Volume 169, number 5
IS June 1990
CHEMICAL PHYSICS LETTERS
TIME-DEPENDENT (WAVEPACKET) QUANTUM APPROACH TO REACTIVE SCATTERING:
VIBRATIONALLY RESOLVED REACTIOiX PROBABILITIES FOR F+H,+ HF+ H
Daniel NEUHAUSER
Dcparlmentof Chemntry, Princeton
Univers~ly. Princeton,
NJ 08544,
USA
Michael BAER
Department
of Theoretical
Physics and Applied Malhematics,
Soreq Nuclear
Research Center,
Yavne 7G600, Israel
Richard S. JUDSON ’ and Donald J. KOURI
Deparfmeni
qfchemlsrry
Received
I5 March I990
and Department
ofPhysrcs.
University
oJHous~on,
Houston,
TX 77204.5641,
LB’,4
We present total and vibrationally resolved reactive scattering probabilities for the F+H,-HF(
0,) +H reaction in three dimensions, for J=O. obtained using a fully quanta1 wavepacket method. The wavepacket propagation is done with an inelastic
scattering method, using the Jacobi coordinates of the initial arrangement and a complex absorbing potential in the product
(HF+H) arrangement. The complex absorbing potential alleviates having to propagate the packet a large distance out into the
product arrangement, but precludes obtaining rotationally resolved probabilities. However,onedoesnot haveto propagateso far
to obtain vibrationallyresolved reaction probabilities. Results for a range of total energies are obtained in a single wavepacket
propagation. Results are compared with those of Bacic, Kress, Parker and Pack using a time-independent method.
1. Introduction
Recently, tremendous progress has been made in the treatment of three-dimensional gas phase atom-diatom
quantum reactive scattering [ l-201. It is now possible to obtain converged quantum mechanical reaction probabilities in full 3-D for a variety of gas phase atom-diatom reactive systems. Up to now, most 3-D reactive
scattering calculations have been done using a variety of time-independent methods [l-17,20]. There has,
however, also been important progress in the adaptation of time-dependent wavepacket methods for 3-D quantum reactive scattering [ 18,191. The techniques being used are combinations of several ideas: (a) the close
coupling wavepacket (CCWP) method [21,22] for inelastic atom-diatom scattering in 3-D, (b) the projection
operator technique to couple the inelastic and the reactive parts of the wavepacket [ 231, (c) the application
of optical potentials located in the product arrangement region, to absorb the portion of the wavepacket describing the reactive products [ 23,241, (d) the use of the Chebychev propagation method of evolving the coupled wavepackets [ 25,261, (e) the use of fast Fourier transforms to evaluate the action of the kinetic energy
of radial motion [ 2 I ,22,27,28], (f ) the possibility of obtaining results for a range of energies in a single wavepacket calculation [ 8,21,22], (g) the calculation of total reaction probabilities using fluxes [ 23,241, (h) the
generalization to obtain vibrationally resolved reactive probabilities using fluxes [ 29 1, and (i ) state-to-state
S-matrix elements by a projection technique [ 19,21,22]. The optical potential technique of ref. [24] is extremely important because it removes any necessity of a detailed propagation of the packet in the product arrangement, so that the packet propagation is purely nonreactive (inelastic), and involves grids only in the iniCurrent address: Sandia National Laboratory, Division 8233, Llvermore, CA 94550, USA.
312
0009-2614/90/$
03.50 0 Elsevier Science Publishers B.V. (North-Holland
)
Volume 169, number5
CHEMICAL PHYSICSLETTERS
15June 1990
tial arrangement coordinates. Depending on where the complex potential is placed, one may obtain either total
reactive probabilities [ 231 or vibrationally resolved probabilities; more details of our procedure for obtaining
the latter probabilities as used in the present calculations are given in refs. [ 18,291, while an alternative procedure enabling the determination of S-matrix elements is described in ref. [ 191, and is based on earlier work
in refs. [21,22].
