G. A. J. Chem. Phys. 1987,87,3888. (4) Launay, J. M.; Le Dourneuf, M. Chem. Phys. Lett. 1989,163,178. (5) (a) Schatz, G. C. Chem. Phys. Lett. 1984,108,532.(b) Colton, M. C.;Schatz,G. C. Chem. Phys. Lett. 1986,124,256.(c) Schatz,G. C. Chem. Phys. Lett. 1988,150,92. (6) (a) Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G.; Zhang, J. Z. H.; Kouri, D. J. J . Phys. Chem. 1986,90,6757.(b) Zhang, J. H. 2.; Kouri,D. J.; Haug,K.;Schwenke,D. W.;Shima,Y.;Truhlar,D. G.J. Chem. Phys. 1988,88,2492.(c) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; Sun, Y.; Zhang, J. 2.H.; Kouri, D. J. J. Phys. Chem. 1987,91,6080. (7) (a) Zhang, J. Z. H.; Miller, W. H. Chem. Phys. Lett. 1987,140,329. (b) Ibid. 1988,153,465. (c) Ibid. 1989,159, 130. (d) Zhang, J. Z. H.; Miller, W. H. J . Chem. Phys. 1988,88,4549. (e) Ibid. 1989,89,4454.(f) Zhang, J. Z. H.; Chu, S.I.; Miller, W. H. J . Chem. Phys. 1989,88, 6233. ( 9 ) Zhang, J. Z. H.; Miller, W. H. J. Chem. Phys. 1989,91,1528. (8) (a) Baer, M. J . Phys. Chem. 1987,91,5846.(b) Baer, M. J. Chem. Phys. 1989,90,3048. (9) (a) Monolopolous, D. E.; Wyatt, R. E. Chem. Phys. Lett. 1988,152, 23. (b) D'Mello, M.; Monolopolous, D. E.; Wyatt, R. E. Chem. Phys. Lea. 1990,168, 113. DMello, M.; Monolopolous, D. E.; Wyatt, R. E.J. Chem. Phys. 1991,94,5985. (10) (a) Neuhauser, D.; Baer, M. J. Chem. Phys. 1989,91,4651. (b) Neuhauser, D.; Baer, M. J. Phys. Chem. 1990,94,185.(c) Neuhauser, D.; Baer,M.J. Chem. Phys. 1990,92,3419.(d)Neuhauser,D.;Baer,M.;Judson, R. S.;Kouri, D. J. J . Chem. Phys. 1990,93,312;Comp. Phys. Comm. 1991, 63, 460. (e) Neuhauser, D.;Judson, R. S.; Jaffe, R.; Baer, M.; Kouri, D. J. J . Chem. Phys. 1991,176,546.( f ) Neuhauser, D.; Judson, R. S.; Kouri, D. J.; Adelman, D. E.; Shafer, N. E.; Kleiner, D. A.; Zare, R. N. Science 1992, 257, 519. (1 1) (a) Baer, M.; Neuhauser, D.; Oreg, Y. J. Chem.Soc.,Furuduy Trans. 1990,86,1721.(b) Last, I.; Neuhauser, D.; Baer, M. J. Chem. Phys. 1992, 96,2017. (c) Last, I.; Baer, M. Chem. Phys. Lett. 1992,189,84.(d) Baer, M.; Nakamura, H. J. Chem. Phys. 1992,96,6565. (e) Last, I.; Baram, A.; Baer,M.Chem.Phys.Lert. 1992,195,435.(f)Szichman,H.;Last,I.;Baram, A.;Baer, M. J . Phys. Chem., in press. (12) (a) Seiderman, T.; Miller, W. H. J . Chem. Phys. 1992,96,4412; 1992,97,2515.(b) Brown, D.; Light, J. C. J. Chem. Phys. 1992,97,5465. (1 3) Zhao, M.; Truhlar, D. G.; Blais, N. C.; Schwenke, D. W.; Kouri, D. J. J . Phys. Chem. 1990,14,6696. (14)Siegbahn, P.; Liu, B. J. Chem. Phys. 1978,64245.Truhlar, D. G.; Horowitz, C. J. Ibid. 1978,68,2466;1979,72, 1514(E). (15)Bowman, J. M. In Adu. in Chem. Phys; Prigogine, I., Rice, S.A., Eds.; J. Wiley and Sons: New York, 1985;Vol. 61,p 115.
3-D Quantum Mechanical Study (16) Miller, W. H.;deHaar, J.O.J. Chem.Phys. 1986,86,6213. Zhang, J. Z. H.; Chu, S.I.; Miller, W. H. J. Chem. Phys. 1988,88, 6233.
Baer, M. Chem. Phys. Lett. 1982,91,43; J . Phys. B 1983, (17) Shima, Y.; 16, 2169. (18) Child, M. S.Mol. Phys. 1991; 72,89. Markovic, N.; Billing, G . D. Chem. Phys. Lett. 1992,195,53. Vibok, A.; Balint-Kurti, G. G. J. Chem. Phys. 1992, 96,7615; J . Phys. Chem. 1992,96, 8712. (19) (a) Sun, Q.;Bowman, J. M.; Schatz, G. C.; Sharp, J. R.; Connor, J. N. L. J . Chem. Phys. 1990,92, 1677. (b) Schatz, G. C.; Sokolovski, D.;
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7041 Connor, J. N. L. J. Chem. Phys. 1991,94,4311. (c) Takada,S.; Ohzaki,A.; Nakamura, H. J. Chem. Phys. 1992,96, 339. (20) Khare, V.; Kouri, D. J.; Baer, M. J. Chem. Phys. 1979, 71, 1188. Nakamura, H.; Ohsaki, A.; Baer, M. J. Phys. Chem. 1986, 90,6176. (21) (a) Mayne, H. R. J. Chem. Phys. 1983, 73, 217. (b) Boonenberg, C. A.; Mayne, H. R. Chem. Phys. Lett. 1984,108, 67. (22) Walker, R. B.; Stechel, E. B.; Light, J. C. J. Chem. Phys. 1978,69,
2922. (23) Last, I.; Ron, S.;Baer, M. Isr. J. Chem. 1989, 29, 451.