In this paper, we report the first results of applying the time-dependent wavepacket procedure to the
F+H,+HF( ur) +H reaction. The present study concentrates on vibrationally resolved (rotationally summed)
and total reaction probabilities, at relatively high total energies, with total angular momentum J= 0. We compare our results to those of Kress et al. [ 16,171. Independent and simultaneousstudies of F+ Hz reactive scattering at different energies by Yu et al, [ 15 1, using the same S5A potential [ 301, were also recently reported.
The paper is organized as follows. In section 2, we briefly summarize the equations solved. In section 3, we
present our results. Finally, in section 4, we give our conclusions and indicate future work.
2. Method
In this section, we shall summarize the approach used in the calculations. The detailed description of the
method is given in refs. [ 18,291. The wavepacket satisfies the time-dependent SchrGdinger equation
in,
av=Htp,
(1)
where
(P-U,
-J+j_
-2J,j,)+
V(R, r, y) .
(2)
Here, R is the (mass-scaled) scattering distance between the diatom center of mass and the incident atom, I
is the (mass-scaled) vibrational distance between the two atoms of the target molecule, y is the polar angle
between the scattering vector R and the diatom vector r, I is the total angular momentum operator, i is the
target molecule’s angular momentum operator, J, and j, are raising and lower operators defined by
Jk =J, 2 iJ, ,
(3)
I+=L+ti.“,
(4)
and P is the system reduced mass,
I/2
P=
hmBmC
fflA+mB+mc
>
’
(5)
with F&, mB, m, being the masses of the three atoms. The collision is treated within a rotating (body) frame
[ 221, and the wavefunction developed in the form
where D”,,Q are the usual Wigner rotation matrices [ 3 11, Y,* are the spherical harmonics, (8, p) are the spherical angular coordinates of R, and (y, <) are the spherical angular coordinates of r, no is the initial diatom
vibrational quantum number,j, is the initial diatom rotational quantum number, m, is the initial z-component
of angular momentum (along a space fixed axis), and Q is the projection of both I and i along the rotating
body frame z-axis. Substitution of eq. (6) into eq. ( 1 ), followed by appropriate projection with basis functions
373
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CHEMICAL
PHYSICSLETTERS
15June 1990
yields the CCWP (in rotations) equations for the expansion coefficient ((Rrt] n,j,m, I@J),
iA~~(Rrt~~,j,m,~j~.l)=T~~~(Rrtl~~~~m~ljnJ)t~~~+,i(Rrtl~~j~~~lj~tlJ)
tT~~_,i(Rrtln,j,m,IjS1-II)+T;
v,Cij’IRr)i(Rrtlnoj,moIj’sZJ),
(7)
where
(8)
(9)
~di’IRr)=2~ s d(cosy)Y,,(%O)[li(R,r,g)-v(r)lY,.,(y,O).
-I
(10)
Here, v(r) is the binding potential of thetarget diatom, and V(Rry) is the full (Born-Oppenheimer) potential
for the three atom system. An essential feature of eq. (7) is that no basis set expansion in terms of target vibrational states has been included; instead, r will be propagated. Although the CCWP formalism in ref. [21]
emphasized basis set expansions, it was also discussed that wavepacket propagation in r could be done, and
that this would permit treatment of more general processes; specifically dissociation. Of course, wavepacket
studies of collinear reaction models also employed wavepacket propagation in both R and r [23,32-361.
The CCWP (in rotational states) equations, eq. (7), are propagated in time using the same basic procedure
developed earlier [2 1,221, employing the Chebychev expansion of the time evolution operator [ 251, adapted
to a matrix Hamiltonian [ 261, and the FFT technique for evaluating spatial derivatives [ 27,28 1. However,
important modifications are made in that two distinct regions are employed; namely, a reagent (RE) region
in which only inelastic scattering can occur, and a strong interaction (IN) region where the possibility of chemical reaction are included [ 231. This procedure greatly enhances the efftciency of the wavepacket propagation
by reducing the region in which both r and R are propagated. In RE, the care further developed in a vibrational
basis { Qnj(r)}, defined by
(11)
so that
(12)
Here, only a relatively small number N, of vibrational states with centrifugal potential j(j+ 1)fi’/2,& is needed.
In the region IN, the [ are written as
(13)
(14)
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Volume 169,number5
CHEMICALPHYSICSLETTERS
15 June 1990
and ~(Rrt 1rr,,j,m, (jU) represents the portion of the packet which can lead to reaction (and also describes the
further distortion of the packet which is not included in the P,<(Rrt 1nOj,,mO1jmJ) terms). The coupled equations for the packet functions q and x are obtained by the projection operator approach described in ref. [ 231.
As noted above, the time evolution of the ~(Rrtl nojomo]jSZJ) involves a grid 60th in R and I [ 17,20,22,3 l353. Finally, in order to make our grid in r, in the region IN, as small as possible, we employ a complex potential
for r, < r< r, + Ar, such that the wavepacket reaching this region is absorbed [ 241. It is found in practice that
r, and Ar, do not have to be so large as to require an inordinate number of grid points. The parameter rl is,
however, large enough that the 4, are essentially zero form r>r,. It is important to emphasize that the parameters of the optical potentials are estimated a priori (prior to calculations) using the inequality relations
derived in ref. [ 241.
We calculate energy resolved reaction probabilities using the following method (see ref. [29] for details).
At the desired set of energies, we Fourier transform the wavepacket from the time to the energy domain, thereby
obtaining the time-independent scattering wavefunctions [ 37,381 (the solutions of the Lippmann-Schwinger
equation):
i(RrElnOjOmO ]jGJ)=aE
s
-co
dtexp(iEt/A)
~(RrtInOjOmOIjQ.T).
(15)
Here the proportionality constant (Yedenotes the weight that energy E ( =fi2k2/2uten410) has in the initial
wavepacket. The calculation of the time to energy Fourier transform is done very accurately and conveniently
by a method based on the Chebychev expansion [ 371. In order to obtain vibrationally resolved reaction probabilities, it is necessary to express the portion of the wavepacket containing the x( RrEl nj,gn,,l jQJ) on a grid
in the product arrangement scattering and vibrational coordinates [ l&l9 1. This is readily achieved using the
Fourier representation of the function [ 191. Having the reactive portion of the wavepacket on the new grid,
one then projects onto the appropriate HF-vibrational-rotational
states, and computes the energy resolved scattered flux into specific nj!2 HF states. The calculation is carried out for several choices of the HF-product scattering distance to verify that they are converged. (Note: the transition probabilities as a function of rotational
state will not be converged. However, the reaction probabilities summed over final HF-rotational states do
converge prior to reaching the region where the complex potential is located.)
Note that if one desires only the total integral reactive probability or cross section, the wavepacket in region
IN need not be expressed in terms of the HF+ H arrangement coordinates. This enables one to reduce the sizes
of grids significantly. Instead, the reactive probabilities are determined by calculating the flux of
~(RrEJn,j,m,,(jSM)
along the line r=r,. Then
(16)
Note that for the total reactive probability, P,,,,(E), we need not calculate and store the full set of time-independent scattering wavefunctions, but only the values off and ax/&, as functions of R, at r=r, are needed
[ 291. Alternatively, we could have calculated the nonreactive probabilities as has been done in previous nonreactive CCWP studies of atom-diatom scattering [ 21,221, then summed over all final Hz+ F quantum states,
and subtracted from one to yield the total reactive probabilities. The alternative approach capable of giving
converged s-matrix elements for reaction into specific HF-vibrational-rotational
stales is given in ref. [ 191.
We now turn to give the results for the F+H2 system.
3. Results
Calculations have been carried out for the F+ HZ (v,,= j,= mO= 0) +HF( vr) + H reaction using the potential
375
CHEMICAL
PHYSICSLETTERS
Volume 169.number 5
15 June 1990
surface SSA due to Truhlar and coworkers [ 301. The first fully converged quantum 3-D results for this potential
were obtained simultaneously and independently by two groups and recently reported [ 15 171. The results of
Yu et al. [ 151 for J=O have been used as benchmarks for other recent studies [ 201: and a comparison explicitly
showing that the results of Yu et al. [ 15 ] and Kress et al. [ 16 ] agree quantitatively over their common energy
range has also been given [ 171. The wavepacket procedure we employ yields results over the range of energies
from 1.80 to 2.26 eV, where we follow Kress et al. [ 16,171 in measuring the energy relative to the bottom of
the HF well. Results may be obtained at any energy contained in the wavepacket. In table I, we give the values
of the parameters used in the wavepacket propagation. The results for a number of energies are given in table
2, along with the corresponding results of Kress et al. [ 1617 1. We concentrate on eight energies for which their
results were tabulated (rather than try to read off values from their plots). By far the largest differences occur
for .I& equal to 2.10 cV. This corresponds to the most problematic energy in the calculations by Kress et al.
[ 16 1,which they find displays resonance characteristics [ 17 ] and threshold effects. We estimate our procedure
for extracting results at a given energy introduces an uncertainty into our energy value of AE= &0.0005 eV.Thus,
we cannot state our energy more precisely than 2.0995 GE< 2.1005, and this will influence the comparison of
results. Kress et al. [ 161 state convergence of P& (total) at 2.10 eV to about 3%, while our results differ by
only 1% from their largest basis set results. For the individual vibrationally resolved reactive probabilities, Kress
et al. [ 161 show changes between their results obtained with 140 and 150 basis functions as follows: P& (z+= 0)
changes by 15.6%, P&(vr= 1) changes by 0.6%, P&(2+=2) changes by O.i%, PFo(vr=3) changes by I%, and
P,“,( v,-= 4) changes by 18%. It is the last probability which changes most rapidly in the vicinity of I&,=2.10
eV. By comparison, our results differ from their N= 150 reactive probabilities by 0% for P,“,( vf =O), 6% for
Table 1
Parametersforthe Ft H2calculations
parameters for total reactive probabilities
R-=2.9
.A at
R,,,=O
r,,=3.0
k.=O.2
jmax=30C)
Vtd)=0.5eV
Rx,“= I ISA
(n j) *‘=(O 0) (0 2) (0 4) (0.6)
(0:8)(0, to;, (;.a;, (;,2;, (;,4)
flux analysis at r= 1.8,1.9, 2.0 A
parameters for vtbrabonally resolved probabilities
F t H1 arrangement
R.,,,=5.0
R,,=O.O
r,,,=s.o
r,,.=a.2
V,=O.3 eV
im.= 54
R,,=20A
(n,j) same as above
H + HF arrangement
flux analysis et I? “)=2.5,2.6, 2.7
r,,,=2.5
&“=a.2
imax= 54
NRb’=45
N,=32
!IR*“Z I A
N,y= 100
N,=54
AR,=2.3A
Ni= 54
propagation time (length of each Chebychev step) At= 5 x 1O-‘5 s
Tmax=4x 10-r3s
initial wavepacket width a=0.4 8,
initial wavepacket center Rlniuat=7 8,
a) All distances are in A. b, NR is the number of R-grid points. ” jm_ is the maximum rotor state in region IN.
d’ VI is the maximum of the imaginary potential. e, AR, is the length of the imaginary potential.
f1 R,, is the maximum R-value in region RE.
‘I (n,j) is the vib-rotational basis used in the projected part of the wavefunctionwhich extends into RE.
h, R is the scattering distance in arrangement HFt H; r 1sthe HF vibrational distance,
376
Volume 169, number 5
15 June 1990
CHEMICAL PHYSICS LETTERS
Table 2
Vibrationally resolved and total reactive probabilities for the F+ H, (0,0,O)+HF( r+) t H reaction
Uf
E (ev)
I .80
1.90
2.00
6.3( -5)
1.29(-4)
3.4( -4)
1
0.006 b,
0.008
25%”
0.0083
0.0090
8%
0.020
0.021
5%
2
0.110
0.118
7%
0.098
0.104
61
3
0.410
0.392
5%
0.602
0.610
1%
a
2.04
2.08
2.10
2.12
2.16
3(-5)
3(-4)
3(-4)
0%
3(-4)
3.6( -4)
0.024
0.013
0.015
0.016
6%
0.156
0.164
5%
0.226
0.115
0.133
0.143
7%
0.454
0.450
1%
0.416
0.383
9%
0.539
0.581
1%
0.440
0.419
5%
0.435
0.438
I%
0.350
0.351
0.2%
0.190
0.125
13%
0.109
0.117
7%
0.104
0.094
10%
0.697
0.704
1%
0.686
0.699
2%
0.602
0.599
0.5%
_a)
4
total
0.526
0.518
2%
0.709
0.723
2%
0.630
0.635
1%
0.666
0.657
1%
0.721
0.713
1%
” A dash means the result was either not calculated or not reported.
b’ The top number if the wavepacket result; the bottom number is taken from tables in ref.
‘) This is the percent difference, calculated as (Pwp-PKBpp) x 100jPkapp.
[ 161
P&( uf = 1 ), 7% for P,“,( vf=2), 5% for P&( uf=3), and 13% for P&,(vf =4). We believe the level ofagreement
at this most difficult energy to be satisfactory. However, we see that the agreement over the rest of the energies
is typically better, with percent differences being largest for the smaller probabilities, which are more difficult
to converge. The absolute differences are similar to what is typical when totally different methods have been
compared. We believe the present wavepacket results at all energies to be accurate to better than 2% for the
total reactive probability, and to within 10% for the other probabilities (excluding perhaps E,,, = 2.10 eV, and
the very small P&0(II,-= 1) probability at I$,,= 1.80 eV). Since our plan is to use this particular method principally for calculating total reactive probabilities, we feel the present results are extremely encouraging. We have
developed a somewhat different wa,wpacket formalism capable of obtaining the actual S-matrix elements for
reaction, and we believe it will yield even more accurate final state resolved reaction probabilities [ 191.
4. Conclusions
In this paper we have tested a time-dependent wavepacket method for calculating vibrationally resolved and
total reactive probabilities by applying it to the extremely challenging F+ H,+HF+ H reaction. The results
have been compared to those of Kress et al. [ 16,171 for a range of relatively high energies. The results for the
P$ are found to agree satisfactorily, with probabilities exceeding 0.1 agreeing to better than lo%, with the
exception ofthe P&(vf=4) result at I&,=2.10. However, Kress et al. [ 161 note that their P&(uf=4) is only
converged to within 18%, at this energy, so our result for this probability agrees within their stated accuracy.
371
Volume 169, number 5
CHEMICAL PHYSICS LETTERS
I5 June 1990
Our results for the total reactive probability agree to within 2% with the results of Kress et al. [ 16,171. With
this method, for J= 0, we are able to calculate total reaction probabilities for about 20 energies over a range
of about 0.2 eV in about 10 min on a CRAY2. Such a calculation takes a little over 1.5 megawords of memory.
The slow scaling of the body frame wavepacket method with numbers of quantum states may make possible
the calculation of total integral reactive cross sections for the Ft H2 system. We are currently carrying out feasibility studies by looking at timings for nonzero .I. In addition, we are beginning calculations on F+ Hz with
the alternative wavepacket method [ 191 in order to test its ability to produce accurate S-matrix elements.
Acknowledgement
RSJ and DJK gratefully acknowledge partial support of this research under NASA-Ames Grant NAG2-503,
and R.A. Welch Foundation Grant E-608. DN is the recipient of a Welzmann Fellowship for 1989-90 and
gratefully acknowledges partial support of this research at Princeton by the Office of Navel Research, and by
N.ASA-Ames Grant NAGZ-503 during a stay in Houston.
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